www.ietdl.org
Published in IET Control Theory and Applications Received on 10th August 2011
Revised on 15th May 2012 doi: 10.1049/iet-cta.2011.0486
ISSN 1751-8644
Brief Paper
Robust controllability of linear systems with multiple
delays in control
S.-H. Chen
1F.-I Chou
2J.-H. Chou
1,31Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, 415 Chien-Kung Road, Kaohsiung 807, Taiwan
2Mechatronics Section, Energy and Agile System Department, Metal Industries Research and Development Centre, 1001 Kaonan Highway, Kaohsiung 811, Taiwan
3Institute of System Information and Control, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchao, Kaohsiung 824, Taiwan
E-mail: choujh@nkfust.edu.tw
Abstract: The considered problem is robust controllability of linear systems with both multiple delays in control and structured
parametric uncertainties. Under the assumption that the linear nominal system with multiple control delays is controllable, a sufficient condition is proposed to preserve the assumed property when system uncertainties are introduced. The application of the proposed sufficient condition is demonstrated in two examples.
1
Introduction
Control processes for dynamic systems are often severely limited. One example of a dynamic system with control delay is a control actuator. Time-delay systems have been studied intensively in the past 20 years, most research works have focused on stability conditions and stabilisation problems [1–8] (see, e.g., and references therein). Although some studies have successfully solved stability and control problems in linear systems with multiple delays in control [9–12], controllability remains an important line of research in control theory because of its essential role in dynamic control systems [13]. Thus, some researchers studied the controllability problem in linear systems with multiple control delays (see, e.g. [14–21] and references therein). Further, obtaining accurate values for some system parameters may be very difficult, if not impossible, because of inaccurate measurements, inaccessible system parameters, or variation of parameters. Such system uncertainties may compromise the controllability property of linear systems with multiple control delays. However, a literature review shows no studies of the issue of robust controllability of uncertain linear systems with multiple delays in control. That is, the controllability problem has been studied in linear systems with multiple delays in control but not in uncertain linear systems with multiple delays in control.
This study therefore developed an approach for investigating robust controllability in linear systems with both multiple delays in control and structured parametric uncertainties. For a linear nominal system in which
multiple delays in control are controllable, a sufficient condition is proposed for preserving the assumed property when system uncertainties are introduced. The proposed sufficient condition preserves the assumed property by revealing the explicit relationships among bounds on system uncertainties. Two numerical examples are given to illustrate the application of the proposed sufficient condition.
2
Controllability robustness
Consider the following linear system with both multiple control delays and system uncertainties
˙x(t) = (A + A)x(t) +
p
i=0
(Bi+ Bi)u(t− hi) (1)
where x(t)∈ Rn is the system state vector; u(t)∈ Rm is the
control input vector; 0= h0<h1<h2< <hpdenote the
time delays in control input where hiis a constant; A and Bi
are the n× n and n × m constant matrices, respectively; and
A and Bi are the uncertain matrices existing in system
matrix A and in the control input matrices Bi, respectively,
because of inaccurate measurements, inaccessible system parameters, or variation in system parameters.
Although many interesting problems involve only a small number of uncertainties, the uncertainties may comprise many entries in the system and in the input matrices [22–24]. For example, consider the two-mass system with an uncertain stiffness given by Sinha [24]. The
1552 IET Control Theory Appl., 2012, Vol. 6, Iss. 10, pp. 1552–1556
www.ietdl.org
ε < (1/( k¯m=1θk)), where |εk| ≤ ε, in order to estimate
the maximal bound on parametric uncertainties that still maintains robust controllability. Additionally, if the uncertain parameters εk(k= 1, 2, . . . , ¯m) are positive, the
statements of the main theorem can be simplified, and Lemma 4 is unnecessary. However, the obtained result is more conservative than the sufficient condition (9) proposed here.
3
Illustrative examples
This section gives two numerical examples to illustrate the application of the proposed sufficient condition.
