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Dynamic Parallel Distributed Compensation for Takagi-Sugeno Fuzzy Systems: An LMI Approach Jing Li Hua O. Wang David Niemann
Laboratory for Intelligent and Nonlinear Control (LINC) Department of Electrical and Computer Engineering Duke University, Durham, NC 27708, U.S.A
Kazuo Tanaka
Department of Mechanical and Control Engineering University of Electro-Communications 1-5-1 Chofugaoka, Chofu, Tokyo 182, Japan
Abstract This paper presents a uni ed systematic framework for designing dynamic feedback controllers for nonlinear systems described by Takagi-Sugeno (T-S) models. Both stabilization and multi-objective control synthesis problems have been addressed. The control laws are in the form of so-called dynamic parallel distributed compensation (DPDC) which are essentially nonlinear dynamic feedback controllers. The associated control synthesis problems are formulated as LMI problems, i.e., the parameters of the DPDC controllers are obtained from a set of LMI conditions. The approach in this paper can also be applied to hybrid or switching systems.
Keywords: Fuzzy control, LMI, dynamic feedback.
Corresponding author. E-mail: [email protected]; Tel: (919) 660-5273; Fax: (919) 660-5293.
1 Introduction Recently Takagi-Sugeno fuzzy models [11] have received a great deal of attention. The appeal of the T-S model lies in that the stability and performance characteristics of the system represented by a T-S model can be analyzed using a Lyapunov function approach [14] [15] [16] [7] [12] [17]. A further and signi cant step has also been taken to utilize Lyapunov-function based control design techniques to the control synthesis problem for T-S models. The so-called parallel distributed compensation (PDC) [15] [16] is one such control design framework that has been proposed and developed over the last few years. It has also been shown that within the framework of T-S fuzzy model and PDC control design, design conditions for the stability and performance of a system can be stated in terms of the feasibility of a set of linear matrix inequalities (LMIs) [15] [16]. This is a signi cant nding in the sense that there exist very ecient numerical algorithms for determining the feasibility of LMIs, so even large scale analysis and design problems are computationally tractable. The PDC control structure [15] [16] utilizes a nonlinear state feedback controller which mirrors the structure of the associated T-S model. The gains of the controller can be determined automatically using an LMI formulation. In this paper, we extend this LMI-based approach to develop dynamic feedback controllers for fuzzy systems described by Takagi-Sugeno (T-S) models. First, we introduce the notion of Dynamic Parallel Distributed Compensation (DPDC) and provide a set of sucient LMI conditions for the existence of a quadratically stabilizing dynamic compensator. Second, we present the performance-oriented controller synthesis of DPDC's to incorporate a number of practical design objectives such as disturbance attenuation, passivity, output constraint and so on. Performance speci cations presented in this paper include L2 gain, general quadratic constraints, generalized H2 performance, output and input constraints. The controller synthesis procedures are formulated as LMI problems. In the case of meeting multiple design objectives, we only need to group these LMI conditions together and nd a feasible solution to the augmented LMI problem [10]. In this paper, the notation M > 0 stands for a positive de nite symmetric matrix. And L(A; P ) = AT P T + PA is de ned as a mapping from Rnn Rnn to Rnn . The same holds for L(AT ; Q) = AQT + QAT . P ?T is the same as (P ?1 )T . R+ = [0; 1). Lp2 (R+ ) is de ned as the set of all p-dimensional vector valued functions u(t); t 2 R+ such that R is de ned as the kuk2 = ( 01 ku(t)k2 dt)1=2 < 1 and Le2 (R+ ) is its extended space which R T + e set of the vector-valued functions u(t), t 2 R such that kuk2 = ( 0 ku(t)k2 dt)1=2 < 1; for all T 2 R+ : The paper is organized as follows: Section 2 describes the Takagi-Sugeno fuzzy model. Section 3 introduces the DPDC controller and presents a set of LMI design conditions which 2
can be used to select the compensator gains so that the closed loop system is globally stable. Section 4 addresses the problem of multi-objective control synthesis for T-S models. Section 5 contains an illustrative application. Concluding remarks are collected in Section 6.
2 The Takagi-Sugeno Model The Takagi-Sugeno fuzzy model consists of a nite set of fuzzy IF : : : THEN rules. Each rule has the following form: Dynamic Part: Rule i: IF p1 (t) is Mi1 and pl (t) is Mil THEN x_ (t) = Ai x(t) + Bi u(t), Output Part: Rule i: IF p1 (t) is Mi1 and pl (t) is Mil THEN y(t) = Ci x(t). The vectors x(t), u(t), and y(t) denote the state, input, and output vectors respectively. Each variable pi (t) is a known parameter. In general, these parameters may be functions of the state variables, external disturbances, and/or time. We will use p(t) to denote the vector containing all the individual parameters, r is the number of IF-THEN rules. The symbols Mij represent membership functions for fuzzy sets. The possibility that the ith rule will re is given by the product of all the membership functions associated with the ith rule. wi (p(t)) = lj=1Mij (pj (t)): P We will assume that the at least one wi (p(t)) is always nonzero so that ri=1 wi (p(t)) 6= 0, and for the remainder of this paper, we will only work with the normalized functions hi (p(t)) = Prwi(wp((t)) : i=1 i p(t)) Using the center of gravity method for defuzzi cation, we can express the T-S model as:
x_ = y =
r X i=1
hi (p)(Ai x + Bi u)
(1)
r X i=1
hi (p)Ci x
where hi (p) is the possibility for the ith rule to re. The assumption that hi (p) have already r X been normalized translates into hi (p) 0 and hi (p) = 1. We will often drop the p and i=1 just write hi , but it should be kept in mind that the hi 's are functions of the fuzzy variable p. The parameter p can be given several di erent interpretations. First, we can assume that the parameter p is a measurable external disturbance signal which does not depend 3
on the state or control input of the system (1). Using this interpretation, equations (1) describes a time-varying linear system. Second, we can assume that the parameter p is a function of the state, p = f (x). using this interpretation, equations (1) describes a nonlinear system. As a slight modi cation to this interpretation we can assume that the parameter p is a function of the measurable outputs of the system, p = f (y). Finally, we can assume that p is an unknown constant value, in which case the equations (1) describes a linear di erential inclusion (LDI). In most cases, we can only derive a bene t from the fuzzy rule based description if we know the values of the parameters, so we will not consider this last interpretation. It is also possible to interpret p using a combination of these approaches. Before the presentation of the main results in this paper, a linear algebra lemma is rst given.
