Journal of the Franklin Institute 343 (2006) 279–294
LQG optimal control of discrete stochastic systems
under parametric and noise uncertainties
Feng-Hsiag Hsiao
a,, Sheng-Dong Xu
b,
Shih-Lin Wu
c, Gwo-Chuan Lee
daDepartment of Electronic Engineering, National University of Tainan, No. 33, Section 2, Shu Lin Street, Tainan 700, Taiwan, R.O.C.
bDepartment of Electrical and Control Engineering, National Chiao Tung University, No. 1001, Ta Hsueh Road, Hsinchu 30010, Taiwan, R.O.C.
c
Department of Computer Science and Information Engineering, Chang Gung University, Kwei San, Taoyuan County 333, Taiwan, R.O.C.
d
Department of Computer Science and Information Engineering, National United University, No. 1, Lien Da, Kung Ching Li, Miaoli 360, Taiwan, R.O.C.
Received 20 July 2005; received in revised form 15 February 2006; accepted 21 February 2006
Abstract
In this paper, the linear-quadratic-Gaussian (LQG) optimal control problem is considered and a robust minimax controller composed of the Kalman filter and the optimal regulator is synthesized to guarantee the asymptotic stability of the discrete time-delay systems under both parametric uncertainties and uncertain noise covariances. Designed procedures are finally elaborated with an illustrative example.
r2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Keywords: LQG optimal control; Minimax controller; Kalman filter
1. Introduction
A well-known property of the discrete linear-quadratic-Gaussian (LQG) optimal control problem is that the optimal regulator, synthesized by the LQ optimal technique, is
www.elsevier.com/locate/jfranklin
0016-0032/$30.00 r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2006.02.038
Corresponding author. Tel.: +886 915321417; fax: +886 33289390. E-mail address:[email protected] (F.-H. Hsiao).
generated from the estimated state which is the output of the Kalman filter. The LQG optimal control design for a linear stochastic dynamic system requires not only an accurate description of the statistical characteristic of noise signal but also an exact system model. Nevertheless, neither noise nor plant parameters may be precisely known in a real control system. Therefore, it is of interest to consider the robust LQG optimal problem for those systems whose noise covariances and plant parameters are known only to be within some classes.
Time delay is commonly encountered in various engineering systems; for example, systems with computer control have delays, as it takes time for the computer to execute numerical operations. Besides, remote working, radar, electric networks, transport process, metal rolling systems, etc. all have delays. The output in these systems responds only to an input after some time interval. The introduction of time-delay factor is often a source of instability and generally complicates the analysis. Hence, the problem of stability analysis of time-delay systems has been one of the main concerns of researchers wishing to inspect the properties of such systems and there have been several research efforts[1–11]on this issue. For instance, Basin et al. proposed the optimal control and filtering algorithms for time-delay systems, and discussed the delay-dependent stability for linear discrete stochastic systems[10–14].
In this paper, the LQG optimal control problem is considered and Minimax theory and Bellman–Gronwall lemma are employed to derive a robust criterion which guarantees the asymptotic stability of the discrete time-delay systems under both parametric uncertainties and uncertain noise covariances. On the basis of this criterion, a robust minimax controller composed of the Kalman filter and the optimal regulator is synthesized to stabilize the uncertain stochastic systems.
The organization of this paper is as follows. The system description is presented in Section 2. The design procedure of a robust minimax controller is proposed in Section 3. An example is provided in Section 4 to illustrate our main results. A conclusion is finally drawn in Section 5.
