Optimization and pole assignment in control system design
Chu, Eric
International Journal of Applied Mathematics and Computer Science, Tome 11 (2001), p. 1035-1053 / Harvested from The Polish Digital Mathematics Library

Some elementary optimization techniques, together with some not so well-known robustness measures and condition numbers, will be utilized in pole assignment. In particular, ''Method 0'' by Kautsky et al. (1985) for optimal selection of vectors is shown to be convergent to a local minimum, with respect to the condition number . This contrasts with the misconception by Kautsky et al. that the method diverges, or the recent discovery by Yang and Tits (1995) that the method converges to stationary points.

Publié le : 2001-01-01
EUDML-ID : urn:eudml:doc:207544
@article{bwmeta1.element.bwnjournal-article-amcv11i5p1035bwm,
     author = {Chu, Eric},
     title = {Optimization and pole assignment in control system design},
     journal = {International Journal of Applied Mathematics and Computer Science},
     volume = {11},
     year = {2001},
     pages = {1035-1053},
     zbl = {1031.93089},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv11i5p1035bwm}
}
Chu, Eric. Optimization and pole assignment in control system design. International Journal of Applied Mathematics and Computer Science, Tome 11 (2001) pp. 1035-1053. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv11i5p1035bwm/

[000] Andry A.N., Shapiro E.Y. and Chung J.C. (1983): Eigenstructure assignment for linear systems. — IEEE Trans. Aerosp. Electr. Syst., Vol.19, pp.711–729.

[001] Bertsekas D.P. (1995): Nonlinear Programming. — Belmont: Athena Scientific.

[002] Bhattacharyya S.P. and De Sousa E. (1982): Pole assignment via Sylvester equations. — Syst. Contr. Lett., Vol.1, pp.261–263. | Zbl 0473.93037

[003] Boyd S., El Ghaoui L., Feron E. and Balakrishnan V. (1994): Linear Matrix Inequalities in System and Control Theory. — Philadelphia: SIAM. | Zbl 0816.93004

[004] Brogan W.L. (1974): Applications of a determinant identity to pole-placement and observer problems. — IEEE Trans. Automat. Contr., Vol.19. | Zbl 0291.93021

[005] Byers R. and Nash S.G. (1989): Approaches to robust pole assignment. — Int. J. Contr., Vol.49, pp.97–117. | Zbl 0666.93042

[006] Cavin K.R. and Bhattacharyya S.P. (1982): Robust and well-conditioned eigenstructure assignment via Sylvester’s equation. — Proc. Amer. Contr. Conf.. | Zbl 0512.93035

[007] Chu E.K. (1986a): A pole-assignment algorithm for linear state feedback. — Syst. Contr. Lett., Vol.7, pp.289–299. | Zbl 0592.93024

[008] Chu E.K. (1986b): Generalizations of the Bauer-Fike theorems. — Numer. Math., Vol.49, pp.85–91.

[009] Chu E.K. (1987): Exclusion theorems and perturbation theory for the generalized eigenvalue problem. — SIAM J. Numer. Anal., Vol.24, pp.1114–1125. | Zbl 0636.15009

[010] Chu E.K. (1988): A controllability condensed form and a state feedback pole assignment algorithm for descriptor systems. — IEEE Trans. Automat. Contr., Vol.33, pp.366–370. | Zbl 0635.93033

[011] Chu E.K. (1993): Approximate pole assignment. — Int. J. Contr., Vol.59, pp.471–484. | Zbl 0781.93036

[012] Chu E.K. (2001a): Optimization and pole assignment in control system design. — Reprint & Preprint Series, Dept. Math. Stat., Monash University.

[013] Chu E.K. (2001b): Pole assignment for second-order systems. — Mech. Syst. Signal Process., (to appear).

[014] Chu E.K. and Datta B.N. (1996): Numerically robust pole assignment for second-order systems. — Int. J. Contr., Vol.64, pp.1113–1127. | Zbl 0850.93318

[015] Chu E.K. and Li N. (1993): Controllability measures and their computation. — CTAC-91, pp.291–298.

