Research Article An Improved Antiwindup Design …downloads.hindawi.com/journals/mpe/2014/316043.pdf · An Improved Antiwindup Design Using an Anticipatory Loop and an Immediate Loop

Research ArticleAn Improved Antiwindup Design Usingan Anticipatory Loop and an Immediate Loop

Qing Wang,1 Maopeng Ran,1 Chaoyang Dong,2 and Maolin Ni3

1 School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China2 School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China3 Chinese Society of Astronautics, Beijing 100048, China

Correspondence should be addressed to Maopeng Ran; [email protected]

Received 13 February 2014; Accepted 13 April 2014; Published 13 May 2014

Academic Editor: Zhengguang Wu

Copyright © 2014 Qing Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We present an improved antiwindup design for linear invariant continuous-time systems with actuator saturation nonlinearities. Inthe improved approach, two antiwindup compensators are simultaneously designed: one activated immediately at the occurrenceof actuator saturation and the other activated in anticipatory of actuator saturation. Both the static and dynamic antiwindupcompensators are considered. Sufficient conditions for global stability and minimizing the induced 𝐿

2gain are established, in

terms of linear matrix inequalities (LMIs). We also show that the feasibility of the improved antiwindup is similar to the traditionalantiwindup. Benefits of the proposed approach over the traditional antiwindup and a recent innovative antiwindup are illustratedwith well-known examples.

1. Introduction

Actuator saturation, which may cause loss in performanceand even instability, is a ubiquitous and inevitable fact inany practical control systems. One general method to reduceadverse effects of saturation is the so-called antiwindup(AW). In AW design, a linear controller which does not takethe saturation nonlinearity into account is first designed.Then, an AW compensator is added to ensure that stabilityis maintained (at least in some region near the origin) andthat less performance degradation occurs than no AW isused [1]. Such an approach has received much attention inrecent several decades due to its intuitive motivation and itseffectiveness in practice (see [1–4] and references therein).The traditional AW scheme is depicted in Figure 1, where P,C, and AW are the plant, the linear controller, and the AWcompensator, respectively. Typically, the linear controller canbe designed using the well-established linear control theory,and various methods for designing the AW compensatorshave been proposed in the literature (see [5–8] for somerepresentative examples).

One of the main features of the traditional AW is thatthe AW compensator activated as soon as the saturation

is encountered (nearly all the AW designs were based onthis paradigm). In a pair of recent papers [9–12], Sajjadi-Kia and Jabbari investigated the effects of deferring theactivation of the AW compensator and a so-called delayedAW was proposed. Based on the assumption that the linearcontroller possesses a reasonable amount of performancerobustness, the motivation of the delayed AW is to applythe AW compensator until the closed-loop performancefaces substantial decrease.With several examples, the authorsshowed that the delayedAWrenders better performance thanthe traditional AW. Motivated by Sajjadi-Kia and Jabbari’swork and considering the dynamical nature of the system,Wuand Lin proposed a new AW which is called as anticipatoryAW [13–16]. The anticipatory AW is opposite to the delayedAW, and its basic idea is to activate the AW compensator inanticipation of actuator saturation. It was shown in [14] thatthe anticipatory AWhas the potential of leading to significantimprovement in the closed-loop performance, in terms ofboth the transient quality in reference tracking and the regionof stability.

A further modified AW is to simultaneously design twoAW loops, one for immediate activation and the other fordelayed activation [17, 18]. The main idea is to separate the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 316043, 9 pageshttp://dx.doi.org/10.1155/2014/316043

2 Mathematical Problems in Engineering

C

+ −

u

q

z

y

P

𝜔

𝜂

u

AW

Figure 1: The traditional AW scheme.

uulim/ga

ulimC

++ +

+

𝜔

𝜂− −

AWa

Pz

yy

u

qa q

ua

AW

Figure 2: The improved AW scheme.

saturated zone and obtain more aggressive and effective AWgains in lower levels of saturation. Using such a multiloopAW, one can obtain better response for moderate levels ofsaturation, but for high saturation levels, the improvementdiminishes (see the numerical example in [18]).

Motivated by Wu and Lin’s work, we are hopeful toachieve a better performance by adding an anticipatoryAW loop to the traditional AW scheme to take a “pre-cautionary” action before actuator saturation occurs. It hasbeen confirmed in [14] that a single anticipatory AW loopcan work better than a single delayed AW loop. The mainobservation, here, is to combine an immediate (traditional)AW loop and an anticipatory AW loop to further improvethe closed-loop performance. The proposed AW scheme isdepicted in Figure 2, where AW

𝑎is the anticipatory AW

compensator. We show that the synthesis results can be castas an optimization over LMIs, and the two sets of AW gainscan be straightforward obtained. Since we focus on the globalresults, we will restrict ourselves to stable plants.

The rest of this paper is organized as follows. In Section 2,we provide a general description of the traditional AW andthe proposed AW. In Section 3, we demonstrate the synthesisresults in detail, with LMIs. We will first focus on staticgains and then extend to dynamic gains. The feasibility ofthe resulting optimization problem will also be examined.In Section 4, we illustrate the benefits of the proposed AWthrough two examples. Finally, we conclude with Section 5.

Notation. The notation in this paper is standard. R is theset of real numbers. 𝐴𝑇 is the transpose of a real matrix 𝐴.The matrix inequality 𝐴 > 𝐵 (𝐴 ≥ 𝐵) means that 𝐴 and 𝐵

are square Hermitian matrices and 𝐴-𝐵 is positive (semi-)definite. A block diagonal matrix with submatrices 𝑋

1,

𝑋2, . . . , 𝑋

𝑝in its diagonal will be denoted by diag{𝑋

1, 𝑋2,

. . . , 𝑋𝑝}. 𝐼 denotes the identity matrix of appropriate dimen-

sions. To reduce clutter, off-diagonal entries in symmetric

matrices are occasionally replaced by “∗”. The sector condi-tion used in this paper is defined as follows.

