Computing, Information and Control ICIC International c⃝2012 ISSN 1349-4198
Volume 8, Number 1(A), January 2012 pp. 403-418
DESIGN OF STABLE AND QUADRATIC-OPTIMAL STATIC OUTPUT FEEDBACK CONTROLLERS FOR TS-FUZZY-MODEL-BASED
CONTROL SYSTEMS: AN INTEGRATIVE COMPUTATIONAL APPROACH
Wen-Hsien Ho1, Shinn-Horng Chen2, I-Te Chen3, Jyh-Horng Chou2,4,∗ and Chun-Chin Shu4
1Department of Medical Information Management 3Center for General Education
Kaohsiung Medical University
No. 100, Shi-Chuan 1st Road, Kaohsiung 807, Taiwan
{ whho; itchen }@kmu.edu.tw
2Department of Mechanical Engineering
National Kaohsiung University of Applied Sciences No. 415, Chien-Kung Road, Kaohsiung 807, Taiwan
4Institute of System Information and Control
National Kaohsiung First University of Science and Technology No. 1, University Road, Yenchao, Kaohsiung 824, Taiwan
∗Corresponding author: [email protected]; [email protected]
Received September 2010; revised January 2011
Abstract. By integrating the stabilizability condition, the orthogonal-functions approach
(OFA) and the hybrid Taguchi-genetic algorithm (HTGA), an integrative computational method is presented in this paper to design the stable and quadratic-optimal static out-put feedback parallel-distributed-compensation (PDC) controller such that (i) the Takagi-Sugeno (TS) fuzzy-model-based control system can be stabilized, and (ii) a quadratic finite-horizon integral performance index for the TS-fuzzy-model-based control system can be minimized. In this paper, the stabilizability condition is proposed in terms of linear matrix inequalities (LMIs). By using the OFA and the LMI-based stabilizability condition, the stable and quadratic-finite-horizon-optimal static output feedback PDC con-trol problem for the TS-fuzzy-model-based dynamic systems is transformed into a static constrained-optimization problem represented by the algebraic equations with constraint of LMI-based stabilizability condition, thus greatly simplifying the optimal static output feedback PDC control design problem. Then, for the static constrained-optimization prob-lem, the HTGA is employed to find the stable and quadratic-optimal static output feedback PDC controllers of the TS-fuzzy-model-based control systems. A design example of sta-ble and quadratic-optimal static output feedback PDC controller for a nonlinear inverted pendulum system controlled by a separately excited direct-current (DC) motor is given to demonstrate the applicability of the proposed integrative computational approach. Keywords: Quadratic optimal control, Static output feedback PDC controller,
Takagi-Sugeno fuzzy model, Orthogonal-functions approach, Hybrid Taguchi-genetic algorithm, Linear matrix inequalities
1. Introduction. Recently, it has been shown that the fuzzy-model-based representa-tion proposed by Takagi and Sugeno [1], known as the TS fuzzy model, is a successful approach for dealing with the nonlinear control systems, and there are many successful applications of the TS-fuzzy-model-based approach to nonlinear control systems [2-15].
Unlike conventional modeling approaches where a single model is used to describe the global behavior of a nonlinear control system, the TS fuzzy modeling approach is essen-tially a multi-model approach in which the simple sub-models (typically linear models) are combined to describe the global behavior of the nonlinear control system. Each fuzzy rule for the TS fuzzy control system has a linear dynamic model as the consequent part that expresses the local dynamics of each fuzzy rule. Then, the overall fuzzy model is achieved by blending these rules. The advantage of controller synthesis for such a fuzzy model is that the linear control methods can be used.
