Chapter 4 LMI-Based Robust Adaptive Control
4.3 Robust Adaptive Control for Mismatched T-S Fuzzy Systems
4.3.5 LMI-based Adaptive Control Design II
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 (4.69) it is reasonable to find a sliding surface such that
P1
[
SB+SHF(β)G]
is nonsingular for any β satisfying βi(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(ΛTYΛ)−1
T
TY B
B B
Y ( ) ]
, −1 −1 −1 =[VT,ST]T,v=[v1T,v2T]T =Mxwhere v1∈Rn−m ,v2∈Rm. By the above transformation, we can see that M−1 =[YΛ,B]and v2 =σ. Then, from system (4.69), we can obtain
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Then from the equivalent control method [33], we can see that the equivalent control is given by ueq(t)=−
∑
ri=1βi(θ)[I+SHF(β)G]−1S(Ai+TiΠi(t))x−h(t,x). By blocks that are readily inferred by symmetry such that the following LMIs holds:i
Suppose that the LMIs (4.74)-(4.76) have a solution vector(Y,c0,c1,c2,δ,η), then there exists a linear sliding surface parameter matrix Ssatisfying P1-P2 and the sliding surface will guarantee that the sliding mode dynamics (4.73) 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 (4.74)-(4.76) , It is clear that if the following inequality (4.79) holds, then
G By using the Schur complement formula, we can see that (4.75) and (4.78) imply
,
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Finally, by using the above inequalities (4.78) and (4.83), we can obtain
I
Now, we will show that Sof (4.77) 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 (4.73) is equivalent to
∑
=The sliding mode dynamics (4.85) 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
Eg = T satisfies for some positive scalarτ
∑
=It should be noted that the inequalities (4.71), (4.78), (4.84) and rI This and (4.86) imply that (4.85) 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
y Using (4.89) and (4.90), we can show that the Lyapunov inequality (4.88) is satisfied if the following inequality holds:
0
Using the Schur complement formula, we can rewrite the above inequality as
0 (3.74)-(3.76), then the above matrix inequality (4.91) can be rewrite as
i
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∆ the following inequalities hold:
1 The previous inequalities (4.94) and (4.95) imply that for all admissible
, )
( Ai
i t A ≤α
∆ the inequality condition (4.93) holds if
1 This implies that (4.92) holds if the following LMI (4.97) holds
0
By using Schur complement formula, the above inequality (4.97) can be rewritten as the LMI (4.74), which implies that the sliding mode dynamics (4.85) is asymptotically stable. Hence, the sliding mode dynamics (4.73) 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 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 (4.77), 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 µi1and ... and θsis µis, THEN The final controller inferred as the weighted average of the each local controller is given by
and we can establish the following theorem.
Theorem 4.6 Consider the closed-loop control system of the uncertain system (4.69)
with control (4.100). Suppose that the LMIs (4.74)-(4.76) has a solution vector ) converges to zero.
Proof: Since Theorem 4.5 implies that the linear sliding surface (4.77) guarantees
P1-P2, we only have to show that σ converges to zero. Define a Lyapunov function as
∑
=172
ω which indicates that
2∩ ∞,
∈L L
Eg E&g∈L∞. Finally, by using Barbalat’s lemma, we can conclude thatσ converges to zero.
Remark 4.3 Theorem 4.5 and 4.6 can be summarized in the form of the following LMI-based design algorithm.
Step 1: Obtain =
∑
ri= BiStep 4: Compute the sliding surface parameter matrixSby using the formula of (4.77).
Step 5: The controller is given by (4.100).
4.3.6 Numerical Examples II
Example 4.5 To demonstrate the performance of the proposed adaptive 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
)]
Note that B1and B2are not matched and almost existing VSS-based fuzzy control design methods cannot be applied to the above system (4.101). Via LMI optimization with (4.101), we can obtain the sliding surface σ =Sx.