Definition 1 (sector condition [19]). A function 𝑓 : R𝑛 → R𝑛is said to belong to the sector [𝐾

1, 𝐾2] with 𝐾 = 𝐾

2− 𝐾1=

𝐾𝑇> 0 if [𝑓(𝜔) − 𝐾

1𝜔]𝑇[𝑓(𝜔) − 𝐾

2𝜔] ≤ 0 for all 𝜔 ∈ R𝑛.

2. Problem Formulation

Consider the following stable plant:

𝑃 :

{{

{{

{

��𝑝= 𝐴𝑝𝑥𝑝+ 𝐵1𝜔 + 𝐵

2��

𝑧 = 𝐶1𝑥𝑝+ 𝐷11𝜔 + 𝐷

12��

𝑦 = 𝐶2𝑥𝑝+ 𝐷21𝜔 + 𝐷

22��,

(1)

where 𝑥𝑝

∈ R𝑛𝑝 is the plant state, 𝜔 ∈ R𝑛𝜔 is the exogenousinput (reference signals, disturbances, and noise), �� ∈ R𝑛𝑢 isthe control input, 𝑧 ∈ R𝑛𝑧 is the controlled output, 𝑦 ∈ R𝑛𝑦 isthemeasurement output, andAp,B1,B2,C1,D11,D12,C2,D21,andD

22are real constantmatrices of appropriate dimensions.

Pairs (Ap, B2) and (C2,Ap) are assumed to be controllable andobservable, respectively. Without loss of generality, we willassume that D

22= 0 henceforth.

Considering plant 𝑃, we assume that an 𝑛𝑐th-order linear

dynamic controller

𝐶 : {��𝑐= 𝐴𝑐𝑥𝑐+ 𝐵𝑐𝑦𝑦 + 𝐵𝑐𝜔𝜔

𝑢 = 𝐶𝑐𝑥𝑐+ 𝐷𝑐𝑦𝑦 + 𝐷

𝑐𝜔𝜔

(2)

has been designed to guarantee that the closed-loop systemis stable and achieve some performance specifications in theabsence of actuator saturation.Here,𝑥

𝑐∈ R𝑛𝑐 is the controller

state, 𝑢 ∈ R𝑛𝑢 is the controller output, andAc,Bcy,𝐵𝑐𝜔,Cc,Dcy,and𝐷

𝑐𝜔are real constantmatrices of appropriate dimensions.

In the absence of actuator saturation, the unconstrainedinterconnection between the plant 𝑃 and linear controller 𝐶is given by

�� = 𝑢. (3)If saturation is present at the input of the plant, the

unconstrained interconnection (3) is no longer guaranteedand it will be replaced by

�� = sat (𝑢) , (4)where sat(⋅) is the standard decentralized saturation functiondefined as

sat (𝑢) = [sat(𝑢1), . . . , sat(𝑢

𝑛𝑢)]𝑇 (5)

with sat(𝑢𝑖) = sign(𝑢

𝑖)min{|𝑢

𝑖|, 𝑢𝑖

lim}; here 𝑢𝑖

lim is the satura-tion bound for the 𝑖th input.

In order to mitigate the undesirable effects caused byactuator saturation, a correction term proportional to 𝑞 =

𝑢 − �� is added to the linear controller; that is,��𝑐= 𝐴𝑐𝑥𝑐+ 𝐵𝑐𝑦𝑦 + 𝐵𝑐𝜔𝜔 + 𝜂1,

𝑢 = 𝐶𝑐𝑥𝑐+ 𝐷𝑐𝑦𝑦 + 𝐷

𝑐𝜔𝜔 + 𝜂2,

(6)

where

𝜂 = [𝜂1

𝜂2

] = −Λ𝑞 = − [Λ1

Λ2

] 𝑞 (7)

Mathematical Problems in Engineering 3

is the traditional static AW compensator. Defining 𝑥 =

[𝑥𝑇

𝑝𝑥𝑇

𝑐]𝑇, the traditional AW closed-loop system can be

written as

�� = 𝐴𝑥 + 𝐵𝜔𝜔 + (𝐵

𝑞− 𝐵𝜂Λ) 𝑞,

𝑧 = 𝐶𝑧𝑥 + 𝐷

𝑧𝜔𝜔 + (𝐷

𝑧𝑞− 𝐷𝑧𝜂Λ) 𝑞,

𝑢 = 𝐶𝑢𝑥 + 𝐷

𝑢𝜔𝜔 − 𝐷

𝑢𝜂Λ𝑞,

(8)

where

𝐴 = [𝐴𝑝+ 𝐵2𝐷𝑐𝑦𝐶2

𝐵2𝐶𝑐

𝐵𝑐𝑦𝐶2

𝐴𝑐

] ,

𝐵𝜔= [

𝐵1+ 𝐵2𝐷𝑐𝜔

+ 𝐵2𝐷𝑐𝑦𝐷21

𝐵𝑐𝜔

+ 𝐵𝑐𝑦𝐷21

] ,

𝐵𝑞= [

−𝐵2

0] , 𝐵

𝜂= [

0 𝐵2

𝐼 0] ,

𝐶𝑧= [𝐶1+ 𝐷12𝐷𝑐𝑦𝐶2

𝐷12𝐶𝑐] ,

𝐷𝑧𝜔

= 𝐷11

+ 𝐷12𝐷𝑐𝑦𝐷21

+ 𝐷12𝐷𝑐𝜔,

𝐷𝑧𝑞

= −𝐷12, 𝐷

𝑧𝜂= [0 𝐷

12] ,

𝐶𝑢= [𝐷𝑐𝑦𝐶2

𝐶𝑐] , 𝐷

𝑢𝜔= 𝐷𝑐𝑦𝐷21

+ 𝐷𝑐𝜔,

𝐷𝑢𝜂

= [0 𝐼] .

