Decentralized Fuzzy Control of Nonlinear Interconnected Dynamic

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 2, APRIL 2011

Decentralized Fuzzy Control of Nonlinear Interconnected Dynamic Delay Systems via Mixed H2/H∞ Optimization With Smith Predictor Chih-Lyang Hwang, Senior Member, IEEE

Abstract—Each subsystem of a nonlinear interconnected dynamic delayed system is approximated by a weighted combination of L transfer function delayed systems (TFDSs). The H2 -norm of the difference between the transfer function of a reference model and the closed-loop transfer function of the kth TFDS of subsystem i is then minimized to obtain a suitable frequency response without incurring oscillating and sluggish phenomena. Because of the existence of the disturbance at the output of the kth TFDS, which is not only large but also contains various frequency components, the H∞ -norm of the weighted sensitivity function between the output disturbance and its corresponding output of the kth TFDS is simultaneously minimized to reduce its effect. Furthermore, with proper selection of weighted sensitivity functions, certain specific modes of the output disturbance can be eliminated. Finally, two simulations are performed; one is the simulation of our designed TFDSs with different delays or nonminimum phases, and the other is the simulation of an internet-based intelligent space for the trajectory tracking of a car-like wheeled robot system. We demonstrate the effectiveness and efficiency of the proposed control. The main contributions of this paper are twofold. First, the control can simultaneously attain robust performance through the mixed H2 /H∞ optimization with a Smith predictor and achieve the robust stability via LN 2 stable with finite gain. Second, fuzzy observer is not needed. Index Terms—Decentralized control, fuzzy linear model, H2 optimization, H∞ -optimization, Internet-based intelligent space, LN 2 -stable, nonlinear interconnected dynamic delayed system (NIDDS), Smith predictor.

I. INTRODUCTION N the past three decades, the performance properties of interconnected systems have been widely studied [1]–[8]. However, the control or modeling of nonlinear interconnected systems has yet to receive sufficient attention. Due to complicated physical configuration and high dimensionality of interconnected systems, the implementation of these systems with centralized control is neither economically plausible nor technically possible [1]. In centralized control scheme, there are drawbacks such as the complexities in the design and debugging process. These, in turn, require scheduling of data collection and

I

Manuscript received April 22, 2010; revised August 8, 2010; accepted November 3, 2010. Date of publication November 29, 2010; date of current version April 4, 2011. The author is with the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei 10607, Taiwan. 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/TFUZZ.2010.2095860

data storage, etc. As a result, a decentralized control scheme is considered to be a more favorable alternative for interconnected systems. However, due to the inherent nonlinear interconnections among subsystems, which do not exist in any efficient decentralized control scheme, in dealing with nonlinear interconnected systems (e.g., the attenuation of uncertainty) [1]–[8], it needs to resort to a new control scheme for such a nonlinear interconnected system. Time delay or dead time is frequently encountered in many dynamic systems. The delay of information transmission between different subsystems, such as the delay caused by variable-speed machines [9], engine idle-speed control [10], damping control in the power system [11], an internet-based control, playout, or monitoring [12]–[17], the image processing for the dynamic target tracking of visual servo system [18], etc., results in a time-delay system. By implementing feedback control in a time-delay system, it always results in certain restrictions on achievable system bandwidth and the maximumavailable gain [10]–[18]. Based on the nominal delay of each subsystem, in this paper, the reduced effect of time-varying delay is also achieved by the Smith predictor. Recently, the use of fuzzy control for nonlinear systems has drawn much attention. Many successful applications have been reported in the literature (for example, the fuzzy controller for the adaptive navigation of a quadruped robot [19], the tracking control for a piezoelectric-actuator system [20], the fuzzy adaptive sliding-mode control for a nonlinear two-axis inverted pendulum servomechanism [21], the internet-based smart-space navigation of a car-like wheeled robot using neural-fuzzy adaptive control [16], etc.). An approach based on the Takagi and Sugeno (T-S) model to utilize heuristics-based fuzzy control in improving the system performance has been reported [16], [19], [22], [23]. In addition, a fuzzy T-S model with delayed control systems has been discussed in [24]–[34]. In many of the literatures on delayed-control systems, N linear (or bilinear) continuous-time (or discrete-time) state-space or transferfunction systems for different fuzzy sets are considered. Then, by the use of parallel-distribution compensation, a linear-state feedback control for every subsystem is adopted to obtain an acceptable performance. After that, an observer is called for if the state is not directly available [22]–[34]. Under this circumstance, however, the control scheme of utilizing closedloop system architecture becomes quite complex. In order to reduce this complexity, an alternative method is considered by using different transfer functions with the same time delay to model the consequent parts of every subsystem. The controller

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HWANG: DECENTRALIZED FUZZY CONTROL OF NONLINEAR INTERCONNECTED DYNAMIC DELAY SYSTEMS

design is therefore directed and simplified. In addition, certain work (e.g., [7] and [24]–[34]) does not attenuate the effect of time delay. Facing these challenges, in this paper, we propose a controller design via mixed H2 /H∞ optimization with Smith predictor in order to obtain a robust and desired system performance. The mixed H2 /H∞ control problem has become a popular research topic in recent years [35], [36]. In engineering practice, certain model matching procedures are applied to shape the response of the closed-loop system, in order to reduce the sluggish and oscillatory effects. In addition, the response shaping based on H2 -optimization is often adopted [37], [38]. On the other hand, H∞ -optimization is usually used as the robust control methodology for diminishing the effect of disturbance [20], [25], [29], [32]. As a result, the mixed H2 /H∞ control problem is developed to provide robust system stability and to improve system performance [39]. The paper is organized as follows: At the outset, a linear time-delay dynamic model, i.e., a specific transfer-functiondelayed system (TFDS) with respect to a specific operating point, is achieved by implementing the least-square parameterestimation technique [40], [41]. Furthermore, in order to enhance system performance, a two-degree-of-freedom (2-DOF) controller with Smith predictor is applied to each TFDS. Subsequently, the H2 -norm of the difference between an assigned reference model and the closed-loop transfer function of the kth TFDS of subsystem i is minimized to obtain a suitable frequency response without causing any oscillating and sluggish effects. However, the system performance may deteriorate or become unstable because of the disturbance at the output caused by the interconnections among subsystems, the interactions from other TFDSs, the error from fuzzy modeling, and/or the uncertainties from nonlinear time-varying subsystems. To prevent this, the H∞ -norm of the weighted sensitivity of the kth TFDS between the output disturbance and its normal output is minimized to attenuate its effect. Moreover, a specific weighted function can be selected to reject certain mode of output disturbance or to attenuate its effect in the system frequency response. In this operation, it does not need to solve the Diophantine equation; it has the computational advantage especially for a low-order system. The stability of the overall system is verified through the criterion of LN 2 -stable with finite gain. II. MATHEMATICAL PRELIMINARIES A polynomial ∈ (s) is defined as A(s) = a0 sn a + a1 sn a −1 + · · · +an a , where ai for i = 0, 1, . . . , na denotes bounded coefficients, na stands for the system degree (i.e., if a0 = 0, deg {A(s)} = na ), and s denotes the differential operator (i.e., sy(t) ≡ dy(t)/dt) or a complex variable in the Laplace transform. The Laplace transform of a continuous-time signal y(t) is denoted by Y (s) = L {y(t)} . If a0 = 1, it is called a monotonic polynomial. The notations of A+ (s) and A− (s) denote, respectively, the stable and unstable parts of A(s). The polynomial A+ (s) is assumed to be monotonic to obtain  ∞ a unique factorization. Define G(s)2 = G(jw)2 = [ −∞ |G(jw)|2 dw/2π]1/2 , where G ∈ H2 , with H2 being the

