Adaptive Variable Structure Control for Linear Systems with Time-varying Multi-delays and Mismatching Uncertainties

Available online at www.sciencedirect.com

Physics Procedia 33 (2012) 1753 – 1761

2012 International Conference on Medical Physics and Biomedical Engineering

Adaptive Variable Structure Control for Linear Systems with Time-varying Multi-delays and Mismatching Uncertainties* MENG Bo1 ,GAO Cunchen2 TANG Shuhong3 ,LIU yunlong3 1 School of Information Science and Engineering School of Mathematics Sciences Ocean University of China Qingdao, 266100, China 3 School of Information Science and Engineering Ocean University of China Qingdao, 266100, China [email protected], [email protected], {wfxytang, fhylren}@163.com3 2

Abstract. This paper discusses the control problems of linear systems with time-varying multi-delays and mismatching uncertainties. A new adaptive variable structure control strategy is given by Lyapunov stability theory. The full invariance of sliding mode with time-varying multi-delays and mismatching uncertainties is proved in the circumstances sliding surface coefficient matrix C to satisfy certain matching conditions. The control strategy ensures that the closed-loop system is globally stable. Simulation results further show that the strategy is feasible and effective.

©2012 2011Published Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [name Committee. organizer] © by Elsevier B.V. Selection and/or peer review under responsibility of ICMPBE International Keywords: Time-varying multi-delays, Linear systems, Adaptive variable structure control, Mismatching uncertainties.

1. Introduction Time-delay commonly exists in various engineering systems, for example, the turbojet engine, aircraft systems, microwave oscillator, nuclear reactor, rolling mill, chemical process, manual control, and long transmission line in pneumatic, hydraulic systems [1-3, 4, 6]. The existence of time delay in a system frequently becomes a source of instability. Since the time-varying uncertainties (structured or unstructured) are inevitable in many practical linear systems, the control of the linear systems with time-varying uncertainties has been an important research topic in the engineering area. The variable structure control *

This work was supported by the National Natural Science Foundation of China (No.60974025).

1875-3892 © 2012 Published by Elsevier B.V. Selection and/or peer review under responsibility of ICMPBE International Committee. doi:10.1016/j.phpro.2012.05.281

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(VSC) has been effectively applied to control the systems with uncertainties because of the intrinsic nature of robustness of the variable structure with sliding mode. The VSC, as an attractive deterministic methodology for system with uncertainties, can provide an easy and effective solution for stabilizing perturbed time-delay dynamic systems [5]. It is well known that VSC is characterized by a discontinuous controller which can change its structure automatically on reaching a switching surface in order to obtain a desired dynamic performance. The advantages of VSC include fast response, good transient performance, insensitive-ness to the matching parameter variations and external disturbances. Refs [6, 7] have also proposed a variable structure model following adaptive control for stabilizing linear systems with time delay. However, in this method the information of upper bound of perturbations is still required. In general, the information of the upper bound of perturbations may not be easily obtained in practice because of the complexity of the structure of the system and the perturbations [8]. Based on the Lyapunov theory, we also develop a simple adaptive variable structure control (AVSC) strategy for stabilizing a class of perturbed time-varying delay systems. The proposed control strategy is composed of three parts, the linear state feedback part, the nonlinear switching part, and the adaptive part. The linear state feedback part is used for assigning the closed-loop eigenvalues. The nonlinear switching part and the adaptive part are used to achieve the robustness of globally asymptotic stability. The adaptive algorithm can effectively adapt the upper bound of perturbations, so that the information of upper bound of perturbations is not required. A practical example is also given for demonstrating the feasibility of the proposed control scheme. 2. The system Model Description In this paper, we consider a class of linear, continuous-time, uncertain dynamic systems with timevarying multi-delays modeled by the following equation: x(t ) [ A

N

A(t )]x(t )

i 1

x(t )

Ahi (t ) x(t hi ) Bu (t )

