880
IEEE TRANSACTIONS ON CYBERNETICS, VOL. 45, NO. 5, MAY 2015
Universal Fuzzy Models and Universal Fuzzy Controllers for Discrete-Time Nonlinear Systems Qing Gao, Member, IEEE, Gang Feng, Fellow, IEEE, Daoyi Dong, Senior Member, IEEE, and Lu Liu, Senior Member, IEEE
Abstract—This paper investigates the problems of universal fuzzy model and universal fuzzy controller for discrete-time nonaffine nonlinear systems (NNSs). It is shown that a kind of generalized T-S fuzzy model is the universal fuzzy model for discrete-time NNSs satisfying a sufficient condition. The results on universal fuzzy controllers are presented for two classes of discrete-time stabilizable NNSs. Constructive procedures are provided to construct the model reference fuzzy controllers. The simulation example of an inverted pendulum is presented to illustrate the effectiveness and advantages of the proposed method. These results significantly extend the approach for potential applications in solving complex engineering problems. Index Terms—Nonaffine nonlinear systems (NNSs), Takagi– Sugeno (T-S) fuzzy models, universal fuzzy controllers.
I. I NTRODUCTION N recent decades, Takagi–Sugeno (T-S) fuzzy models [1], or the so-called fuzzy dynamic models [2], have attracted tremendous attention due to their great advantages in approximating complex nonlinear systems functions and exploiting conventional linear control theory. A large number of theoretical results on systematic stability analysis and controller synthesis have been presented for nonlinear systems using relevant T-S fuzzy approximation models [3]–[17]. However, the results in [18] showed that the commonly used T-S fuzzy models, whose premise variables contains only state variables of the given systems, are only able to approximate affine nonlinear functions to arbitrary level of accuracy on a convex compact set. That is to say, the fuzzy-model-based stabilization approaches based on the commonly used T-S fuzzy models can be only applied to affine nonlinear systems but run into difficulties in dealing with nonaffine nonlinear systems (NNSs).
I
Manuscript received December 17, 2013; revised May 9, 2014; accepted July 8, 2014. Date of publication August 14, 2014; date of current version April 13, 2015. This work was supported in part by the Research Grants Council of the Hong Kong SAR of China under Project CityU/113212 and in part by the Australian Research Council’s Discovery Projects funding scheme under Project DP130101658. This paper was recommended by Associate Editor H. M. Schwartz. Q. Gao is with the School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2600, Australia, and also with the Department of Automation, University of Science and Technology of China, Hefei, Anhui 230026, China (e-mail: [email protected]). G. Feng and L. Liu are with the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong (e-mail: [email protected]; [email protected]). D. Dong is with the School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2600, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2014.2338312
For discrete-time NNSs, Gao et al. [19] recently presented a class of generalized T-S fuzzy models that can be universal function approximators. A fuzzy-model-based approach to semi-global stabilization of discrete-time NNSs was also developed in terms of linear matrix inequalities (LMIs). Nevertheless, some critical issues relevant to the work in [19] still deserve further investigation. For example, [19] only showed the capability of generalized T-S fuzzy models to approximate static nonlinear functions. Are the generalized T-S fuzzy models universal in the sense that the error between the states (or outputs) of the original discrete-time NNS and its generalized T-S fuzzy approximation system can be also arbitrarily small? This is the so-called universal fuzzy model problem which is the first main contribution of this paper. Furthermore, it follows from [19] that the fuzzy stabilization controllers are designed in terms of LMIs whose feasibility is, however, difficult to determine. Does there always exist a fuzzy controller to stabilize a discrete-time NNS if it can be stabilized by an appropriately designed controller? The solution to the universal fuzzy controller problem [20] is another main contribution of this paper. In this sense, this paper can be regarded as a significant extension to the results in [19]. The universal fuzzy controller problem has been investigated in some publications, to mention a few, [20]–[22]. However, to the best of our knowledge, this is the first work on universal fuzzy controller problems in the context of discretetime NNSs. This paper, actually, parallels the recent work [22] on universal fuzzy controllers for continuous-time NNSs. The main ideas of this paper are rooted in those similar ideas in [22]. However, there is significant difference from that in [22] for treating technical details in the discrete-time case. In addition, the capability of generalized T-S fuzzy models to approximate dynamic systems is proved in this paper, which extends the result of static function approximation in [22]. Motivated by the significance and wide applications of discrete-time control systems in various engineering fields, we find it desirable to investigate the problems of universal fuzzy model and universal fuzzy controller for discretetime NNSs. These problems are effectively addressed in this paper. Notations. I and 0 denote, respectively, an identity matrix and a zero matrix of appropriate dimensions. For a real symmetric matrix X, X > 0 means that X is positive definite. x denotes the Euclidean norm of vector x, while A denotes the matrix induced norm of matrix A.
c 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2168-2267 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
GAO et al.: UNIVERSAL FUZZY MODELS AND UNIVERSAL FUZZY CONTROLLERS FOR DISCRETE-TIME NONLINEAR SYSTEMS
II. U NIVERSAL F UZZY M ODELS FOR D ISCRETE -T IME NNS S The following difference equation represents a large class of discrete-time NNSs: x(t + 1) = f (x(t), u(t))
(1)
where x(t) = [x1 (t), . . . , xn (t)]T ∈ X ⊂ n , u(t) = [u1 (t), . . . , um (t)]T ∈ U ⊂ m , X × U is a compact region containing the origin. It is assumed throughout this paper that the continuously differentiable mapping f : n × m → n vanishes at zero, i.e., f (0, 0) = 0. System (1) is called NNS because its system function f is not affine in terms of the system state x nor the control input u. The following generalized T-S fuzzy models [19] can be used to approximate the discrete-time NNS (1): Plant rule Rl : IF x1 (t) is μl1 AND . . . AND xn (t) is μln ; u1 (t) is ν1l AND . . . AND um (t) is νml ; THEN x(t + 1) = Al x(t) + Bl u(t), l ∈ {1, 2, . . . , r}
(2)
which, under the standard fuzzy inference method, can be globally rewritten as x(t + 1) = fˆ (x(t), u(t))
(3)
with ⎧ r ⎪ ⎪ fˆ (x(t), u(t)) = l=1 wl (x(t), u(t))[Al x(t) + Bl u(t)] ⎨ n l μli (xi ) m i=1 j=1 νj (uj ) ⎪ w (x, u) = ⎪ r n ⎩ l μl (x ) m ν l (u ) l=1
i=1
i
i
j=1 j
xˆ (t + 1) = fˆ (ˆx(t), u(t)).
(7)
Let DGF be the set of all discrete-time generalized T-S fuzzy models in the form of (7). Definition 1: Given any discrete-time NNS (1) and an arbitrarily constant ε > 0, if there exists a fˆ ∈ DGF such that under the same initial conditions sup x(t) − xˆ (t) < ε
(8)
t≥0
for the dynamic systems (1) and (7), then DGF is called as a set of universal fuzzy models. Before proceeding, the following lemma regarding the generalized discrete-time Gronwall lemma is presented. Lemma 2 [25]: Suppose y(n) , p(n) , q(n) , and
r(n) , n ∈ N are nonnegative sequences satisfying n−1
q(k)y(k)
(9)
k=0
(4)
then y(n) ≤ p(n) + r(n)
where Rl denotes the lth rule, r the total number of rules, μli and νjl the fuzzy sets, [Al , Bl ] the matrices corresponding to the lth local model, and wl (x, u) the fuzzybasis membership functions satisfying 1 ≥ wl (x, u) ≥ 0 and rl=1 wl (x, u) = 1. For the generalized T-S fuzzy models as in (2) or (3), their capability of universal function approximation is summarized in the following lemma. Lemma 1 [19]: For an arbitrarily given discrete-time NNS in (1) and an arbitrarily positive constant f , there exists a generalized T-S fuzzy model fˆ (x, u) as in (4) such that, for any (x, u) ∈ X × U
n−1
q(k)p(k)
k=0
n−1
(q(j)r(j) + 1). (10)
j=k+1
For any discrete-time NNS (1) Jf |[x,u]=[0,0]
∂f (x, u)
= = [A, B] ∂[xT , uT ]T [x,u]=[0,0]
(11)
is the Jacobian matrix of the function f (x, u) at the origin, where A ∈ n×n and B ∈ n×m . Let b(x, u) = f (x, u) − Ax ˆ u) = fˆ (x, u) − Ax. Then b(·) ˆ and b(x, satisfies the so-called Lipschitz condition, that is ˆ 2 , u) ≤ βb x1 − x2 , ˆ 1 , u) − b(x b(x
x1 , x2 ∈ X (12)
(5)
where the constant βb > 0. Then we have the following result which is the first main contribution of this paper. Theorem 1: DGF is a set of universal fuzzy models for discrete-time NNSs, if the matrix A in (11) and constant βb in (12) satisfy
(6)
βb + A < 1.
l=1
where [Al , Bl ] ≤ f .
of the dynamic systems (1) and (3) also achieves arbitrarily small under the same initial conditions. In fact, the error might grow unbounded as time goes [2]. Therefore, the approximation between these two dynamic systems in (1) and (3), still deserves further investigation. For the sake of convenience, consider the following generalized T-S fuzzy system:
y(n) ≤ p(n) + r(n)
j
f (x, u) = fˆ (x, u) + E(x, u) r = wl (x, u)[(Al + Al )x + (Bl + Bl )]
881
Remark 1: For a given nonaffine nonlinear function, a systematic algorithm was presented in [22] to construct the generalized T-S fuzzy approximation model, which can be used in both discrete-time and continuous-time cases. It follows from [22] that an arbitrarily small error upper bound for approximation can be achieved through using many enough fuzzy rules in the T-S fuzzy model (3). Note that only the approximation between static nonlinear functions f and fˆ are concerned in Lemma 1. However, generally one cannot guarantee that the error between the states
(13)
Proof: First we rewrite the dynamic system (1) as x(t + 1) = Ax(t) + b(x(t), u(t))
(14)
ˆ x(t), u(t)). xˆ (t + 1) = Aˆx(t) + b(ˆ
(15)
and (7) as Denote e(t) = x(t)−ˆx(t). We have the following relationship from (14) and (15): e(t + 1) = Ae(t) + E(t)
(16)
882
IEEE TRANSACTIONS ON CYBERNETICS, VOL. 45, NO. 5, MAY 2015
ˆ x(t), u(t)). where E(t) = b(x(t), u(t)) − b(ˆ Note that e(0) = x(0) − xˆ (0) = 0. Then one can obtain the solution e(t) of (16) as e(t) =
t−1
t−1−k
A
E(k).