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set that consists of all analytic functions G in e {s} ≥ 0. A rational function G(s) in H2 is called inner if |G(jw)| = 1 for almost all w, and is outer if it has no zeros in e {s} ≥ 0. For rational functions G(s) and H(s) in H2 , if G(s) is an inner function and G(s)H(s)2 is well-defined, then G(s)H(s)2 = H(s)2 . When a rational function multiplies by an inner function, it preserves the value of the norm [42]. The rational function A− (s)/A− (−s) is a stable, causal, and all-pass operator, i.e., |A− (jw)/A− (−jw)| = 1 for all w. The notation A(jw)∞ = ess. supw |A(jw)| is defined. The function ∞ ·2 : L2 → + is defined as f (·)2 = [ 0 |f (t)|2 dt]1/2 . The truncation of a function f (t) to the interval [0, T ] is defined as follows [43], [44]: f (t)T = f (t), if 0 ≤ t ≤ T ; f (t)T = 0, if t > T. Then, the resulting lemma and definition about L2 -stable with finite gain can allude to [43] and [44]. In order to deal with multiinput multioutput systems, the set LN 2 consists of N -tuples F = [f1 f2 · · · fN ]T , where fj ∈ L2 for j = 1, 2, . . . , N. The N 2 1/2 norm on LN . Similarly, 2 is defined by F 2 = [ j =1 fj 2 ] N L2 -stable with finite gain for N inputs and N outputs system indicates that N subsystems are all L2 -stable with finite gain. The symbol Aki (s) denotes a polynomial with bounded coefficients, its upper subscript k and lower subscript I, respectively, represent the polynomial of the kth TFDS of subsystem i. The following lemma is given to explain the result of the minimax optimization for the kth TFDS of subsystem i. Lemma 1 [42], [45]:  ∗ ∗ 1)  The optimal J2ki = Wik (s)Vik (s) , which minimizes W k (s)V k (s) , is of an all-pass form i i ∞ [Wik (s)Vik (s)]∗  k k ρi Φi (−s)/Φki (s), = 0,

if nkb− = nkφ i + 1 ≥ 1 i

if nkb− = 0 i

where the polynomial Φki (s) is monic and stable. 2) The constants ρki and ϕkji (1 ≤ j ≤ nkφ i ) are real and are uniquely determined by the interpolation constraints (29) later. Furthermore, the minimized   k as described W (s)V k (s) is given by i i ∞ min Wik (s)Vik (s)∞ = [Wik (s)Vik (s)]∗ ∞ = |ρki |. The following two lemmas discuss the property of the vector inequality and the inverse matrix for a specific positive-definite matrix. Lemma 2: If x ≤ y ∈ n , where xi ≤ yi , ∀i = 1, 2, . . . , n, then Ax ≤ Ay, where A ∈ n ×n , and (A)ij ≥ 0, ∀i, j = 1, 2, . . . , n. Proof: It can be verified by the expansion of the multiplication and the comparison of the ith component. Q.E.D. Lemma 3: Given a matrix A ∈ n ×n > 0 with 0 < (A)ii ≤ 1, 0 ≤ −(A)ij < 1, ∀i = j and i, j = 1, 2, . . . , n, where n ≥ 2. Then B = A−1 , where (B)ii > 0, (B)ij ≥ 0, ∀i = j, and i, j = 1, 2, .., n. Proof: This lemma is proved by the method of induction. In the beginning, n = 2 is considered. Because A > 0 (or |A| > 0), (B)11 = (A)22 /|A| > 0, (B)22 = (A)11 /|A| > 0, (B)12 = −(A)12 /|A| ≥ 0, and (B)21 = −(A)21 /|A| ≥ 0.

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Fig. 1.

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 2, APRIL 2011

Block diagram of the overall closed-loop system of the ith subsystem.

Second, it is assumed that A11 ∈ (n −1)×(n −1) > 0 and B11 = A−1 11 , where (B11 )ii > 0, (B11 )ij ≥ 0, ∀i = j, and i, j = 1, 2, .., n − 1. Then, a positive-definite matrix is supposed as A ∈ n ×n . Based on the result of A11 and B11 ,we want to prove B = A−1 , where (B)ii > 0, (B)ij ≥ 0, ∀i = j, and i, j = 1, 2, .., n. The A and A−1 matrices are partitioned as follows:   A11 A12 B11 B12 −1 A= and A = B = A21 A22 B21 B22 where A11 , B11 ∈ (n −1)×(n −1) , A12 , B12 ∈ (n −1)×1 , A21 , B21 ∈ 1×(n −1) , and A22 , B22 ∈ . In addition, 0 ≤ −(A12 )i1 < 1, i = 1, 2, . . . , n − 1, 0 ≤ −(A21 )1i < 1, i = 1, 2, . . . , n − 1, and A22 > 0. Because AB = In , the following four equations are derived [46]: −1

B11 = A11 − A12 A−1 22 A21

−1 −1 B12 = −A−1 11 A12 A22 − A21 A11 A12

−1 −1 B21 = −A−1 22 A21 A11 − A12 A22 A21 −1

B22 = A22 − A21 A−1 . 11 A12 Because (B11 )ii > 0, (B11 )ij ≥ 0, ∀i = j, i, j = 1, 2, .., n − 1, 0 ≤ −(A21 )1i < 1, i = 1, 2, . . . , n − 1, and A22 > 0, it verifies (B21 )1i ≤ 0, i = 1, 2, . . . , n − 1. Based on matrix inversion lemma (e.g., [46]) −1

B11 = A11 − A12 A−1 22 A21

−1 −1 −1 = A−1 A21 A−1 11 + A11 A12 A22 − A21 A11 A12 11 . Substituting the results (B11 )ii > 0, (B11 )ij ≥ 0, ∀i = j, i, j = 1, 2, .., n − 1, 0 ≤ −(A12 )i1 < 1, i = 1, 2, . . . , n − 1, and 0 ≤ −(A21 )1i < 1, i = 1, 2, . . . , n − 1 into the previous equa−1 > 0. Finally, tion, it verifies B22 = (A22 − A21 A−1 11 A12 ) −1 −1 −1 (B12 )i1 = (−A11 A12 (A22 − A21 A11 A12 ) )i1 ≥ 0, i = 1, 2, . . . , n − 1. The corresponding result is then obtained. Q.E.D. III. PROBLEM FORMULATION Consider the following nonlinear interconnected dynamic delayed system (NIDDS) Σ that contains N subsystems Σi for i = 1, 2, . . . , N : yi (t) = fii [ϕi (t − τi (t))] +

N

fik [ϕk (t − τk (t))]

k =1,k = i

i = 1, 2, . . . , N

(1)

(n )

where ϕi (t) = [yi (t) y˙ i (t) · · · yi (t) ui (t) u˙ i (t) · · · (m ) ui (t)]; yi (t) and ui (t) denote the system output and the (n ) system input of the ith subsystem, respectively; yi (t) = (m ) dn yi (t)/dtn , ui (t) = dm ui (t)/dtm are the nth and mth derivatives of yi (t) and ui (t), respectively; fii [ϕi (t − τi )] denotes the unknown but smooth nonlinear delay function of the ith subsystem; τi (t) ≥ 0, ∀t is the time-varying delay of the ith subsystem; fik [ϕk (t − τk )] , k = i, i, k = 1, 2, . . . , N represents the smooth but unknown interconnections of the ith subsystem Σi , brought about by other subsystems Σm , where m = 1, . . . , N, and m = i. Based on the approximation of the fuzzy linear model [16], [20], [22]–[34], a fuzzy dynamic model to represent the local linear input/output relations of the ith subsystem (see Fig. 1) is described by the fuzzy IF-THEN rules. The kth rule of the fuzzy dynamic model for the ith subsystem Σi with uncertainties is expressed in the following form: k System Rule k: IF z1 i (t) is M1ki . . . and zm¯ i (t) is Mm ¯ i , THEN yi (t) = e−s τ¯i Bik (s)ui (t)/Aki (s) +