(t ),

(t ), t [h , 0].

where x(t ) R n , u (t ) R m are state vector and input vector; constant matrices A R n n , B

known; A R

n n

, Ahi (t ) R

n n

(1)

Rn

m

are

are unknown real-valued functions representing time-varying parameter

uncertainties of the matrices, and A(t ) , Ahi (t ) are mismatched uncertain terms of the system. The unknown scalar function h i (t) denotes bounded and continuous delay satisfying: 0

hi (t )

hi

h

,

h

max1

i N

{h1 , h2 ,

, hN }.

where h is an unknown constant. (t ) R n is an arbitrarily known continuous state vector for specifying initial condition. For completing the description of the uncertain dynamic systems (1), the following assumptions are assumed to be valid: Assume 1 The pair ( A, B ) is controllable, B is full rank, and all the state variables are measurable.

Assume 2 The uncertain matrices piecewise continuous in t.

A R n n , Ahi (t )

Rn

n

are continuously differentiable in x, and

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Further, the state equation (1) is transferred into the regular form with the following state matrix transformation: z (t ) Tx(t ).

where the state transformation matrix T is assumed to be nonsingular and 0

TB

B2

,

(2)

Thus, the regular form of the state equation is z1 (t )

z (t )

Tx(t ) T ( A

z2 (t )

Ahi1 (t )

i 1

Ahi 2 (t )

Ahi (t )

i 1

x(t hi (t )) TBu (t ) T (t ) N

N

A(t )) x(t )

A1

z (t hi (t ))

A2

A11

A12

z1 (t )

A21

A22

z2 (t )

z (t )

1

(t )

0

2

(t )

B2

(3) u (t ).

where TAT TAhi T

1

A11

A12

A21

A22

1

A1 (t ) Ahi1 (t )

, T AT

A1 (t )

1

A2 (t )

Ahi1 (t ) Ahi 2 (t ) , T (t )

R (n R

m) n

(n m) n

,

A2 (t )

, Ahi 2 (t )

Rm n , R

m n

,

1

(t )

2

(t )

, ,

1

(t )

R(n

2

(t )

R .

m)

,

m

It is straightforward to know that the A2 (t ), Ahi 2 (t ) satisfy the conventional match condition with the Lemma 1 given below. Lemma 1 The time-varying uncertainties A2 (t ), 2 (t ) and Ahi 2 (t ), i 1, 2, , N satisfy the match condition: A2 (t )

B2 (t ),

2

(t )

B2 (t ), Ahi 2

B2 i (t ), i 1, 2,

, N.

(4)

Proof: Since matrix B has full column rank, B1 =0 and rank( B2 ) m . B2 is nonsingular. We can always find the matrix (t ) B2 1 A2 (t ) , (t ) B2 1 2 (t ) i (t ) B2 1 Ahi 2 (t ), i 1, 2, , N ; where (t ) R m n , i (t ) R m n , i 1, 2, , N and (t ) R m . Such that (4) is satisfied. Assume 3 The mismatched time-varying uncertainties A1 (t ) , 1 (t ) and Ahi1 (t ), i 1, 2, , N can be assumed to have the structure:

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A1 (t )

D1 F1 (t ) E1 ,

Ahi1 (t )

1

(t )

D3 F3 (t ) E3 ,

D2 F2i (t ) E2 , i 1, 2,

(5)

, N.

where Di , Ei , i 1, 2,3 are the constant matrices and the uncertain matrix Fi (t ) R pi satisfying: F1 (t )T F1 (t )

I , F3 (t )T F3 (t )

I , F2i (t )T F2i (t )

I , i 1, 2,

where I is the identity matrix with corresponding dimension, I matrix.

, i 1, 21,2,

qi

,N

,3

, N.

