Robust sliding mode control approach for systems ...

OPTIMAL CONTROL APPLICATIONS AND METHODS Optim. Control Appl. Meth. (2014) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/oca.2147

Robust sliding mode control approach for systems affected by unmatched uncertainties using H1 with pole clustering constraints Saad Wajdi1,*,†, Sellami Anis1 and Germain Garcia2 1

Unit of research C3S, ENSIT, TUNIS university, 5 Av. Taha Hussein, BP 56, Tunis 1008, Tunisia 2 LAAS, CNRS, Toulouse university, 7 Av. Colonel Roche, F-31400 Toulouse, France

SUMMARY In this paper, the problem of sliding mode control for a class of systems with unmatched parametric uncertainties and external perturbations is considered. LMI technique and polytopic models are used in the design of the switching surface. To achieve some performance requirements and good robustness, in the sliding mode, the H1 norm and the pole clustering method are investigated. Based on the unit vector control approach, a robust control is developed, then, to direct and maintain the system states onto the sliding manifold in finite time. Finally, the validity of the proposed design strategy is demonstrated through the simulation of the quarter-car suspension system. Copyright © 2014 John Wiley & Sons, Ltd. Received 6 March 2014; Revised 16 June 2014; Accepted 28 September 2014 KEY WORDS: sliding mode control; uncertain systems; unmatched uncertainties; multiobjective optimization

constraints

1. INTRODUCTION Robust control has received considerable attention in order to ensure the systems exhibit good performance in the presence of uncertainties. Hence, there have been a number of developments in the search for robust control methods for uncertain models. The sliding mode control (SMC) is one of these methods, and it is widely used in control of dynamic plants operating under uncertainty conditions. It is well-known for its robustness to external disturbances and parameter variations. Generally, the conventional SMC consists of two steps called sliding step and reaching step. First, the determination of sliding surface, which sliding motion must satisfy certain desired properties. Then, design of control law, which will make the switching manifold attractive to the system state. In the literature, much effort has been made, and many criteria for performance and robustness have been used to design the sliding manifold. In [1], the quadratic stability with L2 gain performance measure has been considered. In [2], the quadratic stability and a compensator gain for disturbance attenuation has been used. In [3, 4], the LMI technique and the quadratic stability has been addressed to formulate the sliding surface design problem. To compensate unmatched uncertainties, the H1-norm has been utilized, in [5]. The eigenstructure assignment including pole clustering has been considered by Choi [6]. In [7], the robust sliding hyperplane has been constructed from a Riccati inequality associated with quadratic stabilizability. The problem of determining a switching manifold by minimizing a H2 cost functional has been studied by Edwards [8]. It is important to note that most of the design methods proposed so far have considered only one design objective. *Correspondence to: Saad Wajdi, Unit of research C3S, ENSIT, TUNIS university, 5 Av. Taha Hussein, BP 56, Tunis 1008, Tunisia. † E-mail: [email protected] Copyright © 2014 John Wiley & Sons, Ltd.

