A singularly perturbed linear time-dependent controlled system with multiple point-wise and distributed state delays is considered. The delays in the fast state variable are small of order of the small positive multiplier for a part of the derivatives in the system, which is a parameter of the singular perturbation. The delays in the slow state variable are non-small. Two types of the original singularly perturbed system, standard and nonstandard, are analyzed. For each type, two much simpler parameter-free subsystems (the slow and fast ones) are associated with the original system. It is established in the paper that the approximate state-space controllability of the slow and fast subsystems yields the approximate state-space controllability of the original system robustly with respect to the parameter of singular perturbation for all its sufficiently small values. Illustrative examples are presented.
Z.Artstein and M.Slemrod, On singularly perturbed retarded functional differential equations, J. Differential Equations171(1) (2001), 88–109. doi:10.1006/jdeq.2000.3840.
2.
A.Bensoussan, G.Da Prato, M.C.Delfour and S.K.Mitter, Representation and Control of Infinite Dimensional Systems, Birkhauser, Boston, 2007.
3.
I.M.Cherevko, An estimate for the fundamental matrix of singularly perturbed differential-functional equations and some applications, Differ. Equ.33(2) (1997), 281–284.
4.
R.F.Curtain and H.J.Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer, New York, 1995.
5.
M.C.Delfour, C.McCalla and S.K.Mitter, Stability and the infinite-time quadratic cost problem for linear hereditary differential systems, SIAM J. Control13(1) (1975), 48–88. doi:10.1137/0313004.
6.
M.C.Delfour and S.K.Mitter, Controllability, observability and optimal feedback control of affine hereditary differential systems, SIAM J. Control10(2) (1972), 298–328. doi:10.1137/0310023.
7.
M.C.Delfour and S.K.Mitter, Controllability and observability for infinite-dimensional systems, SIAM J. Control10(2) (1972), 329–333. doi:10.1137/0310024.
8.
M.G.Dmitriev and G.A.Kurina, Singular perturbations in control problems, Autom. Remote Control67(1) (2006), 1–43. doi:10.1134/S0005117906010012.
9.
V.Dragan and A.Ionita, Exponential stability for singularly perturbed systems with state delays, Electron. J. Qual. Theory Differ. Equ.6 (2000), 8p., http://192.43.228.178/journals/EJQTDE/p37.pdf.
10.
E.Fridman, Robust sampled-data control of linear singularly perturbed systems, IEEE Trans. Autom. Control51(3) (2006), 470–475. doi:10.1109/TAC.2005.864194.
11.
R.Gabasov, F.M.Kirillova and V.V.Krakhotko, Controllability of multiloop systems with lumped parameters, Autom. Remote Control32(11) (1971), 1710–1717.
12.
Z.Gajic and M.-T.Lim, Optimal Control of Singularly Perturbed Linear Systems and Applications. High Accuracy Techniques, Dekker, New York, 2001.
13.
V.Y.Glizer, Euclidean space controllability of singularly perturbed linear systems with state delay, Systems Control Lett.43(3) (2001), 181–191. doi:10.1016/S0167-6911(01)00096-2.
14.
V.Y.Glizer, Controllability of singularly perturbed linear time-dependent systems with small state delay, Dynam. Control11(3) (2001), 261–281. doi:10.1023/A:1015276121625.
15.
V.Y.Glizer, Controllability of nonstandard singularly perturbed systems with small state delay, IEEE Trans. Automat. Control48(7) (2003), 1280–1285. doi:10.1109/TAC.2003.814277.
16.
V.Y.Glizer, Euclidean space controllability conditions for a class of singularly perturbed systems with state delays, in: Proc. 18th International Symposium on Mathematical Theory of Networks and Systems, Blacksburg, Virginia, USA, 2008, pp. 1–11, available at Digital Scholar Library of Virginia Technological Institute http://scholar.lib.vt.edu/MTNS/Papers/063.pdf.
17.
V.Y.Glizer, Novel controllability conditions for a class of singularly perturbed systems with small state delays, J. Optim. Theory Appl.137(1) (2008), 135–156. doi:10.1007/s10957-007-9324-8.
