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Series-based approximate approach of optimal tracking control for nonlinear systems with time-delay
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Progress in Natural Science 18 (2008) 1571–1576 www.elsevier.com/locate/pnsc
Series-based approximate approach of optimal tracking control for nonlinear systems with time-delay Gongyou Tang a,*, Mingqu Fan a,b b
a College of Information Science and Engineering, Ocean University of China, Qingdao 266100, China College of Information and Engineering, Shandong University of Science and Technology, Jinan 250031, China
Received 16 February 2008; received in revised form 12 March 2008; accepted 12 March 2008
Abstract The optimal output tracking control (OTC) problem for nonlinear systems with time-delay is considered. Using a series-based approximate approach, the original OTC problem is transformed into iteration solving linear two-point boundary value problems without timedelay. The OTC law obtained consists of analytical linear feedback and feedforward terms and a nonlinear compensation term with an infinite series of the adjoint vectors. By truncating a finite sum of the adjoint vector series, an approximate optimal tracking control law is obtained. A reduced-order reference input observer is constructed to make the feedforward term physically realizable. Simulation examples are used to test the validity of the series-based approximate approach. Ó 2008 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in China Press. All rights reserved. Keywords: Nonlinear systems; Time-delay; Tracking control; Optimal control; Observer
1. Introduction Strictly speaking, almost all the control systems are of nonlinearity. And time-delay is a common phenomenon in real control system. Analysis and synthesis for nonlinear systems with time-delay has been one of the most active research areas in the past decades. Many elegant approaches have been provided and applied to practical engineering. For example, Yue and Lam [1] presented a design approach of reliable memory controller for nonlinear time-delay systems; Cimen and Banks [2] introduced the design of optimal tracking controllers for a general class of nonlinear systems; Shamma and Cloutier [3] gave a state-dependent Riccati equation (SDRE) approach to nonlinear system’s stabilization problems. The optimal control problems for nonlinear systems with time-delay * Corresponding author. Tel.: +86 532 66781230; fax: +86 532 85901980. E-mail address: [email protected] (G. Tang).
generally lead to a nonlinear two-point boundary value (TPBV) problem involving both time-delay and timeadvance terms. With the exception of simplest cases, the analytical solution to this TPBV problem does not exist [4]. Therefore, it is meaningful to obtain some approximate solutions of the optimal control problem. Recently, many good results in the approximate approaches of the optimal control for nonlinear and/or time-delay systems have been obtained. One approach is SDRE that uses direct parameterization to transform the nonlinear system into a linear structure with state-dependent coefficients and to solve the SDRE at each point of the state variable along the trajectory to obtain a nonlinear feedback controller of general form [5,6]. The second type of approach is successive Galerkin approximation (SGA), where an iterative process is used to find a sequence of approximations approaching the solution of the HJB equation [7]. The third type of approach is an approximating sequence of Riccati equations (ASREs) to construct nonlinear time-varying optimal state-feedback controllers for
1002-0071/$ - see front matter Ó 2008 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in China Press. All rights reserved. doi:10.1016/j.pnsc.2008.03.033
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G. Tang, M. Fan / Progress in Natural Science 18 (2008) 1571–1576
_ ¼ AxðtÞ þ A1 xðt sÞ þ fðxðtÞÞ SkðtÞ; 0 < t 6 tf xðtÞ _ ¼ CT QCxðtÞ CT QHzðtÞ þ AT kðtÞ kðtÞ ( AT1 kðt þ sÞ; 0 6 t 6 tf s þ f XT ðxðtÞÞkðtÞ þ 0; tf s 6 t 6 tf
the nonlinear systems [8]. The fourth type of approach is a successive approximate approach (SAA) to solve the optimal control problem for nonlinear systems [4,9–11]. In the recent decades, the OTC problem has been receiving growing attention owing to its wide practical applications, such as the tracking control of spacecraft attitude maneuvers [5], super tanker [3], and the underwater target tracking [12]. The purpose of this paper is to develop a series-based approximate approach solving the OTC problems, where a parameter variable e is introduced and this function is expanded into Maclaurin series with respect to the parameter variable at = 0. The original OTC problem is transformed into a sequence of nonhomogeneous linear TPBV problems without time-delay. This approach only needs to solve a sequence of vector differential equations, instead of solving a sequence of matrix differential equations. That means its computation load is much lower in comparison with the other approximate approaches [5–8].
ð4Þ where S ¼ BR B ; f xT ðxÞ ¼ ofðxÞ=ox . And the OTC law can be written as
2. Problem statement
3. Design of OTC law
Consider nonlinear systems with time-delay as the following:
For the TPBV problem (4), we introduce a real scalar parameter variable ee[0,1]. And a new TPBV problem with parameter variable e is constructed as follows: _ eÞ ¼ Axðt; eÞ þ eA1 xðt s; eÞ þ efðxðt; eÞÞ Skðt; eÞ; xðt;
_ ¼ AxðtÞ þ A1 xðt sÞ þ BuðtÞ þ fðxðtÞÞ; 0 < t 6 tf xðtÞ xðtÞ ¼ /ðtÞ; s 6 t 6 0 yðtÞ ¼ CxðtÞ ð1Þ n
r
q
where x 2 R ; u 2 R , and yðtÞ 2 R are state, control, and output, respectively; A; A1 ; B and C are constant matrices of appropriate dimensions; s > 0 is the time-delay, f : Rn ! U Rn ; fð0Þ ¼ 0, which satisfies the Lipschitz conditions on Rn ; /ðtÞ 2 Rn is the initial state. Assume that ðA þ A1 ; BÞ is completely observable. The reference input ~ y 2 Rq to be tacked by the output y(t) is described by the exosystem z_ ¼ Fz;
~ y ¼ Hz
ð2Þ
m
where z 2 R ; F; H are constant matrices of appropriate dimensions. Assume that (F,H) is completely observable. The quadratic performance index is 1 1 J ¼ eT ðtf ÞQf eðtf Þ þ 2 2 Z tf T e ðtÞQeðtÞ þ uT tÞRuðtÞ dt;
ð3Þ
0
where eðtÞ ¼ ~ yðtÞ yðtÞ is the output tracking error; Qf,Q,R are positive definite matrices of appropriate dimensions. The OTC problem is to find a control law u ðÞ, which minimizes the quadratic performance index (3) subject to constraints (1) and (2). According to the necessary conditions of the OTC problem based on the maximum principle, the following TPBV problem can be obtained:
kðtf Þ ¼ CT Qf Cxðtf Þ CT Qf Hzðtf Þ; xðtÞ ¼ /ðtÞ; s 6 t 6 0 1
T
T
u ðtÞ ¼ R1 BT kðtÞ:
ð5Þ
Note that (4) is a nonlinear TPBV problem with both timedelay and time-advance. Unfortunately, the analytical solution of this nonlinear TPBV problem does not generally exist. Therefore, it is necessary to find approximation approaches for solving the optimal control problem for the discrete-time bilinear systems.
