Computing, Information and Control ICIC International c 2013 ISSN 1349-4198
Volume 9, Number 2, February 2013 pp. 805–819
OBSERVABILITY ROBUSTNESS OF UNCERTAIN FUZZY-MODEL-BASED CONTROL SYSTEMS
Wen-Hsien Ho1, Shinn-Horng Chen2 and Jyh-Horng Chou2,3
1Department of Healthcare Administration and Medical Informatics
Kaohsiung Medical University
No. 100, Shi-Chuan 1st Road, Kaohsiung 807, Taiwan [email protected]
2Department of Mechanical Engineering
National Kaohsiung University of Applied Sciences No. 415, Chien-Kung Road, Kaohsiung 807, Taiwan
{ shchen; choujh }@cc.kuas.edu.tw
3Institute of System Information and Control
National Kaohsiung First University of Science and Technology No. 1, University Road, Yenchao, Kaohsiung 824, Taiwan
Received December 2011; revised April 2012
Abstract. The problem considered in this study is the observability robustness of Takagi
-Sugeno (TS) fuzzy-model-based control systems. Where a nominal TS-fuzzy-model-based control system is locally observable (i.e., where each fuzzy rule in the system has a full row rank for its observability matrix), a sufficient condition is proposed to preserve the assumed property when system uncertainties are considered. The proposed sufficient con-dition preserves the assumed property by indicating the explicit relationships of bounds on system uncertainties. A robustly global observability condition is also presented for uncertain TS-fuzzy-model-based control systems. Finally, the proposed sufficient condi-tions are applied in the example of a nonlinear mass-spring-damper mechanical system with system uncertainties.
Keywords: Fuzzy system models, Fuzzy control, Robust observability, Takagi-Sugeno
(TS) fuzzy model, System uncertainties
1. Introduction. The fuzzy-model-based representation proposed by Takagi and Sugeno [1], known as the TS fuzzy model, has proven effective in many nonlinear control systems ([2-8] and references therein). The robust controllability of the uncertain TS-fuzzy-model-based control systems has also been studied by Chen et al. [9]. On the other hand, most applications of TS fuzzy control systems presented in the literature, however, assume that states are available for controller implementation, which may not be true in prac-tice. Therefore, some researchers have proposed that the nominal TS-fuzzy-model-based control systems should be assumed to be locally observable (i.e., each fuzzy rule for a nominal TS-fuzzy-model-based control system should be assumed to have a full row rank for its observability matrix) when designing observer-based fuzzy parallel-distributed-compensation controllers (see, e.g., [10-16] and references therein).
In practice, however, obtaining accurate values may be difficult, if not impossible, for some system parameters due to inaccurate measurements or due to inaccessible or variable system parameters and sensor and actuator positions. These system uncertainties may negate the observability property of the TS-fuzzy-model-based control systems. The
literature includes many studies of observability problems in various fuzzy systems [17-25]. A recent comprehensive literature review shows that the issue of robustly local and global observability has not been studied in uncertain TS-fuzzy-model-based control systems. Specifically, although the observability problem has been studied in various fuzzy systems, the problem of robustly local and global observability has not been considered in TS-fuzzy-model-based control systems.
This study presents a novel approach for measuring robustly local and global observ-ability in TS-fuzzy-model-based control systems with system uncertainties. Under the assumption that the nominal TS-fuzzy-model-based control systems are locally observ-able, a sufficient condition is proposed to preserve the assumed property when system uncertainties are introduced. The presented approach uses a simple algebraic derivation, and the proposed sufficient condition explicitly indicates how the relationships among bounds on system uncertainties preserve the assumed property. A robustly global observ-ability condition is also presented for uncertain TS-fuzzy-model-based control systems. An application of the proposed sufficient conditions is demonstrated in the example of a nonlinear mass-spring-damper mechanical system with both elemental parameter uncer-tainties and displacement-sensor position variations.
