Robust Anti-Windup Control Considering Multiple Design Objectives

Abstract

A new saturation control technique is proposed to design multiobjective and robust anti-windup controllers for linear systems with input saturations. Based on the characterization of saturation nonlinearities and modeling uncertainties via integral quadratic constraints (IQCs), this method considers a mixed 𝐻₂/𝐻_∞ performance indexes while maintaining dynamic constraints on the controller. The analysis and synthesis conditions are presented in terms of scaled linear matrix inequalities (LMIs). The proposed control algorithm can improve the performance of the input-constrained system while also guaranteeing robustness with respect to the modeling uncertainties. Finally, a numerical example is given to illustrate the effectiveness of the developed techniques.

1. Introduction

Nonlinear control was one of the most active areas of control research. A number of different approaches have recently emerged to discuss this challenging problems, such as the fuzzy control [1–5] and robust sliding mode control [6–8]. Saturation nonlinearities are very common in feedback control systems [9], nearly all physical systems are subjected to some type of control input saturation. If input constraints are not taken into account, harmful effects on system performance and stability may appear. Numerous methods have been proposed to handle such nonlinearities, among which the anti-windup strategy is related to pratical use closely. The basic idea underlining anti-windup designs is to introduce control modifications in order to recover, as much as possible, the performance induced by a previous design carried out on the basis of the unsaturated system. First results on anti-windup consisted on ad hoc methods intended to work with standard PID controllers, which are commonly used in present commercial controllers. Nonetheless, major improvements have been achieved in the last decade as it can be researched rigorousy in theory.

A general framework that unifies a large class of existing anti-windup control schemes in terms of two matrix parameters was proposed in [10]. In [11], a rigorous definition of anti-windup compensation was provided in terms of 𝐿₂ stability and performance. The rigorous stability analysis based on passivity concept was developed in [12]. The synthesis condition of static anti-windup controllers was formulated as an LMI problem in [13]. References [14, 15] further derived the dynamic anti-windup controller synthesis condition with linear matrix inequality (LMI) constraints. In addition, based on the linear fractional transformation (LFT)/linear parameter-varying (LPV) framework, extended anti-windup schemes were introduced in [16, 17]. In these contributions, the saturations are modeled as sector-bounded nonlinearities and the anti-windup control design is recast as a convex optimization problem by absolute stability theory provided that no uncertainty affects the plant.

The problems associated with robustness to plant uncertainty and the problems associated with actuator saturation have often been considered in isolation. There has been little literature which attempts to handle them simultaneously in the anti-windup framework. As noted in [18], nominal linear robustness is only a necessary, but not sufficient condition for the robustness of the overall anti-windup compensated system. Furthermore, [18] introduced an approach to synthesizing anti-windup compensators for input constrained systems subject to additive dynamic uncertainty. Reference [19] considered anti-windup design problem for a closed-loop LFT model whose structured perturbation block contains parametric uncertainties.

In this paper, we propose a unified synthesis method for the construction of multiobjective and robust anti-windup controller for linear systems with actuator saturations, time-varying parametric and dynamic uncertainties. Through an equivalent representation, actuator saturations are treated as sector-bounded nonlinear uncertainty and are included in a block-diagonal operator Δ together with the other uncertainties. Inspired by the research work in [20], the problems associated with robustness are handled within the integral quadratic constraints (IQCs) framework characterizing the properties and structure of Δ. The performance objectives are specified in terms of 𝐻_∞ norm, 𝐻₂ norm, and additional regional constraints on the closed-loop poles. Interestingly, the regional closed-loop poles placement also ensures the pole-placement constraints on the anti-windup controller in that the closed-loop poles exactly consist of the poles of nominal system and those of anti-windup controller. As observed in [21], this helps to prohibit the slow dynamics which remain visible on the plant outputs even when the saturations are no longer active. The overall analysis conditions are cast as an optimization over LMIs using 𝑆-procedure technique and a common quadratic Lyapunov function. The controller synthesis procedure requires solving scaled LMIs with a 𝐷/𝐾-like iteration and provides a full-order dynamic anti-windup controller.