0 < t 6 tf _ kðt; eÞ ¼ CT QCxðt; eÞ CT QHzðtÞ þ AT kðt; eÞ ( eAT1 kðt þ s; eÞ; 0 6 t 6 tf s þ ef xT ðxðt; eÞÞkðt; eÞ þ 0; tf s 6 t 6 tf kðtf ; eÞ ¼ CT Qf Cxðtf Þ CT Qf Hzðtf Þ; xðt; eÞ ¼ /ðtÞ;
s 6 t 6 0
ð6Þ
In the following discussion, we suppose that uðt; eÞ; xðt; eÞ; kðt; eÞ; fðxðt; eÞÞ; f xT ðxðt; eÞÞ are all infinitely differentiable with respect toee[0,1], and their Maclaurin series 1 X i ðiÞ ðÞ ðtÞ ð7Þ ðÞðt; Þ ¼ i! i¼0 ðiÞ
are all convergent at e = 1, where ðÞ ðtÞ ¼ oi ðÞ=oi j¼0 Obviously, when e = 1, the TPBV problem (6) is equal to the TPBV problem (5) with parameter variable e, and the OTC law can be written in the following form: 1 X 1 ðiÞ ð8Þ u ðtÞ u ðtÞ ¼ uðt; 1Þ ¼ i! i¼0
Theorem 1. Consider the OTC problems (1)–(4), the OTC can be described by " # 1 X 1 1 T ð9Þ g ðtÞ u ðtÞ ¼ R B PðtÞxðtÞ þ P1 ðtÞzðtÞ þ i! i i¼0
G. Tang, M. Fan / Progress in Natural Science 18 (2008) 1571–1576
where P(t) is the unique positive semi-definite solution of the following Riccati matrix differential equation: P_ 1 ðtÞ ¼ PðtÞA þ A PðtÞ þ C QC PðtÞSPðtÞ T
T
P_ 1 ðtÞ ¼ P1 ðtÞF þ AT P1 ðtÞ CT QH PðtÞSP1 ðtÞ T
P1 ðtf Þ ¼ C Qf H
ð11Þ
gi(t) is the solution of the ith adjoint vector differential equation g_ i ðtÞ ¼ ðAT PðtÞSÞgi ðtÞ þ iPðtÞA1 xði1Þ ðt sÞ i1 X
ðik1Þ
C ki1 f xT
i AT1 ðPðtÞxði1Þ ðt þ sÞ þ gi1 ðt þ sÞÞ; 0 6 t 6 tf s 0; tf s < t 6 tf
gi ðtf Þ ¼ 0; i ¼ 1; 2;
ð12Þ
x(i)(t)is the solution of the ith state equation
s 6 t 6 0
ð19Þ
Substituting (17) into (18) and comparing with (15), we obtain _ ¼ PðtÞA þ AT PðtÞ þ CT QC PðtÞSPðtÞ; PðtÞ P_ 1 ðtÞ ¼ P1 ðtÞF þ AT P1 ðtÞ CT QH PðtÞSP1 ðtÞ
k¼0
( þ
x_ ðiÞ ðtÞ ¼ ðA SPðtÞÞxðiÞ ðtÞ þ iA1 xði1Þ ðt sÞ þ if ði1Þ SðtÞgi ðtÞ xðiÞ ðtÞ ¼ 0; t0 s 6 t 6 t0 ; i ¼ 1; 2;
iAT1 ðPðtÞxði1Þ ðt þ sÞ þ gi1 ðt þ sÞÞ;0 6 t 6 tf s ð20Þ 0; tf s < t 6 tf
We now ascertain the boundary-value conditions of (20). The third equation of (6) implies kðtf ; 1Þ ¼
x_ ð0Þ ðtÞ ¼ ðA SPðtÞÞxð0Þ ðtÞ SP1 ðtÞzðtÞ; xð0Þ ðtÞ ¼ /ðtÞ
1 X 1 ðiÞ k ðtf Þ i! i¼0
¼ CT Qf Cxðtf Þ CT Qf Hzðtf Þ
ð21Þ
Substituting (17) into (21), it follows ð13Þ
Proof. By substituting (7) into (4) and (6) separately and by comparing the coefficients of the same order terms with respect to ei ði ¼ 0; 1; 2; Þ, we obtain ð0Þ
xð0Þ ðtÞ ¼ /ðtÞ; xðiÞ ðtÞ ¼ 0;
ðPðtÞxðkÞ ðtÞ þ gk ðtÞÞ
k¼0
x_ ð0Þ ðtÞ ¼ Axð0Þ ðtÞ Sk
ð18Þ
g_ i ðtÞ ¼ ðAT PðtÞSÞgi ðtÞ þ iPðtÞA1 xði1Þ ðt sÞ þ iPðtÞf ði1Þ i1 h i X ðik1Þ þi C ki1 f Tx PðtÞxðkÞ ðtÞ þ gkðtÞ
g0 ðtÞ ¼ 0
þ
x_ ðiÞ ðtÞ ¼ ðA SPðtÞÞxðiÞ ðtÞ þ iA1 xði1Þ ðt sÞ þ if ði1Þ Sgi ðtÞ
From (1) and (7), initial conditions of (18) are taken as
P1(t) is the unique solution of the following matrix differential equation:
(
x_ ð0Þ ðtÞ ¼ ðA SPðtÞÞxð0Þ ðtÞ þ P1 zðtÞ
ð10Þ
Pðtf Þ ¼ CT Qf C
þ iPðtÞf ði1Þ þ i
1573
Pðtf Þxð0Þ ðtf Þ þ P1 ðtf Þzðtf Þ ¼ CT Qf Cxðtf Þ CT Qf Hzðtf Þ Pðtf ÞxðiÞ ðtf Þ þ gi ðtf Þ ¼ 0 ð22Þ From (22) the boundary-value conditions of PðtÞ; P1 ðtÞ and gi(t) are taken as
ðtÞ
x_ ðiÞ ðtÞ ¼ AxðiÞ ðtÞ þ iA1 xði1Þ ðt sÞ þ if ði1Þ Sk
ðiÞ
Pðtf Þ ¼ CT Qf C; P1 ðtf Þ ¼ CT Qf H; gi ðtf Þ ¼ 0
ðtÞ ð14Þ
The proof is completed.
