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Stability analysis of sampled-data fuzzy controller for nonlinear systems based on switching T–S fuzzy model
Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Stability analysis of sampled-data fuzzy controller for nonlinear systems based on switching T–S fuzzy model H.K. Lam Department of Electronic Engineering, Division of Engineering, King’s College London, Strand, London, WC2R 2LS, United Kingdom
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Article history: Received 31 August 2008 Accepted 23 February 2009 Keywords: Fuzzy control Stability analysis
abstract This paper investigates the system stability of a sampled-data fuzzy-model-based control system, formed by a nonlinear plant and a sampled-data fuzzy controller connected in a closed loop. The sampled-data fuzzy controller has an advantage that it can be implemented using a microcontroller or a digital computer to lower the implementation cost and time. However, discontinuity introduced by the sampling activity complicates the system dynamics and makes the stability analysis difficult compared with the pure continuous-time fuzzy control systems. Moreover, the favourable property of the continuous-time fuzzy control systems which is able to relax the stability analysis result vanishes in the sampled-data fuzzy control systems. A Lyapunov-based approach is employed to derive the LMI-based stability conditions to guarantee the system stability. To facilitate the stability analysis, a switching fuzzy model consisting of some local fuzzy models is employed to represent the nonlinear plant to be controlled. The comparatively less strong nonlinearity of each local fuzzy model eases the satisfaction of the stability conditions. Furthermore, membership functions of both fuzzy model and sampled-data fuzzy controller are considered to alleviate the conservativeness of the stability analysis result. A simulation example is given to illustrate the merits of the proposed approach. © 2009 Published by Elsevier Ltd
1. Introduction The Takagi–Sugeno (T–S) fuzzy model [1,2] is a powerful mathematical tool to represent the nonlinear system. The T–S fuzzy model gives a general framework to represent the nonlinear plants as an average weighted sum of some linear subsystems. As the linear and nonlinear parts of the nonlinear plant are extracted, the T–S fuzzy model exhibits a semi-linear characteristic to facilitate the system stability analysis and controller design. Stability analysis of fuzzy-model-based control systems has been extensively investigated in the past two decades. Basic stability conditions [3,4] were obtained based on the Lyapunov stability theory. The system stability is guaranteed to be stable if there exists a solution to some Lyapunov inequalities. The stability conditions can be cast as some linear matrix inequalities (LMIs) [5] which can be solved numerically by some convex programming techniques. By sharing the same premise rules between the fuzzy model and fuzzy controller, relaxed stability conditions [6] were obtained. Further relaxed stability conditions were reported in [7–12]. In [13,14], a switching fuzzy model, which consists of some local fuzzy models, was proposed to represent the nonlinear systems. The local fuzzy models switch among each other to represent the dynamics of the nonlinear system based on the information of the system states. A local fuzzy controller is designed corresponding to each local fuzzy model. During the operation, the local fuzzy controllers switch among each other to handle the nonlinear system according to the active local fuzzy model. This property eases the satisfaction of the stability conditions as the nonlinearities of the local fuzzy models are less strong compared with that of the nonlinear system operating in the full operating region. E-mail address: [email protected]. 1751-570X/$ – see front matter © 2009 Published by Elsevier Ltd doi:10.1016/j.nahs.2009.02.011
H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
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When a sampled-data fuzzy controller is employed, compared with a continuous-time fuzzy controller [3,4,7–12], it can be implemented by a microcontroller or a digital computer to lower the implementation cost and time. However, the closedloop control system becomes a sampled-data fuzzy control system of which the control signal is kept constant during the sampling period. In such a case, discontinuity introduced by the sampling activity complicates the system dynamics and makes the analysis difficult. In [15], a linear sampled-data system was investigated using the Lyapunov-based approach. In [16–19], a fuzzy-model-based control approach was employed to study the stability of the sampled-data nonlinear systems. An equivalent jump system was proposed to represent the dynamics of the sampled-data fuzzy control systems at the sampling instant. The closed-loop system is guaranteed to be stable if both the sampled-data fuzzy control system (governing the system dynamics during the sampling period) and the jump system (governing the system dynamics at the sampling instant) are both stable subject to a common time-varying solution to some Lyapunov inequalities. In [20], the linear analysis approach in [15] was employed and extended to analyze the stability of nonlinear sampled-data control system. However, the information of the system nonlinearity was not utilized to produce a less conservative stability analysis result. In [21–23], an intelligent digital redesign approach was proposed. The idea is to approximate the nonlinear plant by a discrete-time fuzzy model. Based on the discrete-time fuzzy model, a discrete-time fuzzy controller is then proposed to close the feedback loop. However, the discretization error due to the discrete-time fuzzy model may be a source of system instability. In this paper, a sampled-data fuzzy controller is proposed to control the nonlinear plants. A switching fuzzy model [13] is employed to represent the nonlinear plant by some local fuzzy models. The nonlinearity of the nonlinear plant is distributed into a number of local fuzzy models. Hence, the nonlinearity of each local fuzzy model is less strong compared with that of the nonlinear plant operating in the full operating domain. This favourable property exhibits potential to produce less conservative stability conditions. Corresponding to each local fuzzy model, a local sampled-data fuzzy controller is designed dedicatedly. However, the discontinuity introduced by the sampling activity complicates the system dynamics to make the stability analysis difficult. Moreover, the discontinuity eliminates the favourable properties given by the continuous-time fuzzy-model-based control systems to yield a relaxed stability result. In order to facilitate the analysis, the Lyapunov–Krasovksii function [15] is employed. The property of the switching fuzzy model is made use of to facilitate the design of a sampled-data fuzzy controller and for stability analysis. To alleviate the conservativeness of the stability analysis result, information of the membership functions between the fuzzy model and the sampled-data fuzzy controller is considered. LMI-based stability conditions are derived to guarantee the stability of the sampled-data fuzzy-model-based control systems. This paper is organized as follows. In Section 2, the switching fuzzy model and the sampled-data fuzzy controller are introduced. In Section 3, stability analysis of the sampled-data fuzzy-model-based control systems is conducted. LMI-based stability conditions are derived. In Section 4, a simulation example is given to illustrate the merits of the proposed approach. A conclusion is drawn in Section 5.
2. Switching T–S fuzzy model and sampled-data fuzzy controller The switching fuzzy model and the sample-data fuzzy controller are discussed in this section. A sampled-data fuzzymodel-based control system is formed by a nonlinear plant represented by the switching fuzzy model and a sampled-data fuzzy controller connected in a closed loop.
2.1. Switching fuzzy model Let the union of the operating domains of Φ local fuzzy models be denoted by D, which is the full operating domain of the nonlinear system to be handled. The rule of the ith local fuzzy model is defined as, Rule j: IF f1i (x(t )) is Mji,1 AND . . . AND fΨi (x(t )) is Mji,Ψ THEN x˙ (t ) = Aij x(t ) + Bij u(t )
for x(t ) ∈ Di ; i = 1, 2, . . . , Φ ; j = 1, 2, . . . , p
(1)
where Mji,α is the fuzzy term of the jth rule of the ith local fuzzy model corresponding to the function fαi (x(t )); α = 1, 2, . . . , Ψ ; i = 1, 2, . . . , Φ ; j = 1, 2, . . . , p; p and Ψ denote the numbers of rules and fuzzy terms respectively; Aij ∈ Rn×n and Bij ∈ Rn×m are the known constant system and input matrices of rule j for the ith local fuzzy model respectively; x(t ) ∈ Rn×1 is the system state vector and u(t ) ∈ Rm×1 is the input vector, Di denotes the operating domain of the ith local SΦ fuzzy model and D = i=1 Di . The inferred ith local fuzzy model is defined as, x˙ (t ) =
p X j =1
wij (x(t )) Aij x(t ) + Bij u(t )
(2)
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H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
where Ψ Q
α=1
wij (x(t )) =
µM i
p Q Ψ P k=1 ζ =1
p X
fαi (x(t ))
j,α
(3)
µM i
fζ (x(t )) i
k,ζ
wij (x(t )) ∈ 0,
wij (x(t )) = 1,
1
for all i and j
(4)
j =1
and µM i (fαi (x(t ))) is the grade of the membership function corresponding to the fuzzy term Mji,α . wij (x(t )) is the normalized j,α
grade of membership which can be regarded as a weighting function. Its value indicates the firing strength of rule j in the ith local fuzzy model. It can be seen from (2) that the value of wij (x(t )) determines the contribution of the jth linear sub-system in the ith local fuzzy model to the nonlinear model. If wij (x(t )) has a large value, the system dynamics of the ith fuzzy model will be dominated by the jth linear sub-system. The initial conditions are assumed to be x(t ) = x(0) for t ∈ [−hs 0]. The switching fuzzy model, which governs the dynamics of the nonlinear plant, is defined as, x˙ (t ) =
Φ X
vl (x(t ))
p X
l =1
wlj (x(t )) Alj x(t ) + Blj u(t )
(5)
j=1
where Φ X
vl (x(t )) = 1
(6)
l =1
vl (x(t )) =
1 when x(t ) ∈ Dl 0 otherwise.
(7)
It should be noted that only one local fuzzy model is active, i.e., if vi = 1, then vl = 0 for all i 6= l. 2.2. Sampled-data fuzzy controller A local sampled-data fuzzy controller can be designed based on the ith local fuzzy model operating in the sub-domain Di . The jth rule has the following format: i Rule j: IF g1i (x(tγ )) is Nji,1 AND . . . AND gΩ (x(tγ )) is Nji,Ω
THEN u(t ) = Gij x(tγ ),
tγ ≤ t < tγ +1 ,
x(t ) ∈ Di ,
i = 1, 2, . . . , Φ ; j = 1, 2, . . . , p
(8)
where Nji,β is the fuzzy term of the jth rule of the ith local sampled-data fuzzy controller corresponding to the function gβi (x(t )); β = 1, 2, . . . , Ω ; i = 1, 2, . . . , Φ ; j = 1, 2, . . . , p; Ω denotes the number of rules; Gij ∈ Rm×n denotes the feedback gain for the jth rule of the ith local sampled-data fuzzy controller, tγ = γ hs , γ = 0, 1, 2, . . . , ∞, denotes a sampling time instant; hs = tγ +1 − tγ denotes the constant sampling period. The output of the ith local sampled-data fuzzy controller is defined as, u(t ) =
p X
mij (x(tγ ))Gij x(tγ )
(9)
j =1
where Ω Q
mij (x(tγ )) =
β=1 p
P
Ω Q
k=1 ζ =1 p X
µN i
mij (x(tγ )) = 1,
j,β
gβi (x(tγ ))
(10)
µN i
k,ζ
gζ (x(tγ )) i
mij (x(tγ )) ∈ 0,
1
for all i and j
(11)
j =1
and µN i (gβi (x(tγ ))) is the grade of the membership function corresponding to the fuzzy term Nji,β , mij (x(tγ )) is the j,β
normalized grade of membership. The sampled-data fuzzy controller is defined as,
H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
421
Fig. 1. Block diagram of a sampled-data fuzzy-model-based control system. Φ X
u(t ) =
p X
vl (x(t ))
l =1
Φ X
=
mlj (x(tγ ))Glj x(tγ )
j =1 p X
vl (x(t ))
l =1
mlj (x(tγ ))Glj x(t − τ (t )),
tγ ≤ t < tγ +1
(12)
j =1
where τ (t ) = t − tγ ≤ hs for tγ ≤ t < tγ +1 . It can be seen from (10) that u(t ) = u(tγ ) which holds a constant value for tγ ≤ t < tγ +1 . 3. Stability analysis In the following, the sampled-data fuzzy-model-based control system is formed by the nonlinear plant in the form of (5) and the sampled-data fuzzy controller of (12) connected in a closed loop. Fig. 1 shows the block diagram of the sampled-data fuzzy-model-based control system. For the sake of simplicity, vi (x(t )), wij (x(t )) and mik (x(t )) are denoted by vi , wij and mik respectively. From (5) and (12), using the property that
PΦ
l=1 vi vl
PΦ
i =1
vi =
Pp
j =1
wij =
Pp
k=1
mik =
PΦ
i =1
Pp Pp
k=1 wij mlk = 1 for all i, we have the following property.
j =1
Φ X Φ X
vi vl
i =1 l =1
p X p X
0 Glk Aij
" wij mlk
j=1 k=1
0 −I Bij
0 0 −I
" # # " #Z p Φ 0 x(t ) 0 t X X u(t ) − x˙ (ϕ)dϕ = 0 . vi mik Gik t −τ (t ) 0 x˙ (t ) 0 i =1 k=1
#"
(13)
From [24], we have the following property to estimate the upper bound of the last term in the left hand side of (13).