(17)
k=0
Based on Lemma 1 and (12), the following relationship can be obtained for an arbitrarily given positive constant εb : ˆ x(t), u(t)) E(t) = b(x(t), u(t)) − b(ˆ ˆ u) + b(x, ˆ u) − b(ˆ ˆ x, u) = b(x, u) − b(x, ˆ u) − b(ˆ ˆ x, u) = f (x, u) − fˆ (x, u) + b(x,
e(t) ≤ ≤
k=0 t−1
A
In other words, the stabilization problem of system (1) can be solved as a problem of robust stabilization for the uncertain T-S fuzzy system (22). In this paper, we employ the following fuzzy controller dynamic feedback:
E(k)
At−1−k (εb + βb e(t))
1 − At + βb At−1 A−k e(k). (19) 1 − A
If condition (13) is satisfied, then by applying Lemma 2 to (19), one has 1 − At + βb At−1 1 − A t−1
A−k εb
k=0
t−1 1 − Ak (A−j βb Aj−1 + 1) 1 − A j=k+1
βb + A t−k−1 1 + εb βb ≤ εb A−k 1 − A 1 − A A k=0 1 1 − (βb + A)t 1 + βb = εb 1 − A 1 − βb − A εb 1 1 1 + βb = . (20) ≤ εb 1 − A 1 − βb − A 1 − βb − A
Therefore, for any positive constant ε, one has e(t) = x(t) − xˆ (t) < ε, t ≥ 0
(21)
by choosing εb < (1 − βb − A)ε. Then one can conclude via Definition 1 that, DGF is a set of universal fuzzy models for discrete-time NNSs, and the proof is thus completed. III. U NIVERSAL F UZZY C ONTROLLERS A. Semi-Global Stabilization of Discrete-Time NNSs Using Generalized T-S Fuzzy Models It follows from Lemma 1 that, in a compact region, a NNS (1) is equivalent to the following uncertain T-S fuzzy system: r
r
wl (¯x(t))[R1 A¯ l + R2 K¯ l + R1 A¯ l ]¯x(t). (25)
¯x(t) ≤ C¯x(0)σ t
wl (x(t), u(t))
l=1
× [(Al + Al )x(t) + (Bl + Bl )u(t)]
(26)
then the closed-loop control system (25) is said to be semiglobally uniformly exponentially stable in a compact region X × U. Theorem 2: If there exist a matrix X > 0, a set of matrices Wl and a set of positive constants εl , l ∈ {1, 2, . . . , r}, such that the linear matrix inequalities ⎡
t−1 At−1
x(t + 1) =
(24)
Next we present the following definition and theorem. Definition 2: If there exist two constants C > 0 and σ ∈ (0, 1), and a compact region X0 × U0 ⊂ X × U around the origin, such that given an arbitrary initial state x¯ (0) ∈ X0 ×U0 , the solution x¯ (t) of (25) exists uniquely and satisfies
k=0
×
wl (x(t), u(t))[Fl x(t) + Gl u(t)].
l=1
t−1
e(t) ≤ εb
r
Denote x¯ (t) = [xT (t), uT (t)]T , R1 = [In , 0n×m ]T , R2 = [0m×n , Im ]T , A¯ l = [Al , Bl ], A¯ l = [Al , Bl ] and K¯ l = [Fl , Gl ]. Then we can express the closed-loop control system consisting of (22) and (24) as x¯ (t + 1) =
k=0
= εb
(23)
l=1
Then from (17) and (18), one has t−1−k
[Al , Bl ] ≤ f .
u(t + 1) =
ˆ u) − b(ˆ ˆ x, u) ≤ f (x, u) − fˆ (x, u) + b(x, < εb + βb e(t). (18) t−1
where
(22)
⎤ X f X (R1 A¯ l X + R2 Wl )T ⎣ ⎦>0 f X εl I 0 R1 A¯ l X + R2 Wl 0 X − εl R1 RT1
(27)
are satisfied, then NNS (1) can be semi-globally exponentially stabilized using the fuzzy controller (24) and moreover, the controller gains can be given as K¯ l = Wl X −1 . Proof: Theorem 1 can be proved using the quadratic Lyapunov function V(t) = x¯ T (t)X −1 x¯ (t) and the proof is thus omitted here. Remark 2: Note that the premise variables of the generalized T-S fuzzy model contain both the control input u and the system state x, thus the well-known parallel distributed compensation (PDC) scheme u(t) = rl=1 wl (x(t), u(t))g(x(t)) cannot be used because it would lead to an algebraic loop. B. Universal Fuzzy Controllers From Theorem 1, it is clear that the proposed approach for controller design of discrete-time NNSs relies on feasibility of a set of LMIs as in (27), which might run into difficulty when a large number of fuzzy rules are involved. Thus the following question is critical and meaningful. Does there exist a dynamic feedback fuzzy controller in the form of (24) to stabilize a discrete-time NNS if it can be stabilized by a smooth dynamic feedback controller u(t + 1) = g(x(t), u(t))? This socalled universal fuzzy controller problem will be addressed subsequently.
GAO et al.: UNIVERSAL FUZZY MODELS AND UNIVERSAL FUZZY CONTROLLERS FOR DISCRETE-TIME NONLINEAR SYSTEMS
Let DGFC be the set of all dynamic feedback fuzzy controllers in the form of (24). The following definitions are introduced first. Definition 3 [24]: Given a discrete-time NNS x(t + 1) = f (x(t), u(t)), if there exists a dynamic feedback controller u(t + 1) = g(x(t), u(t)) such that the closed-loop control system x(t + 1) = f (x(t), u(t)) (28) u(t + 1) = g(x(t), u(t)) is globally uniformly exponentially stable, the NNS is said to be globally uniformly exponentially stabilizable. Definition 4: Given any discrete-time NNS which can be globally uniformly exponentially stabilizable, if there exists a fuzzy control law gˆ belonging to DGFC such that the following closed-loop control system: x(t + 1) = f (x(t), u(t)) (29) u(t + 1) = gˆ (x(t), u(t)) is semi-globally uniformly exponentially stable on a compact set X × U ⊂ n × m , then DGFC is said to be a set of universal fuzzy controllers. Theorem 3: For the class of discrete-time NNSs which are globally uniformly exponentially stabilizable, DGFC is a set of universal fuzzy controllers. Proof: Consider a discrete-time NNS x(t +1) = f (x(t), u(t)) which is globally uniformly exponentially stabilizable. Then there exists a control law u(t + 1) = g(x(t), u(t)), where the mapping g ∈ C 1 , such that the closed-loop control system (28) is globally uniformly exponentially stable. It follows from Lemma 1 that for an arbitrary positive constant g , a dynamic fuzzy controller u(t + 1) = gˆ (x(t), u(t)) ∈ DGFC can be constructed such that gˆ (x(t), u(t)) = g(x(t), u(t)) + (x(t), u(t))
(x, u) ≤ g [xT , uT ]T .
(31)
ˆ x) = R1 f (x, u) + Denote G(¯x) = R1 f (x, u) + R2 g(x, u), G(¯ R2 gˆ (x, u) and ¯ (¯x) = R2 (x, u), where x¯ , R1 and R2 are defined in Section III-A. Then the closed-loop control system (28) may be rewritten as x¯ (t + 1) = G(¯x(t))
V(¯x(t) + d(t)) − V(¯x(t)) ≤ cv d(t)
(33)
Based on the discrete-time Lyapunov converse theorem in [24], the exponential stability of the system (32) is equivalent to that there exist a positive constant c > 1 and a continuous Lyapunov function V(¯x(t)) satisfying
where the constant cv > 0. From (34) and (35), it is clear that the difference of V(¯x(t)) along the trajectories of the system (33) satisfies V(G(¯x(t)) + ¯ (¯x(t))) − V(¯x(t)) = V(G(¯x(t)) + (¯ ¯ x(t))) − V(G(¯x(t))) + V(G(¯x(t))) − V(¯x(t)) ≤ cv ¯ (¯x(t)) − ¯x(t) ≤ (cv g − 1)¯x(t). (36) By choosing a fuzzy controller (24) such that g < 1/cv , one has that along the trajectories of the system (33) V(¯x(t + 1)) − V(¯x(t)) < −˜c¯x(t)
(37)
where c˜ = 1 − cv g > 0. It can be concluded that the closed-loop control system (33), or equivalently, (29) is semi-globally uniformly exponentially stable on the compact set X × U, and furthermore via Definition 3 that DGFC is a set of universal fuzzy controllers. The proof is thus completed. In addition, the following analysis can reveal the rate of the exponential convergence to the origin of the closed-loop control system. It follows from (34) that: 1 V(¯x(t)) (38) V(¯x(t + 1)) ≤ V(¯x(t)) − ¯x(t)2 ≤ 1 − c which implies
1 t ¯x(t)2 ≤ V(¯x(t)) ≤ 1 − V(¯x(0)) c 1 t ≤ c 1− ¯x(0)2 . c
(39)
Thus one has that for the system (32), the state x¯ (t) satisfies t √ 1 1− . (40) ¯x(t) ≤ c¯x(0) c Similarly, the state x¯ (t) of the system (33) satisfies t √ 1 cv g ¯x(t) ≤ c¯x(0) 1− + . c c
(41)
Next an approach to model reference fuzzy controller design is considered. Suppose that x¯ (t + 1) = Gm (¯x(t)) is exponentially stable reference model, where Gm (¯x(t)) = R1 f (x(t), u(t)) + R2 gm (x(t), u(t)) is given. From Theorem 2, one can apply the scheme in [22] to construct the model reference fuzzy controller. To be specific, a dynamic feedback fuzzy controller in the form of (24) can be constructed such that for a small positive constant m > 0
¯x(t)2 ≤ V(¯x(t)) ≤ c¯x(t)2 V(¯x(t + 1)) − V(¯x(t)) = V(G(¯x(t))) − V(¯x(t)) ≤ −¯x(t)2 .
