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A discrete-time iterative learning algorithm for linear time-varying systems
Engineering Applications of Artificial Intelligence 16 (2003) 185–190
Brief Paper
A discrete-time iterative learning algorithm for linear time-varying systems K.K. Tana, S.N. Huanga,b,*, T.H. Leea, S.Y. Limb a
Mechatronics and Automation Lab., Department of Electrical and Computer Engineering, 10 Kent Ridge Crescent, National University of Singapore, Singapore 119260, Singapore b Mechatronics Group, Singapore Institute of Manufacturing Technology, Singapore
Abstract In this paper, an iterative learning algorithm (ILC) is presented for a MIMO linear time-varying system. We consider the convergence of the algorithm. A necessary and sufficient condition for the existence of convergent algorithm is stated. Then, we prove that the same condition is sufficient for the robustness of the proposed learning algorithm against state disturbance, output measurement noise, and reinitialization error. Finally, a simulation example is given to illustrate the results. r 2003 Elsevier Ltd. All rights reserved. Keywords: Iterative learning control; Repetitive systems; Linear discrete systems; Uncertain systems
1. Introduction The iterative learning control (ILC) method has been proposed by Arimoto et al. (1984). Examples of this idea can be found in Arimoto (1990), Bien and Huh (1989), Lucibello (1992), Yamada et al. (1994), Lee et al. (2000) and Tan et al. (2001), including the general area of trajectory following in robotics. The specified task is regarded as the tracking of a given output trajectory for an operation on a specified time interval. The objective of ILC is to use the repetitive nature of the process to improve the accuracy with which the operation is achieved by generating the present control input based on the previous control history and a learning mechanism. The monograph by Moore (1992) contains more detail on the background. A recent book (Chen and Wen, 1999) about ILC surveys the literature until 1998. For real implementation of an iterative learning controller, it is more practical to design and analyze the ILC systems in discrete-time domain. The optimality of the discrete ILC is studied in Togai and Yamano (1985). Subsequently, it has been proven that the tracking error in Togai and Yamano (1985) for the linear time-invariant system will converge to zero if and *Corresponding author. Mechatronics and Automation Lab., Department of Electrical and Computer Engineering, National University of Singapore, Singapore. E-mail address: [email protected] (S.N. Huang). 0952-1976/03/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0952-1976(03)00030-7
only if the input–output coupling matrices are full row rank (Kurek and Zaremba, 1993). In Saab (1995), another necessary and sufficient condition for the linear time-invariant system is proposed. The robustness is also discussed in Saab (1995). In Huang et al. (2001), a sufficient condition for the linear time-varying system is given based on the quadratic criterion. None of the papers involving linear time-varying systems considers the necessary and sufficient condition for convergence. In this paper, we deal with the iterative learning control problem for a multivariable linear time-varying system. We consider the convergence of the learning algorithm, and present a sufficient and necessary condition for the existence of convergent algorithm. Furthermore, this condition proved in this paper is sufficient for the robustness of the proposed iterative learning algorithm. The robustness bounds of the learning system are obtained by using 2-norm rather than traditional lnorm as in Arimoto (1990) for convergence analysis.
2. Necessary and sufficient condition for convergence Consider a repetitive linear discrete-time system as follows: xi ðt þ 1Þ ¼ AðtÞxi ðtÞ þ BðtÞui ðtÞ; yi ðtÞ ¼ CðtÞxi ðtÞ;
ð1Þ
K.K. Tan et al. / Engineering Applications of Artificial Intelligence 16 (2003) 185–190
186
and i denotes the ith repetitive operation of the system; xi ðtÞARn ; ui ðtÞARm ; and yi ðtÞARr are the state, control input, and output of the system, respectively; tA½0; N is the time; and AðtÞ; BðtÞ; and CðtÞ are matrices with appropriate dimensions. Assume that, at each iteration, the control system (1) is exactly re-initialized at x0 ; i.e., xi ð0Þ ¼ x0 :
The following learning law is used Dui ðtÞ ¼ KðtÞei1 ðt þ 1Þ;
t ¼ 0; 1; y; N 1;
where fKð0Þ; Kð1Þ; y; KðN 1Þg is a constant sequence. Lemma 1 (Mosca, 1995). The time-invariant dynamic linear system zðk þ 1Þ ¼ FzðkÞ
Definition 1 (Kurek and Zaremba, 1993). A learning rule is said to be convergent if, and only if (iff) for any x0 ARn it generates a control sequence fui ðkÞ; k ¼ 0; 1; y; N 1g for system (1) such that yi ðtÞ-yd ðtÞ; t ¼ 1; 2; y; N; as i-N: To determine how the error evolves with repetitions, we need to solve Eq. (1), that is, xi ðt þ 1Þ ¼
t Y
AðkÞxi ð0Þ þ
k¼0
t Y
AðkÞBð0Þui ð0Þ
k¼1
þ ? þ AðtÞBðt 1Þui ðt 1Þ þ BðtÞui ðtÞ: Let H0 ðtÞ ¼
t Y
ð2Þ
AðkÞ;
k¼0
H1 ðtÞ ¼
Y t
AðkÞBð0Þ;
k¼1
t Y
AðkÞBð1Þ; y; AðtÞBðt 1Þ ;
Theorem 1. Consider system (1) with a given desired output yd ðkÞ; k ¼ 1; 2; y; N: An iterative learning control law (9) is convergent and is a solution to the problem if and only if each matrix I Cðt þ 1ÞBðtÞKðtÞ where t ¼ 0; 1; y; N 1 is asymptotically stable, i.e., if all of its eigenvalues are inside the unit circle. Proof. Let Eðt þ 1Þ ¼ I Cðt þ 1ÞBðtÞKðtÞ where I is an unit matrix. Considering the assumption and applying the control law (9) to (8) yields ei ðt þ 1Þ ¼ ½I Cðt þ 1ÞBðtÞKðtÞei1 ðt þ 1Þ HðtÞDUi ð0; t 1Þ ¼ Eðt þ 1Þei1 ðt þ 1Þ HðtÞDUi ð0; t 1Þ:
ð3Þ ð4Þ
Thus, we have
For t ¼ 0 at the ith iteration, applying H1 ð0Þ ¼ 0; Ui ð0; 1Þ ¼ 0 to the above equation, we have ei ð1Þ ¼ Eð1Þei1 ð1Þ:
xi ðt þ 1Þ ¼ H0 ðtÞxi ð0Þ þ H1 ðtÞUi ð0; t 1Þ þ BðtÞui ðtÞ; t ¼ 0; 1; y; N 1:
ð10Þ
is asymptotically stable if and only if its state-transition matrix F is a stability matrix, i.e., jli ðFÞjo1 (for the time-invariant case, the state transition matrix is equal to F; see page 19 in the Section 2.4 of Mosca, 1995).
