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# Modern Anti-windup Synthesis: Control Augmentation for Actuator Saturation

Luca Zaccarian
Andrew R. Teel
Pages: 304
Stable URL: http://www.jstor.org/stable/j.ctt7t5z2

1. Front Matter (pp. i-iv)
3. Preface (pp. ix-x)
Luca Zaccarian and Andrew R. Teel
4. Algorithms Summary (pp. xi-xiv)
5. PART 1. PREPARATION
• Chapter One The Windup Phenomenon and Anti-windup Illustrated (pp. 3-22)

Every control system actuator has limited capabilities. A piezoelectric stack actuator cannot traverse an unlimited distance. A motor cannot deliver an unlimited force or torque. A rudder cannot deflect through an unlimited angle. An amplifier cannot produce an unlimited voltage level. A hydraulic actuator cannot change its position arbitrarily quickly. These actuator limitations can have a dramatic effect on the behavior of a feedback control system.

In this book, the term “windup” refers to the degradation in performance that occurs when a saturation nonlinearity is inserted, at the plant input, in an otherwise linear feedback control loop. Usually the term...

• Chapter Two Anti-windup: Definitions, Objectives, and Architectures (pp. 23-47)

The anti-windup problem’s starting point is a plant-controller pair that, when connected in feedback without input saturation, behaves in a satisfying manner. This unconstrained feedback loop then serves as something to emulate when trying to specify and solve the anti-windup problem. Figure 2.1 shows thisunconstrained closed-loop system. Theunconstrained plantis represented byPand theunconstrained controlleris represented byK. For now, think of these systems as being linear systems. Eventually, nonlinear plants and controllers will be considered; most of the ensuing discussion applies equally well to that situation. The plant in the unconstrained closed loop has...

• Chapter Three Analysis and Synthesis of Feedback Systems: Quadratic Functions and LMIs (pp. 48-74)

This chapter starts by addressing the analysis of feedback loops with saturation. In particular, it develops tools for verifying internal stability and quantifyingL₂ external stability for the well-posed feedback interconnection of a linear system with a saturation nonlinearity. Such a system is depicted in Figure 3.1 and has state-space representation${\begin{array}{*{20}{c}} \widetilde{\mathcal{H}} \left\{ {\begin{array}{*{20}{c}} {\dot x = \widetilde Ax + \widetilde B\sigma + \widetilde Ew} \\ {z = \widetilde Cx + \widetilde D\sigma + \widetilde Fw} \\ {u = \widetilde Kx + \widetilde L\sigma + \widetilde Gw} \\ \end{array} } \right.\\ {\sigma = {\text{sat}}(u)}. \\ \end{array} } \caption{(3.1)}$It is more convenient for analysis and synthesis purposes to express this system in terms of thedeadzonenonlinearityq=u― sat(u) as shown in Figure 3.2. The state-space representation of (3.1) using the deadzone nonlinearity is obtained by first solving foruin the equation$u = \widetilde Kx + \widetilde Lu - \widetilde Lq + \widetilde Gw. \caption(3.2)$...

6. PART 2. DIRECT LINEAR ANTI-WINDUP AUGMENTATION
• Chapter Four Static Linear Anti-windup Augmentation (pp. 77-108)

This chapter introduces the first constructive tools for anti-windup augmentation. According to the characterization of Section 2.3, the chapter addresses the simplest possible augmentation scheme that may induce on the closed loop the qualitative objectives discussed in Section 2.2, possibly in addition to some of the quantitative performance objectives of Section 2.4.

This chapter focuses on the “static linear anti-windup” augmentation architecture, wherein the difference between the input and the output of the saturation block drives a static linear system that injects modification signals into the unconstrained controller dynamics. The corresponding anti-windup filter structure is the simplest possible because it...

• Chapter Five Dynamic Linear Anti-windup Augmentation (pp. 109-154)

In the previous chapter, linear static anti-windup designs inducing global and regional guarantees have been formulated and illustrated through several algorithms characterized by both full-authority and external architectures. It has been emphasized, however, that not all anti-windup problems admit such a simple compensation scheme.

Motivated by this fact, this chapter discusses a generalization of the linear static scheme wherein the matrix gains are replaced by a linear dynamic anti-windup compensator$\mathcal{F}$with internal states. The static case will be recovered by this dynamic generalization when the size of the anti-windup filter state is zero. The interconnection of the dynamic anti-windup...

