1. Introduction
The dynamically high-quality response of a controller to the saturation of manipulated variables is an important task in controller design. Since such constraints represent non-linearities, the closed control loop is a non-linear system, even if the controlled system without actuator can be described as a linear, time-invariant system, which is assumed below. In order to avoid stability problems caused by the limitations of the manipulated variables, numerous so-called anti-windup methods are already known
[1] | Hippe, P. Windup in Control. London: Springer; 2010. https://doi.org/10.1007/1-84628-323-X |
[2] | Adamy, J. Nonlinear Systems and Controls. Berlin, Heidelberg: Springer Vieweg; 2022. https://doi.org/10.1007/978-3-662-65633-4 |
[3] | Tarbouriech, S., Garcia, G., Goes da Silva Jr., J. M., Queinnec, I. Stability and Stabilization of Linear Systems with Saturating Actuators. London: Springer; 2011. https://doi.org/10.1007/978-0-85729-941-3 |
[1-3]
. For this purpose, the Popov criterion, the circle criterion, the direct method of Lyapunov or the Kalman-Yakubovich-Popov equations
[2] | Adamy, J. Nonlinear Systems and Controls. Berlin, Heidelberg: Springer Vieweg; 2022. https://doi.org/10.1007/978-3-662-65633-4 |
[3] | Tarbouriech, S., Garcia, G., Goes da Silva Jr., J. M., Queinnec, I. Stability and Stabilization of Linear Systems with Saturating Actuators. London: Springer; 2011. https://doi.org/10.1007/978-0-85729-941-3 |
[4] | Vidyasagar, M. Nonlinear Systems Analysis. 2nd edition. Philadelphia: SIAM; 2002 (unabridged republication). https://doi.org/10.1137/1.9780898719185 |
[2-4]
are often used for stability considerations. In more recent research studies, so-called linear matrix inequalities (LMI) are also used as an alternative for stability studies. However, they often only provide numerical results
[5] | Lerch, S., Dehnert, R., Damaszek, M., Tibken, B. Anti Windup PID Control of Discrete Systems Subject to Actuator Magnitude and Rate Saturation: An Iterative LMI Approach. Proceedings of the 25th International Conference on System Theory, Control and Computing (ICSTCC). Iasi, Romania, 2021, pp. 413–418. https://doi.org/10.1109/ICSTCC52150.2021.9607157 |
[5]
. Other current research work is concerned with the application of the basic principles of anti-windup measures to special controllers such as PI-lead controllers
[6] | Chen, Y., Yang, M., Liu, K., Long, J., Xu, D., Blaabjerg, F. Reversed Structure Based PI-Lead Controller Antiwindup Design and Self-Commissioning Strategy for Servo Drive Systems. IEEE Transactions on Industrial Electronics. 2022, 69 (7), pp. 6586–6599. https://doi.org/10.1109/TIE.2021.3097602 |
[6]
or with switching strategies between different anti-windup measures
. In addition, the use of an
Additional Dynamic Element (ADE) is proposed, with which the stabilization in the limiting case succeeds for any controller stabilizing the unconstrained system
. However, even there a controller must be effective at least at one instance for which the stability in the limiting case can be proven – e.g. with the help of one of the methods listed above.
Stability analysis is especially challenging when the controller contains integral-action components to ensure steady-state accuracy. This is because the controller integral-action components are usually assigned to the controlled system during modeling, which results in an unstable or critically stable system. For this purpose, no positive definite matrices can be found for this, as required or at least aimed for in the Lyapunov theory and in the Kalman-Yakubovich-Popov equations to ensure stability. In
, this problem is overcome by completely avoiding controller integral-action components and instead attempting to ensure steady-state accuracy with the aid of disturbance observers. But this is not always possible when the system parameters are not exactly known. The method of reference variable correction in combination with a special PI-state controller design, as explained for example in
[8] | Nuss, U. Stabilitätsverhalten von zweistufig entworfenen zeitdiskreten PI-Zustandsreglern bei Stellgrößenbegrenzungen [Stability properties of two-stage designed discrete-time PI state controllers considering the limitation of input variables]. at – Automatisierungstechnik. 2017, 65 (10), pp. 705 – 717. https://doi.org/10.1515/auto-2016-0136 (in German) |
[9] | Nuss, U. Zeitdiskrete Regelung [Discrete-time control]. Berlin, Offenbach: VDE; 2020 (in German) |
[8, 9]
for discrete-time systems, provides a help in this respect. However, the stability proof described there has been greatly simplified in the meantime. Furthermore, by means of the above-mentioned procedure, it was also possible to perform the stability verification for such systems in a general way and thus greatly reduce the synthesis effort where the manipulated variables act on the system with dead time.
Due to the significant progress in stability verification for control loops with saturation of the manipulated variables, these new findings are presented in this article. The controlled system and the controller are modeled in state space. In section 2, as an introduction, the methodology of reference variable correction is first briefly explained for continuous-time systems and then the transfer to discrete-time systems is shown. Subsequently, section 3 describes special controller synthesis equations that generate a PI-state controller from an already known P-state controller in a simple way. In addition, both methods are combined in section 3 in order to establish the principles for a Lyapunov-based controller design for systems with an integral-action controller component, taking manipulated variable constraints into account. Using Lyapunov functions, the proof of stability is then provided in section 4. The measures mentioned are carried out for both continuous-time and discrete-time systems. In section 5, the method presented is extended to discrete-time systems with dead time behavior of the actuators. To illustrate the methods described, section 6 deals with an example from the field of electrical drives. A summary concludes the article.
2. Reference Variable Correction in Case of Manipulated Variable Saturation
2.1. State Equations of the Controlled System
The vectorial state differential equation of the controlled system,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFkAAAARCAYAAAC2AACuAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAEGUlEQVR4nO1WWSitbRQ+1+RGhpCS5AYJSZIM5UJEEmUoRRISMkUuzFMiMxnKcGNMbhBxoQwJSYaUOfMcMlt/z6q9s/3ft+19jr3P399+al187/cO633WWs96f5EGKsUv4G878X+HhmQ1QEOyGiBD8tHREQUFBdHp6anaHbm6uqLAwEBaWlpS+9liuLy8pJiYGAoICKCCggLKy8uj1NRUWltbU2ofGZKfn59pYWGBbm5uftzh75Cbm0smJiY0PDyskv17e3uptrZW6XXZ2dmceO/v7/Ty8sJku7u7K7XHf0Iutra2KDw8nLy9vamzs1MlZzQ0NFBWVpbS60JCQqiwsFD63d3dTdbW1krtISV5YGCALCwsqLW1lR4eHigpKYl8fHzo7u5OcOHt7S1VVlZSRUWFoFVXVyvsRGhoKK2srJCvry81NzcLzpmfnycrKyvKzMzkrMIZ9vb2dHx8rNAZv0MyKtrOzk5aXa+vryxp+fn5MvMuLi6YN09PT5qbm+Oxs7MzsrS0ZOllkp+enqisrIyamprI2dmZioqKOGITExP08fEh6MD19TWXeE5OjqChrBTB6OgoJSQk0NvbG2tfSUmJ4Lzy8nIaGRkhAwMD6ujo4O+xsTFR/wDcS2I1NTWUnp4uMyZvLbC6ukqGhoaUmJjIeoykq6urY7I/Y29vj4Pt6upKQ0NDPIbA2Nra8jkycrGzs0O6urrU1tamEEF/ClSMi4sL9wFEPCIigjIyMkTnI4ONjIwUzkhzc3PS19dn09HRIS0tLek39H9zc1Pu+r6+PnJycmI/Hx8fmWh/f3/W5q/Y3d3lytrY2OBvJCDuA59lSMahZmZmnF3f4fz8nKKjoykqKkrQYmNjv92jqqqK4uLiqKWlhc3Ly4vi4+NF50O6bGxslJIiCX5HLlJSUigyMlL6PTU1Rdra2oISOj4+Tm5ubhyM+/t7cnBwoPr6ev7HJB8eHhIsOTmZIwVnlpeXObPFgI3QsXt6egStv79f7gW2t7fJw8OD9UwClDMk4ysQUFhaWhoHxc/Pj8tTon+KQFmSQRb8a2xslI6hB4E8vMK+orS0lHsLsh5KYGxsTIuLi3w/JhkdFJfDG3VwcJBMTU1ZJ6G7qgDexMHBweTo6MgNAjg5OeFMQLP42sxQprgwdG59fZ309PQoLCyMDg4OFD5TGZKh1cha6DGqBr0J/QociQUWryJIEIienJzkYKCvIdmYZAj8/v4+T0aUQDYyR1VA156dnaWZmRl+pQB4+E9PT/PY5+wGUFGQMkmjQpUpQzCAptnV1aXQXJyDZyX8kfiE14/YSwtAg0PmoglK1oPXf2myBqqBhmQ1gEkuLi4mMYNG/zRQ+vLOhLW3t0vn4+UhNg8l+dNAL/rOP7zTJQBH8uaC438AD3GlRlu3QKIAAAAASUVORK5CYII=)
,
(1)serves as the starting point for the following considerations
[2] | Adamy, J. Nonlinear Systems and Controls. Berlin, Heidelberg: Springer Vieweg; 2022. https://doi.org/10.1007/978-3-662-65633-4 |
[10] | Aström, K., Wittenmark, B. Computer-Controlled Systems. 2nd edition. Englewood Cliffs: Prentice-Hall; 1990 |
[2, 10]
. Here
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAsklEQVR4nN2QLQ6DQBSEOQQWRbgHJ8Ah1nMFHJ4EhVmFX4lDEw7Aj0FjwAMBg+BrFhRJW1XVSZ6aL/MmY/BFhtZ/AOd5opTCcRzyPGeeZ4QQBEFwA9u2kaYpSZLgeR5hGFKWJVVVPV+0bYtpmhRF8b5D0zRYlkXXdU9gmiaGYbiiXddFSnnFr+t6A7qQ7/v0fU+WZdi2TRRFHMdxA9oYx/GK3Peduq5ZluWXQ8VxzKfT/gu2S6XV05Kg3gAAAABJRU5ErkJggg==)
denotes the
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAsklEQVR4nN3QQQpFUBjFcRNlxtgmlD1QtmAVVkHKiKktWIChCSO6t5SyBXOS8n+5iuGbvcE7w9Ovzten8SXaf4BhGMjzHNd12baNKIqwbZskSW7Q9z3ruqLrOmmaIoSgKAocx3knmqZRQEqpyizLCMPwBXEc4/v+s+t5HmVZviAIAoWuXHcYhsE8zyzLgnYcB6Zp0nWdAldpWRZ1XVNV1Q3atmXf92diHEemaeI8z1988gN5vurb5LoWpAAAAABJRU5ErkJggg==)
-dimensional state vector,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAvElEQVR4nN2QIQqEUBRFNVtFNyCYxB24F8Vq+MUtWIyCC7BY7LoDixhdgAZBQXEBnsEvjG2mTJoHF967HO6Fp/BllP8A+r4nyzIcx+E4DmlO04RpmvJWuq6Ti6qqzPMsgaIocF33qajrGtu237FhGBJF0QPEcYwQQhrbtmEYBmVZPoDneeR5zr7vpGmKpmmM48iyLDfg+z6WZREEAcMwoOs6TdNQVdUNXLFt27Kuq4y9oEvnef7iUUmS8EkvYGzfyFWvc6oAAAAASUVORK5CYII=)
the
![](data:image/png;base64,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)
-dimensional manipulated variable vector,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA2ElEQVR4nMXRLQqEYBAG4D2KVzCYBLtgEwQRg8XkESwm7yBGjyBYTYqYLAoWQQxiFn/Su8y3sCLr7paFHRgGhifMy9zwpW5Uf0bDMEDXdTbfItd1wXEc6rq+Rl3XwbIsiKKIsiyvkW3bqKqKoSzLXlFRFHAcB+u6QpIkxHF8Rvu+Q9M0pGmKcRwhyzKiKDqjJElgmibCMGQtCAKCIDjQPM9QVRVN0zxvMwwDnucdyPd9FrltW7bs+x48z0NRFCzL8kAUNc9zTNPEEE1KRkG2bfv17+imT03mDlgV2gfBVmYAAAAAAElFTkSuQmCC)
the
![](data:image/png;base64,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)
-dimensional dynamics matrix and
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABCklEQVR4nMXRLauDYBgG4P0GqytWg39AMA22YBEsdhctOliSBYPBpMU4f8WiaxsWg9Hq0ooG8QM/7oOvoCztcMp54WkXz32/PBv84m3+AVVVBV3XIYoiLMsiYxgG4jj+3OT7PgRBQN/36LoOnueB53k0TbOi0+kETdOWiNvtBpZlSQpBk97tdrherwRM21RVxfl8xjAMM3q/36BpGsfjkfSRJAmO46Cu67VTFEVgGAZFUaBtW9LncDigLMsVua6L/X6/9Jl+RVEUsiyb0ZSpKApM01xQEATgOA55ns8oTVNst1tcLhfc73dSXpZlhGGIcRxn9Hq98Hg8yDyfTyRJQrr97SxfkW3b+DY/Zq0L5C9pWUoAAAAASUVORK5CYII=)
the (
![](data:image/png;base64,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)
)-di mensional control input matrix. Disturbance variables are not considered without any generality restriction. Eq. (
1) is supplemented by the output equation
(2)using the
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA00lEQVR4nN3QMQ5FQBAGYLdwAaLRSBRKpVrcRGE7FTUnUegkdAq1SoJCotQQRIj/ZXfzopF3gDfJdN/8mRkBP0qg9T/gOA7EcQzXdUEIgeM4SNOUg/u+EQQBPM/DdV0YhgGiKKJpGg6maYKiKGjblsVWVQVVVbEsCwd5njNAk2j7vg/btnGeJwdJkkCSJGzbhrIsoes6wjB8lhzHEbIsw7IsRFEEwzBQFMUDaGzf96jrmu2haRro0OsfsiyDaZpY1/Ud0HNpQtd172Dfd8zzzC74gg+32aGgpgNd7QAAAABJRU5ErkJggg==)
-dimensional vector
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAv0lEQVR4nN2QIQqEYBSEPYfJI9itFo+gYBIMYhdsYvACRs+gwSZis1i8g8UkgqDB4Lf8bliWX/YAOzDh8ebxZkbhBxSB/xCs64plWTiOw3meiNk0TeI4fguqqqJpGlRVZZ7n+9J1Xbqu+7y4rgvDMGjblmVZ8H2ffd+/PYRhSJ7nlGV5UzJZFAWe5xFFEcdxyIK+79E0jWEYnmPWdU2apnLMcRwRDIKAbdtkga7r2LbNNE3PRYlCkiQhyzKJYv8CHzujv6gBeycAAAAASUVORK5CYII=)
of the control quantities and the (
![](data:image/png;base64,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)
)-dimensional output matrix
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA20lEQVR4nL2QOw5FYBCFbYEFWIHSFrQSC9CoLYJCiEprCTq1nig0KgWFVuVVIB7nxv8nf3Nfuc2dZJrJN2fOHA5fivsjcJ4nsiyD53lwHAeGYSCOYwpc14UoiqBpGpqmwXEcsG0bQRBQoG1bSJKEPM+Z9DRNmOeZApZlQVVVcubJw7IskGUZYRi+NjkMA3ieR5qmr4Gu6yAIAuq6ZsO+77GuKwXGcYQoiiiKggy2bYOu62yBu435vg/TNJEkCTGsKApRYUHt+46qqsibZVniPntn81vUbwHXdfGpH4V35DBxakOEAAAAAElFTkSuQmCC)
. The possibility of the manipulated variables affecting directly the control quantities is disregarded.
The differential or difference equations of the controller integrators are also included in the system description. In this respect, it is assumed below that there are as many controller integrators as controlled variables and as many reference variables as controlled variables. If the output variables of the controller integrators are summarized in the vector
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAARCAYAAAAL4VbbAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA9ElEQVR4nO2Pr4qEYBTFfQofQBA0WwWfQIyC2AWbxWow2HwBX8KgRSaYRWwWg01BxCCC/0DP4rcMs8PuDGyfA5f7Xe7vHL5L4R+iPvBPeNs22LYNjuNQFAX6voemaZAkibyf4DRNcZUgCKQnSYIsyyDLMpqmwRW2rivO8/z+xjAM4HkedV2ThLIs4TgOxnGEoihgWZYYqfuSYRjs+455nuG6LqqqIsYgCGAYxiM5iiKIooiu6wh4zXddsGmaD/h2u4Gmaei6jjiOyeIlvCwL8jz/df2f8Dv5vg9VVXEcx3t4miZyg2VZaNsWlOd5eFVhGD6ZvwCYNZ1/UML+gQAAAABJRU5ErkJggg==)
and the reference variables in the vector
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA6ElEQVR4nOVQPc5EABB1E61GIlH6iQuQUKgdQKJQaBQqCifQaigdQJxAohAVDSIqSiIRb0Pxbb5um612kpnMvLyXmXkEPgziN4lJkoAkSZRl+QBFUUCWZez7jr7vwbIsPM8DsW0bKIpCnucP0XEcMAyDdV2fWZIkVFUF4jgOCIKANE0xjiN0XQfHcRiGAXVdwzRNnOcJ4i6qqiKKIoRhiCzLIIoimqaBZVlo2/b9zK3SNA2u6+LewPM8bNtGHMe4rutN9H0fNE1jmqYHVBQFhmHgvv+fPfM8o+u6P/Dul2X5tuFBEOCTfAG0c1igc+7EqwAAAABJRU5ErkJggg==)
, then the vectorial differential equation of the controller integrators is as follows, provided they operate continuously in time,
![](data:image/png;base64,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)
.
(3)If the controlled system is described in discrete time, the controller design is based on the controlled system state difference equation
[9] | Nuss, U. Zeitdiskrete Regelung [Discrete-time control]. Berlin, Offenbach: VDE; 2020 (in German) |
[10] | Aström, K., Wittenmark, B. Computer-Controlled Systems. 2nd edition. Englewood Cliffs: Prentice-Hall; 1990 |
[9, 10]
(4)instead of Eq. (
1). The indices
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA1ElEQVR4nK3RMQqEMBAFUE+ihaUWnsJcwMpjWNrY2Vl6AauAiFgInsDKGyhqI2hnIYgo/sVZEISwLOxOk0Be+DOJhA8lXfUfMI4jyrKkVQj6voeqqkiSRAzWdYWu62jbVgzquoZhGFiWRQyKogBjDPu+IwxDyLJMPd3A9304jkPAdV1EUYTzPN/g2liWhSAI6DDLsmfEPM+U73keFEXBMAxP0HUdNE3DNE2wbRtxHD9BnucwTRPHcYBzTihNU5qIwPV6TdPQjW3bUFUVxdxNfvUXP4EXWQOmKYrekrMAAAAASUVORK5CYII=)
and
![](data:image/png;base64,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)
(
![](data:image/png;base64,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)
) indicate the sampling time instants
![](data:image/png;base64,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)
respectively
![](data:image/png;base64,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)
of the state variables and the time instants at which the manipulated variables take effect.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABB0lEQVR4nM1RocqDYBT1GQyiwWgcZoNYjKa9gwiaLMqCVmFBMPgClsF8AJcNCj6BNoMGGRM2tiBMz8++Ovx/43/h3HvD4Z5z76WwMah/RDyfz1BVFbZt4/l8rhPrugbHcciy7HviOI6Iogi+78M0TTAMA9d1ydQ8z7EsC6jH4wFZlhHHMV6vF6mapmGaJpRlCZ7nMQwDqDRNIYoi5nkm2O/3CMOQyDVNA5qm0fc9qMvlgt1uR+S7riN9VVW43W7QdR2WZeH9foP6pCRJ4DgODMMAy7LwPA+HwwGn04lY+Nr6eDxCUZTf7/gxLEkSBEFA27brxPv9jqIoCK7X69+fWYvtxCAIsAU/7xBM616OE4wAAAAASUVORK5CYII=)
is the (
![](data:image/png;base64,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)
) -dimensional transition matrix for which
(5)applies
[9] | Nuss, U. Zeitdiskrete Regelung [Discrete-time control]. Berlin, Offenbach: VDE; 2020 (in German) |
[10] | Aström, K., Wittenmark, B. Computer-Controlled Systems. 2nd edition. Englewood Cliffs: Prentice-Hall; 1990 |
[9, 10]
, while
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABEklEQVR4nMWQoeqDUBTG13yARS1iWBusGoeI0SCYrc4o7gEMC3sDu+4RTFYNCiYRt2AVTIJjKmPfHy/My9LW/gc+uPfc3znfOXeFH2P1z+Dj8YBt25AkCZfLhTyUZQlVVYmqqqIdPc/DdrtF13Xk/nq9sN/v4TgOOS/gnDBNc7Hq+x6CICCKImo9DANkWUYQBAuYpik4jkPTNBRs2xY8z+N4POJ0OhEpigJRFHG/3ymYJAk2mw3GcVw6GoYBy7I+tz6fz9B1fUnOBbvdDr7vU/D5fELTNGL3juv1CpZlURQFBeeh1+s1DocDWWqaJriuC4ZhEIYhBeu6RhzHyLKMWM5gnuckd7vdPmf8JX4H3//2TX/cHUm1DxF+3wAAAABJRU5ErkJggg==)
is the discrete-time control input matrix with
[9] | Nuss, U. Zeitdiskrete Regelung [Discrete-time control]. Berlin, Offenbach: VDE; 2020 (in German) |
[10] | Aström, K., Wittenmark, B. Computer-Controlled Systems. 2nd edition. Englewood Cliffs: Prentice-Hall; 1990 |
[9, 10]
![](data:image/png;base64,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)
.
(6)![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAARCAYAAAAG/yacAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABOUlEQVR4nM3SMauCUBQHcEc/gtA3cAucxA/gUuLSKi0uTUG4Vgg6BH2ClnR0sE0Xp7DNRV2baghCWgpRq3944b3He8XzvTe9O13uOT/uPedcCn9Y1D9Hu90OoihiMplgMBig2+3CMAz0+31omobr9fqMbNuGruuoqgrj8Ri9Xo8EFosFRqPR65tOpxPyPEdRFOh0OpjP5ySQJAnW6/X3NWVZBpZlEQRBc01vmzRN0Wq1sN/vf47qGniebwTv6H6/Q1VVDIfDl0nH4xHb7RZ1p+tcgi6XCziOg2VZT6B+tizLWC6XEASBNI2qpeM4oGka0+kUZVl+QmEYot1uY7VaIYoiMjeC4jjGZrMhh1/R7XYjUFEUSJJERtP4jXzfh+u6OBwOYBgG5/O5GdUNmM1mME0Tnud9NOK36wEcwyQ0MFIrRAAAAABJRU5ErkJggg==)
is the sampling time. The output equation is in the discrete-time case
![](data:image/png;base64,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)
.
(7)The vectorial difference equation of the controller integrators is as follows
[9] | Nuss, U. Zeitdiskrete Regelung [Discrete-time control]. Berlin, Offenbach: VDE; 2020 (in German) |
[10] | Aström, K., Wittenmark, B. Computer-Controlled Systems. 2nd edition. Englewood Cliffs: Prentice-Hall; 1990 |
[9, 10]
![](data:image/png;base64,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)
.
(8)2.2. Calculating the Corrected Reference Variables
The reference variable correction method for manipulated variable saturation
[8] | Nuss, U. Stabilitätsverhalten von zweistufig entworfenen zeitdiskreten PI-Zustandsreglern bei Stellgrößenbegrenzungen [Stability properties of two-stage designed discrete-time PI state controllers considering the limitation of input variables]. at – Automatisierungstechnik. 2017, 65 (10), pp. 705 – 717. https://doi.org/10.1515/auto-2016-0136 (in German) |
[9] | Nuss, U. Zeitdiskrete Regelung [Discrete-time control]. Berlin, Offenbach: VDE; 2020 (in German) |
[8, 9]
is based on the assumption that a manipulated variable saturation becomes effective because the setpoint change is too large. If this is the case, the maximum value by which the reference variable may be changed without the manipulated variable constraints becoming effective is calculated. The vector of reference variables corrected in this way is referred to below as
![](data:image/png;base64,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)
. To determine the value of
![](data:image/png;base64,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)
in the limiting case, the control law is specified firstly for the unlimited case and then again for the case of an active manipulated variable limitation. If the matrix of the feedback coefficients of the system state variables to the manipulated variable vector is denoted by
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAARCAYAAADUryzEAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABfklEQVR4nN2SvYrCQBSF8wgWdhLF2kZfwEasoq0oFrGzEEktmBQp1MbGQhDSGlB7fYEgqKBGLAV9AhF/UKPxyNwFIaxLXLbbgYGZO2e+e8+d4fDHwf0TgGVZKJVKiMViaLVadNDv9xGPx5FKpWCapnsFuq4jGAxiu93SfjAYIBqNYjQawbZtd4Cqqshms7Q2DAPFYhGbzeZzC8lkEs1mE+12m+zsdjvXyy/Afr9HIBBAoVAAz/PodDoO0ePxwPV6xeVyocnWLPYCzGYz+Hw+OlQUBYlEwuH7dDqRvUgkgnq9DlEUIcsyaQjQaDQgCAKJl8slNXM+nzuq6Ha7SKfTlPlwOMDj8eB4PIJjlEwmg3K5TEJWBesHy/AOwPQsSSgUIi03nU7h9XqRy+VwPp9xu92Qz+fh9/sxHA4dgHA4jEqlQsnG4/FXD9brNQlZgBHv9zsWiwXFVqvVWwvfXuGToWka/UxW5a8BrFm1Wg2SJGEymTgB1WoVbrPX6/0IfwKyoL3HH0oL3AAAAABJRU5ErkJggg==)
, the matrix of the feedback coefficients of the output variables of the controller integral-action components to the manipulated variable vector by
![](data:image/png;base64,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)
and the matrix of the amplification factors for the reference variables, the so-called pre-filter matrix, by
![](data:image/png;base64,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)
, then the control law in the unlimited case is as follows
![](data:image/png;base64,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)
.
(9)Thereby, the index
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA1ElEQVR4nK3RMQqEMBAFUE+ihaUWnsJcwMpjWNrY2Vl6AauAiFgInsDKGyhqI2hnIYgo/sVZEISwLOxOk0Be+DOJhA8lXfUfMI4jyrKkVQj6voeqqkiSRAzWdYWu62jbVgzquoZhGFiWRQyKogBjDPu+IwxDyLJMPd3A9304jkPAdV1EUYTzPN/g2liWhSAI6DDLsmfEPM+U73keFEXBMAxP0HUdNE3DNE2wbRtxHD9BnucwTRPHcYBzTihNU5qIwPV6TdPQjW3bUFUVxdxNfvUXP4EXWQOmKYrekrMAAAAASUVORK5CYII=)
in brackets in Eq. (
9) only applies to the discrete-time case.