Despite the success of applying the TS-fuzzy-model-based approach to nonlinear con-trol systems, it has become evident that many research issues remain to be addressed. In fact, in many cases, it is very difficult, if not impossible, to obtain a full order output feedback controller of a nonlinear control system. This is due to inaccessible measurement or overly expensive measurement. Therefore, recently, some research studies [16-19] have proposed the linear-matrix-inequality-based (LMI-based) approach to design the static output feedback parallel-distributed-compensation (PDC) controllers of the TS-fuzzy-model-based control systems for the infinite-horizon (i.e., infinite-time) control problems. On the other hand, only robust stability and stabilization are often not enough in control design. In control systems design, it is often of interest to synthesize a quadratic-optimal controller such that the control objective of minimizing a quadratic integral performance criterion is achieved [20]. Hence, recently, some researchers [2,21,22] have proposed some LMI-based approaches to design the quadratic-optimal controllers of TS-fuzzy-model-based control systems. Tanaka and Wang [2], Zheng et al. [21] and Li [22] designed the quadratic-optimal parallel-distributed-compensation (PDC) controllers by minimizing the upper bound of a quadratic infinite-horizon integral performance index. However, under the design consideration of directly minimizing a quadratic infinite-horizon integral per-formance index, it is not easy for the LMI-based approaches presented by Tanaka and Wang [2], Zheng et al. [21] and Li [22] to solve the quadratic-infinite-horizon-optimal PDC control problem of such systems. For some practical problems, we need to deal with the finite-horizon (i.e., finite-time) optimal control problems [23]. However, it is also difficult to apply the LMI-based approaches proposed by Tanaka and Wang [2], Zheng et al. [21] and Li [22] to directly minimize the finite-horizon performance index for solving the quadratic-finite-horizon-optimal PDC control problem of these systems. Besides, for solving the optimal PDC control problems, there are some issues that need to be resolved, such as how to simplify the computation for the above control problem of such systems and also ensure some characteristics of closed-loop systems [24]. Therefore, one of the most important issues is to develop computational methods for designing the quadratic-finite-horizon-optimal PDC controllers where the performance index is directly minimized. Very recently, Ho and Chou [25] have proposed a computational optimization method, which integrates the orthogonal-functions approach (OFA) [26] and the genetic algorithm [27,28], to design quadratic-optimal PDC controllers for the finite-horizon optimal control problem of the TS-fuzzy-model-based control systems where the performance index is di-rectly minimized. Since the method proposed by Ho and Chou [25] only involves algebraic computation and is straightforward and well-adapted to computer implementation, the design procedures of the controllers for these control systems may be either greatly reduced or much simplified accordingly. Ho and Chou [25] have also shown that the computational optimization method integrating the OFA and the genetic algorithm may obtain better results than the LMI-based approaches [2,21,22] for finding the quadratic-optimal PDC controllers of the TS-fuzzy-model-based control systems.
Summing up the above statements and reasons, although the LMI-based approach is successful in designing the static output feedback PDC controllers of the TS-fuzzy-model-based control systems for the infinite-horizon (i.e., infinite-time) control problems proposed by Fang et al. [16], Wu et al. [17], Chung et al. [18] and Huang and Nguang [19], to the authors’ best knowledge, there are no studies investigating the issue of design-ing stable and quadratic-finite-horizon-optimal static output feedback PDC controllers for the TS-fuzzy-model-based control systems by directly minimizing the performance index subject to the constraint of stabilizability. On the other hand, in practice, in order to avoid high gains, the controller gains must be considered to satisfy the constraints. The LMI-based approach proposed by Fang et al. [16], Wu et al. [17], Chung et al. [18] and Huang and Nguang [19] cannot deal with the design problem of the static output feedback PDC controller gains having constraints. Therefore, we can see that it is worth-while to present an efficiently numerical optimization approach accompanied with the stabilizability condition to design the stable and quadratic-finite-horizon-optimal static output feedback PDC controllers having constraints for the TS-fuzzy-model-based con-trol systems, where the performance index subject to the constraint of stabilizability is considered to be directly minimized.
The purpose of this paper is to propose a numerical optimization method accompanied with the stabilizability condition to design stable and quadratic-optimal static output feedback PDC controllers for the finite-horizon optimal control problem of the TS-fuzzy-model-based control systems by integrating the OFA, the hybrid Taguchi-genetic algo-rithm (HTGA) and the LMI technique, where the LMI technique is used to derive the stabilizability condition for ensuring that the closed-loop TS-fuzzy-model-based control systems can be stabilized. The proposed numerical optimization method can not only be applied to find the feedback gain matrices of the stable and quadratic-optimal static output feedback PDC controller for the TS-fuzzy-model-based control system under the minimization of a defined quadratic finite-horizon integral performance index, but also be applied to the case of the elements of the feedback gain matrices having constraints for practical consideration.