[016] Chu E.K. and Li N. (1994): Designing the Hopfield neural network via pole assignment. — Int. J. Syst. Sci., Vol.25, pp.669–681. | Zbl 0805.93020

[017] Cichocki A. and Unbehauen R. (1993): Neural Networks for Optimization and Signal Processing. — London: Wiley. | Zbl 0824.68101

[018] Datta B.N. and Saad Y. (1991): Arnoldi methods for large Sylvester-like observer matrix equations and an associate algorithm for partial pole assignment. — Lin. Alg. Applic., Vol.154–156, pp.225–244.

[019] Davidon W.C. (1975): Optimally conditioned optimization algorithms without line searches. — Math. Prog., Vol.9, pp.1–30. | Zbl 0328.90055

[020] Dennis J.E., Jr., and Schnabel R.B. (1983): Numerical Methods for Unconstrained Optimization and Nonlinear Equations. — Englewood Cliffs: Prentice-Hall.

[021] Dennis J.E. and Wolkowicz H. (1990): Sizing and least change secant methods. — Tech. Rep., Vol.COOR 90–02, Dept. Combinatorics and Optimization, University of Waterloo, Ontario, 1990.

[022] Fahmy M.M. and O’Reilly J. (1982): On eigenstructure assignment in linear multivariable system. — IEEE Trans. Automat. Contr., Vol.27.

[023] Fahmy M.M. and O’Reilly J. (1988): Multistage parametric eigenstructure assignment by output feedback. — Int. J. Contr., Vol.48, pp.97–116. | Zbl 0688.93023

[024] Fahmy M.M. and O’Reilly J. (1988): Parametric eigenstructure assignment by output feedback control. — Int. J. Contr., Vol.48, pp.1519–1535. | Zbl 0658.93043

[025] Fletcher R. (1987): Practical Methods of Optimization, 2-nd Ed. — Chichester: Wiley.

[026] Golub G.H. and Van Loan C.F. (1989): Matrix Computations, 2-nd Ed. — Baltimore: Johns Hopkins University Press.

[027] Gourishanker V. and Ramar K. (1976): Pole assignment with minimum eigenvalue sensitivity to plant variations. — Int. J. Contr., Vol.23, pp.493–504. | Zbl 0317.93031

[028] He C., Laub A. and Mehrmann V. (1995): Placing plenty of poles is pretty. Preposterous. — Vol.SPC 95–17, Forschergruppe “Scientific Parallel Computing”, Fakultät für Mathematik, TU Chemnitz-Zwickau, FRG.

[029] Ho D., Lam J., Xu J. and Tam H.K. (1996): Recurrent neural networks for output feedback robust approximate pole assignment. — Res. Rep., Vol.MA–96–08, Faculty of Science and Technology, City University of Hong Kong.

[030] Joshi S.M. (1989): Control of Large Flexible Space Structures. — Berlin: Springer. | Zbl 0762.93001

[031] Karmarkar N. (1984): A new polynomial-time algorithm for linear programming. — Combinatorica, Vol.4, pp.373–395. | Zbl 0557.90065

[032] Katti S.K. (1983): Pole assignment in multi-input systems via elementary transformations. — Int. J. Contr., Vol.37, pp.315–347. | Zbl 0502.93029

[033] Kautsky J., Nichols N.K. and Van Dooren P. (1985): Robust pole assignment in linear state feedback. — Int. J. Contr., Vol.41, pp.1129–1155. | Zbl 0567.93036

[034] Kimura H. (1975): Pole assignment by gain output feedback. — IEEE Trans. Automat. Contr., Vol.20, pp.509–516. | Zbl 0309.93017

[035] Kimura H. (1977): A further result on the problem of pole assignment by output feedback. — IEEE Trans. Automat. Contr., Vol.22, pp.458–463. | Zbl 0355.93008

[036] Klein G. and Moore B.C. (1977): Eigenvalue-generalized eigenvector assignment with state feedback. — IEEE Trans. Automat. Contr., Vol.22, pp.140–141. | Zbl 0346.93020

[037] Lam J. and Yan W.Y. (1995): A gradient flow approach to robust pole-assignment problem. — Int. J. Robust Nonlin. Contr., Vol.5, pp.175–185. | Zbl 0824.93027

[038] Marcus M. (1962): An inequality connecting the p-condition number and the determinant. — Numer. Math., Vol.4, pp.350–353. | Zbl 0113.32003

[039] Math Works (1995): MATLAB Version 4 Users’ Guide. — Englewood Cliffs: Prentice Hall.