If a manipulated variable saturation now occurs, then instead of the manipulated variable vector
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAvElEQVR4nN2QIQqEUBRFNVtFNyCYxB24F8Vq+MUtWIyCC7BY7LoDixhdgAZBQXEBnsEvjG2mTJoHF967HO6Fp/BllP8A+r4nyzIcx+E4DmlO04RpmvJWuq6Ti6qqzPMsgaIocF33qajrGtu237FhGBJF0QPEcYwQQhrbtmEYBmVZPoDneeR5zr7vpGmKpmmM48iyLDfg+z6WZREEAcMwoOs6TdNQVdUNXLFt27Kuq4y9oEvnef7iUUmS8EkvYGzfyFWvc6oAAAAASUVORK5CYII=)
requested by the controller, only a manipulated variable vector modified by the saturation can act on the system. If this is designated as
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABYAAAARCAYAAADZsVyDAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABqUlEQVR4nO2Sy4uBURjG/SGKnYWVFMqOrKRQWPJPiJSVL+zYsLFRkshSShZipVAuGxElFhTJJdf0TO87NZMmmUaz89TpnO+cvt/7vBcR/kEi0hv8Bv8EN5tNBAIB2Gw2nE4nfgiHw/B6va+B2+02Op0O5HI5drsdP+j1eqRSqdfAdEgkEnA4HHy5XC4hFosxHA5/Dbper9hut/fg2+0Gl8vFcFIul4NMJvtyTz+Mx2NMJhNcLhe+W61WGI1GWK/XDM3n8zCZTBgMBiAegw+HAzQaDUqlEvr9PpxOJ6xWKwMJQLWPx+Ow2+3odruoVqtswuPxwGg0Yr/fIxQKQaVSIZvN4nw+f4LpYDaboVQq4fP5kE6n+TuZTLJLhULBYOrDZrPBdDoFmZnP55BKpey4XC7DYrGw27saLxYL0HRQ9OPxiFarxW5JlJ7b7YbBYMBsNkOj0UAsFoMgCM/Bj0QO/X4/71qtlgOq1WouR71eh0Qi4WbXajXodDr0ej0O9BRMcx2JRHiuM5kM169QKCAYDLJzyqRSqXCm0WgUxWLxu3lU+EeLoH8RcT8ASdMLzAyM0nQAAAAASUVORK5CYII=)
, the control law
(10)is obtained, which can be derived from Eq. (
9), if the possibly constrained manipulated variable vector
![](data:image/png;base64,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)
is used instead of
![](data:image/png;base64,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)
and the corrected reference variable vector
![](data:image/png;base64,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)
is used instead of
![](data:image/png;base64,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)
. The two relationships (
9) and (
10) can now be interpreted in such a way that they apply at the same time. Eq. (
9) generates the manipulated variable vector requested by the controller, while Eq. (
10) describes with which corrected reference variable vector the realizable manipulated variable vector
![](data:image/png;base64,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)
can be generated. If both equations are subtracted from each other and the resulting difference is solved for
![](data:image/png;base64,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)
, the result is as follows
![](data:image/png;base64,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)
.
(11)It specifies how
![](data:image/png;base64,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)
must be modified in order to obtain a realizable reference variable vector. The corrected reference variables are then fed to the setpoint inputs of the controller integrators. This means that instead of Eq. (
3), the following applies for continuous-time control
![](data:image/png;base64,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)
,
(12)whereas for discrete-time control, instead of Eq. (
8),
(13)must be implemented. Thus, Eqs. (
9), (
11) and (
12) or respectively (
13) describe the equations of the controller. The corresponding block diagram is shown in
Figure 1 for the case of discrete-time control, including the discrete-time modeled controlled system. Finally, it should be noted that methods that also calculate corrected reference variables and use the difference between unlimited and limited manipulated variables are sometimes referred to as reverse-correction method
or back-calculation (and tracking) strategy
[5] | Lerch, S., Dehnert, R., Damaszek, M., Tibken, B. Anti Windup PID Control of Discrete Systems Subject to Actuator Magnitude and Rate Saturation: An Iterative LMI Approach. Proceedings of the 25th International Conference on System Theory, Control and Computing (ICSTCC). Iasi, Romania, 2021, pp. 413–418. https://doi.org/10.1109/ICSTCC52150.2021.9607157 |
[6] | Chen, Y., Yang, M., Liu, K., Long, J., Xu, D., Blaabjerg, F. Reversed Structure Based PI-Lead Controller Antiwindup Design and Self-Commissioning Strategy for Servo Drive Systems. IEEE Transactions on Industrial Electronics. 2022, 69 (7), pp. 6586–6599. https://doi.org/10.1109/TIE.2021.3097602 |
[12] | March, P., Turner, C.. Anti-Windup Compensator Designs for Nonsalient Permanent-Magnet Synchronous Motor Speed Regulators. IEEE Transactions on Industry Applications. 2009, 45 (5), pp. 1598–1609. https://doi.org/10.1109/tia.2009.2027157 |
[5, 6, 12]
.
3. Relationship Between the Controller Coefficients of a PI-state Controller and a P-state Controller
In the study, it was shown how the coefficients of a discrete-time PI-state controller can be determined in a very simple way from an already calculated discrete-time P-state controller, provided that the same command response is to apply in both cases
[9] | Nuss, U. Zeitdiskrete Regelung [Discrete-time control]. Berlin, Offenbach: VDE; 2020 (in German) |
[13] | Nuss, U. Ein einfacher Zustandsreglerentwurf im Zuge der Erweiterung der Systemstruktur um Reglerintegratoren und Rechentotzeiten [A simple state space controller design in the context of expanding the control structure by integrators and calculation dead time]. at – Automatisierungstechnik. 2016, 64 (1), pp. 29–40. https://doi.org/10.1515/auto-2015-0058 (in German) |
[9, 13]
. The corresponding relationships for continuous-time state controllers were presented in
[14] | Nuss, U. Regelungstechnik einfach verpackt [control theory simply presented]. forschung im fokus. Offenburg: University of Applied Science Offenburg. 2016, pp. 19 – 22. (in German) |
[14]
– and for single-input single-output systems also in
. With the designation
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABD0lEQVR4nMWRP4qDUBDGcwNLGw9gKZ5Bm2CpWFhKKgsrrayNhV5BCFhZxM4DpNDCRlKk0wOkUBCJiH++xVcsyAY2bLMDw3tv+DHffPMO+CAOW/wz2DQNTqcTjscjyrJE3/cwDAOiKOJyuew72rYNTdMwzzOGYYCqqgiCAK/Xaw/KsowwDAnoui6iKCL3nXTbtuB5Hnmew7IspGmKZVl+zvh4PMCyLHRdB8MwRO6tmev1CkmSME0TOI5DkiTvwc2I4zik6HkeFEXBuq57cBxHCIKAOI5Jsa5rIr+dO/B2u4GiKPi+T1w+n0/QNA3TNL9nJWBVVciyDPf7nYBd15F3URRkn3/7wo/A8/mM33LjvgAD/BCmWAl8VgAAAABJRU5ErkJggg==)
for the already known P-state controller, the following results for continuous-time controllers
![](data:image/png;base64,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)
,
(14)![](data:image/png;base64,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)
,
(15)![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKoAAAAdCAYAAAA6ufdPAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAIQklEQVR4nO2bdagVTRjGv79UbLEDWxELRcUWsQMVuxsTscBuEezGxMTA7kLE7u7u7u56P34Du5y7Z/bE3b333Hs8Dyzcs2fPzLszz7zxzNz/JIII4jj+A6E2IoII/CFC1AjiJD59+iQ/fvwwP0eIGkGcwvfv32XPnj1SqlQpuXDhgnk/QtQI4hR27twpW7ZskRw5csiNGzfM+xGiRqDFo0eP5Nu3byHrv0CBAv8OUWfNmiVfvnwJtRnxCu/evZOlS5dK7ty55enTp9Fq4+PHj7Ju3Tp59eqV13cnT56UQ4cOaa83b96Yz/0zRJ09e7YMHDhQfv36FWpT4hW2b98uJ06ckDRp0sjr16+j3c7hw4elefPmXm0cO3ZM9u3bp708n/0niDpnzhxF0p8/f4balHiLDBkyOCIqYA6mT58e9O/wyHny5FHk/fv3r7oXdkQlXJUrV06ePXvmuC3C1IgRI5wbFQJs3rxZDhw44HV//fr1smbNGu3lSUw3iPr161epXr26nD59OuDf3L17V4YOHaq8ca9evVRxBcKOqGPHjpUmTZo4bgeJpEWLFnL//n0XrIp9QJKePXtK9+7do9y/efOmCqm6y7N4coOooH///op0ThF2RC1TpowcPHjQURt4oly5csnnz59dsio0gHjdunWT1atXB/W7ly9fSurUqVVEcQqKpPz586s2nSCsiEoCT27z+PHjaLfx588fqVy5skyYMMFFy0KHTZs2SbFixeTDhw8BPX/lyhWV7vTu3VuF4P379zvqnzqhdOnSQYV/HUyisiPQsmVLSZYsmUybNi3KQ1u3blUrrHPnzmZyGxdB8l27dm3td9hNvoOHGTlypFSrVk369OnjpQrwOXny5D4LMSa9YMGCsnbtWlftHz9+vKRKlUr69u0rY8aMkUaNGinS6JQLUpKiRYtK9uzZ5dy5c+qZ9u3bS8qUKaVr165mNOB+vnz55N69e67aSi1Qvnx5ZcOoUaNk0KBBUrNmTTl+/LjXsxUqVJBmzZo56i+KR120aJFkzJhRevToYT7ACzOpCRIkUIluXAaJux1RV6xYIRUrVpS3b9+qz3v37lVewwrup0iRwmc/kAidccGCBc6N9sDZs2clc+bMpvcjXGbJkkWOHDmifZ7JHzBggPqbffE2bdrI1KlT5ffv31GeK1SokHTp0sVVWwE8oeABOILJkyerMbZiyJAh0qBBA0d9RSEqcgKr0rPRGTNmSIcOHVSeQYIel1G1alUtUZ8/f64mPJCcC++QNm1a2+9v3bolnTp1krZt28ro0aMd2WsFlXfZsmVVdAPv37+XbNmyqZTGCoiB1rhr1y5F0n79+qnf68AixWY3Qf+QcvHixeY9NljwnlZgo2tEhYQ0tnDhQilZsqTK1R48eKBekBVBJW1dqVZMmjRJrTDdhfe6dOmSI2P9wY6oKAG6AdQBohKCdWBMqKIRrUkhGBc3QYXcsWNHRQLIN3fuXBXNdFuZd+7cUd4XW1q1aqWVogzEBFEJ/Tlz5lRRAHuJVKQCK1eu9HrWVaKy3VWpUiW5fPmyyr/4TDLNANSrVy+g4mLDhg0yb9487TV//vwYl3rsiEoyD1kDgS+iUlhAUPJXFp6bk89kM9G1atVSqQVhnMtzW9ETFEnkh5CbvJatTzvEBFGp5tOnT6/6HzZsmLLdzqO7SlQ6xls8fPhQSpQoIatWrVL5DwNAwu5rxboNcj9jouwuBG0rdESFAEmTJjWFY3+wIypnBurUqaOkHjRHCh6dXnv79m2/tlPEWfHixQuVnhhH23AYefPmta2Whw8fruaLKFekSBGf+XJMEJUt6ho1aijPT6pC0YdD08FVohK2lyxZolYwWiSTfv36dTlz5ozKkwLRwchxGzdurL2aNm0qp06dCsgonrPbPTGuq1evev1OR1QGMlGiRHLx4kXzHsWKUVRZAVE5C2mFMdlG/+3atdN6b0Ryf7Zv27bN63dov0hrhshOtV64cGG1HawDEhppGqCAYjeO1ESHKVOmuE5UFpxRyAHImDhxYq0q5BpRWZXkOZCS1cGEEyrpdNmyZYq4gZxCYrCRsuwuipqYBAWOzqPiqdhpArzr4MGD1eELHVgA1qqf7VikF3Z1DFA4EHrdwsyZM9W4G4UUclK6dOm0hRROI2vWrKa3xT5kKfJFHSi6kOTcAqfveXdOSBkg/NvVAeTdrhAV7Ysts40bN6qbFD14HQaElUvFHx+2Eu10VPLj1q1by+7du03v43kyxxN4siRJkpibBshzyDCEYcI6oBon7KO3uqFPEsUY5/r166t3IK1B08ZWq5fkM0pMwoQJVRrCQsSeTJkyScOGDb0iBbUGqZtbOir9scg5XYWciZzHIiPss1mgAwTWSYHBQBGVCSBHtYZTXpoVzXdOt8BiA+R3VKJISJ5gcElj0CPxQhwKtguTPMsev5FHEkmOHj2qxsEYAxYxbXHvyZMnju2mDjDa40JGs9tdw248J8+Rx2IvHo45og1rUUXR5empncIYS+MMKX3i2OwiLhxCx3VtZypcQJpy/vx5R22wcIsXL67kufgMCjQ8P+dLQwXIjOri9AB72BGVgtB6Yig6IB+tW7eu7NixwwWrYh+EfEjKblEoQcqFQuIUiqjodoFe48aNc8H8mAMrF29o5NQTJ070+T6eBYEVhHoO/vobE6enjFhc/vq4du2aehZvH8g8sVkQ3X8l8QXa9Ne3cVgaBaNKlSq29UAwCDuPCtB8OdAR34/pxXcg4RlnAZwiLIkKkMqoRN0qIiIIDsuXL1cpmFv/DhS2RAUk8hGixj74nyeqfH9nQ4IBPP0f+Z6bg++krIYAAAAASUVORK5CYII=)
,
(16)and for discrete-time systems
![](data:image/png;base64,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)
,
(17)![](data:image/png;base64,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)
,
(18)![](data:image/png;base64,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)
.
(19)With continuous-time control,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAARCAYAAAAG/yacAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABKElEQVR4nNWQMauCUBzF+wzRIPQ13Fr8CG0iuAgJLWKt0u4Q5CwuDS4u+Q1yCHQQwXQr2pqKJkFF1PO4d9AXxHu+8R24cDn8f+d/7xnhjxoR/TOoaRqs12ucTqfhkOd5GI/H2O/3w6A8zyFJEubzOXa73TDItm2Yponlcgld13+Hns8nRFHE6/XCZrOBoijUr+saZVmiqqp3qG1bbLdbuomIbFksFvQeRREYhoGmaTSgg263GwRBQJZldJAE8DzfPWk2myGO434ToVerFRzH6YYMwwDHcW9QkiQ9dDgcMJlM4LouNYuigCzLmE6nuN/vn6E0TeH7Pq7XKzXJp8MwRBAEeDwen6GPvX4TCWFZFsfjcTh0Pp+hqiosy+rbI/X+dC6Xy1sIYb4ALFrUVzMv8ggAAAAASUVORK5CYII=)
describes a diagonal matrix in which the main diagonal contains those control eigenvalues that have been added to the eigenvalues which result from the controlled system without controller integrators.
![](data:image/png;base64,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)
is the corresponding diagonal matrix for discrete-time systems. It should be emphasized in both cases that the control eigenvalues generated by
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABD0lEQVR4nMWRP4qDUBDGcwNLGw9gKZ5Bm2CpWFhKKgsrrayNhV5BCFhZxM4DpNDCRlKk0wOkUBCJiH++xVcsyAY2bLMDw3tv+DHffPMO+CAOW/wz2DQNTqcTjscjyrJE3/cwDAOiKOJyuew72rYNTdMwzzOGYYCqqgiCAK/Xaw/KsowwDAnoui6iKCL3nXTbtuB5Hnmew7IspGmKZVl+zvh4PMCyLHRdB8MwRO6tmev1CkmSME0TOI5DkiTvwc2I4zik6HkeFEXBuq57cBxHCIKAOI5Jsa5rIr+dO/B2u4GiKPi+T1w+n0/QNA3TNL9nJWBVVciyDPf7nYBd15F3URRkn3/7wo/A8/mM33LjvgAD/BCmWAl8VgAAAABJRU5ErkJggg==)
of the non-actuator-saturated P-state-controlled system are not changed by applying Eqs. (
9) and (
14) to (
16) or (
17) to (
19). It should also be noted that Eqs. (
14) to (
16) or (
17) to (
19) lead to
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA00lEQVR4nN3QMQ5FQBAGYLdwAaLRSBRKpVrcRGE7FTUnUegkdAq1SoJCotQQRIj/ZXfzopF3gDfJdN/8mRkBP0qg9T/gOA7EcQzXdUEIgeM4SNOUg/u+EQQBPM/DdV0YhgGiKKJpGg6maYKiKGjblsVWVQVVVbEsCwd5njNAk2j7vg/btnGeJwdJkkCSJGzbhrIsoes6wjB8lhzHEbIsw7IsRFEEwzBQFMUDaGzf96jrmu2haRro0OsfsiyDaZpY1/Ud0HNpQtd172Dfd8zzzC74gg+32aGgpgNd7QAAAABJRU5ErkJggg==)
control eigenvalues that cannot be controlled via
![](data:image/png;base64,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)
(see section 4). Finally, it should be noted that the calculation rules (
16) and (
19) for the pre-filter matrix
![](data:image/png;base64,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)
are the same as the corresponding calculation rules for pure P-state controllers.
Figure 1. Block diagram of discrete-time PI-state control with reference variable correction in the case of manipulated variable saturation
If we now substitute Eq. (
9) into Eq. (
11) and the result for continuous-time control into Eq. (
12), we obtain, taking into account Eq. (
15),
![](data:image/png;base64,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)
.
(20)If this relationship is combined with the controlled system state differential equation (
1), but with
![](data:image/png;base64,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)
instead of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAvElEQVR4nN2QIQqEUBRFNVtFNyCYxB24F8Vq+MUtWIyCC7BY7LoDixhdgAZBQXEBnsEvjG2mTJoHF967HO6Fp/BllP8A+r4nyzIcx+E4DmlO04RpmvJWuq6Ti6qqzPMsgaIocF33qajrGtu237FhGBJF0QPEcYwQQhrbtmEYBmVZPoDneeR5zr7vpGmKpmmM48iyLDfg+z6WZREEAcMwoOs6TdNQVdUNXLFt27Kuq4y9oEvnef7iUUmS8EkvYGzfyFWvc6oAAAAASUVORK5CYII=)
, to form an overall system, the following results
(21)with
![](data:image/png;base64,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)
,
(22)![](data:image/png;base64,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)
,
(23)![](data:image/png;base64,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)
.
(24)In the case of discrete-time systems, by inserting Eq. (
9) into Eq. (
11) and inserting the resulting intermediate result into Eq. (
13) and taking Eq. (
18) into account, the following is obtained
(25)The combination of this relationship with Eq. (
4), again with
![](data:image/png;base64,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)
instead of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAvElEQVR4nN2QIQqEUBRFNVtFNyCYxB24F8Vq+MUtWIyCC7BY7LoDixhdgAZBQXEBnsEvjG2mTJoHF967HO6Fp/BllP8A+r4nyzIcx+E4DmlO04RpmvJWuq6Ti6qqzPMsgaIocF33qajrGtu237FhGBJF0QPEcYwQQhrbtmEYBmVZPoDneeR5zr7vpGmKpmmM48iyLDfg+z6WZREEAcMwoOs6TdNQVdUNXLFt27Kuq4y9oEvnef7iUUmS8EkvYGzfyFWvc6oAAAAASUVORK5CYII=)
, to form an overall system, taking into account Eq. (
22), gives the result
(26)with
![](data:image/png;base64,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)
,
(27)![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFEAAAAfCAYAAACf3SEqAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAFZUlEQVR4nO1ZaUhVXRQNJJB+hGgUiohDlpA4CxqGlP5JoQLJUBF/qAgOIKhIg6SkomlI4hQOFBWYqDiCimKSExg5pQ2iqaWWQ1aUmWY71oZ38T2vn/fa00+fLtjoOffcu89Z5+zh7HeA9vHPOPB/T0ATsE+iGrCnSPz9+zcNDg5ST08PjY2Ncd+3b9+4Dfn8+fOmvrunSFxaWqKoqCjS0dGhmpoa7vv48SM5OjqSra0tjYyMbOq7u5rE169fU11dHQvIkIKsrCw6d+4craysCH1ubm50584dSe93dHQIOl+9esV9u5pEnKqrV6/S8+fPJZkiiAsMDKT4+Hihb2pqikxMTJgcKQBx0Kevr0+RkZHct+tJzM3NlTz++/fvbLoRERH04MEDlps3b5K5uTmTKQfGxsZ7k8QPHz6QkZERBxH8D4mNjSV3d3f6+fOnLN17lsSqqiqyt7enP3/+CH3e3t4UHR0tW7coiYuLi7wbiGAA/AfaEKQGOxFySYT/DAkJEdpIb6ysrKiiokK27jUkgsCAgAA6dOgQ5efn84Ph4WGysLDgwd3d3bKVbAfkkIg1njlzhjIyMoQ+rOvYsWPU398vW7foSSwoKCAnJydaXl4WBnp6elJCQoJsBdsFOSSWl5eTmZkZ+fr68glcWFhgEkBGTk6ObN1rSISPCA8Pp5iYGGHQ3NwcR63m5mbZCrYLcs1ZnVhDInbFxcWFbt26RZ2dnSwI/6ampvT+/fstmQTysosXL9KFCxdERYqz31Ekfvr0iUN/SUkJNTY2sgQHBzOxyK22AtPT09TU1CToU5Wurq4NvyGHxPX0qMrQ0JDwDgIqkmtFsF2NNSTi5VOnTilFYQSasLAwSRNUxbNnz0QVqxtySFRY2EaiKEzgHn3t2jWys7MTzSHXkAgz9vf3FwbAvB0cHOjx48dKL8IEi4uLeXfgnPG8tLSUvnz5wndJtGtrazni3bt37z+jHjbO0tKSN09MfHx8NiRmq8wZ6V1lZSWv9/jx40rBVgElEjHAw8OD0tLShAFv3rwhQ0NDpdQmOzubrl+/Ti9fvqTz589z1h8XF0eurq5MKJ6lp6fzLeDIkSP8HOSuhx8/ftDo6Oi6Mjk5ueFixUi8f/8+nThxgq5cuaJ0gr5+/UqnT5/m4sO7d+82/DaAtUgiEYxra2uzMwcZUIxIraWlRZmZmcJL+BgmjWoHJlNUVES/fv2iGzdukJeXFyesisoITiKi+1ZDjEQcgJMnT3Jpa35+XuhPTU0lPT09SkxM5Dbmi7WISXt7O4+RTCIGgDgko0h1ICBn9e0FMDAwYKe7+hYD0hDFcZJxChU+FSTOzs6qlzERiJGIlAxXOdxEFOUxZBjIBHB5qK+v5z6sTbEWVVGQJplEOROGyeKEwddhx7FreXl5bCrOzs50+/Ztnpy1tTXvNMZsJcRITElJobt373J5C2aLQxEUFMS+WldXlzMRKUBcgJ/HgYBbUw2UmyIRu1FdXc0Bo62tjet38JMNDQ00MzPDNx60QWhfXx8Hma2O0KokgjA/Pz/q7e3ljWxpaaGnT5/y5mM+KD5IBdaI9ShEteqtRGJSUhJJEbmlou2AKolwIZcvX+ZNvXTpEhUWFnKURzs0NHTTKZsYNLYUBrNDfgtfjQXa2NjQo0eP+ISiGPvw4UO16dZYEpHeJCcn8//wi2fPnuVUClnH4cOH2czVBY0iESaKX+5evHjBpS7UC0EcrpXj4+M8Dj7x4MGD9OTJk3/W2drayvqOHj2qGSROTEwIvxkjjcHfgYEBTtFWA9kEnm32J9HVePv2raAT+oFdTeJOwT6JaoDGkFhWViYpVVMIzFJd+AsoFZzNN5MsWAAAAABJRU5ErkJggg==)
.
(28)It can now be seen that both the dynamic matrix
![](data:image/png;base64,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)
and the transition matrix
![](data:image/png;base64,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)
are lower block triangular matrices. Their eigenvalues are identical in their entirety with the eigenvalues of the matrix blocks on the main block diagonal. The eigenvalues of
![](data:image/png;base64,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)
are therefore composed of the eigenvalues of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA2ElEQVR4nMXRLQqEYBAG4D2KVzCYBLtgEwQRg8XkESwm7yBGjyBYTYqYLAoWQQxiFn/Su8y3sCLr7paFHRgGhifMy9zwpW5Uf0bDMEDXdTbfItd1wXEc6rq+Rl3XwbIsiKKIsiyvkW3bqKqKoSzLXlFRFHAcB+u6QpIkxHF8Rvu+Q9M0pGmKcRwhyzKiKDqjJElgmibCMGQtCAKCIDjQPM9QVRVN0zxvMwwDnucdyPd9FrltW7bs+x48z0NRFCzL8kAUNc9zTNPEEE1KRkG2bfv17+imT03mDlgV2gfBVmYAAAAAAElFTkSuQmCC)
and the control eigenvalues on the main diagonal of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAARCAYAAAAG/yacAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABKElEQVR4nNWQMauCUBzF+wzRIPQ13Fr8CG0iuAgJLWKt0u4Q5CwuDS4u+Q1yCHQQwXQr2pqKJkFF1PO4d9AXxHu+8R24cDn8f+d/7xnhjxoR/TOoaRqs12ucTqfhkOd5GI/H2O/3w6A8zyFJEubzOXa73TDItm2Yponlcgld13+Hns8nRFHE6/XCZrOBoijUr+saZVmiqqp3qG1bbLdbuomIbFksFvQeRREYhoGmaTSgg263GwRBQJZldJAE8DzfPWk2myGO434ToVerFRzH6YYMwwDHcW9QkiQ9dDgcMJlM4LouNYuigCzLmE6nuN/vn6E0TeH7Pq7XKzXJp8MwRBAEeDwen6GPvX4TCWFZFsfjcTh0Pp+hqiosy+rbI/X+dC6Xy1sIYb4ALFrUVzMv8ggAAAAASUVORK5CYII=)
caused by the controller integrators. For discrete-time systems, the eigenvalues of the transition matrix
![](data:image/png;base64,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)
are composed of the eigenvalues of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABB0lEQVR4nM1RocqDYBT1GQyiwWgcZoNYjKa9gwiaLMqCVmFBMPgClsF8AJcNCj6BNoMGGRM2tiBMz8++Ovx/43/h3HvD4Z5z76WwMah/RDyfz1BVFbZt4/l8rhPrugbHcciy7HviOI6Iogi+78M0TTAMA9d1ydQ8z7EsC6jH4wFZlhHHMV6vF6mapmGaJpRlCZ7nMQwDqDRNIYoi5nkm2O/3CMOQyDVNA5qm0fc9qMvlgt1uR+S7riN9VVW43W7QdR2WZeH9foP6pCRJ4DgODMMAy7LwPA+HwwGn04lY+Nr6eDxCUZTf7/gxLEkSBEFA27brxPv9jqIoCK7X69+fWYvtxCAIsAU/7xBM616OE4wAAAAASUVORK5CYII=)
and the elements on the main diagonal of
![](data:image/png;base64,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)
, i.e. the control eigenvalues caused by the discrete-time controller integrators.
As the eigenvalues of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAARCAYAAAAG/yacAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABKElEQVR4nNWQMauCUBzF+wzRIPQ13Fr8CG0iuAgJLWKt0u4Q5CwuDS4u+Q1yCHQQwXQr2pqKJkFF1PO4d9AXxHu+8R24cDn8f+d/7xnhjxoR/TOoaRqs12ucTqfhkOd5GI/H2O/3w6A8zyFJEubzOXa73TDItm2Yponlcgld13+Hns8nRFHE6/XCZrOBoijUr+saZVmiqqp3qG1bbLdbuomIbFksFvQeRREYhoGmaTSgg263GwRBQJZldJAE8DzfPWk2myGO434ToVerFRzH6YYMwwDHcW9QkiQ9dDgcMJlM4LouNYuigCzLmE6nuN/vn6E0TeH7Pq7XKzXJp8MwRBAEeDwen6GPvX4TCWFZFsfjcTh0Pp+hqiosy+rbI/X+dC6Xy1sIYb4ALFrUVzMv8ggAAAAASUVORK5CYII=)
and
![](data:image/png;base64,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)
can be assumed to be stable, the dynamic matrix
![](data:image/png;base64,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)
and the transition matrix
![](data:image/png;base64,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)
each describe a stable system, provided the associated controlled system is stable. This is remarkable, especially as the controller integrators included in the model – considered on their own – show critically stable behavior. The reference variable correction according to Eq. (
11), in conjunction with the controller equations (
14) to (
16) respectively (
17) to (
19), has thus succeeded in forming a stable system from a critically stable system – assuming a stable controlled system but an unknown controller matrix
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABD0lEQVR4nMWRP4qDUBDGcwNLGw9gKZ5Bm2CpWFhKKgsrrayNhV5BCFhZxM4DpNDCRlKk0wOkUBCJiH++xVcsyAY2bLMDw3tv+DHffPMO+CAOW/wz2DQNTqcTjscjyrJE3/cwDAOiKOJyuew72rYNTdMwzzOGYYCqqgiCAK/Xaw/KsowwDAnoui6iKCL3nXTbtuB5Hnmew7IspGmKZVl+zvh4PMCyLHRdB8MwRO6tmev1CkmSME0TOI5DkiTvwc2I4zik6HkeFEXBuq57cBxHCIKAOI5Jsa5rIr+dO/B2u4GiKPi+T1w+n0/QNA3TNL9nJWBVVciyDPf7nYBd15F3URRkn3/7wo/A8/mM33LjvgAD/BCmWAl8VgAAAABJRU5ErkJggg==)
. Exactly this is extremely advantageous for the applicability of Lyapunov's direct method in stability analysis and controller synthesis for linear systems with manipulated variable limits (see section 4).
