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Singular perturbations and time-scale methods in control theory: Survey 1976-1983. (English) Zbl 0532.93002

Summary: Recent progress in the use of singular perturbation and two-time-scale methods of modeling and design for control systems is reviewed. Over 350 references are organized into major problem areas. Representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application. The survey is aimed at engineers and applied mathematicians interested in model-order reduction, separation of time scales and allied simplified methods of control system analysis and design. The exposition does not assume prior knowledge of singular perturbation methods.

MSC:

93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
34E15 Singular perturbations for ordinary differential equations
93A15 Large-scale systems
34D15 Singular perturbations of ordinary differential equations
93E20 Optimal stochastic control
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C10 Nonlinear systems in control theory
93B35 Sensitivity (robustness)
93E11 Filtering in stochastic control theory
93C40 Adaptive control/observation systems
93C05 Linear systems in control theory
93B05 Controllability
93B07 Observability
68U20 Simulation (MSC2010)
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References:

[1] (Ardema, M., Singular Perturbations in Systems and Control. Singular Perturbations in Systems and Control, CISM Courses and Lectures, 280 (1983), Springer: Springer New York) · Zbl 0504.00044
[2] Arnold, V. I., (Geometrical Methods in the Theory of Ordinary Differential Equations (1983), Springer: Springer Berlin) · Zbl 0507.34003
[3] Bensoussan, A.; Lions, J. L.; Papanicolaou, G. C., (Asymptotic Analysis for Periodic Structures (1978), North-Holland: North-Holland New York) · Zbl 0411.60078
[4] Blankenship, G. L., Asymptotic analysis in mathematical physics and control theory; some problems with common features, Richerche di Automatica, 10, 265 (1979)
[5] Campbell, S. L., (Singular Systems of Differential Equations (1980), Pitman: Pitman New York) · Zbl 0419.34007
[6] Campbell, S. L., (Singular Systems of Differential Equations II (1982), Pitman: Pitman New York) · Zbl 0482.34008
[7] (Chow, J. H., Time-scale Modeling of Dynamic Networks with Applications to Power Systems. Time-scale Modeling of Dynamic Networks with Applications to Power Systems, Lecture Notes in Control Information Science, 46 (1982), Springer: Springer Berlin) · Zbl 0498.93003
[8] Dontchev, A. L., Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems, (Lecture Notes in Control Information Science, 52 (1983), Springer: Springer Berlin) · Zbl 0512.49001
[9] Eckhaus, W., Formal approximation and singular perturbations, SIAM Review, 19, 593 (1977) · Zbl 0384.35007
[10] Eckhaus, W., (Asymptotic Analysis of Singular Perturbations (1979), North-Holland: North-Holland Amsterdam) · Zbl 0421.34057
[11] Eckhaus, W.; de Jager, E. M., (Theory and Applications of Singular Perturbations (1982), Springer: Springer Berlin) · Zbl 0478.00011
[12] Fossard, A. J.; Magni, J. F., Modelisation commande et applications des systemes a echelles de temps multiples, R.A.I.R.O. Automatique/Systems Analysis and Control, 16, 1 (1982) · Zbl 0486.93006
[13] Frank, P. M., (Introduction to System Sensitivity Theory (1978), Academic Press) · Zbl 0464.93001
[14] Hemker, P. W.; Miller, J. J.H., (The Numerical Analysis of Singular Perturbation Problems (1979), Academic Press: Academic Press New York) · Zbl 0407.00011
[15] Ioannou, P. A.; Kokotovic, P. V., Adaptive Systems with Reduced Models, (Lecture Notes in Control Information Science, 47 (1983), Springer: Springer Berlin) · Zbl 0516.93017
[16] Kevorkian, J.; Cole, J. D., (Perturbation Methods in Applied Mathematics (1981), Springer: Springer Berlin) · Zbl 0456.34001
[17] Kokotovic, P. V.; O’Malley, R. E.; Sannuti, P., Singular perturbations and order reduction in control theory—an overview, Automatica, 12, 123 (1976) · Zbl 0323.93020
[18] Lomov, S. A., (Introduction to the General Theory of Singular Perturbations (1981), Nauka: Nauka Moscow) · Zbl 0514.34049
[19] Meyer, R. E.; Parter, S. V., (Singular Perturbations and Asymptotics (1980), Academic Press: Academic Press New York)
[20] Mischenko, E. F.; Rozov, N. K., (Differential Equations with Small Parameters and Relaxation Oscillations (1980), Plenum Press: Plenum Press New York)
[21] Moiseev, N. N.; Chernousko, F. L., Asymptotic methods in the theory of optimal control, IEEE Trans. Aut. Control, AC-26, 993 (1981) · Zbl 0478.49027
[22] O’Malley, R. E., Book Review, Bulletin AMS, 7, 414 (1982)
[23] Pervozvanskii, A. A.; Gaitsgori, V. G., Suboptimization, decomposition and aggregation, (Proceedings of the 7th IFAC World Congress. Proceedings of the 7th IFAC World Congress, Helsinki, Finland (1978)) · Zbl 0456.93004
[24] Pervozvanskii, A. A.; Gaitsgori, V. G., (Decomposition, Aggregation and Suboptimization (1979), Nauka), (in Russian)
[25] Sandell, N. R.; Varaiya, P.; Athans, M.; Safonov, M. G., Survey of decentralized control methods for large scale systems, IEEE Trans. Aut. Control, AC-23, 108 (1978) · Zbl 0385.93001
[26] Schuss, Z., Singular perturbation methods in stochastic differential equations of mathematical physics, SIAM Review, 22, 119 (1980) · Zbl 0436.60045
[27] Utkin, V. I., (Sliding Modes and their Application to Variable Structure Systems (1977), Mir: Mir Moscow)
[28] Utkin, V. I., Variable structure systems with sliding modes: a survey, IEEE Trans Aut. Control, AC-22, 212 (1977) · Zbl 0382.93036
[29] Utkin, V. I., Variable structure systems: state of the art and perspectives, Aut. & Remote Control, 9, 5 (1983)
[30] Vasileva, A. B., Progress of the theory of o.d.e. with a small parameter multiplying higher derivatives, in the period 1966-1976, Soviet Math. Surveys (Uspehi), XXXI, 6 (192), 102 (1976)
[31] Vasileva, A. B.; Butuzov, V. F., (Singularly Perturbed Equations in Critical Cases (1978), Moscow University Press)
[32] Vasileva, A. B.; Butuzov, V. F., On some results of the theory of singular perturbations in the last five years, Vestnik Mos. Univ. Comp. Math. Cyber. Ser., 15, 3, 35 (1981)
[33] Vasileva, A.; Dmitriev, M., Singular perturbations and some optimal control problems, (Proceedings of the 7th IFAC World Congress. Proceedings of the 7th IFAC World Congress, Paper 23.6 (1978))
[34] Voronov, A. A., Present state and problems of stability theory, Aut. & Remote Control, 5, 5 (1982) · Zbl 0501.93003
[35] Bobisud, L. E.; Christenson, C. O., A singular singularly perturbed system of nonlinear equations from chemical kinetics, J. Math. Anal. & Appl., 74, 296 (1980) · Zbl 0432.35014
[36] Fenichel, N., Geometric singular perturbation theory for ordinary differential equations, J. Diff. Equations, 31, 53 (1979) · Zbl 0476.34034
[37] Glizer, V. J.; Dmitriev, M. G., Singular perturbations and generalized functions, Soviet Dokl, 20, 1360 (1979) · Zbl 0448.34052
[38] Howes, F. A., A boundary layer theory for a class of linear and nonlinear boundary value problems, Rocky Mtn. J. Math., 7, 491 (1977) · Zbl 0368.34023
[39] Howes, F. A., Singularly perturbed semilinear systems, Studies in Appl. Math., 61, 185 (1979) · Zbl 0424.34020