Example 1: Consider a linear uncertain system with multiple
control delays described by ˙x(t) = A+ 3 k=1 εkAk x(t)+ 2 i=0 Bi+ 3 k=1 εkBik u(t− hi) (15) where h0= 0, h1= 1, h2= 2 A= 1 −1 −1 2 0 −3 −4 −3 0 , B0= 0 0 2 , B1= 0 1 0 B2= −1 0 1 , A1= 0 0 0 1 0 −1 −1 −1 0 A2= A3= 0 0 0 0 0 0 0 0 0 , B02= 0 0 1 , B01= B03= 0 0 0 B11= B12= 0 0 0 , B13= 0 1 0 , B21= 0 0 0 B22= −1 0 0 , B23= 0 0 1 , ε1∈ [−1.1 4.4] ε2∈ [−0.3 100] and ε3∈ [−0.3 1.6]
Applying the proposed condition in (9) for robust controllability then obtains
(i) 3 k=1 εkθk ≤ 0.73089 < 1, for ε1∈ [0 4.4], ε2∈ [0 100] and ε3∈ [0 1.6] (16a) (ii) 3 k=1 εkθk ≤ 0.99662 < 1, for ε1∈ [0 4.4], ε2∈ [−0.3 0] and ε3∈ [0 1.6] (16b) (iii) 3 k=1 εkθk ≤ 0.99372 < 1, for ε1∈ [0 4.4], ε2∈ [−0.3 0] and ε3∈ [−0.3 0] (16c) (iv) 3 k=1 εkθk ≤ 0.72798 < 1, for ε1∈ [0 4.4], ε2∈ [0 100] and ε3∈ [−0.3 0] (16d) (v) 3 k=1 εkθk ≤ 0.70383 < 1, for ε1∈ [−1.1 0], ε2∈ [0 100] and ε3∈ [0 1.6] (16e) (vi) 3 k=1 εkθk ≤ 0.96957 < 1, for ε1∈ [−1.1 0], ε2∈ [−0.3 0] and ε3∈ [0 1.6] (16f ) (vii) 3 k=1 εkθk ≤ 0.96666 < 1, for ε1∈ [−1.1 0], ε2∈ [−0.3 0] and ε3∈ [−0.3 0] (16g) (viii) 3 k=1 εkθk ≤ 0.70083 < 1, for ε1∈ [−1.1 0], ε2∈ [0 100] and ε3∈ [−0.3 0] (16h)
The results for (16) confirm that the linear uncertain system with multiple control delays in (15) is robustly controllable.
Example 2: Consider an uncertain linear system with
multiple control delays described by the same equation in Example 1, where |εk| ≤ ε (k = 1, 2, 3).
Applying the proposed sufficient condition in (9) gives a maximal bound of ε < (1/( 3k=1θk))= 0.48335
on parametric uncertainties for ensuring the robust controllability of the uncertain system with multiple delays in control.
4
Conclusions
This study solved the problem of robust controllability of a linear uncertain system with multiple delays in control by converting the rank preservation problem into a non-singularity analysis problem. Assuming that the linear nominal system with multiple delays in control is controllable, a sufficient condition was proposed to preserve the assumed property when system uncertainties are introduced. The proposed sufficient condition preserves the assumed property by revealing the specific relationships of bounds on system uncertainties. Two examples were given to demonstrate the application of the proposed sufficient condition. This study presents a novel approach to robust controllability of uncertain linear systems with multiple control delays. Future works by the authors will consider related problems of robust controllability in linear interval systems with multiple delays in control.
5
Acknowledgment
This work was in part supported by the National Science Council, Taiwan, Republic of China, under grant numbers NSC100-2221-E151-009 and NSC99-2221-E1514-072-MY2.