ð23Þ
h
and k_ ð0Þ ðtÞ ¼ CT QCxð0Þ ðtÞ þ AT kð0Þ ðtÞ CT QHzðtÞ Xi1 ðik1Þ ðkÞ k ðtÞ k_ ðiÞ ðtÞ ¼ CT QCxðiÞ ðtÞ þ AT kðiÞ ðtÞ þ i k¼0 C ki1 f xT ( iAT1 kði1Þ ðt þ sÞ; 0 6 t 6 tf s þ 0; tf s 6 t 6 tf ð15Þ and uðiÞ ðtÞ ¼ R1 BT kðiÞ ðtÞ
ð16Þ
Let kð0Þ ðtÞ ¼ PðtÞxð0Þ ðtÞ þ P1 ðtÞzðtÞ;
kðiÞ ðtÞ
¼ PðtÞxðiÞ ðtÞ þ gi ðtÞ Substituting (17) into (14), one gets
ð17Þ
Remark 1. The result of Theorem 1 can be applied to nonlinear systems with known time-varying delay, since xði1Þ ðt sÞ and gi1 ðt þ sÞ in the ith step iteration of differential equations (13) and (14) are known nonhomogeneous terms. In optimal control law u*(t) (9), the series term cannot be obtained generally. In practice, we can use a finite sum of the series to approximate the series term, i.e., we can obtain an Nth suboptimal tracking control law as follows: " # N X 1 1 T ð24Þ g ðtÞ uN ðtÞ ¼ R B PðtÞxðtÞ þ P1 ðtÞzðtÞ þ i! i i¼0 By (24), we can obtain an iteration algorithm of the Nth suboptimal tracking control law.
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G. Tang, M. Fan / Progress in Natural Science 18 (2008) 1571–1576
u0 ðtÞ ¼ R1 BT ðPðtÞxðtÞ þ P1 ðtÞzðtÞÞ 1 ui ðtÞ ¼ ui1 ðtÞ R1 BT gi ðtÞ; i ¼ 1; 2; ; N i!
^ ^ yðtÞ; _ wðtÞ ¼ FwðtÞ þ H~ ð25Þ
In practice, the iteration times N can be chosen properly according to the demand of control precision. By taking the performance index relative precision as the error control standard, a practical algorithm solution of the Nth suboptimal tracking control law in limited time domain is presented as follows: Algorithm 1 (Find the suboptimal tracking control law un(t))). Step 1: Give0< r < 1, Let i: = 0 and J 1 ¼ 1. Solve P(t) and P1(t) from (11) and (12), respectively. Step 2: Get z(t) from (2) and calculate the reference input ~ yðtÞ. Step 3: Obtain ui(t) from (25). Then get the closed-loop system output y(t) by substituting ui(t) into system (1). Calculate Ji from 1 1 J i ¼ eTi ðtf ÞQf ei ðtf Þ þ 2 2
Z
tf
eTi ðtÞQei ðtÞ þ uTi ðtÞRui ðtÞ dt
0
ð26Þ Step 4: If jðJ i J i1 Þ=J i j < r, then N = i, and output the suboptimal tracking control law uN(t) and stop. Otherwise, let i: = i + 1, and go to step 3.
^z2 ðtÞ ¼ wðtÞ þ K~yðtÞ
is the state vector of the reduced-order obwhere w 2 R ^z2 ðtÞ is the observed value of server, ^ ¼ F2 KF12 ; H ^ ¼ F2 K KF12 K þ F21 KF1 ; K is z2 ðtÞ; F an unknown coefficient matrix. In order to guarantee high convergence speed of the reduced-order observer, we may ^ are asselect a suitable K such that all the eigenvalues of F signed to appointed places. From (27) and (29), we can obtain the observed value of z(t): ^zðtÞ ¼ T2 wðtÞ þ ðT1 þ T2 KÞ~yðtÞ
ð30Þ
And the OTC law with a corresponding reduced-order observer is obtained as follows: ^ ^ yðtÞ _ wðtÞ ¼ FwðtÞ þ H~ u ðtÞ ¼ R1 BT ½PðtÞxðtÞ þ P1 ðtÞT2 wðtÞ 1 X 1 gi ðtÞ þP1 ðtÞðT1 þ T2 KÞ~yðtÞ þ i! i¼0
# ð31Þ
Remark 2. As a result of the OTC law (31) with a corresponding reduced-order observer, the OTC law (31) is not the optimal one. But we can select a suitable observer plus K to approximate the optimal tracking control law according to the preassigned control precision. In practical applications, we generally use a positive integer N to replace the 1 in (31). Consequently, we can obtain a tracking control law as
4. Problem of physical realization The OTC law in (24) contains the state vector z(t) of exosystem (2), which is physically unrealizable. In this section, we will construct a reduced-order reference input observer for it. As we all know, for full rank matrix H, there must be a constant matrix L 2 RðmqÞm that makes the matrix ½HT LT 2 Rmm nonsingular. Denote that 1 F1 F12 H ð27Þ ¼ ½ T1 T2 ; T1 FT ¼ T¼ F21 F2 L
^ ^ yðtÞ _ ¼ FwðtÞ wðtÞ þ H~ " 1
T
uN ðtÞ ¼ R B
PðtÞxðtÞ þ P1 ðtÞT2 wðtÞ þ P1 ðtÞðT1 þ T2 KÞ~yðtÞ þ
N X 1 i¼0
i!
