International Journal of Automation and Power Engineering, 2012, 1: 42-46 Published Online May 2012
- 42 -
www.ijape.org
New Approach for Stabilisation of Continuous Takagi Sugeno Fuzzy System Yassine Manai*, Sami Madssia, Mohamed Benrejeb LA.R. Automatique, National Engineering School of Tunis, Tunis, Tunisia, Email: yacine.manai@gmail.com
(Abstract) This paper deals with the new approach for stabilization of continuous Takagi-Sugeno fuzzy models. Using a new Lyapunov function, new sufficient criteria are established in terms of Linear Matrix Inequality. First, a basic stability criterion is derived for open loop system. Next, a stabilization criterion with PDC controller and relaxed stability criterion are established. Finally, a new approach of PDC controller design is given. Keywords: Takagi-Sugeno Fuzzy Ssystem; Linear Matrix Inequalities; Fuzzy Lyapunov Function; Parallel Distributed Controller PDC
1.
INTRODUCTION
2.
SYSTEM DESCRIPTION AND PRELIMINARIES
Fuzzy control systems have experienced a big growth of industrial applications in the recent decades, because of their Consider a T-S fuzzy continuous model for a nonlinear system as reliability and effectiveness. Many researches are investigated on follows: the Takagi-Sugeno models [4]–[5] which can combine the IF z1 ( t ) is M i1 and and z p ( t ) is M ip (1) flexible fuzzy logic theory and rigorous mathematical theory into 1, , r THEN x ( t ) = Ai x ( t ) + Bi u ( t ) i = a unified framework. Thus, it becomes a powerful tool in where = M ij ( i 1,= 2, , r , j 1, 2, , p ) is the fuzzy set and approximating a complex nonlinear system. Two classes of Lyapunov functions are used to analysis these r is the number of model rules; x ( t ) ∈ ℜn is the state systems: quadratic Lyapunov functions and non-quadratic m n× n n× m Lyapunov ones which are less conservative than first class. Many vector, u ( t ) ∈ ℜ is the input vector, Ai ∈ ℜ , Bi ∈ ℜ , researches are investigated with non-quadratic Lyapunov and z1 ( t ) , , z p ( t ) are known premise variables. functions [3]–[11]. In this paper, a new stability conditions for Takagi Sugeno The final outputs of the fuzzy systems are: sufficient fuzzy models based on the use of fuzzy Lyapunov r function are presented. This criterion is used with PDC= controller (2) x ( t ) ∑ hi ( z ( t ) ) { Ai x ( t ) + Bi u ( t )} to derive stabilization conditions and relaxed stability ones. These i =1 criteria are expressed in terms of Linear Matrix Inequalities where (LMIs) which can be efficiently solved by using various convex z ( t ) = z1 ( t ) z2 ( t ) z p ( t ) optimization algorithms [12]. r The organization of the paper is as follows. In section 2, we hi z ( t ) = wi z ( t ) wi z ( t ) , present the system description and problem formulation and we i =1 give some preliminaries which are needed to derive results. p Section 3 will be concerned to stability analysis for continuous wi z ( t ) = M ij z j ( t ) for all t. T-S fuzzy systems. Stabilization with PDC controller and relaxed j =1 stability criterion are given in section 4 and 5 respectively. Section The term M i1 z j ( t ) is the grade of membership of 6 concerns the proposed approach to design a T-S fuzzy system with Parallel Distributed Controller (PDC). Finally section 7 z j ( t ) in M i1 makes conclusion. r Notation: Throughout this paper, a real symmetric ∑ wi ( z ( t ) ) 0 Since matrix S > 0 denotes S being a positive definite matrix. The i =1 w ( z ( t ) ) ≥ 0, superscript ‘‘T’’ is used for the transpose of a matrix. i= 1, 2, , r
(
)
(
) ∏
(
i
) ∑ (
(
)
(
)
)
- 43 -
∑ hi ( z ( t ) ) = 1 We have i =1 h ( z ( t ) ) ≥ 0, i
where
r
for all t. i= 1, 2, , r
T T = V k ( x (t ) ) x= ,and Pk 0, k 1, 2, , r (t ) Pk x (t ) ; Pk P= k r
∑εk
The PDC fuzzy controller is represented by
∑ wi ( z ( t ) )Fi x ( t )
This candidate Lyapunov function satisfies
i ) V ( x ( t ) ) is C1 ,
r
− = −∑ hi ( z ( t ) ) Fi x ( t ) (3) u (t ) = r i =1 ∑ wi ( z ( t ) ) i =1
=1
k =1
r
ii ) V (= 0 ) 0 and V ( x ( t ) ) ≥ 0 for x ( t ) ≠ 0,
i =1
iii ) x ( t ) → ∞ ⇒ V ( x ( t ) ) → ∞.