LemmaP1 Consider parameter-dependent matrix Pri=1 Prj=1 hi(p)hj (p)Lij , where hi (p) 0 and ri=1 hi (p) = 1 8p is negative de nite for all p if there exist r(r2+1) symmetric matrices Tij with 1 i j r such that the following two conditions hold: 1: Lij + Lji < Tij 8i; j; 1 i j r 2 3
(2)
2:
(3)
T11 : : : T1n 6 7 T = 64 ... . . . ... 75 < 0 T1n : : : Tnn
3 Stabilization of T-S Systems using DPDC In this section we introduce the concept of dynamic parallel distributed compensator (DPDC), and we derive a set of LMI conditions which can be used to design a stabilizing DPDC. In order to derive the LMI design conditions, it is useful to begin with a parameter dependent linear model described by the equations
x_ (t) = A(p)x(t) +B (p)u(t) y(t) = C (p)x(t)
(4)
where x(t) , y(t) and u(t) denote the state, measurement and input vectors, respectively. The variable p(t) is a vector of measurable parameters. In general, these parameters may be functions of the system states, external disturbances, and time. It is noted that T-S models are in this form. A parameter dependent dynamic compensator is a parameter dependent linear system of the form x_ c(t) = Ac(p)xc(t) +Bc(p)y(t) (5) u(t) = Cc (p)xc(t) +Dc(p)y(t): 4
De ning the augmented system matrix
"
Acl (p) = (A(p) + B (p)Dc (p)C (p)) B (p)Cc(p) Bc(p)C (p) Ac(p)
#
and the augmented state vector
xcl (t) = [xT (t) xTc (t)]T ; the resulting closed-loop dynamic equations are described by the equation
x_ cl (t) = Acl (p)xcl (t):
(6)
The system (4) is said to be quadratically stabilizable via an s-dimensional parameter dependent linear compensator if and only if there exist an s-dimensional parameter dependent h 0 i " x(t) # 0 controller and a quadratic Lyapunov function V (x(t); xc (t)) = x (t) xc(t) P xc(t) dV such that dt < 0 or equivalently
PAcl (p) + ATcl (p)P < 0
(7)
Remark: If we x the value of p, equation (7) represents a sucient condition for the
existence of a set of linear, time-invariant controller matrices Ac (p); Bc (p); Cc (p), and Dc (p) which will stabilize the system (4) at the xed value of p. The unknown controller does not enter linearly into equation (7), so this equation does not represent an LMI condition. However, in the paper [10] the authors present a transformation procedure which results in a modi ed set of inequalities which are linear in the unknown data. In what follows, we perform this transformation pointwise with respect to p. We will rst partition the constant matrices P and P ?1 into components.
"
P = P11T P12 P12 P22
#
P ?1 =
The condition that P > 0 is equivalent to
"
"
Q11 Q12 QT12 Q22
#
#
Q11 I > 0 I P11
(8)
Since PP ?1 = I , we have the constraint that P11 Q11 + P12 QT12 = I . We will also de ne the matrices
"
Q11 I 1 = QT12 0
#
"
#
I P11 : 2 = P 1 = 0 P12T 5
Equation (7) will hold if and only if T1 PAcl (p)1 + T1 ATcl (p)P 1 < 0: Writing out the rst term on the left hand side of this equation, we have
"
I 0 P11 P12
#"
#"
(A(p) + B (p)Dc (p)C (p)) B (p)Cc (p) Bc(p)C (p) Ac(p)
#
Q11 I = E (p): QT12 0
and the closed-loop stability condition can be expressed as
E (p) + E T (p) < 0
(9)
We will now assume that the parameter dependent plant can be described by a fuzzy T-S model using r IF : : : THEN rules. In this case, the parameter dependent plant can be described by the equation
"
"
#
r A(p) B (p) = X hi (p) Ai Bi Ci 0 C (p) 0 i=1
#
r X
where h(p) satis es the normalization condition, i.e. hi (p) 0 and hi (p) = 1. i=1 We are now ready to introduce dynamic parallel distributed compensators for this system. In general, the choice of a particular DPDC parameterization will be in uenced by the structure of the T-S subsystems. In this paper, we will only discuss DPDC in the form of:
x_ c =
Ac (p) = Cc (p) =
r X
i=1 j =1
i=1
r X
u = or equivalently that
r r X X
i=1
hi (p)hj (p)Aijc xc +
hi (p)Cci xc + Dcy
r r X X i=1 j =1
r X i=1
hi Bci y
hi (p)hj (p)Aijc
hi(p)Cci
Bc(p) =
(10) (11)
r X i=1
Dc(p) = Dc
hi (p)Bci
(12)