T
a(t ) − 2a(t ) Tik x˙ (ϕ) ≤ ˙ x(ϕ)
T
ˆ ik R
YTik − TTik
Yik − Tik R
a(t ) x˙ (ϕ)
(14)
where
ˆ ik R YTik
Yik R
≥ 0.
(15)
Let x(t ) u(t ) , x˙ (t )
" a(t ) = P1 P2 P4
" P=
#
0 P3 P5
0 Gik , 0
" T
Tik = Yik = P 0 0 , P6
#
T
ˆ ik = Rˆ ik ∈ R(2n+m)×(2n+m) R
for all i and k,
#
P1 = PT1 ∈ Rn×n > 0, P2 ∈ Rm×n , P3 ∈ Rm×m , P4 ∈ Rn×n , P5 ∈ Rn×m
and P6 ∈ Rn×n
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H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
are constant matrices to be determined. From (15), we have,
#T
x(t ) −2 u(t ) x˙ (t )
"
0 Gik x˙ (ϕ) ≤ 0
" PT
x(t ) 2 u(t ) − t −τ (t ) x˙ (t )
Z
−
−
"
t
Φ X
p X
vi
i=1
k=1
Φ X
p X
vi Z
#T
mik
Z mik
#T
x(t ) u(t ) x˙ (t )
"
x(t ) u(t ) + x˙ (ϕ)T Rx˙ (ϕ) x˙ (t )
" ˆ ik R
#
" # " #T # Z t x(t ) x(t ) 0 ˆ u(t ) Rik u(t ) + x˙ (ϕ)T Rx˙ (ϕ) dϕ Gik x˙ (ϕ)dϕ ≤ t −τ (t ) x˙ (t ) x˙ (t ) 0
" PT
Z
k=1
i=1
#
x(t ) 2 u(t ) t −τ (t ) x˙ (t )
#T
x(t ) 2 u(t ) t −τ (t ) x˙ (t )
#T
"
t
"
t
#T
Z t p Φ x(t ) 0 X X u(t ) Gik x˙ (ϕ)dϕ ≤ mik vi t −τ (t ) x˙ (t ) 0 k=1 i=1
" PT
P
#T
p Φ x(t ) 0 X X Gik x˙ (ϕ)dϕ ≤ hs mik u(t ) vi x˙ (t ) 0 k=1 i =1
" T
"
#
"
#
ˆ ik R
#
x(t ) u(t ) x˙ (t )
" ˆ ik R
x(t ) u(t ) + x˙ (ϕ)T Rx˙ (ϕ) dϕ x˙ (t )
"
#
t
+ t −τ (t )
x˙ (ϕ)T Rx˙ (ϕ)dϕ.
(16)
To investigate the system stability of the sampled-data fuzzy-model-based control system formed by (5) and (12), the following Lyapunov function candidate is employed. V (t ) = x(t )T P1 x(t ) +
0
Z
t
Z
−hs
x˙ (ϕ)T Rx˙ (ϕ)dϕ dσ .
(17)
t +σ
From (5), (12) and (13), we have, V˙ (t ) = x(t )T P1 x˙ (t ) + x˙ (t )T P1 x(t ) + hs x˙ (t )T Rx˙ (t ) −
Z
t
x˙ (ϕ)T Rx˙ (ϕ)dϕ t −hs
" =
# T "
x(t ) u(t ) x˙ (t )
P1 0 0
#T "
0 0 0
+ hs x˙ (t )T Rx˙ (t ) −
0 0 0
0 0 0
0 0 0
I 0 0 + 0 0 0
#
"
0 0 0
#T "
I 0 0
P1 0 0
0 0 0
0 x(t ) 0 u(t ) 0 x˙ (t )
# "
#
t
Z
x˙ (ϕ)T Rx˙ (ϕ)dϕ t −h s
" =
#T " 0 PT 0
x(t ) u(t ) x˙ (t )
0
0 0 0
I 0 0 + 0 0 0
#
"
0 0 0
# #T " Z t x(t ) P u(t ) + hs x˙ (t )T Rx˙ (t ) − x˙ (ϕ)T Rx˙ (ϕ)dϕ. t − h s x˙ (t )
I 0 0
(18)
From (13) and (18), we have,
#T " # x(t ) V˙ (t ) = P u(t ) 0 x˙ (t ) " #T " # " #T " # p X p Φ X Φ x(t ) 0 0 0 0 0 0 x(t ) X X T 0 + Glk −I 0 u(t ) + vi vl wij mlk u(t ) P Glk −I P x˙ (t ) Aij Bij −I Aij Bij −I x˙ (t ) i =1 l =1 j=1 k=1 " #T " # Z Z t p Φ x(t ) 0 t X X −2 vi mik u(t ) PT Gik x˙ (ϕ)dϕ + hs x˙ (t )T Rx˙ (t ) − x˙ (ϕ)T Rx˙ (ϕ)dϕ t −τ (t ) t −hs x˙ (t ) 0 i=1 k=1 " #T " # " #T " # p X p Φ X Φ x(t ) 0 0 I 0 0 I x(t ) X X T 0 + Glk −I 0 u(t ) = vi vl wij mlk u(t ) P Glk −I P x˙ (t ) Aij Bij −I Aij Bij −I x˙ (t ) i =1 l =1 j=1 k=1 " #T " # Z Z t p Φ x(t ) 0 t X X −2 vi mik u(t ) PT Gik x˙ (ϕ)dϕ + hs x˙ (t )T Rx˙ (t ) − x˙ (ϕ)T Rx˙ (ϕ)dϕ. t −τ ( t ) t − h s ˙ x ( t ) 0 i=1 k=1 "
#T " 0 PT 0
x(t ) u(t ) x˙ (t )
0 0 0
I 0 0 + 0 0 0
#
"
0 0 0
I 0 0
(19)
H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
By applying the property of (13) to (19) and the fact that hs ≥ τ (t ) which leads to Rx˙ (ϕ)dϕ , we have, V˙ (t ) ≤
Φ X Φ X
p X p X
vi vl
Φ X
+ hs
vl
x(t ) mlk u(t ) x˙ (t ) k=1
Φ X Φ X
p X p X
vi vl
#T
wij mlk
0 Glk Aij
0 −I Bij
" T
× P
Aij
I 0 0 + Glk −I Aij
#
"
0 −I Bij
I 0 −I
x˙ (ϕ)T Rx˙ (ϕ)dϕ ≤
t −hs
x˙ (ϕ)T
#T " # x(t ) P u(t ) x˙ (t )
#
#T
x(t ) u(t ) x˙ (t )
I 0 0 + Glk −I Aij
#
0 −I Bij
t −τ (t )
Rt
Z t Z t x(t ) ˆ lk u(t ) + x˙ (ϕ)T Rx˙ (ϕ)dϕ x˙ (ϕ)T Rx˙ (ϕ)dϕ + hs x˙ (t )T Rx˙ (t ) − R t − h t −τ ( t ) s x˙ (t ) "
"
j=1 k=1
i=1 l=1
"
p X
l =1
≤
wij mlk
j=1 k=1
i=1 l=1
#T " 0 PT Glk
x(t ) u(t ) x˙ (t )
"
423
Rt
0 −I Bij
"
I 0 −I
#T
0 0 0
0 0 0
" ˆ lk + hs P + hs R
0 x(t ) 0 u(t ) . R x˙ (t )
# "
#
(20)
Let X1 X2 X4
0 X3 X5
" −1
X = P
=
0 0 , X6
#
1 n×n X1 = XT1 = P− > 0, X2 ∈ Rm×n , X3 ∈ Rm×m , 1 ∈ R
x(t ) u(t ) . x˙ (t )
" n×n
X4 ∈ R
, X5 ∈ R
n×m
, X6 ∈ R
n×n
and z(t ) = X
−1
#
From (20), we have, V˙ (t ) ≤
Φ X Φ X
vi vl
i=1 l=1
p X p X
wij mlk z(t )T Qijlk z(t )
(21)
j=1 k=1
where
ˆ ik X, Mik = MTik = XT R 0 Glk Aij
" Qijlk =
0 −I Bij
M = MT = R−1 ,
I 0 0 X + X Glk −I Aij
#
0 −I Bij
"
X4 + XT4 = Nlk − X2 + XT5 Aij X1 + Bij X2 − X4 + XT6
I 0 −I
Nik ∈ Rm×n
#T
∗ −X3 − XT3 Bij X3 − X5
0 0 0
" + hs XT Rˆ lk X + hs XT
0 0 0
and 0 0 X R
#
∗ ∗ −X6 − XT6
T
XT4 XT4 + hs Mlk + hs XT5 R XT5 , XT6 XT6
Gik = Nik X−1 ,
i, j = 1, 2, . . . , p.