(9)

In the proposed AW scheme, an artificial saturationelement with a lower saturation bound 𝑢lim/𝑔

𝑎is added; here

𝑔𝑎

> 1 is a design variable specified by designer. We notethat, when the magnitude of the controller output 𝑢 satisfies𝑢lim/𝑔

𝑎≤ 𝑢 < 𝑢lim, only the added (anticipatory) AW loop

is activated, and if the controller output 𝑢 goes beyond 𝑢lim,both the immediate AW loop and the anticipatory AW loopare activated. In Figure 2, signals to motivate the immediateAW loop and the anticipatory AW loop are 𝑞 = 𝑢 − �� and𝑞𝑎

= 𝑢 − 𝑢𝑎, respectively. We first assume that all the AW

gains are static; that is,

AW := −Λ𝑞, AW𝑎:= −Λ

𝑎𝑞𝑎. (10)

Then, the closed-loop system depicted in Figure 2 can bewritten into the following equivalent state-space form:

�� = 𝐴𝑥 + 𝐵𝜔𝜔 + (𝐵

𝑞− 𝐵𝜂Λ) 𝑞 − 𝐵

𝜂Λ𝑎𝑞𝑎,



𝑧𝑞− 𝐷𝑧𝜂Λ) 𝑞 − 𝐷

𝑧𝜂Λ𝑎𝑞𝑎,



𝑢𝜂Λ𝑞 − 𝐷

𝑢𝜂Λ𝑎𝑞𝑎.

(11)

Consider that the anticipatory AW loop has dynamicgains; that is,

��𝑎= 𝐴𝑎𝑥𝑎+ 𝐵𝑎𝑞𝑎,

[𝜂1

𝜂2

] = 𝐶𝑎𝑥𝑎+ 𝐷𝑎𝑞𝑎,

(12)

where 𝑥𝑎

∈ R𝑛𝑎 is the dynamic AW compensator state and𝐴𝑎, 𝐵𝑎, 𝐶𝑎, and 𝐷

𝑎are real constant matrices of appropriate

dimensions. Define 𝑥 = [𝑥𝑇

𝑝𝑥𝑇

𝑐𝑥𝑇

𝑎]𝑇.Then, the closed-loop

system can be described as𝑥 = 𝐴𝑥 + 𝐵

𝜔𝜔 + (𝐵

𝑞− 𝐵𝜂Λ) 𝑞 − 𝐵

𝜂Λ𝑎𝑞𝑎,



𝑧𝑞− 𝐷𝑧𝜂Λ) 𝑞 − 𝐷




𝑢𝜂Λ𝑞 − 𝐷

𝑢𝜂Λ𝑎𝑞𝑎,

(13)

where

𝐴 = [𝐴 𝐵𝜂𝐶𝑎

0 𝐴𝑎

] , 𝐵𝜔= [

𝐵𝜔

0] ,

𝐵𝑞= [

𝐵𝑞

0] , 𝐵

𝜂= [

𝐵𝜂

0] ,

Λ𝑎= −[

𝐵𝑎

𝐷𝑎

] , 𝐵𝜂= [

0 𝐵𝜂

𝐼 0] , 𝐶

𝑧= [𝐶𝑧

𝐷𝑧𝜂𝐶𝑎] ,

𝐷𝑧𝜂

= 𝐷12𝐷𝑢𝜂, 𝐶

𝑢= [𝐶𝑢

𝐷𝑢𝜂𝐶𝑎] ,

𝐷𝑢𝜂

= [0 𝐷𝑢𝜂] .

(14)In this paper, the objective of AWdesign is to compute the

AW gains to meet some performance requirements. Similartomuch of the AWdesign techniques, we choose the induced𝐿2gain as the performance index. The induced 𝐿

2gain from

the exogenous input 𝜔 to the output 𝑧 is defined as [20]

sup‖𝜔‖2 = 0

‖𝑧‖2

‖𝜔‖2

. (15)

3. Main Results

For simplicity, we will first consider the single actuator sys-tem.The results can be readily expanded to the multiactuatorcase, and it will be discussed later.

3.1. Static AW Gains. We state the following lemma thatobtains the stabilizing gains Λ and Λ a.

Lemma 2. The closed-loop system depicted in Figure 2 is stableand the 𝐿

2gain from 𝜔 to 𝑧 is less than 𝛾 if there exist positive

scalars 𝑀 and 𝑀𝑎, symmetric matrix 𝑄 > 0, and matrices 𝑋,

𝑋𝑎such that the following LMI holds:

[[[[[[[[[

[

Φ11

∗ ∗ ∗ ∗

𝐵T𝜔

−𝛾𝐼 ∗ ∗ ∗

𝐶𝑧𝑄 𝐷𝑧𝜔

−𝛾𝐼 ∗ ∗

Φ41

𝐷𝑢𝜔

Φ43

Φ44

∗

Φ51

𝐷𝑢𝜔

Φ53

Φ54

Φ55

]]]]]]]]]

]

< 0, (16)

whereΦ11

= 𝑄𝐴T+ 𝐴𝑄, Φ

41= −𝑋

T𝑎𝐵T𝜂+ 𝐶𝑢𝑄,

Φ43

= −𝑋T𝑎𝐷

T𝑧𝜂, Φ

44= −𝐷𝑢𝜂𝑋𝑎− 𝑋

T𝑎𝐷

T𝑢𝜂

− 2𝑀𝑎,


Φ51

= 𝑀𝐵T𝑞− 𝑋

T𝐵T𝜂+ 𝐶𝑢𝑄, Φ

53= 𝑀𝐷

T𝑧𝑞

− 𝑋T𝐷

T𝑧𝜂,

Φ54

= −𝐷𝑢𝜂𝑋 − 𝑋

T𝐷

T𝑢𝜂, Φ

55= −𝐷𝑢𝜂𝑋 − 𝑋

T𝐷

T𝑢𝜂

− 2𝑀.

(17)

An upper bound on the induced 𝐿2gain can be obtained by

minimizing 𝛾 subject to LMI (16). If the optimization problemis feasible, then the static AW gainsΛ andΛ

𝑎can be calculated

from Λ = 𝑋𝑀−1 and Λ

𝑎= 𝑋𝑎𝑀−1

𝑎.

