×

Sensors and controllers location in distributed systems - A survey. (English) Zbl 0555.93035

A survey of the field of optimal sensors and/or controllers location for dynamical distributed parameter systems modelled by partial differential equations is presented. The recent contributions in this field are grouped according to the main goal for which the location problem is developed, namely: system identification, state estimation, and optimal control. In order to pose the sensors and controllers location problem, the semigroup approach for modelling distributed linear systems is briefly reviewed together with its equivalent (infinite dimensional) and approximate (finite dimensional) Fourier expansion representations. After presenting a concise general review of several methods considered in the current literature, a classification of methods is also proposed. The main classifying factor concerns the use of N-modal approximation schemes, and the different stages of the optimization procedure in which they are required.

MSC:

93C20 Control/observation systems governed by partial differential equations
93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
93B30 System identification
42C15 General harmonic expansions, frames
93C25 Control/observation systems in abstract spaces
93E12 Identification in stochastic control theory
47D03 Groups and semigroups of linear operators
93C05 Linear systems in control theory
93E10 Estimation and detection in stochastic control theory
93E20 Optimal stochastic control
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ahmed, N. U.; Teo, K. L., (Optimal Control of Distributed Parameter Systems (1981), North-Holland: North-Holland Amsterdam) · Zbl 0472.49001
[2] Aidarous, S. E., Optimal allocation of pointwise controllers in stochastic distributed systems, (Proc. 1975 IEEE Control and Decision Conf.. Proc. 1975 IEEE Control and Decision Conf., Houston, Texas (1975)) · Zbl 0312.93048
[3] Aidarous, S. E.; Gevers, M. R.; Installé, M. J., Optimal sensors allocation strategies for a class of stochastic distributed systems, Int. J. Control, 22, 197 (1975) · Zbl 0311.93056
[4] Aidarous, S. E.; Gevers, M. R.; Installé, M. J., Optimal pointwise discrete control and controllers location strategies for stochastic distributed systems, Int. J. Control, 24, 493 (1976) · Zbl 0332.93084
[5] Aidarous, S. E.; Gevers, M. R.; Installé, M. J., On the asymptotic behaviour of sensors’ allocation algorithm in stochastic distributed systems, (Ruberti, A., Distributed Parameter Systems: Modelling and Identification. Distributed Parameter Systems: Modelling and Identification, Lecture Notes in Control & Inf. Sci., vol. 1 (1978), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0379.93056
[6] Akhiezer, N. I.; Glazman, I. M., (Theory of Linear Operators in Hilbert Space, vols. I and II (1981), Pitman: Pitman Boston) · Zbl 0467.47001
[7] Ames, W. F., (Nonlinear Partial Differential Equations in Engineering, vols. I and II (1972), Academic Press: Academic Press New York) · Zbl 0255.35001
[8] Amouroux, M., Sur une méthode de détermination de l’emplacement optimal de points d’action pour une classe de systèmes linéaire à paramètres répartis, C.R. Acad. Sci. Paris, 277, 193 (1973) · Zbl 0274.93017
[9] Amouroux, M.; Babary, J. P., Sur la commandabilité de système linéaire à paramètres répartis et sons utilization pour la determination de points d’action optimaux, R.A.I. R.O., J-3, 120 (1973) · Zbl 0273.93015
[10] Amouroux, M.; Babary, J. P., Optimal pointwise control for a class of distributed parameter systems, (Proc. IFAC 6th World Congress. Proc. IFAC 6th World Congress, Boston/Cambridge, Mass. (1975)), 1-2 · Zbl 0391.49006
[11] Amouroux, M.; Babary, J. P., On the optimal pointwise control and parametric optimization of distributed parameter systems, Int. J. Control, 28, 789 (1978) · Zbl 0391.49006