This paper is organized as follows. Section 2 presents an analysis of robust observability in uncertain TS-fuzzy-model-based control systems and the sufficient criteria for both robustly local and robustly global observability. Section 3 gives an illustrative example to demonstrate the applicability of the proposed sufficient criteria. Finally, Section 4 concludes the study.
2. Robust Observability Analysis. When applying the sector nonlinearity approach to fuzzy model construction, both the fuzzy set of the premise part and the linear dynamic model with system uncertainties of the consequent part in the exact TS fuzzy control model with system uncertainties can be derived from a given nonlinear control model with system uncertainties [2]. The TS-fuzzy-model-based control system with system uncertainties for the nonlinear control system with system uncertainties can be obtained in the following form:
˜
Ri: IF z
1 is Mi1 and . . . and zg is Mig,
THEN ˙x (t) = (Ai+ ∆Ai) x (t) + (Bi+ ∆Bi) u (t) (1) and y (t) = (Ci+ ∆Ci) x (t) , (2) with the initial state vector x (0), where ˜Ri (i = 1, 2, . . . , N ) denotes the i-th implica-tion; 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) , . . . , ym(t)]T denotes the m-dimensional output vector; u (t) = [u1(t) , u2(t) , . . . , up(t)]T denotes the p-dimensional input vector; zi(i = 1, 2, . . . , g) are the premise variables; Ai, Bi and Ci (i = 1, 2, . . . , N ) are the con-sequent constant matrices n × n, n × p and m × n, respectively; ∆Ai, ∆Bi and ∆Ci (i = 1, 2, . . . , N ) are uncertain matrices in the system matrices Ai, the input matrices Bi and the output matrices Ci, respectively, of the consequent part of the i-th rule due to in-accurate measurements, inaccessibility of system parameters, output-sensor measurement variations, or parameter variations, and Mij (i = 1, 2, . . . , N and j = 1, 2, . . . , g) are the fuzzy sets.
Many interesting problems arise from a few uncertainties entering into many entries of the system, input and output matrices [26-29]. The proposed approach presents the
uncertain matrices ∆Ai, ∆Bi and ∆Ci in the forms ∆Ai = ¯ m ∑ k=1 εikAik, ∆Bi = ¯ m ∑ k=1 εikBik, and ∆Ci = ¯ m ∑ k=1 εikCik, (3) respectively, where εik (i = 1, 2, . . . , N and k = 1, 2, . . . , ¯m) are the elemental parametric uncertainties and where Aik, Bik and Cik are the given n× n, n × p and m × n constant matrices, respectively, which are prescribed a priori to denote information that is linearly dependent on the elemental parametric uncertainties εik, in which i = 1, 2, . . . , N and k = 1, 2, . . . , ¯m.
2.1. Robustly local observability. For the uncertain TS-fuzzy-model-based control system in (1) and (2), this subsection assumes that each fuzzy-rule-nominal model ˙x (t) =
Aix (t) + Biu (t) and y (t) = Cix (t) denoted by {Ai, Ci} is observable (i.e., each fuzzy-rule-nominal model {Ai, Ci} has a full row rank for its observability matrix). Due to inevitable uncertainties, each fuzzy-rule-nominal model {Ai, Ci} is perturbed into the fuzzy-rule-uncertain model {Ai+ ∆Ai, Ci+ ∆Ci} . The considered problem is determin-ing the condition under which each fuzzy-rule-uncertain model {Ai+ ∆Ai, Ci+ ∆Ci} for the TS-fuzzy-model-based control system in (1) and (2) remains observable. Before in-vestigating the uncertain TS-fuzzy-model-based control system in (1) and (2) in terms of the robustness of observability, several definition and lemmas must be introduced. Definition 2.1. [3]: The TS-fuzzy-model-based control system is locally observable if each fuzzy-rule model {Ai+ ∆Ai, Ci+ ∆Ci} (i = 1, 2, . . . , N) is observable.