k¼2
Ui ð0; t 1Þ ¼ ½uTi ð0Þ; uTi ð1Þ; y; uTi ðt 1ÞT :
ð9Þ
ð5Þ
For t ¼ 1 at the ith iteration, we have ei ð2Þ ¼ Eð2Þei1 ð2Þ Hð1ÞDui ð0Þ:
ð11Þ
It is obvious that H1 ð0Þ ¼ 0; and Ui ð0; 1Þ ¼ 0: The tracking error ei ðt þ 1Þ at the ith repetition is that ei ðt þ 1Þ ¼ yd ðt þ 1Þ yi ðt þ 1Þ: From this definition, we can reach the following output error equation:
Note that the previous time control Dui ð0Þ is known, and substituting it into the above equation yields
ei ðt þ 1Þ ¼ yd ðt þ 1Þ yi1 ðt þ 1Þ
Letting
ðyi ðt þ 1Þ yi1 ðt þ 1ÞÞ ¼ ei1 ðt þ 1Þ Cðt þ 1Þðxi ðt þ 1Þ xi1 ðt þ 1ÞÞ:
E2;1 ð2Þ ¼ Hð1ÞKð0Þ
ð12Þ ð13Þ
we have ð6Þ
Substituting (5) into (6) yields
ei ð2Þ ¼ Eð2Þei1 ð2Þ þ E2;1 ð2Þei1 ð1Þ: For t ¼ 3 at the ith repetition, we have " # Dui ð0Þ ei ð3Þ ¼ Eð3Þei1 ð3Þ Hð2Þ : Dui ð1Þ
ei ðt þ 1Þ ¼ ei1 ðt þ 1Þ Cðt þ 1ÞH0 ðtÞDxi ð0Þ Cðt þ 1ÞH1 ðtÞDUi ð0; t 1Þ Cðt þ 1ÞBðtÞDui ðtÞ;
ei ð2Þ ¼ Eð2Þei1 ð2Þ Hð1ÞKð0Þei1 ð1Þ:
ð7Þ
where Dxi ð0Þ ¼ xi ð0Þ xi1 ð0Þ; DUi ð0; t 1Þ ¼ Ui ð0; t 1Þ Ui1 ð0; t 1Þ; and Dui ðtÞ ¼ ui ðtÞ ui1 ðtÞ: Letting HðtÞ ¼ Cðt þ 1ÞH1 ðtÞ we have ei ðt þ 1Þ ¼ ei1 ðt þ 1Þ Cðt þ 1ÞH0 ðtÞDxi ð0Þ HðtÞDUi ð0; t 1Þ Cðt þ 1ÞBðtÞDui ðtÞ: ð8Þ
ð14Þ
ð15Þ
With Hð2Þ partitioned into two m r blocks, we have ei ð3Þ ¼ Eð3Þei1 ð3Þ H11 ð2ÞDui ð0Þ H12 ð2ÞDui ð1Þ;
ð16Þ
where ½H11 ð2Þ H12 ð2Þ ¼ Hð2Þ: Substituting the previous time controls Dui ð0Þ ¼ Kð0Þei1 ð1Þ; Dui ð1Þ ¼ Kð1Þei1 ð2Þ into the above equation yields the following closed-loop system at t ¼ 3 of
K.K. Tan et al. / Engineering Applications of Artificial Intelligence 16 (2003) 185–190
the ith repetition
187
ei ðNÞ ¼ EðNÞei1 ðNÞ þ EN;1 ðNÞei1 ð1Þ þ EN;2 ðNÞei1 ð2Þ þ ? þ EN;N1 ðNÞei1 ðN 1Þ: ð18Þ
Remark 2. The theorem guarantees the existence of an iterative learning law that ensures the convergence of the problem. Since a solution to the problem exists only if matrices Cðt þ 1ÞBðtÞ; t ¼ 0; 1; y; N 1; have full row rank, each learning gain KðtÞ (t ¼ 0; 1; y; N 1) can be calculated in a simple way. For example, since Cðt þ 1ÞBðtÞ; t ¼ 0; 1; y; N 1; have full row rank, the invertibility of Cðt þ 1ÞBðtÞ½Cðt þ 1ÞBðtÞT exists (see Anderson and Moore, 1990). One can obtain the desired eigenvalues of I Cðt þ 1ÞBðtÞKðtÞ; t ¼ 0; 1; y; N 1; by calculating
We can write the above equations as a composite form
KðtÞ ¼ ðCðt þ 1ÞBðtÞÞT fCðt þ 1ÞBðt þ 1Þ
ei ð3Þ ¼ Eð3Þei1 ð3Þ H11 ð2ÞKð0Þei1 ð1Þ H12 ð2ÞKð1Þei1 ð2Þ: Define E3;1 ð3Þ ¼ H11 ð2ÞKð0Þ and E3;2 ð3Þ ¼ H12 ð2Þ Kð1Þ and we obtain ei ð3Þ ¼ Eð3Þei1 ð3Þ þ E3;1 ð3Þei1 ð1Þ þ E3;2 ð3Þei1 ð2Þ: ð17Þ Similarly, we have
ei ¼ Eei1 ; where ei ¼ ½eTi ð1Þ; eTi ð2Þ; eTi ð3Þ; y; eTi ðNÞT and 2 Eð1Þ 0 0 y 0 6 Eð2Þ 0 y 0 6 E2;1 ð2Þ 6 6 E ¼ 6 E3;1 ð3Þ E3;2 ð3Þ Eð3Þ y 0 6 ^ ^ ^ ^ ^ 4 EN;1 ðNÞ
½Cðt þ 1ÞBðtÞT g1 ðI Fn Þ;
ð19Þ
EN;2 ðNÞ
ð22Þ
n
3 7 7 7 7: 7 7 5
EN;3 ðNÞ y EðN NÞ
where F is the desired matrix of I Cðt þ 1ÞBðtÞKðtÞ: Thus, we have I Cðt þ 1ÞBðtÞðCðt þ 1ÞBðtÞÞT fCðt þ 1ÞBðt þ 1Þ
½Cðt þ 1ÞBðtÞT g1 ðI Fn Þ ¼ Fn : As long as the eigenvalues of Fn are inside the unit circle, the iterative learning law is convergent.
ð20Þ An important point should be noted here. The matrix E is an Nm Nm constant matrix with respect to the iteration i: Thus, the system becomes a discrete timeinvariant system. According to Lemma 1, ei is convergent iff E is stable, i.e., all jli ½Ejo1: Since the matrix E is a lower block triangular, we have N [ li ½E ¼ flk ½EðjÞg: ð21Þ j¼1
This implies that ei is convergent iff EðjÞ ¼ I CðjÞBðj 1ÞKðj 1Þ for each j ¼ 1; 2; y; N is a stability matrix. This completes the proof. & Remark 1. Theorem 1 shows that the iterative learning algorithm (9) is convergent iff the matrices ½I Cð1ÞBð0ÞKð0Þ; ½I Cð2ÞBð1ÞKð1Þ; y; ½I CðNÞBðN 1ÞKðN 1Þ are stable. For each t; the matrix KðtÞ (t ¼ 0; 1; y; N 1) which stabilizes the corresponding matrix I Cðt þ 1ÞBðtÞKðtÞ; exists if and only if the corresponding pair ðI; Cðt þ 1ÞBðtÞÞ is controllable (see Kurek and Zaremba, 1993; Mosca, 1995). Since the controllability matrix for this pair has the form Cðt þ 1ÞBðtÞ, this property occurs if and only if matrix Cðt þ 1ÞBðtÞ has full row rank, t ¼ 0; 1; y; N 1: Thus, the following theorem can be formulated. Theorem 2. For the repetitive linear time-varying system (1), there exists a convergent iterative learning law (9) which is a solution of the formulated problem if and only if all matrices Cðt þ 1ÞBðtÞ; t ¼ 0; 1; y; N 1 have full row rank.