Proof. Define 𝑞 = 𝜑(𝑢) and 𝑞𝑎= 𝜑𝑎(𝑢). Note that 𝜑 and 𝜑

𝑎

are the dead-zone nonlinearity function. It is straightforwardto show that

𝜑 ∈ [0 1] , 𝜑𝑎∈ [0 1] . (18)

Thus, we have

𝑞𝑇(𝑞 − 𝑢) ≤ 0, 𝑞

𝑇

𝑎(𝑞𝑎− 𝑢) ≤ 0. (19)

We construct a quadratic Lyapunov function in the form𝑉 = 𝑥

T𝑄−1𝑥, where 𝑄 > 0. To guarantee stability of the

closed-loop system and estimate the induced 𝐿2gain from

𝜔 to 𝑧, we require [21]

𝑑

𝑑𝑡(𝑥𝑇𝑄−1𝑥) + 𝛾

−1𝑧𝑇𝑧 − 𝛾𝜔

𝑇𝜔 < 0. (20)

Invoking S-procedure for some positive scalars 𝑊 and𝑊𝑎(for multiactuator case, 𝑊 and 𝑊

𝑎are some positive

definite matrices), we get the following sufficient conditionto guarantee (20):

𝑑



𝑇𝜔 − 2𝑞

𝑇𝑊(𝑞 − 𝑢)

− 2𝑞𝑇

𝑎𝑊𝑎(𝑞𝑎− 𝑢) < 0.

(21)

If all the AW gains are static, then, in view of the closed-loop system equation (11), inequality (21) can be expanded as

[[[

[

Ω11

∗ ∗ ∗

Ω21

Ω22

∗ ∗

Ω31

Ω32

Ω33

∗

Ω41

Ω42

Ω43

Ω44

]]]

]

< 0, (22)

where

Ω11

= 𝐴𝑇𝑄−1

+ 𝑄−1𝐴 + 𝛾−1𝐶𝑇

𝑧𝐶𝑧,

Ω21

= 𝐵𝑇

𝜔𝑄−1

+ 𝛾−1𝐷𝑇

𝑧𝜔𝐶𝑧,

Ω22

= 𝛾−1𝐷𝑇

𝑧𝜔𝐷𝑧𝜔

− 𝛾,

Ω31

= (𝐵𝑞− 𝐵𝜂Λ)𝑇

𝑄−1

+ 𝛾−1(𝐷𝑧𝑞

− 𝐷𝑧𝜂Λ)𝑇

𝐶𝑧

+ 𝑊𝐶𝑢,

Ω32

= 𝛾−1(𝐷𝑧𝑞


𝐷𝑧𝜔

+ 𝑊𝐷𝑢𝜔

,

saulim

−saulim

u

−ulim

ulim

qa

qa= sau

Figure 3: Sector condition for the artificial saturation element.

Ω33

= 𝛾−1(𝐷𝑧𝑞


(𝐷𝑧𝑞

− 𝐷𝑧𝜂Λ) − 𝑊𝐷

𝑢𝜂Λ

− Λ𝑇𝐷𝑇

𝑢𝜂𝑊 − 2𝑊,

Ω41

= −Λ𝑇

𝑎𝐵𝑇

𝜂𝑄−1

− 𝛾−1Λ𝑇

𝑎𝐷𝑇

𝑧𝜂𝐶𝑧+ 𝑊𝑎𝐶𝑢,

Ω42

= −𝛾−1Λ𝑇

𝑎𝐷𝑇

𝑧𝜂𝐷𝑧𝜔

+ 𝑊𝑎𝐷𝑢𝜔

,

Ω43

= −𝛾−1Λ𝑇

𝑎𝐷𝑇

𝑧𝜂(𝐷𝑧𝑞

− 𝐷𝑧𝜂Λ) − 𝑊

𝑎𝐷𝑢𝜂Λ

− Λ𝑇

𝑎𝐷𝑇

𝑢𝜂𝑊,

Ω44

= 𝛾−1Λ𝑇

𝑎𝐷𝑇

𝑧𝜂𝐷𝑧𝜂Λ𝑎− 𝑊𝑎𝐷𝑢𝜂Λ𝑎− Λ𝑇

𝑎𝐷𝑇

𝑢𝜂𝑊𝑇

𝑎

− 2𝑊𝑎.

(23)

Applying the Schur complement and the congruencetransformation diag{𝑄, 𝐼, 𝐼, 𝐼, 𝐼} then a subsequent congru-ence transformation diag{𝐼, 𝐼, 𝐼,𝑊−1

𝑎,𝑊−1} and finally with

the change of variables 𝑀 = 𝑊−1, 𝑀𝑎= 𝑊−1

𝑎, 𝑋 = Λ𝑀, and

𝑋𝑎= Λ𝑎𝑀𝑎leads to (16).

One can obtain a more aggressive and effective anticipa-tory AWcompensator by the following approach.We are nowassuming that the controller output satisfies |𝑢| < 𝑢lim. Undersuch an assumption, the immediate AW loop will never beactivated, and the closed-loop system can be relative writtenas

�� = 𝐴𝑥 + 𝐵𝜔𝜔 − 𝐵

𝜂Λ𝑎𝑞𝑎,


𝑧𝜔𝜔 − 𝐷





(24)

Define 𝑞𝑎

= 𝜑𝑎(𝑢) and 𝑠

𝑎= 1 − 1/𝑔

𝑎. As depicted

in Figure 3, when the magnitude of the controller output isbound with 𝑢lim, the artificial saturation element satisfies thefollowing sector condition:

𝜑𝑎∈ [0, 𝑠

𝑎] . (25)

Thus we now have

𝑞𝑇

𝑎(𝑞𝑎− 𝑠𝑎𝑢) ≤ 0. (26)