S. WAJDI, S. ANIS AND G. GARCIA

In fact, there have been few researches that investigate multiobjective methods for sliding surface design. Reference [9] proposes a multiobjective H2/H1 optimization approach to design the sliding surface for uncertain dynamical systems. Additionally, in [10], by solving the problem of optimal surface design with some quadratic constraints, a mixed H2/H1 approach has been presented to reduce the effect of uncertainties and disturbances on the sliding motion. Feng and Lin [11] proposes a robust control design based on integral sliding mode and parameter dependent Lyapunov function for H2-norm performance to handle a class of time-varying systems with uncertainties. To the best of the authors’ knowledge, SMC for systems with unmatched uncertainties that combine the H1 performance and the pole clustering in LMI region approach for the sliding surface design has not been discussed in the research work, which motivates us for this study. This paper proposes a robust sliding mode control for a class of systems with unmatched parametric uncertainties and mismatched perturbations using the H1 technique with LMI regional pole placement method. It is well-known that H1 theory for uncertain systems has been extensively applied, in the literature, to compact with problems of robust stabilization and disturbance rejection. This theory has also shown the capability of dealing with the model uncertainty problem such as nonlinearities and parameter uncertainties. However, the H1 technique does not give direct control over the shape of the transient response; specifications on damping, rise time, settling time, and overshoot are best expressed in terms of pole locations in LMI regions. One way of simultaneously assuring robust performance and adequate transient behaviors is therefore to combine the H1-norm and pole placement objectives [12, 13]. To this end, starting from a polytopic description, the sliding surface design is formulated as a multiobjective optimization problem in terms of linear matrix inequalities constraints. Based on the ideas of the unit vector control approach [14–17], a robust controller is synthesized, then, to fulfill the reaching and sliding conditions. The major contributions of the current study with respect to the related literature can be summarized as follows: • Considering two types of unmatched uncertainties: parametric uncertainty and external disturbance. • Using H1-norm and regional pole clustering objectives to guarantee the robust stability with disturbance attenuation in the sliding mode. • Designing an efficient SMC law to attain some performance properties and good robustness. • Giving an approximation for the dynamic behavior of the uncertain system during the sliding mode. The paper is structured in the following way: In Section 2, the preliminaries are presented. Section 3 describes the problem to be considered in this paper. In Section 4, we state the sliding surface design method. Section 5 proposes the SMC control synthesis methodology and shows the main results of this work. Section 6 gives the simulation results, and Section 7 concludes the paper. 2. PREMILINARIES Consider an uncertain linear system presented by the following equation

e_ ¼ Ae þ Bue þ D1 w

(1)

z ¼ C1 e þ C 2 ue

where e 2 ℜn, ue 2 ℜm, and z 2 ℜz represent the state, the control input, and the controlled output of the system, respectively. w 2 ℜw is a bounded external disturbance. The H1 synthesis with pole assignment in LMI regions consists of finding a state feedback control ue = Ke such that the closed-loop transfer matrices Gzw from the disturbance w to the output z satisfies the following requirements:

• The closed-loop poles lie in some LMI stability region Ω contained in the left-half plane. • The H1-norm performance is guaranteed ‖Gzw‖1 < γ, γ is a positive scalar. Copyright © 2014 John Wiley & Sons, Ltd.

Optim. Control Appl. Meth. (2014) DOI: 10.1002/oca

ROBUST SLIDING MODE CONTROL APPROACH

Then, the closed-loop system can be written as

e_ ¼ ðA þ BK Þe þ D1 w

(2)

z ¼ ðC1 þ C 2 K Þe

According to [18], a subset Ω of the complex plane is called an LMI region if there exist a symmetric matrix α = [αkl] 2 ℜ p × p and a matrix β = [βkl] 2 ℜ p × p such that n o Ω ¼ z 2 ℂ : α þ zβ þ z βT ¼ ½αkl þ βkl z þ βlk z1k;lp < 0

(3)

Hence, the system (2) is asymptotically stable with disturbance attenuation γ, and the poles lie inside an LMI region Ω, if and only if, there exist a matrix Q and a positive symmetric matrix P 2 ℜ n × n satisfying the following LMI optimization problem Minimize γ subject to 2 6 4

αP þ βU ðP; QÞ þ βU ðP; QÞT < 0

U ðP; QÞ þ U ðP; QÞT

D1

V ðP; QÞT

D1 T

I

Ο

V ðP; QÞ

Ο

γ I

(4)

3 7 5 > > > ¼ þ > < v_ A03 A04 ΔA3 2 " # > v1 > > > > : z ¼ ½C 1 C 2 v2

ΔA2

#!"

ΔA4

v1

#

" þ

v2

0

#

" uþ

B2

D1

# w

D2

(10)

Without loss of generality, the corresponding sliding surface is chosen to be σ ðt Þ ¼ MT T Tx ¼ Sv1 þ v2 ¼ 0 ;

S 2 ℜmðnmÞ

(11)

Then, the sliding motion can be described by the following dynamic representation 8 > < v_ 1 ¼ ðA01 þ ΔA1 Þv1 þ ðA02 þ ΔA2 Þv2 þ D1 w z ¼ C 1 v1 þ C 2 v2 > : v2 ¼ Sv1

(12)

Note that in the sliding mode, the system dynamics will be determined by the choice of the sliding function coefficient S. It should be selected in order: to compensate the non-matched perturbation w, to assure that the dynamic matrix ((A01 + ΔA1) (A02 + ΔA2)S) has (n m) left-half plane eigenvalues despite of the presence of unmatched model parameter uncertainties and to achieve some performance desires. To this end, we choose to investigate the H1 norm with the robust pole clustering approach in the determination of the sliding surface. 4. SLIDING SURFACE DESIGN For reasons of simplicity of synthesis as well as to enjoy the property ‘convexity’, the polytopic representation can be considered one of the most general ways to describe physical parameter uncertainties without any conservatism. Because of this fact, the procedure method presented in this section starts from a polytopic form of the unmatched uncertainties associated with the system matrix. Copyright © 2014 John Wiley & Sons, Ltd.