# gi ðtÞ ð32Þ
As (25), an iteration formula calculating uN(t) in (32) is given as
where T1 2 Rmq ; T2 2 RmðmqÞ ; F1 2 Rqq ; F12 2 Rq ðm qÞ; F21 2 RðmqÞq ; F2 2 RðmqÞðmqÞ are constant matrices. Introducing a nonsingular linear transformation z ¼ Tz, where zT ¼ ½zT1 zT2 ; z1 2 Rq ; z2 2 Rmq , exosystem (2) can be described as z_ 1 ðtÞ ¼ F1z1 ðtÞ þ F12z2 ðtÞ; ~ yðtÞ ¼ z1 ðtÞ
ð29Þ
mq
z_ 2 ðtÞ ¼ F21z1 ðtÞ þ F2z2 ðtÞ ð28Þ
yðtÞ. So, we only need Note that z1 ðtÞ is the reference input ~ to construct a reduced-order observer with respect to z2 ðtÞ. Noting that HT = [Iq, 0] holds and (F,H) is observable, obviously (F2,F21) is also observable. Therefore, we can construct the reduced-order observer as follows: Fig. 1. Simulation curves of the output error e(t).
G. Tang, M. Fan / Progress in Natural Science 18 (2008) 1571–1576
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Table 1 The order of performance and control precision Order i
0
1
2
3
Ji jðJ i J i1 Þ=J i j
23.2608 /
20.6999 0.1101
20.1541 0.0264
20.1372 0.0008385
Table 2 N value and optimal performance index JN at different s Time-delay s 0.1 N JN
Fig. 2. Simulation curves of the control law u(t).
^ ^ yðtÞ _ ¼ FwðtÞ wðtÞ þ H~ u0 ðtÞ ¼ R1 BT ½PðtÞxðtÞ þ P1 ðtÞT2 wðtÞ þ P1 ðtÞðT1 þ T2 KÞ~ yðtÞ 1 1 T ui ðtÞ ¼ ui1 ðtÞ R B gi ðtÞ; i ¼ 1; 2; ;N i ð33Þ
5. Numerical examples Consider a nonlinear system with time-delay described by (1), where " # " # " # " # 3 0 0 1 0 0 ; ; C¼ ; A1 ¼ ; B¼ A¼ 0 1 2 3 0:1 2 " # " # 0:2x1 x2 1 fðxÞ ¼ ð34Þ ; /ðtÞ ¼ 0:3x22 0 The reference input described by the system (2), where 1 1 1 0 F¼ ; H¼ ; z0 ¼ ð35Þ 2 3 0 1 The performance index is selected as Z 1 10 2 J ¼ 2e2 ð10Þ þ ð4e ðtÞ þ u2 ðtÞÞdt 2 0
0.5
1
1.5
2
2.5
3
3 3 3 3 3 3 3 18.6806 17.9340 20.1372 21.1858 21.5409 21.6688 21.7161
Case 2. When time-delay s is taken different values, the order N that satisfies the preassigned control precision and the corresponding JN are listed in Table 2. From Table 2, it can be seen that the performance index values change as time-delay selects different values. But the order for suboptimal tracking control law remains stable, which satisfies the demand of the control precision. It shows that the presented algorithm is effective at both little time-delays and big time-delays. 6. Conclusion This paper has developed an approximate approach solving the two-point boundary value (TPBV) problem with both time-advance and time-delay terms, derived from OTC problem for nonlinear systems with time-delay. By introducing a parameter variable e, we eliminated the advance and delay terms and obtained an approximate approach to the suboptimal tracking control law of nonlinear system with time-delay, namely series-based approximate approach. This algorithm can solve the trouble for obtaining the optimal tracking control law of nonlinear system with time-delay. Simulation examples have shown that the sensitivity approach to designing the suboptimal tracking control law of nonlinear system with time-delay is effective. Acknowledgements
ð36Þ
and the control precision as a = 0.01. When jðJ i J i1 Þ=J i j 6 a, we consider that the tracking control law satisfies the demand of the control precision. Case 1. When time-delay s = 1, the simulation curves of the system tracking error e(t) and the OOTC law u(t) are presented in Figs. 1 and 2, and the performance index values at different orders are listed in Table 1. From Table 1, it can be seen that the performance index values decrease as order increases, and tend to a stable optimal performance index J* ultimately. When i = 3, it can satisfy the demand of the control precision.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 60574023 and 40776051) and the Key Natural Science Foundation of Shandong Province (Grant No. Z2005G01). References [1] Yue D, Lam J. Reliable memory feedback design for a class of nonlinear time-delay systems. Int J Robust Nonlinear Control 2004;14:39–60. [2] Cimen T, Banks SP. Nonlinear optimal tracking control with application to super-tankers for autopilot design. Automatica 2004;40:1845–63. [3] Shamma JS, Cloutier JR. Existence of SDRE stabilizing feedback. IEEE Trans Automatic Control 2003;48:513–7.
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[4] Tang GY. Suboptimal control for nonlinear systems: a successive approximation approach. Syst Control Lett 2005;54:429–34. [5] Sharma R, Tewari A. Optimal nonlinear tracking of spacecraft attitude maneuvers. IEEE Trans Control Syst Technol 2004;12:677–82. [6] Xin M, Balakrishnan SN. Missile longitudinal autopilot design using a new suboptimal nonlinear control method. IEE Proc Control Theory Appl 2003;150:577–84. [7] Beard RW, Saridis GN, Wen JT. Galerkin approximations of the generalized Hamilton–Jacobi–Bellman equation. Automatica 1997;33:2159–77. [8] Cimen T, Banks SP. Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria. Syst Control Lett 2004;53:327–46.