3. Robustness and convergence Robustness in the face of system uncertainty is an important issue in the ILC research. In this section, we will show the uniform convergence of the system against state disturbances, measurement noise at the output, and reinitialization error at each iteration. Consider the following repetitive system with uncertainty and disturbance as follows: xi ðt þ 1Þ ¼ AðtÞxi ðtÞ þ BðtÞui ðtÞ þ wi ðtÞ; yi ðtÞ ¼ CðtÞxi ðtÞ þ vi ðtÞ;
ð23Þ
where wi ðtÞ and vi ðtÞ denote some random state disturbance and output measurement noise; tA½0; N is the time. For the analysis of the ILC process, the following 2-norms are introduced in this paper:
jjf jj ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn f 2; i¼1 i
jjGjj ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lmax ðG T GÞ:
Theorem 3. For system (23) with the assumptions that jjxi ð0Þ xi ð0Þjjpbx0 ; jjwi ðtÞ wi1 ðtÞjjpbw ; jjvi ðtÞ vi1 ðtÞjjpbv ; and Cðt þ 1ÞBðtÞ; t ¼ 0; 1; y; N 1 are full row rank, given the desired trajectory yd ðtÞ over the fixed time interval ½0; N; by using the learning control law (9), then, there exists a constant sequence fKðtÞ; t ¼ 0; 1; y; N 1g such that the tracking error is bounded. Moreover, the tracking error will converge uniformly to zero in ½0; N as i-N when bx0 ; bv ; and bw -0:
K.K. Tan et al. / Engineering Applications of Artificial Intelligence 16 (2003) 185–190
188
Proof. The solution of (23) can be obtained as follows: t t Y Y AðkÞxi ð0Þ þ AðkÞBð0Þui ð0Þ xi ðt þ 1Þ ¼ k¼0
k¼1
k¼1
þ
k¼2
From the definitions of (2)–(4), we have xi ðt þ 1Þ ¼ H0 ðtÞxi ð0Þ þ H1 ðtÞUi ð0; t 1Þ þ BðtÞui ðtÞ t Y þ AðkÞwi ð0Þ þ ? þ wi ðtÞ: ð24Þ k¼1
ei ðt þ 1Þ ¼ ei1 ðt þ 1Þ Cðt þ 1ÞH1 ðtÞDUi ð0; t 1Þ Cðt þ 1ÞBðtÞui ðtÞ Cðt þ 1ÞH0 ðtÞDxi ð0Þ t Y Cðt þ 1Þ AðkÞDwi ð0Þ ? Cðt þ 1ÞDwi ðtÞ Dvi ðtÞ:
ð25Þ
þ DwTiþ1 H% T2 P½H% 2 Dwiþ1 þ 2Dviþ1 þ DvTiþ1 PDviþ1 : T
k¼1
Cðt þ 1ÞDwi ðtÞ Dvi ðtÞ;
ð26Þ
where Eðt þ 1Þ; Etþ1;j ðt þ 1Þ have been given in the proof of Theorem 1. It is convenient to package the information in each trial, that is ei ¼ Eei1 H% 1 Dxi ð0Þ H% 2 Dwi Dvi ; ð27Þ where
ð28Þ
CðNÞH0 ðN 1Þ 6 6 % H2¼6 6 4
Cð1Þ Cð2ÞAð1Þ
0 Cð2Þ
y y
0 0
^ QN1
^ QN1
^
^
CðNÞ
k¼1
AðkÞ
CðNÞ
k¼2
AðkÞ
T
ð33Þ T
Using 2a bpza a þ ð1=zÞb b where z is an arbitrarily positive constant, the following inequalities hold:
Cðt þ 1ÞH0 ðtÞDxi ð0Þ t Y Cðt þ 1Þ AðkÞDwi ð0Þ?
2
¼ jjei jj2 2eTi ET P½H% 1 Dxiþ1 ð0Þ þ H% 2 Dwiþ1 þ Dviþ1 þ 2DxTiþ1 ð0ÞH% T1 P½H% 2 Dwiþ1 þ Dviþ1 þ 1H% 1 Dxiþ1 ð0Þ 2
þ ? þ Etþ1;t ðt þ 1Þei1 ðtÞ
7 7 7; 5
DLiþ1 ¼ Liþ1 Li ¼ eTiþ1 Peiþ1 eTi Pei ¼ eTi ðET PE PÞei 2eTi ET PH% 1 Dxiþ1 ð0Þ
þ DvTiþ1 PDviþ1
ei ðt þ 1Þ ¼ Eðt þ 1Þei1 ðt þ 1Þ þ Etþ1;1 ðt þ 1Þei1 ð1Þ þ Etþ1;2 ðt þ 1Þei2 ð2Þ
Cð2ÞH0 ð1Þ ^
where P is positive-definite matrix, and I is the unit matrix. We consider the Lyapunov function Li ¼ eTi Pei : Then along the solution of (27) we have
þ DwTiþ1 H% T2 PH% 2 Dwiþ1 þ 2DwTiþ1 H% T2 PDviþ1
Applying the control law (9) at 0; 1; y; t to (25), we have a similar result as the proof in Theorem 1
6 6 H% 1 ¼ 6 4
ð32Þ
þ DxTiþ1 ð0ÞH% T1 PH% 1 Dxiþ1 ð0Þ
k¼1
3
ð31Þ
2eTi ET PH% 2 Dwiþ1 2eTi ET PDviþ1 þ 2DxTiþ1 ð0ÞH% T1 PH% 2 Dwiþ1 þ 2DxTiþ1 ð0ÞH% 1 PDviþ1
For the error model, we have
Cð1ÞH0 ð0Þ
Dvi ¼ ½DvTi ð1Þ; DvTi ð2Þ; y; DvTi ðNÞT :
ET PE P ¼ I;
AðkÞwi ð1Þ þ ? þ wi ðtÞ:
2
ð30Þ
Since Cðt þ 1ÞBðtÞ; t ¼ 0; 1; y; N 1 have full row rank, it is shown in Theorem 2 that there exists a sequence fKð0Þ; Kð1Þ; y; KðN 1Þg such that the constant matrix E is stable. Thus, the following Lyapunov equation holds:
þ ? þ AðtÞBðt 1Þui ðt 1Þ t Y þ BðtÞui ðtÞ þ AðkÞwi ð0Þ t Y
Dwi ¼ ½DwTi ð0Þ; DwTi ð1Þ; y; DwTi ðN 1ÞT ;
3 7 7 7; 7 5
y CðNÞ ð29Þ
2eTi ET P½H% 1 Dxiþ1 ð0Þ þ H% 2 Dwiþ1 þ Dviþ1 1 pzeTi ei þ ½H% 1 Dxiþ1 ð0Þ þ H% 2 Dwiþ1 z þ Dviþ1 T ðPEET PÞ
½H% 1 Dxiþ1 ð0Þ þ H% 2 Dwiþ1 þ Dviþ1 1 pzjjei jj2 þ jjPEjj2 jjH% 1 Dxiþ1 ð0Þ þ H% 2 Dwiþ1 þ Dviþ1 jj2 z 1 pzjjei jj2 þ jjPEjj2 ðjjH% 1 jjbx0 z pffiffiffiffiffi pffiffiffiffiffi þ jjH% 2 jj N bw þ N bv Þ2 ; ð34Þ pffiffiffiffiffi pffiffiffiffiffi where jjDxiþ1 ð0Þjjpbx0 ; jjDwi jjp N bw ; jjDvi jjp N bv are used in the proof. Using the definitions of the norm, the following inequalities hold: 2DxTiþ1 ð0ÞH% T1 P½H% 2 Dwiþ1 þ Dviþ1 þ 12 H% 1 Dxiþ1 ð0Þ pffiffiffiffiffi pffiffiffiffiffi p2jjH% T1 Pjjbx0 ðjjH% 2 jj N bw þ N bv þ 12 jjH% 1 jjbx0 Þ; DwTiþ1 H% T2 P½H% 2 Dwiþ1 þ 2Dviþ1 pjjH% T2 PjjNbw ðjjH% 2 jjbw þ 2bv Þ;