The fuzzy controller design is to determine the local feedback gains Fi in the consequent parts. The open-loop system is given by the equation (4)
(
trajectory of the system (6) is given by:
r
x ( t ) = ∑ hi ( z ( t ) )Ai x ( t )
(4)
r
V ( x (t ) ) = ∑ ε kVk ( x (t ) )
i =1
{
r
}
∑∑ hi ( z ( t ) ) h j ( z ( t ) ) Ai − Bi Fj x ( t )
x ( t )
=i 1 =j 1
The equation (10) can be rewritten as,
∑∑ hi ( z (t ) ) ε k x T (t ) × {AiT Pk r
V ( x (t ) ) =
(5)
(10)
k =1
By substituting (3) into (2), the closed-loop fuzzy system can be represented as: r
)
The time derivative of V x ( t ) with respect to t along the
r
k= 1 = i 1
}
+ Pk A i x (t )
(11) Then,
Lemma 1 [2]
r
r
(
)
{
}
If the number of rules that fire for all t is less than or equal to= s, V ( x (t ) ) x T (t ) ∑∑ hi z (t ) ε k × A iT Pk + Pk A i x (t ) k 1= i 1 =
where 1 s ≤ r , then
(
r r 2 i i 1= i 1 i π j
1 ∑ h ( z (t ) ) − s − 1 ∑ ∑ 2hi ( z (t ) )h j ( z (t ) ) ≥ 0, r
where = ∑ hi ( z (t ) ) 1, i =1
3.
completes the proof.
hi ( z ( t ) ) ≥ 0
4.
STABILIZATION WITH PDC CONTROLLER
Consider the closed-loop system (5) which can be rewritten as
BASIC STABILITY CONDITIONS
r
x (t ) = ∑ hi ( z (t ) ) hi ( z (t ) )G ii x (t )
Consider the open-loop system (6).
i =1
r
x ( t ) = ∑ hi ( z ( t ) )Ai x ( t )
G ij + G ji + 2∑∑ hi ( z (t ) ) h j ( z (t ) ) i =1 i j 2 r
(6)
i =1
The aim of the next section is to find conditions for the stability of the unforced T-S fuzzy system by using the Lyapunov theory. Theorem 1 The Takagi Sugeno fuzzy system (6) is stable if there exist positive definite symmetric matrices Pk , k = 1, 2, , r , such that the following LMIs hold. (7) Pk 0, k ∈ {1, , r }
{A
T i
}
1, , r Pj + Pj A i 0, i , j =
where i, j = 1, 2, , r
(8)
G= Ai − Bi Fj . ij In this section we define a fuzzy Lyapunov function and then consider stability conditions. A sufficient stability condition, for ensuring stability is given follows. Theorem 2 The Takagi-Sugeno system (12) is stable if there exist positive definite symmetric matrices Pk , k = 1, 2, , r , and matrices
{G
r
x T (t ) Pk x (t ) ∑ ε kV k ( x (t ) ), ∑=
= k 1= k 1
(12)
where
Pk 0,
Let consider the Lyapunov function in the following form:
= V ( x (t ) )
x (t ) ,
F1 , , Fr such that the following LMIs hold.
Proof r
)
If (8) holds, then V x (t ) 0 and (6) is stable. This
r
(9)
T ii
k ∈ {1, , r }
Pk + Pk G ii } 0, i , k ∈ {1, , r }
G ij + G ji G ij + G ji Pk + Pk ≤ 0, 2 2 for i , j , k = 1, 2, , r such that i j
(13) (14)
T
(15)
- 44 matrices Pk , k = 1, 2, , r and positive semidefinite matrices
where
G= Ai − Bi Fj ij
Q1 , ,Q r such that the following LMIs hols.