(35)
(32)
and (29) may be rewritten as ˆ x(t)) = G(¯x(t)) + ¯ (¯x(t)). x¯ (t + 1) = G(¯
Moreover, from [24] one has that for an arbitrarily bounded disturbance d(t)
(30)
where
883
gˆ m (x, u) = gm (x, u) + m (x, u)
(42)
m (x, u) ≤ m ¯x
(43)
where (34)
884
IEEE TRANSACTIONS ON CYBERNETICS, VOL. 45, NO. 5, MAY 2015
and the closed-loop control system ˆ m (¯x(t)) = R1 f (x(t), u(t)) + R2 gˆ m (x(t), u(t)) (44) x¯ (t + 1) = G is semi-globally uniformly exponentially stable on the compact set X × U.
We have shown that DGFC is a set of universal fuzzy controllers for discrete-time globally uniformly exponentially stabilizable NNSs. Here we will further consider more general discrete-time NNSs which are only globally uniformly asymptotically stabilizable. Definition 5 [24]: Given a NNS x(t + 1) = f (x(t), u(t)), if there exists a dynamic controller u(t + 1) = g(x(t), u(t)) such that for any given initial states x¯ (0), the solution x¯ (t) of (32) satisfies (45)
then the NNS is said to be globally uniformly asymptotically stabilizable. Definition 6 [24]: The closed-loop control system (29), or equivalently (33) is semi-globally input-to-state stable (ISS) on a compact set X × U, if there exist a KL function β and a K function γ , such that given an arbitrarily initial states x¯ (0) ∈ X0 × U0 ⊂ X × U the solution x¯ (t) of (33) satisfies ¯x(t) ≤ β(¯x(0), t) + γ sup ¯ (¯x(τ )) . (46) 0≤τ ≤t
Definition 7: Given any discrete-time NNS which can be globally asymptotically stabilizable, if there exists a fuzzy control law gˆ belonging to DGFC such that the closed-loop control system (33) is semi-globally input-to-state stable on a compact set X × U, then DGFC is called a set of universal practical fuzzy controllers. Theorem 4: For the class of discrete-time NNSs which are globally uniformly asymptotically stabilizable, DGFC is a set of universal practical fuzzy controllers. Proof: Consider a discrete-time NNS x(t +1) = f (x(t), u(t)) which is globally asymptotically stabilizable. Then there exists a dynamic controller u(t + 1) = g(x(t), u(t)), g ∈ C 1 such that the closed-loop control system (32) is globally uniformly asymptotically stable. In addition, from Lemma 1, a dynamic fuzzy controller u(t + 1) = gˆ (x(t), u(t)) ∈ DGFC can be constructed such that (30) and (31) are satisfied. It follows from the discrete-time Lyapunov converse theorem in [24] that, the global uniform asymptotic stability of the closed-loop control system (32) is equivalent to that there exist a continuous Lyapunov function V(¯x(t)), a K∞ function α1 (·), a K∞ function α2 (·) and a K function α3 (·) such that α1 (¯x(t)) ≤ V(¯x(t)) ≤ α2 (¯x(t)) V(¯x(t + 1)) − V(¯x(t)) = V(G(¯x(t))) − V(¯x(t)) ≤ −α3 (¯x(t)).
V(¯x(t) + d(t)) − V(¯x(t)) ≤ cv d(t)
(49)
where the constant c > 0. Along the trajectories of system (33), one has V(¯x(t + 1)) − V(¯x(t)) = V(G(¯x(t)) + ¯ (¯x(t))) − V(¯x(t))
C. Universal Practical Fuzzy Controllers
¯x(t) ≤ β(¯x(0), t)
It also follows from [24] that for any bounded disturbance d(t), V(¯x(t)) satisfies:
(47) (48)
= V(G(¯x(t)) + ¯ (¯x(t))) − V(G(¯x(t))) + V(G(¯x(t))) − V(¯x(t)) ≤ cv ¯ (¯x(t)) − α3 (¯x(t)) ≤ cv g ¯x(t) − α3 (¯x(t)).
(50)
It is noted that there exists a constant σ > 0 such that ¯x(t) ≤ σ , because we assume x¯ (t) is within a compact region around the origin. Therefore, by choosing g small enough such that g < α3 (σ )/cv σ , one has V(G(¯x(t)) + ¯ (¯x(t))) − V(¯x(t)) < 0 in the region {x(t) | σ ≥ ¯x(t) > α3−1 (cv g σ )}, which implies that V(¯x(t)) is an ISS-Lyapunov function candidate for the system (33) [24]. In other words, the system (33) is semi-globally ISS on the compact set X × U. It follows from Definition 6 that DGFC is a set of universal practical fuzzy controllers, and the proof is thus completed. Similarly, the following model reference fuzzy control design approach can be developed. Suppose that x¯ (t + 1) = Gm (¯x(t)) is an asymptotically stable reference model, where Gm (¯x(t)) = R1 f (x(t), u(t)) + R2 gm (x(t), u(t)) is given. From Theorem 2, one can apply the scheme in [22] to construct the model reference fuzzy controller. To be specific, a dynamic feedback fuzzy controller in the form of (24) can be constructed such that for a small positive constant m gˆ m (x, u) = gm (x, u) + m (x, u)
(51)
m (x, u) ≤ m ¯x
(52)
where
and the closed-loop control system ˆ m (¯x(t)) = R1 f (x(t), u(t)) + R2 gˆ m (x(t), u(t)) (53) x¯ (t + 1) = G is semi-globally uniformly input-to-state stable on the compact set X × U. IV. S IMULATION E XAMPLE In this section, we investigate the balancing problem of the following discretized inverted pendulum [19], to illustrate the performance of the proposed method: x1 (t + 1) = x1 (t) + Ts x2 (t) x2 (t + 1) = x2 (t) + Ts f (x(t), u(t)) where f = (g sin(x1 ) − amlx22 sin(2x1 )/2 − a cos(x1 ) (arctan(u) + 0.75u(t)) ∗ 100)/(4l/3 − aml cos2 (x1 )), Ts = 0.01 s is the sampling period, x1 is the angle of pendulum from the vertical, x2 the angular velocity, g = 9.8 m/s2 the gravity constant, m the mass of the pendulum, M the mass of the cart, a = 1/(M + m), and
GAO et al.: UNIVERSAL FUZZY MODELS AND UNIVERSAL FUZZY CONTROLLERS FOR DISCRETE-TIME NONLINEAR SYSTEMS
885
Plant rule Rl : IF |x1 | is μl And |u| is ν l THEN x(t + 1) = (Al + A)x(t) + (Bl + B)u(t)
(54)
where the membership functions are shown in Fig. 1(a) and (b) 1 0.01 A1 = A2 = A3 = 0.1729 1 1 0.01 A4 = A5 = A6 = 0.0585 1 1 0.01 A7 = A8 = A9 = 0.0036 1 0 0 0 , B2 = , B3 = B1 = −0.3089 −0.2301 −0.1854 0 0 0 , B5 = , B6 = B4 = −0.1363 −0.1016 −0.0818 0 0 0 , B8 = , B9 = . B7 = −0.0091 −0.0068 −0.0055 By using the upper bound calculating method in [22], an approximate upper bound of the approximation error can be obtained as ¯ = 0.03. Then the following controller gains can be obtained using Theorem 1: K¯ 1 = 6.0376 5.4999 −1.6829 K¯ 2 = 6.0439 5.5053 −1.2549 K¯ 3 = 5.9793 5.4492 −1.0010 K¯ 4 = 5.4099 5.4957 −0.7423 K¯ 5 = 5.4105 5.4963 −0.5530 K¯ 6 = 5.4097 5.4955 −0.4456 K¯ 7 = 5.1075 5.4923 −0.0496 K¯ 8 = 5.0985 5.4831 −0.0370 K¯ 9 = 5.0686 5.4526 −0.0297 .
Fig. 1. Simulation results. (a) Membership functions of x1 . (b) Membership functions of u. (c) System state trajectories. (d) Control input.
2l the length of the pendulum. In the example, we choose m = 2.0 kg, M = 8.0 kg, 2l = 1.0 m. Following the fuzzy modeling result in [19], we can obtain an uncertain discrete-time dynamic T-S fuzzy model.