4. Controller Synthesis and Stability Verification
The stability analysis of PI-state control with manipulated variable saturation described below is based both on controllability considerations and on Lyapunov's direct method. In combination, both methods are also suitable for controller synthesis. Initially, however, the considerations focus on the stability analysis. In a first step, the system description according to Eq. (
21) respectively (
26) is transformed. The extended state vector
![](data:image/png;base64,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)
is mapped to the state vector
![](data:image/png;base64,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)
for continuous-time control using the transformation rules
(29)![](data:image/png;base64,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)
.
(30)Deriving Eq. (
29) with respect to time, replacing
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABYAAAARCAYAAADZsVyDAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABz0lEQVR4nO2TS6vpURjG94cwNZJPYEoZMmFkYKAkAwMTAzNSInIpE0kGMjFUCsVEDBi5pFxzz6VccokB4TmtV85ub+1zTmc7s/PWaq31tv6/9X+e911v+AfxxuI/+AncbrehVqux2+1eCz4ejyiVSjS/FPzKIPDtdkM0GgWfz0csFsN2u4VSqYRWq/0e+HA4wOfzweVyQSaTwWg0IpvNIpfLfQ/82FQqFXA4HCSTyb+CsaJfLpdncLlcBpfLRbVa/fABU9Tv9zGfz3G9XjEcDtHtdim/WCxoPRqNIBQKkU6nsVqt7uDpdIrBYEAWiMVi+P1+smG/32O9XkOj0SAcDpPnmUwGVquVlLHLgsEgpFIpUqkUeDweWVqr1e5gViiFQoF6vY5QKEQHTCYTzuczAoEA5HI5eW6z2aBSqUguW+v1eni9Xmw2G1ImEAjQbDbfrWDAyWRCiUcvPx6J2WyGTqdDoVCg0Wq1KN9oNCASiWCxWH76+gT+VUGYdIlEQi3IrOl0OqSM2bZcLkkpk8/UMRvj8TjG4/HvwexvEokEnE4nIpEIZrMZ3G43HA4HTqcT7HY7DAYDer0e8vk8PB4PXUhgduirUSwW/6jVPgfj/gDKmAi66SvdjQAAAABJRU5ErkJggg==)
by the right-hand side of Eq. (
21) and finally replacing
![](data:image/png;base64,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)
by Eq. (
29) solved for
![](data:image/png;base64,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)
then leads to the transformed differential state equation
(31)with
(32)and
![](data:image/png;base64,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)
.
(33)If in
![](data:image/png;base64,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)
and
![](data:image/png;base64,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)
the Eqs. (
14) to (
16) are taken into account, then after a few reforming steps the block diagonal matrix
(34)arises and for
![](data:image/png;base64,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)
the result
(35)is obtained. This shows that the lower
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA00lEQVR4nN3QMQ5FQBAGYLdwAaLRSBRKpVrcRGE7FTUnUegkdAq1SoJCotQQRIj/ZXfzopF3gDfJdN/8mRkBP0qg9T/gOA7EcQzXdUEIgeM4SNOUg/u+EQQBPM/DdV0YhgGiKKJpGg6maYKiKGjblsVWVQVVVbEsCwd5njNAk2j7vg/btnGeJwdJkkCSJGzbhrIsoes6wjB8lhzHEbIsw7IsRFEEwzBQFMUDaGzf96jrmu2haRro0OsfsiyDaZpY1/Ud0HNpQtd172Dfd8zzzC74gg+32aGgpgNd7QAAAABJRU5ErkJggg==)
transformed state variables cannot be controlled from
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABYAAAARCAYAAADZsVyDAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABqUlEQVR4nO2Sy4uBURjG/SGKnYWVFMqOrKRQWPJPiJSVL+zYsLFRkshSShZipVAuGxElFhTJJdf0TO87NZMmmUaz89TpnO+cvt/7vBcR/kEi0hv8Bv8EN5tNBAIB2Gw2nE4nfgiHw/B6va+B2+02Op0O5HI5drsdP+j1eqRSqdfAdEgkEnA4HHy5XC4hFosxHA5/Dbper9hut/fg2+0Gl8vFcFIul4NMJvtyTz+Mx2NMJhNcLhe+W61WGI1GWK/XDM3n8zCZTBgMBiAegw+HAzQaDUqlEvr9PpxOJ6xWKwMJQLWPx+Ow2+3odruoVqtswuPxwGg0Yr/fIxQKQaVSIZvN4nw+f4LpYDaboVQq4fP5kE6n+TuZTLJLhULBYOrDZrPBdDoFmZnP55BKpey4XC7DYrGw27saLxYL0HRQ9OPxiFarxW5JlJ7b7YbBYMBsNkOj0UAsFoMgCM/Bj0QO/X4/71qtlgOq1WouR71eh0Qi4WbXajXodDr0ej0O9BRMcx2JRHiuM5kM169QKCAYDLJzyqRSqXCm0WgUxWLxu3lU+EeLoH8RcT8ASdMLzAyM0nQAAAAASUVORK5CYII=)
, neither directly because of the zero matrix in
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAATCAYAAAB7u5a2AAAACXBIWXMAAA7DAAAOwwHHb6hkAAACQElEQVR4nL2Uy0tqURTGz18hgQ5sFCKECA1EAsVJs2riREdShOJEiRTBQQaBUjgRwUdoJOikSTRKEQc2MIgGIj5ARBGVLCvBFz6+y15wDxfu4HavdRdsOHutfX57rW/tvTl8o3H/Bd5ut3FxcYFarYbBYIDhcPg18Ol0iv39fQSDQSiVSrhcLry+vn4NfDKZIJvNkuPj4+NLwDx8PB7DYrFArVbj+PgYJycnMJvNSKfTWCwWy8GZRSIRbGxsYDab0YjH45DJZOh2u8vDHQ4H9vb2+MD9/T3W1tbQarWWgzPNt7a24Pf7yTmfz2G1WnFwcEDNXgre6/UgFAphMBhI793dXdJ+2eNI8MfHR4K/vb1RppeXl9jc3MTz8/NfA19eXqhnPNzn80GlUvELSqUSVlZW8PT0xPvYJWN+9jNLoFgsolwuYzQaoV6v07xarUIqlSKTyVBiHNNXr9fDZrPxoOvra0gkEjSbTZrf3d3BZDIhkUhgZ2eHNtHpdJDL5QSx2+3ESKVSEAgECIfDKBQK4CqVCkQiES1gwWg0Cq1Wi9vbW/6Ms1t7dnZG8e3tbYRCIfT7fRiNRhwdHSEQCNCTwUwsFvNJceyo5XI5Gg8PD8jn86T9r7a6ukrV/FzX6XRo45ubGygUCnqTmAK/wT/TJHb+PR4PNYol02g0qDKv10vfGo0GsViM4uvr61Qh68On4O/v71S62+2mbFnmTI6rqyu6wU6nE4eHh6R/MpnE+fk5vU/c6ekp/jSYvv9i3HfaD6RHwIN1NYSeAAAAAElFTkSuQmCC)
, nor indirectly via the upper
![](data:image/png;base64,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)
state variables because of the zero matrix in the lower block row of
![](data:image/png;base64,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)
. This means that the control eigenvalues contained in
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAARCAYAAAAG/yacAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABKElEQVR4nNWQMauCUBzF+wzRIPQ13Fr8CG0iuAgJLWKt0u4Q5CwuDS4u+Q1yCHQQwXQr2pqKJkFF1PO4d9AXxHu+8R24cDn8f+d/7xnhjxoR/TOoaRqs12ucTqfhkOd5GI/H2O/3w6A8zyFJEubzOXa73TDItm2Yponlcgld13+Hns8nRFHE6/XCZrOBoijUr+saZVmiqqp3qG1bbLdbuomIbFksFvQeRREYhoGmaTSgg263GwRBQJZldJAE8DzfPWk2myGO434ToVerFRzH6YYMwwDHcW9QkiQ9dDgcMJlM4LouNYuigCzLmE6nuN/vn6E0TeH7Pq7XKzXJp8MwRBAEeDwen6GPvX4TCWFZFsfjcTh0Pp+hqiosy+rbI/X+dC6Xy1sIYb4ALFrUVzMv8ggAAAAASUVORK5CYII=)
cannot be controlled via
![](data:image/png;base64,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)
. As these eigenvalues can be considered to be stable, the subsystem described by the lower block row in
![](data:image/png;base64,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)
and
![](data:image/png;base64,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)
is both stable and not controllable and therefore does not need to be considered further in the stability investigations.
If the control is discrete-time, comparable statements can be made. Thus, the application of the transformation
![](data:image/png;base64,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)
,
(36)
(37)to Eq. (
26), taking into account Eqs. (
17) to (
19), leads to the transformed state difference equation
(38)with
![](data:image/png;base64,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)
,
(39)![](data:image/png;base64,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)
.
(40)This also results in a subsystem that cannot be controlled via
![](data:image/png;base64,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)
, which has the eigenvalues contained in
![](data:image/png;base64,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)
that can be assumed as stable. Due to its stability and non-control lability, this subsystem can also be disregarded in the further stability investigations.
The previous explanations have shown that in both continuous-time and discrete-time system description and control, only the transformed first subsystem needs to be considered further for stability analysis after the state transformation has been carried out. However, this is precisely the controlled system with its manipulated variable vector
![](data:image/png;base64,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)
as the input variable. It is therefore sufficient to find a stabilizing control law for the controlled system using a P-state controller with the feedback matrix
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABD0lEQVR4nMWRP4qDUBDGcwNLGw9gKZ5Bm2CpWFhKKgsrrayNhV5BCFhZxM4DpNDCRlKk0wOkUBCJiH++xVcsyAY2bLMDw3tv+DHffPMO+CAOW/wz2DQNTqcTjscjyrJE3/cwDAOiKOJyuew72rYNTdMwzzOGYYCqqgiCAK/Xaw/KsowwDAnoui6iKCL3nXTbtuB5Hnmew7IspGmKZVl+zvh4PMCyLHRdB8MwRO6tmev1CkmSME0TOI5DkiTvwc2I4zik6HkeFEXBuq57cBxHCIKAOI5Jsa5rIr+dO/B2u4GiKPi+T1w+n0/QNA3TNL9nJWBVVciyDPf7nYBd15F3URRkn3/7wo/A8/mM33LjvgAD/BCmWAl8VgAAAABJRU5ErkJggg==)
. The stability of the resulting PI-state control by means of Eqs. (
14) to (
16) respectively (
17) to (
19) is then automatically ensured on the basis of the above explanations. In order to achieve a comparable statement for discrete-time systems, a complex modal transformation of the extended controlled system was carried out in
[8] | Nuss, U. Stabilitätsverhalten von zweistufig entworfenen zeitdiskreten PI-Zustandsreglern bei Stellgrößenbegrenzungen [Stability properties of two-stage designed discrete-time PI state controllers considering the limitation of input variables]. at – Automatisierungstechnik. 2017, 65 (10), pp. 705 – 717. https://doi.org/10.1515/auto-2016-0136 (in German) |
[8]
. However, the above considerations have considerably simplified the proof. Comparable considerations have not yet been made for continuous-time systems.
To find a P-state controller that also stabilizes with manipulated variable limits, the following Lyapunov function is used
![](data:image/png;base64,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)
.
(41)In this,
![](data:image/png;base64,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)
denotes the deviation of the state vector
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAATCAYAAAB7u5a2AAAACXBIWXMAAA7DAAAOwwHHb6hkAAAB+klEQVR4nO2TuasiQRDG379hYiCIoYEYmJgrHrEGirGx4BUKoogHGBh5gKCCgYGZBoK3poKBgooiCCbet99SDc46zApvWTZ7BQ3T01W/rvqq+gv/yb7IfuA/8L+D93o9WCwWdLtdnM9nxGIx+P1+PJ/Pf4OfTid4vV5YrVa2EokEfD4f4vE450iXzOdzHI9HAeRyuWA6neLxeAjh9/sdq9UKjUYDIpEIkUgE1+uVc6LzXC6HVCqF/X6PZrMJvV7PvsmoUjpPJpO8OJ7mo9EIEokEg8GAl0GtVoPL5cLtdmP77XYLrVbLMn5PIBgMolQqCeHkGAgEIJVKUSwWWSUkFwWRVP1+nwtqt9twu904HA4wGo0ol8vs/3A4hNls5qRjcNLX6XQim82yII1Gw/abzYZdIJfLsdvtODj1gmTI5/MoFAqcFARVq9VYLpe/4ZRxOp1mZZM0drsdrVaLOZCuYrGYJ5PJZGKTZbPZBM1VKBSYTCZCzf9kVLpMJmNNI6NqDAYDFosFlEolz5cqUKlUbKq+BacAnU7H9CSjd0A9oF5Q9uFwGOPxmJ3NZjM2RZQAB49Go/i0Op0Ok4wgr8te+pPGBHo9tEwmg1AoxO0ZnEbr06IpIojH40GlUvlYIU2Qw+Hgsv6WLO/y1Ot1rNdrwRlVUq1Wub68w38Bmxb6MaEtzqQAAAAASUVORK5CYII=)
from its stationary position, which is denoted below by
![](data:image/png;base64,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)
. The following therefore applies
![](data:image/png;base64,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)
.
(42)Furthermore, the (
![](data:image/png;base64,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)
)-dimensional matrix
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA1klEQVR4nMXRIQqEQBQGYG/gFUyKwUOY9QBmDyBY9AgGm2Czi2C1m8SqICaxKYKYDeovM8FlYGeXTfvgwYP5+HnME/ClBFJ/QOu6wrZtWJaFIAjgeR6KosB5nmyS4zj08bou1HUNRVGwLAuLTNNEmqZ0nucZkiRhHMcX2rYNqqqi73uKSJIsy3SNB7VtS9E0Tei6DoZhIEkSdvEsy6BpGsIwRBzHTyKDXNeF7/v8L9j3HbquI89zPirLEqIoIooiHMfxHg3DgKqq0DQNH/HO8Tsit/rUxNyTu9xcsj+S+gAAAABJRU5ErkJggg==)
contained in Eq. (
41) must be a positive definite matrix yet to be determined. This means that
![](data:image/png;base64,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)
holds for
![](data:image/png;base64,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)
and
![](data:image/png;base64,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)
only for
![](data:image/png;base64,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)
. If it is now possible to ensure that
![](data:image/png;base64,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)
constantly decreases for
![](data:image/png;base64,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)
and approaches the steady state, i.e.
![](data:image/png;base64,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)
, for
![](data:image/png;base64,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)
then the stability of the system under consideration is proven
[2] | Adamy, J. Nonlinear Systems and Controls. Berlin, Heidelberg: Springer Vieweg; 2022. https://doi.org/10.1007/978-3-662-65633-4 |
[3] | Tarbouriech, S., Garcia, G., Goes da Silva Jr., J. M., Queinnec, I. Stability and Stabilization of Linear Systems with Saturating Actuators. London: Springer; 2011. https://doi.org/10.1007/978-0-85729-941-3 |
[4] | Vidyasagar, M. Nonlinear Systems Analysis. 2nd edition. Philadelphia: SIAM; 2002 (unabridged republication). https://doi.org/10.1137/1.9780898719185 |
[2-4]
.
In order to be able to evaluate whether
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA0ElEQVR4nL2QoQqEYBCEfQrB6rsYxKSYDGIT41/MdqMmu8EXEKv4AoIWo5gFxSboHP8eeBzqXTlu2rLfMjMr4IsErj9DdV1DVVXouo6u62iR5zkURYFlWU9oXVdomgbXdbFtG0HjOEKWZVRV9bLzPA++7x8WWZbRzI8OiDEGx3EImKYJhmFgGIb34GEYkj9XFEVIkuTcLo5jmKaJvu9h2zbmeT5DaZpS+CAIUBTF9Z/4QhRFysbbXkJlWUKSJLRte//xZVnQNA32fb+HPum30AN4NOWACq7mJQAAAABJRU5ErkJggg==)
decreases with a continuous-time system description,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA0ElEQVR4nL2QoQqEYBCEfQrB6rsYxKSYDGIT41/MdqMmu8EXEKv4AoIWo5gFxSboHP8eeBzqXTlu2rLfMjMr4IsErj9DdV1DVVXouo6u62iR5zkURYFlWU9oXVdomgbXdbFtG0HjOEKWZVRV9bLzPA++7x8WWZbRzI8OiDEGx3EImKYJhmFgGIb34GEYkj9XFEVIkuTcLo5jmKaJvu9h2zbmeT5DaZpS+CAIUBTF9Z/4QhRFysbbXkJlWUKSJLRte//xZVnQNA32fb+HPum30AN4NOWACq7mJQAAAABJRU5ErkJggg==)
is derived with respect to time and then the sign of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA5ElEQVR4nK2QrQqEUBCF9ykEq+9iEJNiMohNjBaz3agYjILBFxCr+AAKWoxiFhSboGe5d8HdxZ8te9KdO99wzswDP/Qg+vwIggBlWd5DpmkiiqJ76NKuqioIggBJktC2LW2kaQqe56Gq6gtalgWiKMIwDKzrSqFhGMBxHIqieNuRLLZt7xZJktCaDO2QZVnQdZ0C4zhClmX0ff8d3HVd6k/keR7CMDxu5/s+FEVB13XQNA3TNB2hOI5peMdxkGXZ+Z1Ig2EYmo1sewrleQ6WZdE0zfkxyWOeZ9R1jW3brqE7/Rd6Am1P4NwNt5IyAAAAAElFTkSuQmCC)
is examined. This results in
![](data:image/png;base64,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)
.
(43)![](data:image/png;base64,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)
itself is obtained from the controlled system state differential equation (
1) with
![](data:image/png;base64,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)
instead of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAvElEQVR4nN2QIQqEUBRFNVtFNyCYxB24F8Vq+MUtWIyCC7BY7LoDixhdgAZBQXEBnsEvjG2mTJoHF967HO6Fp/BllP8A+r4nyzIcx+E4DmlO04RpmvJWuq6Ti6qqzPMsgaIocF33qajrGtu237FhGBJF0QPEcYwQQhrbtmEYBmVZPoDneeR5zr7vpGmKpmmM48iyLDfg+z6WZREEAcMwoOs6TdNQVdUNXLFt27Kuq4y9oEvnef7iUUmS8EkvYGzfyFWvc6oAAAAASUVORK5CYII=)
. Here,
![](data:image/png;base64,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)
is also divided into a stationary component
![](data:image/png;base64,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)
and the deviation
(44)of it. With further consideration of
![](data:image/png;base64,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)
, the following follows from the controlled system state differential equation
![](data:image/png;base64,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)
.
However, since
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAxElEQVR4nL2RLwqEcBCF9xBWk3gPT2Az2AxewWYXTBaT3WgziwfwTzGYLNpVtBj8lp+yQXAXFpYdGCbMx7zHmwcf6iHqj0DbtliWxTRN98C6rhRFcczvJPZ9J45jVFUlSRLGccQ0TWzbPoFlWQiCAN/30XUdx3HIsow8z68SVVUhSRJpmt57KMsSWZap6/oKDMNA13XHaU3TCMPwOD/P8wkIQ4Zh0DQNURShKAqu67Jt2wmIRd/3lyxeYf3oF57n8a7F/gl5XaKKmWkzfgAAAABJRU5ErkJggg==)
must be the zero vector in the steady state and
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAxElEQVR4nL2RLwqEcBCF9xBWk3gPT2Az2AxewWYXTBaT3WgziwfwTzGYLNpVtBj8lp+yQXAXFpYdGCbMx7zHmwcf6iHqj0DbtliWxTRN98C6rhRFcczvJPZ9J45jVFUlSRLGccQ0TWzbPoFlWQiCAN/30XUdx3HIsow8z68SVVUhSRJpmt57KMsSWZap6/oKDMNA13XHaU3TCMPwOD/P8wkIQ4Zh0DQNURShKAqu67Jt2wmIRd/3lyxeYf3oF57n8a7F/gl5XaKKmWkzfgAAAABJRU5ErkJggg==)
according to Eq. (
1) is identical with
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFgAAAASCAYAAADfVhk+AAAACXBIWXMAAA7DAAAOwwHHb6hkAAAFA0lEQVR4nO1Y6SumbRR//gMf+KAoKXySbSKlrBEflPnCZIYhW5bJWkJZxzajRhOjUGRLdmEsSdYs2SIZZCuy7/t63n7n7X6ax4PHM5Z5e/OrO+7Lua7rXL9zrt85NxG94lkh+tsO/N/xSvAz45XgZ4YUwVdXV/Tjxw+qqKj4G/48CP39/eTl5UVhYWGUkJBA8fHx7O/Jycnfdk0KUgSPjIyQuro6hYSEvKgjIAdEra+vy7Q9ODggPT09Kigo4Pf5+XnS19enwsLC53ZTbkgQfHp6SsHBweTt7U0fPnx4UUf29/fJzMyMyZKF7e1t0tLSor6+PvHYu3fvKCYm5vkc/ENIENzc3EyRkZGUkZFBDg4Ot04QMi08PJyOjo5oZmaGA9LT0/MoR+QheGxsjFRUVGhzc5PfV1dXycDAgH7+/Clht7y8TFlZWRQREUEXFxc81t7eTlFRUXR+fv4ofx8KMcE44Pv375mw7OxssrGxETv1OxoaGujr16+kqalJlZWVFBAQQHFxcTQxMfEoR+QhOCcnhzQ0NKijo4Nqa2tZj5EUZ2dnEnb19fU0OTnJwdjY2OAx6DZ8fikwwdfX15Sfn0/R0dE8WFVVRcbGxnzom1hbW+PoOzo6koWFBQ0NDf3x5giQqakpPyYmJqSgoEBGRkbisdjYWKk58BWEfvz4kQkuKipiP/D7bWhqauI1UbwRANhizkuBCUZhwYFCQ0O5Kru5udGbN29od3f3zom4Zm/fvn3U5rja0FE8bW1tXKiqq6vFY7hNNwGSDA0NqaSkRDwWGBhITk5Ot+4BeYCcAVNTU5z5o6OjD/YRAf306RMtLCzIebp/IcICnz9/5tZMAHQKRWRlZeXWSdBAZ2dn0tHR4YKDzSEneXl5rM+QmK2tLUpJSSFfX1+WkJ2dnXsdeahEYF1FRUX2AUBmuri4kKenp5QtfLK1teUWDoFBEdTV1RXLRXl5OZMPHxFY2CC7k5KS+HxLS0us68rKyhxE1CgB6GTS0tLYFjd+cXGR1/L392cuBXkVjY+Pk5qaGs3Ozoon4+oqKSnR9PS0hMNYxMfHhzw8PLid09bW5nYOmoh3bIaFcSNwhUE2AgAHQP5TEAxdVVVVZTLQ9YAk3LbBwUEpW9hAHlCEITfp6elkbm5OAwMDNDw8zIVRIKempob29vaorKyMizfsBEJxzl+/fkmsjaDhFl1eXrJMoimAL0gkSC14ZYKxMQiDVgEwQMa5u7vzIshwASgYIBQkIHPQdwok9vb2im8BCEafKmyCw3///l0mwdDH+whG8OA8CIOU4Ybk5ubynN/9FICxlpYW8vPz4y4HfiGLQRyCg4JtbW1Nrq6uXFtgDxlBgkGuGhsb7yQYZ8faQGtrK9nZ2fEaQGZmpriFfLJPZbRvODSqOQLW3d3NEa6rq2Mi5ubm7p2Pw+HqIiNeCiAddQbJlJqaSsXFxXxDETArKysOAAApvFnMUT/Q0iJxIDH4usR8dDWJiYl0eHjIdk/6vwhcSfSmQveBbgMHEDb7rwFFOjk5mezt7amzs5O/DNGelpaWkqWlJROPxAkKCuLOBdLy7ds3blOB4+NjTgr8FN5xXtwOAaIvX76QrAeaBMCJu2yQqc8NyJAsXwWpA2TZoogjGZAAuEGQPbyDVJAljCNRUNTwd5Atz1lFYFvWI+gbNPAum9s+Sp4akA9Zvv7ux0PPJQ+6urrkmvcPLdvLykVEjmMAAAAASUVORK5CYII=)
in the steady state,
![](data:image/png;base64,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)
holds. Overall, this gives the state differential equation
(45)for the deviation of the controlled system state vector from its stationary position. This used in Eq. (
43) then leads to
![](data:image/png;base64,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)
.