We then invoke S-procedure for some positive scalars𝑊𝑎

(formultiactuator case,𝑊𝑎is some positive definitematrices)

to obtain𝑑


−1

𝑎𝑧𝑇𝑧 − 𝛾𝑎𝜔𝑇𝜔 − 2𝑞

𝑇

𝑎𝑊𝑎(𝑞𝑎− 𝑠𝑎𝑢) < 0

(27)

which is a sufficient condition for inequality (20).Expanding inequality (27) with the closed-loop system

matrices in (24), followed by application of Schur comple-ment and then congruent transformations diag{𝑄, 𝐼, 𝐼, 𝐼, 𝐼}

and diag{𝐼, 𝐼, 𝐼,𝑊−1𝑎

}, and finally with the substitution of𝑀𝑎

= 𝑊−1

𝑎and 𝑋

𝑎= Λ𝑎𝑀𝑎, we arrive at the following

equivalent LMI:

[[[[[[

[

Φ11

∗ ∗ ∗

𝐵𝑇

𝜔−𝛾𝑎𝐼 ∗ ∗

𝐶𝑧𝑄 𝐷

𝑧𝜔−𝛾𝑎𝐼 ∗

Ψ41

𝑠𝑎𝐷𝑢𝜔

Φ43

Ψ44

]]]]]]

]

< 0, (28)

where Ψ41

= −𝑋𝑇

𝑎𝐵𝑇

𝜂+ 𝑠𝑎𝐶𝑢𝑄 and Ψ

44= −𝑠

𝑎𝐷𝑢𝜂𝑋𝑎−

𝑠𝑎𝑋𝑇

𝑎𝐷𝑇

𝑢𝜂− 2𝑀𝑎.

To guarantee the uniqueness of Λ𝑎, we use the same 𝑋

𝑎

and𝑀𝑎in (22) and (28); though, to reduce conservatism, we

can use different Lyapunov matrix (denoted by 𝑄𝑎) in (28).

Thus, LMI (28) can be rewritten as

[[[[[[

[

Φ11

∗ ∗ ∗

𝐵𝑇

𝜔−𝛾𝑎𝐼 ∗ ∗

𝐶𝑧𝑄𝑎

𝐷𝑧𝜔

−𝛾𝑎𝐼 ∗

Ψ41


Φ43

Ψ44

]]]]]]

]

< 0, (29)

where Φ11and Ψ

41are counterparts of Φ

11and Ψ

41when 𝑄

is replaced by 𝑄𝑎.

Based on the above analysis, we arrive at the followingtheorem that guarantees global stability and characterizes theinduced L

2gain of the closed-loop system.

Theorem 3. The closed-loop system depicted in Figure 2 isstable and the L

2gain from 𝜔 to 𝑧 is less than 𝛾 if there exist

positive scalars 𝑀 and 𝑀𝑎, symmetric matrices 𝑄 > 0 and

𝑄𝑎> 0, and matrices𝑋,𝑋

𝑎such that LMI (16) and (29) hold.

The optimization problem is

min 𝑐𝛾 + 𝑐𝑎𝛾𝑎

(𝑤𝑖𝑡ℎ 𝑐 > 0, 𝑐𝑎> 0)

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐿𝑀𝐼 (16) 𝑎𝑛𝑑 (29) .

(30)

If the optimization problem is feasible, then the static AW gainsΛ and Λ

𝑎can be calculated from Λ = 𝑋𝑀

−1 and Λ𝑎

=

𝑋𝑎𝑀−1

𝑎.

As the lemma suggests, using LMI (16) alone guaranteesstability and a L

2performance 𝛾. LMI (29) is used to obtain a

more aggressive and effective anticipatory AW compensator.We note that 𝛾 is the only L

2performance of the closed-loop

system. The gain 𝛾𝑎is best described as a measure of the

aggressiveness and effectiveness of the anticipatory AW loop.

3.2. Combination of Static Immediate AW and DynamicAnticipatory AW. The synthesis after letting the anticipatoryAW to be dynamic is parallel to that in previous section.To convexify and simplify the synthesis results, we use thesame change of variable approach in [18], which results in adynamic AW with 𝑛

𝑎= 𝑛𝑝+ 𝑛𝑐.

Theorem 4. The closed-loop system depicted in Figure 2 isstable and the L

2gain from 𝜔 to 𝑧 is less than 𝛾 if there exist

positive scalars𝑀 and𝑀𝑎, symmetric matrices 𝑌 > 0, 𝑌

𝑎> 0,

and 𝑈 > 0, and matrices X, 𝐹1, 𝐹2, 𝐹3, and 𝐹

4such that the

following LMI holds:

[[[[[[[[

[

Δ11

∗ ∗ ∗ ∗ ∗

Δ21

𝐹1+ 𝐹

T1

∗ ∗ ∗ ∗

𝐵T𝜔

0 −𝛾𝐼 ∗ ∗ ∗

Δ41

Δ42

𝐷𝑧𝜔

−𝛾𝐼 ∗ ∗

Δ51

Δ52

𝐷𝑢𝜔

Δ54

Δ55

∗

Δ61

Δ62

𝐷𝑢𝜔

Δ64

Δ65

Δ66

]]]]]]]]

]

< 0, (31)

where

Δ11

= 𝐴𝑌 + 𝑌𝐴T+ 𝐵𝜂𝐹2+ 𝐹

T2𝐵T𝜂,

Δ21

= 𝑈𝐴T+ 𝐹1+ 𝐹

T2𝐵T𝜂,

Δ41

= 𝐶𝑧𝑌 + 𝐷

𝑧𝜂𝐹2, Δ

42= 𝐶𝑧𝑈 + 𝐷

𝑧𝜂𝐹2,

Δ51

= 𝐹T4𝐵T𝜂+ 𝐶𝑢𝑌 + 𝐷

𝑢𝜂𝐹2,

Δ52

= 𝐹T3+ 𝐶𝑢𝑈 + 𝐷

𝑢𝜂𝐹2,

Δ54

= 𝐹4𝐷

T𝑢𝜂𝐷

T12,

Δ55

= 𝐷𝑢𝜂𝐹4+ 𝐹

T4𝐷

T𝑢𝜂

− 2𝑀𝑎,

Δ61

= 𝑀𝐵T𝑞− 𝑋

T𝐵T𝜂+ 𝐶𝑢𝑌 + 𝐷

𝑢𝜂𝐹2,

Δ62

= 𝐶𝑢𝑈 + 𝐷

𝑢𝜂𝐹2, Δ

64= 𝑀𝐷

T𝑧𝑞

− 𝑋T𝐷

T𝑧𝜂,

Δ65

= 𝐷𝑢𝜂𝐹4− 𝑋

T𝐷

T𝑢𝜂, Δ

66= −𝐷𝑢𝜂𝑋 − 𝑋

T𝐷

T𝑢𝜂

− 2𝑀.