6
References
1 Sun, J., Liu, G.P., Chen, J., Rees, D.: ‘Improved stability criteria for linear systems with time-varying delay’, IET Control Theory Appl., 2010, 4, pp. 683–689
2 Sun, M., Jia, Y.: ‘Delay dependent robust H∞ control of time-delay systems’, IET Control Theory Appl., 2010, 4, pp. 1122–1130
IET Control Theory Appl., 2012, Vol. 6, Iss. 10, pp. 1552–1556 1555
www.ietdl.org
3 Wang, W., Zhang, H., Xie, L.: ‘Kalman filtering for continuous-time systems with time-varying delay’, IET Control Theory Appl., 2010, 4, pp. 590–600
4 Donkers, M.C.F., Heemels, W.P.M.H.: ‘Stability analysis of networked control systems using a switched linear systems approach’, IEEE Trans. Autom. Control, 2011, 56, pp. 2101–2115
5 Tong, S., Li, Y., Feng, G., Li, T.: ‘Observer-based adaptive fuzzy backstepping dynamic surface control for a class of non-linear systems’, IET Control Theory Appl., 2011, 5, pp. 1426–1438 6 Zhang, H., Shi, Y.: ‘Robust static output feedback control and remote
PID design for networked motor systems’, IEEE Trans. Ind. Electron., 2011, 58, pp. 5396–5405
7 Zhang, J., Xia, Y., Shi, P., Mahmoud, M.S.: ‘New results on stability and stabilization of systems with interval time-varying delay’, IET Control Theory Appl., 2011, 5, pp. 429–436
8 Yan, X.G., Spurgeon, S.K., Edwards, C.: ‘Global decentralized static output feedback sliding mode control for interconnected time-delay systems’, IET Control Theory Appl., 2012, 6, pp. 192–202
9 Klein, E.J., Ramirez, W.F.: ‘State controllability and optimal regulator control of time-delayed systems’, Int. J. Control, 2001, 74, pp. 281– 289
10 Gu, K., Niculescu, S.I.: ‘Survey on recent results in the stability and control of time-delay systems’, ASME J. Dyn. Syst. Meas. Control, 2003, 125, pp. 158–165
11 Basin, M., Rodriguez-Gonzalez, J.: ‘Optimal control for linear systems with multiple time delays in control input’, IEEE Trans. Autom. Control, 2006, 51, pp. 91–97
12 Wu, H.: ‘Adaptive robust control of uncertain dynamical systems with multiple time-varying delays’, IET Control Theory Appl., 2010, 4, pp. 1775–1784
13 Rosenbrock, H.H.: ‘State-space and multivariable theory’ (John Wiley and Sons, New York, 1970)
14 Chyung, D.H.: ‘Controllability of linear systems with multiple delays in control’, IEEE Trans. Autom. Control, 1970, 15, pp. 694–695 15 Desoer, C.A., Vidyasagar, M.: ‘Feedback systems: input-output
properties’ (Academic Press, New York, 1975)
16 Jiang, W., Song, W., Fei, S., Song, S.: ‘On the controllability and the stabilizability of control delay systems’. Proc. Third World Congress on Intelligent Control and Automation, Hefei, China, 2000, pp. 2846– 2849
17 Klamka, J.: ‘Stochastic controllability and minimum energy control of systems with multiple delays in control’, Appl. Math. Comput., 2008, 206, pp. 704–715
18 Klamka, J.: ‘Stochastic controllability of systems with multiple delays in control’, Int. J. Appl. Math. Comput. Sci., 2009a, 19, pp. 39–47 19 Klamka, J.: ‘Controllability of higher-order linear systems with
multiple delays in control’. Proc. Int. Conf. on Control and Automation, Christchurch, New Zealand, 2009b, pp. 1158–1162 20 Li, J.N., Zhang, Q.L., Li, Y.: ‘Controllability and observability of
networked control systems with time-varying delays’, J. Syst. Eng. Electron., 2009, 20, pp. 800–806
21 Khartovskii, V.E.: ‘A generalization of the problem of complete controllability for differential systems with commensurable delays’, J. Comput. Syst. Sci Int., 2009, 48, pp. 847–855
22 Zhou, K., Khargonekar, P.P.: ‘Stability robustness for linear state-space models with structured uncertainty’, IEEE Trans. Autom. Control, 1987, 32, pp. 621–623
23 Chen, J., Ren, Z.: ‘A Comparison of small gain versus Lyapunov type robust stability bounds’, Int. J. Robust Nonlinear Control, 2001, 11, pp. 1407–1414
24 Sinha, A.: ‘Linear systems: optimal and robust control’ (CRC Press, London, 2007)
25 Olbrot, A.W.: ‘On controllability of linear systems with time delays in control’, IEEE Trans. Autom. Control, 1972, 17, pp. 664–666
1556 IET Control Theory Appl., 2012, Vol. 6, Iss. 10, pp. 1552–1556