[12] Amouroux, M.; Babary, J. P., On optimization of zones of action for an optimal control problem for distributed parameter systems, Int. J. Control, 29, 861 (1979) · Zbl 0444.49004
[13] Amouroux, M.; Babary, J. P.; Malandrakis, C., Optimal location for linear stochastic distributed parameter systems, (Ruberti, A., Distributed Parameter Systems: Modelling and Identification. Distributed Parameter Systems: Modelling and Identification, Lecture Notes in Control & Inf. Sci., vol. 1 (1978), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0392.93030
[14] Amouroux, M.; Di Pillo, G.; Grippo, L., Optimal selection of sensors and control location for a class of distributed parameter systems, Ric. Automatica, 7, 92 (1976)
[15] Athans, M., Toward a practical theory of distributed parameter systems, IEEE Trans Aut. Control, AC-15, 245 (1970)
[16] (Aziz, A. K.; Wingate, J. W.; Balas, M. J., Control Theory of Systems Governed by Partial Differential Equations (1977), Academic Press: Academic Press New York) · Zbl 0409.00018
[17] Balakrishnan, A. V., (Applied Functional Analysis (1980), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0511.46002
[18] Banks, H. T.; Kunisch, K., An approximation theory for nonlinear partial differential equations with applications to identification and control, SIAM J. Control & Optimiz., 20, 815 (1982) · Zbl 0503.93016
[19] Bencala, K. E.; Seinfeld, J. H., Distributed parameter filtering: boundary noise and discrete observations, Int. J. Syst. Sci., 10, 493 (1979) · Zbl 0404.93043
[20] Bensoussan, A., (Filtrage Optimal des Systèmes Lineaires (1971), Dunod: Dunod Paris)
[21] Bensoussan, A., Optimization of sensors location in a distributed parameter system, (Curtain, R. F., Stability of Stochastic Dynamical Systems. Stability of Stochastic Dynamical Systems, Lecture Notes in Mathematics, vol. 294 (1972), Springer-Verlag: Springer-Verlag Berlin)
[22] Bensoussan, A., Control of stochastic partial differential equations, (Ray, W. H.; Lainiotis, D. G., Distributed Parameter Systems (1978), Marcel Dekker: Marcel Dekker New York) · Zbl 0548.35037
[23] Burger, J., Determination de l’emplacement optimal des actioneurs ponctuels dans un système regi par une équation aux derivée partielles linéaire, C.R. Acad. Sci. Paris, 280, 161 (1975) · Zbl 0299.93021
[24] Burger, J., Emplacement optimal des actionneurs ponctuels ou par zones dans un système gouverné par une équation linéaire d’evolution du premier ordre, R.A.I.R.O., J-5, 87 (1976) · Zbl 0332.93049
[25] Burger, J.; Chavent, G., Identification de paramètre repartis dans les équations aux dérivés partielles, R.A.I.R.O. Syst. Anal. & Control, 13, 115 (1979) · Zbl 0405.93017
[26] Butkovskiy, A. G., (Distributed Control System (1969), American Elsevier: American Elsevier New York) · Zbl 0197.42204
[27] Cannon, J. R.; Klein, R. E., Optimal selection of measurement location in a conductor for approximate determination of temperature distribution, (Proc. Joint Aut. Control Conf.. Proc. Joint Aut. Control Conf., Atlanta (1970)), 750-756
[28] Cannon, J. R.; Klein, R. E., Optimal selection of measurement locations in a conductor for approximate determination of temperature distributions, Trans. ASME, J. Dyn. Syst. Meas. & Control, 93-G, 193 (1971)
[29] Caravani, P.; Di Pillo, G.; Grippo, L., Optimal location of a measurement point in a diffusion process, (Proc. IFAC 6th World Congress. Proc. IFAC 6th World Congress, Boston/Cambridge, Mass. (1975)), 3-8
[30] Carotenuto, L.; Raiconi, G., Identifiability and identification of a Galerkin approximation for a class of distributed parameter systems, Int. J. Syst. Sci., 11, 1035 (1980) · Zbl 0494.93013
[31] Carotenuto, L.; Raiconi, G., On the identification of a random parameter function in the one-dimensional diffusion equation and related location, (Proc. 8th IFAC Triennial World Congress, Kyoto, vol. II (1981)), 76-81 · Zbl 0526.93062
