Almost disturbance decoupling and tracking control for multi-input multi-output non-linear uncertain systems: application to a half-car active suspension system

http://pii.sagepub.com/

Control Engineering

Engineers, Part I: Journal of Systems and

http://pii.sagepub.com/content/223/2/215

The online version of this article can be found at:

DOI: 10.1243/09596518JSCE647

2009 223: 215

Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering

T-L Chien, C-C Chen, M-C Tsai and Y-C Chen

systems: Application to a half-car active suspension system

Almost disturbance decoupling and tracking control for multi-input multi-output non-linear uncertain

Published by:

http://www.sagepublications.com

On behalf of:

Institution of Mechanical Engineers

can be found at:

Engineering

Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control

Additional services and information for

http://pii.sagepub.com/cgi/alerts Email Alerts: http://pii.sagepub.com/subscriptions Subscriptions: http://www.sagepub.com/journalsReprints.nav Reprints: http://www.sagepub.com/journalsPermissions.nav Permissions: http://pii.sagepub.com/content/223/2/215.refs.html Citations:

What is This?

- Mar 1, 2009

Version of Record

>>

Almost disturbance decoupling and tracking control for

multi-input multi-output non-linear uncertain systems:

application to a half-car active suspension system

T-L Chien1_{, C-C Chen}2_{*, M-C Tsai}3_{, and Y-C Chen}4 1

Department of Electronic Engineering, Wufeng Institute of Technology, Chia-Yi, Taiwan, Republic of China

2

Department of Electrical Engineering, National Chiayi University, Chiayi City, Taiwan, Republic of China

3

Department of Electrical Engineering, National Formosa University, Yunlin, Taiwan, Republic of China

4

Department of Materials Science and Engineering, National Chiao Tung University, Hsinchu, Taiwan, Republic of China

The manuscript was received on 18 June 2008 and was accepted after revision for publication on 17 September 2008. DOI: 10.1243/09596518JSCE647

Abstract: This study presents a novel feedback linearization control of non-linear multi-input multi-output uncertain systems for the tracking and almost disturbance decoupling performances. The main contribution of this study is to construct a controller, under appropriate conditions, such that the resulting closed-loop system is valid for any initial condition and bounded tracking signal with the following characteristics: input-to-state stability with respect to disturbance inputs and almost disturbance decoupling. In addition, a new theorem on robust stability is proposed in this study to provide a new criterion for closed-loop stability. A typical case, which cannot be solved by any previous study on the almost disturbance decoupling problem, is proposed in this study to exploit the fact that the tracking and the almost disturbance decoupling performances can be easily achieved by the proposed approach. Finally, the proposed control law is simulated in a half-car active suspension system on which the effectiveness of the design is verified.

Keywords: almost disturbance decoupling, multi-input multi-output uncertain system, half-car active suspension system, feedback linearization approach, composite Lyapunov approach

1 INTRODUCTION

Stabilization and tracking are both important tasks in the solution of the control problem. The tracking task is generally more complicated than the stabili-zation task for non-linear control systems. Many approaches to these tasks have been proposed including feedback linearization, variable structure control (sliding mode control), backstepping, regula-tion control, non-linear H‘ control, the internal model principle, and H‘ adaptive fuzzy control. Richter et al. [1] have proposed the use of variable structure control to deal with non-linear systems. However, chattering behaviour caused by discontin-uous switching and imperfect implementation that

can drive the system into unstable regions is inevitable for variable structure control schemes [2]. Backstepping has proven to be a powerful tool for synthesizing controllers for non-linear systems [3]. However, a disadvantage of this approach is an explosion in the complexity which is a result of repeated differentiation of the non-linear functions [4, 5]. An alternative approach is to utilize output regulation control in which the outputs are assumed to be excited by an exosystem [6]. However, this non-linear regulation approach requires the solution of difficult partial differential algebraic equations. Another difficulty is that the exosystem states need to be switched to describe changes in the output and this creates transient tracking errors [7]. In general, non-linear H‘ control requires the solution of the Hamilton–Jacobi equation, which is a difficult non-linear partial differential equation [8–11]. Only for some particular non-linear systems it is possible to

*Corresponding author: Department of Electrical Engineering, National Chiayi University, 300 Syuefu Road, 60004 Chiayi, Taiwan, Republic of China. email: [email protected]

derive a closed-form solution [12]. The control approach that is based on the internal model principle converts the tracking problem into a non-linear output regulation problem [13]. This approach depends on solving a first-order partial differential equation of the centre manifold [6]. For some special non-linear systems and desired trajec-tories, the asymptotic solutions of this equation have been developed using ordinary differential equations [14, 15]. Recently, H‘ adaptive fuzzy control has been proposed to systematically deal with non-linear systems [16]. The drawback with H‘ adaptive fuzzy control is that the complex parameter update law makes this approach impractical in real-world situations. During the past decade significant pro-gress has been made in researching control ap-proaches for non-linear systems based on the feed-back linearization theory [17, 18]. Moreover, the feedback linearization approach has been success-fully applied to many real control systems. These include the control of an electromagnetic suspen-sion system [19], pendulum system [20], spacecraft [21], electrohydraulic servosystem [22], car-pole system [23], bank-to-turn missile system [24], and a compact six-axis magnetic levitation stage [25].

It is difficult to obtain completely accurate mathematical models for many practical control systems. Thus, there are inevitable uncertainties in their models. Therefore, the design of a robust controller that deals with the uncertainties of a control system is of considerable interest. This study presents a systematic analysis and a simple design scheme that guarantees the globally asymptotic stability of a feedback-controlled uncertain system and achieves output tracking and almost distur-bance decoupling performances for a class of non-linear control systems with uncertainties.