(22)
The symbol ‘‘*’’ denotes the transposed element at the corresponding position of the matrix. The information of the membership functions of both the fuzzy model and fuzzy controller is employed to alleviate the conservativeness of the stability analysis. The membership functions of the sampled-data fuzzy controller are designed such that mik − ρi wik > 0 PΦ Pp Pp for all x(t ), x(tγ ), i and k where 0 < ρi < 1. From (21), with the property that j=1 wij = k=1 mik = i=1 vi =
PΦ PΦ i =1
l =1
vi vl
V˙ (t ) ≤
Pp Pp j=1
Φ X Φ X
k=1
vi vl
i=1 l=1
=
Φ X Φ X
wij mlk = 1 for all i, we have,
p X p X
vi vl ρl
p X p X
i=1 l=1
+
wij (mlk + ρl wlk − ρl wlk ) z(t )T Qijlk z(t )
j=1 k=1
Φ X Φ X i=1 l=1
wij wlk z(t )T Qijlk z(t ) +
j=1 k=1
vi vl
p X p X j =1 k =1
Φ X Φ X i=1 l=1
vi vl
p X p X j=1 k=1
wij (mlk − ρl wlk ) z(t )T Λij − Λij z(t )
wij (mlk − ρl wlk ) z(t )T Qijlk z(t )
424
H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
=
Φ X Φ X
vi vl
p X j =1
i =1 l =1
+
Φ X Φ X
vi vl
=
vi vl
p X p X
p X p X
wij (mlk − ρl wlk ) z(t )T Λij z(t ) −
Φ X
vi
i=1
vi wij z(t )T ρi Qijij z(t ) +
Φ X Φ X
vi vl
p X p X
p X Φ X j=1 l=1
wij vl
p X
(mlk − ρl wlk ) z(t )T Λij z(t )
k=1
wij (mlk − ρl wlk ) z(t )T Qijlk + Λij z(t )
j=1 k=1
i=1 l=1
Φ X Φ X
wij (mlk − ρl wlk ) z(t )T Qijlk z(t )
j=1 k=1
i =1 l =1
i =1 j =1
−
Φ X Φ X
j=1 k=1
i =1 l =1 p Φ X X
wij z(t )T ρl Qijij z(t ) +
p
vi vl
X
wij z(t )T (1 − ρl ) Λij z(t )
(23)
j =1
i =1 l =1
where Λij = ΛTij ∈ R(3n+m)×(3n+m) . The proof of the first term of (23) is given in the Appendix. It is due to the property of the switching fuzzy model that only one local fuzzy model is active at any instant (i.e., referring to (8), if vi = 1, vl = 0 for PΦ PΦ PΦ all i 6= l), we have vi = vl = 1 for all i = l leading to i=1 l=1 vi vl = i=1 vi = 1. With this fact and from (21), we have, V˙ (t ) ≤
Φ X
vi
p X
i =1
−
Φ X
j =1
Φ X
vi
Φ X
vi
i =1
vi
i=1
p X
i =1
=
wij z(t )T ρi Qijij z(t ) + wij z(t )T ρi
j =1 p X
p X p X
wij (mik − ρi wik ) z(t )T Qijik + Λij z(t )
j=1 k=1
(1 − ρi ) Λij z(t ) ρi
p X p Φ X X (1 − ρi ) Qijij − Λij z(t ) + vi wij (mik − ρi wik ) z(t )T Qijik + Λij z(t ). (24) ρi i=1 j=1 k=1
wij z(t ) ρi T
j =1
From (24), V˙ (t ) ≤ 0 (equality holds when z(t ) = 0) which implies the asymptotic stability of the sampled-data fuzzy(1−ρ ) model-based control system if Qijij − ρ i Λij < 0 and Qijik + Λij < 0 for all i, j and k. Furthermore, it should be noted that i
hˆ
Rik
the holding of the inequality of (14) requires
h
X 0
0 X1
i
to
X 0
hˆ
Rik
Yik
YTik
0 X1
R
i
YTik
Yik
i
R
≥ 0 for all i and k. Post-multiplying
h
X 0
0 X1
iT
and pre-multiplying
≥ 0, we have,
T
ˆ ik R YTik
Yik R
X 0
0 X1
≥0 =
ˆ ik X XT R X1 YTik X
XT Yik X1 X1 RX1
=
Mik X1 YTik X
∗ X1 M−1 X1
≥0
≥ 0,
i = 1 , 2 , . . . , p.
(25)
It should be noted that (25) is not an LMI due to the existence of the term XM−1 X. With the property that M = MT > 0, we consider the following inequality,
(X1 − ζ M)T M−1 (X1 − ζ M) = XT1 M−1 X1 − ζ XT1 − ζ X1 + ζ 2 M > 0 ⇒ X1 M−1 X1 > 2ζ X1 − ζ 2 M
(26)
where ζ is a non-zero positive scalar. From (25) and (26), it can be seen that the holding of the following LMIs implies the holding of (25).
Mik X1 YTik X
∗ ≥ 0, 2ζ X1 − ζ 2 M
i = 1, 2, . . . , Φ ; k = 1, 2, . . . , p.
(27)
The stability analysis result is summarized in the following theorem. Theorem 1. The sampled-data fuzzy-model-based control systems formed by a continuous-time nonlinear plant in the form of (5) and a sampled-data fuzzy controller of (12) is guaranteed to be asymptotically stable if there exist 0 < ρ i < 1 such that mik (x(tγ )) − ρi wik (x(t )) > 0 for all x(tγ ), x(t ), i and k, a constant non-zero positive sampling period hs , a non-zero positive scalar ζ and matrices X1 = XT1 ∈ Rn×n ,
X2 ∈ Rm×n ,
Mik = MTik ∈ R(2n+m)×(2n+m) ,
X3 ∈ Rm×m ,
X4 ∈ Rn×n ,
M = MT ∈ Rn×n ,
Nik ∈ Rm×n
X5 ∈ Rn×m , and
X6 ∈ Rn×n ,
Λij = ΛTij ∈ R(3n+m)×(3n+m)
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such that the following LMIs are satisfied. X1 > 0; M > 0; (1 − ρi ) Λij < 0, Qijij −
ρi Qijik + Λij < 0,
i = 1, 2, . . . , Φ ; j = 1, 2, . . . , p;
i = 1, 2, . . . , Φ ; j, k = 1, 2, . . . , p; ∗ ≥ 0, i = 1, 2, . . . , Φ ; k = 1, 2, . . . , p 2 2ζ X1 − ζ M
Mik X1 YTik X
where the feedback gains are designed as Gik = Nik X−1 . The above analysis is valid if X is invertible. If there exists a solution to the stability conditions in Theorem 1, we have X1 = XT1 > 0, X3 + XT3 > 0 and X6 + XT6 > 0 which are the sufficient conditions to guarantee that X is invertible. 4. Simulation example A simulation example is given in this section. A sampled-data fuzzy controller of (12) is employed to control a nonlinear plant. Theorem 1 is employed to aid the design of the sampled-data fuzzy controller. Consider the nonlinear plant characterized by three local fuzzy models, each with four fuzzy rules. The rules of the ith local fuzzy model have the following format, Rule j: IF x1 (t ) is Mji,1 AND x2 (t ) is Mji,2 Then x˙ (t ) = Aij x(t ) + Bij u(t ) for x1 (t ) ∈ Di ; i = 1, 2, 3; j = 1, 2, 3, 4
(28)
T where x(t ) = x1 (t ) x2 (t ) . It is assumed that x1 (t ) ∈ −1 1 , x˙ 1 (t ) ∈ −1 1 , x2 (t ) ∈ −2 2 , x˙ 2 (t ) ∈ −2 2 . The operating domain D1 is defined as |x1 (t )| ∈ 0 0.33 , D2 as |x1 (t )| ∈ 0.33 0.66 and D3 as |x1 (t )| ∈ 0.66 1 . The
membership functions are defined as follows.
µM i (x1 (t )) = µM i (x1 (t )) = 0.95e 1,1
−
(|x1 (t )|−ci )2
−
(|x2 (t )|−1)2
22
2,1
µM i (x2 (t )) = µM i (x2 (t )) = 0.95e 1,2
2×22
3,2
,
4,2
3,1
4,1
1,1
,
µM i (x2 (t )) = µM i (x2 (t )) = 1 − µM i (x2 (t )) ; 2,2
µM i (x1 (t )) = µM i (x1 (t )) = 1 − µM i (x1 (t )) ;
1,2
i = 1, 2, 3; c1 = 0.165, c2 = 0.495, c3 = 0.825.
The normalized grades of membership are defined as
µM i (x1 (t )) × µM i (x2 (t )) j,1
wij (x(t )) =
4
P
j,2
,
i = 1, 2, 3; j = 1, 2, 3, 4.
µM i (x1 (t )) × µM i (x2 (t ))
k=1
k,1
k,2
The membership functions of the ith local fuzzy model are shown in Fig. 2. The system and input matrices are defined as
0 ai1
Ai1 = Ai2 =
1 , 0
Ai3 = Ai4 =
0 ai2
1 , 0
Bi1 = Bi3 =
0 bi1
and Bi2 = Bi4 =
0 bi2
where a11 = 11.7707, a12 = 17.2941, b11 = −0.1765, b12 = −0.5117; a21 = 10.9829, a22 = 15.0197, b21 = −0.3298, b22 = −0.2076; a31 = 9.4817, a32 = 13.1953, b31 = −0.5905, b32 = −0.1024. From (5), the switching fuzzy model is defined as, x˙ (t ) =
3 X
vl (x(t ))
l=1
4 X
wlj (x(t )) Alj x(t ) + Blj u(t )
(29)
j =1
where
vi (x1 (t )) =
1 x1 (t ) ∈ Di , i = 1, 2, 3. 0 otherwise.
Based on the switching fuzzy model of (29), a sampled-data fuzzy controller with 3 local sampled-data fuzzy controllers, each with 4 rules, is employed to stabilize the nonlinear plant of (29). Referring to (8), the jth rule of the ith local sampleddata fuzzy controller has the following format:
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(a) Membership functions of the 1st local fuzzy model and sampled-data fuzzy controller. Upper: µM 1 (x1 (t )) = µM 1 (x1 (t )) (solid line), 1,1
2,1
µN 1 (x1 (t )) = µN 1 (x1 (t )) (dotted line). Lower: µM 1 (x1 (t )) 1,1 2,1 3,1 = µM 1 (x1 (t )) (solid line), µN 1 (x1 (t )) = µN 1 (x1 (t )) (dotted line). 4,1
3,1
4,1
(b) Membership functions of the 2nd local fuzzy model and sampled-data fuzzy controller. Upper: µM 2 (x1 (t )) = µM 2 (x1 (t )) (solid line), 2,1
1,1
µN 2 (x1 (t )) = µN 2 (x1 (t )) (dotted line). Lower: µM 2 (x1 (t )) 3,1 1,1 2,1 = µM 2 (x1 (t )) (solid line), µN 2 (x1 (t )) = µN 2 (x1 (t )) (dotted line). 4,1
3,1
4,1
Fig. 2. Membership functions of switching fuzzy model and sampled-data fuzzy controller.
Rule j: IF x1 (t ) is Nji,1 AND x2 (t ) is Nji,2 THEN u(t ) = Gij x(tγ ),
tγ < t ≤ tγ +1 , x1 (t ) ∈ Di , i = 1, 2, 3; j = 1, 2, 3, 4.
(30)
Referring to (12), the sampled-data fuzzy controller is defined as follows. u(t ) =
3 X
vl (x1 (t ))
l =1
=
3 X l =1
4 X
mlj (x(tγ ))Glj x(tγ )
j =1
vl (x(tγ ))
4 X j =1
mlj (x(tγ ))Glj x(t − τ (t )),
tγ < t ≤ tγ +1 .