N

k e−sτ i m (t) Bim (s)um (t)/Akim (s)

m =1,m = i

+

N

Δkim (um (t), t) + wi (t)

(2)

m =1,m = i

where k = 1, 2, . . . , L, i = 1, 2, . . . , N ; m ¯ i ≤ nka i + nkbi + 1, Aki (0) = 1; L is the number of IF–THEN rules; z1 i (t), . . . , zm¯ i (t) are premise variables, which are func(n ) (m ) tions of yi (t), . . . , yi (t), ui (t), . . . , ui (t); τ¯i ≥ 0 is the nominal delay of subsystem i; yi (t) also denotes the output from the kth IF–THEN rules; e−s τ¯i Bik (s)ui (t)/Aki (s) is nominal and known TFDS of subsystem i; Nthe kth −sτ i m (t) e B k (s)um (t) Akim (s) denotes the linear m =1,m = i N im k coupling terms; m =1,m = i Δim (um (t), t) denotes the nonlinear time-varying uncertainties that are relatively bounded  by c0 + N i=1 ci ui (t)2 , where ci ≥ 0, i = 0, 1, . . . , N, [39], and caused by the interconnections of other subsystems and the interactions from other TFDSs; and wi (t) denotes the approximation error of the fuzzy model of the ith subsystem, e.g., time-varying delay, parameter variations. Assume that Aki (s) and Bik (s) for i = 1, 2, . . . , N, k = 1, 2, . . . , L are coprime. One of the main advantages of this form is that no

HWANG: DECENTRALIZED FUZZY CONTROL OF NONLINEAR INTERCONNECTED DYNAMIC DELAY SYSTEMS

state estimator is required; hence, the control design becomes simple and effective. The output of the overall fuzzy system of the ith subsystem is inferred as follows: yi (t) =

L

μki (t){e−¯τ i Bik (s)ui (t)/Aki (s) + d˜ko i (t)}

(3)

k =1

=

L

279

μki (t){e−¯τ i Bik (s)[−Sik (s)yi (t) + Tik (s)ri (t)]/

k =1

[Aki (s)Rik (s)] + dko i (t)}

(10a)

where uki (t) denotes the control input of the kth TFDS of subsystem i [i.e., (7)] ψik (t) = e−¯τ i Bik (s)[ui (t) − uki (t)]/Aki (s)

where N

d˜ko i (t) =

−sτ i m (t)

{e

dko i (t) = d˜ko i (t) + ψik (t).

k Bim (s)um (t)/Akim (s)

m =1,m = i

+ Δkim (um (t), t)} + wi (t) μki (t) = hki (t)/

L

hki (t) and hki (t) =

m ¯i 

(4)

Mik (zi m (t)). (5)

m =1

k =1

It is assumed that hki (t) ≥ 0,

L

μki (t) = 1 for i = 1, 2, . . . , N

k =1

k = 1, 2, . . . , L, and ∀t.

(6)

It is also assumed that the fuzzy controller shares the same fuzzy sets with the fuzzy system (2). k Control Rule j: IF z1 i (t) is M1ki . . . and zm¯ i (t) is Mm ¯ i THEN ui (t) = −Sik (s)y(t)/Rik (s) + Tik (s)ri (t)/Rik (s) − Sik (s)Bik (s)(1 − e−s τ¯i )ui (t)/(Rik (s)Aki (s))

(7)

where the polynomials Rik (s), Sik (s), and Tik (s) are determined to achieve the control of the kth TFDS of subsystem i, and the last term in (7) denotes the Smith predictor to deal with the nominal system delay. The overall fuzzy control of the ith subsystem is then inferred as follows: ui (t) =

L

The signal ψik (t) denotes the interaction on the kth TFDS of subsystem i resulting from other TFDSs, including the error between nominal delay and time-varying delay. In summary, dko i (t) denotes the output disturbance of the kth TFDS of subsystem i caused by the interconnections stemming from other subsystems, the interactions resulting from other TFDSs, the approximation error of fuzzy model, and the nonlinear time-varying uncertainties. In general, the corresponding output disturbance is not small even after including various frequency components. In this situation, an effective controller is required to achieve satisfactory performance [20], [25], [29], [32]. After some mathematical manipulations of (10a), and using (7), the following is achieved:  L μki (t)Akci (s) e−s τ¯i Bik (s)Tik (s) ri (t) yi (t) − k k Akci (s) Ai (s)Ri (s) k =1  Ak (s)Rk (s) k do i (t) = 0. (11) − i k i Ac i (s) Hence, a sufficient condition for the validity of (11) is that the term in the braces {·} of (11) be equal to zero, i.e., yi (t) = e−s τ¯i B−k i (s)Lki (s)ri (t) − Vik (s)dko i (t)

− Sik (s)Bik (s)(1 − e−s τ¯i )ui (t)/(Rik (s)Aki (s))}. (8) In addition, the nominal closed-loop characteristic polynomial of the kth TFDS of subsystem i is described as follows:

(12a)

where Lki (s) = B+k i (s)Tik (s)/Akci (s), Vik (s) = Aki (s)Rik (s)/Akci (s).

μki (t){−Sik (s)y(t)/Rik (s) + Tik (s)ri (t)/Rik (s)

k =1

(10b)

(12b) The aforementioned output disturbance is supposed to be bounded as follows [see (4) and (10b)] N   k k do (t) ≤ α0k + αm ui (t)2 i i i 2

(13)

m =1

Akci (s) = Aki (s)Rik (s) + Bik (s)Sik (s).

(9)

Hence, the closed-loop output response of the ith subsystem, i.e., yi (t), is derived from (3) and (7), i.e., yi (t) =

L

μki (t){e−¯τ i Bik (s)[uki (t) + ui (t) − uki (t)]/

k =1

Aki (s) + d˜ko i (t)} =

L

μki (t){e−¯τ i Bik (s)[−Sik (s)yi (t) + Tik (s)ri (t)]/

k =1

[Aki (s)Rik (s)] + ψik (t) + d˜ko i (t)}

k where α0k i and αm , i, k, m = 1, 2, . . . , N are bounded. In i k , where i, k, m = Section IV, the upper bounds of α0k i and αm i 1, 2, . . . , N, are addressed for the stability of the overall system. Assume that the input/output relationship between ri (t) and y¯ij (t), for a stable reference model of the kth TFDS of subsystem i, is as follows:

(14) y¯ik (t) = e−s τ¯i Gki (s)ri (t) Fik (s)

where Gki (s) and Fik (s) for i = 1, 2, . . . , N, k = 1, 2, . . . , L are coprime, and Fik (s) is a stable monic polynomial with degree nkfi . The purpose of using a reference model is to shape the response of the closed-loop of

the kth TFDS of subsystem i. For tracking a set point, Gki (0) Fik (0) = 1 is assigned.