Fi (t )T Fi (t ) is semi-positive definite

3. Design of Adaptive Variable Structure Controller Let the sliding surface S (t )

Cz (t )

(6)

0

where C [C1 , C2 ] , C1 R m ( n m ) , C2 R m m and C2 B2 is an invertible matrix, Matrix C will be designed in SectionV. It is known that the sliding mode of the variable structure controller is guaranteed if S T S 0 .where the equality only occurs if S 0 . Substituting (3) into (6), S (t )

Cz (t )

C1{ A11 z1 (t ) A12 z2 (t )

A1 (t ) z (t )

N i 1

z (t hi (t )) N i 1

1

(t )} C2 { A21 z1 (t ) A22 z2 (t )

Ahi 2 (t ) z (t hi (t ))

z2 (t ) D1 F1 (t ) E1 z (t )

2

Ahi1 (t )

A2 (t ) z (t )

(t ) B2 u (t )} C1{ A11 z1 (t ) A12

N

(7) D2 F2i (t ) E2 z (t hi (t )) D3 F3 (t )

i 1

E3 } C2 { A21 z1 (t ) A22 z2 (t ) B2 (t ) z (t )

N

B2 i (t )

i 1

z (t hi (t )) B2 (t ) B2 u (t )}.

Assume 4 l0 , l1 are unknown constants, (t , z ) [ h , ] R n (t ) z (t ) l0 l1 z (t ) , (t ) l2 . Assume 5 k i 0 , k i1 are unknown constants, h(t ) [0, h ] hi (t )) ki 0 ki1 z (t ) , i 1, 2, , N . i (t ) z (t According to Assume 4 and Assume 5, we get (t ) z (t )

N i

(t ) z (t hi (t ))

(t )

i i 1

(t ) z (t hi (t ))

satisfying

,

satisfying

(t ) z (t )

i 1 N

,

(8) (t )

g0

g1 z (t ) .

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where g 0 , g1 are unknown constants, let gˆ 0 (t ), gˆ1 (t ) are adaptive parameter estimation of unknown constants g 0 and g1 , then the estimated error is: gˆ 0 (t ) g 0 , g1 (t )

g 0 (t )

gˆ1 (t ) g1 .

(9)

The adaptive law of g 0 , g1 satisfying g 0 (t )

gˆ 0 (t )

r0 1 S T C2 B2 ,

g1 (t )

gˆ1 (t )

r1 1 S T C2 B2 z (t ) .

(10)

where r0 , r1 are positive constants. Taking Lyapunov function 1 T ( S (t ) S (t ) r0 g 02 (t ) r1 g12 (t )) 2 The derivative of Lyapunov equation corresponding to the nominal system (3) is given by V ( S , g 0 , g1 )

V ( S , g 0 , g1 )

S T (t ) S (t ) r0 g 0 (t ) g 0 (t ) r1 g1 (t ) g1 (t )

S T (t ){C1 ( A11 z1

A12 z2 ) C2 ( A21 z1 N

C2 B2 { (t ) z (t )

A22 z2 )} S T (t )

[ i (t ) z (t F2i E2 z (t hi )]}

(t )

i 1

u (t ) r0 g 0 (t ) g 0 (t ) r1 g1 (t ) g1 (t ) A12 z2 ) C2 ( A21 z1 (

N

S T (t ){C1 ( A11 z1

A22 z2 )} S T C2 B2 u (t )

C1 D2 E2 z (t hi (t ))

S

C1 D1 E1 z (t )

(11)

C1 D3

i 1

E3 )

S T (t )C2 B2 {g 0

g1 z (t ) } r0 g 0 (t ) g 0 (t )

r1 g1 (t ) g1 (t )

We propose an AVSC law for the perturbed system (3) as u (t )

(12)

u1 (t ) u2 (t ) u3 (t )

where, u1 (t )

(C2 B2 ) 1{C1 ( A11 z1 (t ) A12 z2 (t )) C2 ( A21 z1 (t )

A22 z2 (t ))}, u2 (t ) N

C1 D2

(C2 B2 )

E2 z (t hi (t ))

1

S { C1 D1 E1 z (t ) S

C1 D3 E3 )},

i 1

u3 (t )

B2T C2T S S T C2 B2

{gˆ 0 (t ) gˆ1 (t ) z (t )

k}.