(46)For
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA0ElEQVR4nL2QoQqEYBCEfQrB6rsYxKSYDGIT41/MdqMmu8EXEKv4AoIWo5gFxSboHP8eeBzqXTlu2rLfMjMr4IsErj9DdV1DVVXouo6u62iR5zkURYFlWU9oXVdomgbXdbFtG0HjOEKWZVRV9bLzPA++7x8WWZbRzI8OiDEGx3EImKYJhmFgGIb34GEYkj9XFEVIkuTcLo5jmKaJvu9h2zbmeT5DaZpS+CAIUBTF9Z/4QhRFysbbXkJlWUKSJLRte//xZVnQNA32fb+HPum30AN4NOWACq7mJQAAAABJRU5ErkJggg==)
to decrease for
![](data:image/png;base64,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)
,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA5ElEQVR4nK2QrQqEUBCF9ykEq+9iEJNiMohNjBaz3agYjILBFxCr+AAKWoxiFhSboGe5d8HdxZ8te9KdO99wzswDP/Qg+vwIggBlWd5DpmkiiqJ76NKuqioIggBJktC2LW2kaQqe56Gq6gtalgWiKMIwDKzrSqFhGMBxHIqieNuRLLZt7xZJktCaDO2QZVnQdZ0C4zhClmX0ff8d3HVd6k/keR7CMDxu5/s+FEVB13XQNA3TNB2hOI5peMdxkGXZ+Z1Ig2EYmo1sewrleQ6WZdE0zfkxyWOeZ9R1jW3brqE7/Rd6Am1P4NwNt5IyAAAAAElFTkSuQmCC)
must be negative. To ensure this even with saturated manipulated variables, the expression
![](data:image/png;base64,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)
is considered separately from the expression
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFUAAAASCAYAAAAqqJKOAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAFc0lEQVR4nO1XWUhVbRT1RfQpo54SNROU8EEsMRIlBA0xc0iNJBKJcHxQybHCqdQc6MGpHFIUJbU0ccwxzcCCUEkEp9QMx0xRs3Js/6wN53DVc2/3/ijB/7vgg3vP/c437L3W2vuq0SH2HWp/+wD/RRx4UDc2NujKlSuUkJBAt27dotDQULpz5w7du3ePtre3D3r7v4IDD+ra2hq5uLjQ5uYmeXl5UUlJCU1PT5O/vz9tbW0d9PZ/BRzUxcVFev36NRUUFFBhYSH19vbS79+/5b7069cvKioqouXl5T9ugHUWFhb4sxBUYU8plJeXU3Z2No+nT59SZ2cnJ2Q3vnz5Qrm5ueLc/Px8ev/+vUL24yyYJ2/v/YLaz58/6dixYxQVFUXz8/PU0NBAJ0+e5AvJQ2lpKWloaFBNTY1Km8kGVR5evnxJWlpazObPnz+zdcAqdgO2YmtrS0FBQZzcoaEhsra2ptTUVLlrt7W1kbq6Ogf2IKEGFiDTkCmATF+8eJEuX74s+QLm46Lu7u508+ZNlTZTJqhQgKWlpfgd6jl37pwkA83MzKisrEz8Hh8fT66urnLXvn79Op/Zzs5OpXOrij2eCmmcOXOGYmNjJV+ANXh6enLWT506xewGIK26ujoKCAhghsEiHj16RI8fP+bfERQPDw/KzMxUeCA/Pz8KCQkR14yMjCRnZ+c98759+0ZHjhyh0dFR/g7FOTk5yWUq5iHg/f39/N7U1BQ/Hx8fp4yMDMrKyhI9vra2lp/920K6I6hYFIFAJufm5iRfCAwMpNbWVvZJY2Njqq6u5uefPn2ipKQklmRMTAzFxcVRenq6aCOQJ9iNAvX161e5BzIxMWFfhbzfvHnDLIXf70ZLSwsdPXqU7Qo2FBwczPt+//5dct0HDx6wbSFQpqamlJeXR+vr69TU1ETNzc109uxZWllZ4bnXrl2jhw8fKhE+aYhBBSueP39Ojo6ONDExITl5aWmJLCws+MI4nI+PD7dJwI8fP5gtkC88uaKiQmGxkwJYr6mpSXfv3qXExERm3cePHyXXQYt26dIl6unpoY6ODrarnJwcyXURaBsbG7FgRkdHk729vfg71INAYp/V1VUyMjKit2/fqnR2WYhBbWxsZPkIspDCkydPdmTw1atXZGhouIN5YJWOjg7LU1WA9bAUZZLh4OBAycnJ4nco4vTp05LvIsGwFEHOSNTx48e5g8Czq1evsncD7e3tpK+vT7Ozs0qfG2yPiIgQ9+agDg4OciaFgKJoQV6ywOaQouxm+IyLwIPwDrwUMtTT02PPhf9CYsoiPDyc/fpPAJtOnDjB7RaAy+BSUgUI5wZZcBYB8HtYABIBdeBeaMdgedgf60B5fX195ObmxgUQtgaALEgmrAaq/fDhAxc/c3Nzfg4Vc/WH5C9cuEC+vr48YOg3btzYcTgwUFtbW5yD4e3tTbq6upxpyA/VdWxsjAOLTRAkZS1gZmaGE4QOQehEpID1qqqq2E8RqOHhYWYZ2ikEYTdgD2AeLi57dkgcbIclIKj4fPv2bQ40uhtYIXzYysqKBgYG2GIAFHKQEHE7f/48vXv3joqLi3l9kakoTjgkJCI7ZDMLjIyM7JkjDNgAmC0wHXZQX1+v1J8DATgo1qqsrFTYnIN5SLCwN87e1dXFzJLC5OSk3HPD8oS9UfBQM+C/uA/qCliHFhAsFeQNNj979oyt0MDAQDqoSt/6fwh0IWA61ABFIqFhYWFc6BBc9Mkgz4sXL9hiQDzMOQyqAnR3d3MXguKMtguAh+LfJ4oTPDQtLY2t6/79+8x2sHVfggqJpaSkSA70vcoAzbe8NTCENg/+pmgeBuSoDCB3RetI9cfKQO0Q+49/AHkSU73lI4vRAAAAAElFTkSuQmCC)
. Because the former expression depends quadratically on
![](data:image/png;base64,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)
, it is generally to be expected that
![](data:image/png;base64,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)
reaches values which, due to the limitation of
![](data:image/png;base64,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)
, lead to such a large amount of
![](data:image/png;base64,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)
that this term determines the sign of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA5ElEQVR4nK2QrQqEUBCF9ykEq+9iEJNiMohNjBaz3agYjILBFxCr+AAKWoxiFhSboGe5d8HdxZ8te9KdO99wzswDP/Qg+vwIggBlWd5DpmkiiqJ76NKuqioIggBJktC2LW2kaQqe56Gq6gtalgWiKMIwDKzrSqFhGMBxHIqieNuRLLZt7xZJktCaDO2QZVnQdZ0C4zhClmX0ff8d3HVd6k/keR7CMDxu5/s+FEVB13XQNA3TNB2hOI5peMdxkGXZ+Z1Ig2EYmo1sewrleQ6WZdE0zfkxyWOeZ9R1jW3brqE7/Rd6Am1P4NwNt5IyAAAAAElFTkSuQmCC)
. The matrix term
![](data:image/png;base64,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)
is therefore assigned a positive definite or at most a positive semi-definite matrix
![](data:image/png;base64,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)
respectively
![](data:image/png;base64,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)
by
(47)or
![](data:image/png;base64,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)
.
(48)It holds that if the matrix product
![](data:image/png;base64,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)
respectively
![](data:image/png;base64,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)
is positive definite, the matrix
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA1klEQVR4nMXRIQqEQBQGYG/gFUyKwUOY9QBmDyBY9AgGm2Czi2C1m8SqICaxKYKYDeovM8FlYGeXTfvgwYP5+HnME/ClBFJ/QOu6wrZtWJaFIAjgeR6KosB5nmyS4zj08bou1HUNRVGwLAuLTNNEmqZ0nucZkiRhHMcX2rYNqqqi73uKSJIsy3SNB7VtS9E0Tei6DoZhIEkSdvEsy6BpGsIwRBzHTyKDXNeF7/v8L9j3HbquI89zPirLEqIoIooiHMfxHg3DgKqq0DQNH/HO8Tsit/rUxNyTu9xcsj+S+gAAAABJRU5ErkJggg==)
is also positive definite if the controlled system is stable, i.e. if
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA2ElEQVR4nMXRLQqEYBAG4D2KVzCYBLtgEwQRg8XkESwm7yBGjyBYTYqYLAoWQQxiFn/Su8y3sCLr7paFHRgGhifMy9zwpW5Uf0bDMEDXdTbfItd1wXEc6rq+Rl3XwbIsiKKIsiyvkW3bqKqKoSzLXlFRFHAcB+u6QpIkxHF8Rvu+Q9M0pGmKcRwhyzKiKDqjJElgmibCMGQtCAKCIDjQPM9QVRVN0zxvMwwDnucdyPd9FrltW7bs+x48z0NRFCzL8kAUNc9zTNPEEE1KRkG2bfv17+imT03mDlgV2gfBVmYAAAAAAElFTkSuQmCC)
only has eigenvalues with a negative real part
. Due to the special choice of the right-hand side of Eq. (
47),
![](data:image/png;base64,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)
can be any
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAsklEQVR4nN3QQQpFUBjFcRNlxtgmlD1QtmAVVkHKiKktWIChCSO6t5SyBXOS8n+5iuGbvcE7w9Ovzten8SXaf4BhGMjzHNd12baNKIqwbZskSW7Q9z3ruqLrOmmaIoSgKAocx3knmqZRQEqpyizLCMPwBXEc4/v+s+t5HmVZviAIAoWuXHcYhsE8zyzLgnYcB6Zp0nWdAldpWRZ1XVNV1Q3atmXf92diHEemaeI8z1988gN5vurb5LoWpAAAAABJRU5ErkJggg==)
-column matrix without violating the positive semi-definiteness of
![](data:image/png;base64,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)
. Only if
![](data:image/png;base64,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)
is to be positive definite, the number of rows of
![](data:image/png;base64,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)
must be at least as large as the number of columns of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABO0lEQVR4nM2RMauCUBiG24XQX2CDYDj2GxyrwSXcnFr6ByLZIEl7UH+gybGlPRRxbygQHByiWoQizfK9nBNXr9zL5Y73hbOc7znn/b73a+APahD9A/D1esHzPMxmM0ynU0wmE0RRVAeLosBqtYKiKAjDEM/nE4vFAr1eD4/HowJJsd1uw/f90i4IAnAchyRJ3iD5zTAMdLtdav8p13XBMEwFpmmKTqeD5XJZG8BxHPA8j9vt9gbJi2azie12WwNHoxEGgwHtl4KXywUsy2K/35fQ6XSCKIpYr9fV1NfrFa1Wi0ZDlOc5xuMxVFXF/X6vQDLAfD7HcDjEZrOBbdvU9ng8fg+cwIfDAf1+H5qm0ex+XSEJXZIkxHGM3W6HLMt+Bs/nM2RZhiAIdJU1kPT09ZimCV3XYVlWeUe4D6aMBms8/O9cAAAAAElFTkSuQmCC)
and, in addition,
![](data:image/png;base64,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)
must apply
[16] | Zurmühl, R., Falk, S. Matrizen und ihre Anwendungen [Matrices and their applications]. 5th edition. Berlin, Heidelberg, New York, Tokyo: Springer; 1984. (in German) |
[16]
. Furthermore, it should be noted that Eqs. (
47) and (
48) are Lyapunov equations. How they can be solved in principle can be read, for example, in
.
If there is a solution for
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA1klEQVR4nMXRIQqEQBQGYG/gFUyKwUOY9QBmDyBY9AgGm2Czi2C1m8SqICaxKYKYDeovM8FlYGeXTfvgwYP5+HnME/ClBFJ/QOu6wrZtWJaFIAjgeR6KosB5nmyS4zj08bou1HUNRVGwLAuLTNNEmqZ0nucZkiRhHMcX2rYNqqqi73uKSJIsy3SNB7VtS9E0Tei6DoZhIEkSdvEsy6BpGsIwRBzHTyKDXNeF7/v8L9j3HbquI89zPirLEqIoIooiHMfxHg3DgKqq0DQNH/HO8Tsit/rUxNyTu9xcsj+S+gAAAABJRU5ErkJggg==)
in Eq. (
47) or (
48), then it can be enforced that the right-hand side of Eq. (
46) does not become positive, provided the controlled system is asymptotically stable or critically stable. To do this,
![](data:image/png;base64,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)
is set in the way
(49)where
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA+0lEQVR4nMWRr4qFUBDG7xv4AjYRiw9wo3aTzSZmr2DSYjLYBF/AKmIzmkyaFUSL2MU/1aB8l3NgkcvCblnYgQMzHz/OfDPzwC/xIPEP0LIsMAwDmqYhCAK4ros0TXGe5+dPlmXBcRya930PQRAwDMMnpCgKsiyj+bquFKrr+oa2bYMoihjHkUJFUeD5fGLf9xtq2xYsy8LzPLxeL5imiWmaPo0Tk6qqUqO6riMMw+/T2bYN3/epmCQJJEnCdV03dBwHZFlGnudU7LoOPM9jnucbKssSDMMgiiLajuyM4zjEcXzviUxUVRWapqEiaUMG+ar/+HbkVj89wrwBionPt6C5zSwAAAAASUVORK5CYII=)
is a positive definite (
![](data:image/png;base64,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)
) -dimensional diagonal matrix with otherwise arbitrary diagonal elements and
![](data:image/png;base64,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)
is an arbitrary matrix of suitable dimension, provided that Eq. (
47) is used as the basis for
![](data:image/png;base64,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)
. When using Eq. (
48),
![](data:image/png;base64,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)
must be selected. The chosen approach based on Eq. (
47) is oriented towards the so-called Kalman-Yakubovich-Popov equations
[2] | Adamy, J. Nonlinear Systems and Controls. Berlin, Heidelberg: Springer Vieweg; 2022. https://doi.org/10.1007/978-3-662-65633-4 |
[3] | Tarbouriech, S., Garcia, G., Goes da Silva Jr., J. M., Queinnec, I. Stability and Stabilization of Linear Systems with Saturating Actuators. London: Springer; 2011. https://doi.org/10.1007/978-0-85729-941-3 |
[4] | Vidyasagar, M. Nonlinear Systems Analysis. 2nd edition. Philadelphia: SIAM; 2002 (unabridged republication). https://doi.org/10.1137/1.9780898719185 |
[2-4]
, which generally results in more degrees of freedom than the more classical approach based on Eq. (
48) with
![](data:image/png;base64,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)
. However, it is often sufficient to work with
![](data:image/png;base64,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)
and the somewhat more simply structured formulas, using either Eq. (
47) or Eq. (
48) as a basis.
Eq. (
49) implies the controller matrix
![](data:image/png;base64,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)
.
(50)With this and Eqs. (
46), (
47) and (
49) we obtain
![](data:image/png;base64,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)
![](data:image/png;base64,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)
.
(51)If
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA+0lEQVR4nMWRr4qFUBDG7xv4AjYRiw9wo3aTzSZmr2DSYjLYBF/AKmIzmkyaFUSL2MU/1aB8l3NgkcvCblnYgQMzHz/OfDPzwC/xIPEP0LIsMAwDmqYhCAK4ros0TXGe5+dPlmXBcRya930PQRAwDMMnpCgKsiyj+bquFKrr+oa2bYMoihjHkUJFUeD5fGLf9xtq2xYsy8LzPLxeL5imiWmaPo0Tk6qqUqO6riMMw+/T2bYN3/epmCQJJEnCdV03dBwHZFlGnudU7LoOPM9jnucbKssSDMMgiiLajuyM4zjEcXzviUxUVRWapqEiaUMG+ar/+HbkVj89wrwBionPt6C5zSwAAAAASUVORK5CYII=)
is now chosen so that
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEUAAAASCAYAAAANBhNmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAEMUlEQVR4nO1WWSh1bRRWLrmQCzdEucKNC0LGkAgRijJFLiSFCHfKLEOhZEjGDBeETJmljCUkyZAMGTKFzNP6etb/793Z52z6nPP/fZ/y1NvZe5/1rvddz3rWel8t+oEEWsCf3sTfhm9Nys3NDQUFBVFeXh5FRUVReno6paWlUW5uLr2/v6vt91uTcnFxQWFhYfTy8sLk9PT00O7uLiUkJNDb25vafkVSLi8vaWxsjOrq6qihoYFWVlZk2cZira2tVFVVxaO2tpYmJibo6elJ/ej+xdnZGVVXV7PfpaUl8fvg4CBtbGzwM9SBNWGDdbFvQCBFiEURiAv29fX1dH9/L67V2dlJDw8P/D43NyfGxKTc3d2Rvr4+ZWRksHFfXx8ZGxszOXJoamoiQ0NDOjk5oYODAwoPD6fk5GSNsgMg4yEhIWRjY8PBA1tbW2RkZESNjY38jjWgBEtLSzo+PhbnKpKijJ2dHdLW1qaamhox0Tk5OWRhYUGnp6f8Dl/goKio6B9Snp+fOUOPj49s8Pr6Sq6urhQQECC7SEVFBfn6+orvyICVlRVdXV1pRAqAnhAYGMjPCCAzM5NMTU0lpPj7+9Ps7Kxk3mekIB5dXV1WFrC3t0cuLi5ka2srkrKwsEB+fn4ELmR7CmoVmUDDUgY2GhkZSfn5+fyO7MbHx1NsbCwvrikKCgo4aGBzc5N9BwcHi6SgFGJiYiSljXWRJMFGGfgfyh8dHeV5xcXFrAhHR0cmBTFERETQzMwM26uQAgeFhYXk7e3NpaQMMGlubk5ZWVlcZvhNSUnhUvovIKgQisjOzmYSQkNDOWAoGerd3t6WzFldXWV1JSYmqvQTAL5QKsPDw1zuIHVxcZEcHByYlJGREUpKSmJyVEgBi+3t7Zyp/f192U3DqZ6eHrNaVlZG9vb2shtZX1+nlpaWT8f09LTKPNS9l5cXBx4XF8cNHKQ0Nzfz3qDQr/Yu2EP5/f39VFJSQr29vbS2tsZKQZzoiVClAAkpAwMDTMjR0dGHC7S1tZGdnR0/ozmZmZnR1NSUit34+Dg3xM+GnNzR3D09PUWVACAFpwKasJx6f4cUa2tr7n0gAGoHKU5OTlRZWcnlpFiOIinIrIeHBx0eHvIfOKpQg8pAjUNqAOSMOeXl5V/e6EcYGhpiqUPiQo8CKc7OznwVUAcI2MfHh9zd3TnxAEiBeiACZaKZFNQS6hgLo2FioHajo6MlxiDMxMRElDWQmppKbm5uamVQDjghdHR0aHJyUvwGUkD+7e2tWj5BCk4W7FMgGqQYGBiw8pXBpMCwq6uLOjo6JGN5eVliDEf43t3dLR6/uEHiEqRYk5rg/Pycs6l4ks3Pz/N9RRPgCBcugMD19TVXgnCZU8S3vub/X/ghRQZfIqW0tJTvMB8NlJUm+My33PidBo+b6lf9av1AHr8AOEpU5EVdjOwAAAAASUVORK5CYII=)
is positive definite – for which the diagonal elements of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA+0lEQVR4nMWRr4qFUBDG7xv4AjYRiw9wo3aTzSZmr2DSYjLYBF/AKmIzmkyaFUSL2MU/1aB8l3NgkcvCblnYgQMzHz/OfDPzwC/xIPEP0LIsMAwDmqYhCAK4ros0TXGe5+dPlmXBcRya930PQRAwDMMnpCgKsiyj+bquFKrr+oa2bYMoihjHkUJFUeD5fGLf9xtq2xYsy8LzPLxeL5imiWmaPo0Tk6qqUqO6riMMw+/T2bYN3/epmCQJJEnCdV03dBwHZFlGnudU7LoOPM9jnucbKssSDMMgiiLajuyM4zjEcXzviUxUVRWapqEiaUMG+ar/+HbkVj89wrwBionPt6C5zSwAAAAASUVORK5CYII=)
only have to be chosen sufficiently large – then
![](data:image/png;base64,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)
is a positive semi-definite matrix term
[16] | Zurmühl, R., Falk, S. Matrizen und ihre Anwendungen [Matrices and their applications]. 5th edition. Berlin, Heidelberg, New York, Tokyo: Springer; 1984. (in German) |
[16]
due to
![](data:image/png;base64,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)
. Furthermore,
![](data:image/png;base64,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)
is also at least positive semi-definite. This ensures that
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA5ElEQVR4nK2QrQqEUBCF9ykEq+9iEJNiMohNjBaz3agYjILBFxCr+AAKWoxiFhSboGe5d8HdxZ8te9KdO99wzswDP/Qg+vwIggBlWd5DpmkiiqJ76NKuqioIggBJktC2LW2kaQqe56Gq6gtalgWiKMIwDKzrSqFhGMBxHIqieNuRLLZt7xZJktCaDO2QZVnQdZ0C4zhClmX0ff8d3HVd6k/keR7CMDxu5/s+FEVB13XQNA3TNB2hOI5peMdxkGXZ+Z1Ig2EYmo1sewrleQ6WZdE0zfkxyWOeZ9R1jW3brqE7/Rd6Am1P4NwNt5IyAAAAAElFTkSuQmCC)
does not become positive. With
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
quadratic and positive definite and
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA+0lEQVR4nMWRr4qFUBDG7xv4AjYRiw9wo3aTzSZmr2DSYjLYBF/AKmIzmkyaFUSL2MU/1aB8l3NgkcvCblnYgQMzHz/OfDPzwC/xIPEP0LIsMAwDmqYhCAK4ros0TXGe5+dPlmXBcRya930PQRAwDMMnpCgKsiyj+bquFKrr+oa2bYMoihjHkUJFUeD5fGLf9xtq2xYsy8LzPLxeL5imiWmaPo0Tk6qqUqO6riMMw+/T2bYN3/epmCQJJEnCdV03dBwHZFlGnudU7LoOPM9jnucbKssSDMMgiiLajuyM4zjEcXzviUxUVRWapqEiaUMG+ar/+HbkVj89wrwBionPt6C5zSwAAAAASUVORK5CYII=)
positive definite in any case,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA5ElEQVR4nK2QrQqEUBCF9ykEq+9iEJNiMohNjBaz3agYjILBFxCr+AAKWoxiFhSboGe5d8HdxZ8te9KdO99wzswDP/Qg+vwIggBlWd5DpmkiiqJ76NKuqioIggBJktC2LW2kaQqe56Gq6gtalgWiKMIwDKzrSqFhGMBxHIqieNuRLLZt7xZJktCaDO2QZVnQdZ0C4zhClmX0ff8d3HVd6k/keR7CMDxu5/s+FEVB13XQNA3TNB2hOI5peMdxkGXZ+Z1Ig2EYmo1sewrleQ6WZdE0zfkxyWOeZ9R1jW3brqE7/Rd6Am1P4NwNt5IyAAAAAElFTkSuQmCC)
is then always negative for
![](data:image/png;base64,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)
and zero for
![](data:image/png;base64,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)
itself.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA5ElEQVR4nK2QrQqEUBCF9ykEq+9iEJNiMohNjBaz3agYjILBFxCr+AAKWoxiFhSboGe5d8HdxZ8te9KdO99wzswDP/Qg+vwIggBlWd5DpmkiiqJ76NKuqioIggBJktC2LW2kaQqe56Gq6gtalgWiKMIwDKzrSqFhGMBxHIqieNuRLLZt7xZJktCaDO2QZVnQdZ0C4zhClmX0ff8d3HVd6k/keR7CMDxu5/s+FEVB13XQNA3TNB2hOI5peMdxkGXZ+Z1Ig2EYmo1sewrleQ6WZdE0zfkxyWOeZ9R1jW3brqE7/Rd6Am1P4NwNt5IyAAAAAElFTkSuQmCC)
is negative definite in this case, while
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA0ElEQVR4nL2QoQqEYBCEfQrB6rsYxKSYDGIT41/MdqMmu8EXEKv4AoIWo5gFxSboHP8eeBzqXTlu2rLfMjMr4IsErj9DdV1DVVXouo6u62iR5zkURYFlWU9oXVdomgbXdbFtG0HjOEKWZVRV9bLzPA++7x8WWZbRzI8OiDEGx3EImKYJhmFgGIb34GEYkj9XFEVIkuTcLo5jmKaJvu9h2zbmeT5DaZpS+CAIUBTF9Z/4QhRFysbbXkJlWUKSJLRte//xZVnQNA32fb+HPum30AN4NOWACq7mJQAAAABJRU5ErkJggg==)
is positive definite. Since these properties apply to the entire state space, the stability condition of Lyapunov's direct method is fulfilled and thus the system under consideration is stable
[2] | Adamy, J. Nonlinear Systems and Controls. Berlin, Heidelberg: Springer Vieweg; 2022. https://doi.org/10.1007/978-3-662-65633-4 |
[3] | Tarbouriech, S., Garcia, G., Goes da Silva Jr., J. M., Queinnec, I. Stability and Stabilization of Linear Systems with Saturating Actuators. London: Springer; 2011. https://doi.org/10.1007/978-0-85729-941-3 |
[4] | Vidyasagar, M. Nonlinear Systems Analysis. 2nd edition. Philadelphia: SIAM; 2002 (unabridged republication). https://doi.org/10.1137/1.9780898719185 |
[2-4]
. The same applies if the considerations are based on Eq. (
48), which with
![](data:image/png;base64,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)
results in
![](data:image/png;base64,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)
.
(52)Both, Eqs. (
51) and (
52), also apply in particular when manipulated variable constraints take effect. In this case, the limitation causes the amount of a certain element of
![](data:image/png;base64,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)
to be smaller than specified by the controller. However, this can be modeled according to Eq. (
49) by a corresponding increase in the relevant diagonal element of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA+0lEQVR4nMWRr4qFUBDG7xv4AjYRiw9wo3aTzSZmr2DSYjLYBF/AKmIzmkyaFUSL2MU/1aB8l3NgkcvCblnYgQMzHz/OfDPzwC/xIPEP0LIsMAwDmqYhCAK4ros0TXGe5+dPlmXBcRya930PQRAwDMMnpCgKsiyj+bquFKrr+oa2bYMoihjHkUJFUeD5fGLf9xtq2xYsy8LzPLxeL5imiWmaPo0Tk6qqUqO6riMMw+/T2bYN3/epmCQJJEnCdV03dBwHZFlGnudU7LoOPM9jnucbKssSDMMgiiLajuyM4zjEcXzviUxUVRWapqEiaUMG+ar/+HbkVj89wrwBionPt6C5zSwAAAAASUVORK5CYII=)
. However, since the diagonal elements of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA+0lEQVR4nMWRr4qFUBDG7xv4AjYRiw9wo3aTzSZmr2DSYjLYBF/AKmIzmkyaFUSL2MU/1aB8l3NgkcvCblnYgQMzHz/OfDPzwC/xIPEP0LIsMAwDmqYhCAK4ros0TXGe5+dPlmXBcRya930PQRAwDMMnpCgKsiyj+bquFKrr+oa2bYMoihjHkUJFUeD5fGLf9xtq2xYsy8LzPLxeL5imiWmaPo0Tk6qqUqO6riMMw+/T2bYN3/epmCQJJEnCdV03dBwHZFlGnudU7LoOPM9jnucbKssSDMMgiiLajuyM4zjEcXzviUxUVRWapqEiaUMG+ar/+HbkVj89wrwBionPt6C5zSwAAAAASUVORK5CYII=)
can be chosen to be arbitrarily large as long as they are only positive and fulfill the condition
![](data:image/png;base64,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)
, this has no negative influence on the definiteness of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA6UlEQVR4nLWQMa5EABCG9ygO4AoKInECDdGJkoaGhIZKotFpRKd0AdGKSqNyBCQ0CsS/4RVvN8u+6k01+eebzPz/A3/U46hXwbIs5Hn+HWIYBkmSfIduz1VVBY7jIAgCpmnCsixwXRc0TSOO4x9onmdQFAXf97Hv+7mdZRlYlsU4jr/neJ5HFEUnsG0bVFVFURTvP0mSBM/zTrGuayiKgnVd3yHDMOA4zinKsoymaT7d2bYN0zRRliV0Xb+OIAgCaJoGURTRdd01dFglCAJhGF7ndDRpmoIkSfR9fw8Nw4C2be8Tv5z8G/QEAbni3ctG3pgAAAAASUVORK5CYII=)
and thus on the stability conclusion.
