Abstract
Our attention turns now to the study of the fundamental question: How does one use input variables in the actual control of a system? Naturally, the use of control functions depends on the desired objectives of performance. Moreover, there are various ways in which the control can be applied. One way is to determine, a priori, a control sequence in the discrete-time case, or a control function in the continuous-time case, and apply it. Thus the control is applied without regard to its lasting effects or to the actual system performance, except insofar as the design goals have been taken into account.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
H. Aling and J.M. Schumacher, “A nine-fold canonical decomposition for linear systems”, Int. J. Control, vol. 39(4), (1984), 779–805.
G. Basile and G. Marro, Controlled and Conditioned Invariants in Linear System Theory, Prentice Hall, Englewood Cliffs, NJ, 1992.
V. D. Blondel, Simultaneous Stabilization of Linear Systems, Lecture Notes in Control and Inf. Sciences, Springer, Heidelberg, 1994.
R.W. Brockett, “The geometry of the set of controllable linear systems”, Research Reports of Autom. Control Lab., Faculty of Engineering, Nagoya University, vol. 24, (1977), 1–7.
R.W. Brockett and C.I. Byrnes, “Multivariable Nyquist criteria, root loci and pole placement: A geometric approach”, IEEE Trans. Autom. Control, vol. 26, (1981), 271–284.
P. Brunovsky, “A classification of linear controllable systems”, Kybernetika, vol. 6.3, (1970), 173–188.
C.I. Byrnes, “Pole assignment by output feedback”, Three Decades of Mathematical Systems Theory, H. Nijmeijer and J.M. Schumacher, Eds., Springer, Heidelberg, 31–87, 1989.
E. Emre and M.I.J. Hautus, “A polynomial characterization of (A, B)-invariant and reachability subspaces”, SIAM J. Control Optim., vol. 18, (1980), 420–436.
P.L. Falb and W.A. Wolovich, “Decoupling in the design and synthesis of multivariable control systems”, IEEE Trans. Autom. Control, vol. 12(6), (1967), 651–659.
B.A. Francis, “The linear multivariable regulator problem”, SIAM J. Control Optim., vol. 15(3), (1977), 486–505.
B.A. Francis and W.M. Wonham, “The internal model principle of control theory.”, Automatica, vol. 12.5, (1976), 457–465.
P.A. Fuhrmann, “Linear feedback via polynomial models”, Int. J. Control 30, (1979), 363–377.
P.A. Fuhrmann, “Duality in polynomial models with some applications to geometric control theory”, IEEE Trans. Autom. Control, 26, (1981), 284–295.
P.A. Fuhrmann and J. C. Willems, “Factorization indices at infinity for rational matrix functions”, Integral Equations and Operator Theory, vol. 2/3, (1979), 287–300.
P.A. Fuhrmann and J.C. Willems, “A study of (A,B)-invariant subspaces via polynomial models”, Int. J. Control, vol. 31, (1980), 467–494.
B.K. Ghosh and C.I. Byrnes, “Simultaneous stabilization and pole-placement by nonswitching dynamic compensation”, IEEE Trans. Autom. Control, vol. 28, (1983), 735–741.
I.C. Gohberg, L. Lerer and L. Rodman, “Factorization indices for matrix polynomials”, Bull. Am. Math. Soc., vol. 84, (1978), 275–277.
E. Gorla and J. Rosenthal, “Pole placement with fields of positive characteristic”, Three Decades of Progress in Control Sciences; X. Hu, U. Jonsson, B. Wahlberg and B. Ghosh, Eds., Springer, 215–231, 2010.
M.L.J. Hautus, “(A, B)-invariant and stabilizability subspaces, a frequency domain description”, Automatica, vol. 16, (1980), 703–707.
M.L.J. Hautus and M. Heymann, “Linear feedback—an algebraic approach”, SIAM J. Control Optim., vol. 16, (1978), 83–105.
M. Hazewinkel and C.F. Martin, “Representations of the symmetric group, the specialization order, systems and Grassmann manifolds”, L’Enseignement Mathematique, vol. 29, (1983), 53–87.
U. Helmke, “The topology of a moduli space for linear dynamical systems”, Comment. Math. Helv., vol. 60, (1985), 630–655.
M. Heymann, Structure and realization problems in the theory of dynamical systems, CISM Courses and Lectures, vol. 204, Springer, 1975.
P.P. Khargonekar and E. Emre, “Further results on polynomial characterizations of (F, G)-invariant and reachability subspaces”, IEEE Trans. Autom. Control, vol. 27 (10), (1982), 352–366.
V. Kucera, “A method to teach the parametrization of all stabilizing controllers”, Proc. 18th IFAC World Congress, Milano, (2011), 6355–6360.
S. Lang, Algebra, Addison-Wesley, Reading, MA, 1965.
A.S. Morse, “Structural invariants of multivariable linear systems”, SIAM J. Control Optim., vol. 11, (1973), 446–465.
A.S. Morse and W.M. Wonham, “Status of noninteracting control.” IEEE Trans. Autom. Control, vol. 16(6), (1971): 568–581.
H. F. Münzner and D. Prätzel-Wolters, “Minimal bases of polynomial modules, structural indices and Brunovsky transformations”, Int. J. Control, vol. 30, (1979), 291–318.
A.B. Özgüler, Linear Multichannel Control, Prentice Hall, 1994.
V.M. Popov, “Invariant description of linear time-invariant controllable systems”, SIAM J. Control Optim., vol. 10, (1972), 252–264.
H.H. Rosenbrock, State-Space and Multivariable Theory, Wiley, New York, 1970.
A. Tannenbaum, Invariance and System Theory: Algebraic and Geometric Aspects, Lecture Notes in Mathematics 845, Springer, Berlin, 1981.
M. Vidyasagar, Control Systems Synthesis – A Factorization Approach, MIT Press, Cambridge, MA, second printing, 1987.
S.H. Wang and E.J. Davison, “Canonical forms of linear multivariable systems”, SIAM J. Control Optim., vol. 14, (1976), 236–250.
X. Wang, “Pole placement by static output feedback”, J. Math. Systems, Estimation, and Control, vol. 2, (1992), 205–218.
J.C. Willems and C. Commault, “Disturbance decoupling by measurement feedback with stability or pole placement”, SIAM J. Matrix Anal. Appl., vol. 19, (1981), 490–504.
H. Wimmer, “Existenzsätze in der Theorie der Matrizen und Lineare Kontrolltheorie”, Monatsh. Math., vol. 78, (1974), 256–263.
W.M. Wonham, Linear Multivariable Control: A Geometric Approach, Springer, 1979.
B.F. Wyman, M.K. Sain, G. Conte and A.M. Perdon, “On the zeros and poles of a transfer function”, Linear Algebra Appl., vol. 122–124, (1989), 123–144.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Fuhrmann, P.A., Helmke, U. (2015). State Feedback and Output Injection. In: The Mathematics of Networks of Linear Systems. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-16646-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-16646-9_6
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16645-2
Online ISBN: 978-3-319-16646-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)