Notation. Let Λ^𝑛×𝑛 denote 𝑛-dimensional diagonal matrix. For compact presentation, given a square matrix 𝑋 we denote He𝑋∶=𝑋+𝑋^𝑇. A block-diagonal structure with sub-blocks 𝑋₁,𝑋₂,…,𝑋_𝑝 in its diagonal will be denoted by diag(𝑋₁,𝑋₂,…,𝑋_𝑝). 𝐿^𝑛_2𝑒 denotes 𝑛-dimensional functional space whose members only need to be square integrable on finite intervals. 𝜖 is a sufficiently small value. Other notations are standard.

2. Problem Statement

The anti-windup control problem is sketched in Figure 1(a). The block 𝑃(𝑠) denotes the stable nominal system and typically includes a model of the plant with uncertainties, nominal controller together with weighing functions specified by the user. Note that Φ=𝑧−Ψ(𝑧), where Ψ denotes the standard saturation operator. For clearness, the anti-windup control diagram in Figure 1(a) is equivalently reformulated as LFT structure in Figure 1(b) with 𝑃(𝑠) described by 𝑥̇_𝑝=𝐴_𝑝𝑥_𝑝+𝐵_𝑟𝑤_𝑟+𝐵_𝑝𝑤_𝑝+𝐵_𝑢𝑧𝑢,_𝑟=𝐶_𝑟𝑥_𝑝+𝐷_𝑟𝑟𝑤_𝑟+𝐷_𝑟𝑝𝑤_𝑝+𝐷_𝑟𝑢𝑧𝑢,_∞=𝐶_∞𝑥_𝑝+𝐷_∞𝑟𝑤_𝑟+𝐷_∞𝑝𝑤_𝑝+𝐷_∞𝑢𝑧𝑢,₂=𝐶₂𝑥_𝑝+𝐷_2𝑟𝑤_𝑟+𝐷_2𝑝𝑤_𝑝+𝐷_2𝑢𝑢,𝑤=𝐷_𝑤𝑟𝑤_𝑟,𝑤_𝑟=Δ𝑧_𝑟. (2.1) Here, 𝑥_𝑝∈ℜ^𝑛 are the states. The input/output channels associated with the robustness are 𝑤_𝑟,𝑧_𝑟∈ℜ^𝑛𝑟. The input/output channels associated with the performance criterion are 𝑤_𝑝∈ℜ^𝑛𝑝, 𝑧_∞∈ℜ^𝑛∞, and 𝑧₂∈ℜ^𝑛2. 𝑢∈ℜ^𝑛𝑢 are the compensated controls, and 𝑤∈ℜ^𝑛𝑤 are the saturation error feedback. For well posedness, we will assume that 𝐷_2𝑝=0.

Figure 1 

(a) Anti-windup control structure; (b) equivalent LFT formulation.

Δ is a causal operator from 𝐿^𝑟_2𝑒[0,∞] to 𝐿^𝑟_2𝑒[0,∞] with its inputs and outputs satisfying the following time-domain integral quadratic constraint ^𝑡₀𝑤_𝑟(𝑡)𝑧_𝑟(𝑡)^𝑇𝑄𝑆^𝑇𝑤𝑆𝑅^𝑟(𝑡)𝑧_𝑟(𝑡)𝑑𝑡≥0,∀𝑡≥0. (2.2) Let 𝑄, 𝑆, 𝑅 be constant scaling matrices such that 𝑄<0, 𝑅>0. We assume that Δ is block diagonal: Δ=diag(Δ₁,…,Δ_𝑟), where Δ_𝑖 denotes a “troublemaking” component. The IQC characterizations for the typical cases considered here are listed in Table 1. Reference [20] provides a fairly complete overview of IQCs. For application, all of the individual IQC are collected in block-diagonal matrices 𝑄=diag(𝑄₁,…,𝑄_𝑟), 𝑅=diag(𝑅₁,…,𝑅_𝑟), and 𝑆=diag(𝑆₁,…,𝑆_𝑟) to characterize the associated composition of Δ.