It is particularly worth mentioning in this context that for the proof of stability described above, both matrices
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA3ElEQVR4nMXQvQmEQBAFYGswM7UCO1AwFrQFG1BwBQMzAzESBGNTmzAxEARrMNRYxD8U3rEbnAind9kNbPbN7Jvh8KU4Wn9A0zTBtm1omoYgCOD7PqIowjiO10lJkkBVVRzHgWEYoCgK0jS9IsuyQAh5f2OaJlzXPdE8z5BlGXmeM7BtGyRJQpZlJ+r7HqIooq5rdF0Hz/NgGAaWZTlRWZYQBIGFjeMYRVFg3/frdmEYss7bE9AOXdfZlFtEc/A8D8dxWIaPqG1bVFWFpmmwruv9xZ/qd0Q3e3rUvACQj9kgWuhvfAAAAABJRU5ErkJggg==)
and
![](data:image/png;base64,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)
must be positive definite if
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA0ElEQVR4nL2QoQqEYBCEfQrB6rsYxKSYDGIT41/MdqMmu8EXEKv4AoIWo5gFxSboHP8eeBzqXTlu2rLfMjMr4IsErj9DdV1DVVXouo6u62iR5zkURYFlWU9oXVdomgbXdbFtG0HjOEKWZVRV9bLzPA++7x8WWZbRzI8OiDEGx3EImKYJhmFgGIb34GEYkj9XFEVIkuTcLo5jmKaJvu9h2zbmeT5DaZpS+CAIUBTF9Z/4QhRFysbbXkJlWUKSJLRte//xZVnQNA32fb+HPum30AN4NOWACq7mJQAAAABJRU5ErkJggg==)
is positive definite and
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA5ElEQVR4nK2QrQqEUBCF9ykEq+9iEJNiMohNjBaz3agYjILBFxCr+AAKWoxiFhSboGe5d8HdxZ8te9KdO99wzswDP/Qg+vwIggBlWd5DpmkiiqJ76NKuqioIggBJktC2LW2kaQqe56Gq6gtalgWiKMIwDKzrSqFhGMBxHIqieNuRLLZt7xZJktCaDO2QZVnQdZ0C4zhClmX0ff8d3HVd6k/keR7CMDxu5/s+FEVB13XQNA3TNB2hOI5peMdxkGXZ+Z1Ig2EYmo1sewrleQ6WZdE0zfkxyWOeZ9R1jW3brqE7/Rd6Am1P4NwNt5IyAAAAAElFTkSuQmCC)
is negative definite. This is possible if the controlled system is stable. However, if the controller integrators had been included in the system model without splitting off the uncontrollable, stable subsystem as described above, then the simultaneous specification of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA3ElEQVR4nMXQvQmEQBAFYGswM7UCO1AwFrQFG1BwBQMzAzESBGNTmzAxEARrMNRYxD8U3rEbnAind9kNbPbN7Jvh8KU4Wn9A0zTBtm1omoYgCOD7PqIowjiO10lJkkBVVRzHgWEYoCgK0jS9IsuyQAh5f2OaJlzXPdE8z5BlGXmeM7BtGyRJQpZlJ+r7HqIooq5rdF0Hz/NgGAaWZTlRWZYQBIGFjeMYRVFg3/frdmEYss7bE9AOXdfZlFtEc/A8D8dxWIaPqG1bVFWFpmmwruv9xZ/qd0Q3e3rUvACQj9kgWuhvfAAAAABJRU5ErkJggg==)
and
![](data:image/png;base64,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)
as positive definite matrices would not have been possible due to the then critically stable system.
If the description of the controlled system is available in discrete-time form, the difference
(53)is formed. If it is negative definite,
![](data:image/png;base64,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)
decreases at each sampling step as long as
![](data:image/png;base64,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)
is not a zero vector. The aim is therefore to ensure that
![](data:image/png;base64,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)
is negative definite in the entire state space. For this purpose,
![](data:image/png;base64,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)
and
![](data:image/png;base64,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)
are replaced in Eq. (
53) according to the right-hand side of Eq. (
41), from which
(54)follows. Furthermore, for
![](data:image/png;base64,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)
according to Eqs. (
4) and (
42), but with
![](data:image/png;base64,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)
instead of
![](data:image/png;base64,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)
, we obtain
(55)In addition, Eq. (
4) leads to the following relationship for the steady state due to
![](data:image/png;base64,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)
and
![](data:image/png;base64,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)
:
From Eq. (
55) therefore arises
![](data:image/png;base64,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)
.
(56)If we now insert Eq. (
56) into Eq. (
54), we finally receive
(57)If we proceed according to Lyapunov's direct method, we must first ensure that the first summand
![](data:image/png;base64,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)
is negative definite or at least negative semi-definite. This is ensured by
(58)respectively
(59)and by setting
![](data:image/png;base64,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)
as a matrix with at least as many rows as columns at maximum rank when applying Eq. (
58) and aiming for positive definiteness of
![](data:image/png;base64,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)
as in the continuous-time system description. If
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABB0lEQVR4nM1RocqDYBT1GQyiwWgcZoNYjKa9gwiaLMqCVmFBMPgClsF8AJcNCj6BNoMGGRM2tiBMz8++Ovx/43/h3HvD4Z5z76WwMah/RDyfz1BVFbZt4/l8rhPrugbHcciy7HviOI6Iogi+78M0TTAMA9d1ydQ8z7EsC6jH4wFZlhHHMV6vF6mapmGaJpRlCZ7nMQwDqDRNIYoi5nkm2O/3CMOQyDVNA5qm0fc9qMvlgt1uR+S7riN9VVW43W7QdR2WZeH9foP6pCRJ4DgODMMAy7LwPA+HwwGn04lY+Nr6eDxCUZTf7/gxLEkSBEFA27brxPv9jqIoCK7X69+fWYvtxCAIsAU/7xBM616OE4wAAAAASUVORK5CYII=)
only has stable eigenvalues, then the positive definiteness of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA1klEQVR4nMXRIQqEQBQGYG/gFUyKwUOY9QBmDyBY9AgGm2Czi2C1m8SqICaxKYKYDeovM8FlYGeXTfvgwYP5+HnME/ClBFJ/QOu6wrZtWJaFIAjgeR6KosB5nmyS4zj08bou1HUNRVGwLAuLTNNEmqZ0nucZkiRhHMcX2rYNqqqi73uKSJIsy3SNB7VtS9E0Tei6DoZhIEkSdvEsy6BpGsIwRBzHTyKDXNeF7/v8L9j3HbquI89zPirLEqIoIooiHMfxHg3DgKqq0DQNH/HO8Tsit/rUxNyTu9xcsj+S+gAAAABJRU5ErkJggg==)
follows from the positive definiteness of
![](data:image/png;base64,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)
respectively
and thus also the positive definiteness of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA00lEQVR4nL2QMQ5EYBSEHcUBXEEhkTiBhuhESfNrFDRUEo1Oo1W6gVZUGpUjIKFRIEb8W2xkyVa7U877Xt6bYfBFzKk/Q1VVQZIkKIqCaZqwLAt834cgCEjT9AXN8wye5xGGIfZ9p9t5nkMURYzj+D4nyzKSJKHAtm0wTRNFUVx/0jQNQRBQs65rGIaBdV2vkG3b8DyPmrquo2maz3Su68JxHJRlCULIfQVRFMGyLKiqiq7r7qEzKsuyiOP4ucwsy8BxHPq+f4aGYUDbts+N305+Bh06PuiAAPQ7GQAAAABJRU5ErkJggg==)
according to Eq. (
41). If, on the other hand,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABB0lEQVR4nM1RocqDYBT1GQyiwWgcZoNYjKa9gwiaLMqCVmFBMPgClsF8AJcNCj6BNoMGGRM2tiBMz8++Ovx/43/h3HvD4Z5z76WwMah/RDyfz1BVFbZt4/l8rhPrugbHcciy7HviOI6Iogi+78M0TTAMA9d1ydQ8z7EsC6jH4wFZlhHHMV6vF6mapmGaJpRlCZ7nMQwDqDRNIYoi5nkm2O/3CMOQyDVNA5qm0fc9qMvlgt1uR+S7riN9VVW43W7QdR2WZeH9foP6pCRJ4DgODMMAy7LwPA+HwwGn04lY+Nr6eDxCUZTf7/gxLEkSBEFA27brxPv9jqIoCK7X69+fWYvtxCAIsAU/7xBM616OE4wAAAAASUVORK5CYII=)
has stable and/or critically stable eigenvalues,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA1klEQVR4nMXRIQqEQBQGYG/gFUyKwUOY9QBmDyBY9AgGm2Czi2C1m8SqICaxKYKYDeovM8FlYGeXTfvgwYP5+HnME/ClBFJ/QOu6wrZtWJaFIAjgeR6KosB5nmyS4zj08bou1HUNRVGwLAuLTNNEmqZ0nucZkiRhHMcX2rYNqqqi73uKSJIsy3SNB7VtS9E0Tei6DoZhIEkSdvEsy6BpGsIwRBzHTyKDXNeF7/v8L9j3HbquI89zPirLEqIoIooiHMfxHg3DgKqq0DQNH/HO8Tsit/rUxNyTu9xcsj+S+gAAAABJRU5ErkJggg==)
can be chosen to be positive definite, but
![](data:image/png;base64,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)
respectively
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABO0lEQVR4nM2RMauCUBiG24XQX2CDYDj2GxyrwSXcnFr6ByLZIEl7UH+gybGlPRRxbygQHByiWoQizfK9nBNXr9zL5Y73hbOc7znn/b73a+APahD9A/D1esHzPMxmM0ynU0wmE0RRVAeLosBqtYKiKAjDEM/nE4vFAr1eD4/HowJJsd1uw/f90i4IAnAchyRJ3iD5zTAMdLtdav8p13XBMEwFpmmKTqeD5XJZG8BxHPA8j9vt9gbJi2azie12WwNHoxEGgwHtl4KXywUsy2K/35fQ6XSCKIpYr9fV1NfrFa1Wi0ZDlOc5xuMxVFXF/X6vQDLAfD7HcDjEZrOBbdvU9ng8fg+cwIfDAf1+H5qm0ex+XSEJXZIkxHGM3W6HLMt+Bs/nM2RZhiAIdJU1kPT09ZimCV3XYVlWeUe4D6aMBms8/O9cAAAAAElFTkSuQmCC)
can then at best be positive semi-definite (see example from section 6).
In order for
![](data:image/png;base64,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)
to be negative (semi-)definite, the sum of all three summands must be negative (semi-)definite in addition to the first summand of Eq. (
57), which can now be represented in the form
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFoAAAATCAYAAAAQ/xqmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAF3ElEQVR4nO1YaUhUbRTub4qifyLKyn5k/UhKE0sNNKw098QFUaNFRZRQVEitaFFT3DKDRFFLFERTRJEi3CjUcmuj1XKpLHHJtVxSO/EcuJeZ8c7YaDZ8ywMXZ96573Kec85zzusa+h+rjjWApg/xX4BGiE5ISKBz587RmTNnKCwsjOLi4igoKIjm5+f/9lH+GjRCtK+vL71//55u3rzJBM/OzpKjoyP9+PFDrXWGh4d5LTjqxIkTFB0dTVFRUZSRkUE/f/5cpdMvDxohurOzkxYWFkSiQQqIV5ecvr4+8vf3p7m5OTpy5AjV1tbS69evKTw8/N9JdH19Pfn4+NDIyIha82SJVoWPHz/S1atX6cqVKxQTE0MVFRU8jkz4/PkzfxaIxlqfPn2SXAfObWxs5HUgX2fPnmXHqALe9/LyoqGhIbVsU8SKiYaunjp1inR0dKipqUmtub9D9Nu3b+ngwYN09+5dlpaBgQHaunUrPX78WO49gWhlwB5FRUXk6upK3d3dfG58x9oTExNKbQsODmbbGhoa1LJNESsm+t27d5y+R48eZY1UByD65MmTSomemZkhd3d3unz5MkejACsrK0pPT5d7dymiu7q6yMjIiB4+fCiOPXv2jNatW0cfPnyQnNPT08OZ6u3tTREREeqYtggrJjo1NZUKCgqosLCQ9u7dS+Pj4zwO3UTKmZqaUn9/Pz158oTMzMwoLy+Pfx8bGyMHBwfavHkzPX/+XHLt5uZmjl5ouiywTnJystyYvb09R70U4EjIBPaTdVhHRwfp6+tTb2+v5LzMzEzKycmh0tJS2rNnjyiNsM3Pz4+MjY1Zul68eEHm5uaUlZWllKcVEQ2yPD09WRNRzEAaCAWqqqo4evbt20dJSUkUEBDA35EBgvHQWDyyxsviwoULdOjQIbluBPMQhSUlJeIY1t20aRN5eHjwmRQxPT3NDr9x44bcOByDeZAjRUxOTrJtcAIiGw5vaWkR5yEILC0tKTExkbPy0aNHot7jvHCGLBYRDd2KjIxU+dy/f5/frampYQ0TCLCzs+M0l8X58+dp586d9OXLF0kylQHrOTk5cbsmi/b2dtLS0hKL4O9gdHSUdHV1xXMLgNQhyiFRinjw4AETKMDFxYWzQhbx8fG0fft27n4EgGR0PYq1ZxHRdXV1lJ2drfJBqsNjgYGBcrqIiNm/fz99//5dJAspvmvXLrmoRRSnpaWxRivTZxQiFKpr167JjUMr3dzcJOcow+DgIOnp6dGbN2/EMcjA7t27ObCk9g4JCaE7d+6IY5A8ZKds4US/vmPHjkUZef36dSouLuasQEbA3mVLBxY4fPiwSCoArVq/fj29fPmS07G6upqjZsuWLewcZIBw+8vNzWUNVIVjx47JRRFkadu2bbyPOgA5OAPSHcAZEADoQKQ6DmTLgQMH6Nu3b+IYZGHjxo18Bth27949bjUNDAx4TLANgYMuDFKKugU5w8Vq2UTDm9gE6e3s7MwPKv/atWv5Wm1iYsJ/YQgi0MLCQi56QkNDueqjB8fBEHWKgCaCjMrKSrp9+zZXfxikLhBx6MORztgPHQv2RH8uBRS1DRs2iHYJtmlra3OkozDi/HAEiEQTcOvWLZ4LJ0Bm0CTk5+fT1NQUjy+baPS36C2lHvS4iB5hE2hYW1ubWNSQBbg6g2gQgN+UAdoOYgwNDbkoLRcg+9WrV0wCWjYpXRaAgq3MNnQqKL5CtON8ra2tLA8ALjhY//jx41RWViauqZEr+NOnT8nGxoasra0XFSgpoKtBd4DI/vr1q1zxURfQTugqohnpLQTDnwIiGbKIzuT06dPsNDhZI0Sj705JSeEURVFcCtC9ixcvcjqj5VpJZOMqjSILstE+ytaYPwHYhL4fhRBXfTQLq0Y0WjBsouyxtbXlSIb8oL2Cvil2H+Xl5XJzLl26xJofGxsrjqG6LwUYqbg/ihjWgvOEMVW3Slmg8KmyTdnlRyMRjXQVWiJElLr/Hv0nAjz/AnFPBDm01JO0AAAAAElFTkSuQmCC)
respectively
![](data:image/png;base64,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)
. To achieve this, starting from the second summand, the control law
(60)or the associated controller matrix
(61)is applied – if applicable with
![](data:image/png;base64,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)
in the case of Eq. (
59). Eq. (
60), after insertion into Eq. (
57) and taking into account Eqs. (
58) and (
61) yields the result
![](data:image/png;base64,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)
.
(62)Using Eq. (
59) as a basis,
![](data:image/png;base64,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)
is then simply specified as the zero matrix and
![](data:image/png;base64,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)
is replaced by
![](data:image/png;base64,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)
. If
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA+0lEQVR4nMWRr4qFUBDG7xv4AjYRiw9wo3aTzSZmr2DSYjLYBF/AKmIzmkyaFUSL2MU/1aB8l3NgkcvCblnYgQMzHz/OfDPzwC/xIPEP0LIsMAwDmqYhCAK4ros0TXGe5+dPlmXBcRya930PQRAwDMMnpCgKsiyj+bquFKrr+oa2bYMoihjHkUJFUeD5fGLf9xtq2xYsy8LzPLxeL5imiWmaPo0Tk6qqUqO6riMMw+/T2bYN3/epmCQJJEnCdV03dBwHZFlGnudU7LoOPM9jnucbKssSDMMgiiLajuyM4zjEcXzviUxUVRWapqEiaUMG+ar/+HbkVj89wrwBionPt6C5zSwAAAAASUVORK5CYII=)
is now chosen so that the bracket expression
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAH4AAAASCAYAAACdFWqpAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAGf0lEQVR4nO2ZV2gUXRTH8+CLJIhIUFFQQ2JXDBFFooJdsSB2MBq7EksUuyYRxS7igyA2LCj2XpJYMC+JQdHYwa4xUWPvvZ2P34F7md1vNtnNLgQxPxiYHWbm3nv+p93ZMCnnnySsrCdQTtnw1wl/6dIlGTp0qCxbtkz69+8vCxculIkTJ8q2bdvKemoh5/nz5zJw4EBZsmSJrnnOnDkyY8YMWb58ufz58yeod/91wp88eVJWrlwpv3//lqioKCkoKJDLly/LvHnzynpqISc/P1/GjRsnv379km7dusnp06fl/v37MmXKFF1/MKjwb968kaysLNm8ebNs3bpVDenLo/bu3Svr1q3TY+PGjXLq1Cn59u1bUJNw8vDhQ1m/fr0eLByYG+Nt375dnj17Jp8/f9brRnjmyhq8effuna7HzJfza9eu+Vzb+fPn7X08C7t379ZrBw8eDDrKWI+Zy+3bt+31Q4cOyaNHj+yczT3Z2dl2HkZ48F7r69ev1V48c/HiRXud+2/cuKHnnz59srZAs7AvX75IlSpVJC0tTV6+fCmZmZlSu3Zt2bBhg+vkDx8+LJUqVZKnT5/qkZSUJMOHDw/aAw0/fvyQtm3bahrnHG7evClVq1aVtWvXeoxjhPcF986cOVPatWsnr169kqtXr0rTpk11DW48ePBA6tSpIytWrLDj7N+/X8LDw+XChQtBrw2H7dixo/To0cM675UrV9T+x48ft3MeNmyYtG7dWl68eGGfdQrvDRkhISFBYmNj5f3793oNR8I+a9as0d84LRrXq1dPgyvs58+f6gUmahm4a9eu0r17d9dBiDqEMRw5ckTq1q2rThMqGjdurBFmwGtZBGnOSUnCw9ixY7UuGsaMGSOTJk1yvffevXvSoEEDGyVAlMTHx4csqw0ZMkQmTJig59h68uTJGmhGeETE/mRdJ8UJDykpKdKzZ089R2T6gJiYGCs8Y9EnENjwvxpPGomLi/NZM8ePHy+zZ8+2k2TAAQMGCA4UCmhoIiIiNEINpLFWrVrZDAAIUatWLY1iX5DeeO7AgQN2vu3bt5elS5e63o9RWrRoYaMRRowYoY4SbJo3IDR1G0jLOGWnTp2s8MeOHdNm1Qm27dChg5ZZX6xatUqdA4hoHJ5MbIQ/e/asZpLv37/rbw/hMQwv6NKliwrgRrNmzWTu3LmSkZGh3SaTLCnqAuHo0aMqKOmWA5FwRAzmhPHJSosXL/bpdHfu3JH69etrBJsa17JlSyksLHS9PzU1VaPbjM2OoXr16lrnQwVjjB49Wm09depUycvLU3sj/NevX6Vz587axzjBQfr06SPTpk2zNd8b+i2eNdGenp6u2Q3hEZtMQ1kxWOF5YN++fZouTFPlDemceof3bNq0Seulsw45oY7u2LGj2MN7gcCWJTk52f5++/atOtuePXuKMac7GLNmzZqyYMECWbRokRrBl+hkECJm586d9hqNYLVq1bT2e3Pu3LkS1+c0tIG5EImISf+B0yI8jkxftXr16oDXCTg12YzaPmrUKF0PwtMXEUxkcJzNYIVnm9SrVy958uSJz5fzgiZNmug5TQTnpCY35s+frwIWdzg7WwNNDUYzYPQaNWrI9evXAzbGrFmzZPDgwX6ladbNeohAA6nV2TA52bVrV4nrc7MNGSwxMdFGOyA8WaV3794eZSYQcFiaWL5v4ESA8GRwart3Vlbhb926pd3m48eP9SLe4tZIYMiRI0fqOZ7ar18/FThU0F9UrlxZU7SBLETDFWhzxf3URTzeH3JycrTjdW6VSK10y6HascCWLVvUmaZPn27fi/CIZhqv0sCWFzvhVCayEZ5GnDLgjXb1RDo30HRwICiR4oS03LBhQ/UeIwI1vnnz5lJUVFTqCRuISlJxhQoVbIr8+PGj9O3bV8cl5QcCe2C2SQjvTHFuUFtZd3R0tM14lIRGjRrJoEGDPJrKYCEyIyMjPcoAwrPOYMah/FasWNEjYBG+TZs2rn1BGEbhAwL7VefBp1EnZAWu0yGbqMBI/HZuf0oL86AmMwb1E+gfzHxMNvIHnCg3N1efO3HihApbHB8+fLDj3L17V6+Z9XLwrSNU4FBnzpzxyCI4qfmAU1r4iIP9nE5OKeEbiBt/3SfbckJDufD/KH4LT/01+1tfh6+tkj/QtZf0fo7iPmIY6Af8eZeBHY0/99MAlhbKiT9jOA++qPoDf1oF+u6wcv5N/gOqYnV8ZfW+TwAAAABJRU5ErkJggg==)
is positive definite, then the entire matrix term in the last row of Eq. (
62) is positive semi-definite due to
![](data:image/png;base64,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)
. Again, this can always be achieved with sufficiently large amounts
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAARCAYAAADtyJ2fAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABIklEQVR4nO2SvYqDQBSF8yhi6ysENJWWIYW12NpYaxXLlD5BHiBNKhHfQBBN6U+jEBADIZoif+pZZpY1EZYUu2y3BwbuwP3OvXOYCX6gCdE/+Beg7/tYLpdQFAVxHEMURcxmM9zv9/dgGIbY7XZgWRar1QpJkmCz2aDve3put9sAfNXDquv1GhzH4XA4jJyzLIMsy7S+Xq+QJAlN03yCxFXTNJimOYLIqgQ8n8/oug55nqOqqufEy+WC6XSK7XY7Auu6hiAI8DyPmui6DsuyniBZj2EYpGk6Ah+PBxaLBfb7Pb0bhgHXdccTgyCgja86nU7geX4wmc/nKIpiHM53ImaqqtKUy7KkIUVRRJN9Cx6PR/om0kwStW0bjuOgbdvf/ZwPpJYyoQDjs3QAAAAASUVORK5CYII=)
(
![](data:image/png;base64,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)
) of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA+0lEQVR4nMWRr4qFUBDG7xv4AjYRiw9wo3aTzSZmr2DSYjLYBF/AKmIzmkyaFUSL2MU/1aB8l3NgkcvCblnYgQMzHz/OfDPzwC/xIPEP0LIsMAwDmqYhCAK4ros0TXGe5+dPlmXBcRya930PQRAwDMMnpCgKsiyj+bquFKrr+oa2bYMoihjHkUJFUeD5fGLf9xtq2xYsy8LzPLxeL5imiWmaPo0Tk6qqUqO6riMMw+/T2bYN3/epmCQJJEnCdV03dBwHZFlGnudU7LoOPM9jnucbKssSDMMgiiLajuyM4zjEcXzviUxUVRWapqEiaUMG+ar/+HbkVj89wrwBionPt6C5zSwAAAAASUVORK5CYII=)
. Furthermore, because the matrix term in the second last row of Eq. (
62) is at least positive semi-definite,
![](data:image/png;base64,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)
cannot become positive. With
![](data:image/png;base64,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)
, quadratic, positive definite specification of
![](data:image/png;base64,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)
and
![](data:image/png;base64,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)
, a stable discrete-time controlled system according to Eq. (
4) can therefore always be stabilized using the control law (
60) via a PI-state controller. This also applies in particular when manipulated variable constraints occur, because when limiting
![](data:image/png;base64,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)
(
![](data:image/png;base64,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)
), only the element
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAARCAYAAADtyJ2fAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABIklEQVR4nO2SvYqDQBSF8yhi6ysENJWWIYW12NpYaxXLlD5BHiBNKhHfQBBN6U+jEBADIZoif+pZZpY1EZYUu2y3BwbuwP3OvXOYCX6gCdE/+Beg7/tYLpdQFAVxHEMURcxmM9zv9/dgGIbY7XZgWRar1QpJkmCz2aDve3put9sAfNXDquv1GhzH4XA4jJyzLIMsy7S+Xq+QJAlN03yCxFXTNJimOYLIqgQ8n8/oug55nqOqqufEy+WC6XSK7XY7Auu6hiAI8DyPmui6DsuyniBZj2EYpGk6Ah+PBxaLBfb7Pb0bhgHXdccTgyCgja86nU7geX4wmc/nKIpiHM53ImaqqtKUy7KkIUVRRJN9Cx6PR/om0kwStW0bjuOgbdvf/ZwPpJYyoQDjs3QAAAAASUVORK5CYII=)
of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA+0lEQVR4nMWRr4qFUBDG7xv4AjYRiw9wo3aTzSZmr2DSYjLYBF/AKmIzmkyaFUSL2MU/1aB8l3NgkcvCblnYgQMzHz/OfDPzwC/xIPEP0LIsMAwDmqYhCAK4ros0TXGe5+dPlmXBcRya930PQRAwDMMnpCgKsiyj+bquFKrr+oa2bYMoihjHkUJFUeD5fGLf9xtq2xYsy8LzPLxeL5imiWmaPo0Tk6qqUqO6riMMw+/T2bYN3/epmCQJJEnCdV03dBwHZFlGnudU7LoOPM9jnucbKssSDMMgiiLajuyM4zjEcXzviUxUVRWapqEiaUMG+ar/+HbkVj89wrwBionPt6C5zSwAAAAASUVORK5CYII=)
needs to be increased in thought until
![](data:image/png;base64,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)
corresponds to the relevant limiting value using Eq. (
60).
5. Discrete-time Systems with Dead Time for Manipulated Variable Determination
With discrete-time systems, it is often the case that the manipulated variables do not act on the system – not even approximately – from the instant at which the state variables from which the respective manipulated variables were determined are sampled. A dead time between the calculation of the manipulated variables and their becoming effective must therefore be taken into account when creating the model. In order to have defined and at the same time easily manageable correlations, a dead time is usually introduced that incorporates exactly one sampling interval
[9] | Nuss, U. Zeitdiskrete Regelung [Discrete-time control]. Berlin, Offenbach: VDE; 2020 (in German) |
[13] | Nuss, U. Ein einfacher Zustandsreglerentwurf im Zuge der Erweiterung der Systemstruktur um Reglerintegratoren und Rechentotzeiten [A simple state space controller design in the context of expanding the control structure by integrators and calculation dead time]. at – Automatisierungstechnik. 2016, 64 (1), pp. 29–40. https://doi.org/10.1515/auto-2015-0058 (in German) |
[9, 13]
. The calculated, if necessary limited manipulated variable vector
![](data:image/png;base64,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)
, which was determined on the basis of the state vector
![](data:image/png;base64,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)
, is then set to a newly introduced vector
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAARCAYAAADtyJ2fAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABFklEQVR4nO3RLauDYBQH8H2IRVkR64oM9jm0muyr24JdsbiPYRjIgjA0+FYUm1EZBpsORItTGf+Lyg2XXbkvcNs9cOLv/5xzngV+WYt/+Bew6zocj0dQFIUgCFCWJXiex3a7RZZl89C2bTiOA5qmYRgGLMuC7/tgGAZxHOPxeLx03/fTqFVVjS8mSTKm3W43HA4H6LqO9XoNWZaxXC4hCAJYlsV+v5/gkEwQBNq2RdM0kCQJURTBdV1cLpcxjCRJpGmK+/0ORVEmeL1esdlsUBTFiM7n88tO7/DDVYc9h1E4joOmaXg+n9+Dw8JhGCLP89krfgq/qrqusVqt4Hnez6BpmtjtdjidThj+fYSiKGKuVVWdDXsDq/Zcwi68CxQAAAAASUVORK5CYII=)
, using the difference equation
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFcAAAASCAYAAAAuXUKzAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAESUlEQVR4nO1WWSu1URQ+F8qFX+DOBXEjuXAhISlFQpnHUlyIjGWIjJkzJDJzYS5KGTKLKPMYMo8RyTxP6+tZ9epwvvc4+pzU13lq1zn73XvttZ+91rOWhFRQGiS/7cD/DBW5SoSKXCVCRa4SoSJXiVCRq0RIXl5eqLGxkXx8fGhra4tub28pNzeXwsPD6erq6rf9e8fR0RGNjY2Jjo2NDdG9d3d31N7eToGBgbS2tsZzuJunpycdHx8rzWfJyMgIZWdnk66uLvX29lJHRwcVFhaSqakp7e/vK+3g76KpqYnMzMxER2pqqujelZUVmpmZIUtLS7YDjI6Oko6ODt3c3CjNZwle9fz8nAwMDGh5eZnu7+9pb2+PgoOD+bVdXFwoOTmZLCwsKCkpiby8vKi1tfXdwOPjIyUmJlJaWhq9vb196/CdnR0KDQ2li4uLn76XDE5OTsjIyIimpqb4f1FREdnZ2dHz8/M/215cXCQHBwdaX1//MM+ai0tqaWlxqoCgsrIyGhwcpLq6OqquruY5DQ0NJhKRXVJS8sFIVVUVO/s3nJ2dUWxsrMw87KSnp5O6ujrb/Qqbm5vU1tYmOmZnZ+Xun56eZnLhz+vrK9nb23OwfAc4B9x8BgLU2dmZpUsaTO7AwAAfjENhAJqLF4Uj0GBAIPfp6YlOT08/GIGWzc/PU1dXF7+g9CGIGFdXVxmHsEb60RS5mKOjo+goKCiQu7+iooKcnJz4TASOpqYmdXd38zfUmpCQEIqLi6OwsDCeGxoaouLiYiYNZ29vb5O1tTVncE5OzgfbyPiAgAB6eHggX19fDjRwyeS2tLSQnp4excTEUEJCAl1fX8s4J0YCNMvb25smJib4UXCQNMTI/cruTyMrK4ulLy8vjzNPW1ubFhYWWINBlpubG8tTeXk5r29oaGAe8vPzmTA8CjItPj5exnZzczPbwCMg0wUdZ3IPDg4oKCiINRbs/w1iJCBizc3Nuaj09/e/z6MDSUlJoejoaNLX1+ffGD09PQrZ/WngjuiAQB6ysbS0lAce//DwkGsJCl5fXx+vR3bCV2QlgkceuYh6BBCyVsh0QOE+FyRIbxQAXc7IyGDdxeGfoUjkooj+JqDHiFrUAUNDQ5YJW1tbzkbIDYhHmuOeERERH/aCExT91dVVsrKy4ocSoBC5tbW1pKamRpmZme9z/v7+1NnZSX5+fhyx6DNtbGyosrKS0DsLECMXKQlHYRd6hVbpt4DWExGKyI6KimJ9NTExYWJRjNGyoaAODw+TsbEx1dTU0O7uLq9ZWlri4ojsg6RGRkbS5OQk25VAi8QGNARANby8vHzXEhhyd3enubk5/iaQie+fUxxz9fX1MhfCOtgUhjKlAXIn754oXtBXDNwFEoCIFO4D/1DgMY81KOrj4+Pk4eHB64WMxjz2CC2pBBorNsR6QByASPtuX/tbwD3k3VM60xQFas1X/fkfG1jPM7o7u2EAAAAASUVORK5CYII=)
.