So the closed-loop system for the T-S model (1) with this controller can be written as:
x_ cl =
r X r X i=1 j =1
hi (p)hj (p)Aijcl xcl
6
(13)
where
Aijcl
Ai + Bi DcCj Bi Ccj Bci Cj Aijc
=
! "
#
E (p) E12 (p) where We can rewrite the equations for the matrix E (p) as E (p) = 11 E21 (p) E22 (p) E11 (p) =
, E12 (p) =
, E21 (p) =
,
r X r X i=1 j =1 r r X X
hi (p)hj (p)E11ij
hi (p)hj (p)((Ai + Bi DcCj )Q11 + Bi Ccj QT12 )
i=1 j =1 r X r X i=1 j =1 r r X X
hi (p)hj (p)E12ij
hi (p)hj (p)(Ai + Bi DcCj )
i=1 j =1
(15)
r r X X i=1 j =1 r X r X
hi (p)hj (p)E21ij
hi (p)hj (p)(P11 (Ai + BiDcCj )Q11
i=1 j =1
+P12 Bci Cj Q11 + P11 Bi Ccj QT12 + P12 Aijc QT12 )
E22 (p) =
,
(14)
(16)
r X r X
i=1 j =1 r r X X i=1 j =1
hi (p)hj (p)E22ij
hi (p)hj (p)(P11 (Ai + BiDcCj ) + P12 Bci Cj )
(17)
And if we de ne
Aij , P11 (Ai + BiDcCj )Q11
+P12 Bci Cj Q11 + P11 Bi Ccj QT12 + P12 Aijc QT12 Bi , P11 BiDc + P12 Bci Ci , DcCi Q11 + CciQT12
(18) (19) (20) (21)
D , Dc
the matrix E (p) then becomes
r r X X
"
hi (p)hj (p) AiQ11 + BiCj Ai + Bi DCj E (p) , Aij P11 Ai + Bi Cj i=1 j =1
#
(22)
In terms of E (p) our stability condition can be written as E (p) + E T (p) < 0 which will 7
hold true according to Lemma 1 if
E11ij + (E11ij )T + E11ji + (E11ji )T E12ij + (E21ij )T + E12ji + (E21ji )T ij )T + (E ij ) + (E ji )T + (E ji ) E ij + (E ij )T + E ji + (E ji )T (E12 21 12 21 22 22 22 22
!
< Tij (23)
2 3 T11 : : : T1n 6 7 T = T0 = 64 ... . . . ... 75 < 0
and
T1n : : : Tnn
(24)
It is noted that the controller (12) can also be simpli ed to
Ac(p) = Cc(p) =
r r X X i=1 j =1
r X i=1
hi (p)hj (p)Aijc
Bc(p) =
r X i=1
hi (p)Bci
Dc(p) = D c
hi(p)Cci
(25)
where Aijc = 12 (Aijc + Ajic ); Bci = Bci ; Cci = Cci ; D c = Dc . So if we de ne variables as Aij , 12 (Aij + Aji); Bi , Bi ; Ci , Ci ; D , D, Aij ; Bi; Ci ; D will depend only on Aijc ; Bci ; Cci; D c . And all the components in the matrix above can be written as:
E11ij + (E11ij )T + E11ji + (E11ji )T = L(Ai ; Q11 ) + L(Aj ; Q11 ) + BiCj + (Bi Cj )T +Bj Ci + (Ci Bj )T E12ij + (E21ij )T + E12ji + (E21ji )T = Ai + Aj + Bi D Cj + Bj D Ci + 2ATij E22ij + (E22ij )T + E22ji + (E22ji )T = L(ATi ; P11 ) + L(ATj ; P11 ) + Bi Cj + Bj Ci +(Bi Cj )T + (Bj Ci )T Therefore, we get the following theorem:
Theorem 2 The fuzzy control system of T-S model (1) is quadratically stabilizable in the large via a DPDC controller (25) if the LMI conditions (8), (24), (23) are feasible with LMI variables Q11 , P11 , Tij , Aij , Bi , Ci and D. And the controller is given by: (26) Aijc = 21 P12?1 (2Aij ? P12 Bci Cj Q11 ? P12 Bcj Ci Q11 ? P11 BiCjcQT12 ?P11 Bj CicQT12 ? P11 (Ai + BiD cCj )Q11 ? P11 (Aj + Bj D cCi)Q11 )Q?121 (27) Bci = P12?1 (Bi ? P11 BiD c) (28) Cci = (Ci ? D c CiQ11 )Q?12T (29) D c = D (30) where P11 , P12 , Q11 and Q12 satisfy the constraint P11 Q11 + P12 QT12 = I
8
4 Performance-Oriented Controller Synthesis of DPDC This section presents LMI conditions which can be used to design DPDC controllers which satisfy a variety of useful performance criteria. As in the previous sections, these results provide sucient conditions for the existence of a satisfactory DPDC controller. The presentation is divided into two subsections. In the rst subsection, we assume only a linear parameter dependent controller structure and derive a collection of parameter dependent conditions expressed in inequalities. Each condition corresponds to a di erent performance criteria. In the second subsection, we restrict our consideration to a DPDC controller structure. This restriction allows us to convert the parameter dependent inequalities into a parameter free LMIs which can be solved numerically.
4.1 Starting from Design Speci cations We will consider the class of systems G which can be described by the equations
x_ cl (t) = Acl (p)xcl (t) +Bcl(p)w(t) z(t) = Ccl (p)xcl (t) +Dcl (p)w(t)
(31)
where x(t) , w(t) and z (t) stands for state, input and performance variable correspondingly. p(t) is the system parameter which may be a ected by both the system states or some exogenous input variables.