Lemma 2.1. The system model ˙x(t) = Ax(t) + Bu(t) and y (t) = Cx (t) is observable if and only if the n2× n(n + m − 1) matrix
Q = In 0 • • • 0 0 • • • 0 CT −AT I n • • • 0 0 • • • CT 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 0 • • • In 0 • • • 0 0 0 0 • • • −AT CT • • • 0 0 (4)
has rank n2, where A∈ Rn×n, C ∈ Rm×n and I
n denotes the n× n identity matrix. Proof: In the above matrix Q of (4), add the product of AT and the first (block) row to the second row. Then add the product of AT and the second row to the third row, and so on. The resulting matrix is
In 0 • • • 0 0 • • • 0 CT 0 In • • • 0 0 • • • CT ATCT • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 0 • • • In 0 • • • • (An−2) T CT 0 0 • • • 0 CT ATCT • • • (An−1)TCT . (5)
The observability matrix [
CT ATCT • • • (An−1)TCT
]
is of rank n if and only if the matrix in (5) has rank n2 (i.e., the matrix in (4) has rank n2) [30]. Therefore, we have the stated result.
Lemma 2.2. [31]: The matrix measures of matrices ¯W and ¯V , namely µ( ¯W ) and µ( ¯V ), respectively, are well defined for any norm and have the following properties:
(i) µ(±I) = ±1, for the identity matrix I;
(ii) − ¯W ≤ −µ(− ¯W ) ≤ Re(λ( ¯W )) ≤ µ( ¯W ) ≤ ¯W , for any norm k•k and any matrix ¯W ∈ Cn×n;
(iii) µ( ¯W + ¯V )≤ µ( ¯W ) + µ( ¯V ), for any two matrices ¯W , ¯V ∈ Cn×n;
(iv) µ(γ ¯W ) = γµ( ¯W ), for any matrix ¯W ∈ Cn×n and any non-negative real number γ; where λ( ¯W ) denotes any eigenvalue of ¯W , and Re(λ( ¯W )) denotes the real part of λ( ¯W ). Lemma 2.3. For any γ < 0 and any matrix ¯W ∈ Cn×n, µ(γ ¯W ) =−γµ(− ¯W ).
Proof: This lemma can be immediately obtained from property (iv) in Lemma 2.2. Lemma 2.4. Let ¯N ∈ Cn×n. If µ(− ¯N ) < 1, then det(I + ¯N )6= 0.
Proof: Since µ(− ¯N ) < 1, property (ii) in Lemma 2.2 gets Re(λ( ¯N ))≥ −µ(− ¯N ) >−1.
This implies that λ( ¯N )6= −1. Therefore, we have the stated result.
According to Lemma 2.1, for the uncertain TS-fuzzy-model-based control system in (1) and (2), each fuzzy-rule-uncertain model {Ai+ ∆Ai, Ci+ ∆Ci} in (1) and (2) is observ-able if and only if the n2× n(n + m − 1) matrix
˜ Qi = Qi + ¯ m ∑ k=1 εikEik (6)
has a full row rank n2, where
Qi = In 0 • • • 0 0 • • • 0 CiT −AT i In • • • 0 0 • • • CiT 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 0 • • • In 0 • • • 0 0 0 0 • • • −ATi CiT • • • 0 0 (7) and Eik = 0 0 • • • 0 0 • • • 0 CikT −AT ik 0 • • • 0 0 • • • C T ik 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 0 • • • 0 0 • • • 0 0 0 0 • • • −AT ik CikT • • • 0 0 . (8)
Let the singular value decomposition of Qi be Qi = Ui [ Si 0n2×n(m−1) ] ViH, (9) where Ui ∈ Rn 2×n2
and Vi ∈ Rn(n+m−1)×n(n+m−1) are the unitary matrices, ViH denotes the complex-conjugate transpose of matrix Vi, Si = diag[σi1, . . . , σin2], and σi1 ≥ σi2 ≥
· · · ≥ σin2 > 0 are the singular values of Qi.
The sufficient criterion presented next ensures that the uncertain TS-fuzzy-model-based control system in (1) and (2) is robustly locally observable.