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To synthesize the problem, two cost functions in the H2 norm and H∞ -norm spaces are defined as follows [38], [39], [42], [45]:  

 J1ki = e−s τ¯i Gki (s) Fik (s) − B−k i (s)Lki (s) 2 and   J2ki = Wik (s)Vik (s)∞ (15) where the weighted function Wik (s) denotes a suitable rational function for the kth TFDS of subsystem i. The selection of the weighted function Wik (s) is discussed in Remark 1. The objectives of this paper are reiterated as follows (cf. Fig. 1): Search for a control function (i.e., the polynomials Rik (s), Sik (s), and Tik (s)) to simultaneously satisfy the following two requirements: 1) For dko i (t) = 0, the cost function J1ki is minimized; 2) The cost function J2ki is simultaneously minimized. Ultimately, the stability of the overall system is also verified by utilizing the concept of LN 2 -stable. IV. DECENTRALIZED FUZZY CONTROL VIA MIXED H2 /H∞ OPTIMIZATION

polynomial Tik (s) and the optimum value of J1ki . Rewrite (20) as ¯ik (s) or B−k i (−s)Gki (s) = B−k i (s)F¯ik (s) + Fik (s)B B−k i (−s)Gki (s) − B−k i (s)F¯ik (s)|s=z 1k ,...,z nk

(21)

where z1k , . . . , znk f are the zeros of Fik (s) = 0.Then, the polyi ¯ k (s)can be obtained from (21) by long division. Subnomial B i stituting (20) into (19) gives ¯ik (s)/B−k (s) − Lki (s)B−k (−s)22 (J1ki )2 = F¯ik (s)/Fik (s) + B i i (22) where the second term on the right-hand side of (22) is unstable, and the remaining terms are stable. By orthogonality, it follows that (J1ki )2 = F¯ik (s)/Fik (s) − Lki (s)B−k i (−s) ¯ik (s)/B−k (s)22 . ¯−k (s)/B−k (s)22 ≥ B + B i i i Hence, the optimal

There are two processes in the design of the decentralized fuzzy controller via mixed H2 /H∞ optimization for the NIDDS (1).

=0 fi

Lki (s)∗

and

(23)

J1ki∗

are achieved as follows:

 Lki (s)∗ = F¯ik (s)/ Fik (s)B−k i (−s) (24)   ¯ik (s)/B−k (s) . (25) J1ki∗ = B i 2

Comparing (24) and (12b) and using (9) gives A. Minimization of J1ki For dko i (t) = 0,the optimal model matching to the cost function J1ji is addressed as follows: Based on the norm-preserving property of the inner function (i.e., e−j w τ¯i  = 1∀w), from (15) J1ki is reduced to the following equation:  

(16) J1ki = Gki (s) Fik (s) − B−k i (s)Lki (s)2 . At the beginning, the polynomial B−k i (s) is factorized as

(17) B−k i (s) = B−k i (−s) · B−k i (s) B−k i (−s)

where B−k i (s) B−k i (−s) is an inner function, and the remaining B−k i (−s) is an outer function. Then J1ki = Gki (s)/Fik (s) − Lki (s)B−k i (−s)B−k i (s)/B−k i (−s)2 = B−k i (s)/B−k i (−s)[B−k i (−s)Gki (s)/(Fik (s)B−k i (s)) − Lki (s)B−k i (−s)]2 .

(18)

Similarly, based on the norm-preserving property of the inner function, (18) is simplified as follows:

Aki (s)Rik (s) + Bik (s)Sik (s) = B+k i (s)B−k i (−s)Fik (s)Xik (s) (26) Tik (s)

(Fik (s)B−k i (s)) − Lki (s)B−k i (−s)22 . (19) Decomposing the first term yields B−k i (−s)Gki (s)/(Fik (s)B−k i (s)) = F¯ik (s)/ ¯ik (s)/B−k (s) Fik (s) + B i

(20)

¯ k (s) are some polynomials to be deterwhere F¯ik (s) and B i mined, which are interrelated, respectively, with the control

(27)

where Xik (s) is a stable polynomial. If the reference model contains the unstable zeros of the nominal system, i.e., Gki (s) = ¯ k (s), for some polynomials G ¯ k (s), then the perfect B−k i (s)G i i j∗ model matching (i.e., J1 i = 0) can be obtained. Then, from (20), we have ¯ k (s) and B ¯ k (s) = 0. F¯ik (s) = B−k i (−s)G i i

(28)

B. Minimization of J2ki In this section, the task is to find the polynomials Rik (s) and Sik (s) such that the optimal J2ki is obtained. From previous studied results [38], [39], [42], [45], the optimal Wik (s)Vik (s)∞ must satisfy the following interpolation constraints: Vik (pki ) = 0, k = 1, 2, . . . , nka −

i

1−

(J1ki )2 = B−k i (−s)Gki (s)/

=

F¯ik (s)Xik (s)

Vik (zik )

= 0, k = 1, 2, . . . , nkb−

i

(29)

where pki (1 ≤ k ≤ nka − ) and zjk (1 ≤ k ≤ nkb− ) denote the zei i ros of Ak−i (s) and B−k i (s), respectively. Based on the result of Lemma 1 and the constraint (29), the following equation is obtained:

  Vik (s) = ρki Wdki (s)Ak−i (s)Φki (−s) Wnki (s)Ak−i (−s)Φki (s) (30)

where Wik (s) = Wnki (s) Wdki (s), Wnki (s) is a stable polynomial, and Wdki (s) contains the zeros in the region e {s} ≥ 0

HWANG: DECENTRALIZED FUZZY CONTROL OF NONLINEAR INTERCONNECTED DYNAMIC DELAY SYSTEMS

to reject the corresponding output disturbance (see Remark 1). Furthermore, the constraint (29) gives the following result: Wnki (s)Ak−i (−s)Φki (s) − ρki Wdki (s)Ak−i (s)Φki (−s) = B−k i (s)Qki (s)

(31)

where nkφ i = nkb− − 1 and nkqi = nkw n + nka − − 1, or we can i i i rewrite (31) as Wnki (zik )Ak−i (−zik )Φki (zik ) −ρki Wdki (zik )Ak−i (zik )Φki (−zik ) = 0.

(32)

By the solutions of ρki and Φki (s) for (32), the following equations are obtained from (12), (31), and (32): Akci (s) = Ak+ i (s)Wnki (s)Ak−i (−s)Φki (s)B+k i (s)Γki (s)

(33)

Rik (s) = ρki Wdki (s)Φki (−s)B+k i (s)Γki (s).

(34)

By comparing (26) and (33) and using (9), we have Γki (s) = B−k i (−s)Fik (s)Ωki (s)

(35)

Xik (s)

(36)

=

Ak+ i (s)Wnki (s)Ak−i (−s)Φki (s)Ωki (s)

where Ωki (s) is a stable polynomial. Then from (33)–(36)

− Sik (s)Bik (s)(1 − e−s τ¯i )ui (t)/(Rik (s)Aki (s))

× Ak−i (−s)Φki (s)Ωki (s)

(37)

Rik (s) = ρki Wdki (s)Φki (−s)B+k i (s)B−k i (−s)Fik (s)Ωki (s). (38)

= −Sik (s)[e−s τ¯i Bik (s)ui (t)/Aki (s) + dko i (t)]/Rik (s) + Tik (s)ri (t)/Rik (s) − Sik (s)Bik (s)(1 − e−s τ¯i )ui (t)/(Rik (s)Aki (s))

Substituting (32), (36), and (37) into (9) yields Sik (s) = Ak+ i (s)Qki (s)B−k i (−s)Fik (s)Ωki (s).

(39)

From (27) and (36), the following polynomial Tik (s) is obtained: =

of the weighted function Wik (s) are summarized as follows: 1) In order to reject the output disturbance, the polynomial Wdki (s)must contain the corresponding modes. For example, Wdki (s) = s[s2 + (wik )2 ] is used to reject the output disturbance dko i (t) = γ1ki + γ2ki sin(wik t + ϕki ), where γ1ki , γ2ki , and ϕki are unknown but bounded constants. 2) In general, an all-pass feature with infinity dc gain is designed to reject an output disturbance that may include the constant and other frequency components. 3) If the output disturbance in a specific frequency range is known, then the magnitude of the weighted function in that specific frequency range must have a larger value. The following theorem discusses the stability of the closedloop interconnected system (1): Theorem 1: Suppose the proposed controller (8) is applied to the NIDDS (1) or an equivalent fuzzy model system (3). Suppose also the inequality (13) with the condition (41) is satisfied. Then, the control system in Fig. 1 is LN 2 −stable, as in (41a) and (41b), shown at the bottom of the page, where 0 < (Γk1 )ii ≤ 1, and 0 ≤ −(Γ)ij < 1, i = j, for i, j = 1, 2, . . . , N. Proof: From (5), (6a), and (8) uki (t) = −Sik (s)yi (t)/Ri (s) + Tik (s)ri (t)/Rik (s)

Akci (s) = B+k i (s)B−k i (−s)Fik (s)Ak+ i (s)Wnki (s)

Tik (s)

281

F¯ik (s)Ak+ i (s)Wnki (s)Ak−i (−s)Φki (s)Ωki (s).

