Robust Sliding Control for Mismatched T-S Fuzzy Time-Delay Systems 85

In this section, system formulation for the uncertain T-S fuzzy time-delay model is described in Section 3.4.1. A robust sliding control method via LMI is proposed in Section 3.4.2. Some examples are used to illustrate the effectiveness of the proposed methods and to compare with the existing methods in Section 3.4.3.

3.4.1 System Formulation

The T-S fuzzy model is described by fuzzy IF-THEN rules, which represent local linear input-output relations of nonlinear systems. The ith rule of the T-S fuzzy time-delay model is of the following form:

Plant Rule i: IFθ₁isµ_i1and … andθ_sisµ_is, THEN ), ( ))

( ( )

( )

(t A x t A x t d t B u t

x& = _i + _τi − + _i x(t) =ψ(t), t∈[−τ,0]

where ψ (t) is the initial condition, x(t)∈Rⁿis the state, u(t)∈R^m is the control input, A_i∈R^n×nare the state matrices, A_τi ∈R^n×n are the delayed state matrices,

m n

i R

B ∈ ^× are the input matrices, θ_j(j=1,⋅ ⋅⋅,s)are the premise variables, sis the number of the premise variables, µ_ij(i=1,⋅ ⋅⋅,r;j=1,⋅ ⋅⋅,s)are the fuzzy sets that are characterized by membership function, r is the number of the IF-THEN rules. The time-varying delayd(t) is bounded as d(t)≤τ.The overall fuzzy model achieved by fuzzy synthesizing of each individual plant rule is given by

)], ( )) ( ( )

( )[

( )

(

1

t u B t d t x A t x A t

x _{i i i}

r

i

i + − +

=

∑

= β θ τ

& x(t)=ψ(t), t∈[−τ,0]

whereθ =[θ₁,⋅ ⋅⋅,θ_s], =

∑

=^r → = ⋅ ⋅⋅

j

s i j i

i(θ) ω(θ)/ ₁ω (θ),ω :R [0,1],i 1, ,r

β is the membership

function of the system with respect to plant rule .i The functionβ (θ ) can be

86

regarded as the normalized weight of each IF-THEN rule and it satisfies thatβ_i(θ)≥0,

∑

_i^r=₁β_i(θ)=1. To take into account parameter uncertainties and external disturbances, we consider the following uncertain T-S fuzzy time-delay model:

))] external disturbances. We will assume that the following assumptions are satisfied:

A1: B₁= B₂ =...=B_r :=Band rank(B)=m.

Using the above assumptions, the uncertain T-S fuzzy model (3.97) can be written as follows: A large number of examples in the literature and various mechanical systems, such as motors and robots, fall into the special cases of the above model (3.98), as reported in

[44], [56-60]. The above model (3.98) also involves the uncertain time-delay system models considered in the previous SMC design methods [44], [56-60]. The symbol ∗ will be used in some matrix expressions to induce a symmetric structure. For given symmetric matrices K and L of appropriate dimensions, the following holds:



3.4.2 Sliding Control Design via LMI

The Sliding Mode Control (SMC) design is decoupled into two independent tasks of lower dimensions. The first is concerned with the design of a sliding surface for the sliding mode such that the reduced-order sliding mode dynamics satisfies the design specifications such as stabilization, tracking, regulation, etc. The second involves choosing a switching feedback control for the reaching mode so that it can drive the system’s dynamics into the switching surface [33]. We first design a sliding surface that guarantees asymptotic stability of the reduced-order sliding mode dynamics using LMIs.

Defining a nonsingular transformation matrix M and the associated vector v=Mxsuch that the above transformation we can obtain, we can transform (3.98) into the following regular form:



88 sliding mode dynamics :

d

Theorem 3.7 Let us consider the sliding mode dynamics (3.101). If the matrix

) variables, and ∗ represents blocks that are readily inferred by symmetry such that the following LMI holds:

,

, then there exists a linear sliding surface parameter matrix Sand the sliding surface

0

will guarantee that the sliding mode dynamics (3.101) is asymptotically stable.