(13)

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Substituting (12), (13) into (11), V ( S , g 0 , g1 ) S T C2 B2 {g 0 (t ) g1 (t ) z (t ) } r0 g 0 (t ) g 0 (t ) r1 g1 (t ) g1 (t ) k S T C2 B2 g1 (t ){r1 g1 (t )

g 0 (t ){r0 g 0 (t )

S T C2 B2 z (t ) } k S T C2 B2 .

V ( S , g 0 , g1 )

where (t )

Substituting (10) into above equation

S T C2 B2 }

(t )

k S T C2 B2

(14)

0.

k S T C2 B2 .

For (14):

t

V (0) V (t )

t

( )d

0

0

( )d , t

0. when t

,

t 0

V (0) .because V (0) is finite positive, It is straightforward to know that when t lim (t ) t

lim k S C2 B2 T

t

( )d

is always less

, we get (t )

0,

0.

Because C2 B2 is invertible, when t . We ensure that the sliding mode S convergence to zero from any initial state and reaching the sliding mode, the system will remain in the sliding surface movement.

4. The Invariant Characteristic of the Sliding Mode Control System In this section, the invariant characteristic of the sliding mode control will be proved in the following part. When (t ) 0 , we discuss the control system: z1 (t ) z2 (t ) N

Ahi1 (t )

i 1

Ahi 2 (t )

A11 A21

A12 A22

z1 (t ) z2 (t )

z (t hi (t ))

A1 z (t ) A2

0 u (t ). B2

(15)

Assume 6 The sliding coefficient matrix C satisfied matching conditions: E1

Ea C , E2

(16)

Eb C

Based on (5) and (6), the state equation (15) can be expressed as: z1 (t ) z2 (t )

A11 A21

A12 A22

N

D2 F2i (t ) Eb C

i 1

Ahi 2 (t )

z1 (t ) z2 (t ) z (t hi (t ))

D1 F1 (t ) Ea C z (t ) A2 0 B2

u (t ).

As 0 h(t ) h , the systems reach the sliding mode about h time, we can grantee: Ahi1 (t ) z (t hi (t )) D2 F2i (t ) Eb Cz (t hi (t )) 0, i 1, 2, , N . In the other part, we have proved and Ahi (t ) match the matching conditions. Moreover, the system can be expressed as:

A2 (t )

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Meng Bo et al. / Physics Procedia 33 (2012) 1753 – 1761

z1 (t )

A11

A12

z1 (t )

0

z2 (t )

A21

A22

z2 (t )

B2

(t ) z (t )

N i

(u (t )

(17)

(t ) z (t hi (t )))

i 1

The system is shown to be invariant on the sliding surface with time-varying multi-delays and mismatching uncertainties.

5. Design of the Sliding Coefficient Matrix C Based on the matching Condition E1 known matrix, rank[ E1T ]

p

m , coefficient matrix C can be selected as

CT

[ E T G C1 , C2 ,

[[e1 , e2 ,

where e1 , e2 ,

, e p , C1 , C2 ,

, Cm

Ea C of Assume 6, we get rank[C T ] rank[C T E1T ] , E1 is

p

, ep ]

, Cm p ] H

C1 , C2 ,

(18)

, Cm p ] H

is m-independent vector, and H

Rm

m

is invertible matrix.

For (17), z1 (t )

(19)

A11 z1 (t ) A12 z2 (t )

In the sliding mode phase S

Cz (t )

C1 z1 (t ) C2 z2 (t )

0

C2 1C1 z1 (t ).

(20)

A12 C2 1C1 ) z1 (t ) .

