Adaptive Control Design via LMI

, 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 .

To assure the effectiveness of our fuzzy controller, we apply the controller to the two-rule fuzzy model (4.34) with nonzero ).d(t We assume that 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 4.4 and Figure 4.5, the proposed controller (4.35) also stabilizes the following two-rule fuzzy model (4.34).

132

134

Figure 4.4 Simulation results withx₁(0)=0.5,x₂(0)=x₃(0)=x₄(0)=0, including amplifying the inputuscale

136

138

140

Figure 4.5 Simulation results withx₁(0)=1,x₂(0)=x₃(0)=x₄(0)=0, including amplifying the inputuscale

4.3 Robust Adaptive Control for Mismatched T-S Fuzzy

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 117-150)