[40] Howes, F. A.; O’Malley, R. E., Singular Perturbations of Second-order Semilinear Systems, (Lecture Notes in Mathematics, 827 (1980), Springer: Springer Berlin), 131 · Zbl 0448.34062
[41] Lakin, W. D.; Van den Driessche, P., Time scales in population biology, SIAM J. Appl. Math., 32, 694 (1977) · Zbl 0358.92016
[42] MacGillivray, A. D., The existence of an overlap domain for a singular perturbation problem, SIAM J. Appl. Math., 36, 106 (1979) · Zbl 0412.76034
[43] MacGillivray, A. D., On the switch back term in the asymptotic expansion of a model singular perturbation problem, J. Math. Anal. & Appl., 77, 612 (1980) · Zbl 0448.34053
[44] Matkowsky, B. J.; Reiss, E. L., Singular perturbations of bifurcations, SIAM J. Appl. Math., 33, 230 (1977) · Zbl 0379.35007
[45] Mika, J., Singularly perturbed evolution equations in banach spaces, J. Math. Anal. & Appl., 58, 189 (1977) · Zbl 0353.34065
[46] Naidu, D. S.; Rajagopalan, P. K., Application of Vasileva’s singular perturbation method to a problem in ecology, Int. J. Syst. Sci., 10, 761 (1979) · Zbl 0414.92027
[47] Nipp, K., An extension of Tichonov’s theorem for the planar case, J. Appl. Math. Phys. (ZAMP), 34, 277 (1983)
[48] O’Malley, R. E., Phase plane solutions to some singular perturbation problems, J. Math. Anal. & Appl., 54, 449 (1976) · Zbl 0334.34050
[49] Reiss, E. L., A modified two time method for the dynamic transitions of bifurcation, SIAM J. Appl. Math., 38, 249 (1980) · Zbl 0448.35010
[50] Sarlet, W., On a common derivation of the averaging method and the two time scale method, Celestial Mechanics, 17, 299 (1978) · Zbl 0389.34038
[51] Sastry, S. S.; Desoer, C. A., Jump behaviour of circuits and systems, IEEE Trans. on Circuits & Syst, CAS-28, 1109 (1981) · Zbl 0476.93036
[52] Sastry, S. S.; Hijab, O., Bifurcation in the presence of small noise, Systems & Control Lett., 1, 159 (1981) · Zbl 0493.93046
[53] Shepherd, J. J., Asymptotic solution of a nonlinear singular perturbation problem, SIAM J. Appl. Math., 35, 176 (1978) · Zbl 0391.34033
[54] Van Harten, A., Nonlinear singular perturbation problems: proofs of correctness of a formal approximation based on a contraction principle in a Banach Space, J. Math Anal. & Appl., 65, 126 (1978) · Zbl 0393.34037
[55] Viktorov, B. V., Application of the singular perturbation method to automatic control systems, Soviet Doklady, 236, 2 (1977) · Zbl 0389.93019
[56] Wollkind, D. J., Singular perturbation techniques: a comparison of the method of matched asymptotic expansions with that of multiple scales, SIAM Review, 19, 502 (1977) · Zbl 0363.34039
[57] Allemong, J. J.; Kokotovic, P. V., Eigensensitivities in reduced order modeling, IEEE Trans. Aut. Control., AC-25, 821 (1980) · Zbl 0446.93029
[58] Anderson, L., Decomposition of two time scale linear systems, (Proc. JACC (1978)), 153
[59] Avramovic, B., Subspace iteration approach to the time scale separation, (18th IEEE Conference on Decision and Control (1979)), 684-687
[60] Chow, J. H., Preservation of controllability in linear time invariant perturbed systems, Int. J. Control, 25, 697 (1977) · Zbl 0356.93006
[61] Eitelberg, E., Model reduction and perturbation structures, Int. J. Control, 35, 1029 (1982) · Zbl 0542.93020
[62] Fernando, K. V.; Nicholson, H., Singular perturbational model reduction of balanced systems, IEEE Trans Aut. Control, AC-27, 466 (1982) · Zbl 0489.93006
[63] Fernando, K. V.; Nicholson, H., Singular perturbational model reduction in the frequency domain, IEEE Trans Aut. Control, AC-27, 969 (1982) · Zbl 0491.93017
[64] Ioannou, P., Robustness of absolute stability, Int. J. Control, 34, 1027 (1981) · Zbl 0471.93029
[65] Javid, S. H., Uniform asymptotic stability of linear time varying singularly perturbed systems, J. Franklin Inst., 305, 27 (1978) · Zbl 0386.93039
[66] Jonckheere, E. A.; Crespo, P.; Silverman, L. M., A closed-loop principal component analysis of singularly perturbed systems, (21st IEEE Conference on Decision and Control (1982)), 1095-1096
[67] Kokotovic, P. V.; Allemong, J. J.; Winkelman, J. R.; Chow, J. H., Singular perturbation and iterative separation of time scales, Automatica, 16, 23 (1980) · Zbl 0427.93007
[68] Litz, L.; Roth, H., State decomposition for singular perturbation order reduction, Int. J. Control, 34, 937 (1981) · Zbl 0468.93010
[69] Luse, D. W.; Khalil, H., A frequency domain approach for systems with slow and fast modes, (Proc. ACC (1983)), 443
[70] Narasimhamurthy, N.; Wu, F. F., On the Riccati equation arising from the study of singularly perturbed systems, (Proc. JACC (1977)), 1244
[71] O’Malley, R. E.; Anderson, R. L., Time-scale decoupling and order reduction for linear time-varying systems, Optimal Control Meth. Appl., 3, 133 (1982) · Zbl 0481.93013
[72] Phillips, R. G., The equivalence of time-scale decomposition techniques used in the analysis and design of linear systems, Int. J. Control, 37, 1239 (1983) · Zbl 0519.93015
[73] Saksena, V. R.; Kokotovic, P. V., Singular perturbation of the Popov-Kalman-Yakubovich Lemma, Systems & Control Lett., 1, 65 (1981) · Zbl 0482.93055
[74] Blankenship, G., Singularly perturbed difference equations in optimal control problems, IEEE Trans Aut. Control, AC-26, 911 (1981) · Zbl 0549.93035
[75] Campbell, S. L., Limit behaviour of solutions of singular difference equations, Linear Algebra & Appl., 23, 167 (1979) · Zbl 0414.65070
[76] Comstock, C.; Hsiao, G. C., Singular perturbation for difference equations, Rocky Mn. J. Math, 6, 561 (1976) · Zbl 0356.65112
[77] Fernando, K. V.; Nicholson, H., Singular perturbational approximations for discrete-time balanced systems, IEEE Trans. Aut. Control, AC-28, 240 (1983) · Zbl 0503.93046
[78] Flaherty, J. E.; O’Malley, R. E., The numerical solution of boundary value problems for stiff differential equations, Math. Comput., 31, 66 (1977) · Zbl 0402.65049
[79] Glizer, V. Ja.; Dmitriev, M. G., Asymptotic solution of some discrete optimal control problems with small step size, Diff. Eqs., 15, 1682 (1979)
[80] Grujic, L. T., Quasi singular perturbations of time discrete systems, (Proc. JACC (1977)), 857
[81] Hoppensteadt, F.; Miranker, W., Multitime methods for systems of difference equations, Stud. Appl. Math., 56, 273 (1977) · Zbl 0365.39002
[82] Javid, S. H., Multitime methods in order reduction and control of discrete systems, (Proceedings of 13th Asilomar Conference on Circuits, Systems & Computers. Proceedings of 13th Asilomar Conference on Circuits, Systems & Computers, Pacific Grove, CA (1979)) · Zbl 0377.49007
[83] Kando, H.; Iwazumi, T., Sub-optimal control of discrete regulator problems via time-scale decomposition, Int. J. Control, 37, 1323 (1983) · Zbl 0516.93005
[84] Kando, H.; Iwazumi, T., Initial value problems of sing. pert. discrete systems via time-scale decomposition, Int. J. System Sci., 14, 555 (1983) · Zbl 0511.93007
[85] Kellogg, R. B.; Tsan, A., Analysis of some difference approximations for a singular perturbation problem without turning points, Maths. Comput., 32, 1025 (1978) · Zbl 0418.65040
[86] Litkouhi, B.; Khalil, H., Infinite time regulators for singularly perturbed difference equations, (20th Allerton Conference on Communication Control and Computers (1982), University of Illinois), 843-854
[87] Locatelli, A.; Schiavon, N., Two-time-scale discrete systems, (First International Conferences on Information Sciences and Systems. First International Conferences on Information Sciences and Systems, Patras, Greece (1976))