2. Problem formulation
A discrete time-delay system is depicted by the following difference equations:
xpðk þ 1Þ ¼ A0xpðkÞ þ DA0ðkÞxpðkÞ þ Xm i¼1 Aixpðk iÞ þ Xm i¼1 DAiðkÞxpðk iÞ
þBpuðkÞ þ DBpðkÞuðkÞ þ vðkÞ, ð2:1aÞ
yðkÞ ¼ CpxpðkÞ þ DCpðkÞxpðkÞ þ eðkÞ (2.1b)
where xpðkÞ is an n 1 state vector; uðkÞ is an r 1 input vector; yðkÞ is a p 1 output
vector; vðkÞ is an n 1 random process vector; eðkÞ is a p 1 random process vector; and A0, Ai, Bp ðrankðBpÞ ¼rÞ and Cp ðrankðCpÞ ¼pÞ are constant matrices with appropriate
dimensions. Moreover, DA0ðkÞ, DAiðkÞ, DBpðkÞ and DCpðkÞ denote linear time-varying
parametric uncertainties with the following upper norm-bounds:
kDA0ðkÞkps; kDAiðkÞkpZi; i ¼ 1; 2; . . . ; m; kDBpðkÞkpd; kDCpðkÞkpr
where s, Zi, d, and r are given constants. The process noise vðkÞ and measurement noise eðkÞ are uncorrelated random sequences with zero mean. Meanwhile, they have no time correlation or are ‘‘white’’ noises, that is,
EfvðiÞvTðjÞg ¼ 0 if iaj, (2.3a) EfeðiÞeTðjÞg ¼ 0 if iaj, (2.3b)
and their covariances or ‘‘noise levels’’ are defined by
EfvðkÞvTðkÞg R1, (2.3c)
EfeðkÞeTðkÞg R2. (2.3d)
In (2.3c)–(2.3d), R1and R2are symmetric, positive definite matrices and have the following
norm-bounds:
kR1R10kp1, (2.3e)
kR2R20kp2 (2.3f)
where 1, 2 are given positive constants and R10, R20 denote the nominal parts of the
actual covariances of the process noise vðkÞ and measurement noise eðkÞ, respectively. By defining an nðm þ 1Þ 1 new state vector
xðkÞ ¼ ½xTpðk mÞ xTpðk m þ 1Þ . . . xTpðk 1Þ xTpðkÞT, (2.4) system (2.1) can then be transformed into the following system:
xðk þ 1Þ ¼ AxðkÞ þ DAðkÞxðkÞ þ BuðkÞ þ DBðkÞuðkÞ þ vðkÞ, (2.5a)
yðkÞ ¼ CxðkÞ þ DCðkÞxðkÞ þ eðkÞ (2.5b) where A ¼ 0 I 0 0 0 0 0 I 0 0 .. . .. . .. . .. . .. . 0 0 0 0 I Am Am1 Am2 A1 A0 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 nn , DAðkÞ ¼ 0 0 0 0 0 0 0 0 .. . .. . .. . .. . 0 0 0 0 DAmðkÞ DAm1ðkÞ DA1ðkÞ DA0ðkÞ 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 nn , ð2:6aÞ
B ¼ 0 0 .. . 0 Bp 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 nr ; DBðkÞ ¼ 0 0 .. . 0 DBpðkÞ 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 nr , (2.6b) C ¼ ½0 0 0 Cppn; DCðkÞ ¼ ½0 0 0 DCpðkÞpn; vðkÞ ¼ GvðkÞ, (2.6c) where n nðm þ 1Þ and G ¼ 0 0 .. . 0 I 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 nn .
According to Eqs. (2.3c) and (2.6c), the covariance of vðkÞ is given by
EfvðkÞvTðkÞg ¼ GR1GTR1 (2.7)
and R1 has the following norm-bound:
kR1R10kp1 (2.8)
where R10 GR10GT and 11kGkkGTk.
Lemma 2.1. If rankðBpÞ ¼r, the pair fA; Bg is controllable.