(31)
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(c) Membership functions of the 3rd local fuzzy model and sampled-data fuzzy controller. Upper: µM 3 (x1 (t )) = µM 3 (x1 (t )) (solid line), 1,1
2,1
µN 3 (x1 (t )) = µN 3 (x1 (t )) (dotted line). Lower: µM 3 (x1 (t )) 1,1 2,1 3,1 = µM 3 (x1 (t )) (solid line), µN 3 (x1 (t )) = µN 3 (x1 (t )) (dotted line). 4,1
3,1
4, 1
(d) Membership functions of the ith local fuzzy model and sampled-data fuzzy controller, i = 1, 2, 3. Upper: µM i (x2 (t )) = µM i (x2 (t )) (solid line), 1, 2
3,2
µN 3 (x2 (t )) = µN 3 (x2 (t )) (dotted line). Lower: µM i (x2 (t )) 2,2 1,2 3,2 = µM i (x2 (t )) (solid line), µN i (x2 (t )) = µN i (x2 (t )) (dotted line). 2,2
4,2
4, 2
Fig. 2. (continued)
The membership functions of the 3 local sampled-data fuzzy controllers are designed as follows.
µN i
1,1
x1 (tγ ) = µN i
2,1
x1 (tγ ) =
x (t ) − c + 0.15 1 γ i 0 . 14 1 x1 (tγ ) − ci − 0.15
µN i
3,1
x1 (tγ ) = µN i
4,1
−0.14
0
x1 (tγ ) = 1 − µN i
1,1
x1 (tγ ) ,
ci − 0.15 ≤ x1 (tγ ) ≤ ci − 0.01 ci − 0.01 < x1 (tγ ) < ci + 0.01 ci + 0.01 ≤ x1 (tγ ) ≤ ci + 0.15 otherwise, c1 = 0.165,
c2 = 0.495,
c3 = 0.825,
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H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
(a) x1 (t ).
(b) x2 (t ). Fig. 3. System responses of the sampled-data fuzzy-model-based control system.
µN i
1,2
µN i
2,2
x2 (tγ ) = µN i
3,2
x2 (tγ ) = µN i
4,2
x2 (tγ ) =
x (t ) − 0.15 1 γ 0.75
0.15 ≤ x1 (tγ ) ≤ 0.9
1
0.9 < x1 (tγ ) < 1.1
0
otherwise,
x1 (tγ ) − 1.85 −0.75
x2 (tγ ) = 1 − µN i
1,2
1.1 ≤ x1 (tγ ) ≤ 1.85
x1 (tγ ) ,
i = 1, 2, 3; j = 1, 2, 3, 4.
The normalized grades of membership are defined as mij (x(tγ )) = P4
µ
Ni j,1
k=1
µ
(x1 (tγ ))×µN i (x2 (tγ ))
Ni k,1
j,2
(x1 (tγ ))×µN i (x2 (tγ ))
, i = 1, 2, 3; j = 1, 2, 3, 4.
k,2
The membership functions of the ith local sampled-data fuzzy controller are shown in Fig. 2. By solving the solution to the stability conditions in Theorem 1 with hs = 0.01s, ζ = 2 and ρi = 0.8, i = 1, 2, 3, using MATLAB LMI toolbox, we obtain the feedback gains as G11 = [102.2848 28.2996], G12 = [69.1148 18.9590], G13 = [124.0610 32.3766], G14 = [76.0269 20.0850]; G21 = [81.0148 22.4583], G22 = [94.5943 26.3441], G23 = [86.5097 23.2746], G24 = [102.3440 27.5273]; G31 = [63.5029 17.6904], G32 = [125.1924 35.9053], G33 = [66.6314 18.1212], G34 = 37.1590]. In this example, the nonlinear plant is assumed to operate in the domain [135.6614 characterized by x˙ 1 (t ) ∈ −1 1 and x˙ 2 (t ) ∈ −2 2 . With this information and considering tγ ≤ t ≤ tγ + hs , we have x1 (t ) = x1 (tγ )+
Rt tγ
x˙ 1 (t )dt which gives the lower and upper bounds as x1 (tγ )−
R tγ +hs tγ
dt = x1 (tγ )− hs = x1 (tγ )− 0.01 and
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(a) u(t ) for 0s ≤ t ≤ 2 s.
(b) u(t ) for 0s ≤ t ≤ 0.5 s. Fig. 4. Control signal of the sampled-data fuzzy controller.
x1 (tγ )+
R tγ +hs
dt = x1 (tγ )+ hs = x1 (tγ )+ 0.01 respectively. Similarly, we have x2 (t ) = x2 (tγ )+
Rt
x˙ 2 (t )dt which gives the t R tγγ+hs lower and upper bounds as x2 (tγ ) − 2 t dt = x2 (tγ ) − 2hs = x2 (tγ ) − 0.02 and x2 (tγ ) + 2 t dt = x2 (tγ ) + 2hs = γ γ x2 (tγ ) + 0.02 respectively. Consequently, for any sampling instant tγ , the values of x1 (t ) and x2 (t ) at or before the next sampling instant are in the range of x1 (tγ ) − 0.01 ≤ x1 (t ) ≤ x1 (tγ ) + 0.01 and x2 (tγ ) − 0.02 ≤ x2 (t ) ≤ x2 (tγ ) + 0.02, respectively, for tγ ≤ t ≤ tγ + hs . It can be seen that there exist ρi = 0.8, i = 1, 2, 3, such that mik (x(tγ )) − ρi wik (x(t )) > 0 for all x(tγ ), x(t ), i and k. The sampled-data fuzzy controller of (31) is applied to control the nonlinear the responses of the plant. Fig. 3 shows system states under different initial system state conditions of x(0) = 1 0 and x(0) = 0.5 0 respectively, and x(t ) = x(0) for t ∈ [−hs 0]. It can be seen that the nonlinear plant can be stabilized successfully. Fig. 4 shows the control tγ
R tγ +hs
signal of the sampled-data fuzzy controller. It can be seen that the control signal is a stepwise function. The amplitude of the control signal is kept constant during the sampling period. The sampled-data fuzzy controller can be implemented using a microcontroller or a digital computer. Furthermore, pure continuous-time theories [3–11] for fuzzy-model-based control systems cannot be applied for design. For comparison purposes, the sampled-data fuzzy controller with hs = 0.005 s (half of the original sampling period) and the continuous-time fuzzy controller using the same feedback gains are employed to handle the nonlinear plant. Fig. 5 shows the system state responses. It can be seen that both fuzzy controllers can stabilize the nonlinear plant. Furthermore, the system state responses are close to those given by the continuous-time fuzzy control system with a smaller value of hs . Considering this figure, it can be seen that the sampled-data fuzzy controller is a good replacement of a continuous-time fuzzy controller to obtain similar system state responses.
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H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
(a) x1 (t ).
(b) x2 (t ). Fig. 5. System responses of the sampled-data fuzzy-model-based control system. Solid line: Sampled-data fuzzy controller with hs = 0.01s, Dotted line: Sampled-data fuzzy controller with hs = 0.005s, Dash-dotted line: Continuous-time fuzzy controller.
5. Conclusion The system stability of the sampled-data fuzzy control systems has been studied. To facilitate the system stability, a switching fuzzy model has been employed to represent the nonlinear system. The switching fuzzy model consists of a number of local fuzzy models which govern the dynamics of the nonlinear plant operating in various sub-domains. By dividing the full operating region into sub-domains, the nonlinear plant can be represented by some local fuzzy models. Corresponding to each local fuzzy model, a local sampled-data fuzzy controller has been proposed to close the feedback loop. A sampled-data fuzzy controller which combines all the local sampled-data fuzzy controllers has been proposed to handle the nonlinear plant. LMI-based stability conditions have been derived based on the Lyapunov stability theory. The favourable property of the switching fuzzy model has been employed to facilitate the stability analysis. As the nonlinearity of each local fuzzy model is less strong, it can ease the satisfaction of the stability conditions. Furthermore, membership functions of the fuzzy model and sampled-data fuzzy controller have been used to alleviate the conservativeness of the stability analysis. A simulation example has been given to illustrate the merits of the proposed approach. Acknowledgements The work described in this paper was supported by King’s College London and an EPSRC grant (Project No. EP/E05627X/1).
H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
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Appendix
The proof of the first term of (23) is given in this appendix. From (22) and (23), we consider the following term, Φ X Φ X
vi vl ρl
i=1 l=1
p X p X
wij wlk z(t )T Qijlk z(t )
j=1 k=1
X4 + XT4 T wij wlk z(t ) vi vl ρl = Nlk − X2 + XT5 j=1 k=1 i=1 l=1 Aij X1 + Bij X2 − X4 + XT6 Φ Φ X X
p X p X
T T T X4 X4 ∗ + hs Mlk + hs XT R XT ∗ z(t ) 5 5 XT6 XT6 −X6 − XT6 ∗ ∗ p X p X T wij wlk −X3 − X3 ∗ j=1 k=1 p p p p XX XX wij wlk Bij X3 − X5 wij wlk −X6 − XT6
∗ −X3 − XT3 Bij X3 − X5
p X p X
wij wlk X4 + XT4 j=1 k=1 p X p Φ X Φ X X wij wlk Nlk − X2 + XT5 = vi vl ρl z(t )T j=1 k=1 i=1 l=1 X p p X wij wlk Aij X1 + Bij X2 − X4 + XT6 j=1 k=1
j=1 k=1
j=1 k=1
T z(t ).
XT4 XT4 + wij wlk hs Mlk + wij wlk hs XT5 R XT5 j=1 k=1 j=1 k=1 XT6 XT6 p X p X
p X p X
By using the property of
Pp
j=1
wij =
Pp
p X
k=1
wlk =
Pp Pp
wij X4 + XT4
j =1
k=1
(A.1)
wij wlk = 1 for all i and l, (A.1) becomes,
∗
j =1 p Φ X Φ X X wlk Nlk − X2 + XT5 vi vl ρl z(t )T k=1 i =1 l =1 X p wij Aij X1 + Bij X2 − X4 + XT6 j =1
∗ T
p
X
wij −X3 − XT3
j =1 p
X
wij Bij X3 − X5
j =1
∗ p X
wij −X6 − X6
j =1
T z(t ).
XT4 XT4 X X T T + wlk hs Mlk + wij hs X5 R X5 k=1 j =1 XT6 XT6 p
p
(A.2)
It is due to the property of the switching fuzzy model that only one local fuzzy model is active at any instant (i.e., referring PΦ PΦ PΦ to (7), if vi = 1, vl = 0 for all i 6= l), we have vi = vl = 1 for all i = l leading to i=1 l=1 vi vl = i=1 vi = 1. Under this condition and from (A.2), we have, p X
Φ X i =1
wij X4 + XT4
j =1 p X T wij Nij − X2 + XT5 vi ρi z(t ) j =1 X p wij Aij X1 + Bij X2 − X4 + XT6 j =1
∗
∗
p
X
wij −X3 − XT3
j =1 p
X j=1
wij Bij X3 − X5
∗ p X j=1
T
wij −X6 − X6
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H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
T p p XT4 X4 X X + wij hs Mij + wij hs XT5 R XT5 z(t ) T T j=1 j=1 X6 X6 T
=
p Φ X X i=1 j=1
=
p Φ X X
vi wij ρi z(t )T
X4 + XT4 Nij − X2 + XT5 Aij X1 + Bij X2 − X4 + XT6
vi wij z(t )T ρi Qijij z(t ).
