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{{Short description|Regulation of nonlinear systems}}
{{more citations needed|date=January 2015}}
'''Linear
==Gain scheduling==
In designing feedback controllers for dynamical systems a variety of modern, [[Multivariable calculus|multivariable]] controllers are used. In general, these controllers are often designed at various operating points using [[Linearization|linearized]] models of the [[Scheduling|system dynamics]] and are scheduled as a function of a [[parameter]] or parameters for operation at intermediate conditions. It is an approach for the control of non-linear systems that uses a family of linear controllers, each of which provides satisfactory control for a different operating point of the system. One or more [[observable]] variables, called the [[Scheduling|scheduling variables]], are used to determine the current operating region of the system and to enable the appropriate linear controller. For example, in case of aircraft control, a set of controllers are designed at different gridded locations of corresponding parameters such as [[Angle of attack|AoA]], [[Mach number|Mach]], [[dynamic pressure]], [[Center of mass|CG]] etc. In brief, gain scheduling is a control design approach that constructs a nonlinear controller for a nonlinear plant by patching together a collection of linear controllers. These linear controllers are blended in real-time via switching or [[interpolation]].▼
Scheduling multivariable controllers can be a very tedious and time
▲In designing feedback controllers for dynamical systems a variety of modern, [[Multivariable calculus|multivariable]] controllers are used. In general, these controllers are often designed at various operating points using [[Linearization|linearized]] models of the [[Scheduling|system dynamics]] and are scheduled as a function of a [[parameter]] or parameters for operation at intermediate conditions. It is an approach for the control of non-linear systems that uses a family of linear controllers, each of which provides satisfactory control for a different operating point of the system. One or more [[observable]] variables, called the [[Scheduling|scheduling variables]], are used to determine the current operating region of the system and to enable the appropriate linear controller. For example in case of aircraft control, a set of controllers are designed at different gridded locations of corresponding parameters such as AoA, [[Mach number|Mach]], [[dynamic pressure]], CG etc. In brief, gain scheduling is a control design approach that constructs a nonlinear controller for a nonlinear plant by patching together a collection of linear controllers. These linear controllers are blended in real-time via switching or [[interpolation]].
▲Scheduling multivariable controllers can be very tedious and time consuming task. A new paradigm is the linear parameter-varying (LPV) techniques which synthesize of automatically scheduled multivariable controller.
===Drawbacks of classical gain scheduling===
* An important drawback of classical gain scheduling approach is that adequate performance and in some cases even stability is not guaranteed at operating conditions other than the design points.<ref>{{cite journal|last=S. Shamma|first=Jeff|title=Gain Scheduling: Potentital Hazards and Possible Remedies|journal=IEEE Control Systems Magazine|year=1992|volume=June|issue=3}}</ref>▼
* Scheduling multivariable controllers is often a tedious and time
▲* An important drawback of classical gain scheduling approach is that adequate performance and in some cases even stability is not guaranteed at operating conditions other than the design points.<ref>{{cite journal|last=S. Shamma|first=Jeff|title=Gain Scheduling: Potentital Hazards and Possible Remedies|journal=IEEE Control Systems|year=1992|volume=June|issue=3}}</ref>
▲* Scheduling multivariable controllers is often a tedious and time consuming task and it holds true especially in the field of aerospace control where the parameter dependency of controllers are large due to increased operating envelopes with more demanding performance requirements.
* 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
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 dependent systems===
In [[control engineering]], a [[state
▲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]], non-[[autonomous]] 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
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
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::<math>\dot{x} =f(x(t),w(t),\dot{w}(t),u(t))</math>
The parameter <math>w</math> is not known but its evolution is measured in real time and used for control. If the above equation of parameter dependent system is linear in time then it is called
::<math>\dot{x}=A(w(t))x(t)+B(w(t))u(t)</math>
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# Parameter Dependent Quadratic Lyapunov Function (PDQLF) to bound the achievable level of performance.
These problems are solved by reformulating the control design into finite-dimensional, [[Convex function|convex]] feasibility problems which can be solved exactly, and infinite-dimensional convex feasibility problems which can be solved approximately
This formulation constitutes a type of gain scheduling problem and contrast to classical gain scheduling, this approach address the effect of parameter variations with assured stability and performance.
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<references/>
==Further
{{refbegin}}
* {{cite book
| author = Briat,
| year =
| title = Linear Parameter-Varying and Time-Delay Systems - Analysis, Observation, Filtering & Control
| publisher = Springer Verlag Heidelberg
| isbn = 978-3-662-44049-0
}}
* {{cite book
| author = Roland, Toth
| year = 2010
| title = Modeling and Identification of Linear Parameter-Varying Systems
| publisher = Springer Verlag Heidelberg
| isbn = 978-3-642-13812-6
}}
* {{cite book
| editor = Javad, Mohammadpour |editor2=Carsten, W. Scherer
| year = 2012
| title = Control of Linear Parameter Varying Systems with Applications
| publisher = Springer Verlag New York
| isbn = 978-1-4614-1833-7
}}
{{refend}}
[[Category:Control theory]]
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