Fuzzy Sets and Systems 146 (2004) 375 – 401 www.elsevier.com/locate/fss
H∞ Tracking-based adaptive fuzzy-neural control for MIMO uncertain robotic systems with time delays Wen-Shyong Yu∗ Department of Electrical Engineering, Tatung Institute of Technology, 40 Chung-Shan North Road 3rd Sec., Taipei 10451, Taiwan Received 14 January 2003; received in revised form 15 July 2003; accepted 17 July 2003
Abstract In this paper, a novel adaptive fuzzy-neural control (AFNC) scheme for multi-input multi-output uncertain robotic systems is proposed for H∞ tracking performance and to suppress the e5ects caused by multiple time-delayed state uncertainties, unmodeled dynamics, and disturbances. Each delayed uncertainty is assumed to be bounded by an unknown gain. A reference model with the desired amplitude and phase properties is given to construct an error model. A fuzzy-neural (FN) system is used to approximate an unknown controlled system from the strategic manipulation of the model following tracking errors. The proposed AFNC scheme uses two on-line estimations, which allows for the inclusion of identifying the gains of the delayed state uncertainties and training the weights of the FN system, simultaneously. Stability and robustness of the AFNC scheme is analyzed in Lyapunov sense. It is shown that the proposed control scheme can guarantee parameter estimation convergence and stability robustness of the closed-loop system with H∞ tracking performance for the overall system without a priori knowledge on the upper bounds of the delayed state uncertainties. The performance of the proposed scheme is evaluated through the simulation results. Simulations are given to show the validity and con9rm the performance of the proposed scheme. c 2003 Elsevier B.V. All rights reserved. Keywords: Adaptive control; Fuzzy-neural system; Model reference; MIMO uncertain dynamical systems; Delays; Robustness; Stability; H∞ tracking performance
1. Introduction Most industrial manipulators in the industry are multivariable multi-input/multi-output (MIMO) systems with nonlinearities and uncertainties (e.g. airplane simulators, mining machines and walking machines in which their loads may vary while performing di5erent tasks, the friction coe=cients ∗
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c 2003 Elsevier B.V. All rights reserved. 0165-0114/$ - see front matter doi:10.1016/j.fss.2003.07.001
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may change in di5erent con9gurations, and some neglected nonlinearity, such as backlash, may appear as a disturbance at the control input) and most of them are usually operated based on human knowledge and expertise. Thus, they pose additional di=culties to the stabilization control design and, especially, the tracking control design. In order to overcome these kinds of di=culties in the design of a controller, various schemes have been developed in the last decades, among which a successful approach is fuzzy logic control with tuning capability in neural weights. Recently, adaptive fuzzy-neural control has attracted increasing attention, essentially because it can provide a powerful learning technique for complex unknown plants to perform complex tasks in highly nonlinear dynamical environment, and can have available quantitative knowledge from repetitive adjustment of the system with better performance than those of fuzzy controls with constant rule bases by using static nonlinear state feedback for decoupling square MIMO systems [13], by tuning consequent membership functions [10], or by using dynamic inversion for robotic manipulators [16]. Since design techniques for dynamical systems are closely related to their stability, robustness and performance properties, this technique including the adaptation capability provides good results to the trajectory tracking problem with good-9tting data by using a small amount of the fuzzy inference mechanisms. Hsu and Fu [5] proposed an adaptive fuzzy variable structure control via backstepping with mth-order B-spline type of membership function and high-gain compensator to improve the tracking performance for non-linear uncertain systems. Furthermore, fuzzy direct/indirect adaptive control algorithm [14,19,20,21,22] and fuzzy adaptive sliding mode control algorithm [29] robust against the reconstruction errors were proposed for achieving H∞ tracking performance for nonlinear dynamical systems with unknown nonlinearities, where the schemes in [19,22] also include high gain to eliminate peaking in the implementation. However, the schemes in [5,13,20] require the assumptions that the dynamics of the system is exactly known and is feedback linearizable with well-de9ned vector relative degree. Furthermore, the inference rules of the fuzzy logic control will typically contain a number of subjectively as well as empirically determined parameters, and in most cases the nonlinearities existing in the dynamical system are not known a priori. Tan et al. [17] presented a multivariable fuzzy controller design optimization in an usually multi-modal multi-dimensional search space to maintain the sight line of a mirror when it is subjected to external disturbances for Gyromirror line-of-sight stabilization platform. Moreover, many researches were performed to improve the performance of the conventional linear control schemes with fuzzy logic controllers and ensure their tracking performance regardless of the complexity of the controlled plants, e.g., an improved fuzzy PI controller with both position and velocity involved in the feedback loop [18], a fuzzy P+ID controller [9], a PD-type fuzzy logic control as the basic control part and learning control as the re9nement part [24], a fuzzy hybrid (analog+fuzzy) control algorithm [15], and a fuzzy model reference adaptive controller using T–S fuzzy controller and PI-type adaptation law with the inclusion of a priori analytic information [3], where the 9rst four are for SISO nonlinear dynamical systems and the last one is for MIMO systems. In addition to the nonlinearities and uncertainties, systems with unknown delayed states are often encountered in practice, such as economic, chemical processes, and hydraulic systems. For this reason, the robust stabilization of uncertain dynamical systems with delayed states has received considerable attention over the years (see, e.g., [1,12,23] and the references therein). For dynamical systems with delayed state uncertainties, where the system state vector is available, the norm bounds of the delayed state uncertainties is generally supposed to be known, and such a bound is employed either to construct some types of stabilizing state feedback controllers [23], or to develop some stability conditions [11]. However, they are all based on the assumption that
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the bounds of the delayed state uncertainties are exactly known. Recently, Cao and Frank [1] and Lee et al. [8] utilize the T–S model with delays for modeling the nonlinear time-delay systems and construct the stability condition which is independent of the delays, where the delayed dependent stability conditions are given by [25]. The former designed the state feedback gain and observer gain of the fuzzy model by solving a set of coupled linear matrix inequalities, and the later designed the output feedback controller based on H∞ and linear matrix inequality. Although, they all assume that the delays of the system are bounded and are not known a priori, the results are conservative. When researchers deal with the problem of dynamical systems with delayed state uncertainties, where the system state vector is available, the parameters of the physical plant are generally supposed to be known [25], which is impractical in real applications, and the key assumption is that the bounds on norm of the delayed state uncertainties are available for design. However, due to the complexity of the structure on uncertainties, delayed uncertainty bounds may not be easily obtained. To relax the assumption as well as attenuate the e5ects caused by unmodeled dynamics, delayed state uncertainties and disturbances, we adopt the H∞ -AFNC scheme in this paper for MIMO uncertain robotic manipulators. Each delayed uncertainty is assumed to be bounded by an unknown gain. A reference model with the desired amplitude and phase properties is given to construct an error model. An FN system is used to represent the unknown controlled system with the desired accuracy to any degree from the strategic manipulation of the model following tracking errors. The AFNC scheme uses two on-line estimations simultaneously for achieving H∞ tracking performance, which allows for the inclusion of identifying the gains of the delayed state uncertainties and training the weights in the FN system. The parameters of the plant model are updated using dynamical adaptation weights and fuzzy inference rules of the FN system, which provides fast parameters update and hence for fast convergence of the tracking errors. The stability and robustness properties of the proposed AFNC scheme are established in the Lyapunov theory framework. The results demonstrate the feasibility of the proposed control scheme, which can guarantee parameter estimation convergence and stability robustness of the closed-loop system with H∞ tracking performance for the overall system without a priori knowledge on the upper bounds of the delayed state uncertainties. The performance is evaluated by simulation studies. The simulation results demonstrate the computational simplicity, tracking performance and robustness by the proposed control scheme. The paper is organized as follows. In Section 2, the characteristics of an n-link robotic manipulator and H∞ tracking performance of the adaptive control with delayed state gain identi9cation but known parameters of the controlled plant are presented. In Section 3, the structure and properties of the AFNC scheme is derived for achieving H∞ tracking performance using two on-line estimations simultaneously with rigorous convergence analysis. In Section 4, simulation work is presented to demonstrate the e5ectiveness of the proposed scheme. Concluding remarks are 9nally made in Section 5.
