ISA Transactions 53 (2014) 802–815

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ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research Article

A novel anti-windup framework for cascade control systems: An application to underactuated mechanical systems Niaz Mehdi a,n, Muhammad Rehan b,1, Fahad Mumtaz Malik a,2, Aamer Iqbal Bhatti c,3, Muhammad Tufail b,4 a

Department of Electrical Engineering, College of Electrical and Mechanical Engineering, National University of Sciences and Technology, Islamabad, Pakistan DTD-Building, Department of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences (PIEAS), P.O. Nilore, Islamabad, Pakistan c Department of Electrical Engineering, Muhammad Ali Jinnah University (MAJU), Rawalpindi, Pakistan b

art ic l e i nf o

a b s t r a c t

Article history: Received 16 July 2013 Received in revised form 7 January 2014 Accepted 23 January 2014 Available online 28 February 2014 This paper was recommended for publication by Dr. Jeff Pieper

This paper describes the anti-windup compensator (AWC) design methodologies for stable and unstable cascade plants with cascade controllers facing actuator saturation. Two novel full-order decoupling AWC architectures, based on equivalence of the overall closed-loop system, are developed to deal with windup effects. The decoupled architectures have been developed, to formulate the AWC synthesis problem, by assuring equivalence of the coupled and the decoupled architectures, instead of using an analogy, for cascade control systems. A comparison of both AWC architectures from application point of view is provided to consolidate their utilities. Mainly, one of the architecture is better in terms of computational complexity for implementation, while the other is suitable for unstable cascade systems. On the basis of the architectures for cascade systems facing stability and performance degradation problems in the event of actuator saturation, the global AWC design methodologies utilizing linear matrix inequalities (LMIs) are developed. These LMIs are synthesized by application of the Lyapunov theory, the global sector condition and the ℒ2 gain reduction of the uncertain decoupled nonlinear component of the decoupled architecture. Further, an LMI-based local AWC design methodology is derived by utilizing a local sector condition by means of a quadratic Lyapunov function to resolve the windup problem for unstable cascade plants under saturation. To demonstrate effectiveness of the proposed AWC schemes, an underactuated mechanical system, the ball-and-beam system, is considered, and details of the simulation and practical implementation results are described. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Anti-windup compensator Decoupled architecture Cascade control systems Linear matrix inequality Underactuated mechanical system

1. Introduction Actuator saturation is a problem which may ruin the performance of a closed-loop system from degraded transients to instability [1]. To ensure stability and performance compliance, such as improved transient response against actuator saturation, anti-windup techniques have been extensively focused by the researchers, see, for example, [2–6] etc. A two-step design approach is promising as it does not require modification of an existing feedback controller, called the nominal or unconstrained controller, designed by ignoring actuator saturation [6]. First step is based on designing a nominal controller to achieve stabilization/tracking objectives; and in the second step, another auxiliary controller, known as anti-windup compensator (AWC), is incorporated to handle adverse effects of the input saturation. Scheme presented in the work [4] was one of the trend setters among the existing AWC design techniques, investigating a decoupled architecture equivalent to the overall closed-loop system with AWC. This novel methodology was widely appreciated due to the ease of applicability and successfully implemented for experimental setups like disk drive [7] and wireless networks [8]. For uncertain systems,

n

Correspondence to: House no. 413, Street 17, I-10/2 Islamabad, Pakistan. Tel.: þ 92 300 5529725. E-mail addresses: [email protected] (N. Mehdi), [email protected] (M. Rehan), [email protected] (F.M. Malik), [email protected] (A.I. Bhatti), [email protected] (M. Tufail). 1 Tel.: þ92 51 2207380 4. 2 House no. 12, Street no. 2, Sector B, Phase-1, Defense Housing Authority, Islamabad, Pakistan. Tel.: þ 92 345 5891417. 3 282, St. 11, Sector I, AECHS, Rawalpindi, Pakistan. Tel.: þ 92 51 2207380 4. 4 Room no. SE-103. Tel.: þ 92 51 2207380 4. http://dx.doi.org/10.1016/j.isatra.2014.01.007 0019-0578 & 2014 ISA. Published by Elsevier Ltd. All rights reserved.

N. Mehdi et al. / ISA Transactions 53 (2014) 802–815

803

robust AWC designs have been reported in the works [9-11]. Recently, a robust anti-windup framework, dealing with disturbances, has been proposed in the work [12]. Moreover, a decoupled architecture based AWC scheme has been proposed to address a class of nonlinear systems [13]. Solutions to the so-called windup problem addressed so far are for individual plants to meet various performance indices; however, sometimes different interconnections of plants [14] aggravate complexity of the problem, requiring more involved solutions. One such example is cascade plants, commonly used in the process industry [15], controlled with cascade controllers, to achieve fast closedloop response and to attain excellent disturbance rejection. To address anti-windup synthesis for cascade control systems, strategies presented for individual plants suffer two major discrepancies, motivating to address the problem separately. First, multi-loop compensation is not possible resulting in poor AWC performance; second, design constraints can be infeasible due to complexity of cascade systems in the presence of saturation (see the recent works [16,17]). A preliminary work on AWC design for cascade plants has been reported in the scheme [16] and, further, robust AWC solutions have been presented in another work [17]. However, the architectures developed in these approaches are based on an analogy from the methodology on AWC design for an individual plant [4]; therefore, these strategies are not much efficient for uncertain systems and unstable plants. The present work is an effort towards AWC synthesis for cascade control systems in order to resolve the industrial issues concerning the control paradigm. The aim of this article is to revisit the problem of AWC design for cascade plants subject to actuator saturation. Two novel performanceoriented full-order AWC architectures using state–space representations are developed for cascade control systems by deriving equivalence between the coupled and the decoupled architectures (instead of analogy). The proposed architectures are more advantageous than the conventional AWC architectures for cascade control systems due to the existence of the equivalent decoupled architectures. Further, these architectures solve anti-windup design problem for both stable and unstable plants under disturbance. It is important to note that the state–space paradigm instead of the transfer function approach has been used for the architectures proposed herein, making it abreast with the modern control state–space theory for cascade plants. Using the architectures and applying the global sector condition, linear matrix inequality (LMI)-based global AWC design methodologies, supportive for a broad range of operating conditions, are developed to ensure ℒ2 gain reduction of the uncertain decoupled nonlinear component. Another important contribution is to devise LMI-based schemes for designing AWC to guarantee local stability for unstable cascade plants by optimizing the region of stability. An application of the new AWC design methodologies has been motivated for a class of industrial systems having great importance in real life, that is, underactuated mechanical systems (UMS) [18]. These systems, including aircrafts, sea ships, helicopters, satellites, submarines and mobile robots, possess fewer actuators than the actual degree of freedom. Generally, these are safety critical systems, therefore it is very important to prevent them from instability, caused by any internal or external factor like actuator saturation. Our proposed AWC for cascade systems can be applied to these UMS with cascade interlinks. Many benchmark UMS like ball-and-beam system and rotating pendulum (Furuta Pendulum) form cascade interconnections, for example, when servo motor dynamics are included in the model. The proposed AWC scheme is applied to a ball-and-beam system, and its simulation results as well as experimental outcomes are provided herein. It is observed that both simulation and testing rig results of the suggested AWC scheme for ball-and-beam system reveal promising performance. The structure of the paper is as follows. Section 2 introduces the system under consideration and poses the problem to be addressed. In Section 3, the proposed architectures have been detailed and equivalence of the coupled and the decoupled architectures has been established. The design of AWC for cascade systems using LMIs has been sorted out in Section 4. Section 5 advances the AWC design results for unstable cascade plants using a local sector condition and a block-diagonal and quadratic Lyapunov functions. In Section 6, simulation and practical implementation results of the proposed methodologies for an underactuated ball-and-beam system are described. Section 7 concludes the article. Notations: Let Γ denote nonlinear operator then ℒ2 gain is defined as jjΓjji;2 : ¼ Sup0 a x A ℒ2 ðjjΓxjj2 Þ=ðjjxjj2 Þ, where jj:jj2 is Euclidean norm of vector x. X 40 and X Z 0 have been used for positive and semi-positive definiteness of a symmetric matrix X respectively. The notation diagðx1 ; x2 ; ::::; xn Þ; where xi defines ith diagonal entry, denotes a block-diagonal matrix X.

