A closed-loop system (with feedback), and "actuator saturation". The poles and zeros of the system change when the system goes into "actuator saturation" behavior?
Thanks a lot for your feedback
While the definition of poles and zeros are well-established for linear differential systems, its 'extension' to the nonlinear case is far from being trivial, so I'm not entirely sure what you mean by 'pole' or 'zero'.
That said, it might be the case that you are designing a model-based controller considering a process represented by a linear system (obtained, for instance, via Jacobian linearisation), while you also want to take into account actuator limitations, in terms of a saturation function. If this is the case, then I suggest you take a look at AntiWind-up techniques (see, for instance, http://cse.lab.imtlucca.it/~bemporad/teaching/ac/pdf/AC2-09-AntiWindup.pdf)
Hope this helps
Hi Nicolas Faedo,
Thanks for your answer. Yes, the case that you explain is my case. I have the poles and zeros of the linearized differential system that I use, but the system has actuator limitations in terms of a saturation function, and I would like to know if the characterization (poles and zeros) of the closed system is modified in this actuator saturation region.
Thanks a lot for the slides about Antiwindup, I'm going to analyze the information.
Jose,
I can see from your profile that you are fluent in Spanish. If that's the case, I can recommend this set of slides on the topic:
https://csd.newcastle.edu.au/SpanishPages/clase_slides_download/C16.pdf
These are based on the book "Control Systems Design" by G. Goodwin and M. Salgado.
Indeed it have a very strong relationship. I think you are after its mathematical concepts or its further relationship to roots, points of inflections, or equilibrium points. I think standard Systems and Control Theory, Dynamical Systems books may solve your problem. Or in short cut way you may follow : https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf
You might also find interesting my 2003 paper "Strong stability of elastic control systems with saturating feedback" Systems and Ctrl. Ltrs. 48, pp. 243--252 for a distributed parameter context.
For this topic, there exist roughly speaking two lines:
- First case: the open-loop system is stable (not necessarily attractive). Then, you are in the case than the one mentionned by Prof. Seidman, and you can hope having a global asymptotic stability result when considering the closed-loop system with the saturation. One would like then whether, when modifying the stabilizing linear feedback law with a saturation, the asymptotic stability is preserved, and how is this stability.
- Second case: the open-loop is unstable. In that case, when closing the loop, the result is local, and there exists a basin of attraction. The challenge reduces then to finding this set or at least having a good approximation of this basin of attraction.
Загарий Г.И. Синтез систем управления на основе критерия
максимальной степени устойчивости / Г.И. Загарий, Ф.М. Шубла-
дзе. – М.: Энергия, 1988. – 99 с.
Linear systems are special. There is a larger encompassing theory for linear systems. For non-linear systems if the critical points are hyperbolic there the concept of stable and unstable manifolds. However, for linear systems there is no over arching general theory. Each non-linear system requires its individual analysis. In the case of non-hyperbolic critical points - the analysis becomes even more subtle. There are some standard methods, such as small perturbations of "well behaved" known systems, stability arguments, fixed point theorems, topological methods, etc. that can be applied but in general - each problem requires a unique approach.
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