[32] Chavent, G., About the stability of the optimal control solution of inverse problems, (Anger, G., Inverse and Improperly Posed Problems in Differential Equations (1979), Academie-Verlag: Academie-Verlag Berlin) · Zbl 0436.93018
[33] Chavent, G., Identification of distributed parameter systems: about the output least squares methods, its implementation, and identifiability, (Proc. 5th IFAC Symp. on Identification & Syst. Parameter Estim.. Proc. 5th IFAC Symp. on Identification & Syst. Parameter Estim., Darmstadt (1979)), 85-97
[34] Chavent, G., Local stability of the output least square parameter estimation technique, Mat. Aplic. Comp., 2, 3 (1983) · Zbl 0567.93017
[35] Chen, W. H.; Seinfeld, J. H., Optimal location of process measurements, Int. J. Control, 21, 1003 (1975) · Zbl 0322.93042
[36] Courant, R.; Friedrichs, K. O., (Supersonic Flow and Shock Waves (1948), Interscience: Interscience New York) · Zbl 0041.11302
[37] Courant, R.; Hilbert, D., (Methods of Mathematical Physics, vol. II (1962), Interscience: Interscience New York) · Zbl 0729.00007
[38] Courdesses, M., Optimal sensors and controllers allocation in output feedback control for a class of distributed parameter systems, (Proc. Symp. on Advances in Measurement & Control. Proc. Symp. on Advances in Measurement & Control, Athens (1978)), 735-740
[39] Courdesses, M.; Amouroux, M., Identifiabilité et identification des systèmes à paramètres repartis, (Proc. 3rd IFAC Symp. on Control of D.P.S.. Proc. 3rd IFAC Symp. on Control of D.P.S., Toulouse (1982)), XXI.1-XXI.6
[40] Courdesses, M.; Amouroux, M.; Polis, M. P., Sur des conditions d’identifiabilité des paramètres d’un système distribué de type parabolique, R.A.I.R.O., Syst., Anal. & Control, 15, 347 (1981) · Zbl 0469.93041
[41] Courdesses, M.; Polis, M. P.; Amouroux, M., On identifiability of parameters in a class of parabolic distributed systems, IEEE Trans Aut. Control, AC-26, 474 (1981) · Zbl 0487.93016
[42] Crouch, P. E.; Kubrusly, C. S.; Pritchard, A. J., Systems identification: some fundamental aspects, (Control Theory Centre Report no. 104 (1982), Univ. of Warwick)
[43] Curtain, R. F., A survey of infinite dimensional filtering, SIAM Rev., 17, 395 (1975) · Zbl 0308.60022
[44] Curtain, R. F., A semigroup approach to the L.Q.G. problem for infinite dimensional systems, IEEE Trans Circ. & Syst., CAS-25, 713 (1978) · Zbl 0402.93045
[45] Curtain, R. F., Compensators for infinite dimensional linear systems, J. Franklin Inst., 315, 331 (1983) · Zbl 0518.93045
[46] Curtain, R. F., Tracking and regulation for distributed parameter systems, Mat. Aplic. Comp., 2, 199 (1983) · Zbl 0535.93036
[47] Curtain, R. F., Finite dimensional compensators for parabolic distributed systems with unbounded control and observation, SIAM J. Control & Optimiz., 22, 255 (1984) · Zbl 0542.93056
[48] Curtain, R. F.; Ichikawa, A., Optimal location of sensors for filtering for distributed systems, (Ruberti, A., Distributed Parameter Systems: Modelling and Identification. Distributed Parameter Systems: Modelling and Identification, Lecture Notes in Control & Inf. Sci., vol. 1 (1978), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0398.93059
[49] Curtain, R. F.; Pritchard, A. J., (Functional Analysis in Modern Applied Mathematics (1977), Academic Press: Academic Press New York) · Zbl 0448.46002
[50] Curtain, R. F.; Pritchard, A. J., Infinite Dimensional Linear Systems Theory, (Lecture Notes in Control & Inf. Sci., vol. 8 (1978), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0352.49003
[51] Davis, M. H.A., (Linear Estimation and Stochastic Control (1977), Chapman and Hall: Chapman and Hall London) · Zbl 0437.60001
[52] Dunford, N.; Schwartz, J. T., (Linear Operators—Part I: General Theory (1958), Interscience: Interscience New York)
[53] El Jai, A., Sur la commande avec estimation de l’etat initial d’une classe de système a paramètres répartis, considérations pratiques, R.A.I.R.O., Syst., Anal. & Control, 11, 313 (1977) · Zbl 0385.93025