Proof. From Eq. (2.6b), obtained here is rankðBÞ ¼ rankðBpÞ. If rankðBpÞ ¼r, the pair
fA; Bg is then controllable if and only if rank½B; AB; . . . ; AnrB rankðUnrÞ ¼n[12]. The
matrix Unr is easily found to have the following form:
Unr ¼ 0 0 0 Bp 0 0 0 ... A0Bp .. . .. . .. . .. . ... ... .. . .. . 0 ... .. . 0 Bp .. . 0 Bp A0Bp .. . Bp A0Bp ðA20þA1ÞBp 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 nðnrþ1Þr (2.9)
and Unr has n linearly independent columns. Thus, rankðUnrÞ ¼n. The proof is then
complete. &
Proof. From Eq. (2.6c), obtained here is rankðCÞ ¼ rankðCpÞ. If rankðCpÞ ¼p, the pair
fA; Cg is then observable if and only if[15]
rank C CA .. . .. . CAnp 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 ðnpþ1Þpn rankðOnpÞ ¼n. (2.10)
The matrix Onp is easily observed to have the following form:
Onp¼ 0 0 0 Cp CpAm CpAm1 CpA1 CpA0 CpA0Am CpAmþCpA0Am1 CpA2þCpA0A1 CpA1þCpA20 .. . .. . .. . .. . 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 ðnpþ1Þpn (2.11) and Onp has n linearly independent rows. Thus, rankðOnpÞ ¼n. The proof is then
complete. &
3. Robust minimax controller
Prior to discussing the design of robust minimax controller for the uncertain stochastic system (2.5), its nominal system is first considered (i.e., DAðkÞ ¼ 0, DBðkÞ ¼ 0, DCðkÞ ¼ 0 and 1¼2¼0):
xðk þ 1Þ ¼ AxðkÞ þ BuðkÞ þ vðkÞ, (3.1a)
yðkÞ ¼ CxðkÞ þ eðkÞ. (3.1b)
The performance index to be minimized is chosen as
J ¼X
1
k¼0
EfxTðkÞQxðkÞ þ uTðkÞWuðkÞg1 (3.2)
where Q ¼ QTX0, W ¼ WT40 and the triple fA, R1=21 , Q1=2g is assumed here to be controllable and observable. The optimal admissible control uoptðtÞ, which minimizes the
performance index J in Eq. (3.2) subject to dynamic system (3.1), is given by
uoptðkÞ ¼ GfxðkÞ^ (3.3a)
where
Gf ¼ ðW þ B T
SBÞ1BTSA (3.3b)
and S is the symmetric positive definite solution of the following discrete Riccati equation: S ¼ ATSA þ Q ðBTSAÞTðW þ BTSBÞ1ðBTSAÞ. (3.3c) Meanwhile, the estimated state ^x is the output of the Kalman filter:
^
xðk þ 1Þ ¼ A ^xðkÞ þ BuðkÞ þ F ½yðkÞ C ^xðkÞ, (3.4a)
F ¼ APCT½R2þCPC T
1 (3.4b)
where the Kalman gain F is chosen to minimize the state reconstruction error ~xðkÞ xðkÞ ^xðkÞ and P is the steady-state solution of the following equation:
Pðk þ 1Þ ¼ APðkÞATþR1APðkÞC T
½R2þCPðkÞC T
1CPðkÞAT. (3.4c) The objective here lies in formulating a robust controller for a given controllable and observable system (3.1) such that the optimal regulator (3.3) still minimizes the performance index J in Eq. (3.2) and the Kalman filter (3.4) also asymptotically tracks the actual states in the presence of parametric uncertainties and uncertain noise covariances.
The approach for the design of a robust controller is divided into two steps. In the first step, we only consider system (3.1) under uncertain noise covariances, i.e.
xðk þ 1Þ ¼ AxðkÞ þ BuðkÞ þ vðkÞ, (3.5a)
yðkÞ ¼ CxðkÞ þ eðkÞ, (3.5b)
with
R12S1¼ fR1: kR1R10kp1; R140g, (3.5c)
R22S2¼ fR2: kR2R20kp2; R240g. (3.5d)
The design of a robust controller for system (3.5) can therefore be considered as a saddlepoint problem which treats the uncertain (but bounded) noise covariances problem. By means of minimax theory, the following lemma is obtained here as:
Lemma 3.1 (Looze et al.[16], Chen and Dong[17]). The robust controller for system (3.5) is a minimax controller that solves the saddlepoint problem with the worst noise covariances, R10þ1I and R20þ2I , i.e. uðkÞ ¼ GfxðkÞ,^ (3.6a) Gf ¼ ðW þ B T SBÞ1BTSA, (3.6b) S ¼ ATSA þ Q ðBTSAÞTðW þ BTSBÞ1ðBTSAÞ, (3.6c) and ^
xðk þ 1Þ ¼ A ^xðkÞ þ BuðkÞ þ F ½yðkÞ C ^xðkÞ, (3.7a)
F ¼ APCT½ðR20þ2I Þ þ CPC T
Pðk þ 1Þ ¼ APðkÞATþ ðR10þ1I Þ APðkÞC T
½ðR20þ2I Þ
þCPðkÞCT1CPðkÞAT. ð3:7cÞ The minimax controller in Eqs. (3.6) and (3.7) may still not be robust if system (3.5) is perturbed not only by noise uncertainties but also by parametric uncertainties, i.e. the uncertain system (2.5) is considered. More constraints must consequently be imposed to let the minimax controller in Eqs. (3.6) and (3.7) become robust under parametric uncertainties.