L2 Gain Performance
De nition 1 [9]: For a causal NLTI (nonlinear time-invariant operator) G : w 2 Le2 (R+ ) ! z 2 Le2 (R+ ) with G(0) = 0, G is L2 stable if w 2 L2 (R) implies z 2 L2(R). G is said to have L2 gain less or equal to 0 if and only if
ZT 0
kz(t)k dt 2
2
ZT 0
kw(t)k2 dt
(32)
for all T 2 R+ .
The well-known Bounded Real Lemma is given below [20].
Lemma 3 For system G : (Acl (p); Bcl (p); Ccl (p); Dcl (p)), the L2 Gain will be less than
> 0 if there exist a matrix P = P T > 0 such that
2 3 T L ( A ( p ) ; P ) PB ( p ) C ( p ) cl cl cl 66 TP T 7 75 < 0 B ( p ) ?
I D ( p ) cl cl 4 Ccl (p)
Dcl (p)
General Quadratic Constraint 9
? I
(33)
De nition 2 [10]: For a causal NLTI G : w ! z with G(0) = 0. Given xed matrices
U = S ?1 S T , V = V T and W , where > 0. z(t) and w(t) need to satisfy the following constraint:
Z T z(t) !0
w(t) for xcl (0) = 0 and w(t) 2 L2 (R+ ) 0
U W WT V
!
z(t) w(t)
!
dt < 0; 8T 0
(34)
Remark [10]: Many performance speci cations (such as L2 gain, passivity and sector
constraint) can be incorporated into this general quadratic constraint framework by choosing di erent U , V and W . De ne the function V (xcl ) = x0cl Pxcl , where P = P T > 0. Suppose
L(Acl (p); P )
BclT (p)P + W T Ccl
PBcl(p) + CclT W DclT W + W T Dcl + V
Then
!
+
CclT (p) DclT (p)
!T
(35)
!0
xcl (t) L(Acl (p); P ) PBcl(p) 0 w(t) BclT (p)P !0 ! ! z (t) U W z (t) < ? w(t) WT V w(t)
d dt V (xcl (t)) =
U Ccl (p) Dcl(p) < 0
!
xcl(t) w(t)
!
(36) (37)
(34) will result by integrating both sides of (37). Applying Schur Complement to (35), we get the following lemma:
Lemma 4 For system G : (Acl (p); Bcl (p); Ccl (p); Dcl (p)), the general quadratic constraint (34) will be satis ed if there exists a matrix P = P T > 0 such that
3 2 T (p)W C T (p)S L ( A ( p ) ; P ) PB ( p ) + C cl cl cl cl 66 T T Ccl W T Dcl + DT W + V DT (p)S 7 75 < 0 ( p ) P + W B cl cl 4 cl T T S Ccl (p)
S Dcl (p)
?
(38)
Generalized H2 Performance De nition 3 [10]: A causal NLTI G : w ! z with G(0) = 0, is said to have generalized
H2 performance less or equal to if and only if
kz(T )k 8T 0
R where xcl (0) = 0 and 0T kw(t)k2 dt 1
10
(39)
De ne the function V (xcl (t)) = x0cl Pxcl , where P > 0. Suppose
!
L(Acl(p); P ) PBcl(p) < 0 BclT (p)P ?I
(40)
Then dtd V (xcl (t)) < w0 (t)w(t). We will suppose Dcl (p) = 0. In this case, if the equation
!
CclT (p) > 0 Ccl (p) I is satis ed, then z 0 (t)z (t) < V (xcl (t)). This leads to the following lemma: P
(41)
Lemma 5 For system G : (Acl (p); Bcl (p); Ccl (p); 0), the generalized H2 performance will be less than if there exists a matrix P = P T > 0 such that (40) and (41) are feasible.
Constraint on System Output De nition 4 [19]: A causal NLTI G : x_ cl = Acl (p)xcl and z = Ccl (p)xcl . is satis es an exponential constraint on the output if
kz(T )k e? T 8T 0
(42)
where xcl (0) = x0
De ne the function V (xcl ) = x0cl Pxcl , where P = P T > 0. Suppose that the equation
L(Acl ; P ) + 2 P < 0
(43)
holds. In this case, the inequality V (xcl (t)) < e?2 t V (xcl (0)) will be satis ed. Furthermore, if the equations ! P Pxcl (0) > 0 (44) x0cl (0)P I and ! P CclT (p) > 0 (45) Ccl (p) I hold, then the inequality
z 0(t)z(t) < (x0cl(t)Pxcl (t)) < e?2 t (x0cl (0)Pxcl (0)) < 2 e?2 t will also be satis ed. Combining these results, we have the following lemma:
Lemma 6 For the system G : x_ cl = Acl (p)xcl and z = Ccl(p)xcl , the exponential constraint kz(T )k e? T 8T 0 will be satis ed if there exists a matrix P = P T > 0 such that
(43), (44) and (45) are feasible.
11
Constraints on Control Input De nition 5 [19]: A causal NLTI G : x_ cl = Acl(p)xcl and u = K (p)xcl with a speci ed initial condition xcl (0) satis es an exponential constraint on the input if
ku(T )k e? T 8T 0:
(46)
Similar to the discussion for exponential constraint on the system output, we have the following lemma:
Lemma 7 For system G : x_ cl = Acl(p)xcl and u = K (p)xcl , the exponential constraint ku(T )k e? T 8T 0 will be satis ed if there exists a matrix P = P T > 0 such that (43), (44) and
P K (p)T K (p) I
!