∗ −X3 − XT3 Bij X3 − X5
T T T ∗ X4 X4 + hs Mij + XT5 R XT5 ∗ z(t ) XT6 XT6 −X6 − XT6 (A.3)
i=1 j=1
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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Stability analysis of sampled-data fuzzy controller for nonlinear systems based on switching T–S fuzzy model H.K. Lam Department of Electronic Engineering, Division of Engineering, King’s College London, Strand, London, WC2R 2LS, United Kingdom
article
info
Article history: Received 31 August 2008 Accepted 23 February 2009 Keywords: Fuzzy control Stability analysis
abstract This paper investigates the system stability of a sampled-data fuzzy-model-based control system, formed by a nonlinear plant and a sampled-data fuzzy controller connected in a closed loop. The sampled-data fuzzy controller has an advantage that it can be implemented using a microcontroller or a digital computer to lower the implementation cost and time. However, discontinuity introduced by the sampling activity complicates the system dynamics and makes the stability analysis difficult compared with the pure continuous-time fuzzy control systems. Moreover, the favourable property of the continuous-time fuzzy control systems which is able to relax the stability analysis result vanishes in the sampled-data fuzzy control systems. A Lyapunov-based approach is employed to derive the LMI-based stability conditions to guarantee the system stability. To facilitate the stability analysis, a switching fuzzy model consisting of some local fuzzy models is employed to represent the nonlinear plant to be controlled. The comparatively less strong nonlinearity of each local fuzzy model eases the satisfaction of the stability conditions. Furthermore, membership functions of both fuzzy model and sampled-data fuzzy controller are considered to alleviate the conservativeness of the stability analysis result. A simulation example is given to illustrate the merits of the proposed approach. © 2009 Published by Elsevier Ltd
1. Introduction The Takagi–Sugeno (T–S) fuzzy model [1,2] is a powerful mathematical tool to represent the nonlinear system. The T–S fuzzy model gives a general framework to represent the nonlinear plants as an average weighted sum of some linear subsystems. As the linear and nonlinear parts of the nonlinear plant are extracted, the T–S fuzzy model exhibits a semi-linear characteristic to facilitate the system stability analysis and controller design. Stability analysis of fuzzy-model-based control systems has been extensively investigated in the past two decades. Basic stability conditions [3,4] were obtained based on the Lyapunov stability theory. The system stability is guaranteed to be stable if there exists a solution to some Lyapunov inequalities. The stability conditions can be cast as some linear matrix inequalities (LMIs) [5] which can be solved numerically by some convex programming techniques. By sharing the same premise rules between the fuzzy model and fuzzy controller, relaxed stability conditions [6] were obtained. Further relaxed stability conditions were reported in [7–12]. In [13,14], a switching fuzzy model, which consists of some local fuzzy models, was proposed to represent the nonlinear systems. The local fuzzy models switch among each other to represent the dynamics of the nonlinear system based on the information of the system states. A local fuzzy controller is designed corresponding to each local fuzzy model. During the operation, the local fuzzy controllers switch among each other to handle the nonlinear system according to the active local fuzzy model. This property eases the satisfaction of the stability conditions as the nonlinearities of the local fuzzy models are less strong compared with that of the nonlinear system operating in the full operating region. E-mail address: [email protected]. 1751-570X/$ – see front matter © 2009 Published by Elsevier Ltd doi:10.1016/j.nahs.2009.02.011
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419
When a sampled-data fuzzy controller is employed, compared with a continuous-time fuzzy controller [3,4,7–12], it can be implemented by a microcontroller or a digital computer to lower the implementation cost and time. However, the closedloop control system becomes a sampled-data fuzzy control system of which the control signal is kept constant during the sampling period. In such a case, discontinuity introduced by the sampling activity complicates the system dynamics and makes the analysis difficult. In [15], a linear sampled-data system was investigated using the Lyapunov-based approach. In [16–19], a fuzzy-model-based control approach was employed to study the stability of the sampled-data nonlinear systems. An equivalent jump system was proposed to represent the dynamics of the sampled-data fuzzy control systems at the sampling instant. The closed-loop system is guaranteed to be stable if both the sampled-data fuzzy control system (governing the system dynamics during the sampling period) and the jump system (governing the system dynamics at the sampling instant) are both stable subject to a common time-varying solution to some Lyapunov inequalities. In [20], the linear analysis approach in [15] was employed and extended to analyze the stability of nonlinear sampled-data control system. However, the information of the system nonlinearity was not utilized to produce a less conservative stability analysis result. In [21–23], an intelligent digital redesign approach was proposed. The idea is to approximate the nonlinear plant by a discrete-time fuzzy model. Based on the discrete-time fuzzy model, a discrete-time fuzzy controller is then proposed to close the feedback loop. However, the discretization error due to the discrete-time fuzzy model may be a source of system instability. In this paper, a sampled-data fuzzy controller is proposed to control the nonlinear plants. A switching fuzzy model [13] is employed to represent the nonlinear plant by some local fuzzy models. The nonlinearity of the nonlinear plant is distributed into a number of local fuzzy models. Hence, the nonlinearity of each local fuzzy model is less strong compared with that of the nonlinear plant operating in the full operating domain. This favourable property exhibits potential to produce less conservative stability conditions. Corresponding to each local fuzzy model, a local sampled-data fuzzy controller is designed dedicatedly. However, the discontinuity introduced by the sampling activity complicates the system dynamics to make the stability analysis difficult. Moreover, the discontinuity eliminates the favourable properties given by the continuous-time fuzzy-model-based control systems to yield a relaxed stability result. In order to facilitate the analysis, the Lyapunov–Krasovksii function [15] is employed. The property of the switching fuzzy model is made use of to facilitate the design of a sampled-data fuzzy controller and for stability analysis. To alleviate the conservativeness of the stability analysis result, information of the membership functions between the fuzzy model and the sampled-data fuzzy controller is considered. LMI-based stability conditions are derived to guarantee the stability of the sampled-data fuzzy-model-based control systems. This paper is organized as follows. In Section 2, the switching fuzzy model and the sampled-data fuzzy controller are introduced. In Section 3, stability analysis of the sampled-data fuzzy-model-based control systems is conducted. LMI-based stability conditions are derived. In Section 4, a simulation example is given to illustrate the merits of the proposed approach. A conclusion is drawn in Section 5.
2. Switching T–S fuzzy model and sampled-data fuzzy controller The switching fuzzy model and the sample-data fuzzy controller are discussed in this section. A sampled-data fuzzymodel-based control system is formed by a nonlinear plant represented by the switching fuzzy model and a sampled-data fuzzy controller connected in a closed loop.
2.1. Switching fuzzy model Let the union of the operating domains of Φ local fuzzy models be denoted by D, which is the full operating domain of the nonlinear system to be handled. The rule of the ith local fuzzy model is defined as, Rule j: IF f1i (x(t )) is Mji,1 AND . . . AND fΨi (x(t )) is Mji,Ψ THEN x˙ (t ) = Aij x(t ) + Bij u(t )
for x(t ) ∈ Di ; i = 1, 2, . . . , Φ ; j = 1, 2, . . . , p
(1)
where Mji,α is the fuzzy term of the jth rule of the ith local fuzzy model corresponding to the function fαi (x(t )); α = 1, 2, . . . , Ψ ; i = 1, 2, . . . , Φ ; j = 1, 2, . . . , p; p and Ψ denote the numbers of rules and fuzzy terms respectively; Aij ∈ Rn×n and Bij ∈ Rn×m are the known constant system and input matrices of rule j for the ith local fuzzy model respectively; x(t ) ∈ Rn×1 is the system state vector and u(t ) ∈ Rm×1 is the input vector, Di denotes the operating domain of the ith local SΦ fuzzy model and D = i=1 Di . The inferred ith local fuzzy model is defined as, x˙ (t ) =
p X j =1
wij (x(t )) Aij x(t ) + Bij u(t )
(2)
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H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
where Ψ Q
α=1
wij (x(t )) =
µM i
p Q Ψ P k=1 ζ =1
p X
fαi (x(t ))
j,α
(3)
µM i
fζ (x(t )) i
k,ζ
wij (x(t )) ∈ 0,
wij (x(t )) = 1,
1
for all i and j
(4)
j =1
and µM i (fαi (x(t ))) is the grade of the membership function corresponding to the fuzzy term Mji,α . wij (x(t )) is the normalized j,α
grade of membership which can be regarded as a weighting function. Its value indicates the firing strength of rule j in the ith local fuzzy model. It can be seen from (2) that the value of wij (x(t )) determines the contribution of the jth linear sub-system in the ith local fuzzy model to the nonlinear model. If wij (x(t )) has a large value, the system dynamics of the ith fuzzy model will be dominated by the jth linear sub-system. The initial conditions are assumed to be x(t ) = x(0) for t ∈ [−hs 0]. The switching fuzzy model, which governs the dynamics of the nonlinear plant, is defined as, x˙ (t ) =
Φ X
vl (x(t ))
p X
l =1
wlj (x(t )) Alj x(t ) + Blj u(t )
(5)
j=1
where Φ X
vl (x(t )) = 1
(6)
l =1
vl (x(t )) =
1 when x(t ) ∈ Dl 0 otherwise.
(7)
It should be noted that only one local fuzzy model is active, i.e., if vi = 1, then vl = 0 for all i 6= l. 2.2. Sampled-data fuzzy controller A local sampled-data fuzzy controller can be designed based on the ith local fuzzy model operating in the sub-domain Di . The jth rule has the following format: i Rule j: IF g1i (x(tγ )) is Nji,1 AND . . . AND gΩ (x(tγ )) is Nji,Ω
THEN u(t ) = Gij x(tγ ),
tγ ≤ t < tγ +1 ,
x(t ) ∈ Di ,
i = 1, 2, . . . , Φ ; j = 1, 2, . . . , p
(8)
where Nji,β is the fuzzy term of the jth rule of the ith local sampled-data fuzzy controller corresponding to the function gβi (x(t )); β = 1, 2, . . . , Ω ; i = 1, 2, . . . , Φ ; j = 1, 2, . . . , p; Ω denotes the number of rules; Gij ∈ Rm×n denotes the feedback gain for the jth rule of the ith local sampled-data fuzzy controller, tγ = γ hs , γ = 0, 1, 2, . . . , ∞, denotes a sampling time instant; hs = tγ +1 − tγ denotes the constant sampling period. The output of the ith local sampled-data fuzzy controller is defined as, u(t ) =
p X
mij (x(tγ ))Gij x(tγ )
(9)
j =1
where Ω Q
mij (x(tγ )) =
β=1 p
P
Ω Q
k=1 ζ =1 p X
µN i
mij (x(tγ )) = 1,
j,β
gβi (x(tγ ))
(10)
µN i
k,ζ
gζ (x(tγ )) i
mij (x(tγ )) ∈ 0,
1
for all i and j
(11)
j =1
and µN i (gβi (x(tγ ))) is the grade of the membership function corresponding to the fuzzy term Nji,β , mij (x(tγ )) is the j,β
normalized grade of membership. The sampled-data fuzzy controller is defined as,
H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
421
Fig. 1. Block diagram of a sampled-data fuzzy-model-based control system. Φ X
u(t ) =
p X
vl (x(t ))
l =1
Φ X
=
mlj (x(tγ ))Glj x(tγ )
j =1 p X
vl (x(t ))
l =1
mlj (x(tγ ))Glj x(t − τ (t )),
tγ ≤ t < tγ +1
(12)
j =1
where τ (t ) = t − tγ ≤ hs for tγ ≤ t < tγ +1 . It can be seen from (10) that u(t ) = u(tγ ) which holds a constant value for tγ ≤ t < tγ +1 . 3. Stability analysis In the following, the sampled-data fuzzy-model-based control system is formed by the nonlinear plant in the form of (5) and the sampled-data fuzzy controller of (12) connected in a closed loop. Fig. 1 shows the block diagram of the sampled-data fuzzy-model-based control system. For the sake of simplicity, vi (x(t )), wij (x(t )) and mik (x(t )) are denoted by vi , wij and mik respectively. From (5) and (12), using the property that
PΦ
l=1 vi vl
PΦ
i =1
vi =
Pp
j =1
wij =
Pp
k=1
mik =
PΦ
i =1
Pp Pp
k=1 wij mlk = 1 for all i, we have the following property.
j =1
Φ X Φ X
vi vl
i =1 l =1
p X p X
0 Glk Aij
" wij mlk
j=1 k=1
0 −I Bij
0 0 −I
" # # " #Z p Φ 0 x(t ) 0 t X X u(t ) − x˙ (ϕ)dϕ = 0 . vi mik Gik t −τ (t ) 0 x˙ (t ) 0 i =1 k=1
#"
(13)
From [24], we have the following property to estimate the upper bound of the last term in the left hand side of (13).