K.K. Tan et al. / Engineering Applications of Artificial Intelligence 16 (2003) 185–190
DvTiþ1 PDviþ1 pjjPjjNb2v :
Li pri L0 þ
Incorporating the above inequalities produces DLiþ1 p ð1 zÞjjei jj2 þ sðbx0 ; bw ; bv Þ;
ð35Þ pffiffiffiffiffi 2 % 2 jj N bwþ jjbffiffiffiffi where x0 ; bw ; bv Þ¼ð1=zÞjjPEjj x0ffi þ jjH pffiffiffiffiffi sðb pffiffiffiffiffi ðjjH% 1p N bv Þ2 þ 2jjH% T1 Pjjbx0 ðjjH% 2 jj N bw þ N bv þ12 jjH% 1 jjbx0 Þþ jjH% T2 PjjNbw ðjjH% 2 jjbw þ 2bv Þ þ jjPjjNb2v ; and 1 > z > 0 constant. Since lmin ðPÞjjei jj2 pLi plmax ðPÞjjei jj2 ; we have Liþ1 Li p
1z Li þ sðbx0 ; bv ; bw Þ: lmax ðPÞ
Rearranging terms, we have 1z Liþ1 p 1 Li þ sðbx0 ; bv ; bw Þ lmax ðPÞ ¼ rLi þ sðbx0 ; bv ; bw Þ;
1 ri sðbx0 ; bw ; bv Þ; 1r
ð38Þ
sðbx0 ; bw ; bv Þ ; lim Li p 1r
ð39Þ
i-N
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sðbx0 ; bw ; bv Þ lim jjei jjp : i-N ð1 rÞlmin ðPÞ
ð40Þ
This implies that the output error is bounded for tA½0; N; and even the uncertainties that exist will converge a residual set ffito pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sðbx0 ; bw ; bv Þ=ð1 rÞlmin ðPÞ whose size will depend on the bounds of bx0 ; bv ; bw : Furthermore, limi-N jjei jj ¼ 0 if bx0 ; bv ; bw -0: &
4. Simulations
where r ¼ 1 ½ð1 zÞ=lmax ðPÞ: The choice for z to make jrjo1 is obvious, i.e., 1 lmax ðPÞozo1 if lmax ðPÞo1;
ð36Þ
0ozo1 if lmax ðPÞX1:
ð37Þ
Consider the example given in Tan et al. (1998). This is a DC motor control problem for velocity tracking. The following dominant model describes the dynamics of the system: x. ¼
Finally, one can easily find that
K1 K2 1 xðtÞ uðtÞ Tl ; ’ þ M M M
ð41Þ
0.4
1.2 1
e1 e2 e3 e20
0.2
0.8
Tracking error
Tracking performance
189
0.6 y1 y2 y3 y28
0.4 0.2
0
-0.2
0 -0.2 0
0.2
0.4 0.6 Time(s)
0.8
-0.4
1
0
0.2
0.4 0.6 Time(s)
0.8
1
0.8
1
Fig. 1. Control performance of plant when K ¼ 2:
0.4
1 0.3 0.8
Tracking error
Tracking performance
1.2
y1 y2 y3 y20
0.6 0.4 0.2
0.2 e1 e2 e3 e20
0.1 0
0 -0.2 0
0.2
0.4 0.6 Time(s)
0.8
1
-0.1 0
0.2
Fig. 2. Control performance of plant when K ¼ 4:
0.4 0.6 Time(s)
190
K.K. Tan et al. / Engineering Applications of Artificial Intelligence 16 (2003) 185–190
where K1 ; K2 ; M are constants, and Tl is the load disturbance. A discrete-system state equation in the time domain may be obtained directly from (41) with the sampling time T ¼ 0:005 s: It can be verified that the product of the output/input coupling matrix CB ¼ 0:0417 is full column rank. To consider the state disturbance, we consider the Stibreck effect 2 ’ 0:02eðx=0:002Þ in the servo system. The output measurement noise is given to be vi ðtÞ ¼ 0:02randomð0; 1Þ: Let the control gain K ¼ 2: The learning law (9) is utilized. The tracking performance is shown in Fig. 1. It can be found that the convergence and robustness are achieved as the iteration number increases. For large K; the learning algorithm can achieve rapid convergence. For example, when we increase the value of K to 4; the tracking performance is shown in Fig. 2. The faster speed of convergence is clearly visible. 5. Conclusions In this paper, an iterative learning algorithm is presented for a linear time-varying multivariable system. We have shown that the output error converges to zero if and only if each product of the output/input coupling matrices at time point is full row rank. This condition is very weak. For example, a convergent learning control always exists for single-input and single-output systems. Then, we prove that the same condition is sufficient for the robustness of the proposed learning algorithm against state disturbance, output measurement noise, and reinitialized error. Moreover, the tracking error on the final iteration is a class K function of the bounds on the uncertainties.
Acknowledgements The authors thank Dr. Y.Q. Chen for his help and valuable discussions of iterative learning algorithms. This work was supported by SimTech project No. U01A-020B.