In this paper, by using the OFA and the LMI-based stabilizability condition, the stable and quadratic-finite-horizon-optimal static output feedback PDC control problem for the TS-fuzzy-model-based control systems is transformed into a static parameter constrained-optimization problem represented by algebraic equations with constraint of LMI-based stabilizability condition, thus greatly simplifying the optimal static output feedback PDC control design problem. The computational complexity for both differential and integral in the optimal static output feedback PDC control design of the original dynamic sys-tems may therefore be reduced remarkably. Then, for the static constrained-optimization problem, the HTGA is employed to find the stable and quadratic-optimal static output feedback PDC controllers of the TS-fuzzy-model-based control systems. The proposed in-tegrative computational method considers directly minimizing the quadratic finite-horizon integral performance index subject to the constraint of stabilizability in designing the sta-ble and quadratic-optimal static output feedback PDC controllers. The reason why the HTGA is applied in this paper is that Tsai et al. [29,30] have shown that the HTGA may obtain better results than those existing improved genetic algorithms reported in the literature. An illustrative example is also given in this paper to demonstrate the applicability of the proposed integrative computational method.
2. Problem Statement. The TS-fuzzy-model-based control system for the nonlinear control system can be obtained as the following form:
˜ Ri : IF z1(t) is Mi1 and . . . and zg(t) is Mig, THEN { ˙x(t) = Aix(t) + Biu(t), y(t) = Cix(t), (1) with the initial state vector x(0), where ˜Ri (i = 1, 2, . . . , N ) denotes the i-th impli-cation, N is the number of fuzzy rules, x(t) = [x1(t), x2(t), . . . , xn(t)]
T
denotes the n-dimensional state vector, y(t) = [y1(t), y2(t), . . . , yr(t)]T denotes the r-dimensional
out-put vector, u(t) = [u1(t), u2(t), . . . , up(t)]T denotes the p-dimensional input vector, zi(t)
(i = 1, 2, . . . , g) are the premise variables, Ai, Biand Ci (i = 1, 2, . . . , N ) are, respectively,
the n× n, n × p and r × n consequent constant matrices, and Mij (i = 1, 2, . . . , N and
j = 1, 2, . . . , g) are the fuzzy sets.
The resulting TS-fuzzy-model-based dynamic system inferred from (1) is represented as
˙x(t) =
N
∑
i=1
hi(z(t))(Aix(t) + Biu(t)), (2a)
y(t) = N ∑ i=1 hi(z(t))Cix(t), (2b) in which z(t) = [z1(t), z2(t), . . . , zg(t)] T
denotes the g-dimensional premise vector, hi(z(t))
= wi(z(t)) /∑N i=1 wi(z(t)), wi(z(t)) = g ∏ j=1
Mij(zj(t)) and Mij(zj(t)) are the grades of
mem-bership of zj(t) in the fuzzy sets Mij (i = 1, 2, . . . , N and j = 1, 2, . . . , g). It can be seen
that, for all t, hi(z(t))≥ 0 and N
∑
i=1
hi(z(t)) = 1.
Before we are able to synthesize a static output feedback controller such that good control performance for a given dynamic system can be efficiently achieved, it is necessary that the given dynamic system can be stabilized by the following static output feedback PDC controller: u(t) =− N ∑ i=1 hi(z(t))Fiy(t) =− N ∑ i=1 N ∑ j=1 hi(z(t))hj(z(t))FiCjx(t), (3)
where Fi (i = 1, 2, . . . , N ) denote the p× r local static output feedback gain matrices.
By substituting (3) into (2a), we can get the closed-loop TS-fuzzy-model-based dynamic system as ˙x(t) = N ∑ i=1 N ∑ j=1 N ∑ k=1 hi(z(t))hj(z(t))hk(z(t))(Ai− BiFjCk)x(t). (4)
Before we investigate the stabilizability condition to design the stable and quadratic-finite-horizon-optimal static output feedback PDC controllers for the TS-fuzzy-model-based dynamic systems (4), the following lemmas need to be introduced first.
Lemma 2.1. [2] If the number of rules that fire for all t is less than or equal to ¯s where 1 < ¯s≤ N, then N ∑ i=1 h2i(z(t))− 1 ¯ s− 1 N ∑ i<j 2hi(z(t))hj(z(t))≥ 0. (5)
For the closed-loop TS-fuzzy-model-based dynamic system (4), the problem of stabiliz-ability analysis is (under the condition that the local static output feedback gain matrices Fi (i = 1, 2, . . . , N ) of the static output feedback PDC controller in (3) have been
speci-fied in advance) to derive a stabilizability criterion for checking whether the closed-loop TS-fuzzy-model-based dynamic system (4) can be stabilized by the specified static output feedback PDC controller or not. Hence, in what follows, we present an LMI-based sta-bilizability criterion to analyze whether the closed-loop TS-fuzzy-model-based dynamic system (4) can be stabilized by the static output feedback PDC controller or not, where the local static output feedback gain matrices Fi have been specified in advance.