[88] Mahmoud, M. S., Order reduction and control of discrete systems, (Proc. IEE, 127D (1982)), 129
[89] Mahmoud, M. S., Structural properties of discrete systems with slow and fast modes, J. Large Scale Syst., 3, 227 (1982) · Zbl 0489.93007
[90] Naidu, D. S.; Rao, A. K., Singular perturbation methods for a class of initial and boundary value problems in discrete systems, Int. J. Control, 36, 77 (1982) · Zbl 0484.93051
[91] O’Malley, R. E.; Flaherty, J. E., Analytical and numerical methods for nonlinear singular singularly perturbed initial value problems, SIAM J. Appl. Math, 38, 225 (1980) · Zbl 0471.65056
[92] Phillips, R. G., Reduced order modeling and control of two time scale discrete systems, Int. J. Control, 31, 765 (1980) · Zbl 0438.93045
[93] Rajagopalan, P. K.; Naidu, D. S., A singular perturbation method for discrete control systems, Int. J. Control, 32, 928 (1980) · Zbl 0453.93039
[94] Rao, A. K.; Naidu, D. S., Singularly perturbed boundary value problems in discrete systems, Int. J. Control, 34, 1163 (1981) · Zbl 0476.93049
[95] Rao, A. K.; Naidu, D. S., Singular perturbation method applied to the open-loop discrete optimal control problem, Optimal Control Appl. & Methods, 3, 121 (1982) · Zbl 0495.93042
[96] Reinhardt, H. J., Singular perturbation of difference methods for linear ordinary differential equations, Applicable Analysis, 10, 53 (1980)
[97] Vasil’eva, A. B.; Faminskaya, M. V., A boundary-value problem for singularly perturbed differential and difference systems when the unperturbed system is on a spectrum, Diff. Eq., 13, 738 (1977)
[98] Asatani, K., Near-optimum control of distributed parameter systems via singular perturbation theory, J. Math. Anal. & Appl., 54, 799 (1976) · Zbl 0334.49024
[99] Asatani, K.; Shiotani, M.; Hattoni, Y., Suboptimal control of nuclear reactors with distributed parameters using singular perturbation theory, Nuclear Sci. & Engng, 62, 9 (1977)
[100] Balas, M. J., Reduced order feedback control of distributed parameter systems via singular perturbation methods, J. Math. Anal. & Appl., 87, 281 (1982) · Zbl 0487.93032
[101] Chow, J. H., Pole-placement design of multiple controllers via weak and strong controllability, Int. J. Syst. Sci., 9, 129 (1978) · Zbl 0369.93006
[102] Chow, J. H.; Kokotovic, P. V., Eigenvalue placement in two time scale systems, (Proceedings of IFAC Symposium on Large Scale Systems. Proceedings of IFAC Symposium on Large Scale Systems, Udine Italy (1976)), 321-326
[103] Chow, J. H.; Kokotovic, P. V., A decomposition of near optimum regulators for systems with slow and fast modes, IEEE Trans. Aut. Control, AC-21, 701 (1976) · Zbl 0332.49034
[104] Dragan, V.; Halanay, A., A singularly perturbed matrix Riccati equation, Rev. Roum. Math. Pures et Appl., 25, 1477 (1980) · Zbl 0464.34042
[105] Dragan, V.; Halanay, A., Suboptimal stabilization of linear systems with several time scales, Int. J. Control, 36, 109 (1982) · Zbl 0487.93041
[106] Gaitsgori, V. G.; Pervozvanski, A. A., Perturbation method in optimal control problems, J. Syst. Sci., 5, 91 (1979)
[107] Kung, C. F., Singular perturbation of an infinite interval linear state regulator problem in optimal control, J. Math. Anal. & Appl., 55, 365 (1976) · Zbl 0346.49011
[108] Longchamp, R., Singular perturbation analysis of a receding horizon controller, Automatica, 19, 303 (1983) · Zbl 0509.93031
[109] Mahmoud, M. S.; Hassan, M. F.; Singh, M. G., Approximate feedback, design for a class of singular perturbed systems, (Proc. IEE, 129D (1982)), 49
[110] Murthy, D. N.P., Solution of linear regulator problem via two parameter singular perturbation, Int. J. Syst. Sci., 9, 1113 (1978) · Zbl 0409.34054
[111] O’Reilly, J., Two time scale feedback stabilization of linear time varying singularly perturbed systems, J. Franklin Inst., 308, 465 (1979) · Zbl 0418.93057
[112] Pervozvanski, A. A.; Gaitsgori, V. G., Control of quasi-conservative linear oscillatory systems, Aut. & Remote Control, 12 (1979)
[113] Pervozvanskii, A. A.; Gaitsgori, V. G., Degenerancy in LQ and LQG problems of optimal control: possibilities to simplify the synthesis, (Proceedings of 8th IFAC Congress, Kyoto, VI (1981)), 155-160
[114] Phillips, R. G., A two stage design of linear feedback controls, IEEE Trans. Aut. Control, AC-26, 1220 (1981)
[115] Porter, B., Design of stabilizing feedback controllers for a class of multivariable linear systems with slow and fast modes, Int. J. Control, 23, 49 (1976) · Zbl 0323.93029
[116] Porter, B., Singular perturbation methods in the design of state feedback controllers for multivariable linear systems, Int. J. Control, 26, 583 (1977) · Zbl 0377.93017
[117] Porter, B., Closed-loop eigenstructure assignment by state feedback in multivariable linear systems with slow and fast modes, Int. J. Control, 28, 93 (1978) · Zbl 0385.93036
[118] Suzuki, M.; Miura, M., Stabilizing feedback controllers for singularly perturbed linear constant systems, IEEE Trans. Aut. Control, AC-21, 123 (1976) · Zbl 0316.93025
[119] Womble, M. E.; Potter, J. E.; Speyer, J. L., Approximations to Riccati equations having slow and fast modes, IEEE Trans. Aut. Control, AC-21, 846 (1976) · Zbl 0343.93086
[120] Balas, M. J., Observer stabilization of singularly perturbed systems, J. Guidance & Control, 1, 93 (1978) · Zbl 0393.93038
[121] Chemouil, P.; Wahdun, A. M., Output feedback control of systems with slow and fast modes, J. Large Scale Syst., 1, 257 (1980) · Zbl 0447.93024
[122] Fossard, A. G.; Magni, J. S., Frequential analysis of singularly perturbed systems with state or output control, J. Large Scale Syst, 1, 223 (1980) · Zbl 0447.93023
[123] Javid, S. H., Observing the slow states of a singularly perturbed system, IEEE Trans. Aut. Control, AC-25, 277 (1980) · Zbl 0432.93049
[124] Javid, S. H., Stabilization of time varying singularly perturbed systems by observer based slow state feedback, IEEE Trans. Aut. Control, AC-27, 702 (1982) · Zbl 0488.93041
[125] Khalil, H. K., On the robustness of output feedback control methods to modeling errors, IEEE Trans. Aut. Control, AC-28, 524 (1981) · Zbl 0474.93024
[126] Mahmoud, M. S., Design of observer-based controllers for a class of discrete systems, Automatica, 18, 323 (1982) · Zbl 0478.93039
[127] O’Reilly, J., Full order observers for a class of singularly perturbed linear time varying systems, Int. J. Control, 30, 745 (1979) · Zbl 0422.93021
[128] O’Reilly, J., Dynamical feedback control for a class of singularly perturbed linear systems using a full order observer, Int. J. Control, 31, 1 (1980) · Zbl 0429.93039
[129] O’Reilly, J., (Observers for Linear Systems (1983), Academic Press: Academic Press London) · Zbl 0513.93001
[130] Özgüner, Ü., Decentralized observers for a large scale system with two time-scales, (Proc. JACC (1977)), 1119
[131] Porter, B., Singular perturbation methods in the design of full order observers for multivariable systems, Int. J. Control, 26, 589 (1977) · Zbl 0377.93018
[132] Saksena, V. R.; Cruz, J. B., Stabilization of singularly perturbed linear time invariant systems using low order observers, IEEE Trans Aut. Control, AC-28, 510 (1981) · Zbl 0486.93048