(21)

z2 (t )

Substituting (20) into (19), z1 (t )

( A11

On Assume 1, ( A, B ) is controllable, so ( A11 , A12 ) are also controllable. If the eigenvalues of A11 A12 C2 1C1 is negative, then the variable structure control system is asymptotically stable on the sliding surface. Theorem 1 Consider the perturbed system (1) with the assume 1-5. Choosing sliding mode (6) and the proposed control law (12) with the adaptation gains (10), then the perturbed system (1) with time-varying state delay is globally asymptotically converging into sliding mode, and always keep movements on the sliding mode. Theorem 2 When the Trajectory of the system got into sliding mode, and the conditions of Assume 6 are meeting, after h times, system in sliding mode for the time-varying uncertainty and multi-delays has full invariance. Sliding surface coefficient matrix C meet (18), the inverse of C2 B2 exists, all the

eigenvalues of A11 A12 C2 1C1 have negative real parts. System in sliding mode is asymptotically stable, and then closed-loop system is global asymptotic stable.

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Meng Bo et al. / Physics Procedia 33 (2012) 1753 – 1761

6. Simulation Results As an example of design of the sliding coefficient matrix C in Section 5, the uncertain linear system is described by (1) with 1 2 0 1 0 3 , A(t ) 0 0 1

A

0 0.1 0.3cos 2 t 0.1 0.3cos 2 t 0 0.2 0.1sin 2 t 0.2 0.1sin 2 t , B 1 0.3sin 2 t 0

Ah1 (t ) h1 (t )

0 0.06sin(0.1t ) 0.06sin(0.1t ) 0 0.03cos(0.1t ) 0.03cos(0.1t ) 0 0 0.03cos(0.1t )

0.1 sin t , h2 (t )

0 0 , 1

0.1 cos t ,

3

0 0.1sin t 0.4 cos3 t 0.1sin 3 t 0.4 cos3 t 0 0.3sin 3 t 0.2 cos3 t 0.3sin 3 t 0.2 cos3 t 1 0.3cos 2 t 0

Ah2 (t )

0.2sin(0.05t ) 0.4 cos(0.05t ) 0.4sin(0.05t ) 0.6 cos(0.05t ) 0.5sin(0.05t )

(t )

The system controller is given as (12) ( r0

1, r1

0.5 ). 0.4

0.25

0.35 0.2

0.3

0.25

g0

g1

0.15

0.1

0.2

0.15

0.1 0.05

0.05

0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time(sec)

Time(sec)

Fig 1 Adaptive Parameter

It is easy to see that

A(t ), Ah1 , Ah2 are mismatched time-varying uncertainties. Let the uncertain linear

system be in the regular form with the state transformation matrix T as unit matrix. Selected sliding surface coefficient matrix C 0 1 1 .It is easy to prove C satisfying E1

Ea C , E2

Eb C , C2 B2

1 ,and A11

A12 C2 1C1

1 1

2 . 3

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Meng Bo et al. / Physics Procedia 33 (2012) 1753 – 1761

0.25

1.5

0.2 0.15 0.1

Sliding mode

System states

s

x1 x2 x3

1

0.5

0

0.05 0 −0.05 −0.1

−0.5

−0.15 −1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

−0.2

0

0.5

1

Time(sec)

1.5

2

2.5

3

3.5

4

4.5

5

Time(sec)

Fig 2 System States and Sliding Mode

7. Conclusion Based on the Lyapunov theorem, we successfully proposed an AVSC scheme in this paper for the linear systems with mismatched time-varying uncertainties and multi-delays. The system is shown to be invariant and stable on the sliding surface when sliding coefficient matching condition. By using the proposed control scheme, the controlled system is guaranteed to have the property of globally asymptotic stability. Simulation results are included to illustrate the effectiveness of the proposed sliding mode controller.

8. Acknowledgment This work was supported by the National Natural Science Foundation of China (No.60974025).

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