The almost disturbance decoupling problem, that is the design of a controller that attenuates the effect of the disturbance on the output terminal to an arbitrary degree of accuracy, was originally developed for linear and non-linear control systems by Willems [26] and Marino et al. [27] respectively. The problem has attracted considerable research attention and many significant results have been developed for both linear and non-linear control systems [28–30]. The almost disturbance decoupling problem of non-linear single-input single-output (SISO) systems was investigated in Marino et al. [27] by using a state feedback approach and solved in terms of sufficient conditions for systems with non-linearities that are not globally Lipschitz and disturbances being linear but possibly actually being

multiples of non-linearities. The resulting state feedback control is constructed following a singular perturbation approach. The sufficient conditions in Marino et al. [27] require that the non-linearities multiplying the disturbances satisfy structural trian-gular conditions. Marino et al. [27] show that for non-linear SISO systems the almost disturbance decoupling problem may not be solvable, as is the case for

_xx1ð Þ~tant {1ð Þzh tx2 ð Þ, _xx2ð Þ~u, y~xt 1

where u and y denote the input and output respectively and h is the disturbance. However, this example can be easily solved via the approach proposed in this paper and this approach has also been successfully used to derive a tracking controller with almost disturbance decoupling for a half-car active suspension system. Throughout the paper, the notation I?I denotes the usual Euclidean norm or the corresponding induced matrix norm.

2 TRACKING AND ALMOST DISTURBANCE DECOUPLING CONTROLLER DESIGN

The following non-linear uncertain control system with disturbances is considered

_xx1 _xx2 : : _xxn 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ~ f1ðx1,x2, . . . ,xnÞ f2ðx1,x2, . . . ,xnÞ : : fnðx1,x2, . . . ,xnÞ 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 z g½ 1ðx1,x2, . . . ,xnÞ g2ðx1,x2, . . . ,xnÞ    gmðx1,x2, . . . ,xnÞ u1ðx1,x2, . . . ,xnÞ u2ðx1,x2, . . . ,xnÞ : : umðx1,x2, . . . ,xnÞ 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 zX p j~1 qjhjdz Df1ðx1,x2, . . . ,xnÞ Df2ðx1,x2, . . . ,xnÞ : : Dfnðx1,x2, . . . ,xnÞ 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ð1aÞ

y1ðx1,x2, . . . ,xnÞ y2ðx1,x2, . . . ,xnÞ : : ymðx1,x2, . . . ,xnÞ 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ~ h1ðx1,x2, . . . ,xnÞ h2ðx1,x2, . . . ,xnÞ : : hmðx1,x2, . . . ,xnÞ 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ð1bÞ that is _XX tð Þ~f X tð ð ÞÞzg X tð ð ÞÞuzX p j~1 qjhjdzDf y tð Þ~h X tð ð ÞÞ

where X(t); [x1(t)x2(t)…xn(t)]TM Rn is the state

vec-tor, u ; [u1u2…um]TM Rm is the input vector,

y ; [y1y2…ym]TM Rm is the output vector,

h_d; [h1d(t)h2d(t)…hpd(t)]Tis a bounded time-varying

disturbances vector, andDf ; [Df1Df2…Dfn]M Rnis an

unknown non-linear function representing uncer-tainty such as modelling error. LetDf be defined as

Df ~X p

i~1 q_ihiu

where hu; [h1u(t)h2u(t)…hpu(t)]Tis a bounded

time-varying vector. f ; [f1f2…fn]TM Rn, g ; [g1g2…gm]

M Rn6m_{, and h; [h}

1h2…hm]TM Rmare smooth vector

fields. The nominal system is then defined as follows:

_XX tð Þ~f X tð ð ÞÞzg X tð ð ÞÞu ð2aÞ

y tð Þ~h X tð ð ÞÞ ð2bÞ

The nominal system of the form (2) is assumed to have the vector relative degree {r1, r2, …, rm} [31], i.e.

the following conditions are satisfied for all XM Rn. 1.

LgjL

k

fhið Þ~0X ð3Þ

for all 1( i ( m, 1 ( j ( m, k , ri2 1, where the

operator L is the Lie derivative [31] and r1+ r2+ ??? + rm5 r. 2. The m6m matrix A: Lg1L r1{1 f h1ð ÞX    LgmL r1{1 f h1ð ÞX Lg1L r2{1 f h2ð ÞX    LgmL r2{1 f h2ð ÞX ... ... Lg1L rm{1 f hmð ÞX LgmL rm{1 f hmð ÞX 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ð4Þ is non-singular.

The desired output trajectory yi_d, 1(i(m and its first ri derivatives are all uniformly bounded and

yi d, yi 1 ð Þ d ,    ,yi ri ð Þ d h i   ¡Bi d, 1¡i¡m ð5Þ where Bi

d is some positive constant. Under the assumption of well-defined vector relative degree, it has been shown [31] that the mapping

w : <n_?<n _ð6Þ defined as j_i: ji 1 ji 2 ... ji ri 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 : wi 1 wi 2 ... wi ri 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 : L0 fhið ÞX L1 fhið ÞX ... Lri{1 f hið ÞX 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 i~1,2, . . . ,m ð7Þ w_kðX tð ÞÞ:gkð Þ, k~rz1,rz2, . . . ,nt ð8Þ and satisfying LgjwkðX tð ÞÞ~0, k~rz1,rz2, . . . ,n, 1¡j¡m ð9Þ is a diffeomorphism onto image, if the following hold.

1. The distribution

G:span gf 1,g2, . . . ,gmg ð10Þ is involutive.