Table 1 

IQC characterization for specified Δ_𝑖.

Considering system (2.1), we assume that a full-order dynamic anti-windup compensator is of the form 𝑥̇_𝑘=𝐴_𝑘𝑥_𝑘+𝐵_𝑘𝑤,𝑢=𝐶_𝑘𝑥_𝑘+𝐷_𝑘𝑤, (2.3) where 𝑥_𝑘∈ℜ^𝑛 is the controller state, and 𝐴_𝑘, 𝐵_𝑘, 𝐶_𝑘, 𝐷_𝑘 are constant matrices of appropriate dimensions. Then, the final closed-loop system admits the realization 𝑥̇_𝑐=𝒜_𝑐𝑥_𝑐+ℬ_𝑟𝑤_𝑟+ℬ_𝑝𝑤_𝑝,𝑧_𝑟=𝒞_𝑟𝑥_𝑐+𝒟_𝑟𝑟𝑤_𝑟+𝒟_𝑟𝑝𝑤_𝑝,𝑧_∞=𝒞_∞𝑥_𝑐+𝒟_∞𝑟𝑤_𝑟+𝒟_∞𝑝𝑤_𝑝,𝑧₂=𝒞₂𝑥_𝑐+𝒟_2𝑟𝑤_𝑟, (2.4) where 𝑥_𝑐=[𝑥^𝑇_𝑝𝑥^𝑇_𝑘]^𝑇 and 𝒜_𝑐ℬ_𝑖𝒞_𝑗𝒟_𝑗𝑖=𝐴_𝑝𝐵_𝑢𝐶_𝑘𝐵_𝑖+𝐵_𝑢𝐷_𝑘𝐷_𝜔𝑖0𝐴_𝑘𝐵_𝑘𝐷_𝜔𝑖𝐶_𝑗𝐷_𝑗𝑢𝐶_𝑘𝐷_𝑗𝑖+𝐷_𝑗𝑢𝐷_𝑘𝐷_𝑤𝑖 (2.5) with 𝑖=𝑟,𝑝 and 𝑗=𝑟,∞,2.

Denoting by 𝑇_∞(𝑠) and 𝑇₂(𝑠) the closed-loop transfer functions from 𝑤_𝑝 to 𝑧_∞ and 𝑧₂ respectively, we consider the following multiobjective synthesis problem: design an dynamic anti-windup controller (2.3) such that as follows.(1)The closed-loop system (2.4) is robustly stable with respect to the perturbation block Δ.(2)Minimize ‖𝑇₂(𝑠)‖₂ subject to ‖𝑇_∞(𝑠)‖_∞<𝛾.(3)The closed-loop poles can be placed in the prescribed complex plane which is described by LMI region.

3. LMI Formulation of System Analysis

In this section, we will provide robust stability and performance analysis conditions for the closed-loop system (2.4) in the LMI framework. The specifications and objectives under consideration include 𝐻_∞ performance, 𝐻₂ performance. Additional regional constraints on the closed-loop poles can also be imposed.

Theorem 3.1 (robust 𝐻_∞ performance). Given the closed-loop system (2.4) with perturbation block Δ satisfying the integral quadratic constraint (2.2) and a scalar 𝛾, if there exist a positive-definite matrix 𝑃_∞ and scaling matrices 𝑄,𝑆,𝑅 such that 𝑃He^∞𝒜_𝑐𝑃_∞ℬ_𝑟+𝒞^𝑇_𝑟𝑆𝑃_∞ℬ_𝑝01002𝑄+𝑆^𝑇𝒟_𝑟𝑟𝑆^𝑇𝒟_𝑟𝑝10000−2𝛾𝐼00𝑅𝒞_𝑟𝑅𝒟_𝑟𝑟𝑅𝒟_𝑟𝑝−12𝒞𝑅0_∞𝒟_∞𝑟𝒟_∞𝑝10−2𝛾𝐼<0 (3.1) then the closed-loop system is robustly stable against the perturbation block Δ, and one has ‖𝑇_∞(𝑠)‖_∞<𝛾 with zero-state initial conditions.