2. System description Consider two systems in the cascade configuration represented by state–space models given by x_ p1 ¼ Ap1 xp1 þ Bp1 usat þ Bw1 w1 ; yp1 ¼ C p1 xp1 þ Dp1 usat þ Dw1 w1 ;

ð1Þ

x_ p2 ¼ Ap2 xp2 þ Bp2 yp1 þBw2 w2 ; yp2 ¼ C p2 xp2 þ Dp2 yp1 þ Dw2 w2 ;

ð2Þ

where xp1 A ℝnp1 ; xp2 A ℝnp2 ; usat A ℝm ; yp1 A ℝq1 ; yp2 A ℝq2 ; w1 A ℝnp1 and w2 A ℝnp2 are the inner plant state, outer plant state, saturated control input, inner plant output, outer plant output, inner plant disturbance and outer plant disturbance, respectively. The disturbances w1 and w2 are bounded in ℒ2 norm sense. The matrices Ap1 ; Ap2 ; Bp1 ; Bp2 ; Bw1 ; Bw2 ; C p1 ; C p2 ; Dp1 ; Dp2 ; Dw1 ; and Dw2 are real constants having appropriate dimensions. Now consider the unconstrained controllers: ) x_ c1 ¼ f 1 ðxc1 ; u2 ; ylin1 Þ; ð3Þ K 1 ðxc1 ; u2 ; ylin1 Þ; ulin ¼ g 1 ðxc1 ; u2 ; ylin1 Þ; x_ c2 ¼ f 2 ðxc2 ; r; ylin2 Þ; u2 ¼ g 2 ðxc2 ; r; ylin2 Þ;

) K 2 ðxc2 ; r; ylin2 Þ;

ð4Þ

804

N. Mehdi et al. / ISA Transactions 53 (2014) 802–815

where xc1 A ℝc1 , xc2 A ℝc2 , r A ℝq2 , ylin1 A ℝq1 , ylin2 A ℝq2 , ulin A ℝm and u2 A ℝq1 are the vectors for inner loop controller state, outer loop controller state, desired reference signal, input to inner controller, input to outer controller, output of inner controller and output of outer controller, respectively. The time-varying vector functions f 1 ð U Þ A ℝc1 , f 2 ð UÞ A ℝc1 , g 1 ð U Þ A ℝm and g 2 ð U Þ A ℝq1 , representing dynamics of the controllers, can be either linear or nonlinear. It is assumed that the controllers (3) and (4), called the unconstrained controllers, are designed to ensure stability and performance requirements for the nominal closed-loop system, that is, when saturation is absent (that is to say, ylin1 ¼ yp1 , ylin2 ¼ yp2 and ulin ¼ usat ). It is worth mentioning that the proposed AWC design techniques can be used along with nonlinear controllers, formulated on the basis of neural networks [19], fuzzy logic [20] and genetic algorithms [21], and a number of cascade controller design techniques [22] are available in the literature. Due to saturation, the real interconnection between the plant (1) and the controller (3) is nonlinear, that is, usat ¼ satðuÞ. A well-known representation for this input saturation is sector bound inequality. Deadzone ðDzÞ and saturation ðsatÞ functions satisfy sector ½0; I as follows (see, for example, [23,24]): DzT ðuÞW½u  DzðuÞ Z0;

ð5Þ

sat T ðuÞW½u satðuÞ Z 0:

ð6Þ

Saturation at the inner plant input can disturb performance of the overall closed-loop system [16,17]; therefore, AWC design in this scenario is an appealing problem from industrial point of view. Following assumption is made on the unconstrained closed-loop system. Assumption 1. Unconstrained closed-loop system is stable and well-posed. Now, we develop two AWC architectures based on deriving equivalent decoupled representations, for clear understanding of the synthesis objectives, separating the unconstrained closed-loop systems with the remaining nonlinear components.

3. Proposed AWC architectures A general architecture of the overall closed-loop cascade control system with AWC block is shown in Fig. 1. The AWC block takes one input and produces three output signals. The input to the AWC block u~ A ℝm is the difference of the unsaturated and the saturated control signals given by u~ ¼ u  usat , and the three outputs ud A ℝm , yd1 A ℝq1 and yd2 A ℝq2 are used to modify the closed-loop system for compensation of performance degradation due to the windup problem. The AWC outputs ud , yd1 and yd2 modify the nominal input ulin as u ¼ ulin  ud , the inner plant output yp1 as ylin1 ¼ yp1 þyd1 and the outer plant output yp2 as ylin2 ¼ yp2 þyd2 to generate the unconstrained signals (in the absence of actuator saturation). The main objective is to design an AWC for cascade plants under actuator saturation, by application of ud ; yd1 and yd2 , to achieve response of the overall closed-loop system closer to the nominal response. Now we propose two AWC architectures for compensation of the windup effects. 3.1. Architecture-I State–space representation of the proposed AWC Architecture-I for cascade systems under actuator saturation is given by 9 x_ aw1 ¼ ðAp1 þ Bp1 F 1 Þxaw1 þ Bp1 u~ þ Bp1 F 2 xaw2 ; > > > > > x_ aw2 ¼ Ap2 xaw2 ; > > > > yd1 ¼ ðC p1 þDp1 F 1 Þxaw1 þ Dp1 u~ þDp1 F 2 xaw2 ; = Γ aw1 ; yd2 ¼ C p2 xaw2 ; > > > > > > ud ¼ F 1 xaw1 þ F 2 xaw2 ; > > > ; u~ ¼ u  usat ¼ DzðuÞ;

r

w1 ulin

K2

K1

_u ud

ylin2

usat

_

x p1 = Ap1 x p1 + B p1usat Bw1w1 y p1

x p2 = Ap2 x p2 + B p2 y p1 Bw2w2

y p1 = C p1x p1 + D p1usat Dw1w1

y p2 = C p2 x p2 + D p2 y p1 Dw2w2

u

ylin1

w2

yd1

Anti-windup Compensator

yd2

Fig. 1. The overall closed-loop system with AWC block.