2. The vector fields Yk

j, 1¡j¡m, 1¡k¡rj ð11Þ

are complete, where Yk j: {1ð Þk{1adk{1~ff ~ggj, 1¡j¡m, 1¡k¡rj ð12Þ ~ff Xð Þ:f Xð Þ{g Xð ÞA{1_{ð Þb XX ð Þ ð13Þ} b Xð Þ: Lr1 f h1ð ÞX Lr2 f h2ð ÞX ... Lrm f hmð ÞX 2 6 6 6 6 6 4 3 7 7 7 7 7 5 ð14Þ

~gg: ~gg½ 1 ~gg2    ~ggm:g Xð ÞA{1ð ÞX ð15Þ adk_fg: f adh k_f{1gi ð16Þ f g ½ :Lg LXf Xð Þ{ Lf LXg Xð Þ ð17Þ

For the sake of convenience, defining the trajec-tory error to be eji:jij{y i jð{1Þ d , i~1,2, . . . ,m, j~1,2, . . . ,ri ð18Þ ei_{: e}i 1e i 2   e i ri h iT [ <ri ð19Þ

and the trajectory error to be multiplied with some adjustable positive constant e

ei j:ej{1eij, i~1, 2, . . . , m, j~1, 2, . . . , ri ð20Þ ei_{: e}i 1e2i   eriið Þt h iT [ <ri ð21Þ ee: e1 e2 ... em 2 6 6 6 6 4 3 7 7 7 7 5[ < r _ð22Þ and j: j1 j₂ ... j_r 2 6 6 6 6 4 3 7 7 7 7 5[ < r _ð23Þ g tð Þ: grz1ð Þgt rz2ð Þ    gt nð Þt  T [ <n{r _ð24Þ q j tð ð Þ,g tð ÞÞ: Lfwrz1ð ÞLt fwrz2ð Þ    Lt fwnð Þt  T : q½ rz1 qrz2    qnT ð25Þ Define a phase-variable canonical matrix Aic to be

Ai_c: 0 1 0    0 0 0 1    0 ... ... 0 0 0    1 {ai 1 {ai2 {ai3    {airi 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 ri|ri 1¡i¡m ð26Þ where ai

1, ai2, . . . , airi are any chosen parameters

such that Ai_cis Hurwitz and the vector Bi will be Bi_{: 0 0    0 1½ }T

ri|1, r¡i¡m ð27Þ

Let Pi be the positive definite solution of the following Lyapunov equation

Ai c  T Pi_zPi_Ai c~{I, 1¡i¡m ð28Þ lmax Pi  

:the maximum eigenvalue of Pi

1¡i¡m ð29Þ

lmin Pi  

:the minimum eigenvalue of Pi

1¡i¡m ð30Þ

l

max:min lmax P1   , lmax P2   , . . . ,lmaxðPmÞg  ð31Þ l

min:min lmin P1   , lmin P2   , . . . , lminðPmÞ   ð32Þ Assumption 1

For all t> 0, g M Rn–rand jM Rr, there exists a positive constant M such that the following inequality holds

q₂₂ðt, g, eeÞ{q22(t, g, 0)

 _{¡M eekkð Þ ð33Þ}

where q22(t, g, e¯); q(j, g).

For the sake of stating precisely the investigated problem, defining

dij:LgjL

ri{1

f hið Þ, 1¡i¡m, 1¡j¡mX ð34Þ

and ei_:ai 1e1iza i 2e2iz    za i rie i ri, 1¡i¡m ð36Þ Definition 1 [32]

Consider the system x˙5 f(t, x, h), where f:[0, ‘]6Rn_6Rn_{R R}n _{is piecewise continuous in t and}

locally Lipschitz in x and h. This system can be regarded as input-to-state stable if there exists a class KL function b, a class K function c, and positive constants k1and k2such that for any initial

state x(t0) with Ix(t0)I , k1and any bounded input

h(t) with supt¢t0kh tð Þkvk2, the state exists and

satisfies x tð Þ k k¡b x tðk ð Þ0 k, t{t0Þzc sup t0¡t¡t h tð Þ k k ð37Þ for all t> t0> 0. The tracking problem with almost

disturbance decoupling is now formulated as follows.

Definition 2 [29]

The tracking problem with almost disturbance decoupling is said to be globally solvable by the state feedback controller u for the transformed-error system by a global diffeomorphism (6), if the controller u enjoys the following properties.

1. It is input-to-state stable with respect to distur-bance inputs.

2. For any initial value x¯e0; [e¯(t0) g(t0)]T, for any

t> t0and for any t0> 0

y tð Þ{ydð Þt j j¡b11ðkx tð Þ0 k, t{t0Þ z 1ffiffiffiffiffiffiffi b₂₂ p b₃₃ sup t0¡t¡t h tð Þ k k ð38Þ and ðt t0 y tð Þ{ydð Þt ½ 2 dt ¡_b1 44 b₅₅ðkxxe0kÞz ðt t0 b₃₃kh tð Þk2 dt 2 4 3 5 _ð39Þ

where b22and b44are positive constants, b33and

b55 are class K functions, and b11 is a class KL

function.

Theorem 1

Suppose that there exists a continuously differenti-able function V:Rn–r_{R R}+ _{such that the following}

three inequalities hold for all gM Rn–r.