Proof. Consider a Lyapunov function 𝑉(𝑥_𝑐)=𝑥^𝑇_𝑐𝑃_∞𝑥_𝑐 for the closed-loop system (2.4). A sufficient condition for the robust 𝐻_∞ performance specification can be established from the inequality ̇𝑤𝑉+^𝑟𝑧_𝑟^𝑇𝑄𝑆^𝑇𝑤𝑆𝑅𝑧^𝑟_𝑟+1𝛾𝑧^𝑇_∞𝑧_∞−𝛾𝑤^𝑇_𝑝𝑤_𝑝<0. (3.2)
First, consider the robust stability with the performance channel removed, the inequality (3.2) is rewritten as 𝑑𝑑𝑡𝑉+^𝑡₀𝑤_𝑟𝑧_𝑟^𝑇𝑄𝑆^𝑇𝑤𝑆𝑅𝑧^𝑟_𝑟𝑑𝑡<0. (3.3) Note that the second term is always nonnegative. According to standard arguments from Lyapunov theory, the closed-loop system is stable. Here, the function 𝑉 decreases to zero, but not necessarily monotonically. Next, consider robust performance, integrating (3.2) from 0 to ∞ with initial condition 𝑥_𝑐(0)=0 yields ‖𝑧_∞‖₂<𝛾‖𝑤_𝑝‖₂. As a result, robust 𝐻_∞ performance can be guaranteed. Inequality (3.2) is equivalent to the LMI condition (3.1) by Schur complement.

Theorem 3.2 (robust 𝐻₂ performance). Given the closed-loop system (2.4) with perturbation block Δ satisfying the integral quadratic constraint (2.2) and a scalar 𝜈, if there exist a positive-definite matrix 𝑃₂ and scaling matrices 𝑄,𝑆,𝑅 such that 𝑃He²𝒜_𝑐𝑃₂ℬ_𝑟+𝒞^𝑇_𝑟01𝑆002𝑄+𝑆^𝑇𝒟_𝑟𝑟00𝑅𝒞_𝑟𝑅𝒟_𝑟𝑟−12𝒞𝑅0₂𝒟_2𝑟10−2𝐼𝑃<0,₂𝑃₂ℬ_𝑝ℬ^𝑇_𝑝𝑃₂𝑊(𝑊)>0,Tr<𝜈² (3.4) then the closed-loop system is robustly stable against the perturbation block Δ, and one has ‖𝑇₂(𝑆)‖₂<𝜈.

Proof. Let {𝑒₁,…,𝑒_𝑛𝑝} be a basis of the input space ℜ^𝑛𝑝. Let 𝑥_𝑐0⋅𝑖=ℬ_𝑝𝑒_𝑖, 𝑖=1,…,𝑛_𝑝 be the initial conditions of the closed-loop system (2.4). Let 𝑧_2⋅𝑖 denote the output response subject to initial condition 𝑥_𝑐0⋅𝑖 and 𝑤_𝑝=0. Then the 𝐻₂ norm ‖𝑇₂(𝑠)‖₂ can be equivalently defined as [22] 𝑇₂(𝑠)²₂∶=^𝑛𝑝_𝑖=1𝑧_2⋅𝑖²₂. (3.5) With these results, a Lyapunov function 𝑉(𝑥_𝑐)=𝑥^𝑇_𝑐𝑃₂𝑥_𝑐 can be constructed to satisfy the following inequality ̇𝑤𝑉+^𝑟𝑧_𝑟^𝑇𝑄𝑆^𝑇𝑤𝑆𝑅𝑧^𝑟_𝑟+𝑧^𝑇₂𝑧₂<0. (3.6)
The robust stability proof is the same as the one in Theorem 3.1. As for robust performance, integrating (3.6) from 0 to ∞ with 𝑥_𝑐(∞)=0 guaranteed by stability, we can obtain ‖𝑧₂‖²₂<𝑉(𝑥_𝑐(0)). As a result, the output energy is bounded by ^𝑛𝑝_𝑖=1𝑧_2⋅𝑖²₂<^𝑛𝑝_𝑖=1𝑒^𝑇_𝑖ℬ^𝑇_𝑝𝑃₂ℬ_𝑝𝑒_𝑖=Trℬ^𝑇_𝑝𝑃₂ℬ_𝑝. (3.7) With an auxiliary parameter 𝑊 such that ℬ^𝑇_𝑝𝑃₂ℬ_𝑝<𝑊, the LMI conditions (9~11) can be obtained by Schur complement.