y p2

ð7Þ

N. Mehdi et al. / ISA Transactions 53 (2014) 802–815

-

xaw1 = (Ap1 + B p1 F1 )xaw1 + B p1 u + B p1 F2 xaw2

ud u

+

xaw2 = Ap2 xaw2 yd1 = (C p1 + D p1 F1 )xaw1 + D p1 u + D p1 F2 xaw2 yd2 = C p2 xaw2

ulin r

ud = F1 xaw1 + F2 xaw2

w2

w1 xn1 = Ap1 x n1 +Bp1ulin Bw1 w1

K1

K2

805

ylin1 = C p1 xn1 + Dp1ulin Dw1 w1

ylin1

+ +

yd2

yd1 xn2 = Ap2 x n2 +B p2 y p1 Bw2 w2

y p1

y lin2 = C p2 x n2 + D p2 y p1 Dw2w2

_

ylin2 +

y p2

ylin1

ylin2

Fig. 2. Equivalent decoupling architecture in Fig. 1 for AWC Architecture-I.

where xaw1 A ℝnp1 and xaw2 A ℝnp2 are AWC state vectors corresponding to inner and outer plants, respectively. The AWC has been parameterized by constant matrices F 1 and F 2 . The AWC design can be accomplished by finding appropriate values of matrices F1 and F2 to ensure closed-loop stability and performance against windup effects. Fig. 2 shows an equivalent decoupled architecture to Fig. 1 by applying AWC block Γ aw1 given by (7). The decoupled architecture comprises of the nominal closed-loop system and the decoupled nonlinear component. Given stability of the nominal closed-loop system, the decoupled architecture facilitates closed-loop stability, by minimizing effect of the decoupled nonlinear component, in addition to robustness of controller K2 against disturbance yd1 . Equivalence in Figs. 1 and 2 can be established using a two-step procedure (see, for example, [13,20,23,24] etc.). In the first step, it can be verified that the mappings Γ 1 : ulin ↦ylin1 and Γ 2 : yp1 ↦ylin2 , given by ) x_ n1 ¼ Ap1 xn1 þ Bp1 ulin þBw1 w1 ; ð8Þ Γ1; ylin1 ¼ C p1 xn1 þ Dp1 ulin þ Dw1 w1 ; x_ n2 ¼ Ap2 xn2 þ Bp2 yp1 þ Bw2 w2 ; ylin2 ¼ C p2 xn2 þ Dp2 yp1 þ Dw2 w2 ;

) Γ2 ;

ð9Þ

are the same in both figures. This can be proved by taking time-derivative of transformations xn1 ¼ ðxp1 þxaw1 Þ and xn2 ¼ ðxp2 þ xaw2 Þ, applying ylin1 ¼ ðyp1 þ yd1 Þ and ylin1 ¼ ðyp2 þ yd2 Þ, and using (1) and (2), (7) and u ¼ ðulin  ud Þ for the architecture in Fig. 1. The second step is to replace the saturation function with the dead-zone nonlinearity using DzðuÞ ¼ u satðuÞ identity, which completes the equivalence proof. Note that the equivalent architecture in Fig. 2 offers the desired design objectives for AWC synthesis. To obtain an appropriate AWC (7), the ℒ2 gain from ulin to yd1 and yd2 can be minimized. Remark 1. Architecture-I is less conservative than the schemes presented in [16,17], owing to guaranteed equivalence between the overall closed-loop system with AWC and the corresponding decoupled architecture than using an analogy from [4]. However, Architecture-I is sensitive to disturbance, and it requires a robust controller K 2 . The works [16,17] have also the same problem. In addition, the proposed technique by Architecture-I and the architectures [16,17] cannot be applied, effectively and reliably, to resolve the windup problem for unstable systems. For instance, if the signal yd1 remains active for a reasonably large time, the output of an unstable outer plant in Fig. 2 can diverge. 3.2. Architecture-II The state–space representation for the proposed Architecture-II to design an AWC for cascade systems with actuator saturation is given by 9 x_ aw1 ¼ ðAp1 þ Bp1 F 1 Þxaw1 þ Bp1 u~ þBp1 F 2 xaw2 ; > > > > > x_ aw2 ¼ Ap2 xaw2 þ Bp2 yd1 ; > > > > yd1 ¼ ðC p1 þ Dp1 F 1 Þxaw1 þ Dp1 u~ þDp1 F 2 xaw2 ; = Γ aw2 ; ð10Þ yd2 ¼ C p2 xaw2 þDp2 yd1 ; > > > > > > ud ¼ F 1 xaw1 þ F 2 xaw2 ; > > > ; ~ u ¼ u  usat ¼ DzðuÞ; where xaw1 A ℝnp1 and xaw2 A ℝnp2 are anti-windup state vectors corresponding to the inner and the outer plants, respectively. Equivalent decoupled architecture in Fig. 1 with AWC block mapping Γ aw2 is provided in Fig. 3. The AWC parameters F1 and F2 can be obtained by solving an optimization problem, in an LMI framework, to minimize effect of the decoupled nonlinear component. Equivalence in Fig. 1 with AWC (10) and Fig. 3 can be established by using the same two-step procedure as for the case of Architecture-I. In the first step, it is proved that nominal closed-loop mapping Γ 3 : ulin ↦ylin1 and Γ 4 : ylin1 ↦ylin2 , given by ) x_ n1 ¼ Ap1 xn1 þ Bp1 ulin þBw1 w1 ; ð11Þ Γ3; ylin1 ¼ C p1 xn1 þ Dp1 ulin þ Dw1 w1 ;

806

N. Mehdi et al. / ISA Transactions 53 (2014) 802–815

ud

xaw1 = (Ap1 + Bp1F1 )xaw1 + Bp1u+ Bp1F2 xaw2 xaw2 = Ap2 xaw2 + Bp2 yd1

-

yd1 = (C p1 + Dp1F1 )xaw1 + Dp1u+ Dp1F2 xaw2

+ u lin

yd2 = C p2 xaw2 + Dp2 yd1 ud = F1 xaw1 + F2 xaw2

r

w2

w1 xn1 = Ap1 x n1 +B p1ulin

K1

K2

Bw1 w1

ylin1

ylin1 = C p1 xn1 + Dp1 ulin + Dw1 w1

xn2 = Ap2 x n2 +B p2 y p1 Bw2 w2 ylin2 = C p2 xn2 + D p2 y p1 Dw2 w2

ylin2 +

yd2

-

y p2

ylin1

ylin2

Fig. 3. Equivalent decoupling architecture in Fig. 1 for AWC Architecture-II.