1: v1k kg 2¡V gð Þ¡v2k kg 2, v1, v2w0 ð40aÞ 2: +tV z + gVTq22ðt, g, 0Þ¡

{2axk k,g 2 axw0 ð40bÞ

3:  ¡$+gV 3k k, $g 3w0 ð40cÞ then the tracking problem with almost disturbance decoupling is globally solvable by the controller defined by u~A{1_{f{bzvg ð41Þ} b: Lr1 f h1 Lrf2h2    Lrfmhm h iT ð42Þ v: v½ 1 v2    vmT ð43Þ vi:yi ri ð Þ d {e{ria i 1 L 0 fhið Þ{yX id h i {e1{riai 2 L 1 fhið Þ{yX i 1 ð Þ d h i {    {e{1_ai ri L ri{1 f hið Þ{yX iðyi{1Þ d h i , 1¡i¡m ð44Þ

Moreover, the influence of disturbances on the L2

-norm of the tracking error can be arbitrarily attenuated by increasing the following adjustable parameter N2. 1. k11:k 2e{ k2 _Q1 j   2 P1 2 e2 {    {k 2 _Qm j   2kPmk2 e2 {4 ð45aÞ k22:2ax{ v2 3M2 16 {v 2 3 wg 2 ð45bÞ N2:min kf 11, k22g ð45cÞ N1:mz1 4 t0sup¡t¡t hdð Þzht uð Þt k k 2 ð45dÞ

wi jð Þ:e e L LXhiq1    e L_LXhiqp ... ... eri L LXLrfi{1hiq  1    eri L LXLrfi{1hiq  q 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 1¡i¡m ð45eÞ wgð Þ:e L LXwrz1q1    L LXwrz1qp ... ... L LXwnq1    L LXwnqq 2 6 6 6 6 6 4 3 7 7 7 7 7 5 ð45fÞ

where k(e):R+_{R R}+ _{is any continuous function}

satisfies lim

e?0k eð Þ~0 and lime?0 e

k eð Þ ~0 ð45gÞ

Proof. Applying the coordinate transformation equation (6) yields _jj1 1~ Lw1 1 LX dX dt ~ Lh1 LX f zg:uz Xp j~1 qjhjdzDf " # ~Lh1 LX f z Xp j~1 Lh1 LX qj hjdzhju   ~j1 2z Xp j~1 Lh1 LX qj hjdzhju   ð46Þ ... _jj1 r1{1~ Lw1 r1{1 LX dX dt ~ LLr1{2 f h1 LX f zg:uzX p j~1 q_jhjdzDf " # ~LL r1{2 f h1 LX f z Xp j~1 LLr1{2 f h1 LX qj hjdzhju   ~Lr1{1 f h1z Xp j~1 LLr1{2 f h1 LX qj hjdzhju   ð47Þ _jj1 r1~ Lw1 r1 LX dX dt ~ LLr1{1 f h1 LX f zg:uz Xp j~1 qjhjdzDf " # ~Lr1 f h1zLg1L r1{1 f h1u1z    zLgmL r1{1 f h1um zX p j~1 LLr1{1 f h1 LX qj hjdzhju   ~c1zd11u1z    zd1mumz Xp j~1 LLr1{1 f h1 LX qj hjdzhju   ð48Þ ... _jjm 1~ Lwm 1 LX dX dt ~ Lhm LX f zg:uz Xp j~1 qjhjdzDf " # ~L1 fhmz Xp j~1 Lh1 LX qj hjdzhju   ~jm 2z Xp j~1 Lhm LX qj hjdzhju   ð49Þ ... _jjm rm{1~ Lwm rm{1 LX dX dt ~ LLrm{2 f hm LX | f zg:uzX p j~1 q_jhjdzDf " # ~Lrm{1 f hmz Xp j~1 LLrm{2 f hm LX qj hjdzhju   ~jm rmz Xp j~1 LLrm{2 f hm LX qj hjdzhju   ð50Þ _jjm rm~ Lwm rm LX dX dt ~ LLrm{1 f hm LX f zg:uz Xp j~1 qjhjdzDf " # ~Lrm f hmzLg1L rm{1 f hmu1z    zLgmL rm{1 f hmum zX p j~1 LLrm{1 f hm LX qj hjdzhju   ~cmzdm1u1z    zdmmum zX p j~1 LLrm{1 f hm LX qj hjdzhju   ð51Þ _ggkð Þ~t Lwk LX dX dt ~ Lwk LX f zg:uz Xp j~1 q_jhjdzDf " #

~Lfwkz Xp j~1 Lwk LX qj hjdzhju   ~qkz Xp j~1 Lwk LX qj hjdzhju   k~rz1, rz2, . . . , n ð52Þ Since ciðj tð Þ, g tð ÞÞ:L_frihiðX tð ÞÞ, 1¡i¡m ð53Þ dij:LgjL ri{1 f hið Þ, 1¡i¡m, 1¡j¡mX ð54Þ q_kðj tð Þ, g tð ÞÞ~Lfwkð Þ, k~rz1, rz2, . . . , nX ð55Þ the dynamic equations of system (1) in the new coordinates are as follows:

_jj1 ið Þ~jt 1 iz1ð Þzt Xp j~1 L LXLif{1h1qj hjdzhju   i~1, 2, . . . , r1{1 ð56Þ _jj1 r1ð Þ~ct 1ðj tð Þ, g tð ÞÞzd11ðj tð Þ, g tð ÞÞu1z    zd1mðj tð Þ, g tð ÞÞum zX p j~1 L LXLrf1{1h1qj hjdzhju   ð57Þ ... _jjm i ð Þ~jt m iz1ð Þzt Xp j~1 L LXLif{1hmqj hjdzhju   i~1, 2, . . . , rm{1 ð58Þ _jjm rmð Þ~ct mðj tð Þ, g tð ÞÞzdm1ðj tð Þ, g tð ÞÞu1z    zdmmðj tð Þ, g tð ÞÞum zX p j~1 L LXLrfm{1hmqj hjdzhju   ð59Þ _ggkð Þ~qt kðj tð Þ, g tð ÞÞ zX p j~1 L LXwkð ÞqX j hjdzhju   , k~rz1, . . . , n ð60Þ yið Þ~jt i1ð Þ, 1¡i¡mt ð61Þ