[133] Smyth, G. W.; D’Azzo, J. J., Digital flight control system design using singular perturbation methods, (21st IEEE Conference Decision and Control (1982)), 1158-1166
[134] Chow, J. H., Asymptotic stability of a class of nonlinear singularly perturbed systems, J. Franklin Inst., 306, 275 (1978)
[135] Chow, J. H.; Kokotovic, P. V., Near-optimal feedback stabilization of a class of nonlinear singularly perturbed systems, SIAM J. Control & Opt., 16, 756 (1978) · Zbl 0392.93022
[136] Chow, J. H.; Kokotovic, P. V., Two time scale feedback design of a class of nonlinear systems, IEEE Trans. Aut. Control, AC-23, 438 (1978) · Zbl 0388.93026
[137] Chow, J. H.; Kokotovic, P. V., A two-stage Lyapunov-Bellman feedback design of a class of nonlinear systems, IEEE Trans. Aut. Control, AC-26, 656 (1981) · Zbl 0487.93025
[138] Ioannou, P.; Kokotovic, P. V., An asymptotic error analysis of identifiers and adaptive observers in the presence of parasites, IEEE Trans Aut. Control, AC-27, 921 (1982) · Zbl 0485.93025
[139] Koda, M., Sensitivity analysis of singularly perturbed systems, Int. J. Syst. Sci., 13, 909 (1982) · Zbl 0488.93018
[140] Kokotovic, P. V.; Ioannou, P. A., Robustness redesign of continuous time adaptive schemes, (20th IEEE Conference on Decision and Control (1981)), 522-527 · Zbl 0516.93017
[141] Kuz’mina, L. K., The stability of solutions of certain systems of differential equations with a small parameter at derivatives, Appl. Math. Mech. (PMM), 41, 567 (1977)
[142] Rohrs, C. E.; co-workers, Analytical verification of undesirable properties of direct model reference adaptive control algorithms, (20th IEEE Conference on Decision and Control (1981)), 1272-1284
[143] Saberi, A.; Khalil, H., Quadratic-type Lyapunov functions for singularly perturbed systems, (20th IEEE Conference on Decision and Control (1981)), 204-205
[144] Sacker, R. J.; Sell, G. R., Singular perturbations and conditional stability, J. Math. Anal. & Appl., 76, 406 (1980) · Zbl 0476.34045
[145] Saberi, A.; Khalil, H., Two time scale feedback design of nonlinear singularly perturbed systems, (Proc. ACC (1983)), 441
[146] Sandell, N. R., Robust stability of systems with applications to singular perturbation, Automatica, 15, 467 (1979) · Zbl 0407.93033
[147] Sannuti, P., On the controllability of singularly perturbed systems, IEEE Trans. Aut. Control, AC-22, 622 (1977) · Zbl 0361.93011
[148] Sannuti, P., On the controllability of some singularly perturbed nonlinear systems, J. Math. Anal. & Appl., 64, 579 (1978) · Zbl 0406.93010
[149] Suzuki, M., Composite controls for singularly perturbed systems, IEEE Trans Aut. Control, AC-26, 505 (1981) · Zbl 0474.93015
[150] Ardema, M. D., Solution of the minimum time-to-climb problem by matched asymptotic expansions, AIAA J., 14, 843 (1976) · Zbl 0338.70005
[151] Ardema, M. D., Characteristics of the boundary-layer equations of the minimum time-to-climb problem, (Proceedings of 14th Allerton Conf. on Circuit and System Theory (1976), University of Illinois), 807-817
[152] Ardema, M. D., Linearization of the boundary layer equations for the minimum time to climb problem, J. Guidance & Control, 2, 434 (1979)
[153] Ardema, M. D., Singular perturbation and the sounding rocket problem, (Proc. JACC (1979)), 901
[154] Ardema, M. D., Nonlinear singularly perturbed optimal control problems with singular arcs, Automatica, 16, 99 (1980) · Zbl 0427.49024
[155] Binding, P., Singularly Perturbed Optimal Control Problems. I: Convergence, SIAM J. Control, 14, 591 (1976) · Zbl 0329.49004
[156] Brauner, C. M., Optimal control of a singularly perturbed system in enzyme kinetics, (Proceedings of 7th IFAC Congress. Proceedings of 7th IFAC Congress, Helsinki (1978)), 945-948 · Zbl 0457.93032
[157] Calise, A. J., Singular perturbation methods for variational problems in aircraft flight, IEEE Trans Aut. Control, AC-21, 345 (1976) · Zbl 0346.49015
[158] Calise, A. J., Energy management and singular perturbations in flight mechanics, (Proceedings of 7th IFAC Congress. Proceedings of 7th IFAC Congress, Helsinki (1978)), 949-955 · Zbl 0457.93031
[159] Calise, A. J., A new boundary layer matching procedure for singularly perturbed systems, IEEE Trans Aut. Control, AC-23, 434 (1978) · Zbl 0378.49026
[160] Calise, A. J., A singular perturbation analysis of optimal aerodynamic and thrust magnitude control, IEEE Trans. Aut. Control, AC-24, 720 (1979) · Zbl 0428.49012
[161] Calise, A. J., A singular perturbation analysis of optimal thrust control with proportional navigation guidance, AIAA J. Guidance & Control, 3, 312 (1980) · Zbl 0438.93053
[162] Calise, A. J., Singular perturbation theory for on-line optimal flight path control, AIAA J. Guidance & Control, 4, 398 (1981) · Zbl 0467.49022
[163] Chow, J. H., A class of singularly perturbed nonlinear, fixed endpoint control problems, J. Opt. Theory & Appl., 29, 231 (1979) · Zbl 0387.49003
[164] Dmitriev, M. G., On a class of singularly perturbed optimal control problems, Appl. Math. Mech. (PMM), 42, 228 (1978)
[165] Dontchev, A. L.; Veliov, V. M., Singular perturbation in Mayer’s problem for linear systems, SIAM J. Control Opt., 21, 566 (1983) · Zbl 0519.49002
[166] Freedman, M. I.; Granoff, B., Formal asymptotic solution of a singularly perturbed nonlinear optimal control problem, J. Opt. Theory & Appl., 19, 301 (1976) · Zbl 0307.49011
[167] Freedman, M. I.; Kaplan, J. L., Singular perturbations of two point boundary value problems arising in optimal control, SIAM J. Control & Opt., 14, 189 (1976) · Zbl 0324.49012
[168] Gičev, T. R.; Dontchev, A. L., Singular perturbation in optimal control problems with fixed final state, C. Re. Acad. bulgare Sci. (Math.), 31, 953 (1978) · Zbl 0402.49030
[169] Gičev, T. R.; Dontchev, A. L., Convergence of the solutions of the singularly perturbed time optimal problem, Appl. Math. Mech. (PMM), 43, 466 (1979)
[170] Glizer, V. Ja.; Dmitriev, M. G., On a connection of singular perturbations with the penalty function method, Soviet Math. Doklady, 17, 1503 (1976) · Zbl 0365.49018
[171] Glizer, V. Ja.; Dmitriev, M. G., On the continuity of the regulator problem with respect to singular perturbations, Appl. Math. Mech. (PMM), 41, 573 (1977)
[172] Glizer, V. Ja.; Dmitriev, M. G., Asymptotic solution of a singularly perturbed Cauchy problem in optimal control, Diff. Eqs., 14, 601 (1978) · Zbl 0389.93021
[173] Halanay, A.; Mirica, S., The time optimal feedback control for singularly perturbed linear systems, Rev. Roum. Math. Pures et Appl., 24, 585 (1979) · Zbl 0415.34050
[174] Howes, F. A., Effective characterization of the asymptotic behaviour of solutions of singularly perturbed boundary value problems, SIAM J. Appl. Math., 30, 296 (1976) · Zbl 0323.34050
[175] Howes, F. A., An approximation method for a class of singularly perturbed second order boundary value problems, J. Math. Anal. & Appl., 58, 653 (1977) · Zbl 0361.34015
[176] Javid, S. H., The time optimal control of a class of nonlinear singularly perturbed systems, Int. J. Control, 27, 831 (1978) · Zbl 0377.49007
[177] Javid, S. H.; Kokotovic, P. V., A decomposition of time scales for iterative computation of time optimal controls, J. Opt. Theory & Appl., 21, 459 (1977) · Zbl 0336.49025
[178] Krikorian, K. V.; Leondes, C. T., Dynamic programming using singular perturbations, J. Opt. Theory Appl., 38, 221-230 (1982) · Zbl 0471.49027
