References
Yu. H. Andreev, “Algebraic methods of the state space in the theory of control of linear objects (review of foreign literature),”Avtomat. Telemekh., No. 3, 5–50 (1977).
I. K. Asmykovich, R. Gabasov, F. M. Kirillova, and V. M. Marchenko, “Control of finite-dimensional systems,”Avtomat. Telemekh., No. 11, 5–29 (1986).
V. E. Belozerov and G. V. Mozhaev, “On the uniqueness of the solutions of problems of decomposition and aggregation of linear automatic control systems,” In:Theory of Complex Systems and Their Simulation Methods [in Russian], VINIISI, Moscow (1982), pp. 4–13.
V. E. Belozerov and G. V. Mozhaev, “On the decomposition of linear stationary automatic control systems,”Kibern. Vyshisl. Tekh., (Kiev), No. 54, 10–16 (1982).
E. B. Vinberg and V. L. Papov, “Invariant theory,” In:Progress in Science and Technology. Series on Contemporary Problems in Mathematics. Fundamental Trends, [in Russian], Vol. 55, VINITI, Akad. Nauk SSSR, Moscow (1989), pp. 137–314.
F. R. Gantmahker,Matrix Theory [in Russian], Nauka, Moscow (1967).
V. V. Krylov, “Construction of models of the internal structure of dynamic systems from input-output relations (abstract realization theory). Parts I, II,”Avtomat. Telemekh., No. 2, 5–15 (1984); No. 3, 5–19 (1984).
N. I. Osetinskii, “On the theory of realization of linear stationary dynamic systems over a field. I, II, III,”Programmirovanie, No. 3, 75–85 (1975); No. 4, 58–68 (1975); No. 1, 70–76 (1976).
N. I. Osetinskii, “Certain results of the modern systems theory: A review,” In:Mathematical Methods of the Systems Theory [in Russian], Mir, Moscow (1979), pp. 271–327.
N. I. Osetinskii, “On the basis of invariants of linear controlled systems,” In:Theory of Complex Systems and Their Simulation Methods [in Russian], VINIISI, Moscow (1988), pp. 76–81.
N. I. Osetinskii, “Certain results and methods of the modern linear systems theory: A review,” In:Theory of Systems. Mathematical Methods and Simulation [in Russian], Mir, Moscow (1989), pp. 328–379.
Ya. I. Roitenberg,Automatic Control [in Russian], Nauka, Moscow (1992).
R. G. Faradzhev,Linear Sequential Machines [in Russian], Sov. Radio, Moscow (1975).
I. R. Shafarevich,Basic Algebraic Geometry [in Russian], Nauka, Moscow (1972).
M. Aigner,Combinatorial Theory, Springer-Verlag, Berlin (1979).
B. D. O. Anderson, M. A. Arbib, and E. G. Manes, “Foundations of system theory: finitary and infinitary conditions,”Lect. Notes Econ. Math. Sys.,115 Springer-Verlag, Berlin (1979).
B. D. O. Anderson, N. K. Bose, and E. I. Jury, “Output feedback stabilization and related problems —solution via decision algebra methods,”IEEE Trans. Autom. Contr.,20, 53–66 (1975).
B. D. O. Anderson and D. G. Luenberger, “Design of multivariable feedback systems,”Proc. IEEE,114, No. 6, 395–399 (1967).
B. D. O. Anderson and R. W. Scott, “Output feedback stabililization—solution by algebraic geometry methods,”Proc. IEEE,65, No. 6, 849–861 (1977).
M. A. Arbib, “Coproducts and group machines,”J. Comp. Syst. Sci.,7, 278–287 (1973).
M. A. Arbib and E. G. Manes, “Foundations of system theory: decomposable systems,”Automatica, No. 10, 285–302 (1974).
M. A. Arbib and E. G. Manes, “Machines in a category: an expository introduction,”SIAM Rev.,16, 163–192 (1974).
M. A. Arbib and H. P. Zeiger, “On the relevance of abstract algebra to control theory,”Automatica,5, 589–606 (1969).
L. Baratchart, “On the parametrization of linear constant systems,”SIAM J. Contr. Optimiz.,23, No. 5, 752–773 (1985).
R. R. Bitmead, S. Y. Kung, B. D. O. Anderson, and T. Kailath, “Greatest common divisors via generalized Sylvester and Besout matrices,”IEEE Trans. Autom. Contr.,23, 1043–1047 (1978).