According to equations (18), (44), (53), and (54), the tracking controller can be rewritten as

u~A{1_{½{bzv ð62Þ}

Substituting equation (62) into equations (57) and (59), the dynamic equations of system (1) can be shown as follows _jji 1ð Þt _jji 2ð Þt ... _jji ri{1ð Þt _jji rið Þt 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 ~ 0 1 0    0 0 0 1 0   0 ... ... 0 0 0    1 0 0 0    0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ji 1ð Þt ji 2ð Þt ... ji ri{1ð Þt ji rið Þt 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 z 0 0 ... 0 1 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 viz Pp j~1 L LXhiq_j hjdzhju   Pp j~1 L LXL1fhiqj hjdzhju   ... Pp j~1 L LXLrfi{1hiqj hjdzhju   2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ð63Þ _gg_rz1ð Þt _gg_rz2ð Þt ... _gg_n{1ð Þt _ggnð Þt 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 ~ qrz1ð Þt qrz2ð Þt ... qn{1ð Þt qnð Þt 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 z Pp j~1 L LXwrz1qj hjdzhju   Pp j~1 L LXwrz2qj hjdzhju   ... Pp j~1 L LXwn{1qj hjdzhju   Pp j~1 L LXwnqj hjdzhju   2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ð64Þ

y_i~ 1 0    0 0½ _r|1 ji 1ð Þt ji 2ð Þt ... ji ri{1ð Þt ji rið Þt 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 r|1 ~ji 1ð Þ, 1¡i¡mt ð65Þ

Combining equations (18), (20), (21), (26), and (44), it can be easily verified that equations (63) to (65) can be transformed into the following form

_gg tð Þ~q j tð ð Þ, g tð ÞÞzwgðhdzhuÞ :q22ðt, g tð Þ, eeÞzwgðhdzhuÞ ð66aÞ e ei : t ð Þ~Ai ceizw i jðhdzhuÞ, 1¡i¡m ð66bÞ y_ið Þ~jt i₁ð Þ, 1¡i¡mt ð67Þ

It is considered that L(e¯, g) defined by a weighted sum of V(g) and W(e¯)

Lðee, gÞ:V gð Þzk eð ÞWð Þee

:V gð Þzk eð Þ W 1 _e1 _{z    zW}m _em _ð68Þ where

W(ee):W1 e1 z    zWm _em ð69Þ is a composite Lyapunov function of the subsystems (66a) and (66b) [33, 34], where W e i _satisfies

Wi ei _:1 2e

iT_Pi_ei _ð70Þ

In view of equations (18), (33), and (40), the derivative of L along the trajectories of equations (66a) and (66b) is given by

_LL~ +tV z + gVT_gg h i zk 2 e 1 . T P1e1_{z e} 1 T_P1 _e1 . z    z e.m T Pmem_{z e} m T_Pm _e.m " # ~ +tV z + gVT_gg h i zk 2 1 eA 1 ce1z 1 e w 1 jðhdzhuÞ T P1e1_{z e} 1 T_P1 1 eA 1 ce1z 1 e w 1 jðhdzhuÞ z    " z 1_eAm cemz 1 e w m jðhdzhuÞ T Pm_emz e m T_Pm 1 eAmc emz 1 e w m j ðhdzhuÞ # ¡ +tV z +gV  T q₂₂ðt, g tð Þ, eÞz + gVTwgðhdzhuÞ h i {k 2e e 1  T e1_{z    z e} m T_em   zk_e kðhdzhuÞk w1j   eP1  1z    z hkð dzhuÞk wmj kPmk e m h i ¡{2axk kg 2zv3k kMg k kzvee 3k kg   hwg kð dzhuÞk {k 2ek kee 2_zk2 e2 w 1 j  2 P1  2 e1   2z1 4kðhdzhuÞk 2_{z    z}k2 e2 w m j  2 Pm k k2 _em 2 z1 4kðhdzhuÞk 2 ¡{2axk kg 2z 1 16v 2 3M2k kg 2_{z4 eek k}2_zv2 3 wg 2 _g k k2_z1 4kðhdzhuÞk 2 {k 2ek kee 2_zk2 e2 w 1 j  2 P1  2 ee k k2_z1 4kðhdzhuÞk 2_{z    z}k2 e2 w m j  2 Pm k k2_{k kee} 2_z1 4kðhdzhuÞk 2 ~{ gk k2 _2a x{ 1 16v 2 3M2{v23 wg 2   { eek k2 k 2e{ k2 e2 w 1 j  2 P1  2 {   k_e22 wm j  2 Pm k k2   zmz1 4 kðhdzhuÞk 2 _ð71Þ

that is

_LL¡{k11k kee 2{k22k kg 2zmz1

4 kðhdzhuÞk

2 _ð72Þ

Using equation (45c) yields _LL¡{N2 k kee 2z gk k2  zmz1 4 kðhdzhuÞk 2 _ð73Þ Define ee: e1 e2 ... em 2 6 6 6 6 4 3 7 7 7 7 5: e1 e1 rem " # , e1 rem[ < r{1 _ð74Þ Hence _LL¡{N2 k kg 2z e 11 2 z e1 rem   2 zmz1 4 kðhdzhuÞk 2 ð75Þ Utilizing equation (75) yields

ðt t0 y1ð Þ{yt 1dð Þt  2 dt ¡L tð Þ0 N2 z mz1 4N2 ðt t0 hdð Þzht uð Þt ð Þ k k2 dt ð76Þ

Similarly, it is easy to prove that ðt t0 yið Þ{yt idð Þt  2 dt¡L tð Þ0 N2 zmz1 4N2 ðt t0 hdð Þzht uð Þt ð Þ k k2_{dt, 2¡i¡m ð77Þ}

so that statement (39) is satisfied. From equation (73) _LL¡{N2 ytotal 2  zmz1 4 kðhdzhuÞk 2 _ð78aÞ where y_total  2 : eek k2_{z gk k}2 _ð78bÞ