x_ n2 ¼ Ap2 xn2 þ Bp2 ylin1 þ Bw2 w2 ; ylin2 ¼ C p2 xn2 þ Dp2 ylin1 þ Dw2 w2 ;

) Γ4;

ð12Þ

are same in both figures by application of xn1 ¼ xp1 þ xaw1 ; xn2 ¼ xp2 þxaw2 ; ylin1 ¼ yp1 þ yd1 , ylin2 ¼ yp2 þ yd2 , u ¼ ulin  ud , (1), (2) and (10). In the second step, saturation nonlinearity is replaced by dead-zone. Note that the AWC (10) can be designed by minimizing the ℒ2 gain only for the mapping from ulin to yd2 in contrast to Architecture-I case. Remark 2. Architecture-II is more general than internal model control (IMC)-based AWC [25] using exact dynamics of plants. By putting F 1 ¼ F 2 ¼ 0, the parameterization (10) becomes 9 ~ > x_ aw1 ¼ Ap1 xaw1 þ Bp1 u; > > > x_ aw2 ¼ Ap2 xaw2 þ Bp2 yd1 ; = Γ IMC ð13Þ ~ yd1 ¼ C p1 xaw1 þ Dp1 u; > > > > y ¼ C p2 xaw2 þ Dp2 y : ; d2

d1

Clearly, IMC-based AWC always exists for asymptotically stable systems, i.e., when Ap1 and Ap2 are Hurwitz. It should be noted that Architecture-I is not a generalization of IMC-based AWC. Remark 3. The signal yd1 coupled with the nominal closed-loop system, as seen in Fig. 2 for the case of Architecture-I, has been uncoupled in the present case by virtue of the terms Bp2 yd1 and Dp2 yd1 in (10). The advantage of uncoupling is threefold as compared to Architecture-I as well as the traditional schemes [16,17]. First, Architecture-II can handle both stable and unstable systems as the undesired signal yd1 is not entering to the nominal closed-loop system. Second, it provides more robust design because K 2 is not needed to be a robust controller. Last, Architecture-II can offer an optimal approach for minimizing the ℒ2 gain from ulin to yd2 rather than from ulin to a weighted sum of yd1 and yd2 . 4. Global AWC design The AWC design comprises of computing parameters F 1 and F 2 by minimizing the ℒ2 gain of the decoupled nonlinear part. Following theorem enables computation of the state-feedback gain matrices F 1 and F 2 for designing global AWC through Architecture-I for cascade systems. Theorem 1. Consider the overall closed-loop system formed by plants under input saturation, controllers and AWC (7) with equivalent architecture shown in Fig. 2. Suppose there exist symmetric matrices Q 1 A ℝnp1np1 and Q 2 A ℝnp2np2 and a diagonal matrix U A ℝmm . The optimization min γ γ 4 0; Q 1 4 0; Q 2 4 0; 1 20 Q 1 ATp1 þ Ap1 Q 1 A 6@ 6 þ LT BT þB L p1 1 6 1 p1 6 6 6n 6 6 6n 6 6n 6 6 6n 4 n

U 40;

3 0

Q 1 C Tp1 þ LT1 Dp1

0

 LT2

0

LT2 DTp1

Q 2 C Tp2

n

 2U

I

UDTp1

0

n

n

 γI

0

0

n

n

n

 W 1 1 γI

0

n

n

n

n

 W 2 1 γI

Bp1 L2

Bp1 U  LT1

Q 2 ATp2 þ Ap2 Q 2

7 7 7 7 7 7 7 7 o 0; 7 7 7 7 7 7 5

ð14Þ

N. Mehdi et al. / ISA Transactions 53 (2014) 802–815

807

pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi γ= W 2 , respectively, for positive scalar ensures the ℒ2 gain of the mappings Γ d1 : ulin ↦yd1 and Γ d2 : ulin ↦yd2 less than scalars Wffiffiffiffiffiffiffi 1 and pffiffiffiffiffiffiffiffi γ= p ffi T T W 2 yd2 . The AWC parameters can be computed design weights W 1 and W 2 , where γ represents the ℒ2 gain of the mapping ulin ↦½ W 1 yd1 by using F 1 ¼ L1 Q 1 1 and F 2 ¼ L2 Q 2 1 . Proof. Consider a quadratic Lyapunov function " #" # P1 0 xaw1 VðxÞ ¼ ½ xTaw1 xTaw2  ¼ xTaw1 P 1 xaw1 þ xTaw2 P 2 xaw2 ; xaw2 0 P2 where P 1 ¼ P T1 ¼ Q 1 1 4 0,P 2 ¼ P T2 ¼ Q 2 1 4 0, and x ¼ ½ xTaw1  pffiffiffiffiffiffiffiffi 2   1 W 1 yd1  _ VðxÞ þ  pffiffiffiffiffiffiffiffi   γ ulin 2 o 0: γ  W 2 yd2 

ð15Þ

xTaw2 T . For a desired ℒ2 gain, we choose the inequality ð16Þ

Using the sector condition (5) for the present case, the inequality ~ Z0 u~ T W½ulin  ud  u

ð17Þ

_ is obtained. The stability immediately follows from the above inequality, which implies that VðxÞ o0 along any closed-loop trajectory when ulin ¼ 0. Then by integrating (16) from t ¼ 0 to t ¼ T, 8 T 4 0, the following inequality is obtained: Z T Z T ðyTd1 W 1 yd1 þ yTd2 W 2 yd2 Þdt γ 2 uTlin ulin dt o 0: ð18Þ Vðx; TÞ  Vðx; 0Þ þ 0

0

As Vðx; 0Þ ¼ 0 (under zero initial condition) and Vðx; TÞ 40, (18) leads to the inequality Using (7), (15) and (17), we obtain zT Φ1 z o 0, where 20 6 6 6 6 6 6 6 6 Φ1 ¼ 6 n 6 6 6 6 6 6n 4

1

T

@ A1 P 1 þ P 1 A1 þ γ  1C

T

P 1 Bp1 þγ  1 C T W 1 Dp1

þγ  1 C T Dp1 W 1 F 2 0

1

ATp2 P 2 þ P 2 Ap2

B C B þ γ  1 F T DT W D F C 1 p1 2 C B 2 p1 @ A þ γ  1 C Tp2 W 2 C p2

n

zT ¼ ½ xTaw1

!