Laboratory for Intelligent and Nonlinear Control (LINC) Department of Electrical and Computer Engineering Duke University, Durham, NC 27708, U.S.A
Kazuo Tanaka
Department of Mechanical and Control Engineering University of Electro-Communications 1-5-1 Chofugaoka, Chofu, Tokyo 182, Japan
Abstract This paper presents a uni ed systematic framework for designing dynamic feedback controllers for nonlinear systems described by Takagi-Sugeno (T-S) models. Both stabilization and multi-objective control synthesis problems have been addressed. The control laws are in the form of so-called dynamic parallel distributed compensation (DPDC) which are essentially nonlinear dynamic feedback controllers. The associated control synthesis problems are formulated as LMI problems, i.e., the parameters of the DPDC controllers are obtained from a set of LMI conditions. The approach in this paper can also be applied to hybrid or switching systems.
Keywords: Fuzzy control, LMI, dynamic feedback.
Corresponding author. E-mail: [email protected]; Tel: (919) 660-5273; Fax: (919) 660-5293.
1 Introduction Recently Takagi-Sugeno fuzzy models [11] have received a great deal of attention. The appeal of the T-S model lies in that the stability and performance characteristics of the system represented by a T-S model can be analyzed using a Lyapunov function approach [14] [15] [16] [7] [12] [17]. A further and signi cant step has also been taken to utilize Lyapunov-function based control design techniques to the control synthesis problem for T-S models. The so-called parallel distributed compensation (PDC) [15] [16] is one such control design framework that has been proposed and developed over the last few years. It has also been shown that within the framework of T-S fuzzy model and PDC control design, design conditions for the stability and performance of a system can be stated in terms of the feasibility of a set of linear matrix inequalities (LMIs) [15] [16]. This is a signi cant nding in the sense that there exist very ecient numerical algorithms for determining the feasibility of LMIs, so even large scale analysis and design problems are computationally tractable. The PDC control structure [15] [16] utilizes a nonlinear state feedback controller which mirrors the structure of the associated T-S model. The gains of the controller can be determined automatically using an LMI formulation. In this paper, we extend this LMI-based approach to develop dynamic feedback controllers for fuzzy systems described by Takagi-Sugeno (T-S) models. First, we introduce the notion of Dynamic Parallel Distributed Compensation (DPDC) and provide a set of sucient LMI conditions for the existence of a quadratically stabilizing dynamic compensator. Second, we present the performance-oriented controller synthesis of DPDC's to incorporate a number of practical design objectives such as disturbance attenuation, passivity, output constraint and so on. Performance speci cations presented in this paper include L2 gain, general quadratic constraints, generalized H2 performance, output and input constraints. The controller synthesis procedures are formulated as LMI problems. In the case of meeting multiple design objectives, we only need to group these LMI conditions together and nd a feasible solution to the augmented LMI problem [10]. In this paper, the notation M > 0 stands for a positive de nite symmetric matrix. And L(A; P ) = AT P T + PA is de ned as a mapping from Rnn Rnn to Rnn . The same holds for L(AT ; Q) = AQT + QAT . P ?T is the same as (P ?1 )T . R+ = [0; 1). Lp2 (R+ ) is de ned as the set of all p-dimensional vector valued functions u(t); t 2 R+ such that R is de ned as the kuk2 = ( 01 ku(t)k2 dt)1=2 < 1 and Le2 (R+ ) is its extended space which R T + e set of the vector-valued functions u(t), t 2 R such that kuk2 = ( 0 ku(t)k2 dt)1=2 < 1; for all T 2 R+ : The paper is organized as follows: Section 2 describes the Takagi-Sugeno fuzzy model. Section 3 introduces the DPDC controller and presents a set of LMI design conditions which 2
can be used to select the compensator gains so that the closed loop system is globally stable. Section 4 addresses the problem of multi-objective control synthesis for T-S models. Section 5 contains an illustrative application. Concluding remarks are collected in Section 6.
2 The Takagi-Sugeno Model The Takagi-Sugeno fuzzy model consists of a nite set of fuzzy IF : : : THEN rules. Each rule has the following form: Dynamic Part: Rule i: IF p1 (t) is Mi1 and pl (t) is Mil THEN x_ (t) = Ai x(t) + Bi u(t), Output Part: Rule i: IF p1 (t) is Mi1 and pl (t) is Mil THEN y(t) = Ci x(t). The vectors x(t), u(t), and y(t) denote the state, input, and output vectors respectively. Each variable pi (t) is a known parameter. In general, these parameters may be functions of the state variables, external disturbances, and/or time. We will use p(t) to denote the vector containing all the individual parameters, r is the number of IF-THEN rules. The symbols Mij represent membership functions for fuzzy sets. The possibility that the ith rule will re is given by the product of all the membership functions associated with the ith rule. wi (p(t)) = lj=1Mij (pj (t)): P We will assume that the at least one wi (p(t)) is always nonzero so that ri=1 wi (p(t)) 6= 0, and for the remainder of this paper, we will only work with the normalized functions hi (p(t)) = Prwi(wp((t)) : i=1 i p(t)) Using the center of gravity method for defuzzi cation, we can express the T-S model as:
x_ = y =
r X i=1
hi (p)(Ai x + Bi u)
(1)
r X i=1
hi (p)Ci x
where hi (p) is the possibility for the ith rule to re. The assumption that hi (p) have already r X been normalized translates into hi (p) 0 and hi (p) = 1. We will often drop the p and i=1 just write hi , but it should be kept in mind that the hi 's are functions of the fuzzy variable p. The parameter p can be given several di erent interpretations. First, we can assume that the parameter p is a measurable external disturbance signal which does not depend 3
on the state or control input of the system (1). Using this interpretation, equations (1) describes a time-varying linear system. Second, we can assume that the parameter p is a function of the state, p = f (x). using this interpretation, equations (1) describes a nonlinear system. As a slight modi cation to this interpretation we can assume that the parameter p is a function of the measurable outputs of the system, p = f (y). Finally, we can assume that p is an unknown constant value, in which case the equations (1) describes a linear di erential inclusion (LDI). In most cases, we can only derive a bene t from the fuzzy rule based description if we know the values of the parameters, so we will not consider this last interpretation. It is also possible to interpret p using a combination of these approaches. Before the presentation of the main results in this paper, a linear algebra lemma is rst given.