F. M. Brasch and J. B. Pearson, “Pole placement using dynamic compensators,”IEEE Trans. Autom. Contr.,15, No. 1, 34–43 (1970).
R. Brockett, “Some geometric questions in the theory of linear systems,”IEEE Trans. Autom. Contr.,21, No. 1, 449–455 (1976).
R. Brockett, “The geometry of the set of controllable systems,”Res. Report Autom. Contr. Lab., Nagoya Univ.,24, 1–7 (1977).
R. Brockett and C. I. Byrnes, “Multivariable Nyquist criteria, root loci and pole placement: A geometric viewpoint,”IEEE Trans. Autom. Contr.,26, No. 1, 271–284 (1981).
P. Brunovsky, “A classification of linear controlled systems,”Kibernetika,6, No. 3, 173–188 (1970).
C. I. Byrnes, “The moduli space for linear dynamical systems,” In:Geometric Control Theory, Math. Sci. Press, Brookline, Massachusetts (1977), pp. 229–276.
C. I. Byrnes, “On the control of certain deterministic infinite dimensional systems by algebro-geometric techniques,”Amer. J. Math.,100, 1333–1381 (1979).
C. I. Byrnes, “On certain problems of arithmetic arising in the realization of linear systems with symmetries,”Asterisque,75,76, 57–65 (1980).
C. I. Byrnes, “Algebraic and geometric aspects of the analysis of feedback systems,” In:Geometric Methods in Linear Systems Theory, North-Holland, Amsterdam (1980), pp. 85–124.
C. I. Byrnes, “Control theory, inverse spectral problems, and real algebraic geometry,” In:Diff. Geom. Methods in Control Theory, Birkhauser, Boston (1982).
C. I. Byrnes, “On the stabilization of the multivariable systems and the Ljusternik-Snirel'mann category of real Grassmannians,”Syst. Contr. Lett.,3, 255–266 (1983).
C. I. Byrnes, “Pole assignment by output feedback,” In:Lect. Notes Contr. Inform. Sci.,135, Springer-Verlag (1989), pp. 31–78.
C. I. Byrnes and B. D. O. Anderson, “Output feedback and generic stabilizability,”SIAM J. Contr. Optimiz. 22, No. 3, 362–380 (1984).
C. I. Byrnes and T. E. Duncan, “On certain topological invariants arising in system theory,” In:New Directions in Appl. Math., Springer-Verlag, New York (1982), pp. 29–71.
C. I. Byrnes and P. Falb, “Applications of algebraic geometry in system theory,”Amer. J. Math.,101, 337–363 (1979).
C. I. Byrnes and U. Helmke, “Intersection theory for linear systems,” In:Proc. 25th IEEE Conf. Decision and Contr., Athens (1986), pp. 1944–1949.
C. I. Byrnes and N. E. Hurt, “On the moduli of linear dynamical systems,”Adv. Math. Studies Anal., No. 4, 83–122 (1979).
C. I. Byrnes and P. K. Stevens, “Global properties of root-locus map,” In:Lect. Notes Contr. Inf. Sci.,39, Springer-Verlag (1982), pp. 9–25.
D. F. Delchamps, “Global structure of families of multivariable linear systems with an application to identification,”Math. Syst. Theory,18, 329–380 (1985).
S. Eilenberg, “Automata, Languages and Machines, Vol. A, Academic Press, New York (1974).
C. Ehresmann, “Sur la topologie de certains espace homogenes,”Ann. Math.,35, 396–443 (1934).
G. D. Forney, “Minimal bases of rational vector spaces with applications to multivariable linear systems,”SIAM J. Contr. Optimiz.,13, 493–520 (1975).
P. Fuhrmann, “Algebraic system theory: An analyst's point of view,”J. Franklin Inst.,301, 521–540 (1976).
B. K. Ghosh, “Transcendental and interpolation methods in simultaneous stabilization and simultaneous partial pole placement problems,”SIAM J. Contr. Optimiz.,24, 1091–1109 (1986).
