Extensions to “Stability Analysis of Fuzzy Control Systems Subject to Uncertain Grades of Membership”

In the December 2005 issue of this journal, Lam and Leung proposed stability results for fuzzy control systems where the membership functions in the controller were not the same as those from the process one, but some multiplicative bounds were known. The main practical context where that situation arises is the uncertain knowledge of the memberships of a Takagi-Sugeno model. This correspondence presents further extensions of those results, allowing for a richer description of the membership uncertainty, in terms of affine inequalities.

cal examples illustrating the possibilities of the approach. A conclusion section closes this correspondence.

II. PRELIMINARIES AND NOTATION
Stability of Closed-Loop TS Fuzzy Systems: Consider a TS fuzzy system [2] of order n with r ruleṡ The aforementioned fuzzy system will be controlled via a statefeedback fuzzy controller (possibly non-PDC, where η i is a membership function that is different to µ i , at least in principle) where z denotes measurable scheduling variables (possibly coincident with z). The controller yields a closed-loop [1] x In the following, µ i and η j will be used as shorthands for µ i (z) and η j (z ), respectively. By considering a Lyapunov function V (x) = x T P x, P > 0, it is straightforward to prove that (3) is a stable system with a decay rate α (i.e., x ≤ Me −αt for some M ) if (dV /dt) + 2αV ≤ 0 for nonzero x [3]. Such inequality, in the case of (3), amounts to the wellknown expression [1], [3] − dV dt General Case: In a general case, many conditions for stability and performance of the closed-loop system (3) may be cast as positivity of fuzzy summations in the form where Q ij is a symmetric R n×n matrix, possibly including unknown decision variables to be found via optimization algorithms (usually LMI [3], [15]). For instance, (4) is a particular case of (5) with ψ = x and As another example of a performance-related condition from [16], which is later used in Section V, we have 1083-4419/$25.00 © 2008 IEEE which may be used to prove that there exists a stabilizing state-feedback controller such that the H ∞ norm (i.e., L 2 -to-L 2 induced norm) of a TS fuzzy system given byẋ is lower than γ. In [3], [9], [10], [12], and [13] other expressions of Q ij for continuous and discrete TS systems are considered.
Remark: Most of the cited literature considers Q ij for cases where i.e., resulting in conditions such as where the only difference with (5) is that µ i µ j appears instead of µ i η j . In order to check the stability and performance of the non-PDC loop (3), it is straightforward to prove that, in most cases, the Q ij proposed in PDC literature appears unaltered in (5). The difference between the PDC and the non-PDC cases lies in the LMIs needed to prove (5), which may be different to those needed for (10). 1) Indeed, in the PDC case (10), widely used sufficient conditions are the adaptation of [4,Th. 2], which may be generalized to a family of asymptotically necessary and sufficient ones [7]. 2) In the non-PDC case (5), if µ i and η j may be arbitrary, (5) holds if and only if the shape-independent conditions do as, for instance, µ 3 = η 5 = 1 (the rest being zero) involves Θ = Q 35 in (5) and the numbers three and five may be arbitrarily replaced by any i or j.
Of course, all LMIs proving (5)-such as the ones in this correspondence-prove (10) as well, but they may be very conservative for the latter case, as numerical examples will show: that is the price to pay for imperfect knowledge of the plant memberships.
Shape-Dependent Cases: There are intermediate cases where the membership functions µ i are not perfectly known (i.e., η i = µ i yielding non-PDC setups) but some knowledge on them is available. In that case, η i may be intentionally designed to be as similar as possible to µ i , and conditions which are less conservative than (11) may be stated. Indeed, Lam and Leung [1] state improved shape-dependent conditions which guarantee closed-loop stability 1 of (3) when given known values of the bounds ρ m i and ρ M i . The following lemma is the generalization of that in [1] removing the need of a symmetric X ij , requiring only X ji = X T ij , and replacing 2X ij by X ij + X ji , in the same way as Liu and Zhang [4] generalize [17]. Details of the proof are omitted for brevity. 1 The cited work discussed only one particular case of Q ij , but it is, trivially, generalizable to any other performance-related expression for Q ij . Lemma 1: Given the bounds (12), (5) holds if there exist P > 0 and matrices X ji = X T ij such that The developments in the next section will present additional results for a more general class of constraints than the "multiplicative uncertainty" (12). Numerical examples in Section V will show that, apart from allowing for more general uncertainty descriptions, the new conditions provide better results than those in [1] with multiplicative uncertainty, at least in some cases.