Pole assignment in convex regions of the left-half plane can be expressed as LMI constraints on the Lyapunov matrix. An LMI region is any region 𝐷 of the complex plane that can be defined as 𝐷=𝑧∈𝐶∶𝐿+𝑀𝑧+𝑀^𝑇𝑧<0 (3.8) with 𝐿=𝐿^𝑇={𝜆_𝑖𝑗}_{1≤𝑖,𝑗≤𝑚} and 𝑀={𝜇_𝑖𝑗}_{1≤𝑖,𝑗≤𝑚} being constant real matrices. Reference [23] gives a thorough discussion for various types of the convex region.

Theorem 3.3 (see [23] (pole placement)). The closed-loop state matrix 𝒜_𝑐 has all its eigenvalues in the LMI region 𝐷 (3.8) if and only if there exists a positive definite matrix 𝑃_pol such that 𝜆_𝑖𝑗𝑃_pol+𝜇_𝑖𝑗𝒜^𝑇_𝑐𝑃_pol+𝜇_𝑗𝑖𝑃_pol𝒜_{𝑐1≤𝑖,𝑗≤𝑚}<0. (3.9)

Note that the closed-loop poles of system (2.4) exactly consist of the poles of system (2.1) and those of controller (2.3); LMI region 𝐷 should include the poles of system (2.1) to ensure the feasibility of the problem. Furthermore, the dynamics of the controller (2.3) can be constrained by the LMI region 𝐷.

4. LMI Approach to Multiobjective Synthesis

Based on the analysis results stated in the above section, in this section we aim to present a constructive procedure to design an anti-windup controller of the form (2.3), satisfying the multiobjective synthesis purposes proposed in Section 2. This procedure relies on a simple change of controller variables to map all LMIs of Section 3 into a set of affine constraints on the new controller variables and the closed-loop Lyapunov matrix.

For tractability in the LMI framework, we must seek a common Lyapunov matrix 𝑃∶=𝑃_∞=𝑃₂=𝑃_pol (4.1) that satisfies Theorems 3.1, 3.2, and 3.3. This restriction has been extensively used in multiobjective control problem such as [23, 24]. Partition 𝑃 and 𝑃⁻¹ as 𝑁𝑃=𝑌𝑁_𝑇∗,𝑃⁻¹=𝑀𝑋𝑀_𝑇∗, (4.2) where 𝑋,𝑌∈ℜ^𝑛×𝑛 are symmetric. Factorizing 𝑃 as 𝑃𝑋₁=𝑋₂,𝑋₁=𝑀𝑋𝐼_𝑇0,𝑋₂=𝐼𝑌0𝑁^𝑇 (4.3) we define the change of controller variables as follows: 𝒜_𝑘∶=𝑌𝐴_𝑝𝑋+𝑌𝐵_𝑢𝐶_𝑘𝑀^𝑇+𝑁𝐴_𝑘𝑀^𝑇,ℬ_𝑘∶=𝑌𝐵_𝑢𝐷_𝑘+𝑁𝐵_𝑘,𝒞_𝑘∶=𝐶_𝑘𝑀^𝑇,𝒟_𝑘∶=𝐷_𝑘. (4.4) For full-order design, one can always assume that 𝑀,𝑁 are 𝑛×𝑛 square and invertible matrices. Hence the controller variables 𝐴_𝑘,𝐵_𝑘,𝐶_𝑘,𝐷_𝑘 can be determined by 𝒜_𝑘,ℬ_𝑘,𝒞_𝑘,𝒟_𝑘,𝑋,𝑌 uniquely. Then through suitable congruence transformation, the analysis results of Section 3 are readily turned into inequality constraints on the variables 𝑋,𝑌,𝒜_𝑘,ℬ_𝑘,𝒞_𝑘,𝒟_𝑘 as well as auxiliary variable 𝑊 and scaling matrices 𝑄,𝑆,𝑅, and we arrive at Theorem 4.1.