C. G. Gibson, Wirthmuller K., et al.,Lect. Notes Math.,552, Springer-Verlag (1977).
J. M. Gracia, I. De Hoyos, and I. Zaballa, “A characterization of feedback equivalence,”SIAM J. Contr. Optimiz. 28, No. 5, 1103–1112 (1990).
M. K. J. Hautus, “Controllability and observability conditions of linear autonomous systems,” In:Proc. Kon. Nederlande Akademie van Wetenschappen, Amsterdam (1969), Ser. A. 72, No. 5, pp. 443–448.
M. K. J. Hautus, “Stabilization, controllability, and observability of linear autonomous systems,” In:Proc. Kon. Nederlande Akademic van Wetenschappen, Amsterdam (1970), Ser. A. 73, No. 5, pp. 448–455.
M. K. J. Hautus and E. Sontag, “New results on pole-shifting for parametrized families of systems,”J. Pure Appl. Algebra,40, 229–244 (1986).
M. Hazewinkel, “Moduli and canonical forms of linear dynamical systems II: The topological case,”Math. Syst. Theory,10, 363–385 (1977).
M. Hazewinkel, “Moduli and canonical forms for linear dynamical systems III: The algebraic geometric case,” In:Proc. NASA-AMS Conf. on Geom. Contr. Theory, Math. Sci. Press (1977), pp. 291–336.
M. Hazewinkel, “On the (internal) symmetry groups of linear dynamical systems,” In:Groups, Systems and Many-Body Physics, Viewe (1979), pp. 362–404.
M. Hazewinkel, “(Fine) moduli (spaces) for linear systems: What are they and what are they good for,” In:Proc. NATO-AMS Adv. Study Inst., Reidel Publ. (1979), pp. 125–193.
M. Hazewinkel, “On families of linear dynamical systems: degeneracy phenomena,” In:Proc. NATO-AMS Conf. Algebraic and Geometric Meth. Linear Syst. Theory (1979), pp. 157–190.
M. Hazewinkel, “A partial survey of the uses of algebraic geometry in systems and control theory,” In:Sym. Math. INDAM (Severi Centennial Conference, 1979), Academic Press (1981), pp. 245–292.
M. Hazewinkel, “Lectures on invariants, representations and Lie algebras in systems and control theory,” In:Lect. Notes Math.,1029, Springer-Verlag (1984), pp. 1–36.
M. Hazewinkel and R. E. Kalman, “On invariants, canonical forms and moduli for linear, constant, finite dimensional dynamical systems,” In:Lect. Notes Econ. Math. Syst. Theory,131, Springer-Verlag (1976), pp. 48–60.
M. Hazewinkel and C. Martin, “Representations of the symmetric group, the specialization order, systems and the Grassmann manifold,”L'Enscign. Math.,29, 53–87 (1983).
M. Hazewinkel and C. Martin, “Symmetric linear systems: an application of algebraic systems theory,”Int. J. Contr.,37, 1371–1384 (1983).
M. Hazewinkel and C. Martin, “Special structure, decentralizations and symmetry for linear systems,”Lect. Notes Contr. Inform. Sci.,58, Springer-Verlag (1984), pp. 437–440.
M. Hazewinkel and A. M. Perdon, “On the theory of families of linear dynamical systems,” In:Proc. MTNS'79 (1979), pp. 155–161.
U. Helmke, “Topology of the, moduli space for reachable linear dynamical systems: The complex case,”Math. Syst. Theory, No. 19, 155–187 (1986).
U. Helmke, “Linear dynamical systems and instantons in Yang-Mills theory,”IMA J. Math. Contr. Inf.,3, 151–166 (1986).
U. Helmke, “A global parametrization of asymptotically stable linear systems,”Syst. Contr. Lett.,13, (1989).
U. Helmke, “The cohomology of moduli spaces of linear dynamical systems,” In:Habilitatiosschrift, Regensburg Univ., (1990), pp. 1–163.
R. Hermann and C. Martin, “Application of algebraic geometry to systems theory: Part, I,”IEEE Trans. Autom. Contr.,22, 19–25 (1977).
R. Hermann and C. Martin, “Application of algebraic geometry to systems theory: Part II: The McMillan degree and Kronecker indices as topological and holomorphic invariants,”SIAM J. Contr. Optimiz.,16, 743–755 (1978).