Because ΔA1 and ΔA2 are affine in δi, the dynamic equation (12) can be easily formulated as 8 > < e_ ¼ Φe þ Ψue þ D1 w z ¼ C 1 e þ C 2 ue > : ue ¼ S e

(13)

where e = v1 is the state of the polytopic system in the sliding mode, and uv = v2 is considered as a fictitious control input for the above equivalent reduced order system. Q Matrices Φ = A01 + ΔA1 and Ψ = A02 + ΔA2 belong to a polytope-type set with known vertices. This set is given by

Y

( ¼

r X

< Φ; Ψ >¼

λj < Φj ; Ψj >;

j¼1

r X

) λj ¼ 1; λj 0 ; r ¼ 2q

(14)

j¼1

Additionally, for each vertex Θj = < Φj, Ψj >, the polytopic coefficient λj is expressed as follows [21] 9 > > > > > > =

δi δi ; i ¼ 1; …; q δi δ i ¯ 8 if δ i ¼ 1 is a coordinate of Θj < λi ¯ eλi ¼ : 1 λi if δi ¼ 1 is a coordinate of Θj λi ¼

> > > > > > ;

⇒ λj ¼

q Y

eλi ; j ¼ 1; …; r

(15)

i¼1

It supposed that the pairs (Φj, Ψj), 8 j 2 I(1, r), are stabilizable for all admissible uncertainties in the parameter box. In this way, the sliding mode existence problem is formulated as a state feedback problem for the reduced order system using a polytopic model. To resolve this problem, it is sufficient to prove that the LMI constraints (4) and (5) are verified at each vertex Θj of the polytope (14) and (15) [18, 22]. Let Ω be any LMI region. If and only if there exist a matrix Q 2 ℜm × (n m) and a positive symmetric matrix P 2 ℜ(n m) × (n m) satisfying the following LMI optimization problem Minimize γ subject to

2 6 4

αP þ βU j ðP; QÞ þ βU j ðP; QÞT

1 j r

U j ðP; QÞ þ U j ðP; QÞT

D1

V ðP; QÞT

D1 T

I

Ο

V ðP; QÞ

Ο

γ I

> > > > > hðt; v1 ; v2 Þ ¼ ΔA1 v1 þ ΔA2 v2 þ D1 w ¼ ½ δ1 > > > > > < > > > > > > > > gðt; v1 ; v2 Þ ¼ ΔA3 v1 þ ΔA4 v2 þ D2 w ¼ ½ δ1 > > > > :

2

A12

A11

" # 7 v1 7 þ D1 w ⋮ 5 v2 Aq2 3 A14 " # 7 v1 þ D2 w ⋮ 7 5 v2 Aq4

6 δq 6 4 ⋮

⋯

2

Aq1 A13

6 δq 6 4 ⋮

⋯

3

Aq3

(21)

After that, we introduce a second transformation T2 : ℜn → ℜn such that y ¼ T 2 v ¼ T 2 Tx ;

yt ¼ yt1

yt2

; y1 2 ℜnm ; y2 2 ℜm

(22)

where T2 ¼

I nm

O

S

Im

the new state variables are then

1

→ T2 ¼

I nm

O

S

Im

y1 ¼ v1

(23)

(24)

y2 ¼ Sv1 þ v2 and the transformed system equations become

y_ 1 ¼ R1 y1 þ R2 y2 þ H ðt; y1 ; y2 Þ

(25)

y_ 2 ¼ R3 y1 þ R4 y2 þ B2 u þ Gðt; y1 ; y2 Þ with R1 ¼ A01 A02 S ; R2 ¼ A02 ; R3 ¼ SR1 A04 S þ A03 ; R4 ¼ A04 þ SA02 and

H ðt; y1 ; y2 Þ ¼ hðt; v1 ; v2 Þ ¼ hðt; y1 ; y2 Sy1 Þ Gðt; y1 ; y2 Þ ¼ Shðt; v1 ; v2 Þ þ gðt; v1 ; v2 Þ ¼ Shðt; y1 ; y2 Sy1 Þ þ gðt; y1 ; y2 Sy1 Þ

(26)

Here, the functions H and G represent the uncertainties in the system dynamics. Copyright © 2014 John Wiley & Sons, Ltd.