2. Problem formulation Consider the dynamical equation of an n-link robotic manipulator described by the following nonlinear di5erential equation with external disturbances: K + C (q; q) ˙ q(t) ˙ + g (q) = (t) + (t); H(q)q(t)
(1)
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where H(q) is the n × n symmetric positive de9nite inertial matrix, C (q; q˙ )˙q is the n × 1 vector of coupled Coriolis and centripetal torques, g (q) is the n × 1 vector of gravitational torques, is the n × 1 vector of joint torques, q is the n × 1 vector of the joint displacement, and qK and q˙ are the n × 1 vectors of the joint acceleration and velocity terms, respectively. Furthermore, (t) ∈ Rn is the external disturbance and is assumed to satisfy ∈ L2 [0; T ]; ∀T ∈ [0; ∞). It is assumed that vectors q˙ ; q are measurable. The following are the properties of the robotic dynamics: (P1) The inertia matrix H(q) is symmetric positive de9nite for every q. Both H(q) and H(q)−1 are uniformly bounded. ˙ (P2) The matrix H(q) − 2C (q; q˙ ) is skew-symmetric for suitable representation of C (q; q˙ ). (P3) C (q; q˙ ) is bounded in q and linear in q˙ . From (P1)–(P3), the equation of the manipulator can be written as N K + C(q; q) ˙ q(t) ˙ + g(q) = B(q)(t) q(t) + (t);
(2)
N = H(q)−1 and (t) = H(q)−1 (t). To where C(q; q˙ ) = H(q)−1 C (q; q˙ ); g(q) = H(q)−1 g (q); B(q) simplify the notation the argument t is in many cases dropped out. Since each link transmitting the energy or the moment to the following links will have some delayed behavior due to inertia e5ect, the states with delayed uncertainties are unavoidable and should be included in the dynamical system. Therefore, Eq. (2) can be written as N K + C(q; q) ˙ q(t) ˙ + g(q) = B(q)(t) q(t) +
r
(t)q(t − hj ) + ;
j=1
q(t) = (t);
t ∈ [−h; 0];
(3)
where (t) is a continuous vector-valued initial function and h = max{hj ; j = 1; 2; : : : ; r}, and dj (t); j = 1; 2; : : : ; r, are nonlinear time-varying continuous functions which represent the gains of the delayed state uncertainties for the system and are assumed to be bounded, i.e., |dj (t)|6 j ; ∀t, where
j ’s, j = 1; 2; : : : ; r, are unknown but positive constants. For convenience, we de9ne #j = j2 ;
j = 1; 2; : : : ; r:
(4)
Hence, #j is obviously an unknown positive constant. Let # = [#1 #2 · · · #r ] . De9ne #ˆ as the estimate of the unknown bound # and the error gain of the delayed state uncertainties as #˜ = #ˆ − #. The reference model for the plant to follow is a linear time invariant stable system with a piecewise continuous and uniformly bounded input rm , and the output qm , related by qKm + M1 q˙m + M0 qm = rm ;
(5)
where M1 and M0 are selected properly such that qm has the desired response of the plant. Let e = q − qm denote the tracking error. In order to design a stable adaptive controller with H∞ tracking performance, we 9rst select a set of parameter matrices F1 ; F0 such that the error matrix polynomial eK + F1 e˙ + F0 e is a Hurwitz polynomial. Then, de9ne z as z = qKm − F1 e˙ − F0 e:
(6)
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Adding −z and subtracting C(q; q˙ )˙q(t) and g(q) from both sides of (3), we have N − z − C(q; q) ˙ q(t) ˙ − g(q) + qK − z = B
r
dj (t)q(t − hj ) + :
(7)
j=1
Substituting (6) in (7), we obtain N − z − C(q; q) ˙ q(t) ˙ − g(q) + eK + F1 e˙ + F0 e = B
r
dj (t)q(t − hj ) + :
(8)
r N − z − C(q; q) ˙eN = eN + B B N − hj ) + B; ˙ q(t) ˙ − g(q) + dj (t)INq(t
(9)
j=1
Therefore, the error dynamics in (8) can be written as
j=1
where
e˙ 0(n) I(n) q˙ N ; eN = ; I= ; qN = q (2n)×1 0(n) 0(n) (2n)×(2n) e (2n)×1 −F0 −F1 I(n) = ; B= I(n) 0(n) (2n)×(2n) 0(n) (2n)×n
(10)
in which 0(n) is an n × n zero matrix. The purpose of the paper is to synthesize an AFNC scheme so that all signals in the overall system are asymptotically stable and its H∞ -norm is less than or equal to a prescribed value in the presence of the unknown uncertain parameters, unknown delayed state uncertainties and external disturbances. 2.1. Adaptive control with delayed state gain identi3cation It is 9rst assumed that the dynamical system (2) is known a priori. Then, the certainty equivalent controller can be de9ned as 1 ˆ † N N 1 −1 N ˙ q(t) ˙ + g(q) − #B II PeN − 2 B PeN ; =B z + C(q; q) (11) 2 where B† denotes the pseudo-inverse of B (i.e., BB† B = B), is any positive constant and P is a symmetric positive de9nite matrix satisfying the following matrix Riccati equation: P + P = −Q
(12)
1 −1 1 2 · · · − in which Q = Q is a symmetric positive de9nite matrix. Furthermore, = [− r ] , 1 where j ’s, j = 1; 2; : : : ; r, are positive constants satisfying the following equation:
Q˜ = Q −
r j=1
j I2n ¿ 0:
(13)
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Substituting (11) into (9) leads to the following: eN˙ =
−
r
1 ˆ N N 1 N − hj ) + : #II P − 2 BB P eN + B dj B† INq(t 2 j=1
(14)
Lemma 1. Consider the system in (2) with given H(q); C (q; q˙ ) and g (q). The adaptive control law for the system is given in (11) in which the gain vector of delayed state uncertainties is updated by #ˆ˙ =
1 2
N eN PININ Pe;
(15)
where P is de3ned in (12) and is any positive de3nite matrix. Then, for any t¿t0 ; eN (t) ˜ and #(t) are uniformly ultimately bounded (UUB), and the H∞ tracking performance for the overall system satis3es the following relationship: J =
T 0
N dt eN (t)Q˜ e(t)
˜ N + #˜ (0)−1 #(0) 6 eN (0)Pe(0) 2 T r N † † N + dt: + 2 #j qN m (t − hj )I (B ) B IqNm (t − hj ) 0
(16)
j=1
Proof. See Appendix A. In practical applications, however, the parameter matrices H(q); C (q; q˙ ) and g (q) are generally uncertain and are con9guration dependent. The controller of (11) is not always obtainable. Therefore, a new controller needs to be designed taking account the unknown nonlinear controlled plant, which will be adequately approximated by an FN system. 2.2. Adaptive fuzzy-neural control with delayed state gain identi3cation In order to apply the FN system with adaptation capability, we 9rst parameterize the dynamical system (2) as follows:
qK1 a2 a1 qK a2n+1 a 2n+2 2 .. + .. .. . . . qKn a2n2 −2n+1 a2n2 −2n+2
· · · an q˙1 q˙ · · · a3n 2 .. .. .. . . . q˙n · · · a2n2 −n
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+
an+1 a3n+1 .. .
