# zbMATH — the first resource for mathematics

$$H_\infty$$ control of linear singularly perturbed systems with small state delay. (English) Zbl 0965.93049
An infinite horizon $$H_\infty$$ state-feedback control problem for singularly perturbed linear systems with a small state delay is considered. The system under consideration is: \begin{aligned} \dot x & = A_1x(t)+ A_2y(t)+ H_1x(t- \varepsilon h)+ H_2y(t- \varepsilon h)+ B_1u(t)+ F_1 w(t),\\ \varepsilon\dot y(t) & = A_3x(t)+ A_4y(t)+ H_3 x(t-\varepsilon h)+ H_4y(t- \varepsilon h)+ B_2u(t)+ F_2 w(t),\\ V(t) & = \text{col}\{C_1x(t)+ C_2y(t), u(t)\},\quad t>0,\end{aligned} where $$x$$, $$y$$ are state variables, $$u$$ is a control, $$w$$ is an exogenous disturbance, $$v$$ is a control output, $$\varepsilon> 0$$ is a small parameter.
An asymptotic solution of the hybrid system of Riccati type algebraic, ordinary differential, and partial differential equations with deviating arguments, associated with this $$H_\infty$$ control problem, is constructed. Based on this asymptotic solution, conditions for the existence of a solution of the original $$H_\infty$$ problem, independent of the singular perturbation parameter, are derived. A simplified controller with parameter-independent gain matrices, solving the original problem for all sufficiently small values of this parameter, is obtained. An illustrative example is presented.

##### MSC:
 93B36 $$H^\infty$$-control 93C23 Control/observation systems governed by functional-differential equations 93C70 Time-scale analysis and singular perturbations in control/observation systems
Full Text:
##### References:
  Basar, T.; Bernard, P., H∞-optimal control and related minimax design problems: A dynamic games approach, (1991), Birkhäuser Boston  Bensoussan, A., Perturbation methods in optimal control, (1988), Wiley New York  Bensoussan, A.; Da Prato, G.; Delfour, M.C.; Mitter, S.K., Representation and control of infinite dimensional systems, (1992), Birkhäuser Boston  Doyle, J.C.; Glover, K.; Khargonekar, P.P.; Francis, B., State-space solution to standard H2 and H∞ control problem, IEEE trans. automat. control, 34, 831-847, (1989) · Zbl 0698.93031  Dragan, V., Asymptotic expansions for game-theoretic Riccati equations and stabilization with disturbance attenuation for singularly perturbed systems, Systems control lett., 20, 455-463, (1993) · Zbl 0784.93040  Fridman, E., Asymptotic of integral manifolds and decomposition of singularly perturbed systems of neutral type, Differential equations, 26, 457-467, (1990) · Zbl 0722.34063  Fridman, E., Decomposition of linear optimal singularly perturbed systems with after-effect, Automat. remote control, 51, 1518-1527, (1990) · Zbl 0733.49033  Fridman, E., Decoupling transformation of singularly perturbed systems with small delays and its application, Z. angew. math. mech., 76, 201-204, (1996) · Zbl 0886.34063  Fridman, E.; Shaked, U., H∞ state-feedback control of linear systems with small state-delay, Systems control lett., 33, 141-150, (1998) · Zbl 0902.93022  Glizer, V.Y., Asymptotic solution of a singularly perturbed set of functional-differential equations of Riccati type encountered in the optimal control theory, Nodea nonlinear differential equations appl., 5, 491-515, (1998) · Zbl 0917.34056  Glizer, V.Y., Stabilizability and detectability of singularly perturbed linear time-invariant systems with delays in state and control, J. dynam. control systems, 5, 153-172, (1999) · Zbl 0943.93045  Glizer, V.Y., Asymptotic solution of a cheap control problem with state delay, Dynam. control, 9, 339-357, (1999) · Zbl 0958.49013  Halanay, A., Differential equations: stability, oscillations, time lags, (1966), Academic Press New York · Zbl 0144.08701  Hale, J., Critical cases for neutral functional equations, J. differential equations, 10, 59-82, (1971) · Zbl 0223.34057  van Keulen, B., H∞-control for distributed parameter systems: A state-space approach, (1993), Birkhäuser Boston · Zbl 0788.93018  Khalil, H.K., Feedback control of nonstandard singularly perturbed systems, IEEE trans. automat. control, 34, 1052-1060, (1989) · Zbl 0695.93030  Khalil, H.K.; Chen, F., H∞ control of two-time-scale systems, Systems control lett., 19, 35-42, (1992) · Zbl 0765.93024  Kokotovic, P.V., Applications of singular perturbation techniques to control problems, SIAM rev., 26, 501-550, (1984) · Zbl 0548.93001  Kokotovic, P.V.; Khalil, H.K.; O’Reilly, J., Singular perturbation methods in control: analysis and design, (1986), Academic Press London · Zbl 0646.93001  Leitmann, G., On one approach to the control of uncertain systems, J. dynamic systems, measurement, and control, 115, 373-380, (1993)  O’Malley, R.E., Singular perturbations and optimal control, Lecture notes in mathematics, (1978), Springer-Verlag New York/Berlin, p. 170-218  Pan, Z.; Basar, T., H∞-optimal control for singularly perturbed systems. part I: perfect state measurements, Automatica J. IFAC, 29, 401-424, (1993) · Zbl 0782.49015  Pan, Z.; Basar, T., H∞-optimal control for singularly perturbed systems. II. imperfect state measurements, IEEE trans. automat. control, 39, 280-300, (1994) · Zbl 0806.93018  P. B. Reddy and P. Sannuti, Optimal control of singularly perturbed time delay systems with an application to a coupled core nuclear reactor, inProceedings, IEEE Conference on Decision Control, 1974, pp. 793-803.  Reddy, P.B.; Sannuti, P., Optimal control of a coupled-core nuclear reactor by a singular perturbation method, IEEE trans. automat. control, 20, 766-769, (1975)  Saksena, V.R.; O’Reilly, J.; Kokotovic, P.V., Singular perturbations and time-scale methods in control theory: survey 1976-1983, Automatica J. IFAC, 20, 273-293, (1984) · Zbl 0532.93002  Tan, W.; Leung, T.; Tu, Q., H∞ control for singularly perturbed systems, Automatica J. IFAC, 34, 255-260, (1998) · Zbl 0911.93032  Vasil’eva, A.B.; Dmitriev, M.G., Singular perturbations in optimal control problems, J. soviet math., 34, 1579-1629, (1986) · Zbl 0595.49019  Vasil’eva, A.B.; Butuzov, V.F.; Kalachev, L.V., The boundary function method for singular perturbation problems, (1995), SIAM Philadelphia · Zbl 0823.34059  Xu, H.; Mizukami, K., Infinite-horizon differential games of singularly perturbed systems: a unified approach, Automatica J. IFAC, 33, 273-276, (1997) · Zbl 0869.90094
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.