Introducing the control law (3.6a) and subtracting (3.7a) from (2.5a) yield
xðk þ 1Þ ^xðk þ 1Þ ~xðk þ 1Þ ¼ ðA F CÞ ~xðkÞ þ ðDAðkÞ DBðkÞGf F DCðkÞÞxðkÞ
þDBðkÞGfxðkÞ þ vðkÞ F eðkÞ.~ ð3:8Þ
Combining Eq. (2.5a) with (3.8), we have
xðk þ 1Þ ¼ AxðkÞ þ DAðkÞxðkÞ þ HnðkÞ (3.9) where xðkÞ ¼ xðkÞ ~ xðkÞ " # , (3.10a) A ¼ A BGf BGf 0 A F C " # , (3.10b) DAðkÞ ¼ DAðkÞ DBðkÞGf DBðkÞGf DAðkÞ DBðkÞGf F DCðkÞ DBðkÞGf " # , (3.10c) H ¼ I 0 I F , (3.10d) nðkÞ ¼ vðkÞ eðkÞ " # . (3.10e)
A robust criterion is derived in the following to guarantee the asymptotic stability of system (3.9). Before proceeding to examine robust stability, Bellman–Gronwall lemma in discrete form which will be used in the proof of the next theorem is given below.
Lemma 3.2 (Desoer and Vidyasagar[18]). Let Zþ denote the set of nonnegative integers:
f0; 1; 2;. . .g and uðkÞ, f ðkÞ, hðkÞ be real-valued sequence on Zþ. Let
hðkÞX0; 8k 2 Zþ.
Under these conditions, if
uðkÞpf ðkÞ þX
k1
i¼0
then uðkÞpf ðkÞ þX k1 i¼0 Y k1 j¼iþ1 ½1 þ hðjÞhðiÞf ðiÞ ( ) ; k ¼ 0; 1; 2; . . . , (3.12)
in whichQk1j¼iþ1½1 þ hðjÞ is set equal to 1 when i ¼ k 1. Note that:
(a) If for some constant hM, hðiÞphM; 8i, then (3.12) becomes
uðkÞpf ðkÞ þ hM
Xk1 i¼0
ð1 þ hMÞki1f ðiÞ. (3.13)
(b) If for some constant fM, f ðiÞpFM; 8i, then (3.12) becomes
uðkÞpfMY
k1
i¼0
½1 þ hðiÞ. (3.14)
Theorem 1. Assume that the matrix A in Eq. (3.10b) is diagonalizable and Hurwitz (i.e., all the eigenvalues of A are inside the unit circle) so that the state transition matrix Aksatisfies the inequality:
kAkkpMrk 2; k ¼ 0; 1; 2; . . . , (3.15) in which MX1 and 0pro1. If the following inequality holds:
rð1 þ hÞo1 (3.16) with h M r 2 s þ Xm i¼1 Zi ! þ4dkGfk þrkF k " # , (3.17)
then system (3.9) is asymptotically stable. Namely, the minimax controller in Eqs. (3.6) and (3.7) is a robust LQG optimal controller under both parametric uncertainties and uncertain noise covariances.
Proof. See Appendix. &
4. Example
A fourth-order model of a fluid catalytic cracking unit (this model was obtained by linearization, followed by normalization, of the Lee and Kugelman[19] model around a nominal stable operating point, as described in Oliveira [20]) is given to illustrate the designed procedures. Using a sampling time T ¼ 0:05 s leads to the following discrete
2If A is a q q diagonalizable and Hurwitz matrix, the following result is obtained: kAkk ¼ kN ^AkN1kpkNkk ^AkkkN1kpMrkwhere ^A ¼ diagðl
1; l2; . . . ; lqÞand M ¼ kNkkkN1k(the condition number of A). The ith column of N is an eigenvector corresponding to li. Moreover, r is chosen as the distance from the origin to the eigenvalue (of the matrix A) nearest to the unit circle.