III. MAIN RESULT
Consider a set of p constraints on the shape of the membership functions of the plant µ i and the controller η i given by the affine inequalities where η and µ denote the membership functions arranged as a column vector, c k and a k are also column vectors, and b k is a scalar. c k , a k , and b k are assumed to be known. Notations a ik and c ik will denote the ith component of vectors a k and c k , respectively. For instance, the constraint µ 2 + µ 1 ≤ 2η 1 + 0.05, in a three-rule fuzzy system, is trivially expressed in (19) If Q ij is linear in some matrix decision variables, then Theorem 1 provides LMI conditions.

Proof: Expression (19) may be written as
Consider now, for a particular fixed k, . Evidently, Γ k ≥ 0 because it is a sum with positive coefficients of positive definite matrices. For a particular k, multiplying (24) by Γ k , we get Subsequently, by using the equalities we get a negative-semidefinite matrix to be denoted as H k given by As (26) holds for each k, By taking H from the aforementioned equation and Θ from (5), it is evident that if Θ + H > 0 can be proved, then Θ > 0. Then, conveniently grouping terms Consider now the variables X ij = X T ji , i, j = 1, . . . , r, which fulfill (20)-(22). By suitably grouping terms and associating (20) to the terms where µ i µ j appears, (21) to those with µ i η j , and (22) to those with η i η j , we have 2 Considering now the original fuzzy summations of matrix Θ in (5) and defining Hence, if (23) holds, ψ T (Θ + H)ψ > 0 for ψ = 0, and therefore, (5) holds.

IV. PARTICULAR CASES
Let us now consider some particular cases of Theorem 1.

A. Multiplicative Uncertainty
Consider now the multiplicative uncertainty case, which is also discussed in [1] Proof: The uncertainty description can be expressed as Hence, the theorem will be proved by using Theorem 1 with 2r constraints, which is divided into two groups (both with b k = 0).

1)
One with a kk = −ρ M k , a ik = 0 for i = k and c kk = +1, c ik = 0 for i = k; Theorem 1 will be applied using M ik and M † ik as relaxation variables. 2) Another group (consider a k = a k+r , c k = c k+r ) with a kk = ρ m k , a ik = 0 for i = k and c kk = −1, c ik = 0 for i = k; Theorem 1 will be applied using T ik and T † ik as relaxation variables.

Note that (20)-(22) in this case reduce to
because a ik , c ik , a ik , and c ik are zero for i = k. The conditions for the theorem being proved arise immediately once the aforementioned particular values of a ik , c ik , a ik , and c ik are substituted in (36)-(38), and the following changes of variable are made:

B. Additive Uncertainty
Consider a set of known additive bounds on the membership function δ k so that, given (5), it is known that Corollary 2: If the membership functions satisfy (39), (5) holds if there exist matrices R ij , N ij , X ij = X T ji , and X i(j+r) = X T (j+r)i , i, j = 1, . . . , r, such that for all i = 1, . . . , r and j = 1, . . . , r and Proof: The uncertainty description can be expressed as Hence, the theorem will be proved by using Theorem 1 with 2r constraints, which is divided into two groups.
1) One with b k = −δ k , a kk = 1, a ik = 0 for i = k and c kk = −1, c ik = 0 for i = k; Theorem 1 will be applied using R ik and R † ik as relaxation variables. 2) Another one with b k = −δ k , a kk = −1, a ik = 0 for i = k and c kk = +1, c ik = 0 for i = k; Theorem 1 will be applied using N ik and N † ik as relaxation variables. By noting that most a ik 's and c ik 's are zero and defining a i(k+r) = a ik , etc., Theorem 1 results in and the corollary results once the following equalities are enforced: R † ij = N ji and N † ij = R ji . The next two lemmas show the following: 1) the PDC case is recovered from the above corollaries under no uncertainty; and 2) as expected, the conditions proposed with the additive or multiplicative uncertainty bounds are less conservative than the trivial ones Q ij > 0.
Lemma 2: When the membership error bound δ i is equal to zero or the multiplicative bounds are equal to one (i.e., µ i = η i ), a feasible set of variables for Corollaries 1 and 2 may be obtained if [4,Th. 2] (which applies to the PDC case) is feasible.
If δ i = 0 for all i's, then (42) can be rewritten as By taking M ij = Q ij /2 and X i(j+r) = 0, which fulfill (48), the inequality (41) gets converted into and the matrix Y 12 is equal to zero. Finally, (43) is  [4,Th. 2]: if the latter is feasible, Corollaries 1 and 2 without uncertainty will be feasible as well.
Proof: Regarding Corollary 2, take all N ij = 0 and all X (i+r)j = X i(j+r) = 0. Then, take R ij = 0 for i = j and R ii = i I for i = 1, . . . , r, choosing a small enough i > 0 so that Q ii 2 i I (I denotes the identity matrix). Then, X ij = 0, i = j, fulfills (41) and (42), and for i = j, X ii = i I fulfills (41). Finally, all X ij 's form a diagonal positive matrix that satisfies (43) if is small enough.
Corollary 1 is also satisfied with the same choice of N ij and R ij plus N † ij = N ij and R † ij = R ij . Details are analogous to those above for Corollary 2.