= Sik (s)Bik (s)ui (t)/(Rik (s)Aki (s)) − Sik (s)dko i (t)/ Rik (s) + Tik (s)ri (t)/Rik (s).

(42)

(40)

That is, the characteristic polynomial of the closed loop and the control polynomials of the kth TFDS of subsystem i for the fuzzy mixed H2 /H∞ optimization in (15) are attained through (37)–(40). Remark 1: Based on the result of the robust control design in the frequency domain [42], [45], the selection procedures

After some mathematical manipulations, (42) is simplified as follows: ui (t) = −Sik (s)Aki (s)dko i (t)/Akci (s) + Aki (s)Tik (s)ri (t)/Akci (s).

1 − αiki Sik (jw)Aki (jw)/Akci (jw)2 > 0, for i = 1, 2, . . . , N, k = 1, 2, . . . , L ⎡ 1 − α1k 1 S1k (jw)Ak1 (jw)/Akc1 (jw)2 −α2k 1 S1k (jw)Ak1 (jw)/Akc1 (jw)2 ⎢ ⎢ −α1k 2 S2k (jw)Ak2 (jw)/Akc2 (jw)2 1 − α2k 2 S2k (jw)Ak2 (jw)/Akc2 (jw)2 ⎢ k Γ1 = ⎢ .. .. ⎢ . . ⎢ ⎣ k k k k k k k −α1 N SN (jw)AN (jw)/Ac N (jw)2 −α2 N SN (jw)AN (jw)/AkcN (jw)2 ···

k −αN S1k (jw)Ak1 (jw)/Akc1 (jw)2 1

··· .. .

k −αN S2k (jw)Ak2 (jw)/Akc2 (jw)2 2 .. .

k k k k −α(N −1) N SN (jw)AN (jw)/Ac N (jw)2

k k 1 − αN SN (jw)AkN (jw)/AkcN (jw)2 N

(43) (41a)

⎤ ⎥ ⎥ ⎥ ⎥ > 0, k = 1, 2, . . . , L ⎥ ⎥ ⎦

(41b)

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 2, APRIL 2011

Taking the L2 norm of (43) [43], [44] and using the results of (13) yields    

ui (t)T 2 ≤ Sik (jw)Aki (jw) Akci (jw)2 dko i (t)T 2  

+ Aki (jw)Tik (jw) Akci (jw)2 ri (t)T  2   N k k k k k ≤ Si (jw)Ai (jw)/Ac i (jw)2 α0 i + αm i ui (t)T 2 m =1

 

+ Aki (jw)Tik (jw) Akci (jw)2 ri (t)T  2 .

(44)

Rearranging the term ui (t)T 2 and moving the other coupled control inputs to the left-hand side of (44), we obtain the following inequalities of total N subsystems:    

1 − αiki Sik (jw)Aki (jw) Akci (jw)2 ui (t)T 2  

≤ Sik (jw)Aki (jw) Akci (jw)2 ⎧ ⎫ N ⎨ ⎬ k × α0k i + αm ui (t)T 2 i ⎩ ⎭ m =1,m = i

 

+ Aki (jw)Tik (jw) Akci (jw)2 ri (t)T 2 for i = 1, 2, . . . , N and k = 1, 2, . . . , L.

(45)

Then, the previous N simultaneous inequalities for the kth TFDS are rewritten in the following matrix form: ¯ (t)T Γk1 U

≤

¯ T Γk2 R(t)

+

Γk3

Based on the Minkowski’s inequality [43], the following result is obtained: (47)

¯ T 2 ≤ c0 , ∀T ≥ 0, where c0 is a bounded posiBecause R(t) tive constant, hence, based on the result of L2 -stable with finite ¯ gain for R(t)or R(t) ∈ LN 2 , the following equation is obtained: k −1 k ¯ ¯ (t)2 ≤ [Γk1 ]−1 Γk2 2 R(t) U 2 + [Γ1 ] Γ3 2

(48a)

k −1 k ¯ U (t)2 ≤ [Γk1 ]−1 Γk2 2 R(t) 2 + [Γ1 ] Γ3 2 .

(48b)

or Similarly, from (6) and ej w τ i  = 1∀w, the system output satisfies the following inequality: Y (t)2 ≤ Γk4 2 R(t)2 + Γk5 2

1/Sik (jw)Aki (jw)/Akci (jw)2 > αiki , for i = 1, 2 2 

 1 − Δkii > Δk2 1 Δk2 1

(49)

where Y (t) = [y1 (t) · · · yN (t)]T , and Γk4 2 , Γk5 2 are proportional to [Γk1 ]−1 2 .Hence, the upper-bounded uncertainty (13) with the condition (41) ensures that the control system in Q.E.D. Fig. 1 is LN 2 −stable with finite gain.

(50)

i=1

 

k  k Si (jw)Aki (jw) Akci (jw)2 < 1, for m, where Δkm i = αm i i = 1, 2, and k = 1, 2, . . . , L. Similarly, the stability for threeinput–three-output system becomes  

1/Sik (jw)Aki (jw) Akci (jw)2 > αiki , for i = 1, 2, 3 2 

(46a)

¯ T = [r1 (t)T  ¯ (t)T = [u1 (t)T  · · · uN (t)T ]T , R(t) where U · · · rN (t)T ]T , Γk2 = diag{Aki (jw)Tik (jw)/Akci (jw)2 }, i = 1, 2, . . . , N, and Γk3 = [α0k 1 S1k (jw)Ak1 (jw)/Akc1 (jw)2 k · · · α0k N SN (jw)AkN (jw)/AkcN (jw)2 ]T . If the upper bound of the uncertainties described by (7) satisfies the condition (41) (i.e., Γk1 > 0, see Remark 2 for the interpretation), based on the results of Lemmas 2 and 3, (46a) can be rewritten as follows:     ¯ (t)T ≤ Γk1 −1 Γk2 R(t) ¯ T + Γk1 −1 Γk3 . U (46b)

¯ (t)T 2 ≤ [Γk1 ]−1 Γk2 2 R(t)T 2 + [Γk1 ]−1 Γk3 2 . U

Remark 2: Condition (41a) denotes the robust stability of each subsystem, which means that the upper bound of the nonlinear time-varying uncertainty for each subsystem of (13) must k , for be bounded, i.e., 1/Sik (jw)Aki (jw)/Akci (jw)2 > αm i k = 1, 2, . . . , L and i, m = 1, 2, . . . , N. The off-diagonal terms of (41b) represent the uncertainties or coupled signals coming from other subsystems. The upper bounds of these uncertainties or coupled signals are also bounded. The positive definite of Γk1 (or the principal minors of Γk1 are positive; also see Remark 3) ensures that LN 2 -stable with finite gain can tolerate these signals. Condition (41) is the same as that obtained with [44]. Remark 3: The stability condition (41) for the two-input–twooutput system is as follows:

 1 − Δkii > Δk2 1 Δk1 2

(51a)

i=1 3 

 

1 − Δkii > Δk1 3 Δk3 2 Δk2 2 + Δk1 2 Δk2 3 Δk3 1 + 1 − Δk1 1

i=1

 



Δk2 3 Δk3 2 + 1 − Δk2 2 Δk1 3 Δk3 1 + 1 − Δk3 3 Δk1 2 Δk2 1 . (51b)

k ∼ If αm = 0 for m, i = 1, 2, . . . , N, k = 1, 2, . . . , L in (13), then i (41) [e.g., N = 2 for (50), N = 3 for (51)] is satisfied.