an+2 a3n+2 .. .
a2n2 −n+1 a2n2 −n+2 +
r
381
q1 1 · · · a2n · · · a4n q2 2 ˆ = B .. .. .. ... . . . · · · a2n2 qn n
dj (t)q(t − hj ) + ;
(17)
j=1
where 1 I(n) Bˆ = √ c1 c2
(18)
for which I(n) is an n × n identity matrix. Let a = [a1 a2 · · · an an+1 · · · a2n a2n+1 · · · a3n · · · a2n2 −n+1 a2n2 −n+2 · · · a2n2 ] be the unknown plant parameter vector. (2n2 )×1 Rewrite (17) in the following form: ˆ + qK + Va = B
r
dj (t)q(t − hj ) + ;
(19)
j=1
where qN 0 · · · 0 0 qN · · · 0 : V= .. . . .. . . . 0 0 · · · qN n×(2n2 )
(20)
Therefore, from (9), the error dynamics in (14) can be written as ˆ − z − Va) + eN˙ = eN + B(B
r
N − hj ) + B; dj (t)INq(t
(21)
j=1
N and B are de9ned for (9). where eN ; I; Let x = [x1 x2 · · · xn xn+1 · · · x2n ] = [˙q q ] . As shown in Fig. 1, the FN system is characterized by fuzzy IF–THEN rules and a fuzzy inference engine. The fuzzy inference engine uses the fuzzy IF–THEN rules to perform a mapping from an input linguistic vector x to an output linguistic variable f(x) ∈ R. Since there is no mature guidance in fuzzy set theory for the determination of the best shapes for fuzzy sets, it is suggested that di5erent shapes for di5erent set points need to be studied to obtain an optimum solution for various ranges of the system states. In addition, the choice of equal-width intervals entails no loss of generality, particularly in applications. The Gaussian membership functions with equal-width intervals of the means are thus proposed to eliminate
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Fig. 1. Con9guration of the FN system.
the sharp boundary and is de9ned as xk − xNk ; k = 1; 2; : : : ; 2n; k (xk ) = exp − 2k
(22)
where xNk and k are the mean and variance of the kth Gaussian membership function k (x), respectively. Before proposing our AFNC scheme, we introduce for (2) the fuzzy basis function expansion. Since a MIMO fuzzy system can always be approximated by a group of multi-input single-output (MISO) fuzzy systems [3], we assume that the fuzzy systems are MISO systems con2n sists of N = k=1 Nk rules in the following form: Ri1 i2 ···i2n : IF x1 is Gi11 AND x2 is Gi22 AND · · · AND xn is Ginn AND xn+1 is Gin+1 n+1 AND · · · AND x2n is Gi2n THEN f(x) is Ci1 i2 :::i2n , 2n i1 = 1; : : : ; N1 ; i2 = 1; : : : ; N2 ; : : : ; i2n = 1; : : : ; N2n , where xk ’s, k = 1; 2; : : : ; 2n, and f(x) denote the linguistic variables associated with the inputs and output of the fuzzy system. Gikk and Ci1 i2 ···i2n are linguistic values of linguistic variables x and f in the universes of discourse U ⊂ R2n and V ⊂ R, respectively, k = 1; 2; : : : ; 2n. By using a center-average defuzzi9er, product inference and singleton fuzzi9er, output f(x) from the FN system can be expressed as f(x) =
N1
···
i1 =1
=
N1 i1 =1
N2n i2n =1
···
N2n i2n =1
2n N1
i1 =1
···
k=1 kik (xk ) N2n 2n i2n =1
%i1 i2 ···i2n (x)ci1 i2 ···i2n ;
k=1
kik (xk )
ci1 i2 ···i2n (23)
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where ci1 i2 ···i2n is the center of the ik th fuzzy set and is the point at which ci1 i2 ···i2n is maximum or equal to 1, and the following a nonlinear mapping of a neural network: 2n k=1 kik (xk ) %i1 i2 ···i2n (x) = N1 (24) ; ik = 1; 2; : : : ; Nk ; k = 1; 2; : : : ; 2n: N2n 2n i1 =1 · · · i2n =1 k=1 kik (xk ) Because Ni11=1 · · · Ni2n2n=1 %i1 i2 ···i2n (x) = 1; %i1 i2 ···i2n (x) can be viewed as a weighting function. For each point x; %i1 i2 ···i2n (x) is the weighting for the (i1 i2 · · · i2n )th fuzzy rule. Hence, these fuzzy inferences can be described in the form of a linear equivalent neural network a = )c;
(25)
where ci1 i2 ···i2n ’s are free (adjustable) parameters and ) = diag{ ; ; : : : ; } with dimension (4n) × (N 4n) for which = [%11···1 %11···2 · · · %11···N2n · · · %N1 N2 ···1 · · · %N1 N2 ···N2n ] is an N × 1 fuzzy basis function vector and %i1 i2 ···i2n ’s are de9ned in (24), and c = [c1 c2 · · · c4n ] for which c‘ = [c11···1‘ c11···2‘ · · · c11···N2n ‘ · · · cN1 N2 ···1‘ · · · cN1 N2 ···N2n ‘ ] ; ‘ = 1; 2; : : : ; 4n. Note from (23) and (25) that the hidden relation is linear in the sense that the unknown parameter vector a depends only on the coe=cients in the fuzzy sets and the coe=cients in the linear combination of the FN system. For control, FN systems are capable of approximating a wide variety of nonlinear systems with known structures but unknown parameters which are dependent on known variables. Based on the universal approximation theorem, the FN system in the form of (25) is capable of uniformly approximating any nonlinear function over Z to any degree of accuracy if Z is compact. 3. Adaptive fuzzy-neural control with H∞ tracking performance Since the parameters of the controlled plant are often unknown, nonlinear and time varying, the FN system is used to approximate the uncertain nonlinear system by using the update laws derived to tune the adjustable parameter vector a. Because the time-varying parameters of the controlled plant represented as the parameter vector a are absorbed partly into the FN system, a can be obtained more accurately by further estimating the unknown but constant weight vector c according the tracking error and the coe=cients of the FN system. Let aˆ = )ˆc be the estimate of a due to cˆ and c˜ = cˆ − c the error vector. Then, the certainty equivalent controller of (11) can be re-de9ned as 1 ˆ † N N 1 −1 ˆ =B z + Vaˆ − #B II PeN − 2 B PeN 2 1 ˆ † N N 1 −1 ˆ =B z + Wcˆ − #B II PeN − 2 B PeN ; (26) 2 where W = V). Substituting (26) into (21) leads to the following: r 1 1 ˆ † ININ PeN − N − hj ) + eN˙ = eN + B Wc˜ − #B B PeN + dj B† INq(t 2 2 j=1 r 1 1 N − hj ) + : = − #ˆ ININ P − 2 BB P eN + B Wc˜ + dj B† INq(t 2 j=1