model: xpðk þ 1Þ ¼ A0xpðkÞ þ A1xpðk 1Þ þ BpuðkÞ þ DA0ðkÞxpðkÞ þ DA1ðkÞxpðk 1Þ þDBpðkÞuðkÞ þ vðkÞ, ð4:1aÞ yðkÞ ¼ CpxpðkÞ þ DCpðkÞxpðkÞ þ eðkÞ, (4.1b) with A0¼ 0:07362 0:1148 0:01044 0:4390 0:03940 0:1492 0:00894 0:8111 0:27940 0:7511 0:003253 6:127 0:04545 0:1559 0:009798 0:8384 2 6 6 6 4 3 7 7 7 5, A1¼ 0:0014 0:0498 0:0085 0:0022 0:0166 0:0058 0:0043 0:0011 0:0027 0:7024 0:0017 5:0004 0:0005 0:0034 0:0202 0:8006 2 6 6 6 4 3 7 7 7 5, Bp¼ 0:2495 0:6030 0:0168 0:2505 5:785 7:2139 0:01157 0:1217 2 6 6 6 4 3 7 7 7 5; Cp¼ 0 1 0 0 0 0 1 0 , and DA0ðkÞ ¼ 0:019 0:01 0:004 0:002 0:004 0:008 sinðkÞ 0 0:0026 0:002 0:003 0:009 0:006 expðkÞ 0:001 cosðkÞ 0:003 0 0:021 2 6 6 6 6 4 3 7 7 7 7 5, DA1ðkÞ ¼ 0:0064 0:01 0:0027 cosðkÞ 0:0041 0:0038 sinðkÞ 0:0035 0 0:0012 0:0015 0:002 0:001 0 0:003 0:004 sinðkÞ 0:005 0:0089 2 6 6 6 6 4 3 7 7 7 7 5, DBpðkÞ ¼ 0:015 cosðkÞ 0:013 0:007 0 0:02 0:011 0 0:013 " #T , DCpðkÞ ¼ 0:01 0:023 sinðkÞ 0 0:014 0:012 0 0:023 cosðkÞ 0:011 " # . ð4:2Þ
In Eq. (4.1), xp ¼ ½Csc Trx Crg TrgTand u ¼ ½Fa FcT. Here Cscdenotes the coke content
in the spent catalyst, Trx the reactor bed temperature, Crg the coke content in the
are Fa (air flow rate) and Fc(catalyst circulation rate). Moreover,
EfvðkÞg ¼ EfeðkÞg ¼ 0; EfvðkÞvTðkÞg R1; EfeðkÞeTðkÞg R2, (4.3a)
where R12S1¼ R1: R10 0:15 0 0 0 0 0:15 0 0 0 0 0:15 0 0 0 0 0:15 2 6 6 6 6 6 4 3 7 7 7 7 7 5 pR1pR10 8 > > > > > < > > > > > : þ 0:15 0 0 0 0 0:15 0 0 0 0 0:15 0 0 0 0 0:15 2 6 6 6 6 6 4 3 7 7 7 7 7 5 9 > > > > > = > > > > > ; , ð4:3bÞ R22S2¼ R2: R20 0:23 0 0 0:23 pR2pR20þ 0:23 0 0 0:23 , (4.3c) and R10 ¼ 0:22 0 0 0 0 0:27 0 0 0 0 0:21 0 0 0 0 0:02 2 6 6 6 4 3 7 7 7 5; R20¼ 0:35 0 0 0:43 . (4.3d)
Meanwhile, the weighting matrices are assumed here to be
Q ¼ 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 5 and W ¼ 12 0 0 12 . (4.4)
It is desired to design a robust LQG optimal controller to stabilize the uncertain stochastic time-delay system (4.1).
Solution: By defining a new state vector
xðkÞ ¼ ½xTpðk 1Þ xTpðkÞT, (4.5) the system (4.1) is transformed into the following system:
xðk þ 1Þ ¼ AxðkÞ þ DAðkÞxðkÞ þ BuðkÞ þ DBðkÞuðkÞ þ vðkÞ, (4.6a) yðkÞ ¼ CxðkÞ þ DCðkÞxðkÞ þ eðkÞ (4.6b)
where A ¼ 0 I A1 A0 " # ¼ 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0:0014 0:0498 0:0085 0:0022 0:07362 0:1148 0:01044 0:4390 0:0166 0:0058 0:0043 0:0011 0:03940 0:1492 0:00894 0:8111 0:0027 0:7024 0:0017 5:0004 0:27940 0:7511 0:003253 6:127 0:0005 0:0034 0:0202 0:8006 0:04545 0:1559 0:009798 0:8348 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 , ð4:7aÞ DAðkÞ ¼ 0 0 DA1ðkÞ DA0ðkÞ " # , B ¼ 0 Bp " # ¼ 0 0 0 0 0:6030 0:2505 7:2139 0:1217 0 0 0 0 0:2495 0:0168 5:785 0:01157 " #T , ð4:7bÞ DBðkÞ ¼ 0 DBpðkÞ " # ; vðkÞ ¼ GvðkÞ; G ¼ 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 2 6 6 6 4 3 7 7 7 5 T , (4.7c) C ¼ ½0 Cp ¼ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 ; DCðkÞ ¼ ½0 DCpðkÞ. (4.7d)
The pairs ðA; BÞ, ðA; CÞ are definitely observed from the fact of Lemmas 2.1 and 2.2 to be controllable and observable, respectively.
