Classic Gain Scheduling

Overview

Gain scheduling has been one of the dominant control strategies for the design of high performance controllers in the industry for the last fifty years at least. The gain scheduling practice can be roughly divided into two basic categories: linearization-based (or classic) and LPV (or modern) gain scheduling. In this chapter the first category is detailed whereas in Chapter 2 some classic results on the second category are mentioned. The classic method can be also divided into two subcategories: approaches that do offer some stability guaranties for the gain-scheduled system and others that do not. This chapter begins with a general introduction to the adaptive control framework (encompassing also gain scheduling) and then proceeds to a detailed citation of all classic gain scheduling methods existing in the bibliography. At the end of the chapter some additional tools used in the gain scheduling context are also detailed.

Adaptive Control Schemes

Adaptive control has risen due to the need of changing/updating afeedback con- Introduction troller K in order to conform to the changing parameters of a process S. As a simple example consider the dynamics of an aircraft: this type of systems operate in different altitudes and with different speeds and thus due to several physical reasons their dynamics change drastically as a function of time. A robust controller designed to cope with the different operating conditions cannot always guarantee, or at least offer some good indications, that the aircraft will behave in a good way for all altitudes and speeds of its flight envelope. To solve this problem an adaptive control system may be used in order to update the controller parameters for changing operating conditions. Three basic types of adaptive control systems exist (see Figure 1.1): Indirect Adaptive Control (IAC): In this control scheme (see Fig. 1.1a) the Indirect Adaptive Control controller parameters (or gains) ϑc are updated in real-time by an autotuner. This auto-tuner is based on an identified process model Sˆ provided by an estimator that uses I/O plant information. This auto-tuner then calculates ϑc as if Sˆ = S. The control scheme has two feedback loops: an internal loop that is fast enough to control the plant and an external one that is slower and detects any potential changes in the system’s model through an estimator. An example of an IAC scheme is adaptive pole placement control: the poles of the closed loop plant are assigned in realtime to a specified location on the complex plane based on the estimate of S and on a given controller structure (e.g. PID). Direct Adaptive Control (DAC): In this control scheme the controller parame- Direct Adaptive Control ters ϑc are estimated directly and the use of a plant parameter estimator is not needed. Take for example a frequently used topology of DAC: the direct Model Reference Adaptive Control (d-MRAC) configuration of Fig. 1.1b. The auto-tuner here computes the difference ed between the outputs of the real plant S and of a target plant model S¯ and tries to find a value for ϑc so that this difference goes to zero. A way to do that is the famous MIT rule [15, 66]. Gain Scheduling Control (GSC): In this control scheme (see Fig. 1.1c) no Gain Scheduling Control complex algorithm is demanded for updating the controller parameters but only a parameter (or scheduling) vector ̺ (that can sufficiently capture the plant’s change of dynamics) and an interpolation method. The controller parameters ϑc are then updated by combining/interpolating different controllers Ki designed for the plant S, for some family of critical values of ̺ 1 . The simplest form of GSC is controller switching where no smooth controller parameter update is performed and a single controller is used, being valid for a pre-defined operating region over ̺In this monograph the latter method will be considered for the control of LPV vs. generic nonlinear parameter/time dependent systems. The gain scheduling con- LBGS trol practice can be further divided into two major categories: the linearizationbased and LPV/q-LPV gain scheduling procedures. The major distinction between these two has to do on the one hand with the approach taken in order to obtain the final nonlinear gain-scheduled controller, and on the other hand on the way that the system nonlinearity is treated. These two methods sometimes overlap and there exist a considerable disagreement over the scientific community on which one is the best suited for a particular problem. The linearization-based gain scheduling procedure (LBGS)2 is mostly based on linearized plants of the initial nonlinear system, calculates a number of controllers of possibly, not the same structure, and finally interpolates them in order to obtain the gain-scheduled controller. The existence of a controller is (almost) always guaranteed but stability issues arise due to the ad-hoc linearizationinterpolation. There exist however some notable exceptions that they do consider stability for the linearized scheduled system, but obviously not on the initial nonlinear one. In this chapter both types of methods will be discussed and some key results as well as references to real world applications will be given. The LPV procedure tries to camouflage the nonlinear dynamics and obtain thus a linear system with time varying state space matrices. These time varying matrices can be treated either as time varying uncertainty thus leading to the so-called LFT formulation, or as parameter-dependent matrices that may form convex hyper-cubes for frozen values of the parameter leading to the Polytopic formulation. In both cases there exist stability guarantees for the overall scheduled system. However, the fact that is not clear enough (see [88], pp. 1012) is for which system the stability guarantees are offered3 . Briefly it can be said that the first class of methods offers a systematic and unconservative design methodology that provides always a controller whereas the second offers a more theoretically sound, yet sometimes conservative in terms of system operation & controller existence, procedure that guarantees global stability of the gain-scheduled plant. In this work the first class of methods will be used and several of its problems addressed. In this chapter the first class of methods (classic) is extensively detailed Contents whereas in the next one the second class ones (modern) are briefly reviewed. The following section (Section 1.2) considers some general results in system modeling whereas the next one (Section 1.3.1) details the LBGS following the famous five (2+3) step procedure (see [88]). Finally, subsections 1.3.2, 1.3.3 consider both the ad-hoc and stability preserving methods existing in the bibliography whereas Section 1.4 presents some related to the gain scheduling practice results concerning interpolation and operating domain triangulation. 

 System Modeling

In this section general modeling issues in the context of gain scheduling are reviewed. Some system equilibrium notions are initially introduced before passing to a citation of various ways to model a physical process. Finally, some material on Jacobian linearization is covered. 1.2.1 Equilibrium Notions Consider a generic non-autonomous4 System forced nonlinear dynamic system S whose modeling state and output dynamics are described by a number of coupled first-order differential equations (see Fig. 1.2): S : x˙(t) = f[x(t),u(t), t] y(t) = h[x(t),u(t),t]. (1.1) The vectors x, u,y represent the states, inputs and outputs of the system with x ∈ Rn ,u ∈ Rnu ,y ∈ Rny respectively5 . The vector-valued functions f, h where f := £ f1(x,u), .. . ,fn(x, u) ¤T and h := £ h1(x, u),. .. , hny (x,u) ¤T perform the following nonlinear mappings: f : R n × R nu × R 7→ R n (1.2) h : R n × R nu × R 7→ R ny . (1.3) The nonlinear system S in fact is a mathematical representation of a physical process and thus for S to provide a valid reproduction of its behavior, several additional hypotheses need to be made. These hypotheses are mostly related to the existence and uniqueness of a solution x ¡ t;t0,x(t0) ¢ given a set of initial state conditions x(t0) and the differentiability of the functions f, h with respect to an equilibrium point or trajectory (see [75], Ch. 3 for more details). The analysis concerning equilibrium notions in the next two subsections considers both autonomous nonlinear and linear systems and their non-autonomous extensions. Another extension is also given for parameter-varying systems used mostly to model processes controlled by gain-scheduled control schemes.