Proof: Let us define a Lyapunov-Krasovskii function (LKF) as

∫

− +

∫ ∫

− + should be noted that a large number of previous methods such as the methods given in [42,43], have used similar Lyapunov-Krasovskii functions to obtain less-conservative stability conditions by exploiting information on the upper bounds of delay and its time derivative. None of the previous SMC design methods [44], [56-60] have used the

term

∫ ∫

− +

& in stability analysis. The time derivative of the Lyapunov-Krasovskii function is given by

∫

− wherexand y are any vectors with appropriate dimensions andH >0,we can obtain

90

By applying the Schur complement formula [48] to (3.102), we can obtain

. the sliding mode dynamics (3.101) is stable.

After the switching surface parameter matrix S is designed so that the reduced-order sliding mode dynamics has a desired response, the next step of the SMC design procedure is to design a switching feedback control law for the reaching mode such that the reachability condition is met [33], [57], [61]. If the switching feedback control law satisfies the reachability condition, it drives the state trajectory to the switching surface σ = xS =0and maintains it there for all subsequent time. We design a sliding fuzzy control law guaranteeing that σ converges to zero. We will use the following nonlinear sliding switching feedback control law as the local controller:

Control Rule i: IF θ is₁ µ_i1and ... and θ_sisµ_is, THEN

andε_i >0. The final controller inferred as the weighted average of the each local controller is given by

∑

= 









 +ΤΠ + +ΤΠ +

−

= ^r

i

i d i i i i

i i

i S A t x S A t x t

t u

1

) ( ))

( (

)) ( (

) ( )

( σ

κ σ θ

β _τ (3.107)

and we can establish the following theorem.

Theorem 3.8 Consider the closed-loop control system of the uncertain system (3.98)

with control (3.107). Suppose that the LMI (3.102) is feasible and the sliding surface is given by (3.103). Then, the switching feedback control law (3.107) induces an ideal sliding motion on the sliding surface σ =0in finite time and the state converges to zero.

Proof: Since Theorem 3.7 implies that the sliding mode dynamics restricted to

=0

σ = Sx is stable, we only have to show that reachability condition σ^Tσ& <−ε σ is satisfied for someε >0.Using SB=Iand the assumption A2, we can obtain

u h

x t A

S x t A

S _{i i d i} ^T

r

i i i i i i

T

Tσ σ β σ

σ =

∑

= ( ( +ΤΠ ( )) + ( _τ +ΤΠ ( )) + )+

& 1

≤

∑

_i^r=₁β_i(κ_i −φ_i u −ς_i) σ ≤ −

∑

_i^r=₁ε_i σ . After all, we can conclude thatσ converges to zero.

Remark 3.4 Theorem 3.7 and 3.8 can be summarized in the form of the following

LMI-based design algorithm.

Step 1: Check that (A_i+A_τi,B)is stabilization. If not, exit.

Step 2: Find a full-rank matrixΛ∈R^n×(n−m)such that B^TΛ=0,Λ^TΛ=I. Step 3: Find a solution vector(Y,c₁,c₂,η) to LMI (3.102).

Step 4: Compute the sliding surface parameter matrixSby using the formula of (3.103).

Step 5: The controller is given by (3.107).

92

3.4.3 Numerical Examples

Example 3.7 To illustrate the performance of the proposed sliding fuzzy control

design method, consider the following T-S fuzzy time-delay model [62] without mismatched parameter uncertainties and external disturbances.

) shows the control results for system (3.108) via the proposed controller (3.107) under the initial condition ϕ(t)=[20]^T. In Figure 3.10, it should be noted that since it is impossible to switch the input u instantaneously, oscillations always occur in the sliding mode of a SMC system.

94

Figure 3.10 Control results for the system (3.108)

Example 3.8 Consider a well-studied example of a continuous-time truck-trailer with time-delay proposed in [63]. The time-delay model is given by

)] angle of trailer (in radians), x₃(t)is the vertical position of rear of trailer (in meters),u(t)is the steering angle (in radians), T =2.0,l=2.8 ,L=5.5, v=−1.0 andt₀ =0.5.The constant parameterais the retarded coefficient satisfying a∈[0,1].