T
a(t ) − 2a(t ) Tik x˙ (ϕ) ≤ ˙ x(ϕ)
T
ˆ ik R
YTik − TTik
Yik − Tik R
a(t ) x˙ (ϕ)
(14)
where
ˆ ik R YTik
Yik R
≥ 0.
(15)
Let x(t ) u(t ) , x˙ (t )
" a(t ) = P1 P2 P4
" P=
#
0 P3 P5
0 Gik , 0
" T
Tik = Yik = P 0 0 , P6
#
T
ˆ ik = Rˆ ik ∈ R(2n+m)×(2n+m) R
for all i and k,
#
P1 = PT1 ∈ Rn×n > 0, P2 ∈ Rm×n , P3 ∈ Rm×m , P4 ∈ Rn×n , P5 ∈ Rn×m
and P6 ∈ Rn×n
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H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
are constant matrices to be determined. From (15), we have,
#T
x(t ) −2 u(t ) x˙ (t )
"
0 Gik x˙ (ϕ) ≤ 0
" PT
x(t ) 2 u(t ) − t −τ (t ) x˙ (t )
Z
−
−
"
t
Φ X
p X
vi
i=1
k=1
Φ X
p X
vi Z
#T
mik
Z mik
#T
x(t ) u(t ) x˙ (t )
"
x(t ) u(t ) + x˙ (ϕ)T Rx˙ (ϕ) x˙ (t )
" ˆ ik R
#
" # " #T # Z t x(t ) x(t ) 0 ˆ u(t ) Rik u(t ) + x˙ (ϕ)T Rx˙ (ϕ) dϕ Gik x˙ (ϕ)dϕ ≤ t −τ (t ) x˙ (t ) x˙ (t ) 0
" PT
Z
k=1
i=1
#
x(t ) 2 u(t ) t −τ (t ) x˙ (t )
#T
x(t ) 2 u(t ) t −τ (t ) x˙ (t )
#T
"
t
"
t
#T
Z t p Φ x(t ) 0 X X u(t ) Gik x˙ (ϕ)dϕ ≤ mik vi t −τ (t ) x˙ (t ) 0 k=1 i=1
" PT
P
#T
p Φ x(t ) 0 X X Gik x˙ (ϕ)dϕ ≤ hs mik u(t ) vi x˙ (t ) 0 k=1 i =1
" T
"
#
"
#
ˆ ik R
#
x(t ) u(t ) x˙ (t )
" ˆ ik R
x(t ) u(t ) + x˙ (ϕ)T Rx˙ (ϕ) dϕ x˙ (t )
"
#
t
+ t −τ (t )
x˙ (ϕ)T Rx˙ (ϕ)dϕ.
(16)
To investigate the system stability of the sampled-data fuzzy-model-based control system formed by (5) and (12), the following Lyapunov function candidate is employed. V (t ) = x(t )T P1 x(t ) +
0
Z
t
Z
−hs
x˙ (ϕ)T Rx˙ (ϕ)dϕ dσ .
(17)
t +σ
From (5), (12) and (13), we have, V˙ (t ) = x(t )T P1 x˙ (t ) + x˙ (t )T P1 x(t ) + hs x˙ (t )T Rx˙ (t ) −
Z
t
x˙ (ϕ)T Rx˙ (ϕ)dϕ t −hs
" =
# T "
x(t ) u(t ) x˙ (t )
P1 0 0
#T "
0 0 0
+ hs x˙ (t )T Rx˙ (t ) −
0 0 0
0 0 0
0 0 0
I 0 0 + 0 0 0
#
"
0 0 0
#T "
I 0 0
P1 0 0
0 0 0
0 x(t ) 0 u(t ) 0 x˙ (t )
# "
#
t
Z
x˙ (ϕ)T Rx˙ (ϕ)dϕ t −h s
" =
#T " 0 PT 0
x(t ) u(t ) x˙ (t )
0
0 0 0
I 0 0 + 0 0 0
#
"
0 0 0
# #T " Z t x(t ) P u(t ) + hs x˙ (t )T Rx˙ (t ) − x˙ (ϕ)T Rx˙ (ϕ)dϕ. t − h s x˙ (t )
I 0 0
(18)
From (13) and (18), we have,
#T " # x(t ) V˙ (t ) = P u(t ) 0 x˙ (t ) " #T " # " #T " # p X p Φ X Φ x(t ) 0 0 0 0 0 0 x(t ) X X T 0 + Glk −I 0 u(t ) + vi vl wij mlk u(t ) P Glk −I P x˙ (t ) Aij Bij −I Aij Bij −I x˙ (t ) i =1 l =1 j=1 k=1 " #T " # Z Z t p Φ x(t ) 0 t X X −2 vi mik u(t ) PT Gik x˙ (ϕ)dϕ + hs x˙ (t )T Rx˙ (t ) − x˙ (ϕ)T Rx˙ (ϕ)dϕ t −τ (t ) t −hs x˙ (t ) 0 i=1 k=1 " #T " # " #T " # p X p Φ X Φ x(t ) 0 0 I 0 0 I x(t ) X X T 0 + Glk −I 0 u(t ) = vi vl wij mlk u(t ) P Glk −I P x˙ (t ) Aij Bij −I Aij Bij −I x˙ (t ) i =1 l =1 j=1 k=1 " #T " # Z Z t p Φ x(t ) 0 t X X −2 vi mik u(t ) PT Gik x˙ (ϕ)dϕ + hs x˙ (t )T Rx˙ (t ) − x˙ (ϕ)T Rx˙ (ϕ)dϕ. t −τ ( t ) t − h s ˙ x ( t ) 0 i=1 k=1 "
#T " 0 PT 0
x(t ) u(t ) x˙ (t )
0 0 0
I 0 0 + 0 0 0
#
"
0 0 0
I 0 0
(19)
H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
By applying the property of (13) to (19) and the fact that hs ≥ τ (t ) which leads to Rx˙ (ϕ)dϕ , we have, V˙ (t ) ≤
Φ X Φ X
p X p X
vi vl
Φ X
+ hs
vl
x(t ) mlk u(t ) x˙ (t ) k=1
Φ X Φ X
p X p X
vi vl
#T
wij mlk
0 Glk Aij
0 −I Bij
" T
× P
Aij
I 0 0 + Glk −I Aij
#
"
0 −I Bij
I 0 −I
x˙ (ϕ)T Rx˙ (ϕ)dϕ ≤
t −hs
x˙ (ϕ)T
#T " # x(t ) P u(t ) x˙ (t )
#
#T
x(t ) u(t ) x˙ (t )
I 0 0 + Glk −I Aij
#
0 −I Bij
t −τ (t )
Rt
Z t Z t x(t ) ˆ lk u(t ) + x˙ (ϕ)T Rx˙ (ϕ)dϕ x˙ (ϕ)T Rx˙ (ϕ)dϕ + hs x˙ (t )T Rx˙ (t ) − R t − h t −τ ( t ) s x˙ (t ) "
"
j=1 k=1
i=1 l=1
"
p X
l =1
≤
wij mlk
j=1 k=1
i=1 l=1
#T " 0 PT Glk
x(t ) u(t ) x˙ (t )
"
423
Rt
0 −I Bij
"
I 0 −I
#T
0 0 0
0 0 0
" ˆ lk + hs P + hs R
0 x(t ) 0 u(t ) . R x˙ (t )
# "
#
(20)
Let X1 X2 X4
0 X3 X5
" −1
X = P
=
0 0 , X6
#
1 n×n X1 = XT1 = P− > 0, X2 ∈ Rm×n , X3 ∈ Rm×m , 1 ∈ R
x(t ) u(t ) . x˙ (t )
" n×n
X4 ∈ R
, X5 ∈ R
n×m
, X6 ∈ R
n×n
and z(t ) = X
−1
#
From (20), we have, V˙ (t ) ≤
Φ X Φ X
vi vl
i=1 l=1
p X p X
wij mlk z(t )T Qijlk z(t )
(21)
j=1 k=1
where
ˆ ik X, Mik = MTik = XT R 0 Glk Aij
" Qijlk =
0 −I Bij
M = MT = R−1 ,
I 0 0 X + X Glk −I Aij
#
0 −I Bij
"
X4 + XT4 = Nlk − X2 + XT5 Aij X1 + Bij X2 − X4 + XT6
I 0 −I
Nik ∈ Rm×n
#T
∗ −X3 − XT3 Bij X3 − X5
0 0 0
" + hs XT Rˆ lk X + hs XT
0 0 0
and 0 0 X R
#
∗ ∗ −X6 − XT6
T
XT4 XT4 + hs Mlk + hs XT5 R XT5 , XT6 XT6
Gik = Nik X−1 ,
i, j = 1, 2, . . . , p.