[179] Kirkorian, K. V.; Leondes, C. T., Application of singular perturbations to optimal control, Control and Dynamic Systems, 18, 131 (1982) · Zbl 0559.93043
[180] Kurina, G. A., Asymptotic solution of a classical singularly perturbed optimal control problem, Soviet Math Dokl, 18, 722 (1977) · Zbl 0374.49002
[181] Kurina, G. A., On a classical singularly perturbed optimal control problem, Diff. Eqs., 19, 710 (1983) · Zbl 0516.49004
[182] Mehra, R. K.; Washburn, R. B.; Sajon, S.; Corell, J. V., A study of the application of singular perturbation theory, NASA CR3167 (1979)
[183] Rozov, N. H.; Gičev, T. R., Singularly perturbed problems with minimal pulse, Diff. Eqs., 19, 259 (1983) · Zbl 0517.49018
[184] Shinar, J., On applications of singular perturbation techniques in nonlinear optimal control problems, Automatica, 19, 203 (1983) · Zbl 0501.93027
[185] Shinar, J. M.; Negrin, M.; Well, K.; Berger, E., Comparison between the exact and an approximate feedback solution for medium range interception problems, (Proceedings of the JACC (1981))
[186] Sridhar, B.; Gupta, N. K., Missile guidance laws based on singular perturbation methodology, J. Guidance & Control, 3, 158 (1980)
[187] Vasil’eva, A. B.; Anikeeva, V. A., Asymptotic expansions of solutions of non-linear problems with singular boundary conditions, Diff. Eq, 12, 1235 (1976) · Zbl 0376.34046
[188] Vasil’eva, A. B.; Dmitriev, M. G., Determination of the structure of generalized solutions of nonlinear optimal control problems, Soviet Math. Dokl., 21, 104 (1980) · Zbl 0452.49004
[189] Vasil’eva, A. B.; Faminskaya, M. V., An investigation of a nonlinear optimal control problem by the methods of singular perturbation theory, Soviet Math. Dokl., 21, 104 (1981) · Zbl 0466.49001
[190] Altshuler, D.; Haddad, A. H., Near optimal smoothing for singularly perturbed linear systems, Automatica, 14, 81 (1978) · Zbl 0376.93031
[191] Bensoussan, A., Singular perturbation results for a class of stochastic control problems, IEEE Trans Aut. Control, AC-26, 1071 (1981) · Zbl 0476.93078
[192] Blankenship, G., On the separation of time scales in stochastic differential equations, (Proceedings of 7th IFAC Congress. Proceedings of 7th IFAC Congress, Helsinki (1978)), 937-944 · Zbl 0457.93055
[193] Blankenship, G. L., Scaling and bifurcations in stochastic differential equations, (21st IEEE Conference Decision and Control (1982)), 925-931
[194] Blankenship, G.; Meyer, D., Linear filtering with wide band noise disturbances, (16th IEEE Conference on Decision and Control (1977)), 580-584
[195] Blankenship, G.; Papanicolaou, G. C., Stability and control of stochastic systems with wide-band noise disturbance, SIAM J. Appl. Math., 34, 437 (1978) · Zbl 0392.93040
[196] SIAM J. Math. Anal., 10, 306 (1979) · Zbl 0406.60052
[197] Bratus, A. S., Asymptotic solutions of some probabilistic optimal control problems, Appl. Math. Mech. (PMM), 41, 13 (1977) · Zbl 0386.93057
[198] Cohen, D. S.; Hoppensteadt, F. C.; Miura, R. M., Slowly modulated oscillations in nonlinear diffusion processes, SIAM J. Appl. Math., 33, 217 (1977) · Zbl 0373.35033
[199] El-Ansary, M.; Khalil, H., Reduced order modeling of nonlinear singularly perturbed systems driven by wide-band noise, (21st IEEE Conference Decision and Control (1982)), 1090-1094
[200] Haddad, A. H., Linear filtering of singularly perturbed systems, IEEE Trans. Aut. Control, AC-31, 515 (1976) · Zbl 0332.93073
[201] Haddad, A. H., On singular perturbations in stochastic dynamic systems, (Proceedings of 10th Asilomar Conference on Circuits, Systems and Computers (1976)), 94-98 · Zbl 0369.93039
[202] Haddad, A. H.; Kokotovic, P. V., Stochastic control of linear singularly perturbed systems, IEEE Trans Aut. Control, AC-22, 815 (1977) · Zbl 0372.93056
[203] Hijab, O.; Sastry, S., Singular perturbation, state aggregation and nonlinear filtering, (Proceedings of IEEE Conference on Decision and Control (1981)), 590-593
[204] Hopkins, W. E.; Blankenship, G. L., Perturbation analysis of a system of quasi-variational inequalities for optimal stochastic scheduling, IEEE Trans Aut. Control, AC-26, 1054 (1981) · Zbl 0469.49009
[205] Khalil, H. K., Control of linear singularly perturbed systems with colored noise disturbances, Automatica, 14, 153 (1978) · Zbl 0389.49010
[206] Khalil, H.; Gajic, Z., Near optimum regulators for stochastic singularly perturbed systems, (21st IEEE Conference Decision and Control (1982)), 1317-1321
[207] Khalil, H. K.; Haddad, A.; Blankenship, G., Parameter scaling and well-posedness of stochastic singularly perturbed control systems, (Proceedings of 12th Asilomar Conference. Proceedings of 12th Asilomar Conference, Pacific Grove, CA (1978)), 407-411
[208] Kuruoglu, N.; Clough, D. E.; Ramirez, W. F., Distributed parameter estimation for systems with fast and slow dynamics, Chem. Engng Sci., 36, 1357 (1981)
[209] Kushner, H. J., A cautionary note on the use of singular perturbation methods for ‘small noise’ models, Stochastics, 6, 117 (1982) · Zbl 0479.60037
[210] Matkowsky, B. J.; Schuss, Z., The exit problem for randomly perturbed dynamical systems, SIAM J. Appl. Math., 33, 365 (1977) · Zbl 0369.60071
[211] Papanicolaou, G., Some probabilistic problems and methods in singular perturbations, Rocky M. J. Math., 6, 653 (1976) · Zbl 0365.60049
[212] Price, D. B., Comments on ‘linear filtering of singularly perturbed systems’, IEEE Trans. Aut. Control, AC-24, 675 (1979) · Zbl 0414.93049
[213] Razvig, V. D., Reduction of stochastic differential equations with small parameters and stochastic integrals, Int. J. Control, 28, 707 (1978) · Zbl 0394.60067
[214] Schuss, Z., (Theory and Applications of Stochastic Differential Equations (1980), John Wiley: John Wiley New York) · Zbl 0439.60002
[215] Sebald, A. V.; Haddad, A. H., State estimation for singularly perturbed systems with uncertain perturbation parameter, IEEE Trans Aut. Control, AC-23, 464 (1978) · Zbl 0387.93048
[216] Singh, Ram-Nandan P., The linear-quadratic-Gaussian problem for singularly perturbed systems, Int. J. Science, 13, 93 (1982) · Zbl 0487.93068
[217] Soliman, M. A.; Ray, W. H., Nonlinear filtering for distributed parameter systems with a small parameter, Int. J. Control, 30, 757-772 (1979) · Zbl 0424.93058
[218] Teneketzis, D.; Sandell, N. R., Linear regulator design for stochastic systems by a multiple time scale method, IEEE Trans on Aut. Control, AC-22, 615-621 (1977) · Zbl 0361.93042
[219] Tsai, C. P., Perturbed stochastic linear regulator problems, SIAM J. Control, 16, 396-410 (1978) · Zbl 0382.93071
[220] Campbell, S. L., Optimal Control of Autonomous Linear Processes with Singular Matrices in the Quadratic Cost Functional, SIAM J. Control, 14, 1092 (1976) · Zbl 0346.49010
[221] Commault, C.; Dion, J. M., Structure at infinity of linear multivariable systems: a geometric approach, IEEE Trans Aut. Control, AC-27, 693 (1982) · Zbl 0485.93041
[222] Dragan, V.; Halanay, A., Cheap control and singularly perturbed matrix Riccati differential equations, Rev. Roum. Math. Pures et Appl., 26, 21 (1981) · Zbl 0463.49005
[223] Francis, B. A., The optimal linear-quadratic time-invariant regulator with cheap control, IEEE Trans Aut. Control, AC-24, 616 (1979) · Zbl 0424.49022