[9] Tang GY, Wang HH. Successive approximation approach of optimal control for nonlinear discrete-time systems. Int J Syst Sci 2005;36:153–61. [10] Tang GY, Wang HH. Suboptimal control for discrete linear systems with time-delay: a successive approximation approach. ACTA Automatica Sinica 2005;31:419–26. [11] Tang GY, Ma H. Successive-approximation approach of optimal control for bilinear discrete-time systems. IEE Proc Control Theory Appl 2005;152:639–44. [12] Carevic D. Tracking target in cluttered environment using multilateral time-delay measurements. J Acoust Soc America 2004;115:1198–206.
Progress in Natural Science 18 (2008) 1571–1576 www.elsevier.com/locate/pnsc
Series-based approximate approach of optimal tracking control for nonlinear systems with time-delay Gongyou Tang a,*, Mingqu Fan a,b b
a College of Information Science and Engineering, Ocean University of China, Qingdao 266100, China College of Information and Engineering, Shandong University of Science and Technology, Jinan 250031, China
Received 16 February 2008; received in revised form 12 March 2008; accepted 12 March 2008
Abstract The optimal output tracking control (OTC) problem for nonlinear systems with time-delay is considered. Using a series-based approximate approach, the original OTC problem is transformed into iteration solving linear two-point boundary value problems without timedelay. The OTC law obtained consists of analytical linear feedback and feedforward terms and a nonlinear compensation term with an infinite series of the adjoint vectors. By truncating a finite sum of the adjoint vector series, an approximate optimal tracking control law is obtained. A reduced-order reference input observer is constructed to make the feedforward term physically realizable. Simulation examples are used to test the validity of the series-based approximate approach. Ó 2008 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in China Press. All rights reserved. Keywords: Nonlinear systems; Time-delay; Tracking control; Optimal control; Observer
1. Introduction Strictly speaking, almost all the control systems are of nonlinearity. And time-delay is a common phenomenon in real control system. Analysis and synthesis for nonlinear systems with time-delay has been one of the most active research areas in the past decades. Many elegant approaches have been provided and applied to practical engineering. For example, Yue and Lam [1] presented a design approach of reliable memory controller for nonlinear time-delay systems; Cimen and Banks [2] introduced the design of optimal tracking controllers for a general class of nonlinear systems; Shamma and Cloutier [3] gave a state-dependent Riccati equation (SDRE) approach to nonlinear system’s stabilization problems. The optimal control problems for nonlinear systems with time-delay * Corresponding author. Tel.: +86 532 66781230; fax: +86 532 85901980. E-mail address: [email protected] (G. Tang).
generally lead to a nonlinear two-point boundary value (TPBV) problem involving both time-delay and timeadvance terms. With the exception of simplest cases, the analytical solution to this TPBV problem does not exist [4]. Therefore, it is meaningful to obtain some approximate solutions of the optimal control problem. Recently, many good results in the approximate approaches of the optimal control for nonlinear and/or time-delay systems have been obtained. One approach is SDRE that uses direct parameterization to transform the nonlinear system into a linear structure with state-dependent coefficients and to solve the SDRE at each point of the state variable along the trajectory to obtain a nonlinear feedback controller of general form [5,6]. The second type of approach is successive Galerkin approximation (SGA), where an iterative process is used to find a sequence of approximations approaching the solution of the HJB equation [7]. The third type of approach is an approximating sequence of Riccati equations (ASREs) to construct nonlinear time-varying optimal state-feedback controllers for
1002-0071/$ - see front matter Ó 2008 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in China Press. All rights reserved. doi:10.1016/j.pnsc.2008.03.033
1572
G. Tang, M. Fan / Progress in Natural Science 18 (2008) 1571–1576
_ ¼ AxðtÞ þ A1 xðt sÞ þ fðxðtÞÞ SkðtÞ; 0 < t 6 tf xðtÞ _ ¼ CT QCxðtÞ CT QHzðtÞ þ AT kðtÞ kðtÞ ( AT1 kðt þ sÞ; 0 6 t 6 tf s þ f XT ðxðtÞÞkðtÞ þ 0; tf s 6 t 6 tf
the nonlinear systems [8]. The fourth type of approach is a successive approximate approach (SAA) to solve the optimal control problem for nonlinear systems [4,9–11]. In the recent decades, the OTC problem has been receiving growing attention owing to its wide practical applications, such as the tracking control of spacecraft attitude maneuvers [5], super tanker [3], and the underwater target tracking [12]. The purpose of this paper is to develop a series-based approximate approach solving the OTC problems, where a parameter variable e is introduced and this function is expanded into Maclaurin series with respect to the parameter variable at = 0. The original OTC problem is transformed into a sequence of nonhomogeneous linear TPBV problems without time-delay. This approach only needs to solve a sequence of vector differential equations, instead of solving a sequence of matrix differential equations. That means its computation load is much lower in comparison with the other approximate approaches [5–8].