LemmaP1 Consider parameter-dependent matrix Pri=1 Prj=1 hi(p)hj (p)Lij , where hi (p) 0 and ri=1 hi (p) = 1 8p is negative de nite for all p if there exist r(r2+1) symmetric matrices Tij with 1 i j r such that the following two conditions hold: 1: Lij + Lji < Tij 8i; j; 1 i j r 2 3
(2)
2:
(3)
T11 : : : T1n 6 7 T = 64 ... . . . ... 75 < 0 T1n : : : Tnn
3 Stabilization of T-S Systems using DPDC In this section we introduce the concept of dynamic parallel distributed compensator (DPDC), and we derive a set of LMI conditions which can be used to design a stabilizing DPDC. In order to derive the LMI design conditions, it is useful to begin with a parameter dependent linear model described by the equations
x_ (t) = A(p)x(t) +B (p)u(t) y(t) = C (p)x(t)
(4)
where x(t) , y(t) and u(t) denote the state, measurement and input vectors, respectively. The variable p(t) is a vector of measurable parameters. In general, these parameters may be functions of the system states, external disturbances, and time. It is noted that T-S models are in this form. A parameter dependent dynamic compensator is a parameter dependent linear system of the form x_ c(t) = Ac(p)xc(t) +Bc(p)y(t) (5) u(t) = Cc (p)xc(t) +Dc(p)y(t): 4
De ning the augmented system matrix
"
Acl (p) = (A(p) + B (p)Dc (p)C (p)) B (p)Cc(p) Bc(p)C (p) Ac(p)
#
and the augmented state vector
xcl (t) = [xT (t) xTc (t)]T ; the resulting closed-loop dynamic equations are described by the equation
x_ cl (t) = Acl (p)xcl (t):
(6)
The system (4) is said to be quadratically stabilizable via an s-dimensional parameter dependent linear compensator if and only if there exist an s-dimensional parameter dependent h 0 i " x(t) # 0 controller and a quadratic Lyapunov function V (x(t); xc (t)) = x (t) xc(t) P xc(t) dV such that dt < 0 or equivalently
PAcl (p) + ATcl (p)P < 0
(7)
Remark: If we x the value of p, equation (7) represents a sucient condition for the
existence of a set of linear, time-invariant controller matrices Ac (p); Bc (p); Cc (p), and Dc (p) which will stabilize the system (4) at the xed value of p. The unknown controller does not enter linearly into equation (7), so this equation does not represent an LMI condition. However, in the paper [10] the authors present a transformation procedure which results in a modi ed set of inequalities which are linear in the unknown data. In what follows, we perform this transformation pointwise with respect to p. We will rst partition the constant matrices P and P ?1 into components.
"
P = P11T P12 P12 P22
#
P ?1 =
The condition that P > 0 is equivalent to
"
"
Q11 Q12 QT12 Q22
#
#
Q11 I > 0 I P11
(8)
Since PP ?1 = I , we have the constraint that P11 Q11 + P12 QT12 = I . We will also de ne the matrices
"
Q11 I 1 = QT12 0
#
"
#
I P11 : 2 = P 1 = 0 P12T 5
Equation (7) will hold if and only if T1 PAcl (p)1 + T1 ATcl (p)P 1 < 0: Writing out the rst term on the left hand side of this equation, we have
"
I 0 P11 P12
#"
#"
(A(p) + B (p)Dc (p)C (p)) B (p)Cc (p) Bc(p)C (p) Ac(p)
#
Q11 I = E (p): QT12 0
and the closed-loop stability condition can be expressed as
E (p) + E T (p) < 0
(9)
We will now assume that the parameter dependent plant can be described by a fuzzy T-S model using r IF : : : THEN rules. In this case, the parameter dependent plant can be described by the equation
"
"
#
r A(p) B (p) = X hi (p) Ai Bi Ci 0 C (p) 0 i=1
#
r X
where h(p) satis es the normalization condition, i.e. hi (p) 0 and hi (p) = 1. i=1 We are now ready to introduce dynamic parallel distributed compensators for this system. In general, the choice of a particular DPDC parameterization will be in uenced by the structure of the T-S subsystems. In this paper, we will only discuss DPDC in the form of:
x_ c =
Ac (p) = Cc (p) =
r X
i=1 j =1
i=1
r X
u = or equivalently that
r r X X
i=1
hi (p)hj (p)Aijc xc +
hi (p)Cci xc + Dcy
r r X X i=1 j =1
r X i=1
hi Bci y
hi (p)hj (p)Aijc
hi(p)Cci
Bc(p) =
(10) (11)
r X i=1
Dc(p) = Dc
hi (p)Bci
(12)