[224] Francis, B. A.; Glover, K., Bounded peaking in the optimal linear regulator with cheap control, IEEE Trans Aut. Control, AC-23, 608 (1978) · Zbl 0391.93019
[225] Grasman, J., On a class of optimal control problems with an almost cost-free solution, IEEE Trans Aut. Control, AC-27, 441 (1982) · Zbl 0483.49034
[226] Grishin, S. A.; Utkin, V. I., On redefinition of discontinuous systems, Diff. Eq., 16, 227 (1980) · Zbl 0443.34014
[227] Hung, Y. S.; MacFarlane, A. G.J., On the relationships between the unbounded asymptote behaviour of multivariable root loci, impulse response and infinite zeros, Int. J. Control, 34, 31 (1981) · Zbl 0469.93038
[228] Izosimov, D. B.; Utkin, V. I., On equivalence of high-gain systems and systems with discontinuous control, Aut. & Remote Control, 11, 189 (1981)
[229] Khalil, H. K.; Saberi, A., Decentralized stabilization of nonlinear interconnected systems using high gain feedback, IEEE Trans. Aut. Control, AC-27, 265 (1982) · Zbl 0469.93063
[230] Kimura, H., A new approach to the perfect regulation and the bounded peaking in linear multivariable control systems, IEEE Trans. Aut. Control, AC-26, 253 (1981) · Zbl 0461.93041
[231] Kimura, H., Perfect and subperfect regulation in linear multivariable control systems, Automatica, 18, 125 (1982) · Zbl 0491.93039
[232] Kouvaritakis, B.; Edmunds, J. M., A multivariable root loci: a unified approach to finite and infinite zeros, Int. J. Control, 29, 393 (1979) · Zbl 0396.93033
[233] Kouvaritakis, B.; Shaked, U., Asymptotic behaviour of root locus of linear multivariable systems, Int. J. Control, 23, 297 (1976) · Zbl 0317.93053
[234] Kwakernaak, H., Asymptotic root loci of multivariable linear optimal regulations, IEEE Trans Aut. Control, AC-21, 378 (1976) · Zbl 0324.49033
[235] O’Malley, R. E., A more direct solution of the nearly singular linear regulator problem, SIAM J. Control & Opt., 14, 1063 (1976) · Zbl 0344.49006
[236] O’Malley, R. E., High-gain feedback systems as singular perturbation problems, (Proc. JACC (1977)), 1278
[237] O’Malley, R. E., Singular perturbations and optimal control, (Mathematical Control Theory. Mathematical Control Theory, Lecture Notes in Mathematics No. 680 (1978), Springer: Springer New York) · Zbl 0403.49002
[238] O’Malley, R. E.; Jameson, A., Singular perturbations and singular arcs II, IEEE Trans Aut. Control, AC-22, 328 (1977) · Zbl 0358.49017
[239] O’Reilly, J., Partial cheap control of the time-invariant regulator, Int. J. Control., 37, 909 (1983) · Zbl 0509.93032
[240] Porter, B., High-gain tracking systems incorporating Lur’e plants with multiple nonlinearities, Int. J. Control, 34, 333 (1981) · Zbl 0474.93038
[241] Sannuti, P., A direct singular perturbation analysis of high-gain and cheap control problems, Automatica, 19, 41 (1983) · Zbl 0505.93024
[242] Sannuti, P.; Wason, H., A singular perturbation canonical form of invertible systems: determination of multivariable root loci, Int. J. Control, 37, 1259 (1983) · Zbl 0525.93012
[243] Sastry, S. S.; Desoer, C. A., Asymptotic unbounded root-loci formulae and computation, IEEE Trans Aut. Control, AC-28, 557 (1983) · Zbl 0526.93044
[244] Shaked, U., The asymptotic behaviour of the root loci of multivariable optimal regulators, IEEE Trans. Aut. Control, AC-23, 425 (1978) · Zbl 0387.93032
[245] Shaked, U.; Bobrovsky, B., The asymptotic minimum variance estimate of stationary linear single output processes, IEEE Trans Aut. Control, AC-26, 498 (1981) · Zbl 0472.93059
[246] Utkin, V. I.; Young, K. D., Synthesis of the switching plane in multivariable systems with variable structure, Aut. & Remote Control, 39, 72 (1978)
[247] Utkin, V. I.; Young, K. D., Methods for construction of discontinuity planes in multidimensional variable structure systems, Aut. & Remote Control, 39, 1466 (1978) · Zbl 0419.93045
[248] Willems, J. C., Almost invariant subspaces; an approach to high gain feedback design, part I: almost controlled invariant subspaces, IEEE Trans Aut. Control, AC-26, 235 (1981) · Zbl 0463.93020
[249] Willems, J. C., Almost invariant subspaces: an approach to high gain feedback design, part II: almost conditionally invariant subspaces, IEEE Trans Aut. Control, AC-27, 1071 (1982) · Zbl 0491.93022
[250] Young, K.-K. D., Asymptotic stability of model reference systems, IEEE Trans Aut. Control, AC-22, 279 (1977) · Zbl 0353.93037
[251] Young, K.-K. D., Multiple time-scales in single-input single-output high-gain feedback systems, J. Franklin Inst., 306, 293 (1978) · Zbl 0393.93022
[252] Young, K.-K. D., Design of variable structure model-following control systems, IEEE Trans Aut. Control, AC-23, 1074 (1978)
[253] Young, K. D., Near insensitivity of linear feedback systems, J. Franklin Inst., 314, 129 (1982) · Zbl 0498.93017
[254] Young, K. D., Disturbance decoupling by high gain feedback, IEEE Trans Aut. Control, AC-27, 970 (1982) · Zbl 0491.93051
[255] Young, K. D.; Kokotovic, P. V., Analysis of feedback loop interactions with actuator and sensor parasitics, Automatica, 18, 577 (1982) · Zbl 0499.93019
[256] Young, K. D.; Kokotovic, P. V.; Utkin, V. I., A singular perturbation analysis of high gain feedback systems, IEEE Trans Aut. Control, AC-22, 931 (1977) · Zbl 0382.49029
[257] Young, K. D.; Kwatny, H. G., A variable structure servomechanism design and application to overspeed protection control, Automatica, 18, 385 (1982) · Zbl 0485.93047
[258] Ahmed-Zaid, S.; Sauer, P. W.; co-workers, Reduced order modeling of synchronous machines using singular perturbations, IEEE Trans Circuits & Syst., CAS-29, 782 (1982) · Zbl 0493.93028
[259] Alden, R. T.H.; Nolan, P. J., Evaluating alternative models for power system dynamic stability studies, IEEE Trans Power Apparatus & Syst., PAS-95, 433 (1976)
[260] Avramovic, B.; Kokotovic, P. V.; Winkelman, J. R.; Chow, J. H., Area decomposition of electromechanical models of power systems, Automatica, 16, 637 (1980) · Zbl 0467.93003
[261] Campbell, S. L.; Rose, N. J., A second-order singular linear system arising in electric power systems analysis, Int. J. Systems Sci., 13, 101 (1982) · Zbl 0474.93032
[262] Chow, J. H.; Allemong, J. J.; Kokotovic, P. V., Singular Perturbation Analysis of Systems with Sustained High Frequency Oscillations, Automatica, 14, 271 (1978) · Zbl 0385.93024
[263] Chow, J. H.; Kokotovic, P. V., Sparsity and time scales, (Proc. ACC (1983)), 656
[264] Desoer, C. A., Distributed networks with small parasitic elements: input-output stability, IEEE Trans Circuits & Syst., CAS-24, 1 (1977) · Zbl 0355.94040
[265] Fossard, A. J.; Berthelot, M.; Magni, J. F., On coherency-based decomposition algorithms, Automatica, 19, 247 (1983) · Zbl 0512.93010
[266] Haller, M. A.; Iung, C., A multiple-scale approach in the study of commutation phenomena: application to a chopper supplying a stepper motor, Electric Machines & Power Syst., 8, 113 (1983)
[267] Kokotovic, P. V.; Avramovic, B.; Chow, J. H.; Winkelman, J. R., Coherency-based decomposition and aggregation, Automatica, 17, 47 (1982) · Zbl 0472.93011
[268] Matsumoto, T.; Chua, L. O.; Kawakami, H.; Ichiraku, S., Geometric properties of dynamic non-linear networks: transversality, local-solvability, and eventual passivity, IEEE Trans Circuits & Syst., CAS-28, 406 (1981) · Zbl 0458.94049
[269] Newcomb, R. W., The semi-state description of non-linear time-variable circuits, IEEE Trans Circuits & Syst., CAS-28, 62 (1981)