Theorem 2.1. The closed-loop TS-fuzzy-model-based dynamic system (4) is stable, if, for
the specified local static output feedback gains matrices Fj (j = 1, 2, . . . , N ) in (3), there
exists a symmetric positive definite matrix P and the symmetric positive semi-definite matrices Qk (k = 1, 2, . . . , N ) such that the following LMIs are simultaneously satisfied:
GTkiiP + P Gkii+ (¯s− 1) Qk < 0, (6a)
and ( Gkij + Gkji 2 )T P + P ( Gkij + Gkji 2 ) − Qk ≤ 0, (6b)
where Gijk = Ai− BiFjCk, i < j, ¯s > 1, and i, j, k = 1, 2, . . . , N . Proof: See Appendix A.
However, only stabilizability is often not enough in control design. The control objective of minimizing a quadratic finite-horizon integral performance criterion for the dynamic systems is also considered in many practical control-engineering applications [20,23]. On the other hand, before we are able to synthesize a controller such that good control performance for a given dynamic system can be efficiently achieved, it is necessary that the given dynamic system can be stabilized by the controller [31,32]. In addition, both optimality and stability should be simultaneously considered in the optimal controllers design [33]. Therefore, the problem considered in this paper is how to specify the local static output feedback gain matrices Fi(i = 1, 2, . . . , N ) of the static output feedback PDC
controller in (3) such that the constraint of the LMI-based stabilizability condition (6) for the closed-loop TS-fuzzy-model-based dynamic system (4) can be satisfied, and such that the optimal control performance for the TS-fuzzy-model-based dynamic system (2) can be achieved by minimizing the following H2 quadratic finite-horizon integral performance index: J = ∫ qtf 0 [ yT(t)Qy(t) + uT(t)Ru(t)]dt = q−1 ∑ k=0 ∫ ktf (k+1)tf[ yT(t)Qy(t) + uT(t)Ru(t)]dt, (7)
where tf denotes a small time interval which is chosen for the independent variable t, q
is a positive integer specified by the designer, Q is a symmetric positive semi-definite matrix, and R is a symmetric positive-definite matrix. Here the time interval of interest is designated as being from t = 0 to t = qtf, where t = 0 is the initial time and t = qtf
is the final time of the control period. The problem to be studied in this paper can be named the mixed H2/LMI static output feedback PDC controllers design problem of the TS-fuzzy-model-based control systems, and the design procedures for the static output feedback PDC controllers can be described as follows:
Step 2: Minimize the H2 quadratic finite-horizon integral performance index (7) for the TS-fuzzy-model-based dynamic system (2).
That is, the design problem of the mixed H2/LMI static output feedback PDC con-trollers for the TS-fuzzy-model-based control systems is a constrained optimization prob-lem. In the next section, we will integrate the OFA, the HTGA and the presented LMI-based stabilizability condition to solve the mixed H2/LMI static output feedback PDC controllers design problem of the TS-fuzzy-model-based control systems, where the per-formance index subject to the constraint of stabilizability condition is considered to be directly minimized.
3. Stable and Quadratic-Optimal Static Output Feedback PDC Controllers Design. Here, consider the time interval ktf ≤ t ≤ (k + 1)tf, where tf is chosen for the
independent variable t, and let us define
t = ktf + η, (8)
and
xk= x(ktf), (9)
in which k = 0, 1, 2, . . . , q− 1 and 0 ≤ η ≤ tf.
The state vector x(t), within ktf ≤ t ≤ (k+1)tf, can be approximated by the truncated
orthogonal-functions (OF) representation as x(t) =
m−1
∑
s=0
x(k)s Ts(t) = ˜x(k)T (t), (10)
where m is the number of terms required for the OF, T (t) = [T0(t), T1(t), . . . , Tm−1(t)]T
denotes the m × 1 OF basis vector, Ti(t) (i = 0, 1, . . . , m− 1) denotes the OF, x
(k)
s
(s = 0, 1, . . . , m− 1) are the n × 1 coefficient vector, and ˜x(k) = [x(k) 0 , x (k) 1 , . . . , x (k) m−1 ] is the n× m coefficient matrix of x(t).
Substituting (3) and the truncated OF representation of x(t) in (10) into the quadratic integral performance index (7), the quadratic integral performance index J becomes the following algebraic form:
J = q−1 ∑ k=0 trace [ W (˜x(k))T (∑N i=1 N ∑ j=1 N ∑ l=1 N ∑ o=1 hi(zk)hj(zk)hl(zk)ho(zk) CiT(Q + FjTRFl)Co ) (˜x(k)) ] , (11)
where the constant matrix W is the product-integration-matrix of two OF basis vectors [25].
