Linear parameter-varying control: Difference between revisions

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* It is also important that the selected scheduling variables reflect changes in plant dynamics as operating conditions change. It is possible in gain scheduling to incorporate linear [[robust control]] methodologies into nonlinear control design; however the global stability, robustness and performance properties are not addressed explicitly in the design process.

Though the approach is simple and the computational burden of linearization scheduling approaches is often much less than for other nonlinear design approaches, its inherent drawbacks sometimes outweigh its advantages and necessitates a new paradigm for the control of dynamical systems. New methodologies such as Adaptive control based on [[Artificial neural networks|Artificial Neural Networks]] (ANN), [[Fuzzy logic]], [[Reinforcement learning|Reinforcement Learning]],<ref>{{Cite journal |last1=Hosseini |first1=Ehsan |last2=Aghadavoodi |first2=Ehsan |last3=Fernández Ramírez |first3=Luis M. |date=September 2020 |title=Improving response of wind turbines by pitch angle controller based on gain-scheduled recurrent ANFIS type 2 with passive reinforcement learning |url=https://linkinghub.elsevier.com/retrieve/pii/S0960148120307588 |journal=Renewable Energy |language=en |volume=157 |pages=897–910 |doi=10.1016/j.renene.2020.05.060|bibcode=2020REne..157..897H |url-access=subscription }}</ref><ref>{{Cite journal |last1=Yeh |first1=Yi-Liang |last2=Yang |first2=Po-Kai |date=2021-11-26 |title=Design and Comparison of Reinforcement-Learning-Based Time-Varying PID Controllers with Gain-Scheduled Actions |journal=Machines |language=en |volume=9 |issue=12 |pages=319 |doi=10.3390/machines9120319 |doi-access=free |issn=2075-1702}}</ref><ref>{{Cite journal |last1=Gutiérrez-Oribio |first1=Diego |last2=Stathas |first2=Alexandros |last3=Stefanou |first3=Ioannis |date=2024-12-17 |title=AI-Driven Approach for Sustainable Extraction of Earth's Subsurface Renewable Energy While Minimizing Seismic Activity |url=https://onlinelibrary.wiley.com/doi/10.1002/nag.3923 |journal=International Journal for Numerical and Analytical Methods in Geomechanics |language=en |doi=10.1002/nag.3923 |issn=0363-9061|arxiv=2408.03664 }}</ref> etc. try to address such problems, the lack of proof of stability and performance of such approaches over entire operating parameter regime requires design of a parameter dependent controller with guaranteed properties for which, a Linear Parameter Varying controller could be an ideal candidate.

==Linear parameter-varying systems==

LPV systems are a very special class of nonlinear systems which appears to be well suited for control of dynamical systems with parameter variations. In general, LPV techniques provide a systematic design procedure for gain-scheduled multivariable controllers. This methodology allows performance, robustness and [[Bandwidth (signal processing)|bandwidth]] limitations to be incorporated into a unified framework.<ref>{{cite web|last=J. Balas|first=Gary|title=Linear Parameter-Varying Control And Its Application to Aerospace Systems|url=http://www.icas.org/ICAS_ARCHIVE/ICAS2002/PAPERS/541.PDF|publisher=ICAS|accessdate=2013-01-29|date=2002}}</ref><ref>{{cite web|last=Wu|first=Fen|title=Control of Linear Parameter Varying systems|url=httphttps://www.maeresearchgate.ncsu.edunet/wupublication/paper/PhDthesis.ps271526928|publisher=Univ. of California, Berkeley|accessdate=20132024-0112-2916|date=1995|url-status=dead|archiveurl=https://web.archive.org/web/20140103153558/httphttps://www.maeresearchgate.ncsu.edunet/wupublication/paper/PhDthesis.ps271526928_Control_of_Linear_Parameter_Varying_Systems|archivedate=2014-01-03}}</ref> A brief introduction on the LPV systems and the explanation of terminologies are given below.

===Parameter dependent systems===

In [[control engineering]], a [[state-space representation]] is a [[mathematical model]] of a physical system as a set of input, <math>u</math> output, <math>y</math> and [[State variable|state]] variables, <math>x</math> related by first-order [[Differential equation|differential]] equations. The dynamic evolution of a [[nonlinear]]{{disambiguation needed|date=April 2024}}, non-[[autonomous]] system is represented by

::<math>\dot{x} = f(x,u,t)</math>

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::<math>x(t_0)=x_0 ,u(t_0)=u_0</math>

The state variables describe the mathematical "state" of a [[dynamical system]] and in modeling large complex [[nonlinear]]{{disambiguation needed|date=April 2024}} systems if such state variables are chosen to be compact for the sake of practicality and simplicity, then parts of dynamic evolution of system are missing. The state space description will involve other variables called exogenous [[Variable (mathematics)|variables]] whose evolution is not understood or is too complicated to be modeled but affect the state variables evolution in a known manner and are measurable in real-time using [[sensors]].

When a large number of sensors are used, some of these sensors measure outputs in the system theoretic sense as known, [[wikt:explicit|explicit]] nonlinear functions of the modeled states and time, while other sensors are accurate estimates of the exogenous variables. Hence, the model will be a time varying, nonlinear system, with the future time variation unknown, but measured by the sensors in real-time.

In this case, if <math>w(t),w</math> denotes the exogenous variable [[Vector (mathematics and physics)|vector]], and <math>x(t)</math> denotes the modeled state, then the state equations are written as