(63)If
![](data:image/png;base64,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)
is now used instead of
![](data:image/png;base64,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)
as the manipulated variable vector acting on the controlled system, the dead time is taken into account in the model. If
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAsklEQVR4nN2QLQ6DQBSEOQQWRbgHJ8Ah1nMFHJ4EhVmFX4lDEw7Aj0FjwAMBg+BrFhRJW1XVSZ6aL/MmY/BFhtZ/AOd5opTCcRzyPGeeZ4QQBEFwA9u2kaYpSZLgeR5hGFKWJVVVPV+0bYtpmhRF8b5D0zRYlkXXdU9gmiaGYbiiXddFSnnFr+t6A7qQ7/v0fU+WZdi2TRRFHMdxA9oYx/GK3Peduq5ZluWXQ8VxzKfT/gu2S6XV05Kg3gAAAABJRU5ErkJggg==)
and
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAARCAYAAADtyJ2fAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABFklEQVR4nO3RLauDYBQH8H2IRVkR64oM9jm0muyr24JdsbiPYRjIgjA0+FYUm1EZBpsORItTGf+Lyg2XXbkvcNs9cOLv/5xzngV+WYt/+Bew6zocj0dQFIUgCFCWJXiex3a7RZZl89C2bTiOA5qmYRgGLMuC7/tgGAZxHOPxeLx03/fTqFVVjS8mSTKm3W43HA4H6LqO9XoNWZaxXC4hCAJYlsV+v5/gkEwQBNq2RdM0kCQJURTBdV1cLpcxjCRJpGmK+/0ORVEmeL1esdlsUBTFiM7n88tO7/DDVYc9h1E4joOmaXg+n9+Dw8JhGCLP89krfgq/qrqusVqt4Hnez6BpmtjtdjidThj+fYSiKGKuVVWdDXsDq/Zcwi68CxQAAAAASUVORK5CYII=)
are then combined to form the overall state vector
(64)this results in the vectorial controlled system state difference equation
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMgAAAASCAYAAAAe/ZHXAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAJJklEQVR4nO2ZdagUXRjG/ct/FEVUBEUUbBAbuxu7A7uwu7u7uxMDuxO7uxsLFbu7X7/fC2eZu3fu7M7e8d77wT4w7Ozszplz3vM+z/ucM/EkjDDCsEU8ENudCCOMuIowQcIIwwFhgoQRhgPCBLHBp0+fZMCAAdKiRQvZtGlTbHcnzuP8+fPSqlUr6dKlizx8+FCvLViwQJo3by7jxo2Tr1+/xnIPQ0eYIFFg1KhRkiFDBnn27FlsdyVGMXr0aLl7966reyBA7ty5ZeDAgb5rp06dkuTJk8uxY8e87mKMIkwQC548eSLLly+XCRMmSPHixSVr1qwyduxYWbdunXz8+DG2uxcjKFu2rJw5c8bVPVTc9OnTy5EjR3zXduzYITlz5pTXr1973cUYhY8glMkGDRrIvn375MePH7JkyRLp2bOnfP78Obb7GBQ+fPggJ0+ejPK4ePGi4/2XLl2SAgUKyJo1a7RqlCtXTm3C06dPpWHDhtKvXz/5/fu3J319+/at9OrVS9X6+/fv2rdGjRrJzZs3PWk/OgiFIDdu3JCUKVPK+/fvfdf69+8vderU8Sxm5OHWrVulXbt2cufOHb3G8+rVqyevXr3y5Bl2UIJACBKgQ4cOUqFCBVm1apV+R0l//fr1zx7uJUiyYsWKSdGiRW0PktwJNWvWlMGDB+s5lQR7ZUg1adIkJQzJ7AVWrFghY8aM0Wfs3LlTWrdurQn1/PlzT9qPDkIhCGLKWIYPH64HcaSikD9e4dq1azofzOWGDRv0GhWL5/zLNY4SBJYzOVeuXJGkSZPqIL16KG0TQBa8bsn24sULVdkLFy540hcnEPj58+fr+fbt2yV79uzy7ds3efnypSbNjBkzPHsWFYq2ixQpIlWqVPEpohdA7EaOHClDhgyRP3/+BPw/ZLAKS5IkSdQame/NmjUL2EabNm1k6NChvu+PHj3SxD106FC0xuIP4sZaxwjX1KlTpXr16p6I+OXLl6VatWqR5iLCGgTlTJcunRw+fDikh5w4cUIT2n9izC5HVMACLV26NMK106dPy/jx4yV16tR6Hgh43W3btmkZtjsOHDjgeP/evXuldu3aSua6detK6dKlZdmyZVpVp02bJl++fAnYByuIQdOmTeXNmzeOv3fs2NFVuwYIT61ateTnz5+RfsMBBKve7969i2BF8+XLJ4sXL/Z9J3EC9YOk3bx5s+8aC3Ty6MGDB+4G9R/2798vkydPtv2N/uTPn18tKs+tWLGiioEb0E8jhFYwv8TTf1PGRxACTVCzZcsm06dP105wgPXr12vZ7Nq1qw6A7wRy5syZGgQUdtCgQTrhqM+wYcMi2JHZs2fLypUr1atWrlw5UgnfvXu3EssKlBs1LFiwYFAlnxLMALFKdkcwich4qVbEgMDTX4hnJTwiwm4NiolKs0ZBeVAygkucOJ83b54kTpxY+vTpY6ukBw8e1P+VLFlS/bVJJj779u2r7SAQ9+7dU9uLRaR/nTp1UgIzyfHjx9c++Mene/fumqTMVdWqVX1br8HArcWCYIzTuvPF2q1w4cIRXAi2kr5269ZNd7YQH5IdQWLthSCNGDFC1y3kFuf+II/q16+v56yVU6RIoWMEKD/bzFhVxg+IMfcgfLgCYlm+fHmNObbZiuvXr0vbtm21slM1yW1IqARhsnv37i0LFy7UG/PmzasPun//viYenSdh+Z4lSxZtBBtG0rFbAaGYZJSrcePGERKK8te+fXutSnPnztVk8S+JdgQxCJYgXoFkwmbMmjXL9vcmTZroOBEUNjV4T0KFLFWqlFYpJpaYEYNUqVJFWldQDalKnTt3luPHj6sVgRBr167V37Fcu3btUkWDQLRJ7Ejco0ePKnGYB5AgQYJI6yLuQ6hQ2ylTpmgFCMZqGbglCJsaCRMmVKIA+gOBK1Wq5FugEx9yiJ1ANkMgAPnCuDk2btyoggKhyBHyxQ6IVq5cuTRH+X/atGk1sVevXq27jcwH/YCgAFFmh43/Y/GJA/MDUf1B/PkfboMKyn1ACTJx4kRlMpOOQsJykpYGuRFFhUSQggUlA2ciYCaMNvbDjiDsNKAUqLuVtQyE77RJ0pUpU0bPIYpJABCTBKHfBJUEI4h2topJIT70FcXCmnEf9o5kJ4kN7AgCKRAf4oxQIC5UbrNbmChRIv3dCBCVgPaxoJkzZ9Z1ooEdQah6rB1QSVTTLdiMOHv2bFD/pcL36NFDXwgyfgAhyRHUmLUIWLRokdownAV5hDiQQ+w8QqSWLVuq6AIngiBeuBiEnLmhOkAUdrEeP36sBIGIpqog6uQxO1/MqRNBTGVGlKw7twHfg5AAqIq1XPIgOkHnWKCxgGXAMJkSaa0QV69e1a04ktyUR38EqiCoYVwBZPdfo7HOQMEQGWO1AOsnK9mDQaZMmTRmVkAm3kjPmTNHJ9GoNcptbLCBscMQHdvrFiQbie8l8P0QwZoX5BDuA7VGIPnkGtYRwrmpeuDcuXMaF56VJ08etXw8k7yjkhrh5gUwry+sgBDk7a1btzTXibdBQIJADCwBvg524yWZBCaSTuAl8aBUA9QN7wlhqEZp0qTRBRcViomsUaOGKia7U1bYEcT4+WTJkqmPZjs0LmDPnj1KAnZQUEYsU4kSJVQZGVeOHDmU1KiXWYQHWuhaQWyptsSJpGHjA1LiiVFkFr/YORQUO4Y1vn37ts4R2598ouasZVjEknBebU+HCvpK0rMeI4ewZVREnAlVgbFhaxFcEpr48T83YD4YO1Ud20ZuFipUSOPIK4uMGTOqWGHxaZ8XwsSInUQsMetI4oSoEFPjWpQgqFNUx5YtWzTZ8Y8cqACkoTxynXNsFCyEoXg3yiXJwos3zo0iESi77WMmmAWVFTyHds3hdhfJDfDITjGwHoybvvDJWKicpn/GNvCdcwJOzKxqiO1xap9xE0vaIXYmDjyLc9M+bfI78Sa+vGhkUumHUWp++5fk4HnBxAwBJFfou8khE0OTTyaGjIv/+PebPHR6Bu9ErDlKO8TRxID2zXX+Q8xwJtgyE3PAde4xc6YEIdBRHaGWWxZPqOj/AQTIKQbWw23p9weJ4tR+KCCxSNbo9s0tgo2b3Va0W5CHTs8I5V0IGwbGrkYF+PEXm6zDhft88FAAAAAASUVORK5CYII=)
,
(65)using the extended transition matrix
(66)and the extended discrete-time control input matrix
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEUAAAAeCAYAAAB6xNMdAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAEW0lEQVR4nO2YWSh9XxTHf2R6kCcRmTJlSOKJJF6QSJSSB55ICC+eCMlUigzxgMxKFJnLkJBMGYvM85ApMmT2/bV23dOPe86998e9/z/X71u77j53n3X2/py111r7/MI/vdEv0v89ia+mf1B4pPRQTk5OMD8/j83NTby+vsp0j1JDmZqaQmhoKJqamhAYGIiKigqZ7vv2UPz9/ZGZmYn8/Pw31x8eHuDr64uCggLWb29vh66uLvOcP7WxsYHs7Gz4+PhwY789FFro/f292PXl5WXo6+tjZGSE9ff396GhoYGOjg5eO1ZWVoiLi2O/lRbK0NAQ1NTUsLi4yPovLy+sX1JSwmvnR0Dp7++HiooKlpaWuGsEpaioiNfOj4BC24a2y8LCAuuTp6irq6OmpobXDi8USlc3Nze4vr7mHvL09MT61Oj3V5QQlPX1dRgaGqK1tZX1t7e3oaWlhZmZGV47vFDIcHh4OLS1tVFfX8/+XFlZgbm5OWxsbFiU/ooSgkIvMTg4GDExMex3ZWUlPD09BWsVwe1TWlrKbnx+fuYGu7m5obi4WN5rkZuEoJB2d3cRHR2NlJQUJCQkYGdnR9CO4PaJiopCamoqN/D4+Ji54NzcnDzXIVdJgvI34oVC8cTV1RVZWVkYHh5mjdKXhYUFLi4uPv1QPpWVlbECS6g1NjZKtaFQKAcHBzAzM0NbWxsHhWIMTY6qQ0WI4pToWXyN3F+a+KDQS+zu7hZsfHZ5ofT09MDZ2ZmlLpGCgoKQnJz814ulwDY4OCjzAewz4oNCXk9ZRqidnZ2J2eGFkpaWhoiICG4QpWFbW1t0dXVx1ygADwwMsOxEKe/w8JBFdTpw3d3doaGhAVVVVairq4OBgQHKy8slvm06c1haWgo2CvwfgfJeubm5sLa2RlJSkuAYMSj0Zr29vVFYWMgNouBKZwdRKqa3np6ejry8PMzOzsLPz49Vi7GxsewkSlssLCwMLS0tWF1dhZGREfb29iRO+OrqCkdHR4KNXow8oIyPj0NHRwenp6eyQ2luboampiYCAgLYRG9vb1k8UVVV5Y7b5CV6enpITExkwdjOzg69vb3MQyIjIxESEoKJiQk2lh5uamr6nxR8skAhz3ZxcXkTGt7rQ2U+QSHPeX/0Jg+hY7uXlxfbPuRRBMXExASPj4+ymP6UZIFC3hwfHy9xzIeg0GKpEMrJyWGZilyS4goVRZ2dnaymMTY2Rm1tLS4vL2Fvb8/iEW0DRUoaFJo3JRBp6V0MytraGmRp9HBaKAVS+sRHcCjQTk9Ps/hRXV3NDlw0kbGxMfT19b2pjhUhaVDoOwp5+NbWlkQ7YlAoe0hrFHe+oqRBoVKDvJbipCT9iE8HImVkZLCDoTSP/TFQaMs4ODjA3d0d5+fnEu0oHRRHR0d4eHiI/UcgRFWs0PYZHR2Fk5MTK0mUBgoVl5QEPvq9h2CJEomo3Pj2UBQhpYZC5zA6X/3ZJicnpd5HTH4DwR3CGGJp3VkAAAAASUVORK5CYII=)
.
(67)For the output equation of the system with dead time, it is correspondingly obtained
(68)with
![](data:image/png;base64,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)
.
(69)If it is now successful to establish a generally valid correlation between the stability behavior of the system extended by dead times and the dead-time-free system with and without manipulated variable constraints, then the effort for stability analysis and controller synthesis can be significantly reduced for systems with manipulated variable saturation and dead times in the manipulated variable paths.
For the difference equation (
65) of the controlled system with dead time, the same stability considerations can now be made as those based on the system state difference equation (
4). Analogous to Eq. (
57), the relationship
then occurs, where first for
![](data:image/png;base64,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)
and then for the remaining terms negative definiteness or negative semi-definiteness must be ensured. With the approaches
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMoAAAATCAYAAADRVJJPAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAJ9ElEQVR4nO2adYxTWxDGN/xFSHAI7hDc3YNbkODu7u5uwd01uLu7BQ/u7u7uzHu/SU5z2213l9suy3v0S2727m17ZM58M99MGyB++OFHsAgI6wX44cd/ASEmSqtWrWTAgAHSoUMH6dy5s/Tr10969OghP378CM31+fEX4OTJk9KyZUsZMmSINGzYUAYNGiQdO3aU9evXh/XSHAgxUYoVKybv3r1TcowYMUKePXsmNWrUkG/fvoXm+vz4C7Bu3ToZNmyYBt3YsWOrb23fvl2GDh0a1ktzIMREuXr1qv41RAE3b96Unz9/hs7K/PhrADFevnyp94Yo379/l1u3boXtwiwIRJTVq1dL3rx5pUmTJvL69etAH7ASxQ5evHih6bVkyZKaYhmvf//+bufyBa5fvy6lSpWSwoULy6VLl0JlDjtg77ly5VIJy/6RGpcvXw7rZYUqTp06JYMHD9ZM0b17dzl48GCg9xiieAN8bPLkySrl+vTpI9OmTZNPnz55NWYgopAlEiRIoIRxB2+JAvr27SuVKlXSVIucK1SokIwZM8arMT2BjFeiRAlp3LjxH5X9du/eLTFjxlTp+uXLF+ndu7eUK1fuj1qjL7Fjxw7Jnz+/BivO/cCBA5I1a1a5d++e0/u8Jcrz58/VjrNnz1ZyvH37VooWLSrz58/3av1KFNLeyJEjpWfPntKmTRuJESOGFuytW7eWTZs2ORXsEMVb7Vi9enWnMRo0aCDt27f3akwrcLxly5ZJ165dNYJFjx5d6tevr42I0aNHy+fPn302l11Mnz5d6z4DDjJz5sz/y+bImzdvJEmSJLJlyxbHswcPHkiyZMlkz549Tu+FKI8ePbI1D7bjzGvWrKnSzYBGQd26dW2NaRAA43LmzClTp05VB5s1a5YUKVJE78+ePSuJEyd2SAI2kDp1asmYMaPcvXvX1oSQMlOmTFqsAZw2Xbp0MmfOHK82YoCBIASGYS7WjvGfPHkiHz9+lIIFC2paDmuwPgIT4ICbN28uderUCZW5cFACk6dr3759oTKvwZQpUyR9+vROQQD/Qbns2rXL8WzJkiUSPnx4DdJ2MitnHDduXKcxQaNGjaRWrVr2N/AvApYvX66bAGzEGu2RYfHixZMLFy54NYkV586dk+TJk6teZXwySbVq1XwW5W/cuCEpU6ZUkoO5c+dqfQLxIVHx4sVl1KhRPpnLLnACoimOweGSXfLkyaNrDw2ENVGQPiYoGJw/f14D2OnTp302z9KlSyVhwoRO9Qi25vyRtt4gAO2YJk0aefr0qWaMDBkyqH5EJyLD0Pa+lCpsJkWKFDJu3DiN7Pv373dKkyEFxiClu+L+/fuqfclY6H/S8MCBA+X9+/c6N/KGtP8roI7i8644fvy4fr8U1OWu9rpz545EiBBBX5s4caK2R5njV8H+KFx/F44cORLsfidMmOD0GRwVKb9y5Uqn5xTYnIW7M7QLZBeNGyvonCVKlEh27tz5y+NR7xjfDCCLENnoQqDlKDBhH92CxYsXq1z5FWAYOjqeuljUPc2aNfP4+Q8fPsi1a9fk9u3b6ggUexSARF6Myj2RlzkY58qVK4HGOHbsmEYwmgbsp2nTpvplKZnk8ePHjvexRtrekAs7kOEY/9WrV2ok7nnGPJCNeyv4f968eUFeVl1uQBZHwnrC169f9YCxA/aHDKyFZxwc9+yb+pEOJdE5qK4OWRWp5+latWqVx89aQQcxuP1u27Yt0F4iRYokR48edTzjjMnsdmpdbI6UcyfNCOyctRUzZ87U7qJpPwMSApIcu7I+Y08SAn5nfIwEQrYliTh1vYjy2bNnd7tAJho/frxujjqGD5NtIBdOTJHMInktSpQo6qiuKZ2onC9fPj04dyCqcnAYnHYpm4Ss8ePHV+c6ceKE1lPc02Kmc7ZhwwaPRuX9UaNGlYcPHwZ6DYMwxsKFC3UfGzdu1AgfK1YsjZxkpBw5csjmzZs1SrFPOlW+AHurV6+e29c4OEhJc4WL97IPDg2ZSk1J94jXcBgIh6NDbk9A5tLF9HRdvHjRJ/tyBxw6VapUsnbtWv2fgETjokCBAuo3BticbioBDVKRAapUqaKqA3VDMwa7tGvXTsej9eu6Z2xStmxZB4kgVe7cuTVjG2zdulXPEnVRsWJFJQWqAxXCPGQlakXmJ8jiz5QeDqLwJpyYNGW+XLSC71VwcFjHBEQOiiYKfw6ShR86dEgNgWO7di5Y/Jo1ayRixIiaDdzJOQhCq5hFzpgxQ0nLeKRtUzuZxgJGpQPnCThct27dJFy4cOpIrhGI7y74ZQFzcThVq1bVaD1p0iQlEN0ys4cWLVooaX0BbJs0aVIpXbq0W9lBbUWzgzVDUmzJ2fA5bD18+HBHIwTpR0axI11/Jzg/nJ4MiNTE6a1KgKBFhiGK85z9k3V69eqlWQKFwPdN2IFSoUyZMm67gwTE8uXLq3OvWLFCA+CiRYucfj1C8CMhMA6koonEOdBMgST4HXMDuGBkuoMoRHOYzEW2cAVFPY6HszIozMT5aMOmTZtW9u7d63ivJ6Jw2Ix/5swZdWRX8DOG2rVrO9aB0/A5xsKpMbaJIsERBeMwD+MgYVyJQvbDMGYuc3DIC5yYTolZI0TBgL4AezFzuqt7+BIuS5YsWiea97EOyNClSxeNkGat/xWiAOoyZDfrdZXzBAQyIwEU/6pcubLuGTlJtK9QoYIjqARFFMD7IGScOHGcfNIA3yS7GdsixfENnmFbiGPGhijIchDin7DQEuZgrECOIbloLdO5MgSj82Cng3P48GHVk0QOjEmHjIXiIKRRjEibD23JnJDK7rft1GVINwyLnCFT4YC0JpEEkAOpw2FBKEjpLtP6GkQwWtjYAgJgA9aHlEWecPhkWuQSgSBbtmz6104z4HeDzBAtWjRdL/UWdSAgKJAtXfeAvyH3ySr4GQ4MUfj+iS6mJzAO6gilwPlaf/GA5DVjYWva1Pz4knm4p0NGFsL2dINRHJA8xEShFoDpSBMkEpug7UcUwLFIl2hnnJzFtG3b1tGiDSmMIcgsSB3IRg1Bc4Fx2QBGo56ALBgCuWcHGJoowlzsh7SNsdgP5KB7wx4Yn0NFiv3qfuyA6AY5OEzWg51ZA9qcoh1nwwZjx47VddI9QmYE5Th/Clgj+p/vOtiPafiwD2qxTp066bkvWLBA5SUtdAIhjhw5cmStUVAHBBLkk7uMbIDcI6tAQAKMAYoEmyFh8WkyPIEQP8PH8DWCpflhJueArwUE1V+3Xug2ojzRjY3h1NzzzNxzcU/aZBNWucMCg5uDVrG3oNAPyX5+h9O7A8VpUOvCSf5PIOBY90dwIzvi9OYZRCeCkwm4kM3UsPiT9R4fxKf461rj0jiyzkPNTI1D59PVj+0gwA8//Age/wAsjhTZLPzR6AAAAABJRU5ErkJggg==)
,
(71)
(72)respectively
![](data:image/png;base64,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)
instead of
![](data:image/png;base64,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)
and at the same time
![](data:image/png;base64,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)
, this is achieved in the same way as previously described in section 4 for discrete-time systems without dead times in the manipulated variable paths. If you now specify
![](data:image/png;base64,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)
as a symmetrical matrix in block matrix notation
![](data:image/png;base64,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)
,
(73)![](data:image/png;base64,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)
as block diagonal matrix
(74)and
![](data:image/png;base64,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)
in block matrix notation
![](data:image/png;base64,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)
,
(75)then it follows from Eq. (
71) by block-by-block writing, taking into account Eq. (
66)
![](data:image/png;base64,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)
,
(76)![](data:image/png;base64,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)
,
(77)![](data:image/png;base64,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)
.