From (26), we can obtain the following condition 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < kH ðt; y1 ; y2 Þk pffiffiqffik h ky1 k2 þ ky2 Sy1 k2 þ k D1 w0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : kGðt; y ; y Þk Gðt; y ; y Þ ¼ G ky k2 þ ky Sy k2 þ G 1 2 1 2 1 2 1 2 1

(27)

such that 8 pffiffiffi G1 ¼ q k S kh þ k g ; G2 ¼ ðk S k D1 þ k D2 Þw0 > > >

2

2 > 3

3

>

A11 A12

A13 A14

> >

>

> >

A

A

> > A A > q1 q2 q3 q4 > >

p ffiffi ffi :

½ δ1 ⋯ δq ¼ σ ð½ δ1 ⋯ δq Þ q ; ð jδi j 1 ; 8i ¼ 1; ::::::; qÞ

(28)

‖ ‖ and σ ðÞ denote, respectively, the two-norm and the largest singular value. Now, based on the ideas of the unit vector approach described in [14], we define the sliding mode control law as

uðxÞ ¼ ul ðxÞ þ un ðxÞ ¼ Lx þ ρðt; xÞ

Nx kF xk þ η

(29)

The general form of this control law consists of two parts: a linear part ul(x) and a nonlinear component un(x), which are added to form u(x). The linear component is a state-dependent feedback controller, whereas the nonlinear part is discontinuous with respect to the state and incorporates all the nonlinear elements of the control law. In order to attain an asymptotically stable sliding mode, it is necessary to force y2 and y_ 2 to become identically zero. To this end, the linear control law part ul(x) can be given by ul ðyÞ ¼ B1 R3 y1 þ R4 R4 y2 2

(30)

where R4 2 ℜmm is any design matrix with stable eigenvalues. In particular, we may set R4 ¼ diagfμi ; i : 1 ::mg = Re ðμi Þ < 0. Transforming back into the original x-space yields ul ðxÞ ¼ B1 R3 2

R4 R4

T 2 Tx

(31)

such that L ¼ B1 R3 2

R4 R4

T 2T

(32)

Before presenting the nonlinear control law part un(x), letting the matrix P2 denote the positive definite unique solution of the Lyapunov function

T

P2 R4 þ R4 P2 ¼ Im

(33)

then P2 y2 = 0 if and only if y2 = 0, and we may take un ðyÞ ¼ ρðt; yÞ

B1 2 P2 y2 kP2 y2 k þ η

(34)

ρðt; yÞ ¼ α1 Gðt; yÞ þ α2 ; α1 > 1; α2 > 0 Copyright © 2014 John Wiley & Sons, Ltd.

(35)



Transforming back into the original x-space, we obtain un ðxÞ ¼ ρðt; xÞ

B1 2 ½O P2 T 2 Tx k½O P2 T 2 Txk þ η

(36)

such that N ¼ B1 2 ½O

P2 T 2 T

(37)

F ¼ ½O P2 T 2 T and ρðt; xÞ ¼ α1 G1 kTxk þ G2 þ α2

(38)

where the modulation function ρ(t, x) is designed to compensate the uncertainties and to introduce, then, a sliding mode on the switching manifold s(x, t) = 0. It is important to note that the major difference between the proposed control in this paper and the original control law is in the structure of ρ(t, x), which has been constructed here to deal with the unmatched uncertainties. Additionally, the parameter η is a small positive constant, which can be used to trade off the requirement of maintaining ideal robustness performance with that of ensuring a smooth control action. Relying on the aforementioned analysis, we state the main result of this paper as follows. Theorem 1 The sliding mode control law gives that