[54] Ewing, D. J.; Higgins, T. J., A sensor location algorithm for distributed parameter systems, (Proc. 9th Annual Allerton Conf. on Circ. and Syst. Theory. Proc. 9th Annual Allerton Conf. on Circ. and Syst. Theory, Allerton (1971))
[55] Friedman, A., (Partial Differential Equations (1969), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York)
[56] Fuhrmann, P. A., (Linear Systems and Operators in Hilbert Space (1981), McGraw-Hill: McGraw-Hill New York) · Zbl 0456.47001
[57] Garabedian, P. R., (Partial Differential Equations (1964), John Wiley: John Wiley New York) · Zbl 0124.30501
[58] Goodson, R. E.; Klein, R. E., A definition and some results of distributed systems observability, IEEE Trans Aut. Control, AC-15, 165 (1970)
[59] Goodson, R. E.; Polis, M. P., Identification of parameters in distributed systems, (Ray, W. H.; Lainiotis, D. G., Distributed Parameter Systems (1978), Marcel Dekker: Marcel Dekker New York) · Zbl 0298.00029
[60] Gustafson, K. E., (Introduction to Partial Differential Equations and Hilbert Space Methods (1980), John Wiley: John Wiley New York) · Zbl 0434.35001
[61] Helton, J. W., Systems with infinite dimensional state space: the Hilbert space approach, (Proc. IEEE, 64 (1976)), 145
[62] Hille, E.; Phillips, R. S., Functional Analysis and Semigroups, AMS Colloq. Publ., vol. XXXI (1957), Providence · Zbl 0078.10004
[63] Ichikawa, A.; Ryan, E. P., Filtering and control of distributed parameter systems with point observations and inputs, (Proc. 2nd IFAC Symp. on Control of D.P.S.. Proc. 2nd IFAC Symp. on Control of D.P.S., Coventry (1977)), 347-357 · Zbl 0408.93045
[64] Ichikawa, A.; Ryan, E. P., Sensor and controller location problems for distributed parameter systems, Automatica, 15, 347 (1979) · Zbl 0408.93045
[65] John, F., (Partial Differential Equations (1981), Springer-Verlag: Springer-Verlag Berlin)
[66] Johnson, T. L., Progress in modelling and control of flexible spacecraft, J. Franklin Inst., 315, 495 (1983) · Zbl 0519.93043
[67] Kaiser, K. W., A method of determining the heat-sensor configuration in temperature distribution controllers, (Proc. 1st IFAC Symp. on Control of D.P.S.. Proc. 1st IFAC Symp. on Control of D.P.S., Banff (1971)), 1-3
[68] Kato, T., (Perturbation Theory for Linear Operators (1980), Springer-Verlag: Springer-Verlag Berlin)
[69] Kitamura, S.; Nakagiri, S., Identifiability of spatially-varying and constant parameters in distributed systems of parabolic type, SIAM J. Control & Optimiz., 15, 785 (1977) · Zbl 0354.93020
[70] Kitamura, S.; Taniguchi, Y., Parameter estimation of one- and two-dimensional heat conduction processes using a minicomputer, Int. J. Control, 34, 955 (1981)
[71] Klein, R. E., Non well-posed problems in the sense of Hadamard and their relationship to distributed parameter state reconstructor problems, (Proc. 1st IFAC Symp. on Control of D.P.S.. Proc. 1st IFAC Symp. on Control of D.P.S., Banff (1971))
[72] Koivo, A. J.; Kruh, P., On the design of approximately optimal feedback controllers for a distributed parameter system, Int. J. Control, 10, 53 (1969) · Zbl 0174.14504
[73] Kubrusly, C. S., Distributed parameter system identification: a survey, Int. J. Control, 26, 509 (1977) · Zbl 0405.93012
[74] Kubrusly, C. S., Linear DPS identification in a stochastic environment, R.A.I.R.O. System Anal. & Control, 14, 343 (1980) · Zbl 0477.93064
[75] Kubrusly, C. S., On identifiability of linear dynamical systems, Mat. Aplic. Comp., 3, 219 (1984) · Zbl 0558.93021
[76] Kumar, S.; Seinfeld, J. H., Optimal location of measurements for distributed parameter estimation, IEEE Trans Aut. Control, AC-23, 690 (1978) · Zbl 0381.93047
[77] Kumar, S.; Seinfeld, J. H., Optimal location of measurements in tubular reactors, Chem. Engng. Sci., 33, 1507 (1978)