(78)While Eq. (
76) represents the discrete-time Lyapunov equation of the dead time-free controlled system, Eqs. (
77) and (
78) directly yield the solutions
![](data:image/png;base64,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)
,
(79)
(80)for the matrix blocks
![](data:image/png;base64,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)
and
![](data:image/png;base64,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)
of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAARCAYAAAA2cze9AAAACXBIWXMAAA7DAAAOwwHHb6hkAAAB5klEQVR4nO2UP8jpYRTHzd7FYmVUiklGpUwyKDZmBgtFiYUMGJDFJHozEROlXlkUKUn+lPwpfzIJA5L/39tz6qp789779upu99Tp1+/3PM/nnPM95/lx8A+N8x/+KfxyucDtdkOlUsHr9ZK7XC4Mh8PX4cyy2SyEQiHO5zOOxyOcTifUajXu9/vr8EAgAK1W+1hIpVIQi8W43W6vwa/XK4H9fj99ZECLxQK9Xv9t8AN+OBwgEAhQKBSw2WwQj8chl8vR7/dfh3e7XXC5XASDQYRCIeRyOex2u28BWXI/+0TwWCxGmX5mTLbFYoHJZIL9fk8+Ho8xnU5pjT3Z+2w2o6FoNptYr9fgsCgGgwFWq/UpmI1pJBKBx+NBNBqF2WxGp9OBQqGARqOhQOy8yWTCx8cH3t7ekEwm0ev1wGm1WuDz+TAajU+lGI1GkEgkyGQyKJVKkEqlaDQaVIlOp4PP58P7+ztOpxPt5/F4JA3Jwkqt1Wqo1+uUxe/WbrchEolQLpdpH/PtdksTxeSUyWQoFosPnX+B/61By+USSqWSsmb6slu7Wq0InE6nMRgMHgFYQDZ11WoV8/n8a/8WBgiHw6R9pVIhPW02G2nMmmm32+FwOKjyfD5Pe1n2HHZx/uSJROIr8Z/aDx7ffHZF8KrSAAAAAElFTkSuQmCC)
. The specification of
![](data:image/png;base64,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)
therefore determines
![](data:image/png;base64,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)
. Because
![](data:image/png;base64,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)
is positive definite if
![](data:image/png;base64,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)
respectively
![](data:image/png;base64,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)
with stable transition matrix
![](data:image/png;base64,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)
is positive definite
and
![](data:image/png;base64,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)
as upper block triangular matrix with stable matrix
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABB0lEQVR4nM1RocqDYBT1GQyiwWgcZoNYjKa9gwiaLMqCVmFBMPgClsF8AJcNCj6BNoMGGRM2tiBMz8++Ovx/43/h3HvD4Z5z76WwMah/RDyfz1BVFbZt4/l8rhPrugbHcciy7HviOI6Iogi+78M0TTAMA9d1ydQ8z7EsC6jH4wFZlhHHMV6vF6mapmGaJpRlCZ7nMQwDqDRNIYoi5nkm2O/3CMOQyDVNA5qm0fc9qMvlgt1uR+S7riN9VVW43W7QdR2WZeH9foP6pCRJ4DgODMMAy7LwPA+HwwGn04lY+Nr6eDxCUZTf7/gxLEkSBEFA27brxPv9jqIoCK7X69+fWYvtxCAIsAU/7xBM616OE4wAAAAASUVORK5CYII=)
fulfills this condition, the positive definiteness of
![](data:image/png;base64,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)
no longer needs to be proven separately. If we also evaluate Eq. (
61) for
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAARCAYAAAA2cze9AAAACXBIWXMAAA7DAAAOwwHHb6hkAAAB5klEQVR4nO2UP8jpYRTHzd7FYmVUiklGpUwyKDZmBgtFiYUMGJDFJHozEROlXlkUKUn+lPwpfzIJA5L/39tz6qp789779upu99Tp1+/3PM/nnPM95/lx8A+N8x/+KfxyucDtdkOlUsHr9ZK7XC4Mh8PX4cyy2SyEQiHO5zOOxyOcTifUajXu9/vr8EAgAK1W+1hIpVIQi8W43W6vwa/XK4H9fj99ZECLxQK9Xv9t8AN+OBwgEAhQKBSw2WwQj8chl8vR7/dfh3e7XXC5XASDQYRCIeRyOex2u28BWXI/+0TwWCxGmX5mTLbFYoHJZIL9fk8+Ho8xnU5pjT3Z+2w2o6FoNptYr9fgsCgGgwFWq/UpmI1pJBKBx+NBNBqF2WxGp9OBQqGARqOhQOy8yWTCx8cH3t7ekEwm0ev1wGm1WuDz+TAajU+lGI1GkEgkyGQyKJVKkEqlaDQaVIlOp4PP58P7+ztOpxPt5/F4JA3Jwkqt1Wqo1+uUxe/WbrchEolQLpdpH/PtdksTxeSUyWQoFosPnX+B/61By+USSqWSsmb6slu7Wq0InE6nMRgMHgFYQDZ11WoV8/n8a/8WBgiHw6R9pVIhPW02G2nMmmm32+FwOKjyfD5Pe1n2HHZx/uSJROIr8Z/aDx7ffHZF8KrSAAAAAElFTkSuQmCC)
according to Eq. (
73) instead of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA1klEQVR4nMXRIQqEQBQGYG/gFUyKwUOY9QBmDyBY9AgGm2Czi2C1m8SqICaxKYKYDeovM8FlYGeXTfvgwYP5+HnME/ClBFJ/QOu6wrZtWJaFIAjgeR6KosB5nmyS4zj08bou1HUNRVGwLAuLTNNEmqZ0nucZkiRhHMcX2rYNqqqi73uKSJIsy3SNB7VtS9E0Tei6DoZhIEkSdvEsy6BpGsIwRBzHTyKDXNeF7/v8L9j3HbquI89zPirLEqIoIooiHMfxHg3DgKqq0DQNH/HO8Tsit/rUxNyTu9xcsj+S+gAAAABJRU5ErkJggg==)
,
![](data:image/png;base64,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)
according to Eq. (
66) instead of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABB0lEQVR4nM1RocqDYBT1GQyiwWgcZoNYjKa9gwiaLMqCVmFBMPgClsF8AJcNCj6BNoMGGRM2tiBMz8++Ovx/43/h3HvD4Z5z76WwMah/RDyfz1BVFbZt4/l8rhPrugbHcciy7HviOI6Iogi+78M0TTAMA9d1ydQ8z7EsC6jH4wFZlhHHMV6vF6mapmGaJpRlCZ7nMQwDqDRNIYoi5nkm2O/3CMOQyDVNA5qm0fc9qMvlgt1uR+S7riN9VVW43W7QdR2WZeH9foP6pCRJ4DgODMMAy7LwPA+HwwGn04lY+Nr6eDxCUZTf7/gxLEkSBEFA27brxPv9jqIoCK7X69+fWYvtxCAIsAU/7xBM616OE4wAAAAASUVORK5CYII=)
,
![](data:image/png;base64,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)
according to Eq. (
67) instead of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABEklEQVR4nMWQoeqDUBTG13yARS1iWBusGoeI0SCYrc4o7gEMC3sDu+4RTFYNCiYRt2AVTIJjKmPfHy/My9LW/gc+uPfc3znfOXeFH2P1z+Dj8YBt25AkCZfLhTyUZQlVVYmqqqIdPc/DdrtF13Xk/nq9sN/v4TgOOS/gnDBNc7Hq+x6CICCKImo9DANkWUYQBAuYpik4jkPTNBRs2xY8z+N4POJ0OhEpigJRFHG/3ymYJAk2mw3GcVw6GoYBy7I+tz6fz9B1fUnOBbvdDr7vU/D5fELTNGL3juv1CpZlURQFBeeh1+s1DocDWWqaJriuC4ZhEIYhBeu6RhzHyLKMWM5gnuckd7vdPmf8JX4H3//2TX/cHUm1DxF+3wAAAABJRU5ErkJggg==)
,
![](data:image/png;base64,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)
according to Eq. (
74) instead of
![](data:image/png;base64,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)
and
![](data:image/png;base64,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)
according to Eq. (
75) instead of
![](data:image/png;base64,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)
, then the following results with
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABgAAAARCAYAAADHeGwwAAAACXBIWXMAAA7DAAAOwwHHb6hkAAACJ0lEQVR4nO1US0uqURT1PziUREGIfkDkVNJBJkITER2J4kAcSSThSCchgZNARDEUIVCQEFEHNRMdVIMsDUIpLTEKxBeopbUue8MVhOBK3ju7Cz443zn7rLUfiyPAP4bgv8BCAuPxGHt7e1AqlYhGo3yQSCSwubkJg8GAh4eH5QQIkUgEa2tr6Ha7/B+Px6FSqXB7e4uvr6/lBVwuFywWC6/Pzs7gcDjQarV+TDwn8P7+jq2tLYTDYRwfH8PtdqPf7y9NPhPodDqQSCSw2WwQiUTIZDJ/hXwmcHl5CalUysOm1tBgPz8/f0T48fGBwWAwL+Dz+bCzs8MbFxcXkMlkuL+/n7tI4vV6HY1GA5PJBG9vb6jVanh9fcVwOOQ1nQeDQej1elSrVTaHgDLV6XTweDxMRMEKhYJFf2M0GnH7AoEA9vf3cXh4iGw2C7FYjKOjIybb2Nhg8t3dXb5PLpxOpxBcXV1BKBTCarVyllSi0WjE6uoqrq+vWYBmIpfLcX5+jtPTU6ysrHALyW1Uudfr5TYTYrEYTCbTzNqCx8dHFItFDiA3UflETHtUMoGcpVarUSgUZrFEQOYwm83Y3t7Gy8vL9wKLDO7m5gbr6+t4fn7mCkulEtrtNpxOJyqVCkKhELRaLZrNJpLJJDQaDcrlMle5kABlk8/nuRV+vx93d3dIpVL8vDw9PSGdTsNut+Pk5AS9Xo/jqH085IODA/zpy+Vyi+TxLX4BJfSzjcHMB0oAAAAASUVORK5CYII=)
instead of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABD0lEQVR4nMWRP4qDUBDGcwNLGw9gKZ5Bm2CpWFhKKgsrrayNhV5BCFhZxM4DpNDCRlKk0wOkUBCJiH++xVcsyAY2bLMDw3tv+DHffPMO+CAOW/wz2DQNTqcTjscjyrJE3/cwDAOiKOJyuew72rYNTdMwzzOGYYCqqgiCAK/Xaw/KsowwDAnoui6iKCL3nXTbtuB5Hnmew7IspGmKZVl+zvh4PMCyLHRdB8MwRO6tmev1CkmSME0TOI5DkiTvwc2I4zik6HkeFEXBuq57cBxHCIKAOI5Jsa5rIr+dO/B2u4GiKPi+T1w+n0/QNA3TNL9nJWBVVciyDPf7nYBd15F3URRkn3/7wo/A8/mM33LjvgAD/BCmWAl8VgAAAABJRU5ErkJggg==)
, taking into account Eq. (
79):
![](data:image/png;base64,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)
.
(81)A comparison with Eq. (
61) shows that for
![](data:image/png;base64,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)
the controller matrix
![](data:image/png;base64,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)
of the system with dead time is associated with the controller matrix
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABD0lEQVR4nMWRP4qDUBDGcwNLGw9gKZ5Bm2CpWFhKKgsrrayNhV5BCFhZxM4DpNDCRlKk0wOkUBCJiH++xVcsyAY2bLMDw3tv+DHffPMO+CAOW/wz2DQNTqcTjscjyrJE3/cwDAOiKOJyuew72rYNTdMwzzOGYYCqqgiCAK/Xaw/KsowwDAnoui6iKCL3nXTbtuB5Hnmew7IspGmKZVl+zvh4PMCyLHRdB8MwRO6tmev1CkmSME0TOI5DkiTvwc2I4zik6HkeFEXBuq57cBxHCIKAOI5Jsa5rIr+dO/B2u4GiKPi+T1w+n0/QNA3TNL9nJWBVVciyDPf7nYBd15F3URRkn3/7wo/A8/mM33LjvgAD/BCmWAl8VgAAAABJRU5ErkJggg==)
of the system without dead time via the relationship
![](data:image/png;base64,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)
.
(82)In
[13] | Nuss, U. Ein einfacher Zustandsreglerentwurf im Zuge der Erweiterung der Systemstruktur um Reglerintegratoren und Rechentotzeiten [A simple state space controller design in the context of expanding the control structure by integrators and calculation dead time]. at – Automatisierungstechnik. 2016, 64 (1), pp. 29–40. https://doi.org/10.1515/auto-2015-0058 (in German) |
[13]
, Eq. (
82) was already derived – without taking manipulated variable limits into account – for the problem that a controller matrix (here
![](data:image/png;base64,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)
) is requested for the system with dead time, which produces the same control behavior as the associated dead time-free system, only delayed by one sampling interval. Interestingly, this controller setting also fulfills the requirement for stability in the case of manipulated variable constraints, provided that the dead time-free system (with the controller matrix
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABD0lEQVR4nMWRP4qDUBDGcwNLGw9gKZ5Bm2CpWFhKKgsrrayNhV5BCFhZxM4DpNDCRlKk0wOkUBCJiH++xVcsyAY2bLMDw3tv+DHffPMO+CAOW/wz2DQNTqcTjscjyrJE3/cwDAOiKOJyuew72rYNTdMwzzOGYYCqqgiCAK/Xaw/KsowwDAnoui6iKCL3nXTbtuB5Hnmew7IspGmKZVl+zvh4PMCyLHRdB8MwRO6tmev1CkmSME0TOI5DkiTvwc2I4zik6HkeFEXBuq57cBxHCIKAOI5Jsa5rIr+dO/B2u4GiKPi+T1w+n0/QNA3TNL9nJWBVVciyDPf7nYBd15F3URRkn3/7wo/A8/mM33LjvgAD/BCmWAl8VgAAAABJRU5ErkJggg==)
) fulfills this requirement. Furthermore, when applying Eq. (
82), the pre-filter matrix ensuring stationary accuracy in the command behavior, which is denoted below as
![](data:image/png;base64,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)
to distinguish it from
![](data:image/png;base64,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)
from Eq. (
19), is identical to
![](data:image/png;base64,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)
, i.e. it holds
![](data:image/png;base64,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)
.
(83)To recognize this, the determination equation for
![](data:image/png;base64,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)
is first written according to Eq. (
19) with
![](data:image/png;base64,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)
instead of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA20lEQVR4nL2QOw5FYBCFbYEFWIHSFrQSC9CoLYJCiEprCTq1nig0KgWFVuVVIB7nxv8nf3Nfuc2dZJrJN2fOHA5fivsjcJ4nsiyD53lwHAeGYSCOYwpc14UoiqBpGpqmwXEcsG0bQRBQoG1bSJKEPM+Z9DRNmOeZApZlQVVVcubJw7IskGUZYRi+NjkMA3ieR5qmr4Gu6yAIAuq6ZsO+77GuKwXGcYQoiiiKggy2bYOu62yBu435vg/TNJEkCTGsKApRYUHt+46qqsibZVniPntn81vUbwHXdfGpH4V35DBxakOEAAAAAElFTkSuQmCC)
,
![](data:image/png;base64,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)
instead of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABB0lEQVR4nM1RocqDYBT1GQyiwWgcZoNYjKa9gwiaLMqCVmFBMPgClsF8AJcNCj6BNoMGGRM2tiBMz8++Ovx/43/h3HvD4Z5z76WwMah/RDyfz1BVFbZt4/l8rhPrugbHcciy7HviOI6Iogi+78M0TTAMA9d1ydQ8z7EsC6jH4wFZlhHHMV6vF6mapmGaJpRlCZ7nMQwDqDRNIYoi5nkm2O/3CMOQyDVNA5qm0fc9qMvlgt1uR+S7riN9VVW43W7QdR2WZeH9foP6pCRJ4DgODMMAy7LwPA+HwwGn04lY+Nr6eDxCUZTf7/gxLEkSBEFA27brxPv9jqIoCK7X69+fWYvtxCAIsAU/7xBM616OE4wAAAAASUVORK5CYII=)
,
![](data:image/png;base64,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)
instead of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABD0lEQVR4nMWRP4qDUBDGcwNLGw9gKZ5Bm2CpWFhKKgsrrayNhV5BCFhZxM4DpNDCRlKk0wOkUBCJiH++xVcsyAY2bLMDw3tv+DHffPMO+CAOW/wz2DQNTqcTjscjyrJE3/cwDAOiKOJyuew72rYNTdMwzzOGYYCqqgiCAK/Xaw/KsowwDAnoui6iKCL3nXTbtuB5Hnmew7IspGmKZVl+zvh4PMCyLHRdB8MwRO6tmev1CkmSME0TOI5DkiTvwc2I4zik6HkeFEXBuq57cBxHCIKAOI5Jsa5rIr+dO/B2u4GiKPi+T1w+n0/QNA3TNL9nJWBVVciyDPf7nYBd15F3URRkn3/7wo/A8/mM33LjvgAD/BCmWAl8VgAAAABJRU5ErkJggg==)
and
![](data:image/png;base64,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)
instead of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABEklEQVR4nMWQoeqDUBTG13yARS1iWBusGoeI0SCYrc4o7gEMC3sDu+4RTFYNCiYRt2AVTIJjKmPfHy/My9LW/gc+uPfc3znfOXeFH2P1z+Dj8YBt25AkCZfLhTyUZQlVVYmqqqIdPc/DdrtF13Xk/nq9sN/v4TgOOS/gnDBNc7Hq+x6CICCKImo9DANkWUYQBAuYpik4jkPTNBRs2xY8z+N4POJ0OhEpigJRFHG/3ymYJAk2mw3GcVw6GoYBy7I+tz6fz9B1fUnOBbvdDr7vU/D5fELTNGL3juv1CpZlURQFBeeh1+s1DocDWWqaJriuC4ZhEIYhBeu6RhzHyLKMWM5gnuckd7vdPmf8JX4H3//2TX/cHUm1DxF+3wAAAABJRU5ErkJggg==)
. It reads
![](data:image/png;base64,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)
.
(84)In this context, taking into account Eqs. (
66) and (
67), it holds
(85)as well as
(86)The easiest way to verify the above relationship is to multiply
![](data:image/png;base64,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)
by
![](data:image/png;base64,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)
to obtain the unit matrix. If
![](data:image/png;base64,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)
is then multiplied from the left by
![](data:image/png;base64,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)
according to Eq. (
69) and from the right by
![](data:image/png;base64,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)
according to Eq. (
67), the following results
![](data:image/png;base64,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)
,
which just equals
![](data:image/png;base64,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)
. The renewed inversion and negation according to Eq. (
84) finally leads to the statement of Eq. (
83).
Finally, it should be noted that the stability statement made in this section also applies if the controlled system includes a dead time in the maniplated variable paths and is to be controlled with a PI-state controller. In this case, as described above, a P-state controller is first designed for the controlled system with dead time using Eqs. (
82) and (
83) (without controller integral-action component) and then
![](data:image/png;base64,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)
from Eq. (
82) is inserted as the controller matrix
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABBklEQVR4nMWRoYqEUBiF5xlsVtsYdZ5AxmIRjDbBMiCCmHwCEa1iMAoWg09g1CDiEwg2wWIRZhTEs3iXRYYNu2FhL/xw/3M//nPuvRf8Yl3+AXo+n7BtG6IoIssychDHMQRBgKZp56QoinC73TDPM+nDMIQsy+j7/oSOSZZlkX2e53AcB9M0nXav1wv3+x1pmpIJQRCQCG+ZxnEEwzB4PB6gaRplWX4Pfogsy2JdV+i6DsMwsO/7O+R5HlRVJUJRFLherxiG4YS2bYOiKPB9nwjH7XieR5IkJ1TXNSiKgmmaxG5ZFkiSBI7j0HXdJ3S8Q1VVaJqGQEe1bUu0L8s//DvXdfFTfQAeHg1GSt+SaAAAAABJRU5ErkJggg==)
in Eq. (
17). The extended system matrices
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
and
![](data:image/png;base64,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)
from Eqs. (
66), (
67) respectively (
69) can be used. The pre-filter matrix
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAARCAYAAAAL4VbbAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABT0lEQVR4nM3RP+tBURzHceUJ3NUidUcZDVhEPACKa1EWDwAbFonJZJWBycZ4p7tdBt1SlEUx4orFn0Hev8695WZj+506yzmvzvl8+rr4Ybn+EX48HlQqFaLRKKPR6ANMp1MSiQS1Ws15udfrIUkS5XL5DcUj6XQat9vNcrl0cL1eJ5vNoijKGw+HQ3K5HLIsc7vdbHy/38lkMnS7XSKRCK/Xi/1+T6FQoFqtkkqleD6fNj6dTsRiMRaLBYFAgPP5TKPRQFVVK0ar1XIKihLFYpHtdkswGGQymVAqlbhcLvh8PjRNc3Cn06Hf72OaJqFQiGQyyWq1wjAMvF4vx+PRxiJLPp9nPp9b7ePxOM1m08o9GAwIh8Ncr1cbz2YzPB4P4/HYOhC5xfeHw8Hq4ff72e12Nt5sNui6znq9/hiGKCm6iLt3jJ/H/TVut9t8u/8AnWN1qtzfcGMAAAAASUVORK5CYII=)
can remain unchanged due to Eq. (
83) and the statements in section 3.
6. Example
The following example from the field of electrical drives is intended to illustrate the methodology described above. It concerns the speed control system of a three-phase drive to be controlled. The associated model consists of the series connection of a dead time element and a P-T
1 element to simulate the subordinate closed current control loop as well as an integrator to model the mechanics (single-mass oscillator).
Figure 2 shows the continuous-time structure of the model. To concentrate on the essentials, the setpoint
![](data:image/png;base64,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)
respectively the actual value
![](data:image/png;base64,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)
of the electric torque are used directly as input and output variables of the subordinate current control loop instead of the corresponding torque-forming current (setpoint) components. The time constant of the closed current control loop is
![](data:image/png;base64,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)
. The dead time element comes from the modeling of the computing time of the signal processor used for control. The difference between the electric torque
![](data:image/png;base64,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)
and the load torque
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAARCAYAAADtyJ2fAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABIUlEQVR4nNWSL6uDYBTGV/c5FkzC4r7DbCpjaYgWo8FiECya1m2yTzCcmASj3ZUVNRo16f647bm8Lwzu8Ipj7Z504PA753ne95ngy5r8I7AoCiyXS1iWBVmWsV6vYds2eJ7HbrcbBj3Pw3a7Rdd1UBQFhmHQgaqq2O/3w2Bd17hcLmiaBovFAr7v00EQBCjLchh8NUQywzA4nU6feXw1cRxjPp+DKPhdj8eDKrper3g+n33QcRysVqve5izLMJvNsNlscD6f38H7/Q5BEOgj/VWiKOJwOPSlVlUFlmURRdEgGIbhO0g8kC+ZTqdwXZde/xhM0xRJkuB4PI6CeZ7jdruNR44Eg+M4CpKlkiRRa6MguaBpGkzTpFHUdR1t234f8h+C0mXdmKfFswAAAABJRU5ErkJggg==)
results in the acceleration torque, which leads to the speed (angular velocity)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA3ElEQVR4nOWQP45FcBSFf/uQtwMRPZugUIlNWIWCQkSsQFiAaCkkSDQSiUqUovOvQZyXuc3LdJPMdHOre/J9xT2X4YfD/rdYFAU8z/sGXddF27YfMcsyqKqKZVlw3zfO8ySYJAkURcF1XWDbtkEURdR1TdBxHOi6Tnvf9+A4Dvu+g6VpCkEQCDzPA03T4Ps+5bIs8Xq9cBwHWBzHkCSJwDzP4HkeeZ5Ttm0bhmHQOWyaJsiyjDAMEQQBTNOEZVmIooikYRg+Zb7kqqowjiMV6boOTdNgXddf/PHPxDcBZXB2mcIK9wAAAABJRU5ErkJggg==)
when integrated via the moment of inertia
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAYAAAARCAYAAAD+H91rAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAn0lEQVR4nK3QKwrEMBQF0C6nVEYUsoJ2B7GB6Oi4uFCI7x66hphSkWUU4qNSSqkI3KHjHjNixFx7eN8GX9I8+TOUUqC1xjAM2LaNVkzThHEccd83BaUUnHO01XVd4JwjhEAhpYSu67DvO4V1XdH3Pc7zpOC9h5SSrltrhRAC8zxTOI4DjDHEGCk8A9u2Rc6ZgjEG1trPlyzL8r725ye+AM1iLEkgZb82AAAAAElFTkSuQmCC)
.
Figure 2. Continuous-time block diagram of the exemplary controlled system.
A discrete-time PI-state controller is to be used as the speed controller. For this purpose, the system model must be discretized beforehand. Using the sampling time
![](data:image/png;base64,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)
and neglecting the load torque, the discrete-time state equations of the dead-time-free system are approximately as follows
[8] | Nuss, U. Stabilitätsverhalten von zweistufig entworfenen zeitdiskreten PI-Zustandsreglern bei Stellgrößenbegrenzungen [Stability properties of two-stage designed discrete-time PI state controllers considering the limitation of input variables]. at – Automatisierungstechnik. 2017, 65 (10), pp. 705 – 717. https://doi.org/10.1515/auto-2016-0136 (in German) |
[17] | Nuss, U. Hochdynamische Regelung elektrischer Antriebe [Highly dynamic control of electrical drives]. 2nd edition. Berlin, Offenbach: VDE; 2017. (in German) |
[8, 17]
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
,
if the state variable
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAARCAYAAADtyJ2fAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABAElEQVR4nO2RsY5EUBSG5yGUWq+g8wAahVB4BoVEItFNoZhapVDTi45oqASVEd10WgWhwr+5spHdrJ1MJtluv/r/7jnnvxe8wYXwL/6FuG0bPM8DwzAIggB930OSJKiq+lwcxxG2beN2u0EURei6jizLkKbpEVyWBeu6nq9aliUoikIURUeAhOu6hiAIiOP4XCyKAjRN436/H4E8z/czWJbdt/gmdl2Hx+MBwzDAcRxc10WSJJjn+QjyPP9TVBQFsiyjbVs4jrOXdL1e97ueik3TgEwlTNOEqqpACvvKqfhr55+Qx8gJvu+DfN1L4jAMsCwLmqbBNE2EYfj6xDOI9wHYrSx3GbqnUAAAAABJRU5ErkJggg==)
is understood to be the actual torque value and the state variable
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAARCAYAAADtyJ2fAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABJ0lEQVR4nO2RMauCUBiG+xGNrv6MfkCr1ODcVrgIQptjiwhODdEkroLgJLTkqKKD4ubmalEaCeJ7+Q5cI+JGXLjbfeGDwznn+d7vvGeEX2hE+gf/Auz7HpZlged5OI6Dqqowm82wWq3eg9frFYZhYLPZQBAEyLIM3/dxPB7Zpa7r0LYtKzJ5GTUMQ4zHY3ieN3Q+nU5YLpfQdR3T6RS2bb+CQRCA4zikaTqAdV0jjmO2pucsFgvmysCyLFEUBRRFwWQywW63w+FwwO12GxrQuHRumubDURRFzOdz5HmO7XbLQlJVlV0mkYPrumzvfr8/wCzLQK6kpmkQRREosG8lSYL1eo3L5fKc6rv/onFpGkp8v99D07TPwPP5DEmSWLJUFNBH4E8i7gv9hihdkVLmuwAAAABJRU5ErkJggg==)
, which is also the controlled variable
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAu0lEQVR4nN2QMQ5EYBCFnUPlCHqtxhFIVBKF6CVaV1A6A4VORKfRuINGJRJCQuETtthsfrsH2Je8YpJvMm+exA9Jl/4DGMcRwzCwLIt937lmXdcJw/AFZFlGURTIskzf9/embdtUVfU+cRwHmqZRliXDMOC6LsuyfGbwfZ84jknT9LYQMkkSHMchCALWdRWBuq5RFIWmaZ7fzPOcKIrEN9u25bLneUzTJAKqqmKaJl3XPRc1zzPbtn1t8gTaoK2aZE2OOAAAAABJRU5ErkJggg==)
, is understood to be the actual speed value. In contrast, the output variable of the dead time element from
Figure 2 serves as the manipulated variable
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAYAAADg1u3YAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAtklEQVR4nN2QIQqFUBBFNVtFNyCYxB24F8VqsLgKo+ACLBa77sAiL74FaBAUFBfgEZ/wjb/98AcGZi6HMzAaX0r7D0AIQVEUeJ7HcRwqnKYJ27bVrg3DoAZd15nnWQFVVeH7/nuibVtc1/1o4zgmSZIXyLKMNE1VsG0blmVR1/ULBEFAWZbs+06e5xiGwTiOLMvyAGEY4jgOURQhpcQ0Tbquo2maB7i1fd+zrqvS3tDd53n+4pMXH7vrCLsFZQIAAAAASUVORK5CYII=)
in the dead time-free system. For
![](data:image/png;base64,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)
, it holds
![](data:image/png;base64,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)
. For the relevant system matrices, this results in the values
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
.