➣ The sliding subspace σ is bounded with respect to the subspace Σ1

1 S ⊂ Σ1 ¼ ðy1 2 ℜnm y2 2 ℜm Þ=V 2 ðy2 Þ ¼ y2 T P2 y2 ε1 2

(39)

with ε1 ¼

η α1 1

2

1 2 λmin ðP2 Þ

(40)

➣ If ðy1 ðt 0 Þ; y2 ðt 0 ÞÞ ¼ y01 ; y02 2 = Σ1 then the time tg required to reach Σ 1 satisfies pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 y02 T P2 y02 2ε1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tg α2 λmin ðP2 Þ Proof ➣ Substituting (29) into the system equation (25), we obtain that

(

y_ 1 ¼ R1 y1 þ R2 y2 þ H ðt; y1 ; y2 Þ

y_ 2 ¼ R4 y2 þ B2 un þ Gðt; y1 ; y2 Þ

(41)

▪ (42)

Consider the Lyapunov function 1 V 2 ðy2 Þ ¼ yT2 P2 y2 2 Copyright © 2014 John Wiley & Sons, Ltd.

(43) Optim. Control Appl. Meth. (2014) DOI: 10.1002/oca


Differentiating (43) and substituting (42) 1 V_ 2 ðy2 Þ ¼ ky2 k2 þ yT2 P2 B2 un þ yT2 P2 G 2

(44)

From (34) yT2 P2 B2 un ¼ ρðyÞ

kP2 y2 k2 kP2 y2 k þ η

(45)

Expressing G from (35) zT2 P2 G GkP2 y2 k ¼

1 ρðt; yÞkP2 y2 k α2 kP2 y2 k α1

(46)

Substituting (46) and (45) into (44) gives 1 1 kP 2 y 2 k V_ 2 ðy2 Þ ky2 k2 α2 kP2 y2 k ρðyÞkP2 y2 k 2 kP 2 y 2 k þ η α 1

(47)

It follows that if kP2 y2 k >

η α1 1

(48)

then V_ 2 ðy2 Þ < 0

(49)

and (48) is satisfied if V 2 ðy2 Þ >

2 η 1 ¼ ε1 α1 1 2 λmin ðP2 Þ

(50)

It may be concluded that the sliding subspace σ is bounded with respect to the subspace ψ 1 defined in (39). ➣ If ðy1 ðt 0 Þ; y2 ðt 0 ÞÞ ¼ y01 ; y02 2 = Σ1 , then (47) leads to V_ 2 ðy2 Þ kP2 k1 V 2 ðy2 Þ α2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2λmin ðP2 ÞV 2 ðy2 Þ

(51)

pffiffiffiffi Noting that if Y_ aY b Y then the time taken for Y to move from Y0 to Y1 is pffiffiffiffiffi 2 a Y0 þ b 2 pffiffiffiffiffi pffiffiffiffiffi Y0 Y1 t 01 ln pffiffiffiffiffi a b a Y1 þ b

(52)

▪

It follows from (51) that the trajectory will reach Σ1 at a finite time tg defined by (41) as required. In order to approximate the dynamic behavior in the sliding mode, let us define a Lyapunov function 1 V 1 ðy1 Þ ¼ yT1 P1 y1 2

(53)

where P1 is the positive definite matrix who is the unique solution of P1 R1 þ RT1 P1 ¼ Inm Copyright © 2014 John Wiley & Sons, Ltd.

(54) Optim. Control Appl. Meth. (2014) DOI: 10.1002/oca


Theorem 2 If rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ffi 2 k h 2 q 1 þ kSk kP 1 k

(55)

= Σ 1 , we have then for every solution (y1, y2) with ðy1 ðt 0 Þ; y2 ðt 0 ÞÞ ¼ y01 ; y02 2 ➣ Every trajectory y1 must finally arrive and remain within the ellipsoid Σ2 ¼ fy1 2 ℜnm =V 1 ðy1 Þ ε2 g

(56)

where ε2 ¼ ε þ

β¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 1 K h kP1 k q 1 þ kSk2 kP1 k3 β2 2 2

(57)

pffiffiffiffiffiffiffi

pffiffiffi

1 2ε1 P2 2 kR2 k þ kh q þ k D1 w0

(58)

and ε > 0 is arbitrary small. R1 ðtt0 Þ 0 y1 in the sense ➣ The uncertain system (6) will approximate the prescribed dynamics yid 1 ðt Þ ¼ e ð Þ t of the motion y (t) from the ideal dynamics yid that every deviation Δy1 ðt Þ ¼ y1 ðt Þ yid 1 1 1 ðt Þ will be bounded with respect to the ellipsoid