P 1 Bp1 F 2

W 1CA

xTaw2

u~ T

F T1 Wτ 0 @

γ  1 F T2 DTp1 W 1 Dp1  F T2 Wτ

1 A

n

γ  1 DTp1 W 1 Dp1  2Wτ

n

n

RT 0

ðyTd1 W 1 yd1 þ yTd2 W 2 yd2 Þdt  γ 2

RT 0

uTlin ulin dt o 0:

3

!

7 7 7 7 7 7 7 7 o 0; 0 7 7 7 7 7 7 Wτ 7 5 γI 0

ð19Þ

uTlin ;

ð20Þ

A1 ¼ ðAp1 þ Bp1 F 1 Þ; C ¼ ðC p1 þDp1 F 1 Þ:

ð21Þ diagðP 1 1 ; P 2 1 ; V  1 ; I; IÞ,

Applying Schur complement to (19) and using congruence transformation [26] with Q 2 ¼ P 2 1 ,L1 ¼ F 1 Q 1 , L2 ¼ F 2 Q 2 and applying (21) lead to LMI (14), which completes the proof of Theorem 1.

□

putting Q 1 ¼ P 1 1 ,

Now we address the global AWC design using Architecture-II by minimizing the ℒ2 gain of the decoupled nonlinear component for architecture shown in Fig. 3. Theorem 2. Consider the overall closed-loop system formed by plants under input saturation, controllers and AWC (10) with equivalent architecture shown in Fig. 3. Suppose there exist symmetric matrices Q 1 A ℝnp1np1 and Q 2 A ℝnp2np2 and a diagonal matrix U A ℝmm . The LMIbased optimization min γ γ 4 0; Q 1 4 0; Q 2 4 0; 1 20 Q 1 ATp1 þ Ap1 Q 1 A 6@ 6 þ L T BT þ B L p1 1 6 1 p1 6 6 6n 6 6 6n 6 6n 4 n

U 4 0; 0 1 Bp1 L2 þ Q 1 C Tp1 BTp2 @ A þL1 DTp1 BTp2

3 Bp1 U LT1

0

Bp1 DP1 U  LT2

0

n

 2U

I

n

n

 γI

n

n

n

Q 2 ATp2 þ Ap2 Q 2

Q 1 C Tp1 DTp2 þ LT1 DTp1 DTp2 7 7 7 7 7 T 7 Q 2 C p2 7o0 7 T T 7 UDp1 Dp2 7 7 0 5 γI

ensures the ℒ2 gain of the mapping less than γ. The AWC parameters can be computed by solving F 1 ¼ L1 Q 1 1 and F 2 ¼ L2 Q 2 1 .

ð22Þ

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N. Mehdi et al. / ISA Transactions 53 (2014) 802–815

_ þ γ  1 jjy jj2  γjjulin jj2 o0 is Proof. Consider the Lyapunov function (15). For stability of the system Γ d2 : ulin ↦yd2 , the inequality VðxÞ d2 required to be satisfied, which along with (17) can be written as _ þ γ  1 jjy jj2 þ u~ T W½ulin ud  u ~  γjjulin jj2 o 0: VðxÞ d2

ð23Þ

The inequalities (17) and (23) imply that the mapping Γ d2 : ulin ↦yd2 is stable in ℒ2 sense (with ℒ2 gain less than γ). Using the same procedure as in the proof of Theorem 1, we obtain LMI (22), which completes the proof of Theorem 2. □ Remark 4. The traditional global AWC design techniques (and the proposed scheme by Theorem 1) for cascade control systems are using bilinear matrix inequalities to provide semi-optimal results, owing to the tuning weights W 1 and W 2 , due to consideration of minimization of the ℒ2 gain for both mappings Γ d1 : ulin ↦yd1 and Γ d2 : ulin ↦yd2 . However, the present scheme by Theorem 2 offers an LMI-based less conservative procedure to obtain the optimal AWC because the decoupled nonlinear component is represented by Γ d2 : ulin ↦yd2 in Fig. 3. 5. Local AWC design The AWC synthesis scheme provided in Section 4 was designed on the basis of the global sector condition (5). A less conservative local sector condition is formulated in the works [27–29], which can be used to develop a local AWC synthesis strategy for dealing with a broad class of cascade control systems. For a region ℑðuÞ ¼ fu A ℝm ;  u r u ω r ug;

ð24Þ

the local sector condition DzðuÞT W½ω  DzðuÞ Z 0

ð25Þ

holds [29]. By selecting ω ¼ ulin  G1 xaw1  G2 xaw2 and using (24) and (25) and u ¼ ulin  F 1 xaw1  F 2 xaw2 , we obtain ℑðuÞ ¼ fu A ℝm ;  u r ðG1  F 1 Þxaw1 þ ðG2 F 2 Þxaw2 rug;

ð26Þ

~ Z 0: u~ T W½ulin  G1 xaw1  G2 xaw2  u

ð27Þ

Now we provide a novel treatment of the local sector condition (26) and (27) for designing a multi-objective local AWC for cascade control systems to offer a large domain of stability using a more general quadratic Lyapunov function. Theorem 3. Consider the overall closed-loop system formed by plants under input saturation, controllers and AWC (10) with equivalent architecture shown in Fig. 3. Suppose there exist symmetric matrices Q 11 A ℝnp1np1 and Q 22 A ℝnp2np2 , a matrix Q 12 A ℝnp1np2 , a diagonal matrix U A ℝmm and scalars γ and s. The LMIs γ 4 0, U 4 0, s 4 0, 2 3 Q 11 Q 12 H 1ðiÞ L1ðiÞ " # Q 11 Q 12 6n Q 22 H 2ðiÞ L2ðiÞ 7 7 Z 0; i ¼ 1; 2; …; m; ð28Þ Q¼ 4 0; 6 4 5 Q T12 Q 22 n n su2ðiÞ 2

0

6 6@ 6 6 LT1 BTp1 þ Bp1 L1 6 6 6 6 6 6 6 6n 6 6 6 6 6 6 6n 6 6n 4

Q 11 ATp1 þAp1 Q 11

1 A

n

0

Ap1 Q 12 þ Bp1 L2

0

1

B T T T T C B þQ 11 C p1 Bp2 þL1 Dp1 Bp2 C @ A T þQ 12 Ap2 0 1 Q 22 ATp2 þ Ap2 Q 22 B C B þB C Q þ Q T C T BT C p2 P1 12 B 12 p1 p2 C @ A þBp2 DP1 L2 þ LT2 DTp1 BTp2

Bp1 U H 1

0

Bp2 DP1 U  H 2

0

n

 2U

n

n

I γI

n

n

n

Q 11 C Tp1 DTp2

13

B C7 B þ LT DT DT C 7 B 1 p1 p2 C 7 @ A7 7 þ Q 12 C Tp2 7 0 T T T 17 7 7 Q 12 C p1 Dp2 B C7 B þ Q CT C7 o0; B C7 22 p2 @ A7 7 T T T þ L2 Dp1 Dp2 7 7 7 7 UDTp1 DTp2 7 7 7 0 5  γI

ð29Þ

ensure the following for all ℒ2 norm bounded signals satisfying jjulin jj22 r λ  1 ¼ γ  1 s  1 : (i) The ℒ2 gain of the mapping Γ d2 : ulin ↦yd2 is less than γ. (ii) The AWC states remain bounded by xTaw sQ  1 xaw o 1. The AWC gains can be computed by solving ½ F 1 F 2  ¼ ½ L1

L2 Q  1 .