The simulation results are shown in Fig. 1(c) and (d) under initial condition x¯ (0) = (80◦ , 0, 0), where all states converge to the origin as time goes to infinity. It noted that when the control input u is around zero, the generalized T-S fuzzy models reduce to the commonly used T-S fuzzy models. Some simulations have been implemented to illustrate this. The results are shown in Table I, where the stabilizable interval indicates that the pendulum can be balanced by corresponding fuzzy controllers under initial conditions (x1 (0), 0, 0) with the maximum x1 (0). It is clear that the approach of control design using generalized T-S fuzzy models has much better performance than the one using commonly used T-S fuzzy models, which is mainly because of their better approximation capability. One can also observe that using more fuzzy rules can achieve better control performance. However, the corresponding computation cost would increase. It has a tradeoff in practical applications for determining the number of fuzzy rules for T-S fuzzy models. V. C ONCLUSION In this paper, we have presented a class of universal generalized T-S fuzzy models for discrete-time NNSs satisfying a sufficient condition. We have also shown that a class
886
IEEE TRANSACTIONS ON CYBERNETICS, VOL. 45, NO. 5, MAY 2015
TABLE I C ONTROL D ESIGN P ERFORMANCE IN D IFFERENT C ASES
of dynamic fuzzy controllers is universal fuzzy controllers for discrete-time globally uniformly exponentially stabilizable NNSs, and universal practical fuzzy controllers for discretetime globally uniformly asymptotically stabilizable NNSs. Constructive procedures are provided to obtain these universal fuzzy controllers. It is noted that the result on the universal fuzzy model problem involves a sufficient condition. An interesting topic for future research is to reduce the conservatism of the sufficient condition or remove the sufficient condition completely. ACKNOWLEDGMENT The authors would like to thank the Associate Editor and the anonymous reviewers for their constructive comments that have been very helpful for improving the original manuscript. R EFERENCES [1] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. SMC-15, no. 1, pp. 116–132, Jan./Feb. 1985. [2] G. Feng, “A survey on analysis and design of model-based fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 14, no. 5, pp. 676–697, Oct. 2006. [3] H. K. Lam, “Stabilization of nonlinear systems using sample-data output-feedback fuzzy controller based on polynomial-fuzzy-modelbased control approach,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 42, no. 1, pp. 258–267, Feb. 2012. [4] S. Tong, T. Wang, and J. T. Tang, “Fuzzy adaptive output tracking control of nonlinear systems,” Fuzzy Sets Syst., vol. 111, no. 2, pp. 169–182, Apr. 2000. [5] Y. J. Liu, S. C. Tong, and W. Wang, “Adaptive fuzzy output tracking control for a class of uncertain nonlinear systems,” Fuzzy Sets Syst., vol. 160, no. 19, pp. 2727–2754, Oct. 2009. [6] Y. J. Liu, S. C. Tong, and C. L. P. Chen, “Adaptive fuzzy control via observer design for uncertain nonlinear systems with unmodeled dynamics,” IEEE Trans. Fuzzy Syst., vol. 21, no. 2, pp. 275–288, Apr. 2013. [7] Y. J. Liu and S. C. Tong, “Adaptive fuzzy control for a class of nonlinear discrete-time systems with backlash,” IEEE Trans. Fuzzy Syst., to be published. [8] J. B. Qiu, G. Feng, and H. J. Gao, “Non-synchronized state estimation of multi-channel networked nonlinear systems with multiple packet dropouts via T-S fuzzy affine dynamic models,” IEEE Trans. Fuzzy Syst., vol. 19, no. 1, pp. 75–90, Feb. 2011. [9] J. B. Qiu, G. Feng, and H. J. Gao, “Fuzzy-model-based piecewise H∞ static output feedback controller design for networked nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 18, no. 5, pp. 919–934, Oct. 2010. [10] S. Tong and H. Li, “Observer-based robust fuzzy control of nonlinear systems with parametric uncertainties,” Fuzzy Sets Syst. vol. 131, no. 2, pp. 165–184, Oct. 2002. [11] C. L. P. Chen, Y. Liu, and G. Wen, “Fuzzy neural network-based adaptive control for a class of uncertain nonlinear stochastic systems,” IEEE Trans. Cybern., vol. 44, no. 5, pp. 583–593, Apr. 2014. [12] D. H. Lee and D. W. Kim, “Relaxed LMI conditions for local stability and local stabilization of continuous-time Takagi–Sugeno fuzzy systems,” IEEE Trans. Cybern., vol. 44, no. 3, pp. 394–405, Mar. 2014. [13] Q. Gao, L. Liu, G. Feng, Y. Wang, and J. Qiu, “Universal fuzzy integral sliding-mode controllers based on T-S fuzzy models,” IEEE Trans. Fuzzy Syst., vol. 22, no. 2, pp. 350–362, Apr. 2014.
[14] Q. Gao, G. Feng, Z. Xi, Y. Wang, and J. Qiu, “Robust H∞ control of T-S fuzzy time-delay systems via a new sliding-mode control scheme,” IEEE Trans. Fuzzy Syst., vol. 22, no. 2, pp. 459–465, Apr. 2014. [15] H. K. Lam and F. H. F. Leung, “Sample-data fuzzy controller for time-delay nonlinear systems: Fuzzy-model-based LMI approach,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 37, no. 3, pp. 617–629, Jun. 2007. [16] S. Tong, Y. Li, Y. Li, and Y. Liu, “Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 6, pp. 1693–1704, Dec. 2011. [17] G. Feng, Analysis and Synthesis of Fuzzy Control Systems: A ModelBased Approach. Boca Raton, FL, USA: CRC Press, 2010. [18] X.-J. Zeng, J. A. Keane, and D. Wang, “Fuzzy systems approach to approximation and stabilization of conventional affine nonlinear systems,” in Proc. IEEE Int. Conf. Fuzzy Syst., Vancouver, BC, Canada, Jul. 2006, pp. 277–284. [19] Q. Gao, X.-J. Zeng, G. Feng, Y. Wang, and J. Qiu, “T-S-fuzzymodel-based approximation and controller design for general nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 42, no. 4, pp. 1131–1142, Aug. 2012. [20] J. J. Buckley, “Universal fuzzy controllers,” Automatica, vol. 28, no. 6, pp. 1245–1248, Nov. 1992. [21] S. G. Cao, N. W. Rees, and G. Feng, “Universal fuzzy controllers for a class of nonlinear systems,” Fuzzy Sets Syst., vol. 122, no. 1, pp. 117–123, Aug. 2001. [22] Q. Gao, G. Feng, Y. Wang, and J. Qiu, “Universal fuzzy controllers based on generalized T-S fuzzy models,” Fuzzy Sets Syst., vol. 201, no. 6, pp. 55–70, Aug. 2012. [23] Q. Gao, X. Zeng, G. Feng, and Y. Wang, “Universal fuzzy models and universal fuzzy controllers based on generalized T-S fuzzy models,” in Proc. IEEE Int. Conf. Fuzzy Syst., Brisbane, QLD, Australia, Jun. 2012. [24] Z. P. Jiang and Y. Wang, “A converse Lyapunov theorem for discretetime systems with disturbances,” Syst. Control Lett., vol. 45, no. 1, pp. 49–58, Jan. 2002. [25] D. S. Clark, “Short proof of a discrete Gronwall inequality,” Discrete Appl. Math., vol. 16, no. 3, pp. 279–281, Mar. 1987. [26] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. Philadelphia, PA, USA: SIAM, 1994.
Qing Gao (M’13) received the B.Eng. and Ph.D. degrees in mechanical and electrical engineering from the University of Science and Technology of China, Hefei, China, in 2008 and 2013, respectively, and the Ph.D. degree in mechatronics engineering from the City University of Hong Kong, Hong Kong, in 2013. Since 2014, he has been a Research Associate with the School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra, ACT, Australia. His current research interests include quantum control, intelligent systems and control, and variable structure control. Dr. Gao was the recipient of the Chinese Academy of Sciences Presidential Scholarship (Special Prize) and the Outstanding Research Thesis Award from the City University of Hong Kong, both in 2013.
GAO et al.: UNIVERSAL FUZZY MODELS AND UNIVERSAL FUZZY CONTROLLERS FOR DISCRETE-TIME NONLINEAR SYSTEMS
Gang Feng (F’09) received the B.Eng. and M.Eng. degrees in automatic control from Nanjing Aeronautical Institute, Nanjing, China, in 1982 and 1984, respectively, and the Ph.D. degree in electrical engineering from the University of Melbourne, Parkville, VIC, Australia, in 1992. He has been with the City University of Hong Kong, Hong Kong, since 2000, where he is currently a Chair Professor of Mechatronic Engineering. He is also a ChangJiang Chair Professor at the Nanjing University of Science and Technology, Nanjing, awarded by Ministry of Education, China. He was a Lecturer/Senior Lecturer with the School of Electrical Engineering, University of New South Wales, Sydney, NSW, Australia, during 1992–1999. His current research interests include hybrid systems and control, modeling and control of energy systems, and intelligent systems and control. Prof. Feng is an Associate Editor of the IEEE T RANSACTIONS ON F UZZY S YSTEMS and Mechatronics, and was an Associate Editor of the IEEE T RANSACTIONS ON AUTOMATIC C ONTROL, IEEE T RANSACTIONS ON S YSTEMS , M AN , AND C YBERNETICS —PART C: A PPLICATIONS AND R EVIEWS, and Journal of Control Theory and Applications. He was awarded an Alexander von Humboldt Fellowship in 1997–1998, and the IEEE T RANSACTIONS ON F UZZY S YSTEMS Outstanding Paper Award in 2007.
Daoyi Dong (SM’11) was born in Hubei, China. He received the B.E. and Ph.D. degrees from the University of Science and Technology of China, Hefei, China, in 2001 and 2006, respectively. He was a Post-Doctoral Fellow with the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China, from 2006 to 2008. Then he joined the Institute of Cyber-Systems and Control, Zhejiang University, Zhejiang, China. He held visiting positions at Princeton University, RIKEN in Japan, University of Hong Kong, and City University of Hong Kong. He is currently a Senior Lecturer at the University of New South Wales, Canberra, ACT, Australia. His current research interests include quantum control, reinforcement learning, and intelligent systems and control. Dr. Dong is a recipient of an International Collaboration Award and an Australian Post-Doctoral Fellowship from the Australian Research Council, and a K.C. Wong Post-Doctoral Fellowship and a President Scholarship from the Chinese Academy of Sciences.
887
Lu Liu (SM’13) received the Ph.D. degree from the Department of Mechanical and Automation Engineering, Chinese University of Hong Kong, Hong Kong, in 2008. From 2009 to 2012, she was an Assistant Professor with the Department of Information Physics and Computing, University of Tokyo, Tokyo, Japan, and then a Lecturer with the Department of Mechanical, Materials and Manufacturing Engineering, University of Nottingham, Nottingham, U.K. She is currently an Assistant Professor with the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong. Her current research interests include networked dynamical systems, control theory and applications, and biomedical devices. Dr. Liu received the Best Paper Award (Guan Zhaozhi Award) in the 27th Chinese Control Conference in 2008.