Σ3 ¼ fΔy1 2 ℜnm =V 1 ðΔy1 Þ ε3 g

(59)

where 8

0 12 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

1

>

>

> 2

2 0

> A > 2kP1 k3 @kh q 1 þ kSk2

>

P1

P1 y1 þ β > <

ε3 ¼

0 12 > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

>

> pffiffiffiffiffiffiffi

> 3 2 >

P 2 2ε2 þ βA @ > > 1

: 2kP1 k kh q 1 þ kSk

y01 2 = Σ2 (60) y01 2 Σ2

▪

Proof ➣ Differentiating (53) and substituting (42) leads to

1 V_ 1 ðy1 Þ ¼ ky1 k2 þ yT1 P1 R2 y2 þ yT1 P1 H 2

(61)

By taking into account the following conditions pffiffiffi kH k k h q

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ky1 k2 þ ky2 Sy1 k2 þ k D1 w0 k h q ky1 k 1 þ kSk2 þ ky2 k þ k D1 w0 (62)

1 1 1

1 1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffi 2 P 2 y ¼ P2

P 2 y ¼ P2 2V ðy Þ P2 2ε ky2 k ¼

P 2 2 1

2 2 2 2

2 2 2

2

Copyright © 2014 John Wiley & Sons, Ltd.

(63)



We obtain that yT1 P1 H

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

ffi

pffiffiffiffiffiffiffiffiffi 2 2

k h kP 1 kky 1 k q 1 þ kSk þ k h kP 1 kky 1 k

P2 2ε1 q þ kP1 kky1 kk D1 w0 2

(64)

and

yT1 P1 R2 y2

1

pffiffiffiffiffiffiffi

kP1 kky1 k R2 P2 2

2ε1

(65)

Equation (61) then gives rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi _V 1 ðy1 Þ 1 ky1 k kP1 k k h ky1 k q 1 þ kSk2 þ β ky1 k 2

(66)

It follows that if rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi2 1 1 2 k h kP 1 k q 1 þ kF k V 1 ðy1 Þ > ε2 ε ¼ kP 1 k3 β 2 2 2

(67)

then V_ 1 ðy1 Þ < 0

(68)

and (68) is satisfied if (55) holds. As a result, every trajectory y1 must ultimately enter and remain within the ellipsoid Σ 2 defined in (56). ➣ Consider the Lyapunov function V1(Δy1), where Δy1 ðt Þ ¼ y1 ðt Þ yid 1 ðt Þ represents the R1 ðtt0 Þ 0 deviation of the motion y1(t) from the ideal dynamics yid y1 . It follows that 1 ðt Þ ¼ e

1 1 V_ 1 ðΔy1 Þ ¼ Δ_y1 T P1 Δy1 þ Δy1 T P1 Δ_y1 2 2 1 2 ¼ kΔy1 k þ Δy1 T P1 R2 y2 þ Δy1 T P1 H 2

(69)

and by using the conditions (62) and (63), we obtain 1

1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

2

P 2 y q 1 þ kSk2 þ β kΔy k _V 1 ðΔy1 Þ 1 kΔy1 k2 þ kP1 k k h P 1

1

1 1

2

(70)

The system uncertainty parameter kh satisfies (55), and as a result, every trajectory y1 must ultimately enter and remain within the ellipsoid Σ2. It also gives that ( V 1 ðy 1 Þ


V 1 y01 ε2

si si

y01 2 = Σ2 y01 2 Σ2

(71)



It is equivalent to

8

1

>

1 > 2 y0

<

P 1 1

P 2 y1

1 > > : pffiffiffiffiffiffiffi 2ε2

= Σ2 si y01 2

(72)

si y01 2 Σ2

▪

equations (71) and (72) then lead to the required result of equations (59) and (60). The sliding mode controller (29) delivers reachability of a bounded region of the sliding subspace in the presence of unmatched parametric uncertainties and disturbances. Also, the deviation from the ideal sliding mode dynamics is bounded. To illustrate the effectiveness of the SMC design method presented in this paper, a quarter-car suspension system will be used in the next section.