(27)
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Theorem 1. Consider the system in (2) with uncertain nonlinear parameter matrices H(q); C (q; q˙ ) and g (q). The adaptive control law for the system is given in (26) in which the weight vector and the gain vector of delayed state uncertainties are updated by ' 1 N cˆ˙ = c˜˙ = c˜ − P1 W Wc˜ − P1 ) V B Pe; 2 2 N #ˆ˙ = 1 eN PININ Pe; 2
respectively, where P1 is de3ned as t t − 1 exp − '()) d) W (s)W(s) ds P1 (t) = 0+
where
' = '0
s
P1 1− k0
(28) (29)
(30)
(31)
˜ are UUB, and for which ' 0 and k 0 are positive constants. Then for any t¿t0 ; eN (t); c˜ (t) and #(t) the H∞ tracking performance for the overall system satis3es the following relationship: T N dt J = eN (t)Q˜ e(t) 0
˜ N + c˜ (0)P1−1 (0)c(0) ˜ + #˜ (0)−1 #(0) 6 eN (0)Pe(0) 2 T r N † † N + dt: #j qN + 2 m (t − hj )I (B ) B IqNm (t − hj )
0
(32)
j=1
Proof. See Appendix B. Remark 1. (a) From (30), since t t exp − '()) d) W (s)W(s) ds 0+
s
(33)
is always positive semide9nite, we have the derivative of P1−1 given by the following: t t −1 ˙ exp − '()) d) (−'(t))W (s)W(s) ds + W (t)W(t) P1 (t) = 0+
−1
s
= −'(t)P1 (t) + W (t)W(t):
(34)
If the system is of persistent excitation, then '(t) in (31) and W (t)W(t) will be a positive number and a positive de9nite matrix, respectively, and the upper bound of P1 (t) is k 0 I and the corresponding lower bound of P1−1 (t) is (1=k 0 )I [26]. (b) It is seen from (30) that the use of the exponential forgetting weighting is to have parameter tracking ability. Further, the estimation aˆ from identifying the parameters of neural network and
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385
fuzzy sets can approximate the unknown plant parameter vector a as precisely as possible. However, in reality estimation error may still occur due to the limitation of the number of fuzzy sets and the computational complexity. Hence, with the exponential forgetting weighting, we can use a fewer number of fuzzy rules and reduce the computational complexity by identifying the weights of the FN system so that the estimation error of the weights approaches in a residual set caused by the model uncertainties. (c) The physical meaning of H∞ tracking performance in (32) is that the e5ect of the disturbance and the gain of the delayed-state uncertainties dj ’s on eN must be attenuated below a desired level from the view point of energy, no matter what and dj ’s are, i.e., the L2 gain from and dj ’s to eN must be equal to or less than a prescribed value 2 . In general, is chosen as a positive small value less than one for attenuation of and dj ’s. (d) From (29), it is seen that by selecting properly the parameters j ; j = 1; 2; : : : ; r, to satisfy ˜ (13) and the attenuation level , the size of the residual set of the bounds on eN (t); c˜ (t), and #(t) can be made as small as possible. In addition, from (29) and the derivations above, the bounds of the delayed state j ; j = 1; 2; : : : ; r, are not required to be known a priori. (e) From Theorem 1, it is seen that the AFNC scheme using the FN system in which the weights as well as the gains of the delayed state uncertainties are identi9ed simultaneously can guarantee that the tracking error e(t) converges to the neighborhood of zero. Also we can conclude that controller (26) in which the estimations for the neural weights (28) and the gains of the delayed state uncertainties (29) will not drift to in9nity. To summarize the above analysis, the design procedure of the AFNC scheme is delineated as follows: Step 1: Specify M1 and M0 for the reference model, F1 and F0 to construct and Q, and then use and Q to solve P in (12). Step 2: Specify the values for ; ; ' 0 ; k 0 and j ; j = 1; 2; : : : ; r, based on the practical constraints. Furthermore, the selections of j ; j = 1; 2; : : : ; r, must satisfy (13). This is to match the magnitude scale of the corresponding delayed state uncertainties so that the designer is free from supplying j ’s at the range bounds of (13). Step 3: De9ne the membership function k (xk ); k = 1; 2; : : : ; 2n, for 2n state variables of the controlled system where premise part is the states within some operating positions and consequent part is the corresponding model parameter. Step 4: Construct V for qN and compute the fuzzy neural network %i1 i2 ···i2n (x) for ik = 1; 2; : : : ; Nk and k = 1; 2; : : : ; 2n to construct ). Then the parameter estimate aˆ is constructed by aˆ = )ˆc. Step 5: Obtain the control and apply to the plant. Then compute the adaptive law (26) to adjust the weights of the neural network, the gains of the delayed state uncertainties and the dynamical weighting matrix P1 from (28), (29) and (34), respectively, and to achieve H∞ tracking control purpose. Remark 2. (a) If the output performance is unsatisfactory, then we can decrease the value in Step 4 to cope with the gains of the disturbances and delayed state uncertainties. In addition, when one decreases the parameter su=ciently, then the upper bound on the steady state errors eN (t) can be made as small as possible.