Theorem 4.1 (multiobjective synthesis for robust anti-windup controller). Given the generalized plant (2.1) with perturbation block Δ satisfies the integral quadratic constraint (2.2) and the LMI region 𝐷 (3.7). There exists a controller (2.3) which robustly stabilizes plant (2.1) and enforces a tight upper bound Tr(𝑊) on ‖𝑇₂(𝑠)‖₂ subject to ‖𝑇_∞(𝑠)‖_∞<𝛾 and closed-loop poles constraints specified by 𝐷, if there exist matrices 𝑋,𝑌,𝒜_𝑘,ℬ_𝑘,𝒞_𝑘,𝒟_𝑘 as well as auxiliary variable 𝑊 and scaling matrices 𝑄,𝑆,𝑅 such that the inequalities hold as shown in (20~22) at the top of the next page, together with 𝐴He^𝑝𝑋+𝐵_𝑢𝒞_𝑘𝐴_𝑝+𝒜^𝑇_𝑘𝐵_𝑟+𝐵_𝑢𝒟_𝑘𝐷_𝑤𝑟+𝑋𝐶^𝑇_𝑟𝑆+𝒞^𝑇_𝑘𝐷^𝑇_𝑟𝑢𝑆𝐵_𝑝000𝑌𝐴_𝑝𝑌𝐵_𝑟+ℬ_𝑘𝐷_𝑤𝑟+𝐶^𝑇_𝑟𝑆𝑌𝐵_𝑝100002𝑄+𝑆^𝑇𝐷_𝑟𝑟+𝑆^𝑇𝐷_𝑟𝑢𝒟_𝑘𝐷_𝑤𝑟𝑆^𝑇𝐷_𝑟𝑝100000−2𝛾𝐼00𝑅𝐶_𝑟𝑋+𝑅𝐷_𝑟𝑢𝒞_𝑘𝑅𝐶_𝑟𝑅𝐷_𝑟𝑟+𝑅𝐷_𝑟𝑢𝒟_𝑘𝐷_𝑤𝑟𝑅𝐷_𝑟𝑝−12𝐶𝑅0_∞𝑋+𝐷_∞𝑢𝒞_𝑘𝐶_∞𝐷_∞𝑟+𝐷_∞𝑢𝒟_𝑘𝐷_𝑤𝑟𝐷_∞𝑝10−2𝐴𝛾𝐼<0,He^𝑝𝑋+𝐵_𝑢𝒞_𝑘𝐴_𝑝+𝒜^𝑇_𝑘𝐵_𝑟+𝐵_𝑢𝒟_𝑘𝐷_𝑤𝑟+𝑋𝐶^𝑇_𝑟𝑆+𝒞^𝑇_𝑘𝐷^𝑇_𝑟𝑢𝑆000𝑌𝐴_𝑝𝑌𝐵_𝑟+ℬ_𝑘𝐷_𝑤𝑟+𝐶^𝑇_𝑟1𝑆00002𝑄+𝑆^𝑇𝐷_𝑟𝑟+𝑆^𝑇𝐷_𝑟𝑢𝒟_𝑘𝐷_𝑤𝑟00𝑅𝐶_𝑟𝑋+𝑅𝐷_𝑟𝑢𝒞_𝑘𝑅𝐶_𝑟𝑅𝐷_𝑟𝑟+𝑅𝐷_𝑟𝑢𝒟_𝑘𝐷_𝑤𝑟−12𝐶𝑅0₂𝑋+𝐷_2𝑢𝒞_𝑘𝐶₂𝐷_2𝑟+𝐷_2𝑢𝒟_𝑘𝐷_𝑤𝑟10−2𝐼𝜆<0,_𝑖𝑗𝑋𝐼𝐼𝑌+𝜇_𝑖𝑗𝐴_𝑝𝑋+𝐵_𝑢𝒞_𝑘𝐴_𝑝𝒜_𝑘𝑌𝐴_𝑝^𝑇+𝜇_𝑗𝑖𝐴_𝑝𝑋+𝐵_𝑢𝒞_𝑘𝐴_𝑝𝒜_𝑘𝑌𝐴_{𝑝1≤𝑖,𝑗≤𝑚}<0,𝑋𝐼𝐵_𝑝𝐼𝑌𝑌𝐵_𝑝𝐵^𝑇_𝑝𝐵^𝑇_𝑝(𝑊).𝑌𝑊>0,MinimizingTr (4.5)