So the closed-loop system for the T-S model (1) with this controller can be written as:
x_ cl =
r X r X i=1 j =1
hi (p)hj (p)Aijcl xcl
6
(13)
where
Aijcl
Ai + Bi DcCj Bi Ccj Bci Cj Aijc
=
! "
#
E (p) E12 (p) where We can rewrite the equations for the matrix E (p) as E (p) = 11 E21 (p) E22 (p) E11 (p) =
, E12 (p) =
, E21 (p) =
,
r X r X i=1 j =1 r r X X
hi (p)hj (p)E11ij
hi (p)hj (p)((Ai + Bi DcCj )Q11 + Bi Ccj QT12 )
i=1 j =1 r X r X i=1 j =1 r r X X
hi (p)hj (p)E12ij
hi (p)hj (p)(Ai + Bi DcCj )
i=1 j =1
(15)
r r X X i=1 j =1 r X r X
hi (p)hj (p)E21ij
hi (p)hj (p)(P11 (Ai + BiDcCj )Q11
i=1 j =1
+P12 Bci Cj Q11 + P11 Bi Ccj QT12 + P12 Aijc QT12 )
E22 (p) =
,
(14)
(16)
r X r X
i=1 j =1 r r X X i=1 j =1
hi (p)hj (p)E22ij
hi (p)hj (p)(P11 (Ai + BiDcCj ) + P12 Bci Cj )
(17)
And if we de ne
Aij , P11 (Ai + BiDcCj )Q11
+P12 Bci Cj Q11 + P11 Bi Ccj QT12 + P12 Aijc QT12 Bi , P11 BiDc + P12 Bci Ci , DcCi Q11 + CciQT12
(18) (19) (20) (21)
D , Dc
the matrix E (p) then becomes
r r X X
"
hi (p)hj (p) AiQ11 + BiCj Ai + Bi DCj E (p) , Aij P11 Ai + Bi Cj i=1 j =1
#
(22)
In terms of E (p) our stability condition can be written as E (p) + E T (p) < 0 which will 7
hold true according to Lemma 1 if
E11ij + (E11ij )T + E11ji + (E11ji )T E12ij + (E21ij )T + E12ji + (E21ji )T ij )T + (E ij ) + (E ji )T + (E ji ) E ij + (E ij )T + E ji + (E ji )T (E12 21 12 21 22 22 22 22
!
< Tij (23)
2 3 T11 : : : T1n 6 7 T = T0 = 64 ... . . . ... 75 < 0
and
T1n : : : Tnn
(24)
It is noted that the controller (12) can also be simpli ed to
Ac(p) = Cc(p) =
r r X X i=1 j =1
r X i=1
hi (p)hj (p)Aijc
Bc(p) =
r X i=1
hi (p)Bci
Dc(p) = D c
hi(p)Cci
(25)
where Aijc = 12 (Aijc + Ajic ); Bci = Bci ; Cci = Cci ; D c = Dc . So if we de ne variables as Aij , 12 (Aij + Aji); Bi , Bi ; Ci , Ci ; D , D, Aij ; Bi; Ci ; D will depend only on Aijc ; Bci ; Cci; D c . And all the components in the matrix above can be written as:
E11ij + (E11ij )T + E11ji + (E11ji )T = L(Ai ; Q11 ) + L(Aj ; Q11 ) + BiCj + (Bi Cj )T +Bj Ci + (Ci Bj )T E12ij + (E21ij )T + E12ji + (E21ji )T = Ai + Aj + Bi D Cj + Bj D Ci + 2ATij E22ij + (E22ij )T + E22ji + (E22ji )T = L(ATi ; P11 ) + L(ATj ; P11 ) + Bi Cj + Bj Ci +(Bi Cj )T + (Bj Ci )T Therefore, we get the following theorem:
Theorem 2 The fuzzy control system of T-S model (1) is quadratically stabilizable in the large via a DPDC controller (25) if the LMI conditions (8), (24), (23) are feasible with LMI variables Q11 , P11 , Tij , Aij , Bi , Ci and D. And the controller is given by: (26) Aijc = 21 P12?1 (2Aij ? P12 Bci Cj Q11 ? P12 Bcj Ci Q11 ? P11 BiCjcQT12 ?P11 Bj CicQT12 ? P11 (Ai + BiD cCj )Q11 ? P11 (Aj + Bj D cCi)Q11 )Q?121 (27) Bci = P12?1 (Bi ? P11 BiD c) (28) Cci = (Ci ? D c CiQ11 )Q?12T (29) D c = D (30) where P11 , P12 , Q11 and Q12 satisfy the constraint P11 Q11 + P12 QT12 = I
8
4 Performance-Oriented Controller Synthesis of DPDC This section presents LMI conditions which can be used to design DPDC controllers which satisfy a variety of useful performance criteria. As in the previous sections, these results provide sucient conditions for the existence of a satisfactory DPDC controller. The presentation is divided into two subsections. In the rst subsection, we assume only a linear parameter dependent controller structure and derive a collection of parameter dependent conditions expressed in inequalities. Each condition corresponds to a di erent performance criteria. In the second subsection, we restrict our consideration to a DPDC controller structure. This restriction allows us to convert the parameter dependent inequalities into a parameter free LMIs which can be solved numerically.
4.1 Starting from Design Speci cations We will consider the class of systems G which can be described by the equations
x_ cl (t) = Acl (p)xcl (t) +Bcl(p)w(t) z(t) = Ccl (p)xcl (t) +Dcl (p)w(t)
(31)
where x(t) , w(t) and z (t) stands for state, input and performance variable correspondingly. p(t) is the system parameter which may be a ected by both the system states or some exogenous input variables.