Since, before the consequent output can be inferred within the small time interval ktf ≤ t ≤ (k + 1)tf, the degree of fulfillment of the antecedent must be computed in
advance, so, as in the studies given by Ho and Chou [34,35], we can let the value of hi(z(t)), within ktf ≤ t ≤ (k + 1)tf, be hi(z(ktf)). Then, integrating (2a) from t = ktf
to t = t within ktf ≤ t ≤ (k + 1)tf, we obtain x(t)− x(ktf) = N ∑ i=1 hi(zk) [ Ai ∫ t ktf x(t)dt + Bi ∫ t ktf u(t)dt ] , (12) where hi(zk) = hi(z(ktf)) and k = 0, 1, 2, . . . , q− 1.
Using the following integral property of the OF: ∫ t
ktf
T (t)dt = HT (t), (13)
and applying (3), (9) and (10), (12) can be cast into the form ˜ x(k)− [xk, 0, 0, . . . , 0] = N ∑ i=1 N ∑ j=1 N ∑ l=1 hi(zk)hj(zk)hl(zk)(Ai− BiFjCl)˜x(k)H, (14)
in which H is the operational matrix of integration for the OF [34,35]. Equation (14) can be rewritten as
˜ x(k)− N ∑ i=1 N ∑ j=1 N ∑ l=1 hi(zk)hj(zk)hl(zk)(Ai− BiFjCl)˜x(k)H = ˜Q(k), (15) where ˜Q(k)= [xk, 0, 0, . . . , 0] is an n× m matrix.
Making use of the Kronecker product, the explicit form for the coefficient matrix ˜x(k) comes directly from (15) as
ˆ x(k) = [ Imn− N ∑ i=1 N ∑ j=1 N ∑ l=1 hi(zk)hj(zk)hl(zk)(HT⊗ (Ai− BiFjCl)) ]−1 ˆ Q(k), (16)
where Imn denotes the mn× mn identity matrix, ˆx(k)=
[
x(k)0 T, x(k)1 T, . . . , x(k)m−1T ]T
, ˆQ(k) = [
xTk, 0T, 0T, . . . , 0T]T, and ⊗ denotes the Kronecker product [36]. This implies that ˜x(k) can be obtained from (16).
Now, if one set of local static output feedback gain matrices {F1, F2, . . . , FN} is given,
then ˜x(k)(k = 0, 1, . . . , q−1) can be calculated from the following algorithm only involving the algebraic computation.
Detailed Steps: Algebraic Algorithm
Step 1: Give a small time interval tf, the specified positive integer q, and the initial state
vector x(0), and set k = 0.
Step 2: Calculate hi(z(k tf)) for i = 1, 2, . . . , N . Step 3: Calculate ˆx(k) from (16).
Step 4: Compute xk+1 by using xk+1= x((k + 1) tf) = ˜x(k)T ((k + 1) tf). Step 5: Set k = k + 1. If k > q− 1, then stop; otherwise go to Step 2.
From the above algorithm, it is obvious that if one set of local static output feedback gain matrices {F1, F2, . . . , FN} is specified, then ˜x(k) (k = 0, 1, . . . , q − 1) can be
deter-mined, and thus the value of the performance index (11) corresponding to this set of {F1, F2, . . . , FN} can be calculated. Given another set of local static output feedback gain
matrices {F1, F2, . . . , FN}, there obtains another value of the performance index (11).
That is, the value of the performance index of algebraic form (11) is actually dependent on the set of local static output feedback gain matrices {F1, F2, . . . , FN}, which means
J = G (f111, f112, . . . , fN pr) (17)
where fijk (i = 1, 2, . . . , N , j = 1, 2, . . . , p and k = 1, 2, . . . , r) denotes the elements of the
local static output feedback gain matrices Fi. Hence, the design problem of the stable and
quadratic-optimal static output feedback PDC controller for the TS-fuzzy-model-based control system is to search for the optimal fijksuch that there exists a symmetric positive
So, from the result mentioned above, we can derive the closed-loop TS-fuzzy-model-based dynamic system (4) is stable if, for the specified static output feedback gains matri-ces Fj (j = 1, 2,· · · , N) in (3), a symmetric positive definite matrix P and the symmetric
positive semi-definite matrices Qk (k = 1, 2, . . . , N ) exist such that the LMIs in (6) are