Based on Lemma 3.1, the minimax controller is described as follows:
uðkÞ ¼ GfxðkÞ,^ (4.8a) ^ xðk þ 1Þ ¼ A ^xðkÞ þ BuðkÞ þ F ½yðkÞ C ^xðkÞ (4.8b) where Gf ¼ 0:0017 0:0545 0:0065 0:5871 0:0402 0:0024 0:0092 0:2879 0:0011 0:0453 0:0061 0:1219 0:0008 0:1053 0:0081 0:6084 (4.8c)
0 50 100 -4 -2 0 2 4 TIME (sec) magnitude 0 50 100 -4 -2 0 2 4 TIME (sec) magnitude 0 50 100 -2 -1 0 1 2 TIME (sec) magnitude 0 50 100 -2 -1 0 1 2 TIME (sec) magnitude x1 ˆ x1 ˆ x2 x2 x3 ˆ x3 ˆ x4 x4 0 50 100 -4 -2 0 2 4 TIME (sec) magnitude 0 50 100 -2 -1 0 1 2 TIME (sec) magnitude 0 50 100 -10 -5 0 5 10 TIME (sec) magnitude 0 50 100 -4 -2 0 2 4 TIME (sec) magnitude x5 ˆ x5 ˆ x6 x6 x7 ˆ x7 x8 ˆ x8
Fig. 1. Simulation of time response for true state x ¼ ½ x1 x2 x3 x4 x5 x6 x7 x8Twith initial condition ½ 4 4 2 2 3 2 8 3 T and state estimate ^x ¼ ½ ^x
1 x^2 x^3 x^4 x^5 x^6 x^7 x^8T with initial condition ½ 4 3 2 2 4 2 7 3 T.
and F ¼ 0:0451 0:0474 0:9732 0:0288 0:0121 0:0237 0:6607 0:0531 0:0356 0:4535 0:0540 0:1063 0:1020 0:1554 1:2869 0:1098 T . (4.8d) The condition number M ¼ 2:77523and r ¼ 0:2926 obtained here through substituting Gf
and F into Eq. (3.10b) to obtain the matrix A and then applying the inequality (3.15). In accordance with Eqs. (2.2) and (4.2), we have s ¼ 0:023, Z1¼0:015, d ¼ 0:032 and r ¼ 0:03. Substituting kGfk and kF k into (3.17) yields h ¼ ðM=rÞ½2ðs þ Z1Þ þ4dkGfk þ
rkF k ¼ 2:1715 and then rð1 þ hÞ ¼ 0:928o1.
The robust stability condition (3.16) is hence satisfied. Namely, the minimax controller in Eq. (4.8) is a robust LQG optimal controller in the presence of parametric uncertainties and uncertain noise covariances. The result of simulation is shown inFig. 1.
5. Conclusion
In this paper, the LQG optimal control problem is considered and Minimax theory and Bellman–Gronwall lemma are employed to derive a robust criterion which guarantees the asymptotic stability of the discrete time-delay systems under both parametric uncertainties and uncertain noise covariances. On the basis of this criterion, a robust minimax controller composed of the Kalman filter and the optimal regulator is synthesized to stabilize the uncertain stochastic systems. Designed procedures are finally elaborated with an illustrative example.
Acknowledgments
The authors wish to express sincere gratitude to Prof. M.F. Aburdene and Prof. J. English for their help and the anonymous reviewers for their constructive comments and helpful suggestions, which lead to substantial improvements of this paper.