C. Proposed Control Algorithm The design procedure of the proposed decentralized fuzzy controller via mixed H2 /H∞ optimization with the Smith predictor is as follows. Step 1: Factorize the nominal subsystem i as Aki (s) = Ak+ i (s)Ak−i (s) and Bik (s) = B+k i (s)B−k i (s). Step 2: Assign a suitable reference model Fik (s) and Gki (s) for the kth TFDS of subsystem i. Step 3: Choose an appropriate weighted function for the second cost function (15), i.e., Wdki (s) and Wnki (s). ¯ k (s) are Step 4: Based on (21), the polynomials F¯ik (s) and B i k∗ k ¯ (s)/B−k (s)2 obtained. Then, the optimal J1 i = B i i is attained. If the matching condition is satisfied, then ¯ k (s), and B ¯ k (s) = 0. F¯ik (s) = B−k i (−s)G i i Step 5: From (32), the scalar ρki and the polynomial Φki (s) ∗ are obtained. Then, the optimal J2ki = ρki is attained. Based on the long division for (31), the polynomial Qki (s) is obtained. Step 6: The characteristic polynomial of the closed loop and the control polynomials of the kth TFDS of subsystem

HWANG: DECENTRALIZED FUZZY CONTROL OF NONLINEAR INTERCONNECTED DYNAMIC DELAY SYSTEMS

  i, i.e., Akci (s) and Rik (s), Sik (s), Tik (s) , for the fuzzy mixed H2 /H∞ optimization in (15) are attained from (37)–(40). V. ILLUSTRATIVE EXAMPLES AND DISCUSSIONS Two examples are considered. The proposed controller is applied to the first example to verify its effectiveness and efficiency. The second example considers an internet-based intelligent space for the trajectory-tracking control of a car-like wheeled robot (CLWR) [16].

In addition, the fuzzy sets of the premise variables z1 i (t) and z2 i (t), i = 1, 2 are defined in the following modified Gaussian membership functions:  −10(z 1 (t)+1) 2 i e , if z1 i (t) ≥ −1 (z ) = M11∼3 1i i 1, otherwise

The kth nominal TFDS of subsystem i (2) is assumed to be Aki (s) = s2 + ak1 i s + ak2 i , Bik (s) = bk0 i s2 + bk1 i s + bk2 i , and τ¯i = τ0 i .The reference model (14) is chosen as Fik (s) = s2 + f1ki s + f2ki , and Gki (s) = g0ki s2 + g1ki s + g2ki . The weighted function for the second cost function (15) is set as Wdki (s) = k k s2 + wd1 s + wd2 and Wnki (s) = wnk 0 i s2 + wnk 1 i s + wnk 2 i . If i i k wd2 = 0, the weighted function contains an integrator to reject i a constant output disturbance. Consider the aforementioned interconnected delay system with the following coefficients:

(z1 i ) = e−10z 1 i (t) M14∼6 i 2

 M17∼9 (z1 i ) i

= 

M21∼3 (z2 i ) = i

A. Mathematical Example

283

e−10(z 1 i (t)−1) ,

if z1 i (t) ≤ 1

1,

otherwise

2

−10(z 2 i (t)+1) 2

e

,

1,

if z2 i (t) ≥ −1 otherwise (z2 i ) = e−10z 2 i (t) M24∼6 i 2

 M27∼9 (z2 i ) = i

e−10(z 2 i (t)−1) ,

if z2 i (t) ≤ 1

1,

otherwise

2

(54)

where z1 i (t) = yi (t), z2 i (t) = yi (t) − yi (t − h), h = 0.01 s denotes the sampling time or control-cycle time, and Mkni∼n +2 (·) denotes the three terms Mkni (·), Mkni+1 (·), and Mkni+2 (·). It is also assumed that the reference inputs to be tracked are assigned as follows: r1 (t) = −1, r2 (t) = 0, as 0 ≤ t ≤ 40

a1∼9 1 1 = 4.0, 3.5, 3.0, 2.5, 2.0, 2.25, 3.25, 3.75, 4.0

r1 (t) = −2, r2 (t) = 1, as 25 < t ≤ 80

a1∼9 2 1 = 8.5, 7.0, 6.5, 6.0, 5.5, 6.0, 6.25, 6.75, 7.25 b1∼9 0 1 = 2.01, 1.93, 1.84, 1.65, 1.57, 1.48, 1.36, 1.27, 1.19 b1∼9 1 1 = −2.02, −2.13, −2.24, −2.35, −2.46, −2.37

r1 (t) = −1, r2 (t) = 2, as 50 < t ≤ 120 and r1 (t) = 0, r2 (t) = 1, as 75 < t ≤ 160.

(55)

Besides the interconnected terms as described in (53), these two subsystems are subjected to the following nonlinear timevarying uncertainties and fuzzy-modeling errors nmi (t) = N j k =1,k = i Δik (uk (t), t) + wi (t):

− 2.28, −2.19, −2.22 b1∼9 2 1 = −4.05, −4.16, −4.27, −4.38, −4.47, −4.36 − 4.25, −4.13, −3.89

nm1 (t) = 0.45 + 0.06 sin(0.2y1 (t) + 0.5u2 (t)y2 (t)

τ¯1 = 1 a1∼9 1 2 = 2.2, 2.5, 3.0, 3.5, 2.5, 3.0, 2.75, 3.5, 4.25

− 0.2 cos(10t))u1 (t) + 0.06 cos(0.5u2 (t)y1 (t)

a1∼9 2 2 = 2.0, 3.5, 2.5, 2.0, 3.5, 4.25, 4.0, 3.5, 4.5

+ 0.2 sin(20t))u2 (t)

b1∼9 0 2 = 1.1, 1.2, 1.4, 1.6, 1.8, 2.2, 1.4, 1.7, 1.9

nm2 (t) = 0.45 + 0.06 cos(−0.5u1 (t)y2 (t)

b1∼9 1 2 = 0.53, 0.64, 0.75, 0.86, 0.93, 1.06, 1.17, 1.24, 1.36 b1∼9 2 2 = −1.54, −1.65, −1.76, −1.87, −1.94, −2.08 − 2.15, −2.26, −2.39 τ¯2 = 1.2.

(52)

 −sτ i m (t) k The interconnected terms N Bim (s)um (t)/ m =1,m = i e k Aim (s)of subsystems 1 and 2 are, respectively, described as follows: a1 1 2 = 20, a2 1 2 = 120, b0 1 2 = 0.52, b1 1 2 = 1.5, b2 1 2 = 1.5 τ12 = 0.5, a1 2 1 = 20, a2 2 1 = 120, b0 2 1 = 0.52 b1 2 1 = 1.5, b2 2 1 = 1.5, τ21 = 0.5.

(53)

− 0.2 sin(20t))u1 (t) − 0.06 sin(0.2y2 (t) + 0.7y1 (t)u2 (t) − 0.2 cos(15t))u2 (t).