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(b) From the above design procedure, the parameters j ; j = 1; 2; : : : ; r, in Step 2 are assigned such that (13) is satis9ed. That is, the system designer can tune the size of the residual set by adjusting properly these parameters which are used in (15) or (29). 4. Design example In this section, we consider a gyroscopic system with single actuating input shown in Fig. 2 which is similar to that of [2] but with some slight di5erence in the assumptions made: the gimbals are rigid but is with a little unbalanced bodies and the conservation of the angular momentum constants are not zero. The gyroscope itself can be mounted on a base (aircraft, missile, etc.) that is moving with respect to the earth. Also, the case of the gyroscope can be mounted on a platform so that it can rotate relative to the base. The inertia of the system is concentrated in the rotor, with J as the radial moment of the inertia and I the axial moment of inertia. This system is acted upon by a single torque input (t) applied along the x-axis (torque axis). From Newton’s law in rotational form, the equations of motion for the gyroscopic system are shown as follows: 2
K = (t) + J ˙ (t − h) sin(,(t − h)) cos(,(t − h)) − E2 (t ˙ − h) sin(,(t − h)) + .(t); J ,(t)
(35)
˙ sin2 (,(t)) + E2 cos(,(t)) = E1 ; J (t)
(36)
˙ cos(,(t))) = E2 ; I ( ˙ (t) + (t)
(37)
where E1 ; E2 are conservation of the angular momentum constants, h is the delay time for the rate ˙ due to the inertias of the radial and axial moments, and . is the bounded disturbance of change, , caused by the unbalanced e5ects. If a torque is applied to a gyroscope, the inner gimbal precesses with respect to inertial space such that (36) and (37) are satis9ed, and, then, the gyroscopic output torque is the reaction torque developed by the angular velocity of precession. Hence, it is desired that the AFNC scheme is described so that all signals in the overall system are bounded, and to keep precession nonrotating and consequently maintain the same direction in inertia space as the desired direction m of a given linear reference model (41) with H∞ tracking performance by means of the force actuated by the torque axis in the presence of the unknown uncertain
β
ψ θ
u X
Fig. 2. A single actuated gyroscopic system.
W.-S. Yu / Fuzzy Sets and Systems 146 (2004) 375 – 401
387
parameters, the unknown delayed state uncertainties and the external disturbances. Without loss of generality, counterclockwise and clockwise rotations are de9ned as positive and negative, respectively. There are three outputs ,; and in this gyroscopic system. From (36) and (37), we have E2 cos(,) − E1
˙ + = 0; J sin2 (,)
(38)
2 ˙ + E1 cos(,) − E2 cos (,) = E2 : I IJ sin2 (,)
(39)
˙ and of ; ˙ , can be seen as in function of ,. From (17) and (35) Hence, the rate changes of ; , –(37), since n = 2 for , and n = 1 for , we can express the plant model as follows: ˙ − h)) + .; ,K + a1 ,˙ + a2 , + a3 = + d(,(t − h); (t
(40)
where ai ; i = 1; 2; 3, are unknown parameters to be estimated. Step 1: A gyroscope having the following parameters is chosen for this simulation: the inertias are taken given by I = 2 and J = 1, the delay time by h = 0:02 s, and the angular momentum constants by E1 = E2 = 1, since (36) and (37) satisfy the conservation laws of the angular momentum. Let the reference model be speci9ed as ,Km + 15,˙m + 75,m + 125
m
= rm ;
(41)
where rm = 125 for unit step tracking. Further, from (8), the parameter matrix F0 is speci9ed by 1024 64 0 F0 = −1 0 0 : 0 0 1
(42)
Therefore, we have = F0 . From (12) for Q = I3×3 , we have 2054:1 192:1 0:0000 0:1 : P = 192:1 24 0:0000 0:1 0:0000
(43)
Step 2: Specify = 2; ' 0 = 1, and k 0 = 1, and = 0:8 (0 = 1:25) to satisfy (13). Further, two values, 0.1 and 0.5, are given for for tracking performance comparisons. In practice, friction and mass unbalance are inevitable for the gyroscope and will cause disturbance torques on the gimbals when the body is accelerating and rotating. Therefore, the sinusoidal disturbances 2 sin(t) are used to simulate these imperfections. Step 3: To minimize the design e5ort and complexity, and reduce the amount of overlap with the fuzzy sets, we try to use as few rules as possible. The fuzzy membership functions for
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x = [x1 x2 x3 ] = [,˙ , ] are given by 1; , ¡ −0:1; (, + 0:1)2 11 (,) = ; , ¿ −0:1; exp − 2(0:03)2 1; , ¿ 0:1; (, − 0:1)2 15 (,) = ; , 6 0:1; exp − 2(0:03)2 (, + 0:05)2 12 (,) = exp − ; 2(0:03)2 (, − 0:05)2 14 (,) = exp − ; 2(0:03)2
1;
˙ = (,˙ + 0:1)2 21 (,) exp − 2(0:03)2 ; 1; ˙ (,˙ − 0:1)2 25 (,) = exp − 2(0:03)2 ; ˙ + 0:05)2 ( , ˙ = exp − ; 22 (,) 2(0:03)2 ˙ − 0:05)2 ( , ˙ = exp − 24 (,) ; 2(0:03)2
1; ( − 0:9)2 31 ( ) = ; exp − 2(0:03)2 1; ( − 1:1)2 35 ( ) = ; exp − 2(0:03)2 ( − 0:95)2 ; 32 ( ) = exp − 2(0:03)2 ( − 1:05)2 34 ( ) = exp − : 2(0:03)2
(44)
(,)2 13 (,) = exp − 2(0:03)2
; (45)
,˙ ¡ −0:1; ,˙ ¿ −0:1; ,˙ ¿ 0:1; (46)
,˙ 6 0:1;
˙ 2 ˙ = exp − (,) 23 (,) 2(0:03)2
; (47)
¡ 0:9; ¿ 0:9; ¿ 1:1;
(48)
6 1:1;
( − 1)2 33 ( ) = exp − 2(0:03)2
; (49)
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As we will see from (35)–(37), there is a corresponding inRuence from the and gimbals on the , gimbal. So, rules (44) and (48) are constructed when the present positions of , and are cross coupling very much and far away from the set point. Therefore, it requires a large drive output to tune the shaft to the set point quickly. In addition, rules (45) and (49) implement the condition when the error starts to decrease and the weights of the FN system are approaching the neighborhood of the true parameters, and the set point is very nearly reached. Thus, a small drive output is given. Because of the inertia of the gyroscope, it is necessary to stop the drive at this instant to keep the overshoot at a minimum. However, rules (46) and (47) deal with the condition when overshoot does occur. A small reverse drive signal is given to bring the shafts to its set point. The rule on the right part of (46) implies the reverse condition of the left. We now establish the center-average formulation for fuzzy logic inference rules. Step 4: From (20), we have ,˙ , 0 0 0 0 V= : (50) 0 0 0 0 ,˙ , To simplify the statement of a rule, and to take into account as well the associated membership functions (44)–(49), we have the following: Let % i 1 i 2 i3 =
3 kik k=1 ; 5 5 5 3 k=1 kik i1 =1 i2 =1
ik = 1; 2; 3; 4; 5; k = 1; 2; 3
i3 =1