6. ILLUSTRATIVE EXAMPLE A car suspension system is the mechanism that really links the body of the car to its wheels. In other meaning, the aim of the suspension system is to improve the road handling, the ride passenger comfort and the stability of vehicles. In the literature, numerous researches were interested in car suspension system design. The main objective of most of those works is to synthesize a control law that guarantees an acceptable performance level of specifications such as holding ability and user comfort. Reference [23] gives a robust strategy in controlling the active suspension system using the proportional integral sliding mode control method. However, the discussion paper [24] (comments on [23]) proves that the influence of the mismatched uncertainties cannot be reduced but amplified when using the proposed method of [23]. In [25], a proportional integral sliding mode control strategy is developed for the half-car model. Reference [26] presents an approach to design a non-fragile H1 controller for active car suspensions. Reference [27] discusses the problem of H1 control for active vehicle suspension systems in finite frequency domain. The approach of output feedback H1 control for the suspension system is considered in [28]. A linear quadratic controller is proposed in [29]. However, all previous works are focused on the vehicle’s response to unmatched road disturbances only. It is important to note that the suspension system has some parameters that are inherently uncertain as dampers and springs, which may lose efficiency very slowly during their lifetimes. That is why, some works have been performed, currently, in the control of active quarter-car system with unmatched parametric uncertainties and road disturbances. For example, the recent work [30] where a robust reliability method for H1 control design of that system is presented. To demonstrate the efficiency of the SMC design method presented in this paper, a comparative study with the aforementioned work will be given. The quarter-car model kg is the body imass and mw = 62 kg is the wheel [30] is given in Figure 1 where mu = 504.5 h

mass. The stiffness of the wheel k 1 2 k1 ¼ 236 ; k 1 ¼ 268 kN=m, the stiffness of the body ¯ h i k 2 2 k 2 ¼ 12:3; k 2 ¼ 13:9 kN=m , and the damping coefficient kc 2 ¯ h i k ¼ 368; k c ¼ 432 Ns=m are chosen to be uncertain parameters and they can be ¯c expressed as k 1 ¼ k 10 þ δ1 dk 1 ; k 2 ¼ k 20 þ δ2 dk 2 ; k c ¼ k c0 þ δ3 dk c k 10 ¼ 252 kN=m ; k 20 ¼ 13:1 kN=m ; k c0 ¼ 400Ns=m ; dk 1 ¼ 16 kN=m ; dk 2 ¼ 0:8 kN=m ; dk c ¼ 32Ns=m Then, the motion equations of the quarter-car model for the active suspension can be written in state space in the form of (6), where u represents the active input of the suspension system generated Copyright © 2014 John Wiley & Sons, Ltd.



Figure 1. Quarter-car model.

Figure 2. Road disturbance w(t).

by means of the hydraulic actuator placed between the two masses, z is the controlled output, and the state vector is x ¼ ½x1 ¼ xu xw ; x2 ¼ xw x0 ; x3 ¼ x_ u ; x4 ¼ x_ w xu and xw are displacements of the body and the wheel masses; x0 = w is the road displacement input. The system matrices are

2

0

6 6 0 6 6 k A0 ¼ 6 20 6 mu 6 4 k 20 mw

0

1

0

0 kc mu kc mw

0

k 10 mw

1

3

2

0

3

2 0 1 6 7 7 1 7 6 0 7 6 7 7 6 1 7 60 kc 7 6 1 7 6 6 7 7; B ¼ 6 7;C¼6 7; D ¼ 6 6 mu 7 4 0 5 40 mu 7 6 7 7 4 5 5 1 kc 0 0 mw mw


2

3

3

0

0 0

1 0

0 07 7 7 1 05

0

0 1



and ΔA = δ1A1 + δ2A2 + δ3A3 is the uncertainty matrix given in the form (7) such that dk1 A1 ¼ zerosð3; 4Þ ; 0 0 0 mw dk2 dk 2 0 A2 ¼ zerosð2; 4Þ ; 0 0 0 ; mw mu dk c dk c A3 ¼ zerosð2; 4Þ ; 0 0 ; 0 0 mu mu