References Anderson, B.D.O., Moore, J.B., 1990. Optimal Control: Linear Quadratic Methods. Prentice-Hall, Englewood Cliffs, NJ. Arimoto, S., 1990. Learning control theory for robotic motion. International Journal of Adaptive Control and Signal Processing 4, 277–293. Arimoto, S., Kawamura, S., Miyazaki, F., 1984. Bettering operating operation of robots by learning. Journal of Robotic Systems 1 (1), 123–140. Bien, Z., Huh, K.M., 1989. High-order iterative control algorithm. IEE Proceedings, Part-D, Control Theory and Applications 13 (3), 105–112. Chen, Y.Q., Wen, C., 1999. Iterative Learning Control: Convergence, Robustness, and Applications. Springer, Berlin. Huang, S.N., Tan, K.K., Lee, T.H., 2001. Iterative learning algorithm with a quadratic criterion for linear time-varying systems. Journal of Systems and Control Engineering (under review). Kurek, J.E., Zaremba, M.B., 1993. Iterative learning control synthesis based on 2-D system theory. IEEE Transactions on Automatic Control AC-38 (1), 121–125. Lee, T.H., Tan, K.K., Lim, S.Y., Dou, H.F., 2000. Iterative learning control of permanent magnet linear motor with relay automatic tuning. Mechatronics 10, 169–190. Lucibello, P., 1992. Learning control of linear systems. Proceedings of American Control Conference, pp. 1888–1892. Moore, K.L., 1992. Iterative Learning Control for Deterministic Systems. Advances in Industrial Contr. Ser. Springer, London, UK. Mosca, E., 1995. Optimal, Predictive and Adaptive Control. PrenticeHall, Englewood Cliffs, NJ. Saab, S.S., 1995. A discrete-time learning control algorithm for a class of linear time-invariant systems. IEEE Transactions on Automatic Control 40 (6), 1138–1142. Tan, K.K., Lee, T.H., Huang, S.N., Lim, S.Y., 1998. Adaptive control of DC permanent magnet linear motor for ultra-precision applications. International Conference on Mectronic Technology, Taiwan, pp. 243–246. Tan, K.K., Dou, H.F., Chen, Y.Q., Lee, T.H., 2001. High precision linear motor control via relay tuning and iterative learning based on zero-phase filtering. IEEE Transactions on Control Systems Technology 9 (2), 244–253. Togai, M., Yamano, O., 1985. Analysis and design of an optimal learning control scheme for industrial robots: a discrete system approach. Proceedings of 24th Conference on Decision Control, Ft. Lauderdale, Fl, pp. 1399–1404. Yamada, K., Watanabe, K., Tsuchiya, M., Kaneko, T., 1994. Robust control design for repetitive control systems with filtered inverse. Proceedings of ASCC, Vol. 1, Tokyo, Japan, pp. 243–246.
Brief Paper
A discrete-time iterative learning algorithm for linear time-varying systems K.K. Tana, S.N. Huanga,b,*, T.H. Leea, S.Y. Limb a
Mechatronics and Automation Lab., Department of Electrical and Computer Engineering, 10 Kent Ridge Crescent, National University of Singapore, Singapore 119260, Singapore b Mechatronics Group, Singapore Institute of Manufacturing Technology, Singapore
Abstract In this paper, an iterative learning algorithm (ILC) is presented for a MIMO linear time-varying system. We consider the convergence of the algorithm. A necessary and sufficient condition for the existence of convergent algorithm is stated. Then, we prove that the same condition is sufficient for the robustness of the proposed learning algorithm against state disturbance, output measurement noise, and reinitialization error. Finally, a simulation example is given to illustrate the results. r 2003 Elsevier Ltd. All rights reserved. Keywords: Iterative learning control; Repetitive systems; Linear discrete systems; Uncertain systems
1. Introduction The iterative learning control (ILC) method has been proposed by Arimoto et al. (1984). Examples of this idea can be found in Arimoto (1990), Bien and Huh (1989), Lucibello (1992), Yamada et al. (1994), Lee et al. (2000) and Tan et al. (2001), including the general area of trajectory following in robotics. The specified task is regarded as the tracking of a given output trajectory for an operation on a specified time interval. The objective of ILC is to use the repetitive nature of the process to improve the accuracy with which the operation is achieved by generating the present control input based on the previous control history and a learning mechanism. The monograph by Moore (1992) contains more detail on the background. A recent book (Chen and Wen, 1999) about ILC surveys the literature until 1998. For real implementation of an iterative learning controller, it is more practical to design and analyze the ILC systems in discrete-time domain. The optimality of the discrete ILC is studied in Togai and Yamano (1985). Subsequently, it has been proven that the tracking error in Togai and Yamano (1985) for the linear time-invariant system will converge to zero if and *Corresponding author. Mechatronics and Automation Lab., Department of Electrical and Computer Engineering, National University of Singapore, Singapore. E-mail address: [email protected] (S.N. Huang). 0952-1976/03/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0952-1976(03)00030-7
only if the input–output coupling matrices are full row rank (Kurek and Zaremba, 1993). In Saab (1995), another necessary and sufficient condition for the linear time-invariant system is proposed. The robustness is also discussed in Saab (1995). In Huang et al. (2001), a sufficient condition for the linear time-varying system is given based on the quadratic criterion. None of the papers involving linear time-varying systems considers the necessary and sufficient condition for convergence. In this paper, we deal with the iterative learning control problem for a multivariable linear time-varying system. We consider the convergence of the learning algorithm, and present a sufficient and necessary condition for the existence of convergent algorithm. Furthermore, this condition proved in this paper is sufficient for the robustness of the proposed iterative learning algorithm. The robustness bounds of the learning system are obtained by using 2-norm rather than traditional lnorm as in Arimoto (1990) for convergence analysis.
2. Necessary and sufficient condition for convergence Consider a repetitive linear discrete-time system as follows: xi ðt þ 1Þ ¼ AðtÞxi ðtÞ þ BðtÞui ðtÞ; yi ðtÞ ¼ CðtÞxi ðtÞ;
ð1Þ
K.K. Tan et al. / Engineering Applications of Artificial Intelligence 16 (2003) 185–190
186
and i denotes the ith repetitive operation of the system; xi ðtÞARn ; ui ðtÞARm ; and yi ðtÞARr are the state, control input, and output of the system, respectively; tA½0; N is the time; and AðtÞ; BðtÞ; and CðtÞ are matrices with appropriate dimensions. Assume that, at each iteration, the control system (1) is exactly re-initialized at x0 ; i.e., xi ð0Þ ¼ x0 :
The following learning law is used Dui ðtÞ ¼ KðtÞei1 ðt þ 1Þ;
t ¼ 0; 1; y; N 1;
where fKð0Þ; Kð1Þ; y; KðN 1Þg is a constant sequence. Lemma 1 (Mosca, 1995). The time-invariant dynamic linear system zðk þ 1Þ ¼ FzðkÞ
Definition 1 (Kurek and Zaremba, 1993). A learning rule is said to be convergent if, and only if (iff) for any x0 ARn it generates a control sequence fui ðkÞ; k ¼ 0; 1; y; N 1g for system (1) such that yi ðtÞ-yd ðtÞ; t ¼ 1; 2; y; N; as i-N: To determine how the error evolves with repetitions, we need to solve Eq. (1), that is, xi ðt þ 1Þ ¼
t Y
AðkÞxi ð0Þ þ
k¼0
t Y
AðkÞBð0Þui ð0Þ
k¼1
þ ? þ AðtÞBðt 1Þui ðt 1Þ þ BðtÞui ðtÞ: Let H0 ðtÞ ¼
t Y
ð2Þ
AðkÞ;
k¼0
H1 ðtÞ ¼
Y t
AðkÞBð0Þ;
k¼1
t Y
AðkÞBð1Þ; y; AðtÞBðt 1Þ ;
Theorem 1. Consider system (1) with a given desired output yd ðkÞ; k ¼ 1; 2; y; N: An iterative learning control law (9) is convergent and is a solution to the problem if and only if each matrix I Cðt þ 1ÞBðtÞKðtÞ where t ¼ 0; 1; y; N 1 is asymptotically stable, i.e., if all of its eigenvalues are inside the unit circle. Proof. Let Eðt þ 1Þ ¼ I Cðt þ 1ÞBðtÞKðtÞ where I is an unit matrix. Considering the assumption and applying the control law (9) to (8) yields ei ðt þ 1Þ ¼ ½I Cðt þ 1ÞBðtÞKðtÞei1 ðt þ 1Þ HðtÞDUi ð0; t 1Þ ¼ Eðt þ 1Þei1 ðt þ 1Þ HðtÞDUi ð0; t 1Þ:
ð3Þ ð4Þ
Thus, we have
For t ¼ 0 at the ith iteration, applying H1 ð0Þ ¼ 0; Ui ð0; 1Þ ¼ 0 to the above equation, we have ei ð1Þ ¼ Eð1Þei1 ð1Þ:
xi ðt þ 1Þ ¼ H0 ðtÞxi ð0Þ þ H1 ðtÞUi ð0; t 1Þ þ BðtÞui ðtÞ; t ¼ 0; 1; y; N 1:
ð10Þ
is asymptotically stable if and only if its state-transition matrix F is a stability matrix, i.e., jli ðFÞjo1 (for the time-invariant case, the state transition matrix is equal to F; see page 19 in the Section 2.4 of Mosca, 1995).