By virtue of Theorem 5.2 of reference [32], equation

(78a) implies the input-to-state stability for the closed-loop system. Furthermore, it is easy to see that

Dmin k kee 2z gk k2  ¡L¡Dmax k kee 2z gk k2  ð79Þ that is Dmin ytotal 2  ¡L¡Dmax ytotal 2  ð80Þ where Dmin:min v1, k 2l  min   and Dmax:max v2,k 2l  max  

Equations (73) and (80) yield, that _LL¡{ N2 DmaxLz mz1 4 t0sup¡t¡t hdð Þzht uð Þt ð Þ k k 2 ð81Þ Hence, L tð Þ¡L tð Þexp {0 N2 Dmaxðt{t0Þ zDmaxðmz1Þ 4N2 sup t0¡t¡t hdð Þzht uð Þt ð Þ k k 2 t¢t0 ð82Þ which implies y1ð Þ{yt 1dð Þt  _¡ ffiffiffiffiffiffiffiffiffiffiffiffiffi2L tð Þ0 kl_min s exp { N2 2Dmax t{t0 ð Þ z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dmaxðmz1Þ 2kl_minN2 s sup t0¡t¡t hdð Þzht uð Þt ð Þ k k ð83Þ Similarly, it is easy to prove that

yið Þ{yt idð Þt  _¡ ffiffiffiffiffiffiffiffiffiffiffiffiffi2L tð Þ0 kl_min s exp { N2 2Dmax t{t0 ð Þ z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dmaxðmz1Þ 2kl_minN2 s sup t0¡t¡t hdð Þzht uð Þt ð Þ k k 2¡i¡m ð84Þ

problem with almost disturbance decoupling is globally solved. This completes the proof.

According to the previous theorems and discus-sions, an efficient algorithm for deriving the almost disturbance decoupling control is proposed as follows:

Step 1. Calculate the vector relative degree r1, r2,

…, rm of the given control system.

Step 2. Choose the diffeomorphism w such that the assumption 1 is satisfied.

Step 3. Adjust some parameters ai

1, ai2, . . . , airi such

that the matrices Ai

c are Hurwitz and calculate the positive definite matrices Pi of the Lyapunov equations (28) by some software package, such as Matlab.

Step 4. Based on the famous Lyapunov approach, design a Lyapunov function to solve the condi-tions (40a) to (40c). If the relative degree r1+ r2+ ??? + rm is equal to the system dimension

n, then this step should be omitted and immedi-ately go to the next step.

Step 5. Appropriately tune the parameters k and e such that NN2. 1 and go to the next step.

Otherwise, go to step 3 and repeat the overall designing procedures.

Step 6. According to equation (41), the desired feedback linearization controller ufeedback can be

constructed such that the uniform ultimate bounded stability is guaranteed. That is, the system dynamics enter a neighbourhood of zero state and remain within it thereafter.

3 ILLUSTRATIVE EXAMPLE

Consider the half-car active suspension system with disturbances shown in Fig. 1. From Huang and Lin [35], the dynamic equations are given as follows

_xx1~x2zDf1 _xx2~ 1 ms { BfzBr ð Þx2z aBð f{bBrÞx4cos x3 ½ {kfx5zBfx6{krx7zBrx8z fðfzfrÞ _xx3~x4 _xx4~1 Jy½ðaBf{bBrÞx2cos x3 { a2_B fzb2Br  

x4cos2x3zakfx5cos x3 {aBfx6cos x3{bkrx7cos x3

zbBrx8cos x3z {afð fzbfrÞcos x3 _xx5~x2{ax4cos x3{x6 _xx6~ 1 muf {Ktf x1zBfx2zaKtfsin x3 ½ {aBfx4cos x3z kð fzKtfÞx5 {Bfx6zKtfzrf{ff _xx7~x2zbx4cos x3{x8 _xx8~ 1 mur {K trx1zBrx2{bKtrsin x3 ½ zbBrx4cos x3z kð rzKtrÞx7 {Brx8zKtrzrr{fr ð85aÞ y1~x1zx2: ~h1 y2~x3zx4: ~h2 ð85bÞ where x15 z is the displacement of the centre of

gravity, x25 z˙ is the payload velocity, msis the mass of

the car body, Bfand Brare the front and rear damping

coefficients, a is the distance between front axle and the centre of gravity, b is the distance between rear axle and the centre of gravity, x35 h is the pitch angle,

x45 _hh is the pitch velocity, kfand krare the front and

rear spring coefficients, zsf and zsr are the front and

rear body displacement, zufand zurare the front and

rear wheel displacements, x55 zsf – zuf is the front

wheel suspension travel, x65 z˙ufis the front unsprung

mass velocity, x75 zsr – zur is rear wheel suspension

travel, ff5 u1and fr5 u2 are the front and rear force

inputs, Jyis its centroidal moment of inertia, mufand

mur are the unsprung masses on the front and rear

wheels, Ktf and Ktrare the front and rear tyre spring

coefficients, zrf and zrrare the front and rear terrain

height disturbances,Df1is the system uncertainty, and

x85 z˙ur is the rear unsprung mass velocity. The

fol-lowing physical parameters are chosen in the present Fig. 1 The half-car active suspension system

simulation: ms5 575 kg, Bf5 Br5 1000 N m/s, a 5

1.38 m, b51.36 m, Jy5 769 kg/m2, muf5 mur5 60 kg,

Ktf5 Ktr5190 000 N/m, kf5 kr5 16 812 N/m, zrf5 zrr5

m_r(1 – cos8pt), Df15 0.1sin t(cos x5), and mr5 0.05 m.