(56)

Each subsystem contains a dc bias and some nonlinear timevarying signals and satisfies the inequality (13). At the beginning, the PID controls to track the reference inputs (55) for the two subsystems (52), which are interconnected with terms (53), are investigated, and the two subsystems are affected by nonlinear time-varying uncertainties and fuzzy-modeling errors (56). With PID controls, many parameter responses are unstable. For simplification, the unstable responses are omitted. This is one of the motivations to design an effective controller for the interconnected system with input delay, which is tackled by a Smith predictor. To begin with, the reference models I with a small bandwidth (57) for these nine TFDSs (54) that have the same

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 2, APRIL 2011

Fig. 2. Frequency response of reference model I (57). (a) Subsystem 1. (b) Subsystem 2.

parameter are assigned as follows: f11∼9 1

=

1.2, f21∼9 1

=

0.35, g01∼9 1

= −0.0219

= 0.1314, g21∼9 = 0.35, f11∼9 = 1.2, f21∼9 = 0.35 g11∼9 1 1 2 2 = −0.04375, g11∼9 = 0.30625, g21∼9 = 0.35. g01∼9 2 2 2

(57)

The frequency responses of the above reference models are depicted in Fig. 2, which possess the bandwidths about 0.2 and 0.3 rad/s. Furthermore, the weighted function for the second cost function is set as follows: 1∼9 1∼9 1∼9 1∼9 = 10, wd2 = 0, wn1∼9 wd1 0 i = 1, wn 1 i = 3.5, wn 2 i = 4.2 i i

i = 1, 2.

(58)

The corresponding frequency response of the aforementioned weighted function is depicted in Fig. 3(a), which has an infinite dc gain and a high-pass feature. It indicates that the steady-state

Fig. 3. Frequency responses with different weighted functions (58) and (60). (a) Weight function with integration (58). (b) Weight function without integration (60).

error is rejected and the effect of the noise with frequency above 2 rad/s is minimized. Then, the corresponding responses of the proposed control are presented in Fig. 4, which are much better than those that use PID control. Although nonlinear time-varying uncertainties and fuzzy-modeling errors (56) or the output disturbances possess high-frequency signals, the response of the control input [i.e., Fig. 4(b)] is attenuated by the proposed fuzzy-decentralized control via mixed H2 /H∞ optimization with Smith predictor. In addition, the response of Fig. 4 has the following optimal values: (J1ki )∗ ∼ = 0 for i = 1, 2, k = 1, 2, . . . , 9, (J2k1 )∗ ∼ = 0.6079, and k ∗ ∼ (J2 2 ) = 0.8762. To address the effects of the design parameters of the proposed control on the system response, different reference models and weighted functions are also examined in the following. The response for the reference model II (59) is described in Fig. 5; it possesses a larger bandwidth about 0.8 and 1 rad/s compared

HWANG: DECENTRALIZED FUZZY CONTROL OF NONLINEAR INTERCONNECTED DYNAMIC DELAY SYSTEMS

Fig. 4. Response of reference model I (57) using weighted function with integration (58). (a) r1 (t)(· · ·), y 1 (t)(−), r2 (t)(· · ·), y 2 (t)(−). (b) u 1 (t)(·−), u 2 (t)(−).

with that of (57) f11∼9 = 2.05, f21∼9 = 1, g01∼9 = −0.0625 1 1 1 = −0.375, f21∼9 = 1, g01∼9 =1 g11∼9 1 1 1 = 2.05, f21∼9 = 1, g01∼9 = −0.125 f11∼9 2 2 2 = −0.875, g21∼9 = 1. g11∼9 2 2

(59)

Although the output response in Fig. 6(a) settles faster compared to the response in Fig. 4(a), it does result in a transient response, i.e., it has an undershoot due to the nonminimum phase feature. However, the response of the corresponding input is still quite smooth as observed by comparing Fig. 6(b) with Fig. 4(b). If high-frequency uncertainties exist in the system, then it often results in a transient response in the wideband reference model. Hence, a suitable selection of the reference model becomes an important issue in order to generate an appropriate closed-loop response. A similar situation happens for

285

Fig. 5. Frequency response of reference model II (59). (a) Subsystem 1. (b) Subsystem 2.

a weighted function without integration (60); it has a response shown in Fig. 7 and appears to have error in its steady-state response. The corresponding frequency response of this weighted function is depicted in Fig. 3(b), which only possesses a finite dc gain. Hence, the steady-state error occurs when the system encounters constant uncertainty (56). In summary, it is also an important task to make an appropriate choice of the weighted function to attain an acceptable response 1∼9 1∼9 1∼9 1∼9 = 10, wd2 = 10, wn1∼9 wd1 0 i = 1, wn 1 i = 3.5, wn 2 i = 4.2 i i

i = 1, 2.

(60)

Based on the previous investigations, our proposed control is effective for a class of nonlinear interconnected dynamic delayed systems.

286

Fig. 6. Response of reference model II (59) using weighted function with integration (58). (a) r1 (t)(· · ·), y 1 (t)(−), r2 (t)(· · ·), y 2 (t)(−). (b) u 1 (t)(·−), u 2 (t)(−).

B. Internet-Based Intelligent Space for the Trajectory Tracking of a CLWR Fig. 8 shows the block diagram of a CLWR in an internetbased intelligent space [16]. The CLWR consists of two dc motors, one microprocessor, one driver a wireless local-area network (WLAN) device, and a mechanism. The overall control system includes a CLWR, two charge-coupled devices (CCDs), and two personal computers connected in an Internet. The Internet sever computer consists of an image processing card and a WLAN device and performs path planning and computation of the proposed control in the client computer. The symbols (ˆ xw , yˆw , ρˆw ), yi , ui , i = 1, 2 in Fig. 8 represent, respectively, the posture of the CLWR in the 2-D world coordinate, the angular position of the front wheel, the angular velocity of the rear wheel, and the control inputs to the front and rear wheels. The objective of this control problem is to make the angular position of the front wheel (with the unit radian) and angular velocity of

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 2, APRIL 2011

Fig. 7. Response of reference model I (57) using weighted function without integration (60). (a) r1 (t)(· · ·), y 1 (t)(−), r2 (t)(· · ·), y 2 (t)(−). (b) u 1 (t)(·−), u 2 (t)(−).

the rear wheel (with the unit radian/second) track their desired values (i.e., θd and vd ), respectively. Because the diameter of the wheel is d = 12.7cm, the translation velocity of the CLWR is the product of the wheel radius and the angular velocity of the rear wheel and is given by v(t) = d · ω(t)/2 = d · y2 (t)/2. In this example, the reference inputs are r1 (t) = θd (t) and r2 (t) = ωd (t), where θd (t) and ωd (t) are the desired angular position of the front wheel and the desired angular velocity of the rear wheel, respectively. At the beginning, the front and rear wheels of the CLWR are separately identified by suitable step responses without considering any time delay. After successful identification of the motor dynamics, the time of the signal transmitting from the client computer through the server computer, the image processing time in the server computer, and the time of calculation of the control law in the client computer are all estimated, and the total time is found to be around 0.26 s. The dynamics of

HWANG: DECENTRALIZED FUZZY CONTROL OF NONLINEAR INTERCONNECTED DYNAMIC DELAY SYSTEMS

Fig. 8.

287

Block diagram of the overall system.

the ith operating condition of the CLWR can be expressed by the following two-input and two-output transfer function matrix [16]: Yi (s) = Gi (s)Ui (s)

(61)

where Yi (s), Ui (s) ∈ (s)2×1 , and Gi (s) ∈ (s)2×2 . Define Gji k (s)as the transfer function of the jth row and the kth column for Gi (s), or the relation between the kth input and the jth ouput of subsystem i. Because the angular velocity of the rear wheel does not have any influence on the output of the front wheel, we get the result G12 i (s) = 0; i.e., the system output 1 (i.e., y1 (t)) is not affected by the system input 2 (i.e., u2 (t)). Nine subsystems are approximated from the following transfer function matrices obtained from their corresponding step responses:

2  G11 1 (s) = 250.457/ s + 22.857s + 250.457

2  G22 1 (s) = 19.377/ s + 7.273s + 19.377

2  G21 1 (s) = −0.18/ s + 7.273s + 19.377

2  11 22 G11 2 (s) = G1 (s), G2 (s) = 32.915/ s + 8s + 32.915

2  G21 2 (s) = −0.315/ s + 8s + 32.915 11 22 22 21 G11 3 (s) = G1 (s), G3 (s) = G2 (s), G3 (s) = 0

 2 G11 4 (s) = 557.546/ s + 30.189s + 557.546

2  22 21 G22 4 (s) = G1 (s), G4 (s) = −0.12/ s + 7.273s + 19.377 11 22 22 G11 5 (s) = G4 (s), G5 (s) = G2 (s)

 2 G21 5 (s) = −0.25/ s + 8s + 32.915 11 22 22 21 G11 6 (s) = G4 (s), G6 (s) = G3 (s), G6 (s) = 0 11 22 22 21 G11 7 (s) = G4 (s), G7 (s) = G1 (s), G7 (s) = 0 11 22 22 21 G11 8 (s) = G4 (s), G8 (s) = G2 (s), G8 (s) = 0 11 22 22 21 G11 9 (s) = G4 (s), G9 (s) = G3 (s), G9 (s) = 0.