The limits 1 and 0 correspond to a no-delay term and to a completed-delay term. We assume that the disturbance input h(t)is unknown but bounded as h(t) ≤1.By using the fact thatsin(x)≈xifx≈0,we can represent the above model as the following two-rule T-S fuzzy model, including parameter uncertainties and external disturbances:

Plant Rule 1: IFθ(t)is about 0, THEN

96

t We can obtain the following fuzzy controller:

Control Rule 1: IFθ(t)is about 0, THEN

The final controller inferred as the weighted average of each local controller is given by

. )]

sgn(

2 . 1 )) ( (

)) ( (

[ )

( ²

1

σ

β +ΤΠ + _τ +ΤΠ +

−

=

∑

=

d i i i i

i i i

i S A t x S A t x

t

u (3.111)

To demonstrate the controller ability, we apple the above fuzzy controller (3.111) to the system model (3.110) withh(t)=sintandd(t)=τ =0.1. Figure 3.11 shows the closed-loop system responses of (3.110) and the proposed controller (3.111) with the initial conditionψ(t)=[0.4π,0.8π,−4]^T. Moreover, the closed-loop system responses of the truth model (3.109) and the proposed controller (3.111) with the initial conditionψ(t)=[0.4π,0.8π,−4]^Tare also shown in Figure 3.12. In Figure 3.11 and Figure 3.12, it should be noted that since it is impossible to switch the input u instantaneously, oscillations always occur in the sliding mode of a SMC system. From Figure 3.11 and Figure 3.12, the proposed controller is applicable to T-S fuzzy time-delay systems with mismatched parameter uncertainties in the state matrix and external disturbances and the nonlinear truth model. The control performances of the two-rule T-S fuzzy model (3.110) and the nonlinear truth model (3.109) are satisfactory.

98

100

Figure 3.11 Simulation results with the proposed method on the two-rule T-S fuzzy model (3.110).

102

Figure 3.12 Simulation results with the proposed method on the nonlinear truth model (3.109).

104

Chapter 4 LMI-Based Robust Adaptive Control

In this chapter, LMI-based robust adaptive control methods are developed for distinct uncertain Takagi-Sugeno fuzzy models/time-delay models. The introduction of this chapter is introduced in Section 4.1. In Section 4.2, a robust adaptive control method is proposed for T-S fuzzy systems. Section 4.3 presents two kinds of robust adaptive control methods for mismatched T-S fuzzy systems. A robust adaptive control method is presented for mismatched T-S fuzzy time-delay systems in Section 4.4.

4.1 Introduction

Fuzzy techniques have been widely and successfully applied to nonlinear system modeling and control for over two decades. The feedback stabilization problem of a nonlinear system in the Takagi-Sugeno (T-S) model [5] has been studied extensively. In the T-S model, local models are combined to describe the global behavior of the nonlinear system. Some authors [23-29] have studied to solve the feedback stabilization problem based on the assumption that the local model can be described by a simple linear system. In practice, the inevitable uncertainties may enter a nonlinear system model in a very complicated way. The uncertainty may include modeling errors, parameter variations, external disturbances, and fuzzy approximation errors. In such a situation, the fuzzy feedback control design methods of [23-29] may not work well anymore. To deal with the problem, some authors [30,31] have exploited the variable structure system (VSS) theory which has proposed an effective method to design robust controllers for uncertain nonlinear systems where external disturbances are bounded by known upper norm bounds.

Some authors [36-40] have relaxed the assumption and they have proposed adaptive laws to estimate the upper norm bounds. However, the previous VSC-based fuzzy control methods have considered the problem of adaptive control design and stability analysis for uncertain T-S fuzzy models where the input matrices of the local system models satisfy the assumption that each nominal local system shares the same input channel. It is practically difficult to satisfy this assumption. Moreover, these years, other authors [44-46] have exploited the SMC approach theory which has provided an effective means to design robust controllers for uncertain fuzzy time-delay systems where external disturbances are bounded by known upper norm bounds.

In this chapter, we propose robust adaptive control design methods for different uncertain T-S fuzzy models with matched/mismatched parameter uncertainties and external disturbances which are bounded by unknown upper norm bounds. As the local controller, we use an adaptive controller with a nonlinear switching feedback control term and an adaptation law to specify unknown upper norm bounds. We derive LMI conditions for existence of linear sliding surfaces guaranteeing asymptotic stability of the reduced order equivalent sliding mode dynamics, and we give an explicit formula of the switching surface parameter matrix in terms of the solution of the LMI existence conditions. We also design the nonlinear switching feedback control term and an adaptation law to drive the system trajectories so that a stable sliding motion is induced in finite time on the switching surface and the state converges to zero. Moreover, a robust adaptive control design method is also presented for the uncertain T-S time-delay model with mismatched parameter uncertainties and external disturbances. Finally, some examples are used to illustrate the effectiveness of the proposed methods for distinct uncertain T-S fuzzy models and to compare with the existing methods in each final subsection.