7. PART 3. MODEL RECOVERY ANTI-WINDUP AUGMENTATION
• Chapter Six The MRAW Framework (pp. 157-173)

All of the anti-windup algorithms presented in this part of the book share a common, external anti-windup structure. The term “model recovery anti-windup” (MRAW) will be used for these algorithms because the structure they use recovers the unconstrained plant model as seen from the viewpoint of the unconstrained controller. This feature keeps the unconstrained controller from misbehaving in the constrained closed loop with anti-windup augmentation, and it gives this closed loop some information about how it would be performing in the ideal case without constraints. Equipped with this information, the degrees of freedom that remain within the MRAW structure can...

• Chapter Seven Linear MRAW Synthesis (pp. 174-199)

Figure 7.1 represents the model recovery anti-windup (MRAW) scheme introduced in Chapter 6. In that scheme, the upper block$\mathcal{P}$represents the plant, the lower block$\mathcal{K}$represents the unconstrained controller and the intermediate block$\widehat{\mathcal{P}}$, which corresponds to a model of the plant, contains the dynamical states of the anti-windup augmentation. This structure permits recovering information about the unconstrained response, so that unconstrained response recovery is possible through the extra degree of freedom represented by the unspecified compensation signalν.

Throughout this chapter, it will be useful to refer to suitable state-space representations of the plant and of...

• Chapter Eight Nonlinear MRAW Synthesis (pp. 200-225)

In this chapter the degrees of freedom available within the model recovery antiwindup (MRAW) structure, which was introduced in Chapter 6, are exploited using nonlinear control ideas that account for actuator constraints. Recall that the MRAW structure turns the unconstrained response recovery objective into the task of choosing a feedbackνto drive to zero the state and performance variable of the system$\begin{array}{*{20}c} {\dot x_{aw} = A_p x_{aw} + B_{p,u} [{\text{sat}}(\nu + y_c) - y_c ]} \\ {z_{aw} = C_{p,z} x_{aw} + D_{p,zu} [{\text{sat}}(\nu + y_c ) - y_c ]}, \\ \end{array}$, at least for signalsycthat satisfy$\parallel {y_c}(\cdot) - {\text{sat}}({y_c}(\cdot)){\parallel _2} < \infty$or$\parallel {y_c}(\cdot) - {\text{sa}}{{\text{t}}_\delta }({y_c}(\cdot)){\parallel _2} < \infty$for some δ > 0. Thus, the MRAW structure makes direct designs for actuator saturation applicable to the anti-windup problem. Here, three particular approaches will be discussed for...

• Chapter Nine The MRAW Structure Applied to Other Problems (pp. 226-244)

When dealing with plants with both input rate and magnitude saturation, the model recovery anti-windup (MRAW) scheme can be employed to recover the performance of a controller synthesized for the system without input magnitude and rate saturation. From a mathematical viewpoint, input rate saturation can be modeled by augmenting the plant equations, reported here for completeness,\mathcal{P}\left\{ \begin{align*} {\dot x}_p &= {A_p}{x_p} + B_{p,u}\sigma + B_{p,w}w \\ y &= C_{p,y}{x_p} + D_{p,yu}\sigma + D_{p,yw}w \\ z &= C_{p,z}{x_p} + D_{p,zu}\sigma + D_{p,zw}w, \end{align*} \right.\caption{(9.1)}, with an additional state equation having the same size as the control input and associated with the following discontinuous dynamics for the new state variable σ:$\dot{\sigma} = R\,{\text{sign}}({\text{sat}}(u) - \sigma)), \caption{(9.2)}$whereuis the input before saturation, sat(·) denotes the input saturation, and σ is the...

• Chapter Ten Anti-windup for Euler-Lagrange Plants (pp. 245-268)

This chapter focuses on a specific case study where the model recovery anti-windup (MRAW) construction described in Section 6.3 is applied to a class of nonlinear plants consisting of all fully actuated Euler-Lagrange systems. In particular, for all such plants, an anti-windup construction is given here that, provided that some feasibility conditions are satisfied, is able to recover global asymptotic stability of the closed loop with saturation as long as the unconstrained controller guarantees global asymptotic stability and local exponential stability of the unconstrained closed loop.

This case study can be understood as an example of how far the techniques...

• Chapter Eleven Annotated Bibliography (pp. 269-284)

The history of anti-windup research ranges from the early approaches dating back to the 1950s (where fixes to malfunctioning analog controllers typically corresponded to ad hoc solutions proposed and implemented by industrialists) to the numerous recently reported results (where systematic solutions to suitable formalizations of the problem provide constructive tools with guaranteed performance and stability properties).

In the past fifty years, anti-windup research migrated from the initial ad hoc attempts to more general and systematic constructions and further, to formal definitions of the underlying problem and modern high-performance solutions. Many instances of the anti-windup problem are still unsolved and are...

8. Index (pp. 285-288)