IEEE TRANSACTIONS ON CYBERNETICS, VOL. 45, NO. 5, MAY 2015
Universal Fuzzy Models and Universal Fuzzy Controllers for Discrete-Time Nonlinear Systems Qing Gao, Member, IEEE, Gang Feng, Fellow, IEEE, Daoyi Dong, Senior Member, IEEE, and Lu Liu, Senior Member, IEEE
Abstract—This paper investigates the problems of universal fuzzy model and universal fuzzy controller for discrete-time nonaffine nonlinear systems (NNSs). It is shown that a kind of generalized T-S fuzzy model is the universal fuzzy model for discrete-time NNSs satisfying a sufficient condition. The results on universal fuzzy controllers are presented for two classes of discrete-time stabilizable NNSs. Constructive procedures are provided to construct the model reference fuzzy controllers. The simulation example of an inverted pendulum is presented to illustrate the effectiveness and advantages of the proposed method. These results significantly extend the approach for potential applications in solving complex engineering problems. Index Terms—Nonaffine nonlinear systems (NNSs), Takagi– Sugeno (T-S) fuzzy models, universal fuzzy controllers.
I. I NTRODUCTION N recent decades, Takagi–Sugeno (T-S) fuzzy models [1], or the so-called fuzzy dynamic models [2], have attracted tremendous attention due to their great advantages in approximating complex nonlinear systems functions and exploiting conventional linear control theory. A large number of theoretical results on systematic stability analysis and controller synthesis have been presented for nonlinear systems using relevant T-S fuzzy approximation models [3]–[17]. However, the results in [18] showed that the commonly used T-S fuzzy models, whose premise variables contains only state variables of the given systems, are only able to approximate affine nonlinear functions to arbitrary level of accuracy on a convex compact set. That is to say, the fuzzy-model-based stabilization approaches based on the commonly used T-S fuzzy models can be only applied to affine nonlinear systems but run into difficulties in dealing with nonaffine nonlinear systems (NNSs).
I
Manuscript received December 17, 2013; revised May 9, 2014; accepted July 8, 2014. Date of publication August 14, 2014; date of current version April 13, 2015. This work was supported in part by the Research Grants Council of the Hong Kong SAR of China under Project CityU/113212 and in part by the Australian Research Council’s Discovery Projects funding scheme under Project DP130101658. This paper was recommended by Associate Editor H. M. Schwartz. Q. Gao is with the School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2600, Australia, and also with the Department of Automation, University of Science and Technology of China, Hefei, Anhui 230026, China (e-mail: [email protected]). G. Feng and L. Liu are with the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong (e-mail: [email protected]; [email protected]). D. Dong is with the School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2600, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2014.2338312
For discrete-time NNSs, Gao et al. [19] recently presented a class of generalized T-S fuzzy models that can be universal function approximators. A fuzzy-model-based approach to semi-global stabilization of discrete-time NNSs was also developed in terms of linear matrix inequalities (LMIs). Nevertheless, some critical issues relevant to the work in [19] still deserve further investigation. For example, [19] only showed the capability of generalized T-S fuzzy models to approximate static nonlinear functions. Are the generalized T-S fuzzy models universal in the sense that the error between the states (or outputs) of the original discrete-time NNS and its generalized T-S fuzzy approximation system can be also arbitrarily small? This is the so-called universal fuzzy model problem which is the first main contribution of this paper. Furthermore, it follows from [19] that the fuzzy stabilization controllers are designed in terms of LMIs whose feasibility is, however, difficult to determine. Does there always exist a fuzzy controller to stabilize a discrete-time NNS if it can be stabilized by an appropriately designed controller? The solution to the universal fuzzy controller problem [20] is another main contribution of this paper. In this sense, this paper can be regarded as a significant extension to the results in [19]. The universal fuzzy controller problem has been investigated in some publications, to mention a few, [20]–[22]. However, to the best of our knowledge, this is the first work on universal fuzzy controller problems in the context of discretetime NNSs. This paper, actually, parallels the recent work [22] on universal fuzzy controllers for continuous-time NNSs. The main ideas of this paper are rooted in those similar ideas in [22]. However, there is significant difference from that in [22] for treating technical details in the discrete-time case. In addition, the capability of generalized T-S fuzzy models to approximate dynamic systems is proved in this paper, which extends the result of static function approximation in [22]. Motivated by the significance and wide applications of discrete-time control systems in various engineering fields, we find it desirable to investigate the problems of universal fuzzy model and universal fuzzy controller for discretetime NNSs. These problems are effectively addressed in this paper. Notations. I and 0 denote, respectively, an identity matrix and a zero matrix of appropriate dimensions. For a real symmetric matrix X, X > 0 means that X is positive definite. x denotes the Euclidean norm of vector x, while A denotes the matrix induced norm of matrix A.
c 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2168-2267 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
GAO et al.: UNIVERSAL FUZZY MODELS AND UNIVERSAL FUZZY CONTROLLERS FOR DISCRETE-TIME NONLINEAR SYSTEMS
II. U NIVERSAL F UZZY M ODELS FOR D ISCRETE -T IME NNS S The following difference equation represents a large class of discrete-time NNSs: x(t + 1) = f (x(t), u(t))
(1)
where x(t) = [x1 (t), . . . , xn (t)]T ∈ X ⊂ n , u(t) = [u1 (t), . . . , um (t)]T ∈ U ⊂ m , X × U is a compact region containing the origin. It is assumed throughout this paper that the continuously differentiable mapping f : n × m → n vanishes at zero, i.e., f (0, 0) = 0. System (1) is called NNS because its system function f is not affine in terms of the system state x nor the control input u. The following generalized T-S fuzzy models [19] can be used to approximate the discrete-time NNS (1): Plant rule Rl : IF x1 (t) is μl1 AND . . . AND xn (t) is μln ; u1 (t) is ν1l AND . . . AND um (t) is νml ; THEN x(t + 1) = Al x(t) + Bl u(t), l ∈ {1, 2, . . . , r}
(2)
which, under the standard fuzzy inference method, can be globally rewritten as x(t + 1) = fˆ (x(t), u(t))
(3)
with ⎧ r ⎪ ⎪ fˆ (x(t), u(t)) = l=1 wl (x(t), u(t))[Al x(t) + Bl u(t)] ⎨ n l μli (xi ) m i=1 j=1 νj (uj ) ⎪ w (x, u) = ⎪ r n ⎩ l μl (x ) m ν l (u ) l=1
i=1
i
i
j=1 j
xˆ (t + 1) = fˆ (ˆx(t), u(t)).
(7)
Let DGF be the set of all discrete-time generalized T-S fuzzy models in the form of (7). Definition 1: Given any discrete-time NNS (1) and an arbitrarily constant ε > 0, if there exists a fˆ ∈ DGF such that under the same initial conditions sup x(t) − xˆ (t) < ε
(8)
t≥0
for the dynamic systems (1) and (7), then DGF is called as a set of universal fuzzy models. Before proceeding, the following lemma regarding the generalized discrete-time Gronwall lemma is presented. Lemma 2 [25]: Suppose y(n) , p(n) , q(n) , and
r(n) , n ∈ N are nonnegative sequences satisfying n−1
q(k)y(k)
(9)
k=0
(4)
then y(n) ≤ p(n) + r(n)
where Rl denotes the lth rule, r the total number of rules, μli and νjl the fuzzy sets, [Al , Bl ] the matrices corresponding to the lth local model, and wl (x, u) the fuzzybasis membership functions satisfying 1 ≥ wl (x, u) ≥ 0 and rl=1 wl (x, u) = 1. For the generalized T-S fuzzy models as in (2) or (3), their capability of universal function approximation is summarized in the following lemma. Lemma 1 [19]: For an arbitrarily given discrete-time NNS in (1) and an arbitrarily positive constant f , there exists a generalized T-S fuzzy model fˆ (x, u) as in (4) such that, for any (x, u) ∈ X × U
n−1
q(k)p(k)
k=0
n−1
(q(j)r(j) + 1). (10)
j=k+1
For any discrete-time NNS (1) Jf |[x,u]=[0,0]
∂f (x, u)
= = [A, B] ∂[xT , uT ]T [x,u]=[0,0]
(11)
is the Jacobian matrix of the function f (x, u) at the origin, where A ∈ n×n and B ∈ n×m . Let b(x, u) = f (x, u) − Ax ˆ u) = fˆ (x, u) − Ax. Then b(·) ˆ and b(x, satisfies the so-called Lipschitz condition, that is ˆ 2 , u) ≤ βb x1 − x2 , ˆ 1 , u) − b(x b(x
x1 , x2 ∈ X (12)
(5)
where the constant βb > 0. Then we have the following result which is the first main contribution of this paper. Theorem 1: DGF is a set of universal fuzzy models for discrete-time NNSs, if the matrix A in (11) and constant βb in (12) satisfy
(6)
βb + A < 1.
l=1
where [Al , Bl ] ≤ f .
of the dynamic systems (1) and (3) also achieves arbitrarily small under the same initial conditions. In fact, the error might grow unbounded as time goes [2]. Therefore, the approximation between these two dynamic systems in (1) and (3), still deserves further investigation. For the sake of convenience, consider the following generalized T-S fuzzy system:
y(n) ≤ p(n) + r(n)
j
f (x, u) = fˆ (x, u) + E(x, u) r = wl (x, u)[(Al + Al )x + (Bl + Bl )]
881
Remark 1: For a given nonaffine nonlinear function, a systematic algorithm was presented in [22] to construct the generalized T-S fuzzy approximation model, which can be used in both discrete-time and continuous-time cases. It follows from [22] that an arbitrarily small error upper bound for approximation can be achieved through using many enough fuzzy rules in the T-S fuzzy model (3). Note that only the approximation between static nonlinear functions f and fˆ are concerned in Lemma 1. However, generally one cannot guarantee that the error between the states
(13)
Proof: First we rewrite the dynamic system (1) as x(t + 1) = Ax(t) + b(x(t), u(t))
(14)
ˆ x(t), u(t)). xˆ (t + 1) = Aˆx(t) + b(ˆ
(15)
and (7) as Denote e(t) = x(t)−ˆx(t). We have the following relationship from (14) and (15): e(t + 1) = Ae(t) + E(t)
(16)
882
IEEE TRANSACTIONS ON CYBERNETICS, VOL. 45, NO. 5, MAY 2015
ˆ x(t), u(t)). where E(t) = b(x(t), u(t)) − b(ˆ Note that e(0) = x(0) − xˆ (0) = 0. Then one can obtain the solution e(t) of (16) as e(t) =
t−1
t−1−k
A
E(k).