ð4Þ where S ¼ BR B ; f xT ðxÞ ¼ ofðxÞ=ox . And the OTC law can be written as
2. Problem statement
3. Design of OTC law
Consider nonlinear systems with time-delay as the following:
For the TPBV problem (4), we introduce a real scalar parameter variable ee[0,1]. And a new TPBV problem with parameter variable e is constructed as follows: _ eÞ ¼ Axðt; eÞ þ eA1 xðt s; eÞ þ efðxðt; eÞÞ Skðt; eÞ; xðt;
_ ¼ AxðtÞ þ A1 xðt sÞ þ BuðtÞ þ fðxðtÞÞ; 0 < t 6 tf xðtÞ xðtÞ ¼ /ðtÞ; s 6 t 6 0 yðtÞ ¼ CxðtÞ ð1Þ n
r
q
where x 2 R ; u 2 R , and yðtÞ 2 R are state, control, and output, respectively; A; A1 ; B and C are constant matrices of appropriate dimensions; s > 0 is the time-delay, f : Rn ! U Rn ; fð0Þ ¼ 0, which satisfies the Lipschitz conditions on Rn ; /ðtÞ 2 Rn is the initial state. Assume that ðA þ A1 ; BÞ is completely observable. The reference input ~ y 2 Rq to be tacked by the output y(t) is described by the exosystem z_ ¼ Fz;
~ y ¼ Hz
ð2Þ
m
where z 2 R ; F; H are constant matrices of appropriate dimensions. Assume that (F,H) is completely observable. The quadratic performance index is 1 1 J ¼ eT ðtf ÞQf eðtf Þ þ 2 2 Z tf T e ðtÞQeðtÞ þ uT tÞRuðtÞ dt;
ð3Þ
0
where eðtÞ ¼ ~ yðtÞ yðtÞ is the output tracking error; Qf,Q,R are positive definite matrices of appropriate dimensions. The OTC problem is to find a control law u ðÞ, which minimizes the quadratic performance index (3) subject to constraints (1) and (2). According to the necessary conditions of the OTC problem based on the maximum principle, the following TPBV problem can be obtained:
kðtf Þ ¼ CT Qf Cxðtf Þ CT Qf Hzðtf Þ; xðtÞ ¼ /ðtÞ; s 6 t 6 0 1
T
T
u ðtÞ ¼ R1 BT kðtÞ:
ð5Þ
Note that (4) is a nonlinear TPBV problem with both timedelay and time-advance. Unfortunately, the analytical solution of this nonlinear TPBV problem does not generally exist. Therefore, it is necessary to find approximation approaches for solving the optimal control problem for the discrete-time bilinear systems.
0 < t 6 tf _ kðt; eÞ ¼ CT QCxðt; eÞ CT QHzðtÞ þ AT kðt; eÞ ( eAT1 kðt þ s; eÞ; 0 6 t 6 tf s þ ef xT ðxðt; eÞÞkðt; eÞ þ 0; tf s 6 t 6 tf kðtf ; eÞ ¼ CT Qf Cxðtf Þ CT Qf Hzðtf Þ; xðt; eÞ ¼ /ðtÞ;
s 6 t 6 0
ð6Þ
In the following discussion, we suppose that uðt; eÞ; xðt; eÞ; kðt; eÞ; fðxðt; eÞÞ; f xT ðxðt; eÞÞ are all infinitely differentiable with respect toee[0,1], and their Maclaurin series 1 X i ðiÞ ðÞ ðtÞ ð7Þ ðÞðt; Þ ¼ i! i¼0 ðiÞ
are all convergent at e = 1, where ðÞ ðtÞ ¼ oi ðÞ=oi j¼0 Obviously, when e = 1, the TPBV problem (6) is equal to the TPBV problem (5) with parameter variable e, and the OTC law can be written in the following form: 1 X 1 ðiÞ ð8Þ u ðtÞ u ðtÞ ¼ uðt; 1Þ ¼ i! i¼0
Theorem 1. Consider the OTC problems (1)–(4), the OTC can be described by " # 1 X 1 1 T ð9Þ g ðtÞ u ðtÞ ¼ R B PðtÞxðtÞ þ P1 ðtÞzðtÞ þ i! i i¼0
G. Tang, M. Fan / Progress in Natural Science 18 (2008) 1571–1576
where P(t) is the unique positive semi-definite solution of the following Riccati matrix differential equation: P_ 1 ðtÞ ¼ PðtÞA þ A PðtÞ þ C QC PðtÞSPðtÞ T
T
P_ 1 ðtÞ ¼ P1 ðtÞF þ AT P1 ðtÞ CT QH PðtÞSP1 ðtÞ T
P1 ðtf Þ ¼ C Qf H
ð11Þ
gi(t) is the solution of the ith adjoint vector differential equation g_ i ðtÞ ¼ ðAT PðtÞSÞgi ðtÞ þ iPðtÞA1 xði1Þ ðt sÞ i1 X
ðik1Þ
C ki1 f xT
i AT1 ðPðtÞxði1Þ ðt þ sÞ þ gi1 ðt þ sÞÞ; 0 6 t 6 tf s 0; tf s < t 6 tf
gi ðtf Þ ¼ 0; i ¼ 1; 2;
ð12Þ
x(i)(t)is the solution of the ith state equation
s 6 t 6 0
ð19Þ
Substituting (17) into (18) and comparing with (15), we obtain _ ¼ PðtÞA þ AT PðtÞ þ CT QC PðtÞSPðtÞ; PðtÞ P_ 1 ðtÞ ¼ P1 ðtÞF þ AT P1 ðtÞ CT QH PðtÞSP1 ðtÞ
k¼0
( þ
x_ ðiÞ ðtÞ ¼ ðA SPðtÞÞxðiÞ ðtÞ þ iA1 xði1Þ ðt sÞ þ if ði1Þ SðtÞgi ðtÞ xðiÞ ðtÞ ¼ 0; t0 s 6 t 6 t0 ; i ¼ 1; 2;
iAT1 ðPðtÞxði1Þ ðt þ sÞ þ gi1 ðt þ sÞÞ;0 6 t 6 tf s ð20Þ 0; tf s < t 6 tf
We now ascertain the boundary-value conditions of (20). The third equation of (6) implies kðtf ; 1Þ ¼
x_ ð0Þ ðtÞ ¼ ðA SPðtÞÞxð0Þ ðtÞ SP1 ðtÞzðtÞ; xð0Þ ðtÞ ¼ /ðtÞ
1 X 1 ðiÞ k ðtf Þ i! i¼0
¼ CT Qf Cxðtf Þ CT Qf Hzðtf Þ
ð21Þ
Substituting (17) into (21), it follows ð13Þ
Proof. By substituting (7) into (4) and (6) separately and by comparing the coefficients of the same order terms with respect to ei ði ¼ 0; 1; 2; Þ, we obtain ð0Þ
xð0Þ ðtÞ ¼ /ðtÞ; xðiÞ ðtÞ ¼ 0;
ðPðtÞxðkÞ ðtÞ þ gk ðtÞÞ
k¼0
x_ ð0Þ ðtÞ ¼ Axð0Þ ðtÞ Sk
ð18Þ
g_ i ðtÞ ¼ ðAT PðtÞSÞgi ðtÞ þ iPðtÞA1 xði1Þ ðt sÞ þ iPðtÞf ði1Þ i1 h i X ðik1Þ þi C ki1 f Tx PðtÞxðkÞ ðtÞ þ gkðtÞ
g0 ðtÞ ¼ 0
þ
x_ ðiÞ ðtÞ ¼ ðA SPðtÞÞxðiÞ ðtÞ þ iA1 xði1Þ ðt sÞ þ if ði1Þ Sgi ðtÞ
From (1) and (7), initial conditions of (18) are taken as
P1(t) is the unique solution of the following matrix differential equation:
(
x_ ð0Þ ðtÞ ¼ ðA SPðtÞÞxð0Þ ðtÞ þ P1 zðtÞ
ð10Þ
Pðtf Þ ¼ CT Qf C
þ iPðtÞf ði1Þ þ i
1573
Pðtf Þxð0Þ ðtf Þ þ P1 ðtf Þzðtf Þ ¼ CT Qf Cxðtf Þ CT Qf Hzðtf Þ Pðtf ÞxðiÞ ðtf Þ þ gi ðtf Þ ¼ 0 ð22Þ From (22) the boundary-value conditions of PðtÞ; P1 ðtÞ and gi(t) are taken as
ðtÞ
x_ ðiÞ ðtÞ ¼ AxðiÞ ðtÞ þ iA1 xði1Þ ðt sÞ þ if ði1Þ Sk
ðiÞ
Pðtf Þ ¼ CT Qf C; P1 ðtf Þ ¼ CT Qf H; gi ðtf Þ ¼ 0
ðtÞ ð14Þ
The proof is completed.