k¼2
Ui ð0; t 1Þ ¼ ½uTi ð0Þ; uTi ð1Þ; y; uTi ðt 1ÞT :
ð9Þ
ð5Þ
For t ¼ 1 at the ith iteration, we have ei ð2Þ ¼ Eð2Þei1 ð2Þ Hð1ÞDui ð0Þ:
ð11Þ
It is obvious that H1 ð0Þ ¼ 0; and Ui ð0; 1Þ ¼ 0: The tracking error ei ðt þ 1Þ at the ith repetition is that ei ðt þ 1Þ ¼ yd ðt þ 1Þ yi ðt þ 1Þ: From this definition, we can reach the following output error equation:
Note that the previous time control Dui ð0Þ is known, and substituting it into the above equation yields
ei ðt þ 1Þ ¼ yd ðt þ 1Þ yi1 ðt þ 1Þ
Letting
ðyi ðt þ 1Þ yi1 ðt þ 1ÞÞ ¼ ei1 ðt þ 1Þ Cðt þ 1Þðxi ðt þ 1Þ xi1 ðt þ 1ÞÞ:
E2;1 ð2Þ ¼ Hð1ÞKð0Þ
ð12Þ ð13Þ
we have ð6Þ
Substituting (5) into (6) yields
ei ð2Þ ¼ Eð2Þei1 ð2Þ þ E2;1 ð2Þei1 ð1Þ: For t ¼ 3 at the ith repetition, we have " # Dui ð0Þ ei ð3Þ ¼ Eð3Þei1 ð3Þ Hð2Þ : Dui ð1Þ
ei ðt þ 1Þ ¼ ei1 ðt þ 1Þ Cðt þ 1ÞH0 ðtÞDxi ð0Þ Cðt þ 1ÞH1 ðtÞDUi ð0; t 1Þ Cðt þ 1ÞBðtÞDui ðtÞ;
ei ð2Þ ¼ Eð2Þei1 ð2Þ Hð1ÞKð0Þei1 ð1Þ:
ð7Þ
where Dxi ð0Þ ¼ xi ð0Þ xi1 ð0Þ; DUi ð0; t 1Þ ¼ Ui ð0; t 1Þ Ui1 ð0; t 1Þ; and Dui ðtÞ ¼ ui ðtÞ ui1 ðtÞ: Letting HðtÞ ¼ Cðt þ 1ÞH1 ðtÞ we have ei ðt þ 1Þ ¼ ei1 ðt þ 1Þ Cðt þ 1ÞH0 ðtÞDxi ð0Þ HðtÞDUi ð0; t 1Þ Cðt þ 1ÞBðtÞDui ðtÞ: ð8Þ
ð14Þ
ð15Þ
With Hð2Þ partitioned into two m r blocks, we have ei ð3Þ ¼ Eð3Þei1 ð3Þ H11 ð2ÞDui ð0Þ H12 ð2ÞDui ð1Þ;
ð16Þ
where ½H11 ð2Þ H12 ð2Þ ¼ Hð2Þ: Substituting the previous time controls Dui ð0Þ ¼ Kð0Þei1 ð1Þ; Dui ð1Þ ¼ Kð1Þei1 ð2Þ into the above equation yields the following closed-loop system at t ¼ 3 of
K.K. Tan et al. / Engineering Applications of Artificial Intelligence 16 (2003) 185–190
the ith repetition
187
ei ðNÞ ¼ EðNÞei1 ðNÞ þ EN;1 ðNÞei1 ð1Þ þ EN;2 ðNÞei1 ð2Þ þ ? þ EN;N1 ðNÞei1 ðN 1Þ: ð18Þ
Remark 2. The theorem guarantees the existence of an iterative learning law that ensures the convergence of the problem. Since a solution to the problem exists only if matrices Cðt þ 1ÞBðtÞ; t ¼ 0; 1; y; N 1; have full row rank, each learning gain KðtÞ (t ¼ 0; 1; y; N 1) can be calculated in a simple way. For example, since Cðt þ 1ÞBðtÞ; t ¼ 0; 1; y; N 1; have full row rank, the invertibility of Cðt þ 1ÞBðtÞ½Cðt þ 1ÞBðtÞT exists (see Anderson and Moore, 1990). One can obtain the desired eigenvalues of I Cðt þ 1ÞBðtÞKðtÞ; t ¼ 0; 1; y; N 1; by calculating
We can write the above equations as a composite form
KðtÞ ¼ ðCðt þ 1ÞBðtÞÞT fCðt þ 1ÞBðt þ 1Þ
ei ð3Þ ¼ Eð3Þei1 ð3Þ H11 ð2ÞKð0Þei1 ð1Þ H12 ð2ÞKð1Þei1 ð2Þ: Define E3;1 ð3Þ ¼ H11 ð2ÞKð0Þ and E3;2 ð3Þ ¼ H12 ð2Þ Kð1Þ and we obtain ei ð3Þ ¼ Eð3Þei1 ð3Þ þ E3;1 ð3Þei1 ð1Þ þ E3;2 ð3Þei1 ð2Þ: ð17Þ Similarly, we have
ei ¼ Eei1 ; where ei ¼ ½eTi ð1Þ; eTi ð2Þ; eTi ð3Þ; y; eTi ðNÞT and 2 Eð1Þ 0 0 y 0 6 Eð2Þ 0 y 0 6 E2;1 ð2Þ 6 6 E ¼ 6 E3;1 ð3Þ E3;2 ð3Þ Eð3Þ y 0 6 ^ ^ ^ ^ ^ 4 EN;1 ðNÞ
½Cðt þ 1ÞBðtÞT g1 ðI Fn Þ;
ð19Þ
EN;2 ðNÞ
ð22Þ
n
3 7 7 7 7: 7 7 5
EN;3 ðNÞ y EðN NÞ
where F is the desired matrix of I Cðt þ 1ÞBðtÞKðtÞ: Thus, we have I Cðt þ 1ÞBðtÞðCðt þ 1ÞBðtÞÞT fCðt þ 1ÞBðt þ 1Þ
½Cðt þ 1ÞBðtÞT g1 ðI Fn Þ ¼ Fn : As long as the eigenvalues of Fn are inside the unit circle, the iterative learning law is convergent.
