LMI-based Adaptive Control Design I

, 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 (4.36) can be written as follows.

∑

= + Π + + +

= _i^r _i A_i T_{i i} t x t B HF G u ht x t

x_&() 1β (θ)( ( )) ( ) [ (β) ][ (, )] (4.37) 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,

[

^{I I}

]

diag

F(β)= (1−2β₁(θ)) ,⋅ ⋅⋅,(1−2β_r(θ)) . (4.38) It should be noted that the system (4.36) 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 (4.36). 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 ≤ (4.39) 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].

4.3.2 LMI-based Adaptive Control Design I

The Sliding Mode Control (SMC) design is decoupled into two independent tasks of lower dimensions: The first involves the design of m(n− )1 −dimensional switching 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 = xS =0where Sis a m×nmatrix.

Referring to the previous results [33], [51], we can see that for the system (4.37) 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

∑

^r= =

i 1βi(θ) 1.

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

=0 x

S 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 ofSB.

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

Then from the equivalent control method [33], we can see that the equivalent control is given by u_eq(t)=−

∑

_i^r₌₁^β_i(^θ)[I+SHF(^β)G]⁻¹S(A_i+T_iΠ_i(t))x−h(t,x). By setting

144 by symmetry such that the following LMIs holds:

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

will guarantee that the sliding mode dynamics (4.41) 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 (4.42)-(4.44)

, It is clear that if the following inequality (4.47) holds, then

G

By using the Schur complement formula, we can see that (4.43) and (4.46) imply ,

Finally, by using the above inequalities (4.46) and (4.51), we can obtain

I

Now, we will show that Sof (4.45) 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 (4.41) is equivalent to

146

The sliding mode dynamics (4.53) is asymptotically stable if there exists a positive definite matrixP₀∈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

∑

=

It should be noted that the inequalities (4.39), (4.46), (4.52) and rI This and (4.54) imply that (4.53) is asymptotically stable if there exists a positive definite matrixP such that ₀

0

i Using (4.57) and (4.58), we can show that the Lyapunov inequality (4.56) is satisfied if the following inequality holds:

0

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

0 (4.42)-(4.44), which implies that the sliding mode dynamics (4.53) is asymptotically stable. Hence, the sliding mode dynamics (4.41) 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 (4.45), 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_s is µ_is, THEN

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

∑

= 

and we can establish the following theorem.

Theorem 4.4 Consider the closed-loop control system of the uncertain system (4.37)

with control (4.62).Suppose that the LMIs (4.42)-(4.44) has a solution vector )

Proof: Since Theorem 4.3 implies that the linear sliding surface (4.45) guarantees

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

∑

=

This implies that ≤−(1− )

∑

=₁ ( ) ² −

∑

=₁ ( ) ≤0

r i

r

i i i

i i

E^&g ω β θ χ σ β θ α σ which indicates

that ,E_g∈L₂ ∩L_∞ E&_g∈L_∞. Finally, by using Barbalat’s lemma, we can conclude thatσ converges to zero.

Remark 4.2 Theorem 4.3 and 4.4 can be summarized in the form of the following LMI-based design algorithm.

Step 1: Obtain =

∑

^r_i= B_i B r

1

1 and

[

( ), ,( )

]

2 1

1 B Br

B B

H = − ⋅⋅⋅ − for givenB . _i

Step 2: Check that (A_i,B)is stabilization. If not, exit.

Step 3: Find a solution vector(Y,c₁,c₂,η) to LMI (4.42)-(4.44).

Step 4: Compute the sliding surface parameter matrixSby using the formula of (4.45).

Step 5: The controller is given by (4.62).

4.3.3 Numerical Examples I

Example 4.3 To illustrate the performance of the proposed adaptive fuzzy control design method, consider the following two-rule fuzzy model from a vertical take-off and landing (VTOL) helicopter model [55]

Plant Rule 1: IF x₁is about 0, THEN

)]

, ( [ )) (

(A₁ T_{1 1} t x B₁ u h t x

x&= + Π + +

Plant Rule2: IF x₁is about ± THEN 2,

)]

, ( [ ))

(

(A₂ T_{2 2} t x B₂ u h t x

x& = + Π + +

150

It should be noted that T₁and T₂are not matched and thus the previous VSS-based fuzzy control design methods cannot be applied to the above system (4.63). Via LMI optimization with (4.63), we can obtain the sliding surface σ =Sx.

By setting ^hˆ(^t,^x)=

[

0.9sin3^t 0.9sin3^t

]

^T and χ_i =1,α_i =0.01,r=2,l=1, ε_k =1, and

t , 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

[ ]

^{ˆsgn( ) .}

To assure the effectiveness of our fuzzy controller, we apply the controller to the two-rule fuzzy model (4.63) with nonzerod(t).We assume thatd(t)

[

x₁^sin2t−^0.5sgn(x₄⁾ x₁^sin2t−^0.5sgn(x₄⁾

]

^T.

= The time histories of the state, ρˆ_k, the

sliding variable σ, and the input (4.64) are shown in Figure 4.6 when ,

0 ) 0 ( ) 0 ( ) 0

( _{2 4}

1 =x =x =

x x₃(0)=10.

From Figure 4.6, the proposed controller is applicable to T-S fuzzy systems with mismatched parameter uncertainties in the state matrix and unknown norm-bounded external disturbances. The control performances are satisfactory. Besides, in Figure 4.6, since it is impossible to switch the input u instantaneously, oscillations on control input u always occur in the sliding mode of an SMC system. It should be noted that all existing VSS-based fuzzy control system design methods cannot be applied to the two-rule fuzzy model (4.63) because B₁is not in the range space ofB₂.

152

154

156

Figure 4.6 Simulation results withx₁(0)=x₂(0)=x₄(0)=0, x₃(0)=10.

Example 4.4 For the special case of Π t_i( )≡0, the robust adaptive controller design is proposed in [64]. Consider the following inverted pendulum on a cart

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

),

ρ1 are unknown constants. Here, we approximate the system (4.65) by the following two-rule fuzzy model.

Plant Rule 1: IF x₁is about 0, THEN

158

Because B₁is not in the range space of B₂, all existing VSS-based fuzzy control system design methods cannot be applied to the above system (4.66). Via LMI optimization with (4.66), we can obtain the sliding surface σ =Sx.

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 (4.66) with nonzero d(t). We assume that

).

x In Figure 4.7, 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 4.7, the control performances of the proposed controller are also satisfactory for the two-rule fuzzy model (4.66).

160

162

Figure 4.7 Simulation results withx₁(0)=40^o(2π/9rad),x₂(0)=x₃(0)=x₄(0)=0.

4.3.4 System Formulation II

Consider the following uncertain T-S fuzzy model [49], including parameter uncertainties and unknown norm-bounded external disturbances:

)]) matrices of appropriate dimensions, ∆A_i(t)represents the parameter uncertainties in A_i,h(t,x)∈R^m denotes external disturbances. θ =[θ₁,⋅ ⋅⋅,θ_s],θ_j(j=1,.⋅ ⋅⋅s) are the

ω 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=₁β (θ)=1

i i . We will assume that the followings are satisfied:

A1: The n×m matrix B defined by =

∑

^r_i= B_i B 1r ₁

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

A3: )∆A_i(t is of the form T_iΠ_i(t) where )Π_i(t is unknown,

The system (4.68) does not have to satisfy the restrictive assumption that all the input

164

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 written as follows.

)] It should be noted that the system (4.68) 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 (4.68). Since β_i(θ)≥0 and The following lemma will be used to establish our main results.

Lemma 4.1 For any vectors a and b with appropriate dimensions, the following

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]

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 152-175)