for which = [%111 %112 %113 %114 %115 · · · %551 %552 %553 %554 %555 ] . Therefore, )=
0
0
125
:
(51)
3×375
Therefore, each element of the vector aˆ = [aˆ 1 aˆ 2 aˆ 3 ] can be expressed as aˆ‘ =
5 5 5
%i1 i2 i3 cˆi1 i2 i3 ‘ ;
‘ = 1; 2; 3:
(52)
i1 =1 i2 =1 i3 =1
Step 5: Let the initial conditions be given by P1 (0)−1 = I375×375 ; cˆ (0) = [10 10· · · 10] , and
ˆ #(0) = 0, and let Bˆ = I(3) ;
375
01 B = [1 0 1] ; IN = 0 0 : 01
(53)
Update the estimates of the vectors cˆ and #ˆ and obtain the weighting matrix P1 from (28), (29) and (34), respectively. Then the adaptive controllers will be derived from (11) and (26), respectively,
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W.-S. Yu / Fuzzy Sets and Systems 146 (2004) 375 – 401
for comparison depending on di5erent assumptions on the plant parameter matrices. From (11) and according to Lemma 1, adaptive controller can be obtained as = ,Km − 15(,˙ − ,˙m ) − 75(, − ,m ) − 125( −
m)
−
1 ˆ † N N 1 #B II PeN − 2 B PeN 2
(54)
and from (26) and according to Theorem 1, adaptive controller can be obtained as follows: = ,Km − 15(,˙ − ,˙m ) − 75(, − ,m ) − 125( −
m)
+ Wcˆ −
1 ˆ † N N 1 N #B II PeN − 2 B Pe; 2
(55)
where B† = [0:5 0 0:5]:
(56)
Then, the adaptive controllers (54) and (55) are used for stabilizing the gyroscopic system, respectively, for comparison. Figs. 3 and 4 show the simulation results with disturbances 2 sin(t) as follows: (a) the output angle of the gyroscope and the reference model output m , (b) the tracking error e = − m , (c) ˙ (e) the outputs , ˙ (f) the estimate #, ˆ (g) the estimates aˆ 1 ; aˆ 2 and the outputs ,, (d) the outputs ,, aˆ 3 , and (h) the input torque (t). Fig. 3 shows the responses for = 0:5. From Fig. 3(a) and (b) in the H∞ tracking purpose with the FN system (i.e., without a priori knowledge on plant dynamics) can be achieved e5ectively than the case without (i.e., with a priori knowledge on plant dynamics), and the maximum steady state tracking error for the former case is 1:8% and the latter is 4:1%. The responses of the learning gain of delayed state uncertainties are quite satisfactory for both cases (refer to Fig. 3(f)). The proposed AFNC (26) and the updating law for weight (28) and for delayed state gain (29) demonstrate an e5ective tool for the learning of delayed state uncertainties. On the other hand, Fig. 4 shows the responses for = 0:1. From Fig. 4(a) and (b) the H∞ tracking purpose with the FN system (i.e., without a priori knowledge on plant dynamics) can be achieved e5ectively than the case without (i.e., with a priori knowledge on plant dynamics), and the maximum steady state tracking error for the former case is 0:86% and the latter is 2:15%. As demonstrated in Fig. 4(h), the control input is signi9cantly reduced while maintaining satisfactory tracking responses. As compared with Figs. 3 and 4, the time-varying disturbances are absorbed partly into the FN system and the estimates can be obtained more accurately. The responses of the estimates of the plant model and delayed state uncertainties for the uncertain plant converge and do not drift to in9nity due to strong compensation of the estimation algorithms for the weights of the neural network and the gains of the delayed state uncertainties (see Fig. 3(f) and (g) and Fig. 4(f) and (g)). The proposed AFNC scheme derived from these estimates iteratively for achieving model following purpose has the ability to stabilize the controlled plant and has better performance than that without using the FN system. In other words, the use of FN system with adaptation weights can indeed improve the performance of the closed-loop system (see Figs. 3 and 4(a), (b) and (h)). The main reason of the result is that the only use of updating law for delayed state uncertainties cannot achieve an excellent model following result as the system is subject to the controlled plant with plant uncertainties (since the plant dynamics are con9guration dependent) and time-varying disturbances. This is one of the important motivation for the study. The results reveal that the proposed AFNC scheme indeed improves the system performances including convergence of the estimations and tracking errors, the smoothness of the control inputs and easy selection of the parameters of the estimations. It seems
W.-S. Yu / Fuzzy Sets and Systems 146 (2004) 375 – 401
391
0.02 1 0 0.8
error output e
-0.02 0.6
-0.04
0.4 -0.06
0.2
-0.08
ψm ψ ψa
e ea
0 0
0.5
1
1.5
2
(a)
2.5
3
3.5
4
4.5
-0.1
5
0
0.5
1
1.5
2
(b)
Time (sec)
2.5
2.5
3
3.5
4
5
4.5
Time (sec)
14
θ θa
θ θa
12 2
10 8
1.5 6 4 1 2 0
0.5
-2 0
(c)
0
0.5
1
1.5
2
2.5 Time (sec)
3
3.5
4
4.5
5
-4
(d)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time (sec)
Fig. 3. Step responses of the gyroscopic system with disturbance 2 sin(t) for = 0:5, where the subscript a denotes the responses without using the FN system. (a) The output trajectories m ; , and a . (b) The tracking errors e and ea . (c) The output trajectories , and ,a . (d) The output trajectories ,˙ and ,˙a . (e) The output trajectories ˙ and ˙a . (f ) Time responses of the estimates #ˆ and #ˆa . (g) Time responses of the estimates aˆ1 ; aˆ2 , and aˆ3 . (h) Time responses of the input torques and a .
that the robustness of the proposed control scheme is excellent. In summary, the control input is bounded, and the estimations of the FN system and the gain of the delayed state uncertainty will gradually approach a steady state in which the plant follows the reference model with faster rising time, little oscillations and tracking errors, and has a rather good dynamical performance.
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2
60 β βa
50
40 delayed estimate
1.5
30
1
20
10 ˆ ϑ ˆϑa 0.5
0 0
0.5
1
1.5
2
(e)
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
(f )
Time (sec)
2
2.5 3 Time (sec)
3.5
4
4.5
150
25
5
τ τa
100 20 50
15 control input
0
10
-50
-100 5 aˆ 1
-150
aˆ 2 aˆ 3
0
(g)
0
0.5
1
1.5
2
2.5 Time (sec)
3
3.5
4
4.5
5
-200
0
(h)
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time (sec)
Fig. 3. continued
Remark 3. The above simulations indicate that the control scheme by properly selecting the parameter and by using 9ve fuzzy rules in this example for each state can guarantee that the tracking error e(t) converges to the neighborhood of zero if the disturbances and uncertainties exist. Using the FN system with dynamical adaptation capability for weights can signi9cantly reduce the amount of the fuzzy inference rules with excellent robustness for achieving H∞ tracking performance.