L2 Gain Performance
De nition 1 [9]: For a causal NLTI (nonlinear time-invariant operator) G : w 2 Le2 (R+ ) ! z 2 Le2 (R+ ) with G(0) = 0, G is L2 stable if w 2 L2 (R) implies z 2 L2(R). G is said to have L2 gain less or equal to 0 if and only if
ZT 0
kz(t)k dt 2
2
ZT 0
kw(t)k2 dt
(32)
for all T 2 R+ .
The well-known Bounded Real Lemma is given below [20].
Lemma 3 For system G : (Acl (p); Bcl (p); Ccl (p); Dcl (p)), the L2 Gain will be less than
> 0 if there exist a matrix P = P T > 0 such that
2 3 T L ( A ( p ) ; P ) PB ( p ) C ( p ) cl cl cl 66 TP T 7 75 < 0 B ( p ) ?
I D ( p ) cl cl 4 Ccl (p)
Dcl (p)
General Quadratic Constraint 9
? I
(33)
De nition 2 [10]: For a causal NLTI G : w ! z with G(0) = 0. Given xed matrices
U = S ?1 S T , V = V T and W , where > 0. z(t) and w(t) need to satisfy the following constraint:
Z T z(t) !0
w(t) for xcl (0) = 0 and w(t) 2 L2 (R+ ) 0
U W WT V
!
z(t) w(t)
!
dt < 0; 8T 0
(34)
Remark [10]: Many performance speci cations (such as L2 gain, passivity and sector
constraint) can be incorporated into this general quadratic constraint framework by choosing di erent U , V and W . De ne the function V (xcl ) = x0cl Pxcl , where P = P T > 0. Suppose
L(Acl (p); P )
BclT (p)P + W T Ccl
PBcl(p) + CclT W DclT W + W T Dcl + V
Then
!
+
CclT (p) DclT (p)
!T
(35)
!0
xcl (t) L(Acl (p); P ) PBcl(p) 0 w(t) BclT (p)P !0 ! ! z (t) U W z (t) < ? w(t) WT V w(t)
d dt V (xcl (t)) =
U Ccl (p) Dcl(p) < 0
!
xcl(t) w(t)
!
(36) (37)
(34) will result by integrating both sides of (37). Applying Schur Complement to (35), we get the following lemma:
Lemma 4 For system G : (Acl (p); Bcl (p); Ccl (p); Dcl (p)), the general quadratic constraint (34) will be satis ed if there exists a matrix P = P T > 0 such that
3 2 T (p)W C T (p)S L ( A ( p ) ; P ) PB ( p ) + C cl cl cl cl 66 T T Ccl W T Dcl + DT W + V DT (p)S 7 75 < 0 ( p ) P + W B cl cl 4 cl T T S Ccl (p)
S Dcl (p)
?
(38)
Generalized H2 Performance De nition 3 [10]: A causal NLTI G : w ! z with G(0) = 0, is said to have generalized
H2 performance less or equal to if and only if
kz(T )k 8T 0
R where xcl (0) = 0 and 0T kw(t)k2 dt 1
10
(39)
De ne the function V (xcl (t)) = x0cl Pxcl , where P > 0. Suppose
!
L(Acl(p); P ) PBcl(p) < 0 BclT (p)P ?I
(40)
Then dtd V (xcl (t)) < w0 (t)w(t). We will suppose Dcl (p) = 0. In this case, if the equation
!
CclT (p) > 0 Ccl (p) I is satis ed, then z 0 (t)z (t) < V (xcl (t)). This leads to the following lemma: P
(41)
Lemma 5 For system G : (Acl (p); Bcl (p); Ccl (p); 0), the generalized H2 performance will be less than if there exists a matrix P = P T > 0 such that (40) and (41) are feasible.
Constraint on System Output De nition 4 [19]: A causal NLTI G : x_ cl = Acl (p)xcl and z = Ccl (p)xcl . is satis es an exponential constraint on the output if
kz(T )k e? T 8T 0
(42)
where xcl (0) = x0
De ne the function V (xcl ) = x0cl Pxcl , where P = P T > 0. Suppose that the equation
L(Acl ; P ) + 2 P < 0
(43)
holds. In this case, the inequality V (xcl (t)) < e?2 t V (xcl (0)) will be satis ed. Furthermore, if the equations ! P Pxcl (0) > 0 (44) x0cl (0)P I and ! P CclT (p) > 0 (45) Ccl (p) I hold, then the inequality
z 0(t)z(t) < (x0cl(t)Pxcl (t)) < e?2 t (x0cl (0)Pxcl (0)) < 2 e?2 t will also be satis ed. Combining these results, we have the following lemma:
Lemma 6 For the system G : x_ cl = Acl (p)xcl and z = Ccl(p)xcl , the exponential constraint kz(T )k e? T 8T 0 will be satis ed if there exists a matrix P = P T > 0 such that
(43), (44) and (45) are feasible.
11
Constraints on Control Input De nition 5 [19]: A causal NLTI G : x_ cl = Acl(p)xcl and u = K (p)xcl with a speci ed initial condition xcl (0) satis es an exponential constraint on the input if
ku(T )k e? T 8T 0:
(46)
Similar to the discussion for exponential constraint on the system output, we have the following lemma:
Lemma 7 For system G : x_ cl = Acl(p)xcl and u = K (p)xcl , the exponential constraint ku(T )k e? T 8T 0 will be satis ed if there exists a matrix P = P T > 0 such that (43), (44) and
P K (p)T K (p) I
!