106

4.2 Robust Adaptive Control for T-S Fuzzy Systems

In this section, system formulation for the uncertain T-S fuzzy model is described in Section 4.2.1. A robust adaptive control method via LMI is proposed in Section 4.2.2.

Some examples are used to illustrate the effectiveness of the proposed methods and to compare with the existing methods in Section 4.2.3.

4.2.1 System Formulation

Consider the following uncertain T-S fuzzy model [49]:

[ ]

∑

= + +

= ^r

i i Aix t Biu t Bih t x

t

x_&( ) ₁β (θ) ( ) ( ) ( , ) (4.1) where x(t)∈Rⁿ is the state, u(t)∈R^m is the control input, A ,_i B_i are constant matrices of appropriate dimensions, θ =[θ₁,⋅ ⋅⋅,θ_s],θ_j(j =1,⋅ ⋅⋅,s) are the premise variables, s is the number of the premise variables, β_i(θ)=

∑

= r

j j

i(θ)/ 1ω (θ),

ω ω_i:R^s →[0,1] ,i=1,⋅ ⋅⋅,r is the membership function of the

system with respect to plant rule ri, is the number of the IF-THEN rules, β_i can be regarded as the normalized weight of each IF-THEN rule and it satisfies thatβ_i(θ)≥0,

∑

^r_i=₁β_i(θ)=1,h(t,x)∈R^m represents the lumped nonlinearities or uncertainties. We will assume that the followings are satisfied:

A1: The n× matrix m Bdefined by B= r

∑

^r_i= B_i / 1

1 satisfies the rank constraint rank(B)=m, i.e., the matrix B has full column rankm.

A2: The functionh( xt, )is unknown but bounded as h(t,x)−hˆ(t,x) ≤

∑

^l_k=₀ρ_k x ^k where ρl

ρ₀,⋅ ⋅⋅, are unknown constants, hˆ x(t, )is an estimate of h( xt, ), and lis a known positive integer.

The system (4.1) does not have to satisfy the restrictive assumption that all the input

matrices of the local system models are in the same range space. It should be noted that the assumption A1 implies that rank(B_i)≤mand each nominal local system model may not share the same input channel. The assumption A2 with l=1 and

0 ) , ˆ(t x =

h has been used in the literature [50]. We can set hˆ x(t, )as the nominal value of h( xt, ).Using the above assumptions, the uncertain T-S fuzzy model (4.1) can be written as follows:

∑

= + + +

= _i^r _i A_ix t B HF G u h t x

t

x_&( ) ₁β (θ) ( ) [ (β) ][ ( , )] (4.2) whereβ=[β₁(θ),⋅ ⋅⋅,β_r(θ)],and the matrices H,G,F(β) are defined by

[

( ), ,( )

]

2 1

1 B Br

B B

H= − ⋅ ⋅⋅ − ,^G=

[

^I,⋅ ⋅⋅,^I

]

^T,

[

(1 2 ( )) , ,(1 2 ( ))

]

. )

( diag ₁ I I

F β = − β θ ⋅⋅⋅ − β_r θ (4.3)

It should be noted that the system (4.1) does not have to satisfy

2 ,

1 B Br

B = =⋅ ⋅⋅= which is used in almost all published results on VSS design methods including the VSS-based fuzzy control design methods of [33,34]. Hence the methods [30,31] cannot be applied to the above model (4.1). Since β_i(θ)≥0 and

∑

^r= =

i 1β(θ) 1,we can see that the following inequality always holds:

. ) ( ) ( ) ( )

( F F F I

F^T β β = β ^T β ≤ (4.4) Many examples in the literature and various mechanical systems such as motors and robots do not satisfy the restrictive assumptions that each nominal local system model shares the same input channel and they fall into the special cases of the above model [49].

4.2.2 Adaptive Control Design via LMI

The Sliding Mode Control (SMC) design is decoupled into two independent tasks

−

− )

108

surfaces for the sliding mode such that the reduced order sliding mode dynamics satisfies the design specifications such as stabilization, tracking, regulation, etc. The second is concerned with the selection of a switching feedback control for the reaching mode so that it can drive the system’s dynamics into the switching surface [33]. We first characterize linear sliding surfaces using LMIs.