W.-S. Yu / Fuzzy Sets and Systems 146 (2004) 375 – 401
393
0.02 1 0 0.8
error output e
-0.02 0.6
-0.04
0.4 -0.06
0.2
-0.08
ψm ψ ψa
e ea -0.1
0 0
0.5
1
1.5
2
(a)
2.5
3
3.5
4
4.5
0
5
0.5
1
2
1.5
(b)
Time (sec)
2.5
2.5
3
3.5
4
4.5
5
Time (sec)
14
θ θa
θ θa
12 2 10
8
1.5
6 1
4
2 0.5 0
0
0
0.5
1
(c)
-2 1.5
2
2.5 3 Time (sec)
3.5
4
4.5
5
(d)
0
0.5
1
1.5
2
2.5 3 Time (sec)
3.5
4
4.5
5
Fig. 4. Step responses of the gyroscopic system with disturbance 2 sin(t) for = 0:1, where the subscript a denotes the responses without using the FN system. (a) The output trajectories m ; , and a . (b) The tracking errors e and ea . (c) The output trajectories , and ,a . (d) The output trajectories ,˙ and ,˙a . (e) The output trajectories ˙ and ˙a . (f ) Time responses of the estimates #ˆ and #ˆa . (g) Time responses of the estimates aˆ1 ; aˆ2 , and aˆ3 . (h) Time responses of the input torques and a .
5. Conclusions In this paper, an AFNC scheme for MIMO uncertain robotic systems has been developed based on the H∞ tracking design technique and the general idea that appropriate estimation of the
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W.-S. Yu / Fuzzy Sets and Systems 146 (2004) 375 – 401
2.4
60 β βa
2.2
50 2 1.8 delayed estimate
40 1.6 1.4 1.2
30
20 1 0.8 10
ˆ ϑ ˆ ϑa
0.6 0.4
0
0.5
1
1.5
2
(e)
2.5
3
3.5
4
4.5
0
5
0
0.5
1
1.5
2
(f)
Time (sec)
2.5
3
3.5
4
4.5
150
25
5
Time (sec)
τ τa
100 20 50
control input
15
0
-50
10
-100 5 a ˆ1
-150
aˆ 2 aˆ 3 -200
0 0
(g)
0.5
1
1.5
2
2.5 3 Time (sec)
3.5
4
4.5
5
0
(h)
0.5
1
1.5
2
2.5 3 Time (sec)
3.5
4
4.5
5
Fig. 4. continued
adaptation process should provide a satisfactory basis for the control. Nonideal e5ects such as system with unknown parameters and nonlinearities and multiple delayed state uncertainties are considered from a practical point of view. The considered delayed state uncertainties are assumed to be bounded by some unknown gains. As well, an FN system is used to represent the unknown controlled system according to the Stone Weierstrass theorem. A reference model with the desired amplitude and phase properties is given to construct an error model. The AFNC scheme uses two on-line estimations operating simultaneously, which allows for the inclusion of identifying the gains
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395
of the delayed state uncertainties and training the weights of the FN system simultaneously. It is shown that, in the sense of Lyapunov-type stability, the proposed control scheme can guarantee estimation convergence and stability robustness of the closed-loop system with H∞ tracking performance. Furthermore, the constraint demanding prior knowledge on upper bounds of the delayed state uncertainties is removed through the design algorithm of the proposed scheme. As demonstrated in the illustrated example, the AFNC scheme proposed in this paper can achieve a better model following tracking performance over that without using the FN system with adaptation weights. Appendix A. Proof of Lemma 1 Choose the Lyapunov–Krasovskii functional candidate as ˜ = eN PeN + #˜ −1 #˜ + N c; ˜ #) V (e;
r
j
j=1
t t − hj
N d&; eN (&)e(&)
(A.1)
where P is de9ned in (12) and is any positive de9nite matrix. Taking the derivative of V in (A.1) along the trajectory (14), we have r
r
j=1
j=1
N − N − hj ) V˙ = eN˙ PeN + eN PeN˙ + 2#˜ −1 #˜˙ + j eN (t)e(t) j eN (t − hj )e(t r 2 N N ˆ N − hj ) = eN PeN + eN PeN − #eN PII PeN − 2 eN PBB PeN + 2 eN Pdj INq(t j=1
+ 2 B PeN +
r
N − j eN (t)e(t)
j=1
6 eN
P + P +
r j=1
−
r
N − hj ) + 2#˜ −1 #˜˙ j eN (t − hj )e(t
j=1
j I2n eN −
1 B PeN −
r 1 N N e PBB P e + 2 #j eN PINqNm (t − hj ) 2 j=1
+2
r
N − hj ) − #ˆ eN PININ PeN #j eN PINe(t
j=1 r
N − hj ) + 2#˜ −1 #˜˙ − j eN (t − hj )e(t j=1
1 B PeN − + 2
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1 1 B PeN − B PeN − + 2 P + P + = eN j=1 r r 1 1 #j B† INqNm (t − hj ) B PeN − #j B† INqNm (t − hj ) − B PeN − j=1 j=1 2 r ˆ N N N † † N + 2 #j qN m (t − hj )I (B ) B IqNm (t − hj ) − #eN PII PeN
r
j I2n eN −
j=1 r r N − hj ) + 2#˜ −1 #˜˙ − N − hj ): #j eN PINe(t j eN (t − hj )e(t +2 j=1
(A.2)
j=1
ˆ˙ we have the following Applying (12), (13), and (15) to (A.2), and using the fact that #˜˙ = #, relationship: V˙ = −eN Q˜ eN −
1 B PeN −
1 B PeN − + 2
r 1 #j B† INqNm (t − hj ) − B PeN − j=1 r 1 × B PeN − #j B† INqNm (t − hj ) j=1 2 r N † † + 2 #j qN m (t − hj )I (B ) B IqNm (t − hj )
(A.3)
j=1
− #ˆ eN PININ PeN + 2
r
r
N − hj ) + 2#˜ −1 #˜˙ − N − hj ) #j eN PINe(t j eN (t − hj )e(t
j=1
j=1
2 r ˆ N N N † † N 6 −eN Q˜ eN + 2 + 2 #j qN m (t − hj )I (B ) B IqNm (t − hj ) − #eN PII PeN j=1
+2
r j=1
r
N − hj ) + 2#˜ −1 #˜˙ − N − hj ) #j eN PINe(t j eN (t − hj )e(t
= −eN Q˜ eN + 2 + 2
r j=1
2
j=1
ˆ N N N † † N #j qN m (t − hj )I (B ) B IqNm (t − hj ) − #eN PII PeN