18.
V.Y.Glizer, -stabilizability conditions for a class of nonstandard singularly perturbed functional-differential systems, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms16(2) (2009), 181–213.
19.
V.Y.Glizer, Controllability conditions of linear singularly perturbed systems with small state and input delays, Math. Control Signals Syst.28(1) (2016), 1–29. doi:10.1007/s00498-015-0152-3.
20.
A.Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York, 1966.
21.
J.K.Hale and S.M.Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.
22.
V.U.Hoang Linh, On the robustness of asymptotic stability for a class of singularly perturbed systems with multiple delays, Acta Math. Vietnam.30(2) (2005), 137–151.
23.
T.Kaczorek, Linear Control Systems, Research Studies Press and John Wiley, New York, 1993.
24.
R.E.Kalman, Contributions to the theory of optimal control, Bol. Soc. Mat. Mexicana5(2) (1960), 102–119.
25.
J.Klamka, Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht, 1991.
26.
J.Klamka, Controllability of dynamical systems. A survey, Bull. Pol. Acad. Sci.: Tech.61(2) (2013), 335–342.
27.
P.V.Kokotovic, A.Bensoussan and G.L.Blankenship, eds, Singular Perturbations and Asymptotic Analysis in Control Systems, Lecture Notes in Control and Information Sciences, Vol. 90, Springer-Verlag, Berlin, 1987.
28.
P.V.Kokotovic and A.H.Haddad, Controllability and time-optimal control of systems with slow and fast modes, IEEE Trans. Automat. Control20(1) (1975), 111–113. doi:10.1109/TAC.1975.1100852.
29.
P.V.Kokotovic, H.K.Khalil and J.O’Reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, London, 1986.
30.
A.N.Kolmogorov and S.V.Fomin, Introductory Real Analysis, Dover, New York, 2007.
31.
T.B.Kopeikina, Controllability of singularly perturbed linear systems with delay, Differ. Equ.25(9) (1989), 1055–1064.
32.
T.B.Kopeikina, Unified method of investigating controllability and observability problems of time-variable differential systems, Funct. Differ. Equ.13(3–4) (2006), 463–481.
33.
C.Kuehn, Multiple Time Scale Dynamics, Springer, New York, 2015.
34.
G.A.Kurina, Complete controllability of singularly perturbed systems with slow and fast modes, Math. Notes52(3–4) (1992), 1029–1033.
35.
A.Manitius and R.Triggiani, Function space controllability of linear retarded systems: A derivation from abstract operator conditions, SIAM J. Control Optim.16(4) (1978), 599–645. doi:10.1137/0316041.
36.
A.Manitius and R.Triggiani, Sufficient conditions for function space controllability and feedback stabilizability of linear retarded systems, IEEE Trans. Automat. Control23(4) (1978), 659–665. doi:10.1109/TAC.1978.1101796.
37.
C.D.Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, 2000.
38.
P.Sannuti, On the controllability of singularly perturbed systems, IEEE Trans. Automat. Control22(4) (1977), 622–624. doi:10.1109/TAC.1977.1101568.
39.
P.Sannuti, On the controllability of some singularly perturbed nonlinear systems, J. Math. Anal. Appl.64(3) (1978), 579–591.
40.
J.L.Soule, Linear Operators in Hilbert Space, Gordon and Breach, New York, 1968.
41.
A.B.Vasil’eva, V.F.Butuzov and L.V.Kalachev, The Boundary Function Method for Singular Perturbation Problems, SIAM, Philadelphia, 1995.
42.
R.B.Vinter and R.H.Kwong, The infinite time quadratic control problem for linear systems with state and control delays: An evolution equation approach, SIAM J. Control Optim.19(1) (1981), 139–153. doi:10.1137/0319011.
43.
Y.Zhang, D.S.Naidu, C.Cai and Y.Zou, Singular perturbations and time scales in control theories and applications: An overview 2002–2012, Int. J. Inf. Syst. Sci.9(1) (2014), 1–36.