EXAMPLES
This section presents several examples which illustrate the possibilities of the methodology with the additive and multiplicative uncertainties, in decay-rate and H ∞ settings.
Example 1: Let us consider a three-rule TS systemẋ = is an approximation of µ j (x) fulfilling (39), for a shared δ k = δ. Several values of the uncertainty δ will be tested, ranging from δ = 0 (which is the well-known PDC case η i = µ i ) to δ = 1 (indicating an absolute ignorance on the shape of µ i ).
The control objective will be to find the F j maximizing the achievable quadratic decay rate α by checking (5) with arising from (4) with the usual change of variables Y = P −1 and M j = F j Y . The sufficient conditions provided in Corollary 2 will be used, searching for the maximum value of α for which a feasible LMI solution exists. The maximum α achieved for different values of δ appears in Table I. The results in the referred table show that the more precise the knowledge of µ is (i.e., the lower δ is), the faster decay rates can be achieved. The results for δ = 0 are coincident with those [4, Th. 2] for the PDC case, as discussed in Lemma 2. Furthermore, the results for δ = 1 are coincident with the ones obtained by using a plain, nonfuzzy, and linear regulator u = −Kx robustly stabilizing a polytopic system via the LMI conditions [15] Such conditions are, in fact, equivalent to the shape-independent ones Q ij > 0 (indeed, conditions for M 1 in (51) are the same as those for M 2 , etc.; therefore, there is no loss of generality by assuming that In summary, with the methodology in this correspondence, a smooth transition between a full-PDC fuzzy controller and a robust linear one is achieved: as uncertainty increases, the feasible performance decreases. If the uncertainty in memberships is greater than 0.3, the performance of fuzzy and nonfuzzy (i.e., plain linear) controllers is the same. For lower uncertainty levels, the fuzzy control outperforms linear regulators, as expected.
The aforementioned model has also been tested with the multiplicative uncertainty bounds. In particular, with ρ M j = 3 and ρ m j = 1/3, application of the procedure in [1] yields an achievable decay rate of 0.304, application of Lemma 1 yields a decay rate of 0.308 (i.e., the additional decision variables achieve a marginal improvement), and Corollary 1 produces the best result, proving that a decay 0.322 is achievable.
Example 2: The same model from the previous example, with the multiplicative uncertainty bounds ρ M j = 3 and ρ m j = 1/3, has been used for the state-feedback H ∞ control, with Q ij as given in (7). If a disturbance input with B 1i = (−1 1 − 1) T is considered and an output y = (0 1 0)x, the following results are obtained: 1) a robust linear regulator obtains an induced-norm bound of 6.988; 2) a full PDC results in 6.437, i.e., a better disturbance rejection, as expected; and 3) uncertainty in memberships yields intermediate values, which are better than the linear controller but worse than the PDC one: 6.726 for the original result in [1], 6.699 for Lemma 1, and 6.593 for Corollary 1.
Example 3: Let us now discuss the same example as in [1] regarding the multiplicative uncertainty. Consider a TS fuzzy system with two rules, with matrices Analogously to [1], a two-rule state-feedback fuzzy controller is built by designing the feasible for a particular value of were also feasible for lower values of it.
Clearly, Corollary 1 in this correspondence achieves better results than [1] in all tested uncertain cases, i.e., it finds a larger set of values for a and b yielding a stable closed loop. As expected, for no uncertainty in memberships (ρ M i = ρ m i = 1), the PDC case is recovered in both cases (denoted with • in the figure), with results coincident to those from [4, Th. 2].

V. CONCLUSION
This correspondence presents an extension of the methodology in [1] to consider arbitrary affine constraints in the shape of uncertain membership functions in a non-PDC fuzzy control setup. The proposed extensions apply to various stability and performance requirements in continuous and discrete systems by making different choices for Q ij .
With the same type of restrictions than [1], numerical examples illustrate that performance improvements over [1] may also be achieved, at least in some cases. The examples in this correspondence also illustrate the gradual loss of performance from a "full-PDC" fuzzy controller to a "robust linear" one as uncertainty in the memberships increases.