Due to the fact that the matrix variables 𝑋,𝑌,𝒜_𝑘,ℬ_𝑘,𝒞_𝑘,𝒟_𝑘 and scaling matrices 𝑄,𝑆,𝑅 enter the inequalities (21~22) in nonlinear fashion, synthesis conditions are no longer convex optimization problem. In order to overcome this difficulty, one will resort to the following iterative scheme based on LMI.

Step 1. Initialize scaling matrices 𝑄,𝑆,𝑅.

Step 2. With fixed 𝑄,𝑆,𝑅, perform control synthesis according to Theorem 4.1. Compute two invertible matrices 𝑀,𝑁∈ℜ^𝑛×𝑛 such that 𝑀𝑁^𝑇=𝐼−𝑋𝑌. (4.6) Equation (4.4) can be solved for 𝐷_𝑘,𝐶_𝑘,𝐵_𝑘,𝐴_𝑘 in this order.

Step 3. Apply Theorems 3.1, 3.2, 3.3, and (4.1) to the closed-loop system (2.4) to solve scaling matrices 𝑄,𝑆,𝑅 minimizing Tr(𝑊).

Step 4. Iterate over Step 2 to Step 3 until Tr(𝑊) cannot be decreased significantly.

It is important to mention that the previously described iterative scheme, although not guaranteeing a global solution theoretically, has proven very efficient in practice.

5. Application Example

As an application, a missile benchmark problem [25] will be used to demonstrate the effectiveness of the results discussed. The model is linearized at 𝛼=10deg (angle of attack) and 𝑀𝑎=3 (Mach number), and admits the realization 𝛼̇𝑞̇𝜗̇=𝑍_𝛼𝑀10_𝛼𝛼𝑞𝜗+𝑍00010^𝛿𝑀_𝛿0𝛿, (5.1) where 𝑞, 𝜗, and 𝛿 denote pitch rate, pitch angle, and elevator deflection, respectively. The measurement outputs are the flight path angle 𝑟=𝜗−𝛼 and the pitch rate 𝑞. The parametric uncertainties originate from the aerodynamic force 𝑍 and moment 𝑀 with uncertainty level of ±20%. The actuator dynamics are given by 𝐺_act(𝑠)=150²/(𝑠²+210𝑠+150²) with saturation limit 𝛿∈[−15,15] deg.

Ignoring the saturation, a PID controller can be designed as 𝛿_𝑐=[1.5∫(𝑟−𝑟_𝑐)𝑑𝑡+2𝑟+0.3𝑞]. 𝛿_𝑐 and 𝑟_𝑐 denote the commanded signal to the actuator and the commanded flight path angle, respectively. According to the analysis results in Section 3, the PID controller can guarantee global stability for the saturated plant with 𝐾_Φ=1. Although the PID controller provides adequate stability and nominal performance, the tracking trajectory of the nominal system under saturation deteriorates and exhibits great overshoot (see Figure 4). This clearly necessitates the anti-windup compensation scheme.