Hence the mathematical model can be rewritten as _xx1~x2z0:1 sin t cos xð 5Þ _xx2~{3:478x2z0:035x4cos x3{29:238x5 z1:739x6{29:238x7z1:739x8 z 0:0017uð 1z0:0017u2Þ _xx3~x4 _xx4~3:563x2cos x3{4:881x4cos2x3 z30:17x5cos x3{1:794x6cos x3 {29:732x7cos x3 z1:768x8cos x3

z {0:0018uð 1z0:00176u2Þcos x3 _xx5~x2{1:38x4cos x3{x6 _xx6~{3166:667x1z16:667x2z4370 sin x3 {23x4cos x3z3446:867x5{16:667x6 z158:33 1{cos 8ptð Þ{0:0167u1 _xx7~x2z1:36x4cos x3{x8 _xx8~{3166:667x1z16:667x2 {4306:667 sin x3z22:667x4cos x3 z3446:067x7 {16:667x8z158:33 1{cos 8ptð Þ {0:0167u2 ð86aÞ y1~x1zx2:~h1 y2~x3zx4:~h2 ð86bÞ Now it is shown how to explicitly construct a controller that tracks the desired signals y1

d~yd2~0 and attenu-ates the disturbance’s effect on the output terminal to an arbitrary degree of accuracy. Let us arbitrarily choose a1

1~a21~0:06, A1c~A2c~{0:06, P

1_{5 P}2_{5 25/3,}

and lmin~lmax~25=3. From equation (41), the de-sired tracking controllers are obtained

u1~{1381:25x4cos x3z16 977:16x5 {1009:63x6z150:69x7 {9:07x8z1198:00x2{523:44x1 z505:62x3ðcos x3Þ{1z786:52x4ðcos x3Þ{1 ð87Þ u2~1360:66x4cos x3z220:63x5 {13:244x6z17047:1x7 {1013:81x8{799:21x2{535:32x1 {505:62x3ðcos x3Þ{1{786:52x4ðcos x3Þ{1 ð88Þ It can be verified that the relative conditions of Theorem 1 are satisfied with e5 0.03, B1

d~B2d~0, M~pffiffiffi3, v15 v25 1, ax5 1, v35 2, k115 24.86, k225 1.25, N15 79, N25 1.25, and k~10 ffiffi e p . Hence the tracking controllers will steer the output tracking errors of the closed-loop system, starting from any initial value, to be asymptotically attenuated to zero by virtue of Theorem 1. The complete trajectories of the outputs are depicted in Fig. 2 and Fig. 3.

4 COMPARATIVE EXAMPLE TO EXISTING APPROACH

Marino et al. [27] exploits the fact that for a non-linear SISO system the almost disturbance decou-pling problem can not be solved, as the following example shows: _xx1ð Þt _xx2ð Þt   ~ tan{1ð Þx2 0   z 0 1   uz 1 0   h tð Þ ð89aÞ y tð Þ~x1ð Þt ð89bÞ

where u and y denote the input and output respectively, h(t) :5 0.5sin t is the disturbance. The feedback control algorithm proposed in this paper will solve it perfectly. Applying the same design

Fig. 2 The output trajectory x1 of the half-car active

procedures of Theorem 1 yields the desired tracking and almost disturbance decoupling controller as follows u~ 1zx2 2   {sin t{ 0:03ð Þ{2ðx1{sin tÞ h { 0:03ð Þ{1 _tan{1_x 2{cos t  i ð90Þ The output trajectory of the feedback-controlled system for (89) is depicted in Fig. 4. From Fig. 4, it is obvious to see that the desired tracking and almost disturbance decoupling performance are achieved.

It is worth noting that the sufficient conditions given in Marino et al. [27] (in particular the structural conditions on non-linearities multiplying distur-bances) are not necessary in this study where a non-linear state feedback control is explicitly de-signed which solves the almost disturbance decou-pling problem. For instance, the almost disturbance

decoupling problem is solvable for the system (89) by a non-linear state feedback control, according to the current proposed approach, while the sufficient con-ditions given in Marino et al. [27] fail when applied to the system (89). The design techniques in this study are also entirely different than those in Marino et al. [27] since the singular perturbation tools are not used.

5 CONCLUSION

A novel feedback control to globally solve the tracking problem with almost disturbance decou-pling for multi-input multi-output non-linear un-certain system has been proposed. A discussion and a practical application of feedback linearization of non-linear control systems using a parameterized coordinate transformation have been presented. One comparative example is proposed to show the significant contribution of this paper with respect to existing approaches. A practical example of a half-car active suspension system has been used to demonstrate the applicability of the proposed feed-back linearization approach and the composite Lyapunov approach. Simulation results have been presented to show that the proposed methodology can be successfully applied to the feedback linear-ization problem and is able to achieve the desired tracking and almost disturbance decoupling perfor-mances of the controlled system.

REFERENCES

1 Richter, H., O’Dell, B. D., and Misawa, E. A. Robust positively invariant cylinders in constrained variable structure control. IEEE Trans. Autom. Control, 2007, 52(11), 2058–2069.

2 Liang, Y. W., Xu, S. D., and Tsai, C. L. Study of VSC reliable designs with application to spacecraft attitude stabilization. IEEE Trans. Control Syst. Technol., 2007, 15(2), 332–338.

3 Semsar-Kazerooni, E., Yazdanpanah, M. J., and Lucas, C. Nonlinear control and disturbance decoupling of HVAC systems using feedback linearization and backstepping with load estima-tion. IEEE Trans. Control Syst. Technol., 2008, 16(5), 918–929.

4 Yip, P. P. and Hedrick, J. K. Adaptive dynamic surface control: a simplified algorithm for adaptive backstepping control of nonlinear systems. Int. J. Control, 1998, 71(5), 959–979.