Proof r
(16)
k =1
with
T = V k ( x (t ) ) x= (t ) Pk x (t ) , k 1, 2,, r
(
G ij + G ji G ij + G ji Pk + Pk − Q k ≤ 0, 2 2 for k = 1, 2, , r and i j
(22)
where
s 1 G= Ai − B i Fj ij
where = Pk P ,and Pk ≥ 0, = k 1, 2, , r . T k
(21)
T
Let consider the Lyapunov function in the following form:
V ( x (t ) ) = ∑ ε kV k ( x (t ) )
G iiT Pk + Pk G ii + ( s − 1)Q k 0, i , k ∈ {1, , r }
)
The time derivative of V x ( t ) is given by:
Proof
r
V ( x (t ) ) = ∑ ε kVk ( x (t ) )
Let consider the Lyapunov function in the following form:
k =1
∑ ε {x (t ) P x (t ) + x (t ) P x (t )} r
T
k =1
(
k
k
= ϒ1 ( x , z ) x
where
r
T
r
(t ) ∑∑ h ( z (t ) ) ⋅ ε k k 1= i 1 =
× {G iiT Pk + Pk G ii } x (t ) r
r
G + G ij ji × 2
G ij + G ji Pk + Pk 2
)
The time derivative of V x ( t ) is given by: r
(19)
V ( x (t ) ) = ∑ ε kVk ( x (t ) ) k =1
= ∑ ε k ( x T (t ) Pk x (t ) + x T (t ) Pk x (t ) )
x (t )
(24)
k =1
= k 1= i 1 ij
T
(
k =1
r
ϒ2 ( x , z ) = x (t ) ∑∑∑ hi ( z (t ) ) h j ( z (t ) ) ε k T
x T (t ) Pk x (t ) (23) r
(18)
2 i
k
T where Pk P = = Pk 0, k 1,= 2, , r , ∑ ε k 1 . k ,and
By substituting (12) into (17), we obtain,
V ( x (t ) ) = ϒ1 ( x , z ) + ϒ 2 ( x , z )
r
k k k 1= k 1 =
T
k
r
ε V ( x (t ) ) ∑ ε ) ∑=
(17) = V x (t )
(20)
By substituting (12) into (24), we obtain,
V ( x (t ) ) = ϒ1 ( x , z ) + ϒ 2 ( x , z )
(25)
where r
r
∑∑ (
)
{
}
If (14) and (15) holds, the time derivative of the fuzzy hi2 z (t ) ε k × G iiT Pk + Pk G ii x (t ) = ϒ1 ( x , z ) x T (t ) Lyapunov function is negative. Consequently, we have k 1= i 1 = r r (26) 2 T T = V ( x (t ) ) x (t ) ∑∑ hi ( z (t ) ) ⋅ ε k × (G ii Pk + Pk G ii ) r r T k=1 i=1 = ϒ 2 ( x , z ) x (t ) hi z ( t ) h j z ( t ) ⋅ ε k
{
r
}
+∑∑∑ hi ( z (t ) ) h j ( z (t ) )ε k k=1 i=1 i j
G ij + G ji T G ij + G ji × Pk + Pk x (t ) 2 2 0 and the closed loop fuzzy system (5) is stable. This completes proof.
5.
∑∑∑ (
r
= k 1= i 1ij
G + G ij ji × 2
T
Pk + Pk
G ij + G ji 2
) (
)
x (t )
(27)
If (22) holds, and from Lemma 3, we have r r V ( x (t ) ) ≤ x T (t ) ∑∑ hi2 ( z (t ) ) ⋅ ε k × (G iiT Pk + Pk G ii k=1 i=1
{
r
)} x (t )
r
+∑∑∑ 2hi ( z (t ) ) h j ( z (t ) ) ⋅ ε k x T (t )Q k x (t )
RELAXED STABILITY CONDITIONS
k=1 i=1 i j
≤ ∑∑ hi2 ( z (t ) ) ⋅ ε k × x T (t ) {G iiT Pk + Pk G ii } x (t ) r
In this section we define a fuzzy Lyapunov function and then consider relaxed stability conditions. A sufficient stability condition, for ensuring relaxed stability of closed-loop system is given follows.
r
k=1 i=1
r
r
+ ( s − 1) ∑∑ hi2 ( z (t ) ) ⋅ ε k x T (t )Q k x (t ) k=1 i=1
Theorem 3
= ∑∑ hi2 ( z (t ) ) ⋅ ε k × x T (t ) {G iiT Pk + Pk G ii + ( s − 1)Q k } x (t ) Assume that the number of rules that fire for all t is less than or k=1 i=1 equal to s, where 1 s ≤ r , then, the Takagi-Sugeno system (12) If (21) holds then the time derivative of the fuzzy Lyapunov is stable if there exist positive definite symmetric r
r
function is negative and the closed loop fuzzy system (5) is stable. This terminates the proof.
6.
DESIGN OF PDC CONTROLLER
- 45 Consider relaxed condition stability given by Theorem 3. The design problem to determine gains Fi for continuous fuzzy system is given as X k 0, Y k ≥ 0, and M ik ( i , k = 1, , r ) satisfying
−X k A iT − A i X k + M ikT B iT + B i M ik − ( s − 1)Y k 0,
This section describes the PDC controller design for system (2) with control law(3), by determine Fi gains.
(36)
6.1. Main Design Approach Consider condition stability given by Theorem 2. We define X k = Pk−1 by multiplying the inequality on the left and right by X k = Pk−1 , we obtain the following LMIs conditions that constitute a stable fuzzy controller design problem
X k 0,
k ∈ {1, , r }
(28)
2Y k − X k A iT − A i X k − X k A Tj − A j X k + M Tjk B iT + B i M
jk
+ M ikT B Tj + B j M ik ≥ 0.