(17)
k=0
Based on Lemma 1 and (12), the following relationship can be obtained for an arbitrarily given positive constant εb : ˆ x(t), u(t)) E(t) = b(x(t), u(t)) − b(ˆ ˆ u) + b(x, ˆ u) − b(ˆ ˆ x, u) = b(x, u) − b(x, ˆ u) − b(ˆ ˆ x, u) = f (x, u) − fˆ (x, u) + b(x,
e(t) ≤ ≤
k=0 t−1
A
In other words, the stabilization problem of system (1) can be solved as a problem of robust stabilization for the uncertain T-S fuzzy system (22). In this paper, we employ the following fuzzy controller dynamic feedback:
E(k)
At−1−k (εb + βb e(t))
1 − At + βb At−1 A−k e(k). (19) 1 − A
If condition (13) is satisfied, then by applying Lemma 2 to (19), one has 1 − At + βb At−1 1 − A t−1
A−k εb
k=0
t−1 1 − Ak (A−j βb Aj−1 + 1) 1 − A j=k+1
βb + A t−k−1 1 + εb βb ≤ εb A−k 1 − A 1 − A A k=0 1 1 − (βb + A)t 1 + βb = εb 1 − A 1 − βb − A εb 1 1 1 + βb = . (20) ≤ εb 1 − A 1 − βb − A 1 − βb − A
Therefore, for any positive constant ε, one has e(t) = x(t) − xˆ (t) < ε, t ≥ 0
(21)
by choosing εb < (1 − βb − A)ε. Then one can conclude via Definition 1 that, DGF is a set of universal fuzzy models for discrete-time NNSs, and the proof is thus completed. III. U NIVERSAL F UZZY C ONTROLLERS A. Semi-Global Stabilization of Discrete-Time NNSs Using Generalized T-S Fuzzy Models It follows from Lemma 1 that, in a compact region, a NNS (1) is equivalent to the following uncertain T-S fuzzy system: r
r
wl (¯x(t))[R1 A¯ l + R2 K¯ l + R1 A¯ l ]¯x(t). (25)
¯x(t) ≤ C¯x(0)σ t
wl (x(t), u(t))
l=1
× [(Al + Al )x(t) + (Bl + Bl )u(t)]
(26)
then the closed-loop control system (25) is said to be semiglobally uniformly exponentially stable in a compact region X × U. Theorem 2: If there exist a matrix X > 0, a set of matrices Wl and a set of positive constants εl , l ∈ {1, 2, . . . , r}, such that the linear matrix inequalities ⎡
t−1 At−1
x(t + 1) =
(24)
Next we present the following definition and theorem. Definition 2: If there exist two constants C > 0 and σ ∈ (0, 1), and a compact region X0 × U0 ⊂ X × U around the origin, such that given an arbitrary initial state x¯ (0) ∈ X0 ×U0 , the solution x¯ (t) of (25) exists uniquely and satisfies
k=0
×
wl (x(t), u(t))[Fl x(t) + Gl u(t)].
l=1
t−1
e(t) ≤ εb
r
Denote x¯ (t) = [xT (t), uT (t)]T , R1 = [In , 0n×m ]T , R2 = [0m×n , Im ]T , A¯ l = [Al , Bl ], A¯ l = [Al , Bl ] and K¯ l = [Fl , Gl ]. Then we can express the closed-loop control system consisting of (22) and (24) as x¯ (t + 1) =
k=0
= εb
(23)
l=1
Then from (17) and (18), one has t−1−k
[Al , Bl ] ≤ f .
u(t + 1) =
ˆ u) − b(ˆ ˆ x, u) ≤ f (x, u) − fˆ (x, u) + b(x, < εb + βb e(t). (18) t−1
where
(22)
⎤ X f X (R1 A¯ l X + R2 Wl )T ⎣ ⎦>0 f X εl I 0 R1 A¯ l X + R2 Wl 0 X − εl R1 RT1
(27)
are satisfied, then NNS (1) can be semi-globally exponentially stabilized using the fuzzy controller (24) and moreover, the controller gains can be given as K¯ l = Wl X −1 . Proof: Theorem 1 can be proved using the quadratic Lyapunov function V(t) = x¯ T (t)X −1 x¯ (t) and the proof is thus omitted here. Remark 2: Note that the premise variables of the generalized T-S fuzzy model contain both the control input u and the system state x, thus the well-known parallel distributed compensation (PDC) scheme u(t) = rl=1 wl (x(t), u(t))g(x(t)) cannot be used because it would lead to an algebraic loop. B. Universal Fuzzy Controllers From Theorem 1, it is clear that the proposed approach for controller design of discrete-time NNSs relies on feasibility of a set of LMIs as in (27), which might run into difficulty when a large number of fuzzy rules are involved. Thus the following question is critical and meaningful. Does there exist a dynamic feedback fuzzy controller in the form of (24) to stabilize a discrete-time NNS if it can be stabilized by a smooth dynamic feedback controller u(t + 1) = g(x(t), u(t))? This socalled universal fuzzy controller problem will be addressed subsequently.
GAO et al.: UNIVERSAL FUZZY MODELS AND UNIVERSAL FUZZY CONTROLLERS FOR DISCRETE-TIME NONLINEAR SYSTEMS
Let DGFC be the set of all dynamic feedback fuzzy controllers in the form of (24). The following definitions are introduced first. Definition 3 [24]: Given a discrete-time NNS x(t + 1) = f (x(t), u(t)), if there exists a dynamic feedback controller u(t + 1) = g(x(t), u(t)) such that the closed-loop control system x(t + 1) = f (x(t), u(t)) (28) u(t + 1) = g(x(t), u(t)) is globally uniformly exponentially stable, the NNS is said to be globally uniformly exponentially stabilizable. Definition 4: Given any discrete-time NNS which can be globally uniformly exponentially stabilizable, if there exists a fuzzy control law gˆ belonging to DGFC such that the following closed-loop control system: x(t + 1) = f (x(t), u(t)) (29) u(t + 1) = gˆ (x(t), u(t)) is semi-globally uniformly exponentially stable on a compact set X × U ⊂ n × m , then DGFC is said to be a set of universal fuzzy controllers. Theorem 3: For the class of discrete-time NNSs which are globally uniformly exponentially stabilizable, DGFC is a set of universal fuzzy controllers. Proof: Consider a discrete-time NNS x(t +1) = f (x(t), u(t)) which is globally uniformly exponentially stabilizable. Then there exists a control law u(t + 1) = g(x(t), u(t)), where the mapping g ∈ C 1 , such that the closed-loop control system (28) is globally uniformly exponentially stable. It follows from Lemma 1 that for an arbitrary positive constant g , a dynamic fuzzy controller u(t + 1) = gˆ (x(t), u(t)) ∈ DGFC can be constructed such that gˆ (x(t), u(t)) = g(x(t), u(t)) + (x(t), u(t))
(x, u) ≤ g [xT , uT ]T .
(31)
ˆ x) = R1 f (x, u) + Denote G(¯x) = R1 f (x, u) + R2 g(x, u), G(¯ R2 gˆ (x, u) and ¯ (¯x) = R2 (x, u), where x¯ , R1 and R2 are defined in Section III-A. Then the closed-loop control system (28) may be rewritten as x¯ (t + 1) = G(¯x(t))
V(¯x(t) + d(t)) − V(¯x(t)) ≤ cv d(t)
(33)
Based on the discrete-time Lyapunov converse theorem in [24], the exponential stability of the system (32) is equivalent to that there exist a positive constant c > 1 and a continuous Lyapunov function V(¯x(t)) satisfying
where the constant cv > 0. From (34) and (35), it is clear that the difference of V(¯x(t)) along the trajectories of the system (33) satisfies V(G(¯x(t)) + ¯ (¯x(t))) − V(¯x(t)) = V(G(¯x(t)) + (¯ ¯ x(t))) − V(G(¯x(t))) + V(G(¯x(t))) − V(¯x(t)) ≤ cv ¯ (¯x(t)) − ¯x(t) ≤ (cv g − 1)¯x(t). (36) By choosing a fuzzy controller (24) such that g < 1/cv , one has that along the trajectories of the system (33) V(¯x(t + 1)) − V(¯x(t)) < −˜c¯x(t)
(37)
where c˜ = 1 − cv g > 0. It can be concluded that the closed-loop control system (33), or equivalently, (29) is semi-globally uniformly exponentially stable on the compact set X × U, and furthermore via Definition 3 that DGFC is a set of universal fuzzy controllers. The proof is thus completed. In addition, the following analysis can reveal the rate of the exponential convergence to the origin of the closed-loop control system. It follows from (34) that: 1 V(¯x(t)) (38) V(¯x(t + 1)) ≤ V(¯x(t)) − ¯x(t)2 ≤ 1 − c which implies
1 t ¯x(t)2 ≤ V(¯x(t)) ≤ 1 − V(¯x(0)) c 1 t ≤ c 1− ¯x(0)2 . c
(39)
Thus one has that for the system (32), the state x¯ (t) satisfies t √ 1 1− . (40) ¯x(t) ≤ c¯x(0) c Similarly, the state x¯ (t) of the system (33) satisfies t √ 1 cv g ¯x(t) ≤ c¯x(0) 1− + . c c
(41)
Next an approach to model reference fuzzy controller design is considered. Suppose that x¯ (t + 1) = Gm (¯x(t)) is exponentially stable reference model, where Gm (¯x(t)) = R1 f (x(t), u(t)) + R2 gm (x(t), u(t)) is given. From Theorem 2, one can apply the scheme in [22] to construct the model reference fuzzy controller. To be specific, a dynamic feedback fuzzy controller in the form of (24) can be constructed such that for a small positive constant m > 0
¯x(t)2 ≤ V(¯x(t)) ≤ c¯x(t)2 V(¯x(t + 1)) − V(¯x(t)) = V(G(¯x(t))) − V(¯x(t)) ≤ −¯x(t)2 .