Proof. Rewriting (10) in an appropriate form given by ~ x_ aw ¼ ðA þ BFÞxaw þ Bu; ~ yd ¼ ðC þDFÞxaw þ Du; ud ¼ Fxaw ; u~ ¼ u  usat ¼ DzðuÞ;

ð30Þ

N. Mehdi et al. / ISA Transactions 53 (2014) 802–815

where

"

A¼

Ap1

0

Bp2 C p1

Ap2

#

" ; B¼

F 2 ; C ¼ ½ Dp2 C p1

F ¼ ½ F1

Bp1 Bp2 Dp1

#

" ; xaw ¼

xaw1 xaw2

809

# ;

C p2 ; D ¼ Dp2 Dp1 :

Consider a quadratic Lyapunov function as " #" # P 11 P 12 xaw1 ¼ xaw Pxaw ; VðxÞ ¼ ½ xTaw1 xTaw2  T P 12 P 22 xaw2

ð31Þ

ð32Þ

_ þ γ  1 jjy jj  γjjulin jj o0 where P is a symmetric positive definite matrix. It is seen in the proof of Theorem 2 that the inequality VðxÞ d2 RT guarantees the ℒ2 gain of the mapping Γ d2 : ulin ↦yd2 less than γ. Under zero initial condition, (18) can be written as Vðx; TÞ o γ 0 uTlin ulin dt, R T which further implies that xTaw1 P 11 xaw1 þ xTaw1 P 12 xaw2 þxTaw2 P 21 xaw1 þ xTaw2 P 22 xaw2 o s  1 (because 0 uTlin ulin dt rjjulin jj22 rλ  1 and

s  1 ¼ γλ  1 ). Consequently, the AWC states remain bounded by xTaw1 sP 11 xaw1 þ xTaw1 sP 12 xaw2 þ xTaw2 sP T12 xaw1 þ xTaw2 sP 22 xaw2 o 1 for all time. To satisfy (27), we can include the region xTaw1 sP 11 xaw1 þ xTaw1 sP 12 xaw2 þ xTaw2 sP T12 xaw1 þ xTaw2 sP 22 xaw2 o 1 into ℑðuÞ by means of the

inequality: 2 P 11 6n 6 4 n

G1ðiÞ  F 1ðiÞ

P 12

3

G2ðiÞ  F 2ðiÞ 7 7 Z 0; i ¼ 1; 2; …; m: 5 su2ðiÞ

P 22 n

Applying congruence transform, diagðP 1 1 ; P 2 1 ; V  1 ; I; IÞ, to (33) and substituting " # " # " #1 Q 11 Q 12 P 11 P 12 H1 L ¼ ½ L1 L2 ; H ¼ ; ; ¼ T T Q 12 Q 22 P 12 P 22 H2

ð33Þ

ð34Þ

_ þ γ  1 jjy jj γjjulin jj o 0 is under Q ¼ P  1 , ω ¼ ulin  Gxaw , u ¼ ulin  Fxaw , H ¼ GQ , and L ¼ FQ ; we obtain the LMI (28). Further, the VðxÞ d2 satisfied if the inequality _ ~ o0 VðxÞ þγ  1 jjyd2 jj  γjjulin jj þ u~ T W½ulin  G1 xaw1  G2 xaw2  u

ð35Þ

~ Z0). The LMI (29) is obtained by using (35) in the same way as for the proof of Theorem 2, holds (because u~ W½ulin  G1 xaw1 G2 xaw2  u which completes the proof of Theorem 3. □ T

The architectures and the AWC design methodologies developed in the present study can be extended to achieve robustness against parametric uncertainties present in the cascade plants (1) and (2). A framework for designing a robust AWC for cascade plants is provided in Appendix A.

6. Application to underactuated ball-and-beam system Mechanical systems having more degrees of freedom than actuators are called underactuated mechanical systems. Many benchmark systems including the ball-and-beam, inertial wheel pendulum, cart-pole, rotating pendulum and Acrobot systems belong to UMS. All these complex systems, posing challenges from control perspective, are widely used to validate different control strategies. These UMS, having two generalized coordinates, can further be divided into Classes I and II, based upon the shape variable actuation. Shape variables are those generalized coordinates which appear in the inertia matrix of a system, remaining are known external variables. The inertial wheel pendulum, Acrobot and TORA systems are Class-I examples due to actuated shape variables. On the other hand the rotating pendulum, cart-pole and ball-and-beam systems belong to Class-II as the shape variables are not actuated. Class-II systems can be transformed into cascade systems using change of coordinates for testing of the cascade AWC schemes [30–33]. 6.1. Ball-and-beam system description The ball-and-beam system (Quanser Model SRV02 and BB01) used for testing of the proposed AWC schemes is shown in Fig. 4. It is a benchmark for studying the control problems like rocket toppling control system and vehicle suspension control system. It has two degrees of freedom (DOF): first is free rolling of the ball along the tilted beam while second is rotation of the beam along the rotation axis of motor. The system comprises of a mechanical structure of a ball, a beam and a DC servo motor as shown in the schematic diagram in Fig. 5. An encoder is attached with the DC servo motor to measure shaft angular position θðtÞ and ball position xðtÞ. On the steel beam, a nickel–chromium wire-wound resistor forms a track, upon which the ball can freely move. The rolling ball on the biased wire-wound resistor acts as a wiper of potentiometer, and, consequently, ball position is obtained by measuring the voltage at the rod end. It is worth mentioning that the system is unstable in open-loop as ball position changes in an uncontrollable fashion in the absence of edge-limits for a fixed value of the beam angle input. The setup has an in-built property that it can be controlled using hardware-in-loop (HIL) simulations in Simulinks. 6.2. System modeling The ball-and-beam system can be modeled either by coupled or uncoupled dynamics [32]. Uncoupled dynamics give rise to two cascade models, hence, preferred for the present scenario. Schematic diagram of the system shown in Fig. 5, with system parameters

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N. Mehdi et al. / ISA Transactions 53 (2014) 802–815

Power Supply and Power Amplifier

Potentio Meter Steel Ball Steel Beam

Servo Motor and Gear Train

Fig. 4. The ball-and-beam system to validate the proposed AWC scheme.

L

Frx

in( ) m Bg s

J B mBx 2 R m Bx Ftx

Ball

x

Lever Arm r

( ) m B g cos mB g Mot Tor or que

Output Gear

Fig. 5. Schematic diagram of the ball-and-beam system.