M. Heymann, “Pole assignment in multi-input linear systems,”IEEE Trans. Autom. Contr.,13, No. 6, 748–749 (1968).
H. I. Hiller, “On the height of the first Stiefel-Whitney class,”Proc. Amer. Math. Soc.,79, 495–498 (1980).
D. Hinrichsen and D. Prätzel-Wolters, “Generalized Hermite matrices and complete invariants of strict system equivalence,”SIAM J. Contr. Optimiz.,21, 298–305 (1983).
D. Hinrichsen and D. Prätzel-Wolters, “A canonical form for static linear output feedback,”Lect. Notes Contr. Inf. Sci.,58, 441–462 (1984).
D. Hinrichsen and D. Prätzel-Wolters, “A Jordan canonical form for reachable linear systems,”Linear Algebra Appl.,122–124, 143–175 (1989).
B. L. Ho and R. E. Kalman, “Effective construction of linear state variable models from input/output functions,”Regelungstechnik,14, 545–548 (1966).
T. Kailath,Linear Systems, Prentice-Hall, Englewood Cliffs, New Jersey (1980).
R. E. Kalman, “Canonical structure of linear dynamical systems,”Proc. Natl. Acad. Sci. USA,48, No. 4, 596–600 (1962).
R. E. Kalman, “Mathematical description of linear dynamical systems,”SIAM J. Contr. Optimiz,1, 152–192 (1963).
R. E. Kalman, “Kronecker invariants and feedback,” In:Proc. Conf. Ordinary Diff. Eqs., Washington (1971), pp. 459–471.
R. E. Kalman, “Algebraic geometric description of the class of linear systems of constant dimension”, In:8th Annual Princeton Conf. Inform. Sci. Systs., Princeton, New Jersey (1974).
R. E. Kalman, P. Falb, and M. A. Arbib,Topics in Mathematical System Theory, McGraw Hill, New York (1965).
H. Kimura, “Pole assignment by gain output feedback,”IEEE Trans. Autom. Contr.,20, 509–516 (1975).
H. Kimura, “A further result on the problem of pole assignment by output feedback,”IEEE Trans. Autom. Contr.,22, 509–516 (1977).
H. Kraft,Geometrische Methoden in der Invariaten Theorie, Vieweg, Braunschweig-Wiesbaden (1984).
P. S. Krishnaprasad and C. Martin, “On families of systems and deformations”Int. J. Contr.,38, No. 5, 1055–1079 (1983).
C. E. Langenhop, “On the stabilization of linear systems,”Proc. Amer. Math. Soc.,15, No. 5, 735–742 (1964).
V. G. Lomadze, “Finite dimensional time invariant linear dynamical systems. Algebraic theory,”Acta Appl. Math.,19, 149–201 (1990).
D. G. Luenberger, “Canonical forms for linear multivariable systems,”IEEE Trans. Autom. Contr.,12, No. 3, 290–293 (1967).
A. G. J. MacFarlane, “Linear multivariable feedback theory: a survey,”Automatica,8, 455–492 (1972).
A. G. J. MacFarlane, “Trends in linear multivariable feedback control theory,”Automatica,2, 273–277 (1973).
W. Massey,Homology and Cohomology Theory, Marcel Dekker, New York (1978).
D. Q. Mayne, “A canonical model for identification of multivariable systems,”IEEE Trans. Autom. Contr.,17, 728–729 (1972).
B. McMillan, “Introduction to formal realizability theory,”Bell System Tech. J.,31, 217–279, 591–600 (1952).
A. S. Morse, “Ring models for delay-differential systems,”Automatica,12, 529–531 (1976).
D. Mumford,Algebraic Geometry. I: Complex Projective Varieties, Springer-Verlag, New York (1976).
D. Mumford and J. Fogarty,Geometric Invariant Theory, Springer-Verlag, Berlin (1982).
P. E. Newstead,Introduction to Moduli Problems and Orbit Spaces, Tata Inst. of Fund. Research, Bombay (1978).
V. M. Popov, “Hyperstability and optimality of automatic systems with several control functions,”Rev. Roum. Sci. Electrotech. Energ.,9, 629–690 (1964).