(37)
for i j
where
−1 X k P= = F= X k Qk X k k , M ik i X k ,Y k −1 (29) for α , β = 1, , r M i α = M i β ⋅ X β ⋅ X α The feedback gains Fi , matrices Pk , and Q k can be obtained as −1 −1 (39) Pk X= M ik X= PkY k Pk − B j Fi X k ≤ 0. = k , Fi k , Qk
X k A iT + A i X k − X k FiT B iT − B i Fi X k 0,
i , k ∈ {1, , r }
X k A iT + A i X k + X k A Tj + A j X k − X k F jT B iT − B i F j X k − X k FiT B Tj for i , j , k = 1, 2, , r such that (30)
i j
6.3. Decay Rate
Let define M ik = Fi X k , and suppose that we have the following assumption (31) M i α = M i β ⋅ X β−1 ⋅ X α for α , β = 1, , r The Fi gains are deduced through equation Fi = M ik X k−1 for i = 1, , r . Substituting into the above inequalities, we obtain:
X k A iT + A i X k − M ikT B iT − B i M ik 0,
i , k ∈ {1, , r }
(32)
X k A iT + A i X k + X k A Tj + A j X k − M Tjk B iT − B i M
jk
− M ikT B Tj − B j M ik ≤ 0.
for i , j , k = 1, 2, , r such that
(33)
i j
+ M ikT B Tj + B j M ik ≥ 0.
(40)
For all i , k and T
G ij + G ji G ij + G ji Pk + Pk + 2α Pk ≤ 0 2 2 for i j ,α 0
(41)
maximize α
X 1 ,, X r , M 1 ,, M r
Subject to X k 0,
− X k A iT − A i X k + M ikT B iT + B i M ik − 2α X k 0,
(35)
i j
where , M ik Fi X k , = X k P= −1 for α , β = 1, , r M i α = M i β ⋅ X β ⋅ X α The feedback gains Fi and Pk matrix can be obtained as −1 k
−1 = Pk X= M ik X k−1 k , Fi
From the solutions X k and M ik .
6.2. Fuzzy Controller Design Using Relaxed Stability Conditions
(42)
−X k A iT − A i X k − X k A Tj − A j X k + M Tjk B iT + B i M
−X k A iT − A i X k − X k A Tj − A j X k jk
equivalent to G iiT Pk + Pk G ii + 2α Pk 0
The largest lower bound on the decay rate can be found by resolve the problem
The stable fuzzy controller design problem in term of LMI conditions is given as Find X k 0 and M ik ( i , k = 1, , r ) satisfying (34) −X k A iT − A i X k + M ikT B iT + B i M ik 0,
+ M Tjk B iT + B i M
The condition that V ( x (t ) ) ≤ αV ( x (t ) ) for all trajectories is
jk
+ M ikT B Tj + B j M ik − 4α X k ≥ 0. (43)
i j where −1 X k P= Fi X k = k , M ik −1 M i α = M i β ⋅ X β ⋅ X α
for α , β = 1, , r
(44)
6.4. Decay Rate Controller Design Using Relaxed Stability Conditions The condition that V ( x (t ) ) ≤ αV ( x (t ) ) for all trajectories is equivalent to G iiT Pk + Pk G ii + ( s − 1)Q k + 2α Pk 0 For all i , k and
(45)
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G ij + G ji 2
T
G ij + G ji Pk + Pk 2
− Q k + 2α Pk ≤ 0 (46)
The largest lower bound on the decay rate can be found by resolve the problem maximize α
[8]
[9]
X 1 ,, X r , M 1 ,, M r
Subject to [10]
X k 0,Y k ≥ 0 − X k A iT − A i X k + M ikT B iT + B i M ik − ( s − 1)Y k − 2α X k 0,
(47) (48) where −1 X k P= Fi= X k , Y k X k Qk X k = k , M ik (49) −1 for α , β = 1, , r M i α = M i β ⋅ X β ⋅ X α Remark: The condition M i α = M i β ⋅ X β−1 ⋅ X α , for α , β = 1, , r is necessary to determine the Fi gains, another approach take proportionality between state vectors in order to compute these gains [8].
7.
CONCLUSION
This paper provided a new non-quadratic fuzzy Lyapunov function for the stability and stabilization of Takagi-Sugeno fuzzy systems in terms of a combination of the LMI approach and Lyapunov theory. A relaxed stability conditions are derived by the use of the proposed approach. In addition, a new approach to design a T-S fuzzy system with Parallel Distributed Controller (PDC) is presented which is based on relation of proportionality between M αβ matrices in order to determine closed loop gains.
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