[270] Peponides, G.; Kokotovic, P. V.; Chow, J. H., Singular perturbations and time scales in nonlinear models of power systems, IEEE Trans Circuits & Syst., CAS-29, 758 (1982) · Zbl 0496.93035
[271] Peponides, G.; Kokotovic, P., Weak connections, time scales and aggregation of nonlinear systems, IEEE Trans Aut. Control, AC-28, 729 (1983)
[272] Sannuti, P., Use of singular perturbation methods to formulate electrical network equations, Rocky Mn J. Math., 6, 709 (1976)
[273] Sannuti, P., Singular perturbations in the state space approach of linear electrical networks, Circuit Theory & Appl., 9, 47 (1981) · Zbl 0455.94028
[274] Sastry, S.; Varaiya, P., Hierarchical stability and alert state steering control of power systems, IEEE Trans Circuits & Syst., CAS-27, 1102 (1980) · Zbl 0463.93063
[275] Sastry, S.; Varaiya, P., Coherency for interconnected power systems, IEEE Trans Aut. Control, AC-26, 218 (1981) · Zbl 0464.93016
[276] Winkelman, J. R.; Chow, J. H.; Allemong, J. J.; Kokotovic, P. V., Multi-time-scale analysis of a power system, Automatica, 16, 35 (1980) · Zbl 0429.93023
[277] Winkelman, J. R.; Chow, J. H.; Bowler, B. C.; Avramovic, B.; Kokotovic, P. V., An analysis of interarea dynamics of multimachine systems, IEEE Trans Power App. & Syst., PAS-100, 754 (1981)
[278] Delebecque, F., A reduction process for perturbed Markov chains, SIAM J. Appl. Math., 43, 325 (1983) · Zbl 0518.60080
[279] Delebecque, F.; Quadrat, J. P., Contribution of stochastic control singular perturbation averaging and team theories to an example of large scale systems: management of hydropower production, IEEE Trans Aut. Control, AC-23, 209 (1978) · Zbl 0379.93055
[280] Delebecque, F.; Quadrat, J. P., The optimal cost expansion of finite controls, finite state Markov chains with weak and strong interactions, (Analysis and Optimization of Systems. Analysis and Optimization of Systems, Lecture Notes in Control and Information Sciences, 28 (1980), Springer: Springer Berlin) · Zbl 0516.93004
[281] Delebecque, F.; Quadrat, J. P., Optimal control of Markov chains admitting strong and weak interactions, Automatica, 17, 281 (1981) · Zbl 0467.49020
[282] Delebecque, F.; Quadrat, J. P.; Kokotovic, P. V., Aggregability of dynamic systems and lumpability of Markov chains, (20th IEEE Conference on Decision and Control (1981)), 199-203 · Zbl 0516.93004
[283] Gaitsgori, V. G.; Pervozvanskii, A. A., Aggregation of states in a Markov chain with weak interactions, Kibernetika, 3, 91 (1975)
[284] Gaitsgori, V. G.; Pervozvanskii, A. A., On the optimization of weakly controlled stochastic systems, Sov. Math. Dokl., 21, 408 (1980)
[285] Javid, S. H., Nested optimization of weakly coupled Markov chains, (Proceedings of 18th Allerton Conference on Communication, Control and Computing (1980), University of Illinois), 881-890
[286] Pervozvanskii, A. A.; Gaitsgori, V. G., On aggregation of linear control systems, Aut. & Remote Control, 8, 88 (1980)
[287] Phillips, R. G.; Kokotovic, P. V., A singular perturbation approach to modelling and control of Markov chains, IEEE Trans Aut. Control, AC-26, 1087 (1981) · Zbl 0475.93079
[288] Quadrat, J. P.; Viot, M., Product form and optimal local feedback for multi-index Markov chain, (Proceedings of 18th Allerton Conference on Communication, Control and Computing (1980), University of Illinois), 870-880
[289] Sauer, C. H.; Chandy, K. M., Approximate solution of queueing models, Computer, 25 (1980)
[290] Stewart, G. W., Computable error bounds for aggregated Markov chains, J. Assoc. Computing Machinery, 30, 271 (1983) · Zbl 0523.60062
[291] Teneketzis, D.; Javid, S. H.; Sridhar, B., Control of weakly coupled Markov chains, (19th IEEE Conference on Decision and Control (1980)), 137-142
[292] IEEE Trans Aut. Control, AC-28, 1017 (1983) · Zbl 0522.93009
[293] Grujic, L. T., Vector Lyapunov functions and singularly perturbed large scale systems, (Proc. J ACC (1976)), 409
[294] Grujic, L. T., Converse lemma and singularly perturbed large scale systems, (Proc. J ACC (1977)), 1107
[295] Grujic, L. T., Singular perturbations, uniform asymptotic stability and large scale systems, (Proc. J ACC (1978)), 339
[296] Grujic, L. T., Singular perturbations, large scale systems and asymptotic stability of invariant sets, Int. J. Systems Sci., 12, 1323 (1979) · Zbl 0419.93066
[297] Grujic, L. T., Uniform asymptotic stability of nonlinear singularly perturbed general and large-scale systems, Int. J. Control, 33, 481 (1981) · Zbl 0464.93068
[298] Ioannou, P.; Kokotovic, P., Decentralized adaptive control in the presence of multiparameter singular perturbations and bounded disturbances, (Proc. ACC (1983)), 553
[299] Khalil, H. K., Stabilization of multiparameter singularly perturbed systems, IEEE Trans on Aut. Control, AC-24, 790 (1979) · Zbl 0416.93074
[300] Khalil, H. K., Multimodel design of a Nash strategy, J. Opt. Theory and Appl., 31, 553 (1980) · Zbl 0416.90093
[301] Khalil, H. K., Asymptotic stability of a class of nonlinear multiparameter singularly perturbed systems, Automatica, 17, 797 (1981) · Zbl 0485.93053
[302] Khalil, H. K.; Kokotovic, P. V., Control strategies for decision makers using different models of the same system, IEEE Trans Aut. Control, AC-23, 289 (1978) · Zbl 0384.93007
[303] Khalil, H. K.; Kokotovic, P. V., Control of linear systems with multiparameter singular perturbations, Automatica, 15, 197 (1979) · Zbl 0409.93019
[304] Khalil, H. K.; Kokotovic, P. V., D-stability and multiparameter singular perturbations, SIAM J. Control & Opt., 17, 56 (1979)
[305] Khalil, H. K.; Kokotovic, P. V., Decentralized stabilization of systems with slow and fast modes, J. Large Scale Systems, 1, 141 (1980) · Zbl 0461.93048
[306] Kokotovic, P. V., Subsystems, time-scales and multimodeling, Automatica, 17, 789 (1981) · Zbl 0485.93007
[307] Khalil, H. K., On the existence of positive diagonal \(P\) such that \(PA + A^TP < 0\), IEEE Trans Aut. Control, AC-27, 181 (1982) · Zbl 0474.49033
[308] Ladde, G. S.; Šiljak, D. D., Multiparameter singular perturbations of linear systems with multiple time scales, Automatica, 19, 385 (1983) · Zbl 0513.93004
[309] Özgüner, Ü., Near-optimal control of composite systems: the multi-time scale approach, IEEE Trans Aut. Control, AC-24, 652 (1979) · Zbl 0412.93006
[310] Saksena, V. R.; Başar, T., A multimodel approach to stochastic team problems, Automatica, 18, 713 (1982) · Zbl 0496.93048
[311] Saksena, V. R.; Cruz, J. B., Nash strategies in decentralized control of multiparameter singularly perturbed large-scale systems, J. Large Scale Systems, 2, 219 (1981) · Zbl 0482.93010
[312] Saksena, V. R.; Cruz, J. B., A multimodel approach to stochastic Nash games, Automatica, 18, 295 (1982) · Zbl 0495.93048
[313] Singh, Y. P., Multiple time analysis of coupled non-linear systems, Int. J. Control, 36, 99 (1982) · Zbl 0486.93033
[314] Young, K. K.D., State-space decompositions for linear singularly perturbed systems, (21st IEEE Conference Decision and Control (1982)), 1084-1089
[315] Cruz, J. B., On order reduction for models of Nash and Stackelberg differential games, (1979 International Symposium on Circuits and Systems. 1979 International Symposium on Circuits and Systems, Tokyo, Japan (1979)), 435-438