Theorem 2.1. Suppose that each fuzzy-rule-nominal model {Ai, Ci} is observable. The uncertain TS-fuzzy-model-based control system in (1) and (2) is robustly locally observable if the following conditions simultaneously hold true:
¯ m ∑ k=1 εikφik < 1, (10) where i = 1, 2, . . . , N ; φik = { µ(−Si−1UH i EikVi[In2, 0n2×n(m−1)]T ) , for εik ≥ 0; −µ(Si−1UiHEikVi[In2, 0n2×n(m−1)]T ) , for εik < 0;
the matrices Eik, Si, Ui and Vi (i = 1, 2, . . . , N ) are defined by (8) and (9), respectively, and In2 denotes the n2× n2 identity matrix.
Proof: Since each fuzzy-rule-nominal model {Ai, Ci} (i = 1, 2, . . . , N) is observable, matrix Qi in (7) has a full row rank (i.e., rank(Qi) = n2) according to Lemma 2.1. We know that rank(Qi) = rank ( Si−1UiHQiVi ) . (11)
Thus, instead of rank( ˜Qi), we can discuss the rank of [ In2 0n2×n(m−1) ] + ¯ m ∑ k=1 εikRik, (12) where Rik = Si−1UiHEikVi, for i = 1, 2, . . . , N and k = 1, 2, . . . , ¯m. Since a matrix has rank of at least n2 if it has at least one nonsingular n2× n2 submatrix, a sufficient condition for the matrix in (12) to have rank n2 is the nonsingularity
Gi = In2 + ¯ m ∑ k=1 εikR¯ik, (13) where ¯Rik = Si−1UiHEikVi[In2, 0n2×n(p−1)]T, for i = 1, 2, . . . , N .
According to Lemmas 2.2 and 2.3 and (10),
µ ( − ¯ m ∑ k=1 εikR¯ik ) = µ ( − ¯ m ∑ k=1 εik ( Si−1UiHEikVi [ In2, 0n2×n(p−1) ]T)) ≤ ¯ m ∑ k=1 µ ( −εik ( Si−1UiHEikVi [ In2, 0n2×n(p−1) ]T)) = ¯ m ∑ k=1 εikφik < 1. (14)
Thus, from Lemma 2.4, we have
det(Gi) = det ( In2 + ¯ m ∑ k=1 εikR¯ik ) 6= 0. (15)
Hence, matrix Gi in (13) is nonsingular. That is, matrix ˜Qi in (6) has a full row rank of n2. According to the above results and Lemma 2.1, the local observability of the uncertain TS-fuzzy-model-based control system in (1) and (2) is ensured. Thus, the proof is completed.
robustly and globally observable, the data obtained data in (34)-(38) show that the cri-terion for robustly global observability is more conservative than that for robustly local observability.
4. Conclusions. In this study of the robust observability problem for uncertain TS-fuzzy-model-based control systems, the problem of rank preservation for robust observ-ability of uncertain TS-fuzzy-model-based control systems is converted to a nonsingularity analysis problem. Under the assumption that each fuzzy rule of a nominal TS-fuzzy-model-based control system has a full row rank for its observability matrix, a sufficient criterion was proposed for preserving the assumed property when the system uncertain-ties are included in the nominal TS-fuzzy-model-based control systems. The proposed sufficient criterion indicates the explicit relationships of bounds on system uncertainties that are needed to preserve the assumed property. The criterion for the robustly global observability of the TS-fuzzy-model-based control system is also presented. A nonlinear mass-spring-damper mechanical system with both elemental parameter uncertainties and displacement-sensor measurement variations is also given to illustrate the application of the proposed sufficient criteria.
Acknowledgment. This work was in part supported by the National Science Coun-cil, Taiwan, under grant numbers NSC96-2628-E-327-004-MY3, NSC 97-2221-E-037-003, NSC98-2221-E-151-048 and NSC 99-2320-B-037-026-MY2.
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