Table 1 Ball-and-beam system parameters. Symbol

Quantity

Value

g mB mb R l d Km Kg Rm η Jeq JB Beq

Gravity acceleration Ball mass Beam mass Ball radius Beam length Lever length Motor back EMF constant Gear ratio Motor resistance Equivalent efficiency Equivalent load inertia Ball inertia Equivalent viscous friction

9.8 (m/s2) 0.064 (kg) 0.65 (kg) 0.0254 (m) 0.425 (m) 0.12 (m) 0.00767 (V-s/rad) 14 2.6 (Ω) 0.7395 0.0023(Kg-m2) 0.00064 (Kg-m2) 4  10  3 (Nm/(rad/s))

defined in Table 1, can be modeled using the force balance equation, given by   J mB þ B2 x€ mB α_ 2 x  mB g sin ðαÞ ¼ 0; R

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811

_ small, the square term can be omitted. where α is the beam angle and xðtÞ is the ball position. By assuming the beam angular velocity ðαÞ For a small range of beam angle, we obtain sin ðαÞ  α. As a result, the ball-and-beam system transfer function becomes XðsÞ mB g ¼ αðsÞ ðmB þ J B =R2 Þs2 The relation between beam angle and motor pinion gear angle is α ¼ ðd=lÞθ, which can be represented by the transfer function Gp2 ¼ XðsÞ=θðsÞ ¼ 0:418=s2 after substitution of the parameter values. Servo motor transfer function with voltage as input and position as output is given by ηK m K g =Rm J eq θðsÞ ¼ VðsÞ s½s þðBeq =J eq þ ηK 2m K 2g =Rm J eq Þ By substituting the coefficients, the servo motor transfer function becomes Gp1 ¼ 160=sðs þ 1Þ. Clearly Gp1 is marginally stable while Gp2 is unstable. 6.3. Nominal cascade controllers and AWC design Many linear [32] and nonlinear [33] controllers have been applied to the ball-and-beam system, however, two proportional-integralderivative (PID) controllers K 1 ðsÞ and K 2 ðsÞ are used for the inner and the outer plants, respectively. First PID controller is tuned using Simulink (Matlab) response optimization block for inner loop to control motor position to meet 5:0% overshoot and peak time t p ¼ 0:15 s design specifications, given by   1:25 þ 0:2s : K 1 ðsÞ ¼ 6:5 þ s The outer loop PID controller is tuned using Simulink response optimization block to control the ball position on the beam for 41% overshoot and t p ¼ 4 s, given by   0:75 þ 2s : K 2 ðsÞ ¼ 2:5 þ s Note that a high overshoot is resulted due to fast closed-loop response. A first-order low pass filter has also been applied to remove noise at the ball position sensor output, given by FilterðsÞ ¼

11 : s þ 10

The filter improves performance of the system drastically and removes chattering. By solving the LMIs of Theorem 3 for the unstable cascade plants of the ball-and-beam system for s ¼ 5000, we obtain the AWC gain matrix as F ¼ ½  479:2

 2765:7

 063:2

 0112:7 :

6.4. Simulation and experimental implementation results Simulation results of the closed-loop system without saturation, with saturation, with the IMC-based AWC and with the proposed AWC are shown in Fig. 6. The nominal response without saturation, having an overshoot, is tracking the step reference signal. The closed-loop system response with saturation reveals unstable oscillations with increasing amplitude due to the windup effect. To overcome these oscillations, first IMC-based scheme is employed, which is failed as shown in Fig. 6, owing to the fact that the system is

Fig. 6. Normalized simulation response of the closed-loop ball-and-beam system.

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N. Mehdi et al. / ISA Transactions 53 (2014) 802–815

Fig. 7. Normalized experimental (practical) response of the closed-loop ball-and-beam system.

Fig. 8. Effect of saturation on expected value of the ℒ2 gain of the decoupled nonlinear component.

unstable. Next scheme presented in [16] was also tried, leading to infeasible results. However, the proposed AWC response, by application of Theorem 3, is closer to the nominal closed-loop response, as it brings out the system from instability, showing strength of the scheme in comparison to that of IMC-based methodology and the approach presented in [16]. The proposed AWC and cascade controllers have been successfully implemented on the ball-and-beam system through HIL with similar results to that of simulation. HIL related experimental setup comprises of QuaRCs 2.0 software which attains soft real-time environment on WindowssXP operating system. It can be integrated with Simulinks seamlessly for rapid prototyping and hardwarein-loop testing using Q8 data acquisition board. The experimental implementation results for the nominal closed-loop system, the closed-loop system under saturation and the overall closed-loop system with AWC are shown in Fig. 7. Normalized results are shown by dividing the outputs with the corresponding constant reference signals. When the input saturates, the system starts bounded oscillations, in contrast to simulation results, due to obstacles at edges of the finite steel rods acting as output-limiters. In contrast, both responses without saturation and with AWC are following the reference signal. Note that the overshoots in both the cases are different in comparison to the simulation results. It is due to the fact that the nominal response is generated by applying a low reference signal (to avoid saturation) and the response with AWC is generated by using a high reference signal (to test the AWC under saturation). As actual system is nonlinear, different reference signals can result in different overshoots. To sum up, the proposed methodology has satisfactory simulation as well as experimental implementation results for the ball-and-beam system and, hence, can be applied to control unstable industrial processes. To investigate frequency domain effects of saturation on the decoupled nonlinear component, sine wave signals of different frequencies are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RT 2 RT 2 injected at ulin , output yd is measured, and the expected ℒ2 gain γ n is calculated using γ n ¼ 0 yd dt = 0 ulin dt at different frequencies. This expected ℒ2 gain in dB, that is 20 log γ n ; is plotted against corresponding frequency, ranging from 10  2 rad/s to 102 rad/s with 0.1 rad/s increment. In the presence of saturation nonlinearity, the response found is reasonably similar to that of a low pass filter as shown in Fig. 8.

N. Mehdi et al. / ISA Transactions 53 (2014) 802–815

813

Moreover the gain γ n remains high around 220 dB for frequencies up to 6 rad/s then reduces drastically to as low as  150 dB with a sharp roll off. From here, it can be concluded that the effect of decoupled nonlinear component, containing the saturation block, is higher at low frequencies.