By setting hˆ(t,x) =
[
0.9sin3t 0.9sin3t]
Tandχi =1,αi =0.0001,r=2,l =1, εk =2,and sectsampling =0.01 , we can obtain the following nonlinear controller:
Control Rule 1: IFx1is about 0, THEN
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[
t t]
Tt
u( )= −0.9sin3 −0.9sin3 ˆ sgn( ).
1 ) 1
( 2 2 2 δ2 σ
σ ω
− − +
−
− S A TTT x
The final controller inferred as the weighted average of each local controller is given by
[
t t]
Tt
u( )= −0.9sin3 −0.9sin3 ˆ sgn( ) . 1
) 1 (
) (
∑
1=
+ − +
+
− r
i
i T
i i i
i S A TT x δ σ
σ ω θ
β (4.102)
To assure the effectiveness of our fuzzy controller, we apply the controller to the two-rule fuzzy model (4.101) with nonzerod(t).We assume thatd(t)
[
0.251sin2 0.1sgn( 4) 0.251sin2 0.1sgn( 4)]
.x T
t x x
t
x − −
= π π The time histories of the state, ρˆk,
the sliding variable ,σ and the input (4.102) are shown in Figure 4.8 when ,
0 ) 0 ( ) 0 ( ) 0
( 2 4
1 =x =x =
x x3(0)=10. In Figure 4.8, 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 4.8, the proposed controller is applicable to uncertain fuzzy systems with mismatched parameter uncertainties in the state matrix and unknown norm-bounded external disturbances. The control performances of the proposed controller are satisfactory for the two-rule fuzzy model (4.101). Note that almost existing VSS-based fuzzy control system design methods cannot be applied to the two-rule fuzzy model (4.101) because B1 is distinct fromB2.
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178
Figure 4.8 Simulation results withx1(0)=x2(0)=x4(0)=0, x3(0)=10.
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Example 4.6 For the special case of ∆ tAi( )≡0, the robust adaptive controller design is proposed in [64]. Consider the following inverted pendulum on a cart
2,
1 x
x& = 1 (3 sin 3 cos [ ( ) ]),
1 1
2 φ
ψ − + +
= g x a x u d t
x& l x&3 =x4,
]) ) ( [ 4 2 sin 5
. 1 1 (
1
4 φ
ψ − + +
−
= mag x au d t
x& (4.103)
where x1is the angle (rad ) of the pendulum from the vertical, x2=x&1, x is the 3 displacement (m) of the cart, x4= x&3, ψ=4−3macos2x1,φ =mlx22sinx1,uis the input, and )d(t is related to external disturbances which may be caused by the frictional force.
), /(
1 m M
a= + mis the mass of the pendulum, M is the mass of the cart, 2lis the length of the pendulum, g =9.8m/s2 is the gravity constant. We set M =9kg
kg m 1
, = ,l =1m.We assume that d(t)is bounded as d(t)≤ρ0+ρ1 x where ρ0and ρ1 are unknown constants. To design the fuzzy controller (40), we must have a fuzzy model. Here, we approximate the system (4.103) by the following two-rule fuzzy model.
Plant Rule 1: IF x1is about 0, THEN
)]
, (
1[
1x B u h t x
A
x&= + +
Plant Rule2: IF x2is about ±60o(±π/3 rad),THEN )]
, (
2[
2x B u h t x
A
x& = + +
where ,
Via LMI optimization with (4.104), we can obtain the sliding surface σ =Sx. By setting ,ˆ(, ) sin 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 (4.104) with nonzero d(t). We assume that variable σ , and the input (4.105) are shown in Figure 4.9 when
), since it is impossible to switch the input u instantaneously, oscillations always occur in
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the sliding mode of a SMC system.From Figure 4.9, the control performances of the proposed controller are also satisfactory for the two-rule fuzzy model (4.104).
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Figure 4.9 Simulation results withx1(0)=60o(2π/9 rad), x2(0)=x3(0)=x4(0)=0.
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