(35)
(32)
and (29) may be rewritten as ˆ x(t)) = G(¯x(t)) + ¯ (¯x(t)). x¯ (t + 1) = G(¯
Moreover, from [24] one has that for an arbitrarily bounded disturbance d(t)
(30)
where
883
gˆ m (x, u) = gm (x, u) + m (x, u)
(42)
m (x, u) ≤ m ¯x
(43)
where (34)
884
IEEE TRANSACTIONS ON CYBERNETICS, VOL. 45, NO. 5, MAY 2015
and the closed-loop control system ˆ m (¯x(t)) = R1 f (x(t), u(t)) + R2 gˆ m (x(t), u(t)) (44) x¯ (t + 1) = G is semi-globally uniformly exponentially stable on the compact set X × U.
We have shown that DGFC is a set of universal fuzzy controllers for discrete-time globally uniformly exponentially stabilizable NNSs. Here we will further consider more general discrete-time NNSs which are only globally uniformly asymptotically stabilizable. Definition 5 [24]: Given a NNS x(t + 1) = f (x(t), u(t)), if there exists a dynamic controller u(t + 1) = g(x(t), u(t)) such that for any given initial states x¯ (0), the solution x¯ (t) of (32) satisfies (45)
then the NNS is said to be globally uniformly asymptotically stabilizable. Definition 6 [24]: The closed-loop control system (29), or equivalently (33) is semi-globally input-to-state stable (ISS) on a compact set X × U, if there exist a KL function β and a K function γ , such that given an arbitrarily initial states x¯ (0) ∈ X0 × U0 ⊂ X × U the solution x¯ (t) of (33) satisfies ¯x(t) ≤ β(¯x(0), t) + γ sup ¯ (¯x(τ )) . (46) 0≤τ ≤t
Definition 7: Given any discrete-time NNS which can be globally asymptotically stabilizable, if there exists a fuzzy control law gˆ belonging to DGFC such that the closed-loop control system (33) is semi-globally input-to-state stable on a compact set X × U, then DGFC is called a set of universal practical fuzzy controllers. Theorem 4: For the class of discrete-time NNSs which are globally uniformly asymptotically stabilizable, DGFC is a set of universal practical fuzzy controllers. Proof: Consider a discrete-time NNS x(t +1) = f (x(t), u(t)) which is globally asymptotically stabilizable. Then there exists a dynamic controller u(t + 1) = g(x(t), u(t)), g ∈ C 1 such that the closed-loop control system (32) is globally uniformly asymptotically stable. In addition, from Lemma 1, a dynamic fuzzy controller u(t + 1) = gˆ (x(t), u(t)) ∈ DGFC can be constructed such that (30) and (31) are satisfied. It follows from the discrete-time Lyapunov converse theorem in [24] that, the global uniform asymptotic stability of the closed-loop control system (32) is equivalent to that there exist a continuous Lyapunov function V(¯x(t)), a K∞ function α1 (·), a K∞ function α2 (·) and a K function α3 (·) such that α1 (¯x(t)) ≤ V(¯x(t)) ≤ α2 (¯x(t)) V(¯x(t + 1)) − V(¯x(t)) = V(G(¯x(t))) − V(¯x(t)) ≤ −α3 (¯x(t)).
V(¯x(t) + d(t)) − V(¯x(t)) ≤ cv d(t)
(49)
where the constant c > 0. Along the trajectories of system (33), one has V(¯x(t + 1)) − V(¯x(t)) = V(G(¯x(t)) + ¯ (¯x(t))) − V(¯x(t))
C. Universal Practical Fuzzy Controllers
¯x(t) ≤ β(¯x(0), t)
It also follows from [24] that for any bounded disturbance d(t), V(¯x(t)) satisfies:
(47) (48)
= V(G(¯x(t)) + ¯ (¯x(t))) − V(G(¯x(t))) + V(G(¯x(t))) − V(¯x(t)) ≤ cv ¯ (¯x(t)) − α3 (¯x(t)) ≤ cv g ¯x(t) − α3 (¯x(t)).
(50)
It is noted that there exists a constant σ > 0 such that ¯x(t) ≤ σ , because we assume x¯ (t) is within a compact region around the origin. Therefore, by choosing g small enough such that g < α3 (σ )/cv σ , one has V(G(¯x(t)) + ¯ (¯x(t))) − V(¯x(t)) < 0 in the region {x(t) | σ ≥ ¯x(t) > α3−1 (cv g σ )}, which implies that V(¯x(t)) is an ISS-Lyapunov function candidate for the system (33) [24]. In other words, the system (33) is semi-globally ISS on the compact set X × U. It follows from Definition 6 that DGFC is a set of universal practical fuzzy controllers, and the proof is thus completed. Similarly, the following model reference fuzzy control design approach can be developed. Suppose that x¯ (t + 1) = Gm (¯x(t)) is an asymptotically stable reference model, where Gm (¯x(t)) = R1 f (x(t), u(t)) + R2 gm (x(t), u(t)) is given. From Theorem 2, one can apply the scheme in [22] to construct the model reference fuzzy controller. To be specific, a dynamic feedback fuzzy controller in the form of (24) can be constructed such that for a small positive constant m gˆ m (x, u) = gm (x, u) + m (x, u)
(51)
m (x, u) ≤ m ¯x
(52)
where
and the closed-loop control system ˆ m (¯x(t)) = R1 f (x(t), u(t)) + R2 gˆ m (x(t), u(t)) (53) x¯ (t + 1) = G is semi-globally uniformly input-to-state stable on the compact set X × U. IV. S IMULATION E XAMPLE In this section, we investigate the balancing problem of the following discretized inverted pendulum [19], to illustrate the performance of the proposed method: x1 (t + 1) = x1 (t) + Ts x2 (t) x2 (t + 1) = x2 (t) + Ts f (x(t), u(t)) where f = (g sin(x1 ) − amlx22 sin(2x1 )/2 − a cos(x1 ) (arctan(u) + 0.75u(t)) ∗ 100)/(4l/3 − aml cos2 (x1 )), Ts = 0.01 s is the sampling period, x1 is the angle of pendulum from the vertical, x2 the angular velocity, g = 9.8 m/s2 the gravity constant, m the mass of the pendulum, M the mass of the cart, a = 1/(M + m), and
GAO et al.: UNIVERSAL FUZZY MODELS AND UNIVERSAL FUZZY CONTROLLERS FOR DISCRETE-TIME NONLINEAR SYSTEMS
885
Plant rule Rl : IF |x1 | is μl And |u| is ν l THEN x(t + 1) = (Al + A)x(t) + (Bl + B)u(t)
(54)
where the membership functions are shown in Fig. 1(a) and (b) 1 0.01 A1 = A2 = A3 = 0.1729 1 1 0.01 A4 = A5 = A6 = 0.0585 1 1 0.01 A7 = A8 = A9 = 0.0036 1 0 0 0 , B2 = , B3 = B1 = −0.3089 −0.2301 −0.1854 0 0 0 , B5 = , B6 = B4 = −0.1363 −0.1016 −0.0818 0 0 0 , B8 = , B9 = . B7 = −0.0091 −0.0068 −0.0055 By using the upper bound calculating method in [22], an approximate upper bound of the approximation error can be obtained as ¯ = 0.03. Then the following controller gains can be obtained using Theorem 1: K¯ 1 = 6.0376 5.4999 −1.6829 K¯ 2 = 6.0439 5.5053 −1.2549 K¯ 3 = 5.9793 5.4492 −1.0010 K¯ 4 = 5.4099 5.4957 −0.7423 K¯ 5 = 5.4105 5.4963 −0.5530 K¯ 6 = 5.4097 5.4955 −0.4456 K¯ 7 = 5.1075 5.4923 −0.0496 K¯ 8 = 5.0985 5.4831 −0.0370 K¯ 9 = 5.0686 5.4526 −0.0297 .
Fig. 1. Simulation results. (a) Membership functions of x1 . (b) Membership functions of u. (c) System state trajectories. (d) Control input.
2l the length of the pendulum. In the example, we choose m = 2.0 kg, M = 8.0 kg, 2l = 1.0 m. Following the fuzzy modeling result in [19], we can obtain an uncertain discrete-time dynamic T-S fuzzy model.