(32)

The optimization problem is

min 𝑐𝛾 + 𝑐𝑎𝛾𝑎

(𝑤𝑖𝑡ℎ 𝑐 > 0, 𝑐𝑎> 0)

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐿𝑀𝐼 (31) ,

[[[[[[[

[

Δ11

∗ ∗ ∗ ∗

Δ21

𝐹1+ 𝐹𝑇

1∗ ∗ ∗

𝐵𝑇

𝜔0 −𝛾

𝑎𝐼 ∗ ∗

Δ41

Δ42

𝐷𝑧𝜔

−𝛾𝑎𝐼 ∗

Γ51

Γ52


Δ54

Γ55

]]]]]]]

]

< 0,

(33)

where Δ11

and Δ41

are counterparts of Δ11

and Δ41

when Yis replaced with 𝑌

𝑎, Γ51

= 𝐹𝑇

4𝐵𝑇

𝜂+ 𝑠𝑎𝐶𝑢𝑌𝑎+ 𝑠𝑎𝐷𝑢𝜂𝐹2, Γ52

=

𝐹𝑇

3+ 𝑠𝑎𝐶𝑢𝑈+ 𝑠𝑎𝐷𝑢𝜂𝐹2, and Γ

55= 𝑠𝑎𝐷𝑢𝜂𝐹4+ 𝑠𝑎𝐹𝑇

4𝐷𝑇

𝑢𝜂− 2𝑀𝑎.

If the optimization problem is feasible, then the AW gains canbe calculated from Λ = 𝑋𝑀

−1, 𝐴𝑎

= 𝐹1𝑈−1, 𝐵𝑎

= 𝐹3𝑀−1

𝑎,

𝐶𝑎= 𝐹2𝑈−1, and 𝐷

𝑎= 𝐹4𝑀−1

𝑎.


Proof. We rely on the Lyapunov matrix 𝑉 = 𝑥𝑇𝑄−1𝑥 with

𝑄 > 0 and partition 𝑄 as follows:

𝑄 = [𝑌 𝑈

𝑈 𝑈] , (34)

where 𝑌 = 𝑌𝑇∈ R𝑛𝑎×𝑛𝑎 and 𝑈 = 𝑈

𝑇∈ R(𝑛𝑝+𝑛𝑐)×(𝑛𝑝+𝑛𝑐). Define

a set of variables 𝐹1

= 𝐴𝑎𝑈, 𝐹2

= 𝐶𝑎𝑈, 𝐹3

= 𝐵𝑎𝑀𝑎, 𝐹4

=

𝐷𝑎𝑀𝑎, and𝑋 = Λ𝑀. Expanding (21) (where 𝑥 is replaced by

𝑥) in terms of closed-loop system matrices in (13) and withsome proper congruence transformations leads to (31).

As before, we hope to obtain a more aggressive andeffective anticipatory AW compensator. Assuming that |𝑢| <𝑢lim, the closed-loop system can be written as

𝑥 = 𝐴𝑥 + 𝐵𝜔𝜔 − 𝐵

𝜂Λ𝑎𝑞𝑎,


𝑧𝜔𝜔 − 𝐷





(35)

Note that the sector condition (26) is also guaranteed ifthe magnitude of the controller output satisfies |𝑢| < 𝑢lim.Thus, inequality (27) (where 𝑥 is replaced by 𝑥) still holdtrue. Followed by the substitution of the closed-loop systemmatrices in (35), application of some proper congruencetransformations yields

[[[[[[

[

Δ11

∗ ∗ ∗ ∗

Δ21

𝐹1+ 𝐹𝑇

1∗ ∗ ∗

𝐵𝑇

𝜔0 −𝛾

𝑎𝐼 ∗ ∗

Δ41

Δ42

𝐷𝑧𝜔

−𝛾𝑎𝐼 ∗

Γ51

Γ52


Δ54

Γ55

]]]]]]

]

< 0, (36)

where Γ51

= 𝐹𝑇

4𝐵𝑇

𝜂+𝑠𝑎𝐶𝑢𝑌+𝑠𝑎𝐷𝑢𝜂𝐹2. To reduce conservative,

we use a different Y in (36) (denoted by 𝑌𝑎). Thus, we

complete the proof of Theorem 4.

As before, LMI (31) alone guarantees stability and a L2

performance 𝛾. LMI (33) is used to obtain a more aggressiveand effective dynamic AW compensator.

Remark 5. The parameters 𝑔𝑎, 𝑐, and 𝑐

𝑎will indeed affect the

obtained closed-loop performance (in general, we can fix c =1 and adjust 𝑐

𝑎). It is straightforward to see that a larger 𝑐

𝑎will

lead to a smaller 𝛾𝑎.The values of 𝑔

𝑎and 𝑐𝑎can be determined

by a trial and error procedure, based on the computationalresults. Take 𝑔

𝑎as an example; we set the initial value of

𝑔𝑎as 1 + 𝛿 for a small scalar 𝛿 > 0. Then adjust 𝛿 to 1.1𝛿

or 0.1𝛿 iteratively until a desired closed-loop performance isachieved [14].