(62)

Two typical step responses derived in [16] are shown in Fig. 9 and assume that their system outputs have the following

Fig. 9. Two typical responses. (a) The rear wheel with step input v d = 20.4 cm/s at t = 0 s and θd = 15 ◦ at t = 20 s from the CLWR (· · ·) and 1 (s), G 2 2 the mathematical models G 25 ∼6 2 , 3 , 5 , 8 (s)(−). (b) The front wheel with step input v d = 37 cm/s at t = 0 s and θd = 15 ◦ at t = 0.2 s from the CLWR 1 (s) (−). (· · ·) and the mathematical model G 14 ∼9

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 2, APRIL 2011

Fig. 10. Response of internet-based control of a CLWR using the proposed control. (a) r1 (t)(· · ·), and y 1 (t)(−). (b) r2 (t)(· · ·), and y 2 (t)(−). (c) u 1 (t)(· · ·), and u 2 (t)(−). (d) r1 (t)(· · ·), and y 1 (t + 0.26)(−). (e) r2 (t)(· · ·), and y 2 (t + 0.26)(−).

HWANG: DECENTRALIZED FUZZY CONTROL OF NONLINEAR INTERCONNECTED DYNAMIC DELAY SYSTEMS

disturbances:

VI. CONCLUSION

nm1 (t) = 0.25 + 0.05 sin(0.2y1 (t) + 0.5u2 (t)y2 (t) +0.2 cos(1000t)) nm2 (t) = 0.75 − 0.1 sin(0.2y2 (t) + 0.1u1 (t)y1 (t) −0.2 cos(1500t)).

(63)

In addition, the fuzzy sets of the premise variables z1 i (t) and z2 i (t), i = 1, 2 are defined as the following modified Gaussian membership functions:  (z1 i ) M11∼3 i

=

e−10(z 1 i (t)+−0.7g i ) , if z1 i (t) ≥ −0.7gi 2

1, −10z 12 i

M14∼6 (z1 i ) = e i  M17∼9 (z1 i ) = i  M21∼3 (z2 i ) i

=

M24∼6 (z2 i ) = e i  =

otherwise (t)

e−10(z 1 i (t)−0.7g i ) ,

if z1 i (t) ≤ 0.7gi

1,

otherwise

2

e−10(z 2 i (t)+70g i ) ,

if z2 i (t) ≥ −70gi

1,

otherwise,

2

−10z 22 i

(z2 i ) M27∼9 i

289

(t)

,

e−10(z 2 i (t)−70g i ) , if z2 i (t) ≤ 70gi 2

1,

otherwise

(64)

A decentralized control of nonlinear interconnected dynamic delayed systems using fuzzy mixed H2 /H∞ optimization with Smith predictor is established. The past effort on the use of fuzzy T-S delayed control did not consider the robust tracking performance (e.g., the attenuation of uncertainty and delay). In this paper, a 2-DOF optimization technique with a Smith predictor was employed to control each TFDS. A suitable frequency response without incurring the oscillating and sluggish phenomena was obtained. In addition, the effect of output disturbance caused by interconnections coming from other subsystems, due to the approximation error of the ith subsystem and from the interactions resulting from other TFDSs, was minimized. Furthermore, the control implementation did not need to calculate the Diophantine equation. There were also computational advantages, especially for low-order systems. The stability of the overall system was proven via LN 2 -stability with finite gain. Finally, the simulation of the TFDS with a nonminimum phase was compared with that of the PID control. The usefulness and effectiveness of our proposed control were confirmed. The second example was an internet-based intelligent space for the trajectory tracking of a CLWR. It appeared that they were quite consistent with the predicted results. Consequently, it is believed that the control design is a promising method that is suitable for a wide class of nonlinear interconnected dynamic delayed systems in the presence of nonlinear time-varying uncertainties. REFERENCES

where z1 i (t) = yi (t), z2 i (t) = yi (t) − yi (t − h), h = 0.01 s denotes the sampling time (or control cycle time), g1 = 1, g2 = 10, and Mkni∼n +2 (·) denotes consecutive functions Mkni (·), Mkni+1 (·), and Mkni+2 (·). It is also assumed that the reference inputs to be tracked for a period of 60 s are assigned as follows: r1 (t) = π/6, r2 (t) = 0.8025, as 0 ≤ t ≤ 15 r1 (t) = −π/6, r2 (t) = 0.8025, as 15 < t ≤ 30 r1 (t) = 0, r2 (t) = 5.83, as 30 < t ≤ 60.

(65)

The response using the weighted function (58) and the reference model II (59) is shown in Fig. 10 and considered to be acceptable. When certain high-frequency disturbance components are added to the outputs, as depicted in Fig. 10 (a) and (b), this output disturbance (63) is almost attenuated by the proposed control because the control inputs become quite smooth as shown in Fig. 10(c). Because the Internet-based intelligent space possesses a large delay (i.e., 0.26 s), it is better to use a reference model with a larger bandwidth, i.e., reference model II. The response of Fig. 10(d) and (e) also indicates that the tracking performance is satisfactory for a nonlinear interconnected delayed system. In addition, the resulting control parameters are almost the same as those derived in Section V-B. Our proposed control has demonstrated its effectiveness and efficiency for a large class of uncertain nonlinear interconnected dynamic delayed systems.

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Chih-Lyang Hwang (SM’08) received the B.E. degree in aeronautical engineering from Tamkang University, Taipei, Taiwan, in 1981 and the M.E. and Ph.D. degrees in mechanical engineering from Tatung Institute of Technology, Taipei, in 1986 and 1990, respectively. From 1990 to 2006, he had been with the Department of Mechanical Engineering, Tatung Institute of Technology (or Tatung University), where during 1996–2006, he was a Professor of mechanical engineering involved in teaching and research in the area of servo control and control of manufacturing systems and robotic systems. During 1998–1999, he was a Research Scholar with the George W. Woodruf School of Mechanical Engineering, Georgia Institute of Technology, Atlanta. Since 2006, he has been a Professor with the Department of Electrical Engineering, Tamkang University. In 2011, he will be a Professor with the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei. He is the author or coauthor of about 115 journal and conference papers. His current research interests include navigation of mobile robot, fuzzy (or neural network) modeling and control, variable structure control, robotics, visual tracking system, network-based control, and distributed sensor network. Prof. Hwang received a number of awards, including the Excellent Research Paper Award from the National Science Council of Taiwan and the Hsieh-Chih Industry Renaissance Association from the Tatung Company. From 2003 to 2006, he was on the Committee of the Automation Technology Program of the National Science Council of Taiwan. He was a member of the Technical Committee of the 28th Annual Conference of the IEEE Industrial Electronics Society. During August 31–September 12, 2008, he was also a member of the Visiting Program of Automation, Machinery, and Solid Mechanics of the National Science Council.