The simulation results are shown in Fig. 1(c) and (d) under initial condition x¯ (0) = (80◦ , 0, 0), where all states converge to the origin as time goes to infinity. It noted that when the control input u is around zero, the generalized T-S fuzzy models reduce to the commonly used T-S fuzzy models. Some simulations have been implemented to illustrate this. The results are shown in Table I, where the stabilizable interval indicates that the pendulum can be balanced by corresponding fuzzy controllers under initial conditions (x1 (0), 0, 0) with the maximum x1 (0). It is clear that the approach of control design using generalized T-S fuzzy models has much better performance than the one using commonly used T-S fuzzy models, which is mainly because of their better approximation capability. One can also observe that using more fuzzy rules can achieve better control performance. However, the corresponding computation cost would increase. It has a tradeoff in practical applications for determining the number of fuzzy rules for T-S fuzzy models. V. C ONCLUSION In this paper, we have presented a class of universal generalized T-S fuzzy models for discrete-time NNSs satisfying a sufficient condition. We have also shown that a class
886
IEEE TRANSACTIONS ON CYBERNETICS, VOL. 45, NO. 5, MAY 2015
TABLE I C ONTROL D ESIGN P ERFORMANCE IN D IFFERENT C ASES
of dynamic fuzzy controllers is universal fuzzy controllers for discrete-time globally uniformly exponentially stabilizable NNSs, and universal practical fuzzy controllers for discretetime globally uniformly asymptotically stabilizable NNSs. Constructive procedures are provided to obtain these universal fuzzy controllers. It is noted that the result on the universal fuzzy model problem involves a sufficient condition. An interesting topic for future research is to reduce the conservatism of the sufficient condition or remove the sufficient condition completely. ACKNOWLEDGMENT The authors would like to thank the Associate Editor and the anonymous reviewers for their constructive comments that have been very helpful for improving the original manuscript. R EFERENCES [1] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. SMC-15, no. 1, pp. 116–132, Jan./Feb. 1985. [2] G. Feng, “A survey on analysis and design of model-based fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 14, no. 5, pp. 676–697, Oct. 2006. [3] H. K. Lam, “Stabilization of nonlinear systems using sample-data output-feedback fuzzy controller based on polynomial-fuzzy-modelbased control approach,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 42, no. 1, pp. 258–267, Feb. 2012. [4] S. Tong, T. Wang, and J. T. Tang, “Fuzzy adaptive output tracking control of nonlinear systems,” Fuzzy Sets Syst., vol. 111, no. 2, pp. 169–182, Apr. 2000. [5] Y. J. Liu, S. C. Tong, and W. Wang, “Adaptive fuzzy output tracking control for a class of uncertain nonlinear systems,” Fuzzy Sets Syst., vol. 160, no. 19, pp. 2727–2754, Oct. 2009. [6] Y. J. Liu, S. C. Tong, and C. L. P. Chen, “Adaptive fuzzy control via observer design for uncertain nonlinear systems with unmodeled dynamics,” IEEE Trans. Fuzzy Syst., vol. 21, no. 2, pp. 275–288, Apr. 2013. [7] Y. J. Liu and S. C. Tong, “Adaptive fuzzy control for a class of nonlinear discrete-time systems with backlash,” IEEE Trans. Fuzzy Syst., to be published. [8] J. B. Qiu, G. Feng, and H. J. Gao, “Non-synchronized state estimation of multi-channel networked nonlinear systems with multiple packet dropouts via T-S fuzzy affine dynamic models,” IEEE Trans. Fuzzy Syst., vol. 19, no. 1, pp. 75–90, Feb. 2011. [9] J. B. Qiu, G. Feng, and H. J. Gao, “Fuzzy-model-based piecewise H∞ static output feedback controller design for networked nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 18, no. 5, pp. 919–934, Oct. 2010. [10] S. Tong and H. Li, “Observer-based robust fuzzy control of nonlinear systems with parametric uncertainties,” Fuzzy Sets Syst. vol. 131, no. 2, pp. 165–184, Oct. 2002. [11] C. L. P. Chen, Y. Liu, and G. Wen, “Fuzzy neural network-based adaptive control for a class of uncertain nonlinear stochastic systems,” IEEE Trans. Cybern., vol. 44, no. 5, pp. 583–593, Apr. 2014. [12] D. H. Lee and D. W. Kim, “Relaxed LMI conditions for local stability and local stabilization of continuous-time Takagi–Sugeno fuzzy systems,” IEEE Trans. Cybern., vol. 44, no. 3, pp. 394–405, Mar. 2014. [13] Q. Gao, L. Liu, G. Feng, Y. Wang, and J. Qiu, “Universal fuzzy integral sliding-mode controllers based on T-S fuzzy models,” IEEE Trans. Fuzzy Syst., vol. 22, no. 2, pp. 350–362, Apr. 2014.
[14] Q. Gao, G. Feng, Z. Xi, Y. Wang, and J. Qiu, “Robust H∞ control of T-S fuzzy time-delay systems via a new sliding-mode control scheme,” IEEE Trans. Fuzzy Syst., vol. 22, no. 2, pp. 459–465, Apr. 2014. [15] H. K. Lam and F. H. F. Leung, “Sample-data fuzzy controller for time-delay nonlinear systems: Fuzzy-model-based LMI approach,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 37, no. 3, pp. 617–629, Jun. 2007. [16] S. Tong, Y. Li, Y. Li, and Y. Liu, “Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 6, pp. 1693–1704, Dec. 2011. [17] G. Feng, Analysis and Synthesis of Fuzzy Control Systems: A ModelBased Approach. Boca Raton, FL, USA: CRC Press, 2010. [18] X.-J. Zeng, J. A. Keane, and D. Wang, “Fuzzy systems approach to approximation and stabilization of conventional affine nonlinear systems,” in Proc. IEEE Int. Conf. Fuzzy Syst., Vancouver, BC, Canada, Jul. 2006, pp. 277–284. [19] Q. Gao, X.-J. Zeng, G. Feng, Y. Wang, and J. Qiu, “T-S-fuzzymodel-based approximation and controller design for general nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 42, no. 4, pp. 1131–1142, Aug. 2012. [20] J. J. Buckley, “Universal fuzzy controllers,” Automatica, vol. 28, no. 6, pp. 1245–1248, Nov. 1992. [21] S. G. Cao, N. W. Rees, and G. Feng, “Universal fuzzy controllers for a class of nonlinear systems,” Fuzzy Sets Syst., vol. 122, no. 1, pp. 117–123, Aug. 2001. [22] Q. Gao, G. Feng, Y. Wang, and J. Qiu, “Universal fuzzy controllers based on generalized T-S fuzzy models,” Fuzzy Sets Syst., vol. 201, no. 6, pp. 55–70, Aug. 2012. [23] Q. Gao, X. Zeng, G. Feng, and Y. Wang, “Universal fuzzy models and universal fuzzy controllers based on generalized T-S fuzzy models,” in Proc. IEEE Int. Conf. Fuzzy Syst., Brisbane, QLD, Australia, Jun. 2012. [24] Z. P. Jiang and Y. Wang, “A converse Lyapunov theorem for discretetime systems with disturbances,” Syst. Control Lett., vol. 45, no. 1, pp. 49–58, Jan. 2002. [25] D. S. Clark, “Short proof of a discrete Gronwall inequality,” Discrete Appl. Math., vol. 16, no. 3, pp. 279–281, Mar. 1987. [26] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. Philadelphia, PA, USA: SIAM, 1994.
Qing Gao (M’13) received the B.Eng. and Ph.D. degrees in mechanical and electrical engineering from the University of Science and Technology of China, Hefei, China, in 2008 and 2013, respectively, and the Ph.D. degree in mechatronics engineering from the City University of Hong Kong, Hong Kong, in 2013. Since 2014, he has been a Research Associate with the School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra, ACT, Australia. His current research interests include quantum control, intelligent systems and control, and variable structure control. Dr. Gao was the recipient of the Chinese Academy of Sciences Presidential Scholarship (Special Prize) and the Outstanding Research Thesis Award from the City University of Hong Kong, both in 2013.
GAO et al.: UNIVERSAL FUZZY MODELS AND UNIVERSAL FUZZY CONTROLLERS FOR DISCRETE-TIME NONLINEAR SYSTEMS
Gang Feng (F’09) received the B.Eng. and M.Eng. degrees in automatic control from Nanjing Aeronautical Institute, Nanjing, China, in 1982 and 1984, respectively, and the Ph.D. degree in electrical engineering from the University of Melbourne, Parkville, VIC, Australia, in 1992. He has been with the City University of Hong Kong, Hong Kong, since 2000, where he is currently a Chair Professor of Mechatronic Engineering. He is also a ChangJiang Chair Professor at the Nanjing University of Science and Technology, Nanjing, awarded by Ministry of Education, China. He was a Lecturer/Senior Lecturer with the School of Electrical Engineering, University of New South Wales, Sydney, NSW, Australia, during 1992–1999. His current research interests include hybrid systems and control, modeling and control of energy systems, and intelligent systems and control. Prof. Feng is an Associate Editor of the IEEE T RANSACTIONS ON F UZZY S YSTEMS and Mechatronics, and was an Associate Editor of the IEEE T RANSACTIONS ON AUTOMATIC C ONTROL, IEEE T RANSACTIONS ON S YSTEMS , M AN , AND C YBERNETICS —PART C: A PPLICATIONS AND R EVIEWS, and Journal of Control Theory and Applications. He was awarded an Alexander von Humboldt Fellowship in 1997–1998, and the IEEE T RANSACTIONS ON F UZZY S YSTEMS Outstanding Paper Award in 2007.
Daoyi Dong (SM’11) was born in Hubei, China. He received the B.E. and Ph.D. degrees from the University of Science and Technology of China, Hefei, China, in 2001 and 2006, respectively. He was a Post-Doctoral Fellow with the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China, from 2006 to 2008. Then he joined the Institute of Cyber-Systems and Control, Zhejiang University, Zhejiang, China. He held visiting positions at Princeton University, RIKEN in Japan, University of Hong Kong, and City University of Hong Kong. He is currently a Senior Lecturer at the University of New South Wales, Canberra, ACT, Australia. His current research interests include quantum control, reinforcement learning, and intelligent systems and control. Dr. Dong is a recipient of an International Collaboration Award and an Australian Post-Doctoral Fellowship from the Australian Research Council, and a K.C. Wong Post-Doctoral Fellowship and a President Scholarship from the Chinese Academy of Sciences.
887
Lu Liu (SM’13) received the Ph.D. degree from the Department of Mechanical and Automation Engineering, Chinese University of Hong Kong, Hong Kong, in 2008. From 2009 to 2012, she was an Assistant Professor with the Department of Information Physics and Computing, University of Tokyo, Tokyo, Japan, and then a Lecturer with the Department of Mechanical, Materials and Manufacturing Engineering, University of Nottingham, Nottingham, U.K. She is currently an Assistant Professor with the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong. Her current research interests include networked dynamical systems, control theory and applications, and biomedical devices. Dr. Liu received the Best Paper Award (Guan Zhaozhi Award) in the 27th Chinese Control Conference in 2008.