Let us define the linear sliding surface as σ = Sx=0 where Sis a m×nmatrix.

Referring to the previous results [33], [51], we can see that for the system (4.2) it is reasonable to find a sliding surface such that

P1

[

SB+SHF(β)G

]

is nonsingular for any β satisfying β_i(θ)≥0,i=1,⋅ ⋅⋅,r, and

∑

_i^r=₁β_i(θ)=1.

P2 The reduced (n−m)order sliding mode dynamics restricted to the sliding surface

=0

Sx is asymptotically stable for all admissible uncertainties.

It should be noted that P1 is necessary for the existence of the unique equivalent control [33] and the assumption A1 is necessary for the nonsingularity of SB.

Define a transformation matrix and the associated vector v as M =[Λ(Λ^TYΛ)⁻¹

From the equivalent control method [33], we can see that the equivalent control is given by u_eq(t)= ^r₁ ( )[I SHF( )G] ¹SA_ix h(t,x).

i i + −

−

∑

₌ ^{β θ β −} By setting σ_& =σ =0and substituting u(t)with u_eq(t),we can show that the reduced (n−m)order sliding mode dynamics restricted to the switching surface σ = Sx=0is given by

∑

= by symmetry such that the following LMIs holds:

i a linear sliding surface parameter matrix Ssatisfying P1-P2 and the sliding surface

0

will guarantee that the sliding mode dynamics (4.6) is asymptotically stable.

Proof: By using Schur complement formula [48], we can easily show that in fact the

following LMIs are incorporated in the LMIs (4.7)-(4.9) , It is clear that if the following inequality (4.12) holds, then

G

110

By using the Schur complement formula, we can see that (4.8) and (4.11) imply ,

Finally, by using the above inequalities (4.11) and (4.16), we can obtain

I

Now, we will show that Sof (4.10) guarantees P2. Using the matrix inversion lemma:

B

where A and B are compatible constant matrices such that (I +AB)is nonsingular, we can show that the sliding mode dynamics (4.6) is equivalent to

∑

=

The sliding mode dynamics (4.18) is asymptotically stable if there exists a positive definite matrix P₀∈R^{(n−m)×(n−m)}such that the time derivative of the Lyapunov function

1 0

) 1

(t v Pv

E_g = ^T satisfies for some positive scalarτ

∑

= Ζ ≤− This and (4.19) imply that (4.18) is asymptotically stable if there exists a positive definite matrixP such that ₀

0

where ∗ represents blocks that are readily inferred by symmetry.

Let z be _i z_i =[I −N(β)D₀]⁻¹N(β)C_i0ywherey∈R^(n−m). Then z can be rewritten _i Using (4.22) and (4.23), we can show that the Lyapunov inequality (4.21) is satisfied if the following inequality holds:

.

Using the Schur complement formula, we can rewrite the above inequality as

112 (4.7)-(4.9), which implies that the sliding mode dynamics (4.18) is asymptotically stable. Hence, the sliding mode dynamics (4.6) is asymptotically stable.

After the switching surface parameter matrix Sis designed so that the reduced )

(n−m order sliding mode dynamics has a desired response, the next step of the SMC design procedure is to design a switching feedback control law for the reaching mode such that the reachability condition is met. If the switching feedback control law satisfies the reachability condition, it drives the state trajectory to the switching surface

=0

= xS

σ and maintains it there for all subsequent time. With σ of (4.10), we design an adaptive fuzzy control law guaranteeing that σ converges to zero. We will use the following nonlinear adaptive switching feedback control law as the local controller.

Control rule i: IF θ₁is µ_i1and ... and θ_sis µ_is, THEN controller inferred as the weighted average of the each local controller is given by

∑

= 

and we can establish the following theorem.

Theorem 4.2 Consider the closed-loop control system of the uncertain system (4.2)

with control (4.27). Suppose that the LMIs (4.7)-(4.9) has a solution vector )

Proof: Since Theorem 4.1 implies that the linear sliding surface (4.10) guarantees

P1-P2, we only have to show that σconverges to zero. Define a Lyapunov function as

∑

= σ converges to zero.