[316] Farber, N.; Shinar, J., Approximate solution of singularly perturbed nonlinear pursuit-evasion games, J. Opt. Theory & Appl., 32, 39 (1980) · Zbl 0416.90100
[317] Farber, N.; Shinar, J., Approximate solution of singularly perturbed nonlinear pursuit-evasion games between two airplanes in a horizontal plane, (Proceedings of the Atmospheric Flight Mechanics Conference. Proceedings of the Atmospheric Flight Mechanics Conference, Danvers, MA. Proceedings of the Atmospheric Flight Mechanics Conference. Proceedings of the Atmospheric Flight Mechanics Conference, Danvers, MA, AIAA Paper No. 80-1597 (1980)), 337-347
[318] Farber, N.; Shinar, J., A variable modeling approach for singularly perturbed pursuit-evasion problems, (23rd Israel Annual Conference on Aviation and Astronautics (1981)) · Zbl 0416.90100
[319] Gardner, B. F., Zero sum strategy for systems with fast and slow modes, (Proceedings of 15th Allerton Conference on Communication, Control and Computing (1977), University of Illinois: University of Illinois Urbana), 96-103
[320] Gardner, B. F.; Cruz, J. B., Well-posedness of singularly perturbed Nash games, J. Franklin Inst., 306, 355 (1978) · Zbl 0402.90100
[321] Khalil, H. K.; Gardner, B. F.; Cruz, J. B.; Kokotovic, P. V., Reduced-order modelling of closed-loop Nash games, (Proceedings of IRIA/IFAC Symposium on Systems Optimization and Analysis. Proceedings of IRIA/IFAC Symposium on Systems Optimization and Analysis, Springer Lectures Notes in Control and Information Science, 14 (1978)), 119-126
[322] Khalil, H. K.; Kokotovic, P. V., Feedback and Well-Posedness of Singularly Perturbed Nash Games, IEEE Trans. Aut. Control, AC-24, 699 (1979) · Zbl 0419.49015
[323] Khalil, H. K.; Medanic, J. V., Closed-loop Stackelberg strategies for singularly perturbed linear quadratic problems, IEEE Trans Aut. Control, AC-25, 66 (1980) · Zbl 0432.49007
[324] Özgüner, U., Near-Nash feedback control of a composite system with a time-scale hierarchy, IEEE Trans Systems, Man & Cybern, SMC-12, 62 (1982)
[325] Salman, M. A.; Cruz, J. B., Well posedness of linear closed Stackelberg strategies for singularly perturbed systems, J. Franklin. Inst., 308, 25 (1979) · Zbl 0408.90096
[326] Salman, M. A.; Cruz, J. B., Team-optimal Stackelberg strategies for systems with slow and fast modes, Int. J. Control, 37, 1401 (1983) · Zbl 0515.90092
[327] Shinar, J., Solution techniques for realistic pursuit-evasion games, (Leondes, C. T., Advances in Control and Dynamic Systems, Vol. 17 (1981), Academic Press: Academic Press New York), 63-124 · Zbl 0547.90106
[328] Shinar, J.; Farber, N.; Negrin, M., A three dimensional air combat game analysis by forced singular perturbations, (9th Flight Mechanics Conference. 9th Flight Mechanics Conference, San Diego, CA. 9th Flight Mechanics Conference. 9th Flight Mechanics Conference, San Diego, CA, AIAA Paper No. 82-1327 (1982))
[329] Campbell, S. L., Linear systems of differential equations with singular coefficients, SIAM J. Math. Anal., 6, 1057 (1977) · Zbl 0379.34009
[330] Campbell, S. L., Singular perturbation of autonomous linear systems II, J. Diff. Eqns., 29, 362 (1978) · Zbl 0354.34056
[331] Campbell, S. L., Nonregular singular dynamic Leontief systems, Econometrica, 47, 1565 (1979) · Zbl 0418.90024
[332] Campbell, S. L., On a singularly perturbed autonomous linear control problem, IEEE Trans Aut. Control, AC-24, 115 (1979) · Zbl 0398.93024
[333] Campbell, S. L., Singular linear systems of differential equations with delays, Applicable Analysis, 2, 129 (1980) · Zbl 0444.34062
[334] Campbell, S. L., A more singular singularly perturbed linear system, IEEE Trans Aut. Control, AC-26, 507 (1981)
[335] Campbell, S. L., A procedure for analyzing a class of nonlinear semistate equations that arise in circuit and control problems, IEEE Trans Circuits & Syst., CAS-28, 256 (1981)
[336] Campbell, S. L., On an assumption guaranteeing boundary layer convergence of singularly perturbed systems, Automatica, 17, 645 (1981) · Zbl 0465.93034
[337] Campbell, S. L.; Clark, K., Order and index of singular time-invariant linear systems, Systems and Control Lett., 1, 119 (1981) · Zbl 0474.93031
[338] Campbell, S. L.; Rose, N. J., Singular perturbation of autonomous linear systems III, Houston J. Math., 4, 527 (1978) · Zbl 0387.34041
[339] Campbell, S. L.; Rose, N. J., Singular perturbation of autonomous linear systems, SIAM J. Math. Anal., 10, 542 (1979) · Zbl 0413.34055
[340] Cobb, D., Feedback and pole placement in descriptor variable systems, Int. J. Control, 33, 1135 (1981) · Zbl 0464.93039
[341] Cobb, D., On the solution of linear differential equations with singular coefficients, J. Diff. Eqs., 46, 310 (1982) · Zbl 0489.34006
[342] Cobb, D., Descriptor variable systems and optimal state regulation, IEEE Trans Aut. Congrol, AC-27, 601 (1982) · Zbl 0522.93036
[343] Gear, C. W.; Petzold, L. R., Differential/algebraic systems and matrix pencils, (Lecture Notes in Mathematics, 973 (1982), Springer: Springer Berlin), 75-89 · Zbl 0494.65038
[344] Francis, B. A., Convergence in the boundary layer for singularly perturbed equations, Automatica, 18, 57 (1982) · Zbl 0468.49003
[345] O’Malley, R. E.; Flaherty, J. E., Singular singular perturbation problems, (Singular Perturbations and Boundary Layer Theory, 594 (1977), Springer: Springer Berlin), 422-436
[346] O’Malley, On singular singularly-perturbed initial value problems, Applicable Anal., 8, 71 (1978) · Zbl 0397.34068
[347] O’Malley, R. E., Partially singular control problems as singular singular perturbation problems, (Proceedings of 7th IFAC Congress. Proceedings of 7th IFAC Congress, Helsinki (1978)), 957-961
[348] O’Malley, R. E., A singular singularly-perturbed linear boundary value problem, SIAM J. Math. Anal., 10, 695 (1979) · Zbl 0413.34021
[349] Pandolfi, L., On the regulator problem for linear degenerate control systems, J. Opt. Theory & Appl., 33, 241 (1981) · Zbl 0421.93036
[350] Sincovec, R. F.; co-workers, Analysis of descriptor systems using numerical algorithms, IEEE Trans Aut. Control, AC-26, 139 (1981)
[351] Vasil’eva, A. B., Singularly perturbed systems with an indeterminacy in their degenerate equations, Diff. Eqs., 12, 1227 (1976) · Zbl 0374.34040
[352] Verghese, G.; Van Dooren, P.; Kailath, T., Properties of the system matrix of a generalized state-space system, Int. J. Control, 30, 235 (1979) · Zbl 0418.93016
[353] Verghese, G. C.; Levy, B. C.; Kailath, T., A Generalized state space for singular systems, IEEE Trans Aut. Control, AC-26, 811 (1981) · Zbl 0541.34040
[354] Wilkinson, J. H., Note on the practical significance of the Drazin inverse, (Campbell, S. L., Recent Applications of Generalized Inverses (1982), Pitman) · Zbl 0491.65025
[355] Young, K. D., Feedback design of singularly perturbed systems, (Proceedings of 17th Allerton Conference on Communication, Control and Computing. Proceedings of 17th Allerton Conference on Communication, Control and Computing, Illinois (1979)), 449-454
[356] Young, K. D., Analysis of singular singularly-perturbed systems, (Proceedings of 18th Allerton Conference on Communication, Control and Computing. Proceedings of 18th Allerton Conference on Communication, Control and Computing, Illinois (1980)), 116-123
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