(22)
The symbol ‘‘*’’ denotes the transposed element at the corresponding position of the matrix. The information of the membership functions of both the fuzzy model and fuzzy controller is employed to alleviate the conservativeness of the stability analysis. The membership functions of the sampled-data fuzzy controller are designed such that mik − ρi wik > 0 PΦ Pp Pp for all x(t ), x(tγ ), i and k where 0 < ρi < 1. From (21), with the property that j=1 wij = k=1 mik = i=1 vi =
PΦ PΦ i =1
l =1
vi vl
V˙ (t ) ≤
Pp Pp j=1
Φ X Φ X
k=1
vi vl
i=1 l=1
=
Φ X Φ X
wij mlk = 1 for all i, we have,
p X p X
vi vl ρl
p X p X
i=1 l=1
+
wij (mlk + ρl wlk − ρl wlk ) z(t )T Qijlk z(t )
j=1 k=1
Φ X Φ X i=1 l=1
wij wlk z(t )T Qijlk z(t ) +
j=1 k=1
vi vl
p X p X j =1 k =1
Φ X Φ X i=1 l=1
vi vl
p X p X j=1 k=1
wij (mlk − ρl wlk ) z(t )T Λij − Λij z(t )
wij (mlk − ρl wlk ) z(t )T Qijlk z(t )
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H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
=
Φ X Φ X
vi vl
p X j =1
i =1 l =1
+
Φ X Φ X
vi vl
=
vi vl
p X p X
p X p X
wij (mlk − ρl wlk ) z(t )T Λij z(t ) −
Φ X
vi
i=1
vi wij z(t )T ρi Qijij z(t ) +
Φ X Φ X
vi vl
p X p X
p X Φ X j=1 l=1
wij vl
p X
(mlk − ρl wlk ) z(t )T Λij z(t )
k=1
wij (mlk − ρl wlk ) z(t )T Qijlk + Λij z(t )
j=1 k=1
i=1 l=1
Φ X Φ X
wij (mlk − ρl wlk ) z(t )T Qijlk z(t )
j=1 k=1
i =1 l =1
i =1 j =1
−
Φ X Φ X
j=1 k=1
i =1 l =1 p Φ X X
wij z(t )T ρl Qijij z(t ) +
p
vi vl
X
wij z(t )T (1 − ρl ) Λij z(t )
(23)
j =1
i =1 l =1
where Λij = ΛTij ∈ R(3n+m)×(3n+m) . The proof of the first term of (23) is given in the Appendix. It is due to the property of the switching fuzzy model that only one local fuzzy model is active at any instant (i.e., referring to (8), if vi = 1, vl = 0 for PΦ PΦ PΦ all i 6= l), we have vi = vl = 1 for all i = l leading to i=1 l=1 vi vl = i=1 vi = 1. With this fact and from (21), we have, V˙ (t ) ≤
Φ X
vi
p X
i =1
−
Φ X
j =1
Φ X
vi
Φ X
vi
i =1
vi
i=1
p X
i =1
=
wij z(t )T ρi Qijij z(t ) + wij z(t )T ρi
j =1 p X
p X p X
wij (mik − ρi wik ) z(t )T Qijik + Λij z(t )
j=1 k=1
(1 − ρi ) Λij z(t ) ρi
p X p Φ X X (1 − ρi ) Qijij − Λij z(t ) + vi wij (mik − ρi wik ) z(t )T Qijik + Λij z(t ). (24) ρi i=1 j=1 k=1
wij z(t ) ρi T
j =1
From (24), V˙ (t ) ≤ 0 (equality holds when z(t ) = 0) which implies the asymptotic stability of the sampled-data fuzzy(1−ρ ) model-based control system if Qijij − ρ i Λij < 0 and Qijik + Λij < 0 for all i, j and k. Furthermore, it should be noted that i
hˆ
Rik
the holding of the inequality of (14) requires
h
X 0
0 X1
i
to
X 0
hˆ
Rik
Yik
YTik
0 X1
R
i
YTik
Yik
i
R
≥ 0 for all i and k. Post-multiplying
h
X 0
0 X1
iT
and pre-multiplying
≥ 0, we have,
T
ˆ ik R YTik
Yik R
X 0
0 X1
≥0 =
ˆ ik X XT R X1 YTik X
XT Yik X1 X1 RX1
=
Mik X1 YTik X
∗ X1 M−1 X1
≥0
≥ 0,
i = 1 , 2 , . . . , p.
(25)
It should be noted that (25) is not an LMI due to the existence of the term XM−1 X. With the property that M = MT > 0, we consider the following inequality,
(X1 − ζ M)T M−1 (X1 − ζ M) = XT1 M−1 X1 − ζ XT1 − ζ X1 + ζ 2 M > 0 ⇒ X1 M−1 X1 > 2ζ X1 − ζ 2 M
(26)
where ζ is a non-zero positive scalar. From (25) and (26), it can be seen that the holding of the following LMIs implies the holding of (25).
Mik X1 YTik X
∗ ≥ 0, 2ζ X1 − ζ 2 M
i = 1, 2, . . . , Φ ; k = 1, 2, . . . , p.
(27)
The stability analysis result is summarized in the following theorem. Theorem 1. The sampled-data fuzzy-model-based control systems formed by a continuous-time nonlinear plant in the form of (5) and a sampled-data fuzzy controller of (12) is guaranteed to be asymptotically stable if there exist 0 < ρ i < 1 such that mik (x(tγ )) − ρi wik (x(t )) > 0 for all x(tγ ), x(t ), i and k, a constant non-zero positive sampling period hs , a non-zero positive scalar ζ and matrices X1 = XT1 ∈ Rn×n ,
X2 ∈ Rm×n ,
Mik = MTik ∈ R(2n+m)×(2n+m) ,
X3 ∈ Rm×m ,
X4 ∈ Rn×n ,
M = MT ∈ Rn×n ,
Nik ∈ Rm×n
X5 ∈ Rn×m , and
X6 ∈ Rn×n ,
Λij = ΛTij ∈ R(3n+m)×(3n+m)
H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
425
such that the following LMIs are satisfied. X1 > 0; M > 0; (1 − ρi ) Λij < 0, Qijij −
ρi Qijik + Λij < 0,
i = 1, 2, . . . , Φ ; j = 1, 2, . . . , p;
i = 1, 2, . . . , Φ ; j, k = 1, 2, . . . , p; ∗ ≥ 0, i = 1, 2, . . . , Φ ; k = 1, 2, . . . , p 2 2ζ X1 − ζ M
Mik X1 YTik X
where the feedback gains are designed as Gik = Nik X−1 . The above analysis is valid if X is invertible. If there exists a solution to the stability conditions in Theorem 1, we have X1 = XT1 > 0, X3 + XT3 > 0 and X6 + XT6 > 0 which are the sufficient conditions to guarantee that X is invertible. 4. Simulation example A simulation example is given in this section. A sampled-data fuzzy controller of (12) is employed to control a nonlinear plant. Theorem 1 is employed to aid the design of the sampled-data fuzzy controller. Consider the nonlinear plant characterized by three local fuzzy models, each with four fuzzy rules. The rules of the ith local fuzzy model have the following format, Rule j: IF x1 (t ) is Mji,1 AND x2 (t ) is Mji,2 Then x˙ (t ) = Aij x(t ) + Bij u(t ) for x1 (t ) ∈ Di ; i = 1, 2, 3; j = 1, 2, 3, 4
(28)
T where x(t ) = x1 (t ) x2 (t ) . It is assumed that x1 (t ) ∈ −1 1 , x˙ 1 (t ) ∈ −1 1 , x2 (t ) ∈ −2 2 , x˙ 2 (t ) ∈ −2 2 . The operating domain D1 is defined as |x1 (t )| ∈ 0 0.33 , D2 as |x1 (t )| ∈ 0.33 0.66 and D3 as |x1 (t )| ∈ 0.66 1 . The
membership functions are defined as follows.
µM i (x1 (t )) = µM i (x1 (t )) = 0.95e 1,1
−
(|x1 (t )|−ci )2
−
(|x2 (t )|−1)2
22
2,1
µM i (x2 (t )) = µM i (x2 (t )) = 0.95e 1,2
2×22
3,2
,
4,2
3,1
4,1
1,1
,
µM i (x2 (t )) = µM i (x2 (t )) = 1 − µM i (x2 (t )) ; 2,2
µM i (x1 (t )) = µM i (x1 (t )) = 1 − µM i (x1 (t )) ;
1,2
i = 1, 2, 3; c1 = 0.165, c2 = 0.495, c3 = 0.825.
The normalized grades of membership are defined as
µM i (x1 (t )) × µM i (x2 (t )) j,1
wij (x(t )) =
4
P
j,2
,
i = 1, 2, 3; j = 1, 2, 3, 4.
µM i (x1 (t )) × µM i (x2 (t ))
k=1
k,1
k,2
The membership functions of the ith local fuzzy model are shown in Fig. 2. The system and input matrices are defined as
0 ai1
Ai1 = Ai2 =
1 , 0
Ai3 = Ai4 =
0 ai2
1 , 0
Bi1 = Bi3 =
0 bi1
and Bi2 = Bi4 =
0 bi2
where a11 = 11.7707, a12 = 17.2941, b11 = −0.1765, b12 = −0.5117; a21 = 10.9829, a22 = 15.0197, b21 = −0.3298, b22 = −0.2076; a31 = 9.4817, a32 = 13.1953, b31 = −0.5905, b32 = −0.1024. From (5), the switching fuzzy model is defined as, x˙ (t ) =
3 X
vl (x(t ))
l=1
4 X
wlj (x(t )) Alj x(t ) + Blj u(t )
(29)
j =1
where
vi (x1 (t )) =
1 x1 (t ) ∈ Di , i = 1, 2, 3. 0 otherwise.
Based on the switching fuzzy model of (29), a sampled-data fuzzy controller with 3 local sampled-data fuzzy controllers, each with 4 rules, is employed to stabilize the nonlinear plant of (29). Referring to (8), the jth rule of the ith local sampleddata fuzzy controller has the following format:
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(a) Membership functions of the 1st local fuzzy model and sampled-data fuzzy controller. Upper: µM 1 (x1 (t )) = µM 1 (x1 (t )) (solid line), 1,1
2,1
µN 1 (x1 (t )) = µN 1 (x1 (t )) (dotted line). Lower: µM 1 (x1 (t )) 1,1 2,1 3,1 = µM 1 (x1 (t )) (solid line), µN 1 (x1 (t )) = µN 1 (x1 (t )) (dotted line). 4,1
3,1
4,1
(b) Membership functions of the 2nd local fuzzy model and sampled-data fuzzy controller. Upper: µM 2 (x1 (t )) = µM 2 (x1 (t )) (solid line), 2,1
1,1
µN 2 (x1 (t )) = µN 2 (x1 (t )) (dotted line). Lower: µM 2 (x1 (t )) 3,1 1,1 2,1 = µM 2 (x1 (t )) (solid line), µN 2 (x1 (t )) = µN 2 (x1 (t )) (dotted line). 4,1
3,1
4,1
Fig. 2. Membership functions of switching fuzzy model and sampled-data fuzzy controller.
Rule j: IF x1 (t ) is Nji,1 AND x2 (t ) is Nji,2 THEN u(t ) = Gij x(tγ ),
tγ < t ≤ tγ +1 , x1 (t ) ∈ Di , i = 1, 2, 3; j = 1, 2, 3, 4.
(30)
Referring to (12), the sampled-data fuzzy controller is defined as follows. u(t ) =
3 X
vl (x1 (t ))
l =1
=
3 X l =1
4 X
mlj (x(tγ ))Glj x(tγ )
j =1
vl (x(tγ ))
4 X j =1
mlj (x(tγ ))Glj x(t − τ (t )),
tγ < t ≤ tγ +1 .