In the anti-windup design, firstly parametric uncertainties in 𝑍 and 𝑀 are extracted from the plant in a linear fractional way and rescaled to [−1,1]. Secondly, to avoid excitation of unmodeled high-frequency dynamics, a multiplicative input uncertainty Δ_𝑑(𝑠) weighted by 𝑊_𝑑(𝑠)=1.5[(𝑠+2)/(𝑠+80)] is placed at the actuator. Finally, we end up with the control interconnection as shown in Figure 2. Constant weights 𝑊_𝑒=1 and 𝑊_𝑛=0.001 are used to reflect the tracking performance and measurements with noise.

Figure 2 

Interconnection structure for anti-windup design.

We combine the sector-bounded nonlinearity Φ=𝐼−Ψ with the modeling uncertainties as a block-diagonal uncertainty structure given by Δ=diag(Φ,𝛿_𝑧𝐼,𝛿_𝑚𝐼,Δ_𝑑). Then, the anti-windup control diagram in Figure 2 is equivalently reformulated as LFT structure in Figure 1(b) for design. For low-order compensator, the actuator dynamics are ignored in design. This is justified by the fact that the bandwidth of the system is far below that of the actuator. The LMI region 𝐷 specified in Figure 3 is used to constrain the dynamics of the compensator. We choose to minimize ‖𝑇₂(𝑠)‖₂ subject to ‖𝑇_∞(𝑠)‖_∞<𝛾. 𝐾_Φ=0.8 is used to allocate the partial design freedom for coping with robustness and performance at the cost of global stability. As a result, we achieve 𝛾=38.6 and ‖𝑇₂(𝑠)‖₂=4.2. The control deflection should satisfy the condition |𝛿|≤(1/(1−𝐾_Φ))15 deg. The distribution of the poles of the compensator is shown in Figure 3.

Figure 3 

LMI region with poles [−1346,−61,−1,−9.8±10.2𝑖].

(a)

(b)

Figure 4 

Time-domain responses to a double pulse reference.

For numerical simulations, the measurement noise is chosen as band-limited white noise of power 10⁻⁶ passed through a zero order holder with sampling time 10⁻³ s. The resulting anti-windup response almost coincides with the linear response (see Figure 4). We can see that the designed anti-windup controlled guarantees the stability and recovers the nominal performance when the actuator is saturated deeply. For comparison, the anti-windup response of the nondynamically constrained compensator become worse because of the existence of a slow compensator mode −0.002. Figure 5 shows the time-domain robust performance behaves. As expected from previous results, Figure 5 illustrated that the anti-windup performance of the obtained controller is robust with respect to the error in model parameters.

Figure 5 

Time-domain responses for all combinations of perturbed aerodynamics.

6. Conclusion

This paper presents a unified synthesis method for the construction of multiobjective and robust anti-windup compensator for linear systems with actuator saturations, time-varying parametric and dynamic uncertainties. Motivated by the capability of integral quadratic constraints in characterizing saturation nonlinearities and modeling uncertainties, the concerned anti-windup and robustness problems are addressed in the framework of IQCs. The performance objectives are specified in terms of a mixed 𝐻₂/𝐻_∞ norm and additional constraints on the poles of the controller. The controller synthesis procedure requires solving scaled LMIs with a 𝐷/𝐾-like iteration and provides dynamically constrained anti-windup compensators. Finally, simulation example demonstrates the effectiveness of the results.

Acknowledgments

This research was partially supported by the National Natural Science Foundation of China (No. 61104112), the Fundamental Research Funds for the Central Universities (No. HIT. NSRIF. 201162), and the China Postdoctoral Science Foundation (No. 2012M510960).