Appendix
The solution xðkÞ to Eq. (3.9) is expressed as
xðkÞ ¼ Akxð0Þ þX k1 j¼0 Akj1DAðjÞxðjÞ þX k1 j¼0 Akj1HnðjÞ. (A.1)
Taking norms on both sides of Eq. (A.1), we obtain
kxðkÞk ¼ Akxð0Þ þX k1 j¼0 Akj1DAðjÞxðjÞ þX k1 j¼0 Akj1HnðjÞ pkAkkkxð0Þk þX k1 j¼0 kAkj1kkDAðjÞkkxðjÞk þX k1 j¼0 kAkj1kkHkknðjÞk. ðA:2Þ
Similarly, taking norms on both sides of Eqs. (3.10c)–(3.10e), we have the following inequalities: kDAðkÞkpkDAðkÞ DBðkÞGfk þ kDAðkÞ DBðkÞGf F DCðkÞk þ 2kDBðkÞkkGfk pkDAðkÞk þ kDBðkÞkkGfk þ kDAðkÞk þ kDBðkÞkkGfk þ kF kkDCðkÞk þ2kDBðkÞkkGfk p2kDAðkÞk þ 4kDBðkÞkkGfk þ kF kkDCðkÞk p2X m i¼0 kDAiðkÞk þ 4kDBpðkÞkkGfk þ kF kkDCpðkÞk p2 s þX m i¼1 Zi ! þ4dkGfk þrkF k, ðA:3Þ kHkp2 þ kFk, (A.4) knðkÞkpkvðkÞk þ keðkÞkpkvðkÞk þ keðkÞkp½trðR10þ1I Þ1=2þ ½trðR20þ2I Þ1=2. (A.5) Substituting Eq. (3.15) and inequalities (A.3)–(A.5) into Eq. (A.2) yields
kxðkÞkpMrkkxð0Þk þX k1 j¼0 Mrkj1 2 s þX m i¼1 Zi ! þ4dkGfk þrkF k " # kxðjÞk þX k1 j¼0 Mrkj1ð2 þ kF kÞf½trðR10þ1I Þ1=2þ ½trðR20þ2I Þ1=2g. ðA:6Þ
Multiplying both sides of Eq. (A.6) by rkleads to
kxðkÞkrkpMkxð0Þk þX k1 j¼0 Mrj1 2 s þX m i¼1 Zi ! þ4dkGfk þrkF k " # kxðjÞk þX k1 j¼0 Mrj1ð2 þ kF kÞf½trðR10þ1I Þ1=2þ ½trðR20þ2I Þ1=2g. ðA:7Þ
Inequality (A.7) can be changed to
kxðkÞkrkpMkxð0Þk þ M1 r k r 1 ð2 þ kF kÞf½trðR10þ1I Þ 1=2 þ ½trðR20þ2I Þ1=2g þX k1 j¼0 Mr1 2 s þX m i¼1 Zi ! þ4dkGfk þrkF k " # kxðjÞkrj. ðA:8Þ
Applying Lemma 3.2 to Eq. (A.8), we obtain the following inequality: kxðkÞkrkpMkxð0Þk þ M1 r k r 1 ð2 þ kF kÞf½trðR10þ1I Þ 1=2þ ½trðR 20þ2I Þ1=2g þMkxð0Þk½ð1 þ hÞk1 þ hMð2 þ kF kÞf½trðR10þ1I Þ1=2 þ ½trðR20þ2I Þ1=2g ð1 þ hÞk1 hðr 1Þ rð1 þ hÞkr1k ðr 1Þ½rð1 þ hÞ 1 ( ) , ðA:9Þ where h ¼M r 2 s þ Xm i¼1 Zi ! þ4dkGfk þrkF k " # .
Multiplying rkto both sides of Eq. (A.9), we get the following result:
kxðkÞkpMrkkxð0Þk þ Mr k1 r 1ð2 þ kF kÞf½trðR10þ1I Þ 1=2þ ½trðR 20þ2I Þ1=2g þMkxð0Þk½rð1 þ hÞkMrkkxð0Þk þ Mð2 þ kF kÞf½trðR10þ1I Þ1=2 þ ½trðR20þ2I Þ1=2g ½rð1 þ hÞkrk r 1 Mð2 þ kF kÞf½trðR10þ1I Þ 1=2 þ ½trðR20þ2I Þ1=2g hf½rð1 þ hÞkr rg ðr 1Þ½rð1 þ hÞ 1. ðA:10Þ Since 0pro1 and rð1 þ hÞo1, kxðkÞk will approach to a certain value
Mð2 þ kF kÞ 1 r 1 þ M½2ðs þPmi¼1ZiÞ þ4dkGfk þrkF k r þ M½2ðs þPmi¼1ZiÞ þ4dkGfk þrkF k 1 ( ) f½trðR10þ1I Þ1=2 þ ½trðR20þ2I Þ1=2g, ðA:11Þ
as k ! 1. Thus, system (3.9) is asymptotically stable.