(31)
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427
(c) Membership functions of the 3rd local fuzzy model and sampled-data fuzzy controller. Upper: µM 3 (x1 (t )) = µM 3 (x1 (t )) (solid line), 1,1
2,1
µN 3 (x1 (t )) = µN 3 (x1 (t )) (dotted line). Lower: µM 3 (x1 (t )) 1,1 2,1 3,1 = µM 3 (x1 (t )) (solid line), µN 3 (x1 (t )) = µN 3 (x1 (t )) (dotted line). 4,1
3,1
4, 1
(d) Membership functions of the ith local fuzzy model and sampled-data fuzzy controller, i = 1, 2, 3. Upper: µM i (x2 (t )) = µM i (x2 (t )) (solid line), 1, 2
3,2
µN 3 (x2 (t )) = µN 3 (x2 (t )) (dotted line). Lower: µM i (x2 (t )) 2,2 1,2 3,2 = µM i (x2 (t )) (solid line), µN i (x2 (t )) = µN i (x2 (t )) (dotted line). 2,2
4,2
4, 2
Fig. 2. (continued)
The membership functions of the 3 local sampled-data fuzzy controllers are designed as follows.
µN i
1,1
x1 (tγ ) = µN i
2,1
x1 (tγ ) =
x (t ) − c + 0.15 1 γ i 0 . 14 1 x1 (tγ ) − ci − 0.15
µN i
3,1
x1 (tγ ) = µN i
4,1
−0.14
0
x1 (tγ ) = 1 − µN i
1,1
x1 (tγ ) ,
ci − 0.15 ≤ x1 (tγ ) ≤ ci − 0.01 ci − 0.01 < x1 (tγ ) < ci + 0.01 ci + 0.01 ≤ x1 (tγ ) ≤ ci + 0.15 otherwise, c1 = 0.165,
c2 = 0.495,
c3 = 0.825,
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H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
(a) x1 (t ).
(b) x2 (t ). Fig. 3. System responses of the sampled-data fuzzy-model-based control system.
µN i
1,2
µN i
2,2
x2 (tγ ) = µN i
3,2
x2 (tγ ) = µN i
4,2
x2 (tγ ) =
x (t ) − 0.15 1 γ 0.75
0.15 ≤ x1 (tγ ) ≤ 0.9
1
0.9 < x1 (tγ ) < 1.1
0
otherwise,
x1 (tγ ) − 1.85 −0.75
x2 (tγ ) = 1 − µN i
1,2
1.1 ≤ x1 (tγ ) ≤ 1.85
x1 (tγ ) ,
i = 1, 2, 3; j = 1, 2, 3, 4.
The normalized grades of membership are defined as mij (x(tγ )) = P4
µ
Ni j,1
k=1
µ
(x1 (tγ ))×µN i (x2 (tγ ))
Ni k,1
j,2
(x1 (tγ ))×µN i (x2 (tγ ))
, i = 1, 2, 3; j = 1, 2, 3, 4.
k,2
The membership functions of the ith local sampled-data fuzzy controller are shown in Fig. 2. By solving the solution to the stability conditions in Theorem 1 with hs = 0.01s, ζ = 2 and ρi = 0.8, i = 1, 2, 3, using MATLAB LMI toolbox, we obtain the feedback gains as G11 = [102.2848 28.2996], G12 = [69.1148 18.9590], G13 = [124.0610 32.3766], G14 = [76.0269 20.0850]; G21 = [81.0148 22.4583], G22 = [94.5943 26.3441], G23 = [86.5097 23.2746], G24 = [102.3440 27.5273]; G31 = [63.5029 17.6904], G32 = [125.1924 35.9053], G33 = [66.6314 18.1212], G34 = 37.1590]. In this example, the nonlinear plant is assumed to operate in the domain [135.6614 characterized by x˙ 1 (t ) ∈ −1 1 and x˙ 2 (t ) ∈ −2 2 . With this information and considering tγ ≤ t ≤ tγ + hs , we have x1 (t ) = x1 (tγ )+
Rt tγ
x˙ 1 (t )dt which gives the lower and upper bounds as x1 (tγ )−
R tγ +hs tγ
dt = x1 (tγ )− hs = x1 (tγ )− 0.01 and
H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
429
(a) u(t ) for 0s ≤ t ≤ 2 s.
(b) u(t ) for 0s ≤ t ≤ 0.5 s. Fig. 4. Control signal of the sampled-data fuzzy controller.
x1 (tγ )+
R tγ +hs
dt = x1 (tγ )+ hs = x1 (tγ )+ 0.01 respectively. Similarly, we have x2 (t ) = x2 (tγ )+
Rt
x˙ 2 (t )dt which gives the t R tγγ+hs lower and upper bounds as x2 (tγ ) − 2 t dt = x2 (tγ ) − 2hs = x2 (tγ ) − 0.02 and x2 (tγ ) + 2 t dt = x2 (tγ ) + 2hs = γ γ x2 (tγ ) + 0.02 respectively. Consequently, for any sampling instant tγ , the values of x1 (t ) and x2 (t ) at or before the next sampling instant are in the range of x1 (tγ ) − 0.01 ≤ x1 (t ) ≤ x1 (tγ ) + 0.01 and x2 (tγ ) − 0.02 ≤ x2 (t ) ≤ x2 (tγ ) + 0.02, respectively, for tγ ≤ t ≤ tγ + hs . It can be seen that there exist ρi = 0.8, i = 1, 2, 3, such that mik (x(tγ )) − ρi wik (x(t )) > 0 for all x(tγ ), x(t ), i and k. The sampled-data fuzzy controller of (31) is applied to control the nonlinear the responses of the plant. Fig. 3 shows system states under different initial system state conditions of x(0) = 1 0 and x(0) = 0.5 0 respectively, and x(t ) = x(0) for t ∈ [−hs 0]. It can be seen that the nonlinear plant can be stabilized successfully. Fig. 4 shows the control tγ
R tγ +hs
signal of the sampled-data fuzzy controller. It can be seen that the control signal is a stepwise function. The amplitude of the control signal is kept constant during the sampling period. The sampled-data fuzzy controller can be implemented using a microcontroller or a digital computer. Furthermore, pure continuous-time theories [3–11] for fuzzy-model-based control systems cannot be applied for design. For comparison purposes, the sampled-data fuzzy controller with hs = 0.005 s (half of the original sampling period) and the continuous-time fuzzy controller using the same feedback gains are employed to handle the nonlinear plant. Fig. 5 shows the system state responses. It can be seen that both fuzzy controllers can stabilize the nonlinear plant. Furthermore, the system state responses are close to those given by the continuous-time fuzzy control system with a smaller value of hs . Considering this figure, it can be seen that the sampled-data fuzzy controller is a good replacement of a continuous-time fuzzy controller to obtain similar system state responses.
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H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
(a) x1 (t ).
(b) x2 (t ). Fig. 5. System responses of the sampled-data fuzzy-model-based control system. Solid line: Sampled-data fuzzy controller with hs = 0.01s, Dotted line: Sampled-data fuzzy controller with hs = 0.005s, Dash-dotted line: Continuous-time fuzzy controller.
5. Conclusion The system stability of the sampled-data fuzzy control systems has been studied. To facilitate the system stability, a switching fuzzy model has been employed to represent the nonlinear system. The switching fuzzy model consists of a number of local fuzzy models which govern the dynamics of the nonlinear plant operating in various sub-domains. By dividing the full operating region into sub-domains, the nonlinear plant can be represented by some local fuzzy models. Corresponding to each local fuzzy model, a local sampled-data fuzzy controller has been proposed to close the feedback loop. A sampled-data fuzzy controller which combines all the local sampled-data fuzzy controllers has been proposed to handle the nonlinear plant. LMI-based stability conditions have been derived based on the Lyapunov stability theory. The favourable property of the switching fuzzy model has been employed to facilitate the stability analysis. As the nonlinearity of each local fuzzy model is less strong, it can ease the satisfaction of the stability conditions. Furthermore, membership functions of the fuzzy model and sampled-data fuzzy controller have been used to alleviate the conservativeness of the stability analysis. A simulation example has been given to illustrate the merits of the proposed approach. Acknowledgements The work described in this paper was supported by King’s College London and an EPSRC grant (Project No. EP/E05627X/1).
H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
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Appendix
The proof of the first term of (23) is given in this appendix. From (22) and (23), we consider the following term, Φ X Φ X
vi vl ρl
i=1 l=1
p X p X
wij wlk z(t )T Qijlk z(t )
j=1 k=1
X4 + XT4 T wij wlk z(t ) vi vl ρl = Nlk − X2 + XT5 j=1 k=1 i=1 l=1 Aij X1 + Bij X2 − X4 + XT6 Φ Φ X X
p X p X
T T T X4 X4 ∗ + hs Mlk + hs XT R XT ∗ z(t ) 5 5 XT6 XT6 −X6 − XT6 ∗ ∗ p X p X T wij wlk −X3 − X3 ∗ j=1 k=1 p p p p XX XX wij wlk Bij X3 − X5 wij wlk −X6 − XT6
∗ −X3 − XT3 Bij X3 − X5
p X p X
wij wlk X4 + XT4 j=1 k=1 p X p Φ X Φ X X wij wlk Nlk − X2 + XT5 = vi vl ρl z(t )T j=1 k=1 i=1 l=1 X p p X wij wlk Aij X1 + Bij X2 − X4 + XT6 j=1 k=1
j=1 k=1
j=1 k=1
T z(t ).
XT4 XT4 + wij wlk hs Mlk + wij wlk hs XT5 R XT5 j=1 k=1 j=1 k=1 XT6 XT6 p X p X
p X p X
By using the property of
Pp
j=1
wij =
Pp
p X
k=1
wlk =
Pp Pp
wij X4 + XT4
j =1
k=1
(A.1)
wij wlk = 1 for all i and l, (A.1) becomes,
∗
j =1 p Φ X Φ X X wlk Nlk − X2 + XT5 vi vl ρl z(t )T k=1 i =1 l =1 X p wij Aij X1 + Bij X2 − X4 + XT6 j =1
∗ T
p
X
wij −X3 − XT3
j =1 p
X
wij Bij X3 − X5
j =1
∗ p X
wij −X6 − X6
j =1
T z(t ).
XT4 XT4 X X T T + wlk hs Mlk + wij hs X5 R X5 k=1 j =1 XT6 XT6 p
p
(A.2)
It is due to the property of the switching fuzzy model that only one local fuzzy model is active at any instant (i.e., referring PΦ PΦ PΦ to (7), if vi = 1, vl = 0 for all i 6= l), we have vi = vl = 1 for all i = l leading to i=1 l=1 vi vl = i=1 vi = 1. Under this condition and from (A.2), we have, p X
Φ X i =1
wij X4 + XT4
j =1 p X T wij Nij − X2 + XT5 vi ρi z(t ) j =1 X p wij Aij X1 + Bij X2 − X4 + XT6 j =1
∗
∗
p
X
wij −X3 − XT3
j =1 p
X j=1
wij Bij X3 − X5
∗ p X j=1
T
wij −X6 − X6
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H.K. Lam / Nonlinear Analysis: Hybrid Systems 3 (2009) 418–432
T p p XT4 X4 X X + wij hs Mij + wij hs XT5 R XT5 z(t ) T T j=1 j=1 X6 X6 T
=
p Φ X X i=1 j=1
=
p Φ X X
vi wij ρi z(t )T
X4 + XT4 Nij − X2 + XT5 Aij X1 + Bij X2 − X4 + XT6
vi wij z(t )T ρi Qijij z(t ).
∗ −X3 − XT3 Bij X3 − X5
T T T ∗ X4 X4 + hs Mij + XT5 R XT5 ∗ z(t ) XT6 XT6 −X6 − XT6 (A.3)
i=1 j=1
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