Remark 4.1 Theorem 4.1 and 4.2 can be summarized in the form of the following LMI-based design algorithm.

Step 1: Obtain =

∑

^r=

114

Step 4: Compute the sliding surface parameter matrixSby using the formula of (4.10).

Step 5: The controller is given by (4.27).

4.2.3 Numerical Examples

Example 4.1 Consider the following inverted pendulum on a cart [49]

2, and )d(t is related to external disturbances which may be caused by the frictional force.

), are unknown constants. Here, we approximate the system (4.28) by the following two-rule fuzzy model.

Plant Rule 1: IF x₁is about 0, THEN

,

The inverted pendulum on a cart (4.28) can be cast as (4.2) with data (4.29).

Because B₁is not in the range space of B₂and the previous adaptive fuzzy control system design methods cannot be applied to the above system (4.29). Via LMI optimization with (4.29), we can obtain the sliding surface σ =Sx.

By setting ˆ( , ) sin , and sect_sampling =0.01 , we can obtain the following nonlinear controller:

Control Rule 1: IFx₁is about 0, THEN

The final controller inferred as the weighted average of each local controller is given by .

To assure the effectiveness of our fuzzy controller, we apply the controller to the two-rule fuzzy model (4.29) with nonzero ).d(t We assume that d(t)=

116

. 0 ) 0

4( =

x Figure 4.3 shows the time histories of the state, ρˆ_k,the sliding variable σ , and the input (4.30) when x₁(0)=60^o(π/3 rad), x₂(0)= x₃(0)=

. 0 ) 0

4( =

x In Figure 4.1, Figure 4.2, and Figure 4.3, it should be noted that since it is impossible to switch the input u instantaneously, oscillations always occur in the sliding mode of a SMC system. It is observed that in our simulations the proposed controller (4.30) stabilizes the following two-rule fuzzy model (4.29).

118

120

Figure 4.1 Simulation results withx₁(0)=20^o(π/9 rad), x₂(0)=x₃(0)=x₄(0)=0.

122

124

Figure 4.2 Simulation results withx₁(0)=40^o(2π/9 rad), x₂(0)=x₃(0)=x₄(0)=0.

126

128

Figure 4.3 Simulation results withx₁(0)=60^o(π/3 rad), x₂(0)=x₃(0)=x₄(0)=0.

Example 4.2 Consider the following example of a ball and beam system [52], whose dynamic equations are described as follows:

, 0

sin − ² =

 +



 



 + M r&& MG θ Mrθ&

R

J_b

(

^{Mr2 +J +J}b

)

^θ&&^+2Mrr&^θ&^{+MGr cosθ =τ} (4.31) where r is the ball position, θ is the beam angle, J is the moment of inertia of the beam, M , J , and R are the mass, the moment of inertia, and the radius of the ball _b respectively, G is the acceleration of gravity, and τ is the torque applied to the beam.

Define B=M/(J_b/R² +M) and change the coordinates in the input space by using the invertible transformation

u J J Mr MGr

r

Mr cos _b)

2 + + ² + +

= θ θ

τ & & (4.32)

where u is the new input, then the ball and beam system can be written in the following state-space form:

2,

1 x

x& = x&₂ =B(x₁x₄² −Gsinx₃), x&₃ = x₄, x&4 =u+d⁽t⁾ (4.33)

where

[

1 2 3 4

] [ ]

^.

T T

r r x

x x x

x= = _& θ θ^& The system parameters used for simulation areM =0.05kg, R=0.01m, J =0.02kgm², J_b =2×10⁻⁶kgm², G=9.81m/s² andB=0.7143.We assume that d(t)is bounded as d(t)≤ρ₀+ρ₁ x where ρ₀and ρ₁ are unknown constants. Then, we approximate the system by the following two-rule fuzzy model:

Plant rule 1: IF x₁ is greater than 0, THEN

)]

, (

1[

1x B u h t x

A

x& = + + .

Plant rule 2: IF is smaller than 0, THEN

130 obtain the following nonlinear controller:

Control Rule 1: IFx₁is greater than 0, THEN

The final controller inferred as the weighted average of each local controller is given by

The final controller inferred as the weighted average of each local controller is given by

在文檔中 線性矩陣不等式的強健適應滑差控制應用於T-S模糊系統 (頁 95-0)