V. M. Popov, “Invariant description of linear time invariant controllable systems,”SIAM J. Contr. Optimiz.,10, 252–264 (1972).
I. Postlethwaite and A. G. J. MacFarlane, “A complex variable approach to the analysis of linear multivariable feedback systems,”Lect. Notes Contr. Inform. Sci.,12, Springer-Verlag (1979).
I. Postlethwaite, A. G. J. MacFarlane, and J. M. Edmunds, “Principal gains and principal phases in the analysis of linear multivariable feedback systems,”IEEE Trans. Autom. Contr.,26, 32–46 (1981).
C. Processi, “The invariant theory of n×n-matrices,”Adv. Math.,19, 306–381 (1976).
R. Rado, “A theorem on independence relations,”Q. J. Math. 13, 83–89 (1962).
H. H. Rosenbrock,State Space and Multivariable Theory, Wiley, New York (1970).
J. Rosenthal, “Tuning natural frequencies of output feedback,” In:Computation and Control, Birkhäuser, Boston (1989), pp. 276–282.
J. Rosenthal, “New results in pole assignment by real output feedback,”SIAM J. Contr. Optimiz.,30, No. 1, 203–211 (1992).
J. Rosenthal, “A compactification of the space of multivariable systems using geometric invariant theory,”J. Math. Systems, Estimation, Contr.,1, 111–121 (1992).
Y. Rouchaleau and E. Sontag, “On the existence of minimal realizations of linear dynamical systems over Noetherian integral domains,”J. Comp. Syst. Sci.,18, 65–75 (1979).
Y. Rouchaleau and B. Wyman, “Linear dynamical systems, over integral domains,”J. Comp. Syst. Sci.,9, No. 2, 129–142 (1974).
Y. Rouchaleau and B. Wyman, and R. E. Kalman, “Algebraic structure of linear dynamical systems III. Realization theory over commutative rings,”Proc. Natl. Acad. Sci. USA.,69, 3404–3406 (1972).
H. Seraji, “Design of pole placement compensators for multivariable systems,”Automatica,16, 335–338 (1980).
E. Sontag, “Linear systems over commutative rings: A survey,”Ric. Autom.,7, 1–34 (1976).
T. A. Springer, “The unipotent variety of a semisimple group,” In:Proc. Bombay Coll. Algebr. Geometry (1969), pp. 379–391.
R. Steinberg, “Desingularization on the unipotent variety,”Invent. Math., 209–224 (1976).
A. Tannenbaum, “Invariance and system theory: Algebraic and geometric aspects,”Lect. Notes Math.,845, Springer-Verlag (1981).
M. Vidyasagar, H. Schneider, and B. A. Francis, “Algebraic and topological aspects of feedback stabilization,”IEEE Trans. Autom. Contr.,27, No. 4, 880–894 (1982).
X.-C. Wang, “Géometric inverse eigenvalue problems,” In:Computation and Control, Birkhäuser, Boston (1983).
H. Whitney,Complex Analytic Varieties, Addison-Wesley, Reading, Massachusetts (1972).
J. C. Willems, “From time series to linear systems,”Automatica,22 (1986); No. 5, 561–580 (1986); No. 6, 675–694 (1986);23, No. 1, 87–115 (1987).
J. C. Willems and W. H. Hesselink, “Generic properties of the pole-placement, problem,” In:Proc. 7th IFAC Congr. (1978), pp. 1725–1729.
W. M. Wonham,Linear Multivariable Control: A Geometric Approach, Springer-Verlag, New York (1979).
W. M. Wonham and A. S. Morse, “Decoupling and pole assignment in linear multivariable systems: a geometric approach,”SIAM J. Contr.,8, No. 1, 1–18 (1970).
W. M. Wonham and A. S. Morse, “Feedback invariants of linear multivariable systems,”Automatica,8, No. 1, 93–100 (1972).
B. F. Wymen, “Pole placement over integral domains,”Commun. Algebra,6, 969–993 (1976).
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 29, Optimizatsiya i Upravlenie-1, 1996.
Rights and permissions
About this article
Cite this article
Osetinskii, N.I. Methods of invariant analysis for linear control systems. J Math Sci 88, 587–657 (1998). https://doi.org/10.1007/BF02364664
Issue Date:
DOI: https://doi.org/10.1007/BF02364664