Remark 6. The results can be readily extended to multi-input plants. In the multi-input case, the design variable 𝑔

𝑎

is replaced by a diagonal matrix 𝐺𝑎

= diag{𝑔𝑎𝑖}; here, 𝑔

𝑎𝑖is

the design point chosen for the 𝑖th input. In addition, thepositive scalars 𝑊, 𝑊

𝑎, 𝑀, and 𝑀

𝑎in single-input case are

now diagonal positive definite matrices.

3.3. Feasibility of the Improved AW. In [18], the authorspointed out that their modified AW, which contains animmediate activation compensator and a delayed activationcompensator, is feasible if the traditional AW has a solution.In this subsection, we will show that similar feasibilitycondition can be obtained for the modified AW proposed inthis paper.

In traditional AW synthesis, the condition for stabilityand a L

2performance level of 𝛾 is as follows [22]:

𝑑



𝑇𝜔 − 2𝑞

𝑇𝑊(𝑞 − 𝑢) < 0. (37)

We assume that there exists a pair of solutions (𝑄∗, 𝛾∗,𝑊∗) satisfying the above inequality; that is,𝑑

𝑑𝑡(𝑥𝑇𝑄∗−1

𝑥) + 𝛾∗−1

𝑧𝑇𝑧 − 𝛾∗𝜔𝑇𝜔 − 2𝑞

𝑇𝑊∗(𝑞 − 𝑢) < 0.

(38)

We can always find a small enough 𝑊∗

𝑎(𝑊∗𝑎

> 0) suchthat (20) holds:

𝑑

𝑑𝑡(𝑥𝑇𝑄∗−1

𝑥) + 𝛾∗−1

𝑧𝑇𝑧 − 𝛾∗𝜔𝑇𝜔 − 2𝑞

𝑇𝑊∗(𝑞 − 𝑢)

− 2𝑞𝑇

𝑎𝑊∗

𝑎(𝑞𝑎− 𝑢) < 0.

(39)

Now consider (27). As 0 < 𝑠𝑎< 1, a sufficient condition

to ensure (27) is𝑑


−1

𝑎𝑧𝑇𝑧 − 𝛾𝑎𝜔𝑇𝜔 − 2𝑞

𝑇

𝑎𝑊𝑎(𝑞𝑎− 𝑢) < 0.

(40)

We note that if (39) holds then (𝑄∗, 𝛾∗,𝑊∗𝑎) is a solution

of (40) (i.e., a solution of (27)). Thus, (𝑄∗, 𝛾∗, 𝑊∗, 𝑊∗𝑎) is a

solution of inequalities (20) and (27). Finally, we can concludethat the proposed AW is guaranteed to have a solution if thetraditional AW counterpart is feasible.

4. Numerical Examples

For ease of comparison, we choose the two examples that arealso considered in [18].

Example 1. The plant and linear controller are as follows:

[

[

𝐴𝑝

𝐵2

𝐵1

𝐶1

𝐷12

𝐷11

𝐶2

𝐷22

𝐷21

]

]

=

[[[[[

[

−10.6 −6.09 −0.9 1 0

1 0 0 0 0

0 1 0 0 0

−1 −11 −30 0 −1

−1 −11 −30 0 0

]]]]]

]

,

[𝐴𝑐

𝐵𝑐𝑦

𝐵𝑐𝜔

𝐶𝑐

𝐷𝑐𝑦

𝐷𝑐𝜔

] = [

[

−80 0 1 −1

1 0 0 0

20.25 1600 80 −80

]

]

.

(41)

The saturation limit is 𝑢lim = 1.In [10], a static AW compensator

Λ = [−0.1968 0.0025 − 0.9860]𝑇 (42)

which guarantees a performance level of 𝛾 = 85.78 wasdesigned through traditional approach for this example. In[18], two AW compensators

Λ = [−1512.0 18.9 79.5]𝑇,

Λ = [−97.3641 1.2207 0.9435]𝑇

(43)


0 2 4 6 10 12 16 18 20−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time (s)

Plan

t out

put

Absence of saturationI-AWID-AW

IA-AWWithout AW

14

−0.09

−0.1

0

10 14

8

8

0.150.1520.154

(a)

0 2 4 6 8 10 12 14 16 18 20Time (s)

−1.5

−0.5−1

−2

00.5

1.51

2

Plan

t out

put


IA-AWWithout AW

(b)

Figure 4: Transient performance: (a) small reference signal and (b) large reference signal.

Plan

t inp

ut 9.29

15 15.080 0.15−1

1

−0.99−0.98−0.97−0.96−0.95

−1−0.99−0.98−0.97−0.96−0.95

0.95

0 2 4 6 8 10 12 14 16 18 20Time (s)

−1.5

−0.5

−1

0

0.5

1.5

1

uuau

(a)

0 2 4 6 8 10 12 14 16 18 20Time (s)

Plan

t inp

ut

−1.5

−0.5

−1

0

0.5

1.5

10.95

15.6 15.7

0.970.99

uuau

(b)

Figure 5: Time history of control commands: (a) small reference signal and (b) large reference signal.

were designed for the immediate AW loop and the delayedAW loop, respectively. We select 𝑔

𝑎= 1.05 for this example.

Using the aforementioned technique with 𝑐 = 1 and 𝑐𝑎= 20,

we obtain 𝛾 = 85.1305 and 𝛾𝑎= 1.0002, and the resulting AW

compensators are as follows:

Λ = [−1.3340 × 100

1.6833 × 10−2

− 2.2484 × 10−1]𝑇

,

Λ𝑎= [6.3135 × 10

−1− 7.8859 × 10

−36.4954 × 10

−4]𝑇

.

(44)

Simulation results for a small and a large reference signals(which are the same as that in [18]) are depicted in Figure 4.We note that I-AW represents the traditional AW which hasa single immediate AW loop, ID-AW represents the modifiedAW which consists of an immediate AW loop and a delayedAW loop, and IA-AW represents the proposed modified AWwhich consists of an immediate AW loop and an anticipatoryAW loop. As Figure 4 suggests, system with IA-AW achievesthe best performance, no matter the reference signal is smallor large. In addition, the performance obtained by IA-AW is

much better than that obtained by a single delayed AW loopor a single anticipatory AW loop (see the numerical examplein [14]). The time histories of u, 𝑢

𝑎, and �� are illustrated in

Figure 5. We can see that both the anticipatory AW loop andthe immediate AW loop are activated.