ð20Þ An important point should be noted here. The matrix E is an Nm Nm constant matrix with respect to the iteration i: Thus, the system becomes a discrete timeinvariant system. According to Lemma 1, ei is convergent iff E is stable, i.e., all jli ½Ejo1: Since the matrix E is a lower block triangular, we have N [ li ½E ¼ flk ½EðjÞg: ð21Þ j¼1
This implies that ei is convergent iff EðjÞ ¼ I CðjÞBðj 1ÞKðj 1Þ for each j ¼ 1; 2; y; N is a stability matrix. This completes the proof. & Remark 1. Theorem 1 shows that the iterative learning algorithm (9) is convergent iff the matrices ½I Cð1ÞBð0ÞKð0Þ; ½I Cð2ÞBð1ÞKð1Þ; y; ½I CðNÞBðN 1ÞKðN 1Þ are stable. For each t; the matrix KðtÞ (t ¼ 0; 1; y; N 1) which stabilizes the corresponding matrix I Cðt þ 1ÞBðtÞKðtÞ; exists if and only if the corresponding pair ðI; Cðt þ 1ÞBðtÞÞ is controllable (see Kurek and Zaremba, 1993; Mosca, 1995). Since the controllability matrix for this pair has the form Cðt þ 1ÞBðtÞ, this property occurs if and only if matrix Cðt þ 1ÞBðtÞ has full row rank, t ¼ 0; 1; y; N 1: Thus, the following theorem can be formulated. Theorem 2. For the repetitive linear time-varying system (1), there exists a convergent iterative learning law (9) which is a solution of the formulated problem if and only if all matrices Cðt þ 1ÞBðtÞ; t ¼ 0; 1; y; N 1 have full row rank.
3. Robustness and convergence Robustness in the face of system uncertainty is an important issue in the ILC research. In this section, we will show the uniform convergence of the system against state disturbances, measurement noise at the output, and reinitialization error at each iteration. Consider the following repetitive system with uncertainty and disturbance as follows: xi ðt þ 1Þ ¼ AðtÞxi ðtÞ þ BðtÞui ðtÞ þ wi ðtÞ; yi ðtÞ ¼ CðtÞxi ðtÞ þ vi ðtÞ;
ð23Þ
where wi ðtÞ and vi ðtÞ denote some random state disturbance and output measurement noise; tA½0; N is the time. For the analysis of the ILC process, the following 2-norms are introduced in this paper:
jjf jj ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn f 2; i¼1 i
jjGjj ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lmax ðG T GÞ:
Theorem 3. For system (23) with the assumptions that jjxi ð0Þ xi ð0Þjjpbx0 ; jjwi ðtÞ wi1 ðtÞjjpbw ; jjvi ðtÞ vi1 ðtÞjjpbv ; and Cðt þ 1ÞBðtÞ; t ¼ 0; 1; y; N 1 are full row rank, given the desired trajectory yd ðtÞ over the fixed time interval ½0; N; by using the learning control law (9), then, there exists a constant sequence fKðtÞ; t ¼ 0; 1; y; N 1g such that the tracking error is bounded. Moreover, the tracking error will converge uniformly to zero in ½0; N as i-N when bx0 ; bv ; and bw -0:
K.K. Tan et al. / Engineering Applications of Artificial Intelligence 16 (2003) 185–190
188
Proof. The solution of (23) can be obtained as follows: t t Y Y AðkÞxi ð0Þ þ AðkÞBð0Þui ð0Þ xi ðt þ 1Þ ¼ k¼0
k¼1
k¼1
þ
k¼2
From the definitions of (2)–(4), we have xi ðt þ 1Þ ¼ H0 ðtÞxi ð0Þ þ H1 ðtÞUi ð0; t 1Þ þ BðtÞui ðtÞ t Y þ AðkÞwi ð0Þ þ ? þ wi ðtÞ: ð24Þ k¼1
ei ðt þ 1Þ ¼ ei1 ðt þ 1Þ Cðt þ 1ÞH1 ðtÞDUi ð0; t 1Þ Cðt þ 1ÞBðtÞui ðtÞ Cðt þ 1ÞH0 ðtÞDxi ð0Þ t Y Cðt þ 1Þ AðkÞDwi ð0Þ ? Cðt þ 1ÞDwi ðtÞ Dvi ðtÞ:
ð25Þ
þ DwTiþ1 H% T2 P½H% 2 Dwiþ1 þ 2Dviþ1 þ DvTiþ1 PDviþ1 : T
k¼1
Cðt þ 1ÞDwi ðtÞ Dvi ðtÞ;
ð26Þ
where Eðt þ 1Þ; Etþ1;j ðt þ 1Þ have been given in the proof of Theorem 1. It is convenient to package the information in each trial, that is ei ¼ Eei1 H% 1 Dxi ð0Þ H% 2 Dwi Dvi ; ð27Þ where
ð28Þ
CðNÞH0 ðN 1Þ 6 6 % H2¼6 6 4
Cð1Þ Cð2ÞAð1Þ
0 Cð2Þ
y y
0 0
^ QN1
^ QN1
^
^
CðNÞ
k¼1
AðkÞ
CðNÞ
k¼2
AðkÞ
T
ð33Þ T
Using 2a bpza a þ ð1=zÞb b where z is an arbitrarily positive constant, the following inequalities hold:
Cðt þ 1ÞH0 ðtÞDxi ð0Þ t Y Cðt þ 1Þ AðkÞDwi ð0Þ?