As can easily be seen, the transition matrix
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABB0lEQVR4nM1RocqDYBT1GQyiwWgcZoNYjKa9gwiaLMqCVmFBMPgClsF8AJcNCj6BNoMGGRM2tiBMz8++Ovx/43/h3HvD4Z5z76WwMah/RDyfz1BVFbZt4/l8rhPrugbHcciy7HviOI6Iogi+78M0TTAMA9d1ydQ8z7EsC6jH4wFZlhHHMV6vF6mapmGaJpRlCZ7nMQwDqDRNIYoi5nkm2O/3CMOQyDVNA5qm0fc9qMvlgt1uR+S7riN9VVW43W7QdR2WZeH9foP6pCRJ4DgODMMAy7LwPA+HwwGn04lY+Nr6eDxCUZTf7/gxLEkSBEFA27brxPv9jqIoCK7X69+fWYvtxCAIsAU/7xBM616OE4wAAAAASUVORK5CYII=)
has an eigenvalue at
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAARCAYAAADKZhx3AAAACXBIWXMAAA7DAAAOwwHHb6hkAAABUUlEQVR4nO2TMYuCcBjG/SIufYnwCxhEk6tjRXtTQ4JD0GaDNDVJtAhuDQ3pKCiCRA3h0BAkDaEgLkb5HP6huCjvziM6uLsHHvgv7/t73+flT+GHRP2Dfw04TVMsFgu0Wi0cDofXgDebDfr9PsrlMiiKgu/7rwEbhgHP8zAajR6De70eqtUqmS6LpFQqwXGcmybr9RqSJOVaUZTcASaTyT34dDqhXq9jv99jt9uBYRjIskxu817L5RKiKOZ6OBwWA18eQRCAZVl0u12cz+dPYyyiXHAcx+B5Hs1mE0mSPCy2LAuNRoOk88iCIBQDH49HtNttcByHKIrgui7m8/ld8Xa7haqquZ7NZrng8XhMwNkpr+DBYACapqFpGimuVCrQdf2D4L6u7N9Op1PUajUC7nQ6ZKnslNRqtYJpmlfbtk2if4bCMLzpfTEBP4XwDf098BveRmfjy/BYJAAAAABJRU5ErkJggg==)
, which is why the controlled system is not asymptotically stable. It can therefore be assumed that
![](data:image/png;base64,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)
respectively
![](data:image/png;base64,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)
will not be a positive definite matrix term. The calculation of
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEcAAAASCAYAAAAJ88NbAAAACXBIWXMAAA7DAAAOwwHHb6hkAAADL0lEQVR4nO2X3yt7YRzH/QFa2S5248KiVsqN4m41KSkXpFwoJUliy4S2CBd+TIncICJFKOVGylK7mdLKxZqStflVIvkRIfPb+9vno41xdnbO106+9d2rns55zp7zvJ/n/Tyf53OWhASCJBG/PYh/FVnmlJSUoLe3F9XV1WhpaUFrays6Ozvx+vqq5Bh/DVnmFBUV4fn5GVVVVZidncXx8THMZvP/Zc7a2hrGxsbYgGAwGH5+cXHB15A5xOXl5bdOX15eMD8/z31QmZqawsbGBj8XI5quktC4QuMcHx/H0tISbm5u+DdBczweDzQaDUZHRwV3xWdzhHh7e8Pk5CR0Oh1OT08RCASQl5eHkZER0YHG0lWC/f19pKWlYXFxkRe/q6sL+fn5eHx8/DDn6ekJm5ubmJ6ehtVqhVqtRltbGxYWFnB+fh7RYSxziKGhIRQXF4frPT09KC0tZeM+I0dXCXZ3d5GZmQmfz8d1uqpUKlxdXb2b8/DwgMbGRtTU1GB7e5snbjQa2cmJiQnk5OSEtxqtaHl5OYaHh6MKhtrY7fZwvbKyEiaTKaKdHF2lcDgcyM7Oxu3tbbiempqKu7u7d3OcTicyMjJ4ILSytDM6Ojq48cHBAVJSUvhK7OzsoKysDHV1dTg7OxMUvL+/59UgIbpfXl5Gbm4uvF5vRDs5ukpBehUVFbyAFGKUdAYGBrjO5lC8kXuUiWgr073L5eIDtLu7m1+gGJQKiSQnJ6O9vZ1TP4WY3+//FlI/1V1ZWcHc3JxoibaABPVdWFjI4d7X18ehv7q6yqFOsDmUGWgiTU1NsNls0Gq17GhzczO/IDf2Z2ZmeKJfzfjKT3X7+/vR0NAgWvb29qK+f3JyAr1ej/X1dcHfwwcyTYQcGxwchMFg4POAVjTWBIWora2FxWKR1DaeunJxu91IT0+PursiUvn19TUKCgqQlZUluh3FoA9DOtDooJUaivHQlQstQn19PZ95R0dHgm0izKGB0TlA5fDw8K9Et7a2+H36mKJJSyEeunKh7BTSpEwpROKPpwgJc0Rgc+jUl1KkQGlbrA9KvyGk6tLBGW/oGyuWblICcf4Ah+8Vjx1PRkoAAAAASUVORK5CYII=)
according to Eq. (
58) or (
59) shows this immediately. Because with the symmetrical matrix
and the abbreviation
![](data:image/png;base64,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)
you get
Because of the zero element on the main diagonal,
![](data:image/png;base64,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)
respectively
![](data:image/png;base64,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)
can at best be positive semi-definite, and only if the secondary diagonal elements are zero. From this follows directly
![](data:image/png;base64,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)
.
Since it is sufficient for the example to carry out the controller calculation with
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAARCAYAAAC1tw6GAAAACXBIWXMAAA7DAAAOwwHHb6hkAAACXUlEQVR4nO2Vz0tiURTHBTetlP6E9uJGEGtXZGIoulDaRAiuhJAKfwSCulMUQhGSInFjQYIuFDe2MvwBWqhLCQkEEVyICoo/0u9wr/RmJseGIaMW84Xz4Nz3zj2fe865PBa+mVhfDfBW/4H+Jlaj0YBarYZUKsX9/T1dHI1GODk5QSKRoH6n04FWq8Xm5iZub2+Xkvjh4QF2ux0WiwV+vx+DwWAGNJ1OodPpIBaL8fLyQhczmQxWV1dxdXXFbGC1WrG3t8cEfkS1Wg0ikQjpdBqtVgt8Ph+Xl5czIPI4PT2FUqmkC6Q6R0dHEAqFDBCB2NraQrVa/TAM0fn5OQQCAdrtNvUPDw+xsbFBc1Mgj8cDmUxGXxYKBRgMBhwcHDBAoVAINpsNpJpvFY/HcXZ2ttAeHx/nYjQaDe3IZDKhvtfrBZfLRb/fnwGRHpIZGo/HOD4+RqlUokCBQADdbhe7u7toNpt/PG04HKazsMhyudxcDMklkUgY/+LiAmw2G71ebwZEEu/s7KBYLNJ2ERGgYDAIn8/32ywtQ3K5fA5oZWXlZ4Wur6+xvb0No9HIlJgAuVwu7O/v01u2SKS65AYusmQyORej1+uxvr5OK0LkdDrB4/EwHA5nQHd3d1hbW6PD9TonBIgMXiwWe/e02WyWtm2RVSqVuZhoNErzPT8/U5/cXpPJRHNToFQqBQ6Hg3w+zwQRIIVCQcu4bJG5JNUzm824ubmBSqVCvV6n71ivH5BWvU490dPTE/PRZ4gctFwu07y/jsT3+3V8NcBbsRwOB/7F3G73hxJGIpF39/8BSKCSDN99rEwAAAAASUVORK5CYII=)
and thus use Eq. (
48) as a basis, it applies that
![](data:image/png;base64,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)
.
Finally, the condition
must be fulfilled so that
![](data:image/png;base64,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)
can be positive semi-definite. For the matrix
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAARCAYAAAAPFIbmAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA1klEQVR4nMXRIQqEQBQGYG/gFUyKwUOY9QBmDyBY9AgGm2Czi2C1m8SqICaxKYKYDeovM8FlYGeXTfvgwYP5+HnME/ClBFJ/QOu6wrZtWJaFIAjgeR6KosB5nmyS4zj08bou1HUNRVGwLAuLTNNEmqZ0nucZkiRhHMcX2rYNqqqi73uKSJIsy3SNB7VtS9E0Tei6DoZhIEkSdvEsy6BpGsIwRBzHTyKDXNeF7/v8L9j3HbquI89zPirLEqIoIooiHMfxHg3DgKqq0DQNH/HO8Tsit/rUxNyTu9xcsj+S+gAAAABJRU5ErkJggg==)
, it follows under the above conditions
![](data:image/png;base64,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)
.
It has positive definiteness for
![](data:image/png;base64,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)
and
![](data:image/png;base64,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)
. Furthermore, the positive semi-definiteness of
![](data:image/png;base64,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)
must also be fulfilled, which leads in the example to the condition
using
![](data:image/png;base64,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)
. In the next step, the P-state controller matrix
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABD0lEQVR4nMWRP4qDUBDGcwNLGw9gKZ5Bm2CpWFhKKgsrrayNhV5BCFhZxM4DpNDCRlKk0wOkUBCJiH++xVcsyAY2bLMDw3tv+DHffPMO+CAOW/wz2DQNTqcTjscjyrJE3/cwDAOiKOJyuew72rYNTdMwzzOGYYCqqgiCAK/Xaw/KsowwDAnoui6iKCL3nXTbtuB5Hnmew7IspGmKZVl+zvh4PMCyLHRdB8MwRO6tmev1CkmSME0TOI5DkiTvwc2I4zik6HkeFEXBuq57cBxHCIKAOI5Jsa5rIr+dO/B2u4GiKPi+T1w+n0/QNA3TNL9nJWBVVciyDPf7nYBd15F3URRkn3/7wo/A8/mM33LjvgAD/BCmWAl8VgAAAABJRU5ErkJggg==)
is calculated by means of Eq. (
61). Because
![](data:image/png;base64,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)
was selected, the result is
![](data:image/png;base64,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)
.
If, for example,
![](data:image/png;base64,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)
holds and
![](data:image/png;base64,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)
is chosen without restricting the generality, then
![](data:image/png;base64,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)
and
![](data:image/png;base64,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)
, for example, fulfill the above conditions. For
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABD0lEQVR4nMWRP4qDUBDGcwNLGw9gKZ5Bm2CpWFhKKgsrrayNhV5BCFhZxM4DpNDCRlKk0wOkUBCJiH++xVcsyAY2bLMDw3tv+DHffPMO+CAOW/wz2DQNTqcTjscjyrJE3/cwDAOiKOJyuew72rYNTdMwzzOGYYCqqgiCAK/Xaw/KsowwDAnoui6iKCL3nXTbtuB5Hnmew7IspGmKZVl+zvh4PMCyLHRdB8MwRO6tmev1CkmSME0TOI5DkiTvwc2I4zik6HkeFEXBuq57cBxHCIKAOI5Jsa5rIr+dO/B2u4GiKPi+T1w+n0/QNA3TNL9nJWBVVciyDPf7nYBd15F3URRkn3/7wo/A8/mM33LjvgAD/BCmWAl8VgAAAABJRU5ErkJggg==)
and
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAwAAAARCAYAAADpPU2iAAAACXBIWXMAAA7DAAAOwwHHb6hkAAABRElEQVR4nM2RMcuCUBSG+wetbi0OgVtLa7hI4BQRpIFzhItOCZK4RNBPaGtzzqYQkYiICBoVsb+gkxjK+3HvENyt75u+A5cL55znnvO+t4VfRIvEPwWKosB8PsdgMMD9fv80NE2DzWaD4XCI6/XKTnAcB+12G/v9/gM8n0/wPA9JklBVFQssFgvIskxfJPF+v2GaJp1q2za7Up7nmE6nWC6X0HWdFqMogmVZ6Pf78H2fBdI0haIo8DwPo9EIZVlC0zTcbjd0u11aZ4DD4QDXdXE+nyGKItWx2+0QBAF6vR7dgAFWqxWOxyOSJIEgCBiPx1Qk0aOqKmsrETebzRDHMbW30+ngdDqhrmtMJhNst1sWIOJIUxiGNEn8Jv6Tm+M4GIZBp30AIuhyuSDLMuZXX68XzT8eD2oxo+Hb+BuwXq/x7SH9P6WnedaLMMqYAAAAAElFTkSuQmCC)
, it follows from this, but for general
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAARCAYAAAAR3bZVAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAeElEQVR4nNXQsQnAIBQEUNexdAUncBNXcANXEKwVXMDSBezUHWxs5YKmSCFpA7n2fY7jE7yErPwLU0qQUkJrjRACKKWw1t6Yc4b3HpxzGGMQY0Rr7alVSkEIcdbOOTc4507svYMxhlLKibXWjevowDEG1qhV/9GHLpmlewmU1i9mAAAAAElFTkSuQmCC)
,
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
.
If the controlled system model is then extended by the dead time element, Eq. (
82) immediately provides the extended controller matrix
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUMAAAAtCAYAAAA0ugmZAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAQa0lEQVR4nO2dB6wUxR/HjYlEhBBClIQQExAMFooYaQJCRAGNAqGIIL0jEEAgoKCISJCi0ruK5dF77x3pvaPSpIsKAiqgzj+f+Wcu+/Z293Zv9/bdvZtvsnn3bve2zPzmO786e5/Q0NDQSHPcB7L6JjQ0NDSyGpoM0ww///yzOHXqlNx+++23TPv+++8/cePGDXH27Flx/vx5ce/evSy6S2f88ssvlvf2559/Rp6N7Z9//smCu9NIVWgyTDM8/vjj4oknnhDFixcX48aNy7Tv9OnTolOnTuK7774T9erVE5MnT86iu7QGRJ2RkSGKFSsm/vjjj6j933//vXwuthw5coirV69mwV1qpCo0GaYZIEO0Pitcv35dalRg+vTp4q233pLaYrJg9uzZYtOmTSJnzpzizp07jscWKlRIk6GGJ2gyTDM4kaECJmjXrl2lhpiMeOihhzQZagQOTYZphlhkiCY4Y8YMMWjQoJiEk1XQZKiRCGgyTDPEIkPM0A8++EDcvn07kOtBSAsWLBDLly+XPr8goMlQIxHQZJhmcCJDosuvvvqq1AoJrnz44Ye+fIZErl988UWxbt068emnn4rXX389KoLtFZcuXRIPPvigOHnypONxmgw1vEKTYZrBiQwvXLggunfvLv2FbN9++23c14FE33nnHdGoUSP5P9csWrSor3Pu3r1bEjT3hva6Y8cO22M1GWp4hSbDNIObAEoQIMfv6aefFoMHD5b/kwNYuXJl0a5du4RfG2gy1PCKtCTDv//+W9y6dUtuf/31V1bfTmBQz+Tk7wuLDO/evSseeOABMXbs2Mh3mOCvvPJKwq8NkpkM0ZpVXzFJaIQP2l31gXIFZRsyZPDhm9qzZ0/MY1u0aCEKFiwoRowYIebNmxe1/99//xUHDhyQTn8jaDT8YKScrFmzJrB7N4OOmjZtmrhy5Yqn30E8Q4cOFY888ojtMXZkiCbH9cybVXKzG9Af999/f2BkyKRldX92k1nYZEg60ubNm8X27dtjHkv700fI36xZs2yPQw5xBXz22Wfi888/F0ePHo3s47nJu0QewwLPuHXrVrmFBZ5z5syZ4uLFi5Hv+DxlyhTxySefiC+++CIyTpB95eJhW716te15yaOl/fPlyxeRk2xBhggEDfHoo4+KJUuWxDweMnzmmWcs95F4PGHCBFGiRAkxZMiQyPcIJoJLIx8/flz+nwhAwn379pURU69kCJjp4iFDqk8KFy4ctb377rue7wEwcBA0AieA9qpZs6Zo3rx5XOebM2eO5f3xvRXCJEOCOjwn98PAjQXan36IhY0bN8oJ5Pfff5cyx2d+e/jwYekzzZMnTyRJPtFAFiHlIkWKhJZ/evDgQdGvXz+RO3fuiMySRVCxYkWxfv16KWMjR44Ub775plRUatWqJUaNGiUrp6pVqya2bNkS8xpGOckWZIhAoMHUqVPHNxkSRKD29b333pMEq7Bv3z5Ro0YNce3atcDu2wrM/mhVaK5hkmHQgPyqVq0qOnToIP9nknnuuefEmDFjEn5tECYZMpGQNtS4cWNHTU/BLRl+9NFHkcmI9mSwr1q1ShIjrh7OwbXDwJkzZ2QfMnbCIsNjx45J8qMvlcyiKVJOyoRAm2DZEaSDDCFP/sIFr732mqva+mxHhgpBkKGCmQz79OkjmjVrJtVz9s2dOzehCwGkOhmCqVOniqeeekpOLtQNlytXTg6qMJAVPsOgyRDNhgll7dq1YsWKFbItly1bFtkfJhkqhEmGCkYyhOy++uor8fLLL4v+/fuLl156SWzbti3T8ZjOxrEb69yaDD2SIQsXYDYzI6OJPvnkk9I8ShSyAxkyq0+aNElqNwMHDpSpMWEhO5AhAx/tCJMQ3+ELL7wg9u7dG9mfjmSITFWqVElOFMpMbtq0aSQIwvgka+Gnn35yfe4oMuTEOBwXLlwYCULgv+J/tkSbh0EAMly0aFHM49ySIQ5aBfLvxo8fLz8rVT2RpAIZxkO2fsgQ3xRmLJMAq9egzSUTfvzxR+n0/vjjj0Xnzp0dgwdZRYY45mPBLRkqMNAh2ZYtW2aqvOEctEmYYOx88803oV6Tvjx37pz8TKYEYwN3FtiwYYPUEhUZMv4bNGggXU1uzx1Fhph8VB4wkNQgQMt6+OGHpToab1QxDEAAEydOlM7dunXrykivU+VELDIkklumTBlRvnx5aQ5zLnx5rVq1kqYfvhySfxNRu8tagjiqcRr36NFD7N+/39Pv/ZAhJqwy/3EH4BZIplVr8JcxQXN/bdq0iUxOVgiTDNFGcNqztBi+Kkxap3bzQob4wUaPHi0nAPU8EAGTQt68eeWkEIbGjaxjfmKqkxGAuZ5o2WCyI4pOoIjAJVoxfkIm627dukkNlUnb6Dqg/d1o5wq2ZjINzIOqRT7ffvttWbSf7ItkotUaF/WMpVHFIkNMD3UuZiTV6TiQ2YcwJmrhU8jM+Cxey9eCMJMRQjThxYsXe7p2WMB5zvJiBBLsECYZMj6MfWZMA7GCFzJEW8ddYhyDaEfG64VhtUFCxmsqzSyRMD/nr7/+Kr+nLWhjxiLfGUkZ2fWipFiSIQ+LJoBpiMMb5iW0n6gUkqyEGzM5VeGXDBGmnj17Ssc0WkmyAWc5EWpcIk4TRTInXXs1kzUSB0syZBCVLl1avPHGG6Js2bIJTSpWwBGaK1cuyw3V2E0wJB5oMnTWDJlpWagBH1UymckKaAbkKxKcsYMbMsTsxh1hJ4O4SRIBTYbJA0syxBwkaRl1lKiVSpZNdjD4CXaYN/x+drAjw507d1qey24jsBQPvFzDabOK6vkhQ0wPqgsgwC+//FJ07NgxrudLFKjuQHPFWmnfvr34+uuvbY8NSzPEJLPrHztz2YkMhw0b5lsu4h279HkQcok/PR6QQB/U2HDbJpZkyI2QBwbI42FWjDdocuTIEcsyNzOw961KrNSGYzoWKF3D12neli5davsbOzI8dOiQ5bnsNsr/4oGXazhtVj49P2QI0RAkGjBggDRF3aYnhAUih5AggSUit06+OTdkCJE5yZ/yUTmBqKVd/9hd34kMISS/ckGQLx4wZoOQS2MJphegqQc1Nty2iSUZ4iciMgPQEomMkd9kxK5du8Tw4cPl7EXkEQJFOIlAEuInssWFKY+BTIlOO4GMeqsSKzaWe3KqLfQDbSaHk2eYCCCTBHeIKDqV9rkhQ8j1scces5VByt8SAW0mJw+iyBDtCq0QogOYIZS4QBoqXweyI5mR4Ar+RASFzxAfoX8c29QRMpviyyESnYz+JqDJMDXJkIDJ888/L2UMuTQmIJuhAygabhBFhvi+EDLWmiOlhpA2Wh6rFKuVW8gxevbZZ2W+Ez4B8nkgTcwJkh7xLzEQQXYhQ7Tfy5cvR/6nGgC/FUvj+1mkVIG0CXMaEKYfGrjxul6QjGSIuyPWytRugOsD892NXKUyGeKeIp+S/EHSuQAKC7Jn3NSKPcgpMuN3FXEF+ou0FWM7c08UY/zwww8JH9dMdFzfmE5k1SbgxIkTsi1YMchoCrttk7jK8cg3rF+/fqZUGxoFjRCNkn0QJ9+RgEpSbCqTIdqHeWEBMtuZIFgRA3dAvKCzcQFwLuO7ifG1MongfmCJ/HhyuZKJDJEV5INSxgoVKvg+H9UFTNKcF3+mUz5ZKpMh5j/Ej+JBEQHjCNkgwInsValSRf6eiZMCiS5dushE49q1a/sqjqBdSadq27atHCOKjCBHfMnIJRkG5qXtggLPyaSJTxhly1hF0rp1a3lduIaxobiFUjw1JlXFmJc2iYsMYVhyu1jeio7CeU/qC5FnBi3aIJ01f/58mZ9Ih7kpTQoCJEDTACTholFRAeD08iE3ZIggQvBGMqS+NgiCZ4ajjUhjIlil0Lt3b6mBAzoT94MZCAhRb9reqi41WciQgUUlDRtEGIRbgmfDD43PGmJ0KrkKO+ka7QTC4porV6501EhikSEZHcgZydQFChSQbUllkgoK4h7o1auXvC5uKhZy4BgmHcZfvMDK4xpUfRjfmY1bjOsA2h1fvxUYf8glGlk8gLSwUjMyMiTBGRUvpanSDqwLoPYxTs250F7aJIoMSUOJtfGA3CxlaaimqO1EHxEAPtPBRGMJvtBJMHxY9cyo7qwjR6IwswfRLCczMxYZcg7qL3HUG8kQsqKzIDK3tY9OYAY2kiH3r/I7KUDnHSJGMEO///778jfsJxfQjCDIkIHMhEKb0u/xJN4juGqhiazw0YZJhpAXAw7FADKgNNSpbtqtz5B+Rksyg+9UcJFAo6pPRl6QC79AozKSIQuUIPuA8V+qVKkomWC84D5TloAfkN5lJkNAiShaIW4qBRYBgaAJrClLwUubRJGhseTFbkv2hRqIDGJSuCnFcRqcNAymKtomVTgsFokmhmBAEJA/JgMC7xdmMiQJXUXwIWHI2AiIEhOKyYYJ6ObNm1Hn9EuGaB2YHggcZIuG6rcKKbuTIUBGWLzWTTqYGzJk/UxMQ/MrHHAPGN8yiDWmLAS0RT/uGwUzGZJqRa4eQBEqWbJkJn8efm/MVMURBFb9wI4M4SHuRb1kDJCOo9xLaLTA3CaeyNDXnScJUJfJ0XIDp8GJuU3HY44RVELwMPWMwPwPYhEDMxmSO4eJADDJzWuycV9OicbADxnimGZAI2AA7VcFxfwgHciQgWjl1rBCLDKEXJiQrHxdvGCLhUIUkFGVKE8AFN++X5jJEBmlDwHugOrVq2c6HiUBWQ6qdNeODAEymj9//qj1EtAO0Ui5Zy9tku3IkEHbpEkTOTu7gZvBSUOiAZIuBIhisRw/MzWRc7eCbwc0O5y7EJzqWBzUDAJmWBbMMC/prnxw/JbomZWp7ocM0UpZrSeoF8grBEGGVNtYlczZra0Xts8QWaFP3CAWGdJeaIVM8EzGihR4HqwHY804miBmM9oYC8H6jSgrXy/nUn53yJn+I2DDeDBnUmAaQ55YZVgUfoM4kBfap9HVRbwCIlywYEHEZ4nJjhmMvNIGKoDipU2yHRmSYkADWpmNViD4g+bHZle2h7+L/cz4pMBAMryAB/8JmpOfVWsgMXV9NoImCAEEzLkRRtJ4zIAEORafqNW7L9Ak1T3bwYkMESKCOsb7DCJglN01Q9oJa8EpaGcEZKH63krTp//UfqPZCxmhpRknQeSGogeOC+LlUNyPUTbVxIg8UtJmVwiBr5RIr5sXYjmBKhjj9RUhsrgt1+f51T0xHjieMckyXqpd3LQJVqT5GbMFGWq4hxMZMjHwIiyiggSJ8Le4nWDsgIA2bNhQFC9ePNTXYiZzao1GckKTYZoh1krX5IdSikmgxu+S8qQ3EASCYDkn/hvlE/ULNHO3+WMaGm6gyTDNkMrleABfGtFE/EZ+l/DS0DBCk2GaIdXJEI2VXE+ihyTg2kGToYZXaDJMM6Q6GRrTSpygyVDDKzQZphlSnQzxaRIB5D3W2meoESQ0GaYZUp0MITgi1OS1Ob2yUpOhhldoMkwzpDIZUnalEmhZiNipJFKToYZXaDJMM1DVQKkfpqax4D0VwGIIRJOpNCCirBaCUCARXSXSUpngN0dSI72gyVAjZUBlAYnbVAMl6r3VGumL+zQ0NDQ0/o//ASI1kI0GioboAAAAAElFTkSuQmCC)
.
Finally, if a controller integral-action component with the eigenvalue
![](data:image/png;base64,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)
assigned to it is added, it follows from Eqs. (
17) and (
18)
![](data:image/png;base64,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)
,
![](data:image/png;base64,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)
.
Figure 3 shows the transient response that is achieved. Here, a speed setpoint step from 0 to
![](data:image/png;base64,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)
is specified at time
![](data:image/png;base64,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)
with vanishing initial state variable values. The step height was intentionally chosen so large that the manipulated variable of the speed controller, i.e. the torque setpoint
![](data:image/png;base64,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)
, reaches the limit
![](data:image/png;base64,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)
at the start of the transient response. In addition to the uncorrected speed setpoint
![](data:image/png;base64,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)
, the actual speed value
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAARCAYAAADkIz3lAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAA3ElEQVR4nOWQP45FcBSFf/uQtwMRPZugUIlNWIWCQkSsQFiAaCkkSDQSiUqUovOvQZyXuc3LdJPMdHOre/J9xT2X4YfD/rdYFAU8z/sGXddF27YfMcsyqKqKZVlw3zfO8ySYJAkURcF1XWDbtkEURdR1TdBxHOi6Tnvf9+A4Dvu+g6VpCkEQCDzPA03T4Ps+5bIs8Xq9cBwHWBzHkCSJwDzP4HkeeZ5Ttm0bhmHQOWyaJsiyjDAMEQQBTNOEZVmIooikYRg+Zb7kqqowjiMV6boOTdNgXddf/PHPxDcBZXB2mcIK9wAAAABJRU5ErkJggg==)
and the unlimited torque setpoint
![](data:image/png;base64,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)
,
Figure 3 also shows the corrected speed setpoint
![](data:image/png;base64,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)
, the saturated torque setpoint
![](data:image/png;base64,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)
and the output variable
![](data:image/png;base64,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)
of the speed controller integrator. With regard to the latter, it should be noted that
Figure 3 is based on the fact that the control difference is first multiplied by
![](data:image/png;base64,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)
and only then integrated and fed to the manipulated variable determination with a positive sign. In addition to the aforementioned value
![](data:image/png;base64,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)
for the weighting factor,
Figure 3 also shows curves for other values of the weighting factor
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAARCAYAAAAR3bZVAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAeElEQVR4nNXQsQnAIBQEUNexdAUncBNXcANXEKwVXMDSBezUHWxs5YKmSCFpA7n2fY7jE7yErPwLU0qQUkJrjRACKKWw1t6Yc4b3HpxzGGMQY0Rr7alVSkEIcdbOOTc4507svYMxhlLKibXWjevowDEG1qhV/9GHLpmlewmU1i9mAAAAAElFTkSuQmCC)
in order to demonstrate its influence on the control quality. All diagrams are based on the moment of inertia
![](data:image/png;base64,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)
and the sampling time
![](data:image/png;base64,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)
.
As can be clearly seen in
Figure 3, the control loop dynamics continue to increase as the weighting factor
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAARCAYAAAAR3bZVAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAAeElEQVR4nNXQsQnAIBQEUNexdAUncBNXcANXEKwVXMDSBezUHWxs5YKmSCFpA7n2fY7jE7yErPwLU0qQUkJrjRACKKWw1t6Yc4b3HpxzGGMQY0Rr7alVSkEIcdbOOTc4507svYMxhlLKibXWjevowDEG1qhV/9GHLpmlewmU1i9mAAAAAElFTkSuQmCC)
decreases. At
![](data:image/png;base64,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)
, however, there appears a clear torque ripple. But, at
![](data:image/png;base64,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)
, the previously mentioned sufficient stability condition is no longer fulfilled. Since this is only a sufficient condition, stable operation cannot be excluded, which is also shown in principle in
Figure 3d.
Figure 3. Transient response of the relevant variables of the speed control loop when a speed setpoint step is specified with a limited torque setpoint; a) r=10, b) r=5, c) r=0,5, d) r=0,05.