7. Conclusion This paper provided a comprehensive study on decoupled architecture based full-order anti-windup design for stable and unstable cascade systems under actuator saturation. Two novel decoupled architectures were developed by ensuring the equivalence between the coupled and the corresponding decoupled architectures. The proposed architectures are less conservative in terms of AWC performance and stability in contrast to the traditional coprime factor analogy-based architectures for cascade systems. By utilizing these architectures, a number of global and local LMI-based conditions, by addressing their advantages and shortcomings, were synthesized to compute the AWC gains for stable and unstable cascade systems. The work was augmented by presenting the AWC for unstable cascade plants by incorporating a more general sector condition to enlarge the domain of stability. The proposed AWC can be computed numerically using convex optimization routines to bestow certain stability and performance properties on the complete nonlinear closed-loop system. The proposed AWC scheme is simulated as well as applied practically to a ball-and-beam system, a benchmark for underactuated mechanical systems, to cope with input saturation. Encouraging results were obtained that depicts effectiveness and potential applications of the proposed AWC schemes for industrial cascade control systems.

Appendix A. Robust AWC design Consider the cascade systems with parametric norm bounded uncertainty characterized by state–space model as x_ p1 ¼ ðAp1 þΔAp1 Þxp1 þBp1 usat þ Bw1 w1 ; yp1 ¼ ðC p1 þ ΔC p1 Þxp1 þDp1 usat þ Dw1 w1 ;

ðA:1Þ

x_ p2 ¼ ðAp2 þΔAp2 Þxp2 þBp2 yp1 þ Bw2 w2 ; yp2 ¼ ðC p2 þ ΔC p2 Þxp2 þDp2 yp1 þ Dw2 w2 ; np1Xnp1

np2Xnp2

ðA:2Þ q1Xnp1

q2Xnp2

where ΔAp1 A R , ΔAp2 A R , ΔC p1 A R and ΔC p2 A R are inner plant state perturbation, outer plant state perturbation, inner plant output perturbation and outer plant output perturbation matrices, respectively, having bounded matrix norms. To synthesize the AWC, the generalized Architecture-II with modified local sector condition is used. K 1 ðxc1 ; u2 ; ylin1 Þ and K 2 ðxc2 ; r; ylin2 Þ are norm bounded robust controllers as described earlier. Fig. A.1 depicts the decoupled architecture established using the same two-step procedure described in Section 3.2 with the modified disturbance terms given by the relations ϕ11 ¼ Bw1 w1  ΔAp1 xaw1 , ϕ21 ¼ Bw2 w2  ΔAp2 xaw2 , ϕ12 ¼ Dw1 w1  ΔC p1 xaw1 and ϕ22 ¼ Dw2 w2  ΔC p2 xaw2 . To design an AWC, the following objective function is employed:  pffiffiffiffiffiffiffiffi 2   1 W p yd2  _ ðA:3Þ VðxÞ þ  pffiffiffiffiffiffiffi   γ μlin 2 o 0 γ  W r xaw  where W P and W r are the performance and the robustness weighting scalars, respectively. The term containing W r in the objective function introduces the robustness against the plant uncertainty. To minimizing the effect of perturbation terms, the ℒ2 gain of the mapping Γ r : ulin ↦xaw must be minimized (see Fig. A.1). The performance weight W P can be used to achieve AWC performance by minimizing the ℒ2 gain of the mapping Γ r : ulin ↦yd2 . Incorporating the local sector conditions (26) and (27) provides Theorem A.1 for AWC gain calculation. .

xaw1 = (A p1 + B p1 F1 )xaw1 + B p1 u∼ + B p1 F2 xaw2

ud

.

xaw2 = A p2 xaw2

u∼

+

yd1 = (C p1 + D p1 F1 )xaw1 + D p1 u∼ + D p1 F2 xaw2 yd2 = C p2 xaw2

u lin _

Ap1

+

ud = F1 xaw1 + F2 xaw2 xaw1

Bw2

Bw1 Dw1

w1

_

+

_

C p1

+ Dw2

w2

Ap2

_

xaw2

C p2

+

yd2

r 11

12

.

K2 y lin2

xn1 = Ap1 x n1 +B p1 ulin

K1

ylin1 = C p1 xn1 + D p1 ulin

ylin1

11 12

+ +

yd1

21

22

.

xn2 = Ap2 x n2 +B p2 y p1

y p1

ylin2 = C p2 xn2 + D p2 y p1

ylin1 Fig. A.1. Decoupling architecture for designing robust AWC against perturbations.

21 22

ylin2 +

_

y p2

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Theorem A.1. Consider the overall closed-loop system formed by the plants under input saturation, controllers and AWC (10) with equivalent architecture shown in Fig. A.1. Suppose there exist symmetric matrices Q 11 A ℝnp1np1 and Q 22 A ℝnp2np2 , a matrix Q 12 A ℝnp1np2 , a diagonal matrix U A ℝmm and scalars γ and s. The LMIs γ 4 0, U 4 0, s 40; 2 3 Q 11 Q 12 H 1ðiÞ  L1ðiÞ " # Q 11 Q 12 6n Q 22 H 2ðiÞ  L2ðiÞ 7 7 Z 0; i ¼ 1; 2; …; m; ðA:4Þ Q¼ 4 0; 6 4 5 Q T12 Q 22 n n su2ðiÞ 2

0

6 6@ 6 6 LT1 BTp1 þ Bp1 L1 6 6 6 6 6 6 6 6n 6 6 6 6 6 6 6n 6 6n 6 6 6n 4

Q 11 ATp1 þAp1 Q 11

n

1 A

0

0

1

Ap1 Q 12 þ Bp1 L2

B T T T T C B þQ 11 C p1 Bp2 þL1 Dp1 Bp2 C @ A T þQ 12 Ap2 0 1 Q 22 ATp2 þ Ap2 Q 22 B C B þB C Q þ Q T C T BT C B p2 P1 12 12 p1 p2 C @ A þBp2 DP1 L2 þ LT2 DTp1 BTp2

Q 11 C Tp1 DTp2

1

B C B þ LT DT DT C B 1 p1 p2 C @ A þ Q 12 C Tp2 0 T T T 1 Q 12 C p1 Dp2 B C B þ Q CT C B C 22 p2 @ A T T T þ L2 Dp1 Dp2

3

Bp1 U H 1

0

Bp2 DP1 U  H 2

0

n

 2U

I

UDTp1 DTp2

n

n

γI

0

0

n

n

n

 γW p 1 I

0

n

n

n

n

 γW r 1 I

I

0

0

7 7 7 7 7 7 7 7 7 7 7 7 7o0 7 7 7 7 7 7 7 7 7 7 7 5

ðA:5Þ

ensure the following for all ℒ2 norm bounded signals satisfying jjulin jj22 rλ  1 ¼ γ  1 s  1 : pffiffiffiffiffiffiffiffi (i) The ℒ2 gain of the mapping Γ d2 : ulin ↦yd2 is less than γ= W p . pffiffiffiffiffiffiffi (ii) The ℒ2 gain of the mapping Γ r : ulin ↦xaw is less than γ= W r . (iii) The AWC states remain bounded by xTaw sQ  1 xaw o 1. The AWC gain matrices can be computed by solving ½ F 1 Proof. The proof is similar to that of Theorem 3.

F 2  ¼ ½ L1

L2 Q  1 .

&

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