Example 2. Consider the following example taken from [22]with actuator saturation 𝑢lim = 5:

[

[

𝐴𝑝

𝐵1

𝐵2

𝐶1

𝐷11

𝐷12

𝐶2

𝐷21

𝐷22

]

]

=

[[[[[[[[[

[

0 1 0 0 0 0

−330.46 −12.15 −2.44 0 0 2.71762

0 0 0 1 0 0

−812.61 −29.87 −30.10 0 −15.61 6.68268

0 0 1 0 0 0

1 0 0 0 0 0

0 0 1 0 0 0

]]]]]]]]]

]

.

(45)


0 1 1.5 2 2.5 3 3.5 4 4.5 5Time (s)

−0.1

−0.05

0

0.05

0.5

0.15

0.1


IA-AWWithout AW

z

(a)

0

5


IA-AWWithout AW

−15

−10

−5

u

0 1 1.5 2 2.5 3 3.5 4 4.5 5Time (s)

0.5

(b)

Figure 6: System response: (a) controlled output and (b) control command.

For this plant, static AW is not feasible. In [22], a dynamicAW with 𝑛

𝑎= 4 has been designed. In [18], an immediate

static AW compensator and a delayed dynamic AW compen-sator were simultaneously designed. Using Theorem 4 with

𝑐 = 1, 𝑐𝑎

= 20, and 𝑔𝑎

= 1.1, we obtain the immediatestatic AW compensator and the anticipatory dynamic AWcompensator as follows:

Λ = [−3.2177 × 100

6.9510 × 101

2.2583 × 100

1.2101 × 103

1.0537 × 102]𝑇

,

𝐴𝑎=

[[[[[[[[[[[[[[[

[

9.4734 × 100

1.8697 × 101

1.0377 × 101

−6.2122 × 100

−7.2457 × 10−1

3.2341 × 10−1

5.1008 × 100

1.2722 × 10−1

−7.7848 × 102

−6.1629 × 102

−3.2932 × 102

2.0814 × 102

1.7320 × 101

−1.3155 × 101

−1.9018 × 102

−5.1783 × 100

3.2894 × 101

5.1818 × 101

−2.7946 × 101

−1.7101 × 101

−2.0763 × 100

9.7751 × 10−1

1.5257 × 101

3.8451 × 10−1

−1.9162 × 103

−1.5092 × 103

−8.2887 × 102

5.0887 × 102

4.2449 × 101

−3.2502 × 101

−4.6938 × 102

−1.2794 × 101

−4.1933 × 101

−6.2229 × 101

−7.1688 × 102

−1.0255 × 102

−7.8405 × 102

−6.5210 × 101

4.5811 × 101

−2.5154 × 101

−1.2631 × 103

−1.8707 × 103

−2.4179 × 103

5.8860 × 102

−6.0947 × 102

−2.8116 × 102

−1.8892 × 103

−1.1105 × 102

−2.5919 × 103

−3.2742 × 103

−2.6531 × 103

1.2413 × 103

−2.1609 × 102

−4.2016 × 102

−5.1487 × 103

−1.6566 × 102

−6.3470 × 102

−1.9625 × 103

−2.2511 × 103

6.5502 × 102

−1.6360 × 102

−1.8693 × 102

−1.8126 × 103

−7.6273 × 101

]]]]]]]]]]]]]]]

]

,

𝐵𝑎= [−7.3023 × 10

−51.8710 × 10

−3−2.3074 × 10

−44.6813 × 10

−34.6430 × 10

−36.6275 × 10

−37.1269 × 10

−32.4370 × 10

−3]𝑇,

𝐶𝑎=

[[[[[[[

[

3.2144 × 101

1.3465 × 102

−6.1002 × 102

−1.7056 × 102

−7.2678 × 102

−6.1800 × 101

1.0299 × 102

−2.3419 × 101

−1.3026 × 103

6.4224 × 103

−4.6565 × 101

−2.3379 × 103

8.1685 × 102

2.4871 × 102

2.0358 × 103

9.7648 × 101

−2.7566 × 103

−3.4909 × 103

−2.9198 × 103

1.3161 × 103

−2.2227 × 102

−4.2649 × 102

−5.0892 × 103

−1.6916 × 102

−2.7529 × 104

−3.7637 × 104

−3.1202 × 104

1.2980 × 104

8.7481 × 101

2.6140 × 102

3.3648 × 103

1.0113 × 102

−1.8205 × 103

−1.4185 × 103

−1.0956 × 103

4.8082 × 102

−7.3344 × 101

1.2916 × 102

3.7709 × 102

5.2704 × 101

]]]]]]]

]

,

𝐷𝑎= [4.2089 × 10

−3−8.1195 × 10

−38.1115 × 10

−3−1.0155 × 10

−2−4.8913 × 10

−3]𝑇

.

(46)

The two compensators guarantee performance levels 𝛾 =

186.3811 and 𝛾𝑎= 23.2318. Simulation results for a step input

of duration 0.1 s and magnitude 0.5 are depicted in Figure 6.We can see that the proposed improved AW achieves the bestsystem response by forcing the system to leave the saturationzone earlier than both the I-AW and the ID-AW.

5. ConclusionsWe have proposed an improved AW design approach forstable linear systems subject to actuator saturation. In the pro-posed approach, two AW compensators were simultaneouslycomputed, one for immediate activation at the occurrenceof saturation and the other for anticipatory activation. Using


the induced 𝐿2gain as the performance index, the synthesis

results were formulated and solved as optimization problemsover LMIs. Numerical examples confirmed the effectivenessof the proposed AW design method.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (Grant nos. 61273083 and 61074027).

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