ð23Þ
h
and k_ ð0Þ ðtÞ ¼ CT QCxð0Þ ðtÞ þ AT kð0Þ ðtÞ CT QHzðtÞ Xi1 ðik1Þ ðkÞ k ðtÞ k_ ðiÞ ðtÞ ¼ CT QCxðiÞ ðtÞ þ AT kðiÞ ðtÞ þ i k¼0 C ki1 f xT ( iAT1 kði1Þ ðt þ sÞ; 0 6 t 6 tf s þ 0; tf s 6 t 6 tf ð15Þ and uðiÞ ðtÞ ¼ R1 BT kðiÞ ðtÞ
ð16Þ
Let kð0Þ ðtÞ ¼ PðtÞxð0Þ ðtÞ þ P1 ðtÞzðtÞ;
kðiÞ ðtÞ
¼ PðtÞxðiÞ ðtÞ þ gi ðtÞ Substituting (17) into (14), one gets
ð17Þ
Remark 1. The result of Theorem 1 can be applied to nonlinear systems with known time-varying delay, since xði1Þ ðt sÞ and gi1 ðt þ sÞ in the ith step iteration of differential equations (13) and (14) are known nonhomogeneous terms. In optimal control law u*(t) (9), the series term cannot be obtained generally. In practice, we can use a finite sum of the series to approximate the series term, i.e., we can obtain an Nth suboptimal tracking control law as follows: " # N X 1 1 T ð24Þ g ðtÞ uN ðtÞ ¼ R B PðtÞxðtÞ þ P1 ðtÞzðtÞ þ i! i i¼0 By (24), we can obtain an iteration algorithm of the Nth suboptimal tracking control law.
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G. Tang, M. Fan / Progress in Natural Science 18 (2008) 1571–1576
u0 ðtÞ ¼ R1 BT ðPðtÞxðtÞ þ P1 ðtÞzðtÞÞ 1 ui ðtÞ ¼ ui1 ðtÞ R1 BT gi ðtÞ; i ¼ 1; 2; ; N i!
^ ^ yðtÞ; _ wðtÞ ¼ FwðtÞ þ H~ ð25Þ
In practice, the iteration times N can be chosen properly according to the demand of control precision. By taking the performance index relative precision as the error control standard, a practical algorithm solution of the Nth suboptimal tracking control law in limited time domain is presented as follows: Algorithm 1 (Find the suboptimal tracking control law un(t))). Step 1: Give0< r < 1, Let i: = 0 and J 1 ¼ 1. Solve P(t) and P1(t) from (11) and (12), respectively. Step 2: Get z(t) from (2) and calculate the reference input ~ yðtÞ. Step 3: Obtain ui(t) from (25). Then get the closed-loop system output y(t) by substituting ui(t) into system (1). Calculate Ji from 1 1 J i ¼ eTi ðtf ÞQf ei ðtf Þ þ 2 2
Z
tf
eTi ðtÞQei ðtÞ þ uTi ðtÞRui ðtÞ dt
0
ð26Þ Step 4: If jðJ i J i1 Þ=J i j < r, then N = i, and output the suboptimal tracking control law uN(t) and stop. Otherwise, let i: = i + 1, and go to step 3.