[78] Le Pourhiet, A.; Le Letty, L., Optimization of sensors location in distributed parameter system identification, (Proc. 4th IFAC Symp. on Identification & System Parameter Estimation. Proc. 4th IFAC Symp. on Identification & System Parameter Estimation, Tibilisi (1976)), 1581-1591
[79] Lee, M. H.; Koppel, L. B.; Lim, H. C., Optimal sensor locations and controller settings for class of counter current processes, (Ind. Eng. Chem. Proc. Des. Devl., 12 (1973)), 36
[80] Leigh, J. R., (Functional Analysis and Linear Control Theory (1980), Academic Press: Academic Press New York) · Zbl 0479.49001
[81] Lions, J. L., (Controle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles (1968), Dunod: Dunod Paris) · Zbl 0179.41801
[82] Lions, J. L., Some Aspects of the Optimal Control of Distributed Parameter Systems, (CBMS Regional Conf. Series in Appl. Math., vol. 6 (1972), SIAM: SIAM Philadelphia) · Zbl 0275.49001
[83] Lions, J. L., Optimal control of deterministic distributed parameter system, (Ray, W. H.; Lainiotis, D. G., Distributed Parameter Systems (1978), Marcel Dekker: Marcel Dekker New York) · Zbl 0275.49001
[84] Lions, J. L., On the foundations of the optimal control of distributed systems, (Irack, K.; Malanowski, K.; Walukiewicz, S., Optimization Techniques—Part I. Optimization Techniques—Part I, Lecture Notes in Control & Inf. Sci., vol. 22 (1980), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0451.49009
[85] Malandrakis, C. G., Optimal sensor and controller allocation for a class of distributed parameter systems, Int. J. Syst. Sci., 10, 1283 (1979) · Zbl 0439.93034
[86] Mikhlin, S. G., (Mathematical Physics: An Advanced Course (1970), North-Holland: North-Holland Amsterdam) · Zbl 0202.36901
[87] Morari, M.; O’Dowd, J., Optimal sensors location in the presence of nonstationary noise, Automatica, 16, 463 (1980) · Zbl 0441.93038
[88] Nakagiri, S-I., Identifiability of linear systems in Hilbert spaces, SIAM J. Control & Optimiz., 21, 501 (1983)
[89] Nakamori, Y.; Miyamoto, S.; Ikeda, S.; Sawaragi, Y., Measurement optimization with sensitivity criteria for distributed parameter systems, IEEE Trans Aut. Control, AC-25, 889 (1980) · Zbl 0444.93027
[90] Naylor, A. W.; Sell, G. R., (Linear Operator Theory in Engineering and Science (1982), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0497.47001
[91] Omatu, S.; Koide, S.; Soeda, T., Optimal sensor location problem for linear distributed parameter systems, IEEE Trans Aut. Control, AC-23, 665 (1978) · Zbl 0381.93045
[92] Omatu, S.; Seinfeld, J. H., Optimization of sensor and actuator location in a distributed parameter system, J. Franklin Inst., 315, 407 (1983) · Zbl 0528.93068
[93] Phillipson, G. A., (Identification of Distributed Systems (1971), American Elsevier: American Elsevier New York) · Zbl 0221.93003
[94] Polis, M. P., The distributed system parameter identification problem: a survey of recent results, (Proc. 3rd IFAC Symp. on Control of D.P.S.. Proc. 3rd IFAC Symp. on Control of D.P.S., Toulouse (1982)), SP.45-SP.58
[95] Polis, M. P.; Goodson, R. E., Parameter identification in distributed systems: a synthesizing overview, (Proc. IEEE, 64 (1976)), 45
[96] Polis, M. P.; Goodson, R. E.; Wozny, M. J., On parameter identification of distributed systems using Galerkin’s criterion, Automatica, 9, 56 (1973)
[97] Pritchard, A. J., Infinite-dimensional linear system theory, (Proc. IEE, 126 (1979)), 641
[98] Qureshi, Z. H.; Ng, T. S.; Goodwin, G. C., Optimum experimental design for identification of distributed parameter systems, Int. J. Control, 31, 21 (1980) · Zbl 0431.93017
[99] Rafajlłowicz, E., Design of experiments for eignvalue identification in distributed-parameter systems, Int. J. Control, 34, 1079 (1981)
[100] Rafajlłowicz, E., Optimal experimental design for identification of linear distributed-parameter systems: frequency domain approach, IEEE Trans Aut. Control, AC-28, 806 (1983)
[101] Ray, W. H., Distributed parameter state estimation algorithms and applications—a survey, (Proc. IFAC 6th World Congress. Proc. IFAC 6th World Congress, Boston/Cambridge, Mass. (1975)), 1-8