2
¼ jjei jj2 2eTi ET P½H% 1 Dxiþ1 ð0Þ þ H% 2 Dwiþ1 þ Dviþ1 þ 2DxTiþ1 ð0ÞH% T1 P½H% 2 Dwiþ1 þ Dviþ1 þ 1H% 1 Dxiþ1 ð0Þ 2
þ ? þ Etþ1;t ðt þ 1Þei1 ðtÞ
7 7 7; 5
DLiþ1 ¼ Liþ1 Li ¼ eTiþ1 Peiþ1 eTi Pei ¼ eTi ðET PE PÞei 2eTi ET PH% 1 Dxiþ1 ð0Þ
þ DvTiþ1 PDviþ1
ei ðt þ 1Þ ¼ Eðt þ 1Þei1 ðt þ 1Þ þ Etþ1;1 ðt þ 1Þei1 ð1Þ þ Etþ1;2 ðt þ 1Þei2 ð2Þ
Cð2ÞH0 ð1Þ ^
where P is positive-definite matrix, and I is the unit matrix. We consider the Lyapunov function Li ¼ eTi Pei : Then along the solution of (27) we have
þ DwTiþ1 H% T2 PH% 2 Dwiþ1 þ 2DwTiþ1 H% T2 PDviþ1
Applying the control law (9) at 0; 1; y; t to (25), we have a similar result as the proof in Theorem 1
6 6 H% 1 ¼ 6 4
ð32Þ
þ DxTiþ1 ð0ÞH% T1 PH% 1 Dxiþ1 ð0Þ
k¼1
3
ð31Þ
2eTi ET PH% 2 Dwiþ1 2eTi ET PDviþ1 þ 2DxTiþ1 ð0ÞH% T1 PH% 2 Dwiþ1 þ 2DxTiþ1 ð0ÞH% 1 PDviþ1
For the error model, we have
Cð1ÞH0 ð0Þ
Dvi ¼ ½DvTi ð1Þ; DvTi ð2Þ; y; DvTi ðNÞT :
ET PE P ¼ I;
AðkÞwi ð1Þ þ ? þ wi ðtÞ:
2
ð30Þ
Since Cðt þ 1ÞBðtÞ; t ¼ 0; 1; y; N 1 have full row rank, it is shown in Theorem 2 that there exists a sequence fKð0Þ; Kð1Þ; y; KðN 1Þg such that the constant matrix E is stable. Thus, the following Lyapunov equation holds:
þ ? þ AðtÞBðt 1Þui ðt 1Þ t Y þ BðtÞui ðtÞ þ AðkÞwi ð0Þ t Y
Dwi ¼ ½DwTi ð0Þ; DwTi ð1Þ; y; DwTi ðN 1ÞT ;
3 7 7 7; 7 5
y CðNÞ ð29Þ
2eTi ET P½H% 1 Dxiþ1 ð0Þ þ H% 2 Dwiþ1 þ Dviþ1 1 pzeTi ei þ ½H% 1 Dxiþ1 ð0Þ þ H% 2 Dwiþ1 z þ Dviþ1 T ðPEET PÞ
½H% 1 Dxiþ1 ð0Þ þ H% 2 Dwiþ1 þ Dviþ1 1 pzjjei jj2 þ jjPEjj2 jjH% 1 Dxiþ1 ð0Þ þ H% 2 Dwiþ1 þ Dviþ1 jj2 z 1 pzjjei jj2 þ jjPEjj2 ðjjH% 1 jjbx0 z pffiffiffiffiffi pffiffiffiffiffi þ jjH% 2 jj N bw þ N bv Þ2 ; ð34Þ pffiffiffiffiffi pffiffiffiffiffi where jjDxiþ1 ð0Þjjpbx0 ; jjDwi jjp N bw ; jjDvi jjp N bv are used in the proof. Using the definitions of the norm, the following inequalities hold: 2DxTiþ1 ð0ÞH% T1 P½H% 2 Dwiþ1 þ Dviþ1 þ 12 H% 1 Dxiþ1 ð0Þ pffiffiffiffiffi pffiffiffiffiffi p2jjH% T1 Pjjbx0 ðjjH% 2 jj N bw þ N bv þ 12 jjH% 1 jjbx0 Þ; DwTiþ1 H% T2 P½H% 2 Dwiþ1 þ 2Dviþ1 pjjH% T2 PjjNbw ðjjH% 2 jjbw þ 2bv Þ;
K.K. Tan et al. / Engineering Applications of Artificial Intelligence 16 (2003) 185–190
DvTiþ1 PDviþ1 pjjPjjNb2v :
Li pri L0 þ
Incorporating the above inequalities produces DLiþ1 p ð1 zÞjjei jj2 þ sðbx0 ; bw ; bv Þ;
ð35Þ pffiffiffiffiffi 2 % 2 jj N bwþ jjbffiffiffiffi where x0 ; bw ; bv Þ¼ð1=zÞjjPEjj x0ffi þ jjH pffiffiffiffiffi sðb pffiffiffiffiffi ðjjH% 1p N bv Þ2 þ 2jjH% T1 Pjjbx0 ðjjH% 2 jj N bw þ N bv þ12 jjH% 1 jjbx0 Þþ jjH% T2 PjjNbw ðjjH% 2 jjbw þ 2bv Þ þ jjPjjNb2v ; and 1 > z > 0 constant. Since lmin ðPÞjjei jj2 pLi plmax ðPÞjjei jj2 ; we have Liþ1 Li p
1z Li þ sðbx0 ; bv ; bw Þ: lmax ðPÞ
Rearranging terms, we have 1z Liþ1 p 1 Li þ sðbx0 ; bv ; bw Þ lmax ðPÞ ¼ rLi þ sðbx0 ; bv ; bw Þ;
1 ri sðbx0 ; bw ; bv Þ; 1r
ð38Þ
sðbx0 ; bw ; bv Þ ; lim Li p 1r
ð39Þ
i-N
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sðbx0 ; bw ; bv Þ lim jjei jjp : i-N ð1 rÞlmin ðPÞ
ð40Þ
This implies that the output error is bounded for tA½0; N; and even the uncertainties that exist will converge a residual set ffito pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sðbx0 ; bw ; bv Þ=ð1 rÞlmin ðPÞ whose size will depend on the bounds of bx0 ; bv ; bw : Furthermore, limi-N jjei jj ¼ 0 if bx0 ; bv ; bw -0: &
4. Simulations
where r ¼ 1 ½ð1 zÞ=lmax ðPÞ: The choice for z to make jrjo1 is obvious, i.e., 1 lmax ðPÞozo1 if lmax ðPÞo1;
ð36Þ
0ozo1 if lmax ðPÞX1:
ð37Þ
Consider the example given in Tan et al. (1998). This is a DC motor control problem for velocity tracking. The following dominant model describes the dynamics of the system: x. ¼
Finally, one can easily find that
K1 K2 1 xðtÞ uðtÞ Tl ; ’ þ M M M
ð41Þ
0.4
1.2 1
e1 e2 e3 e20
0.2
0.8
Tracking error
Tracking performance
189
0.6 y1 y2 y3 y28
0.4 0.2
0
-0.2
0 -0.2 0
0.2
0.4 0.6 Time(s)
0.8
-0.4
1
0
0.2
0.4 0.6 Time(s)
0.8
1
0.8
1
Fig. 1. Control performance of plant when K ¼ 2:
0.4
1 0.3 0.8
Tracking error
Tracking performance
1.2
y1 y2 y3 y20
0.6 0.4 0.2
0.2 e1 e2 e3 e20
0.1 0
0 -0.2 0
0.2
0.4 0.6 Time(s)
0.8
1
-0.1 0
0.2
Fig. 2. Control performance of plant when K ¼ 4:
0.4 0.6 Time(s)
190
K.K. Tan et al. / Engineering Applications of Artificial Intelligence 16 (2003) 185–190
where K1 ; K2 ; M are constants, and Tl is the load disturbance. A discrete-system state equation in the time domain may be obtained directly from (41) with the sampling time T ¼ 0:005 s: It can be verified that the product of the output/input coupling matrix CB ¼ 0:0417 is full column rank. To consider the state disturbance, we consider the Stibreck effect 2 ’ 0:02eðx=0:002Þ in the servo system. The output measurement noise is given to be vi ðtÞ ¼ 0:02randomð0; 1Þ: Let the control gain K ¼ 2: The learning law (9) is utilized. The tracking performance is shown in Fig. 1. It can be found that the convergence and robustness are achieved as the iteration number increases. For large K; the learning algorithm can achieve rapid convergence. For example, when we increase the value of K to 4; the tracking performance is shown in Fig. 2. The faster speed of convergence is clearly visible. 5. Conclusions In this paper, an iterative learning algorithm is presented for a linear time-varying multivariable system. We have shown that the output error converges to zero if and only if each product of the output/input coupling matrices at time point is full row rank. This condition is very weak. For example, a convergent learning control always exists for single-input and single-output systems. Then, we prove that the same condition is sufficient for the robustness of the proposed learning algorithm against state disturbance, output measurement noise, and reinitialized error. Moreover, the tracking error on the final iteration is a class K function of the bounds on the uncertainties.
Acknowledgements The authors thank Dr. Y.Q. Chen for his help and valuable discussions of iterative learning algorithms. This work was supported by SimTech project No. U01A-020B.
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