^z2 ðtÞ ¼ wðtÞ þ K~yðtÞ
is the state vector of the reduced-order obwhere w 2 R ^z2 ðtÞ is the observed value of server, ^ ¼ F2 KF12 ; H ^ ¼ F2 K KF12 K þ F21 KF1 ; K is z2 ðtÞ; F an unknown coefficient matrix. In order to guarantee high convergence speed of the reduced-order observer, we may ^ are asselect a suitable K such that all the eigenvalues of F signed to appointed places. From (27) and (29), we can obtain the observed value of z(t): ^zðtÞ ¼ T2 wðtÞ þ ðT1 þ T2 KÞ~yðtÞ
ð30Þ
And the OTC law with a corresponding reduced-order observer is obtained as follows: ^ ^ yðtÞ _ wðtÞ ¼ FwðtÞ þ H~ u ðtÞ ¼ R1 BT ½PðtÞxðtÞ þ P1 ðtÞT2 wðtÞ 1 X 1 gi ðtÞ þP1 ðtÞðT1 þ T2 KÞ~yðtÞ þ i! i¼0
# ð31Þ
Remark 2. As a result of the OTC law (31) with a corresponding reduced-order observer, the OTC law (31) is not the optimal one. But we can select a suitable observer plus K to approximate the optimal tracking control law according to the preassigned control precision. In practical applications, we generally use a positive integer N to replace the 1 in (31). Consequently, we can obtain a tracking control law as
4. Problem of physical realization The OTC law in (24) contains the state vector z(t) of exosystem (2), which is physically unrealizable. In this section, we will construct a reduced-order reference input observer for it. As we all know, for full rank matrix H, there must be a constant matrix L 2 RðmqÞm that makes the matrix ½HT LT 2 Rmm nonsingular. Denote that 1 F1 F12 H ð27Þ ¼ ½ T1 T2 ; T1 FT ¼ T¼ F21 F2 L
^ ^ yðtÞ _ ¼ FwðtÞ wðtÞ þ H~ " 1
T
uN ðtÞ ¼ R B
PðtÞxðtÞ þ P1 ðtÞT2 wðtÞ þ P1 ðtÞðT1 þ T2 KÞ~yðtÞ þ
N X 1 i¼0
i!
# gi ðtÞ ð32Þ
As (25), an iteration formula calculating uN(t) in (32) is given as
where T1 2 Rmq ; T2 2 RmðmqÞ ; F1 2 Rqq ; F12 2 Rq ðm qÞ; F21 2 RðmqÞq ; F2 2 RðmqÞðmqÞ are constant matrices. Introducing a nonsingular linear transformation z ¼ Tz, where zT ¼ ½zT1 zT2 ; z1 2 Rq ; z2 2 Rmq , exosystem (2) can be described as z_ 1 ðtÞ ¼ F1z1 ðtÞ þ F12z2 ðtÞ; ~ yðtÞ ¼ z1 ðtÞ
ð29Þ
mq
z_ 2 ðtÞ ¼ F21z1 ðtÞ þ F2z2 ðtÞ ð28Þ
yðtÞ. So, we only need Note that z1 ðtÞ is the reference input ~ to construct a reduced-order observer with respect to z2 ðtÞ. Noting that HT = [Iq, 0] holds and (F,H) is observable, obviously (F2,F21) is also observable. Therefore, we can construct the reduced-order observer as follows: Fig. 1. Simulation curves of the output error e(t).
G. Tang, M. Fan / Progress in Natural Science 18 (2008) 1571–1576
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Table 1 The order of performance and control precision Order i
0
1
2
3
Ji jðJ i J i1 Þ=J i j
23.2608 /
20.6999 0.1101
20.1541 0.0264
20.1372 0.0008385
Table 2 N value and optimal performance index JN at different s Time-delay s 0.1 N JN
Fig. 2. Simulation curves of the control law u(t).
^ ^ yðtÞ _ ¼ FwðtÞ wðtÞ þ H~ u0 ðtÞ ¼ R1 BT ½PðtÞxðtÞ þ P1 ðtÞT2 wðtÞ þ P1 ðtÞðT1 þ T2 KÞ~ yðtÞ 1 1 T ui ðtÞ ¼ ui1 ðtÞ R B gi ðtÞ; i ¼ 1; 2; ;N i ð33Þ
5. Numerical examples Consider a nonlinear system with time-delay described by (1), where " # " # " # " # 3 0 0 1 0 0 ; ; C¼ ; A1 ¼ ; B¼ A¼ 0 1 2 3 0:1 2 " # " # 0:2x1 x2 1 fðxÞ ¼ ð34Þ ; /ðtÞ ¼ 0:3x22 0 The reference input described by the system (2), where 1 1 1 0 F¼ ; H¼ ; z0 ¼ ð35Þ 2 3 0 1 The performance index is selected as Z 1 10 2 J ¼ 2e2 ð10Þ þ ð4e ðtÞ þ u2 ðtÞÞdt 2 0
0.5
1
1.5
2
2.5
3
3 3 3 3 3 3 3 18.6806 17.9340 20.1372 21.1858 21.5409 21.6688 21.7161
Case 2. When time-delay s is taken different values, the order N that satisfies the preassigned control precision and the corresponding JN are listed in Table 2. From Table 2, it can be seen that the performance index values change as time-delay selects different values. But the order for suboptimal tracking control law remains stable, which satisfies the demand of the control precision. It shows that the presented algorithm is effective at both little time-delays and big time-delays. 6. Conclusion This paper has developed an approximate approach solving the two-point boundary value (TPBV) problem with both time-advance and time-delay terms, derived from OTC problem for nonlinear systems with time-delay. By introducing a parameter variable e, we eliminated the advance and delay terms and obtained an approximate approach to the suboptimal tracking control law of nonlinear system with time-delay, namely series-based approximate approach. This algorithm can solve the trouble for obtaining the optimal tracking control law of nonlinear system with time-delay. Simulation examples have shown that the sensitivity approach to designing the suboptimal tracking control law of nonlinear system with time-delay is effective. Acknowledgements
ð36Þ
and the control precision as a = 0.01. When jðJ i J i1 Þ=J i j 6 a, we consider that the tracking control law satisfies the demand of the control precision. Case 1. When time-delay s = 1, the simulation curves of the system tracking error e(t) and the OOTC law u(t) are presented in Figs. 1 and 2, and the performance index values at different orders are listed in Table 1. From Table 1, it can be seen that the performance index values decrease as order increases, and tend to a stable optimal performance index J* ultimately. When i = 3, it can satisfy the demand of the control precision.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 60574023 and 40776051) and the Key Natural Science Foundation of Shandong Province (Grant No. Z2005G01). References [1] Yue D, Lam J. Reliable memory feedback design for a class of nonlinear time-delay systems. Int J Robust Nonlinear Control 2004;14:39–60. [2] Cimen T, Banks SP. Nonlinear optimal tracking control with application to super-tankers for autopilot design. Automatica 2004;40:1845–63. [3] Shamma JS, Cloutier JR. Existence of SDRE stabilizing feedback. IEEE Trans Automatic Control 2003;48:513–7.
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