0

0

dkc mw

dk c mw

In order to provide adequate rates of damping and oscillatory behavior to the car-quarter system, we consider that the closed-loop poles are placed inside the disk region Ω(μ, τ) centered in μ = 12 with radius τ = 8 in the complex plane. Therefore, we obtain the following values: S ¼ ½ 0:3406 31:4773 2:6702 ; M ¼ ½ 2:6087 31:4773 0:7808 L ¼ 10 ½ 0:1294 2:2606 0:0019 5

F ¼ ½ 1:3043 15:7387

0:0175 ; N ¼ ½ 80:2658

0:3904

968:5105

0:9116 24:0236

28:0476

0:4558 ; γ ¼ 3:8561

The road profile w(t) representing a single bump that acts as disturbance is described by the following equation ( w0 w ðt Þ ¼

2 0

ð1 cosð5πtÞÞ ; ;

0 t 0:4 0:4 t

where w0 = 10 cm is the height of the bump. The disturbance input w(t) is shown in Figure 2. Taking R4 ¼ diagf1g ; η ¼ 0:05; α1 ¼ 1:2; α2 ¼ 0:1 and considering, first, two cases: (s0) the nominal system; (s1) the uncertain plant with δ1 = + 0.5 , δ2 = 0.5 , δ3 = 0.5 yield the following simulation result Figure 3 shows the evolution responses of the sliding function s(t) for nominal and uncertain systems. It indicates that the trajectory reaches in a finite time tg and stays within a certain bounded motion (the subspace Σ 1) around the nominal sliding surface σ in the presence of the mismatched uncertainty. So, it is clear that the proposed SMC design strategy provides good robustness and performance qualities for the quarter-car system in the presence of unmatched uncertainties.

Figure 3. Switching function s(t). Copyright © 2014 John Wiley & Sons, Ltd.



Figure 4. Bump responses and the corresponding input control forces of the system with uncertain parameters generated randomly. Table I. Comparison between our method and the method of [30]. 2 x€m u ðm=s Þ

xm 1 ðcmÞ

xm 2 ðcmÞ

um (kN)

0.79 1.7

0.58 4

0.18 0.5

0.38 1.1

Our method Method of [30]

Now, as in [30], the uncertain parameters are generated randomly in their admissible ranges. Twenty times simulations results of the bump responses and the corresponding control forces input are displayed in Figure 4. The performance of a suspension system is essentially evaluated in terms of ride quality and road holding ability. Ride comfort, associated with the body acceleration and the suspension deflection, and holding capacity, associated with the tyre deflection, are important goals that must be taken into account in the design of the car suspension system. In other meaning, a good suspension system should provide small amplitude values for car body acceleration, suspension deflection and tyre deflection. In this context, the performance characteristics and robustness of quarter-car model under the proposed method are evaluated and compared with the design method developed by [30]. Taking into consideration the simulation results displayed in Figure 4 and [30] Figure 2, the comparison can be summarized in Table I. m m The comparative table shows that our method gives lower amplitude values (€x m u , x1 , and x2 ) for car body acceleration €x u ðt Þ, suspension deflection x1 (t), and tyre deflection x2 (t). It indicates, also, that the active quarter-car system with sliding mode controller needs less power (input control u(t)) than the system with the method of [30]. In addition, it is easy to see (from Figure 4 and [30] Figure 2) that the SMC method presented in this paper can reduce the oscillations and provide fast responses. So, compared with the control scheme of [30], we can conclude that our method has a better control performance. Also, it achieves good rejection of the non-matched perturbation, manifests suitable robustness against the unmatched parametric uncertainty and guarantees some performance needs. Hence, the sliding mode control strategy proposed in this study seems to be a good choice for the control design of the quarter-car system with unmatched uncertainties. 7. CONCLUSION In this paper, based on LMI technique and polytopic representation of the parametric uncertainties, we have shown that the sliding surface design problem for systems affected by mismatched Copyright © 2014 John Wiley & Sons, Ltd.



uncertainties can be formulated as a multiobjective optimization problem. In which, the H1 scheme and the pole assignments in LMI regions method are used to guarantee the system robustness and some desired performances in spite of unmatched parameter uncertainties and external disturbances. An efficient sliding mode control law is synthesized, then, to satisfy the reaching and sliding conditions. Finally, the simulation results prove the effectiveness of the proposed approach.

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