5 Swaroop, D., Hedrick, J. K., Yip, P. P., and Gerdes, J. C. Dynamic surface control for a class of nonlinear systems. IEEE Trans. Autom. Control, 2000, 45(10), 1893–1899.

Fig. 3 The output trajectory x3of the half-car active

suspension system

Fig. 4 The output trajectory of the feedback-controlled system for system (89)

6 Isidori, A. and Byrnes, C. I. Output regulation of nonlinear systems. IEEE Trans. Autom. Control, 1990, 35, 131–140.

7 Peroz, H., Ogunnaike, B., and Devasia, S. Output tracking between operating points for nonlinear processes: van de Vusse example. IEEE Trans. Control Syst. Technol., 2002, 10(4), 611–617. 8 Ball, J. A., Helton, J. W., and Walker, M. L. H‘

control for nonlinear systems with output feedback. IEEE Trans. Autom. Control, 1993, 38, 546–559. 9 Isidori, A. and Kang, W. H‘control via

measure-ment feedback for general nonlinear systems. IEEE Trans. Autom. Control, 1995, 40, 466–472.

10 Van der Schaft, A. J. L2-gain analysis of nonlinear

systems and nonlinear state feedback H‘ control. IEEE Trans. Autom. Control, 1992, 37, 770–784. 11 Litrico, X. and Fromion, V. H‘ control of an

irrigation canal pool with a mixed control politics. IEEE Trans. Control Syst. Technol., 2006, 14(1), 99–111.

12 Isidori, A. H‘ control via measurement feedback for affine nonlinear systems. Int. J. Robust Non-linear Control, 1994, 4(4), 521–552.

13 Pisu, P. and Serrani, A. Attitude tracking with adaptive rejection of rate gyro disturbances. IEEE Trans. Autom. Control, 2007, 52(12), 2374–2379. 14 Huang, J. and Rugh, W. J. On a nonlinear

multi-variable servomechanism problem. Automatica, 1990, 26, 963–992.

15 Gopalswamy, S. and Hedrick, J. K. Tracking non-linear nonminimum phase systems using sliding control. Int. J. Control, 1993, 57, 1141–1158. 16 Chen, B. S., Lee, C. H., and Chang, Y. C. H‘

tracking design of uncertain nonlinear SISO sys-tems: adaptive fuzzy approach. IEEE Trans. Fuzzy Syst., 1996, 4(1), 32–43.

17 Nijmeijer, H. and Van der Schaft, A. J. Nonlinear dynamical control systems, 1990 (Springer Verlag, New York, NY).

18 Slotine, J. J. E. and Li, W. Applied nonlinear control, 1991 (Prentice-Hall, New York, NY). 19 Joo, S. J. and Seo, J. H. Design and analysis of the

nonlinear feedback linearizing control for an electromagnetic suspension system. IEEE Trans. Autom. Control, 1997, 5(1), 135–144.

20 Corless, M. J. and Leitmann, G. Continuous state feedback guaranteeing uniform ultimate bounded-ness for uncertain dynamic systems. IEEE Trans. Autom. Control, 1981, 26(5), 1139–1144.

21 Sheen, J. J. and Bishop, R. H. Adaptive nonlinear control of spacecraft. In Proceedings of the American

Control Conference, Baltimore. MD, June 1994, pp. 2867–2871.

22 Alleyne, A. A systematic approach to the control of electrohydraulic servosystems. In Proceedings of the American Control Conference, Philadelphia, PA, June 1998, pp. 833–837.

23 Bedrossian, N. S. Approximate feedback lineariza-tion: the car-pole example. In Proceedings of the 1992 IEEE International Conference on Robotics and automation, Nice, France, May 1992, pp. 1987–1992 (IEEE, Piscataway, NJ).

24 Lee, S. Y., Lee, J. I., and Ha, I. J. A new approach to nonlinear autopilot design for bank-to-turn mis-siles. In Proceedings of the 36th Conference on Decision and control, San Diego, CA, December 1997, pp. 4192–4197 (IEEE, Piscataway, NJ). 25 Zhang, Z. and Menq, C. H. Six-axis magnetic

levitation and motion control. IEEE Trans. Robot., 2007, 23(2), 196–205.

26 Willems, J. C. Almost invariant subspace: an approach to high gain feedback design – part I: almost controlled invariant subspaces. IEEE Trans. Autom. Control, 1981, 26(1), 235–252.

27 Marino, R., Respondek, W., and Van der Schaft, A. J. Almost disturbance decoupling for single-input single-output nonlinear systems. IEEE Trans. Autom. Control, 1989, 34(9), 1013–1017.

28 Weiland, S. and Willems, J. C. Almost disturbance decoupling with internal stability. IEEE Trans. Autom. Control, 1989, 34(3), 277–286.

29 Marino, R. and Tomei, P. Nonlinear output feed-back tracking with almost disturbance decoupling. IEEE Trans. Autom. Control, 1999, 44(1), 18–28. 30 Qian, C. and Lin, W. Almost disturbance

decou-pling for a class of high-order nonlinear systems. IEEE Trans. Autom. Control, 2000, 45(6), 1208–1214.

31 Isidori, A. Nonlinear control systems, 1989 (Springer Verlag, New York, NY).

32 Khalil, H. K. Nonlinear systems, 1996 (Prentice-Hall, Englewood Cliffs, NJ).

33 Khorasani, K. and Kokotovic, P. V. A corrective feedback design for nonlinear systems with fast actuators. IEEE Trans. Autom. Control, 1986, 31, 67–69.

34 Marino, R. and Kokotovic, P. V. A geometric approach to nonlinear singularly perturbed sys-tems. Automatica, 1988, 24(1), 31–41.

35 Huang, C. J. and Lin, J. S. Nonlinear backstepping control design of half-car active suspension sys-tems. Int. J. Vehi. Des., 2003, 33(4), 332–350.