References
[1] T. Mori, Criteria for asymptotic stability of linear time delay systems, IEEE Trans. Autom. Control 30 (2) (1985) 158–161.
[2] F.-H. Hsiao, J.-G. Hsieh, M.S. Wu, Determination of the tolerable sector of series nonlinearities in uncertain time-delay systems under dynamical output feedback, ASME J. Dyn. Syst. Meas. Control 113 (1991) 525–531.
[3] C. Santacesaria, R. Scattolini, Easy tuning of Smith predictor in presence of delay uncertainty, Automatica 29 (1993) 1595–1597.
[4] T.H. Lee, Q.G. Wang, K.K. Tan, Robust Smith-predictor controller for uncertain delay systems, AIChE J. 42 (1996) 1033–1040.
[5] L. Dugard, E.I. Verriest, Stability and Control of Time-Delay Systems, Springer, Berlin, 1998.
[6] K.R. Lee, J.H. Kim, E.T. Jeung, H.B. Park, Output feedback robust H1control of uncertain fuzzy dynamic systems with time-varying delay, IEEE Trans. Fuzzy Syst. 8 (2000) 657–664.
[7] K. Gu, V.L. Kharitonov, J. Chen, Stability of Time-Delay Systems, Birkhau¨ser, Basel, 2003.
[8] N.-J. Su, H.-Y. Su, J. Chu, Dealy-dependent robust H1control for uncertain time-delay systems, IEE Proc. Control Theory Appl. 150 (2003) 489–492.
[9] R.-J. Wang, W.-W. Lin, W.-J. Wang, Stabilizability of linear quadratic state feedback for uncertain fuzzy time-delay systems, IEEE Trans. Syst. Man Cybern. Part-B 34 (2004) 1288–1292.
[10] M.V. Basin, A.E. Rodkina, Delay-dependent stability conditions for a class of nonlinear stochastic multiple state delay systems, Proceedings of the Fifth IFAC Workshop on Time Delay Systems, Leuven, Belgium, 2004, pp. 175–180.
[11] M.V. Basin, A.E. Rodkina, On delay-dependent stability for a class of nonlinear stochastic delay-difference equations, Dyn. Continuous, Discrete Impulsive Syst. 12A (2005) 663–675.
[12] M.V. Basin, J.G. Rodriguez-Gonzalez, R. Martinez-Zuniga, Optimal filtering for linear state delay systems, IEEE Trans. Autom. Control 50 (2005) 684–690.
[13] M.V. Basin, J.G. Rodriguez-Gonzalez, A closed-form optimal control for linear systems with equal state and input delays, Automatica 41 (2005) 915–921.
[14] M.V. Basin, J.G. Rodriguez-Gonzalez, R. Martinez-Zuniga, Optimal controller for linear systems with time delays in input and observations, Dyn. Continuous, Discrete and Impulsive Systems 12B (2005) 1–11. [15] C.-T. Chen, Linear System Theory and Design, Saunders College Publishing, Orlando, FL, 1984. [16] D.P. Looze, H.V. Poor, K.S. Vastola, J.C. Darragh, Minimax control of linear stochastic systems with noise
uncertainty, IEEE Trans. Autom. Control 28 (9) (1983) 882–888.
[17] B.-S. Chen, T.-Y. Dong, LQG optimal control system design under plant perturbation and noise uncertainty: a state-space approach, Automatica 25 (1989) 431.
[18] C.A. Desoer, M. Vidyasagar, Feedback Systems: Input–output Properties, Academic Press, New York, 1975. [19] W. Lee, A.M. Kugelman, Number of steady-state operating points and local stability of open-loop fluid
catalytic cracker, Ind. Eng. Chem. Process Des. Dev. 12 (2) (1973) 197.
[20] N.M.C. Oliveira, Newton-type algorithms for nonlinear constrained chemical process control, Ph.D. Thesis, Carnegie Mellon University, 1994.