[040] Mayne D. and Murdoch P. (1970): Model control of linear time invariant systems. — Int. J. Contr., Vol.11. | Zbl 0186.48204

[041] Miminis G.S. (1981): Numerical Algorithms for Controllability and Eigenvalue Allocation. — Ph.D. Thesis, School Comp. Sci., McGill University.

[042] Miminis G.S. and Paige C.C. (1982a): An algorithm for pole assignment of time invariant multi-input linear systems. — Proc. IEEE Conf. Decision Control, pp.62–67. | Zbl 0478.93022

[043] Miminis G.S. and Paige C.C. (1982b): An algorithm for pole assignment of time-invariant linear systems. — Int. J. Contr., Vol.35, pp.341–354. | Zbl 0478.93022

[044] Miminis G.S. and Paige C.C. (1988): A direct algorithm for pole assignment of time-invariant multi-input linear systems using state feedback. — Automatica, Vol.24. | Zbl 0652.93015

[045] Paige C.C. (1981): Properties of numerical algorithms related to computing controllability. — IEEE Trans. Automat. Contr., Vol.26, pp.130–138. | Zbl 0463.93024

[046] Petkov P.Hr., Christov N.D. and Konstantinov M.M. (1986): A computational algorithm for pole assignment of linear multi-input systems. — IEEE Trans. Automat. Contr., Vol.31, pp.1044–1047. | Zbl 0607.93020

[047] Rosenbrock H.H. (1970): State-Space and Multivariable Theory. — London: Nelson. | Zbl 0246.93010

[048] Saad Y. (1988): Projection and deflation methods for partial pole assignment in linear state feedback. — IEEE Trans. Automat. Contr., Vol.33, pp.290–297. | Zbl 0641.93031

[049] Stewart G.W. and Sun J.G. (1990): Matrix Perturbation Theory. — San Diego: Academic Press.

[050] Varga A. (1981a): A Schur method for pole assignment. — IEEE Trans. Automat. Contr., Vol.26, pp.517–519. | Zbl 0475.93040

[051] Varga A. (1981b): Numerical stable algorithm for standard controllability form determination. — Electr. Lett., Vol.17, pp.74–75.

[052] Wolkowicz H. (1990): Measures for symmetric rank-one updates. — Tech. Rep., Vol.CORR 90–03, Dept. Combinat. Optim., University of Waterloo, Ontario.

[053] Wonham W.M. (1979): Linear Multivariable Control: A Geometric Approach, 2-nd Ed. — Berlin: Springer. | Zbl 0424.93001

[054] Xiao Y., Crusca F. and Chu E.K. (1996): Bilinear matrix inequalities in robust control: phase I—Problem formulation. — Tech. Rep., Vol.TR–96–3, Dept. Electr. Comp. Syst. Eng., Monash University, Caulfield 3145, Australia.

[055] Yang Y. (1989): A new condition number of eigenvalue and its application in control theory. — J. Computat. Math., Vol.7, pp.15–21. | Zbl 0665.15006

[056] Yang Y. (1997): Robust System Design: Pole Assignment Approach, Ph.D. Thesis, Dept. Electr. Eng., University of Maryland at College Park, MD 20742, 1997.

[057] Yang Y. and Tits A.L. (1995): Globally convergent algorithms for robust pole assignment by state feedback. — Tech. Rep., Dept. Electr. Eng. and Inst. Syst. Res., University of Maryland at College Park, MD 20742.

[058] Yang Y. and Tits A.L. (1993): On robust pole assignment by state feedback. — Proc. Amer. Contr. Conf., San Francisco, pp.2765–2766.

[059] Zhao Q. (1996): Measures for Least Change Secant Methods. — M.Sc. Thesis, Dept. Combinat. Optim., University of Waterloo, Ontario.