[102] Ray, W. H., Some recent applications of distributed parameter systems theory—a survey, Automatica, 14, 281 (1978) · Zbl 0375.93030
[103] (Ray, W. H.; Lainiotis, D. G., Distributed Parameter Systems—Identification, Estimation and Control (1978), Marcel Dekker: Marcel Dekker New York) · Zbl 0476.00031
[104] Robinson, A. C., A survey of optimal control of distributed parameter systems, Automatica, 7, 371 (1971) · Zbl 0218.49001
[105] (Ruberti, A., Distributed Parameter Systems: Modelling and Identification. Distributed Parameter Systems: Modelling and Identification, Lecture Notes in Control & Inf. Sci., vol. 1 (1978), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0366.00020
[106] Sakawa, Y., Observability and related problems of partial differential equations of parabolic type, SIAM J. Control, 13, 14 (1975) · Zbl 0296.93006
[107] Sawaragi, Y.; Soeda, T.; Omatu, S., Modelling, Estimation and their Applications for Distributed Parameter Systems, (Lecture Notes in Control & Inf. Sci., vol. 11 (1978), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0392.93001
[108] Schechter, M., (Modern Methods in Partial Differential Equations (1977), McGraw-Hill: McGraw-Hill New York) · Zbl 0377.35003
[109] Seinfeld, J. H.; Chen, W. H., Estimation of parameters in partial differential equations from noisy experimental data, Chem. Engng. Sci., 26, 753 (1971)
[110] Showalter, R. E., (Hilbert Space Methods for Partial Differential Equations (1977), Pitman: Pitman New York) · Zbl 0364.35001
[111] Sokollik, F., Identifikation von Systemen mit verteilten Parametern, Messen. Steuern. Regeln., 17, 271 (1974)
[112] Sokollik, F., Zur Parameterschätzung von Systemen mit Verteilten Parametern mittels on-Line-Verfahren, Messen. Steuern. Regeln., 19, 84 (1976)
[113] Sokollik, F., Zur Optimalen Wahl der Meßorte bei der Parameterschätzung von Systemen mit verteilten Parametern, Messen. Steuern. Regeln., 19, 194 (1976)
[114] (Stavroulakis, P., Distributed Parameter Systems Theory: Control and Estimation. Distributed Parameter Systems Theory: Control and Estimation, Benchmark Papers in Elec. Eng. & Comp. Sci., vol. 26/27 (1983), Hutchinson Ross: Hutchinson Ross Stroudsburg), Parts I/II
[115] Treves, F., (Basic Linear Partial Differential Equations (1975), Academic Press: Academic Press New York) · Zbl 0305.35001
[116] Tzafestas, S. G., Distributed parameter state estimation, (Ray, W. H.; Lainiotis, D. G., Distributed Parameter Systems (1978), Marcel Dekker: Marcel Dekker New York) · Zbl 0275.93053
[117] (Tzafestas, S. G., Simulation of Distributed Parameter and Large-Scale Systems (1980), North-Holland: North-Holland Amsterdam) · Zbl 0424.00031
[118] (Tzafestas, S. G., Distributed Parameter Control Systems (1982), Pergamon Press: Pergamon Press Oxford) · Zbl 0539.00030
[119] Tzafestas, S. G.; Stavroulakis, P., Recent advances in the study of distributed parameter systems, J. Franklin Inst., 315, 285 (1983) · Zbl 0517.93037
[120] Wang, P. K.C., Control of distributed parameter systems, (Leondes, C. T., Advances in Control Systems, vol. 1 (1964), Academic Press: Academic Press New York) · Zbl 0204.17202
[121] Weidmann, J., (Linear Operators in Hilbert Spaces (1980), Springer-Verlag: Springer-Verlag Berlin)
[122] Weinberger, H. F., (A First Course in Partial Differential Equations (1965), Ginn-Xerox: Ginn-Xerox Lexington) · Zbl 0127.04805
[123] Yosida, K., (Functional Analysis (1980), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0152.32102
[124] Yu, T. K.; Seinfeld, J. H., Observability of a class of hyperbolic distributed parameter systems, IEEE Trans Aut. Control, AC-16, 495 (1971)
[125] Yu, T. K.; Seinfeld, J. H., Observability and optimal measurement location in linear distributed parameter systems, Int. J. Control., 18, 785 (1973) · Zbl 0263.93010
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.