Linear parameter-varying control (LPV control) deals with the control of linear parameter-varying systems, a class of nonlinear systems which can be modelled as parametrized linear systems whose parameters change with their state.
Gain scheduling In designing feedback controllers for dynamical systems a variety of modern, multivariable controllers are used. In general, these controllers are often designed at various operating points using linearized models of the 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 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, 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 a 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. 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 (ANN), Fuzzy logic, Reinforcement Learning, 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 limitations to be incorporated into a unified framework. 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, u {\displaystyle u} output, y {\displaystyle y} and state variables, x {\displaystyle x} related by first-order differential equations. The dynamic evolution of a nonlinear, non-autonomous system is represented by
x ˙ = f ( x , u , t ) {\displaystyle {\dot {x}}=f(x,u,t)}
If the system is time variant
x ˙ = f ( x ( t ) , u ( t ) , t ) , x ( t 0 ) {\displaystyle {\dot {x}}=f(x(t),u(t),t),x(t_{0})}
x ( t 0 ) = x 0 , u ( t 0 ) = u 0 {\displaystyle x(t_{0})=x_{0},u(t_{0})=u_{0}}
… excerpt ends here. Continue reading the full article.