W.-S. Yu / Fuzzy Sets and Systems 146 (2004) 375 – 401 r
1 N − h j ) − − + 2#˜ −1 #˜˙ − j e(t j
397
! ! −1 N N − hj ) − j e(t #j I PeN #j IN PeN
j=1
+ #eN PININ PeN
2 r ˆ N N N † † N 6 −eN Q˜ eN + 2 + 2 #j qN m (t − hj )I (B ) B IqNm (t − hj ) − #eN PII PeN j=1
N + 2#˜ −1 #˜˙ + #eN PININ Pe:
(A.4)
ˆ 0 ) is 9nite, we have From (29) and since #˜ = #ˆ − #, where #(t 2 r N † † N : #j qN V˙ 6 −eN Q˜ eN + 2 + m (t − hj )I (B ) B IqNm (t − hj )
(A.5)
j=1
From (P1), since H(q) is nonsingular, H(q)−1 is bounded and, thus, (t) is also bounded. Therefore,
(t) = H(q)−1
(t) is bounded, and we have
V˙ 6 −eN Q˜ eN + 2 H(q)−1 2 (t) 2 +
r j=1
2
N 2 qNm (t − hj ) 2 #j B† I
2 r ˜ e N 2 qNm (t − hj ) 2 ; N 2 + 2 H(q)−1 2 (t) 2 + 6 −4min (Q) #j B† I
(A.6)
j=1
˜ denotes the minimal eigenvalue of Q. ˜ Therefore, whenever where 4min (Q)
" N 2 qNm (t − hj ) 2 H(q)−1 2 (t) 2 + ( rj=1 #j )2 B† I
N ¿ ;
e ˜ 4min (Q)
(A.7)
V˙ 60. In light of Lyapunov stability theory of the retarded functional di5erential equation, [4,7] since is the design constant serving as an attenuation level, it can be concluded from (A.5) that ˜ are UUB in the presence of the delayed state uncertainties dj q(t N − for any t¿t0 ; eN (t); c˜ (t) and #(t) hj ); j = 1; 2; : : : ; r, and the external disturbance [6]. By assumption that ∈ L2 [0; T ]; ∀T ∈ [0; ∞), we integrate (A.5) from t = 0 to T , and obtain
T 0
N dt eN (t)Q˜ e(t)
6 V (0) + 2
T 0
+
2
r
N † † N dt: #j qN m (t − hj )I (B ) B IqNm (t − hj )
j=1
(A.8)
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W.-S. Yu / Fuzzy Sets and Systems 146 (2004) 375 – 401
Substituting (A.1) into (A.8), we have the H∞ tracking performance satisfying T N dt eN (t)Q˜ e(t) J = 0
˜ N + c˜ (0)P1−1 (0)c(0) ˜ + #˜ (0)−1 #(0) 6 eN (0)Pe(0) 2 T r † †N N + dt: #j qN + 2 m (t − hj )I (B ) B IqNm (t − hj )
0
(A.9)
j=1
This completes the proof. Appendix B. Proof of Theorem 1 With the bounds of P1 (t) given, choose the Lyapunov–Krasovskii functional candidate as t r −1 −1 ˜ ˜ ˜ N d; N c; ˜ #) = eN PeN + c˜ P1 c˜ + # # + j (B.1) eN ()e() V (e; t − hj
j=1
where P and P1 are de9ned in (12) and (30), respectively, and is any positive de9nite matrix. Taking the derivative of V in (B.1) and using (27) and (34), we have r
−1 N V˙ = eN˙ PeN + eN PeN˙ + 2c˜ P1−1 c˜˙ + c˜ P˙ 1 c˜ + 2#˜ −1 #˜˙ + j eN (t)e(t) j=1
−
r
N − hj ) j eN (t − hj )e(t
j=1
2 = eN PeN + eN PeN − #ˆ eN PININ PeN − 2 eN PBB PeN + 2c˜ W B PeN r r r N N − N − hj ) N − hj ) + 2 B PeN + +2 j eN (t)e(t) j eN (t − hj )e(t eN Pdj Iq(t j=1
+ c˜
j=1
!
j=1
−'P1−1 + W W c˜ + 2c˜ P1−1 c˜˙ + 2#˜ −1 #˜˙
2 eN PBB PeN + 2c˜ W B PeN 2 r r r N − N − hj ) N − hj ) + 2 B PeN + +2 j eN (t)e(t) j eN (t − hj )e(t eN Pdj INq(t
= eN PeN + eN PeN − #ˆ eN PININ PeN −
j=1
!
j=1
j=1
+ c˜ −'P1−1 + W W c˜ + 2c˜ P1−1 c˜˙ + 2#˜ −1 #˜˙ r ! 6 eN P + P + j I2n eN + c˜ 2P1−1 c˜˙ + 2W B PeN − 'P1−1 c˜ + W Wc˜ j=1
W.-S. Yu / Fuzzy Sets and Systems 146 (2004) 375 – 401
399
1 B PeN − + 2 r 1 #j eN PINqNm (t − hj ) − 2 eN PBB PeN + 2 j=1
−
1 B PeN −
r
+2
r
N − hj ) − #ˆ eN PININ PeN + 2#˜ −1 #˜˙ − N − hj ) #j eN PINe(t j eN (t − hj )e(t
j=1
= eN P + P +
r
j=1
! j I2n eN + c˜ 2P1−1 c˜˙ + 2W B PeN − 'P1−1 c˜ + W Wc˜
j=1
1 1 B PeN − B PeN − + 2 − r 1 #j B† INqNm (t − hj ) − B PeN − j=1 r 1 #j B† INqNm (t − hj ) × B PeN − j=1 2 r N † † N + 2 #j qN m (t − hj )I (B ) B IqNm (t − hj )
j=1
− #ˆ eN PININ PeN + 2
r
N − hj ) + 2#˜ −1 #˜˙ #j eN PINe(t
j=1
−
r
N − hj ): j eN (t − hj )e(t
(B.2)
j=1
Applying (12), (13), and (28) to (B.2), using the fact that cˆ˙ = c˜˙, and following the same procedures as with (A.3) and (A.4), we have the following relationship: 2 r N † † N : #j qN (B.3) V˙ 6 −eN Q˜ eN + 2 + m (t − hj )I (B ) B IqNm (t − hj ) j=1
Therefore, whenever
" N 2 qNm (t − hj ) 2 H(q)−1 2 (t) 2 + ( rj=1 #j )2 B† I
N ¿ ;
e ˜ 4min (Q)
(B.4)
V˙ 60. In light of Lyapunov stability theory of retarded functional di5erential equation, [4,7] since is the design constant serving as an attenuation level, it can be concluded from (B.3) that for
400
W.-S. Yu / Fuzzy Sets and Systems 146 (2004) 375 – 401
˜ N − any t¿t0 ; eN (t); c˜ (t) and #(t) are UUB in the presence of the delayed state uncertainties dj q(t hj ); j = 1; 2; : : : ; r, and the external disturbance [6]. By assumption that ∈ L2 [0; T ]; ∀T ∈ [0; ∞), we integrate (B.3) from t = 0 to t = T , and obtain T N dt eN (t)Q˜ e(t) 0
6 V (0) + 2
T 0
+
2
r
N † † N dt: #j qN m (t − hj )I (B ) B IqNm (t − hj )
(B.5)
j=1
Substituting (B.1) into (B.5), we have the H∞ tracking performance satisfying T N dt eN (t)Q˜ e(t) J = 0
˜ N + c˜ (0)P1−1 (0)c(0) ˜ + #˜ (0)−1 #(0) 6 eN (0)Pe(0) 2 T r † †N N + dt: #j qN + 2 m (t − hj )I (B ) B IqNm (t − hj )
0
(B.6)
j=1
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