Abstract

A representation formula for all controllers that satisfy an $\mathcal{L}^\infty $-type constraint is derived for time-varying systems. It is now known that a formula based on two indefinite algebraic Riccati equations may be found for time-invariant systems over an infinite time support (see [J. C. Doyle et al., IEEE Trans. Automat. Control, AC-34 (1989), pp. 831–847]; [K. Glover and J. C. Doyle, Systems Control Lett., 11 (1988), pp. 167–172]; [K. Glover et al., SIAM J. Control Optim., 29 (1991), pp. 283–324]; [M. Green et al., SIAM J. Control Optim., 28 (1990), pp. 1350–1371]; [D. J. N. Limebeer et al., in Proc. IEEE conf. on Decision and Control, Austin, TX, 1988]; [G. Tadmor, Math. Control Systems Signal Processing, 3 (1990), pp. 301–324]). In the time-varying case, two indefinite Riccati differential equations are required. A solution to the design problem exists if these equations have a solution on the optimization interval. The derivation of the representation formula illustrated in this paper makes explicit use of linear quadratic differential game theory and extends the work in [J. C. Doyle et al., IEEE Trans. Automat. Control, AC-34 (1989), pp. 831–847] and [G. Tadmor, Math. Control Systems Signal Processing, 3 (1990), pp. 301–324]. The game theoretic approach is particularly simple, in that the background mathematics required for the sufficient conditions is little more than standard arguments based on “completing the square.”

Keywords

  1. ℋ∞-optimal control
  2. game theory
  3. indefinite Riccati equations
  4. four-block general distance problems
  5. worst-case design

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References

1.
Brian D. O. Anderson, John B. Moore, Linear optimal control, Prentice-Hall Inc., Englewood Cliffs, N.J., 1971xiv+399
2.
Joseph A. Ball, Nir Cohen, Sensitivity minimization in an $H\sp \infty$ norm: parametrization of all suboptimal solutions, Internat. J. Control, 46 (1987), 785–816
3.
M. Banker, Linear stationary quadratic games, Proc. CDC, 1973, 193–197
4.
Tamer Basar, G. J. Olsder, Dynamic noncooperative game theory, Mathematics in Science and Engineering, Vol. 160, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982xii+430
5.
Tamer Basar, A dynamic games approach to controller design: disturbance rejection in discrete time, Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 1–3 (Tampa, FL, 1989), IEEE, New York, 1989, 407–414
6.
Leonard D. Berkovitz, A variational approach to differential gamesAdvances in game theory, Princeton Univ. Press, Princeton, N.J., 1964, 127–174
7.
Dennis S. Bernstein, Wassim M. Haddad, LQG control with an $H\sb \infty$ performance bound: a Riccati equation approach, IEEE Trans. Automat. Control, 34 (1989), 293–305
8.
R. Brockett, Finite Dimensional Linear Systems, John Wiley, New York, 1970
9.
John C. Doyle, Keith Glover, P. Khargonekar, B. Francis, State-space solutions to standard ${\scr H}\sb 2$ and ${\scr H}\sb \infty$ control problems, IEEE Trans. Automat. Control, 34 (1989), 831–847
10.
Keith Glover, John C. Doyle, State-space formulae for all stabilizing controllers that satisfy an $H\sb \infty$-norm bound and relations to risk sensitivity, Systems Control Lett., 11 (1988), 167–172
11.
K. Glover, D. J. N. Limebeer, J. Doyle, E. Kasenally, M. G. Safonov, A characterization of all solutions to the four block general distance problem, SIAM J. Control Optim., 29 (1991), 283–324
12.
G. H. Golub, C. F. Van Loan, Matrix computations, North Oxford Academic, Oxford, UK, 1983
13.
Michael Green, Keith Glover, D. J. N. Limebeer, J. C. Doyle, A J-spectral factorization approach to $\scr H\sb \infty$ control, SIAM J. Control Optim., 28 (1990), 1350–1371
14.
P. R. Halmos, Introduction to Hilbert Space and the theory of Spectral Multiplicity, Chelsea Publishing Company, New York, N. Y., 1951, 114–
15.
Y. C. Ho, A. E. Bryson, S. Baron, Differential games and optimal pursuit-evasion strategies, IEEE Trans. Automatic Control, AC-10 (1965), 385–389
16.
Y. C. Ho, Differential games, dynamic optimization, and generalized control theory, J. Optimization Theory Appl., 6 (1970), 179–209
17.
David H. Jacobson, Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games, IEEE Trans. Automatic Control, AC-18 (1973), 124–131
18.
Pramod P. Khargonekar, Ian R. Petersen, Mario A. Rotea, $H\sb \infty$-optimal control with state-feedback, IEEE Trans. Automat. Control, 33 (1988), 786–788
19.
Pramod P. Khargonekar, Ian R. Petersen, Kemin Zhou, Robust stabilization of uncertain linear systems: quadratic stabilizability and $H\sp {\infty}$ control theory, IEEE Trans. Automat. Control, 35 (1990), 356–361
20.
H. Kimura, R. Kawatani, Synthesis of reduced-order $H\sp \infty$ controllers based on conjugation, Internat. J. Control, 50 (1989), 525–541
21.
D. J. N. Limebeer, E. M. Kasenally, E. Jaimouka, M. G. Safonov, A characterization of all solutions to the four block general distance problem, Proc. IEEE CDC, Vol. 1, Austin, TX, 1988, 878–880
22.
I. R. Petersen, D. J. Clements, J-spectral factorization and Riccati equations in problems of H∞ optimization via state feedback, 1988, preprint
23.
Ian R. Petersen, Disturbance attenuation and $H\sp \infty$ optimization: a design method based on the algebraic Riccati equation, IEEE Trans. Automat. Control, 32 (1987), 427–429
24.
M. G. Safonov, D. J. N. Limebeer, Simplifying the H∞ theory via loop shifting, Proc. IEEE CDC, Vol. 2, Austin, TX, 1988, 1399–1404
25.
W. E. Schmitendorf, S. J. Citron, A conjugate-point condition for a class of differential games, J. Optimization Theory Appl., 4 (1969), 109–121
26.
William E. Schmitendorf, Differential games with open-loop saddle point conditions, IEEE Trans. Automatic Control, AC-15 (1970), 320–325
27.
Gilead Tadmor, Worst-case design in the time domain: the maximum principle and the standard $H\sb \infty$ problem, Math. Control Signals Systems, 3 (1990), 301–324
28.
S. Wetland, Linear quadratic games, ℋ∞ the Riccati equation, Workshop on the Riccati Equation in Control, Systems and Signals, Como, Italy, 1989
29.
Leonard Weiss, P. L. Falb, Doležal's theorem, linear algebra with continuously parametrized elements, and time-varying systems, Math. Systems Theory, 3 (1969), 67–75
30.
P. Whittle, Risk-sensitive linear/quadratic/Gaussian control, Adv. in Appl. Probab., 13 (1981), 764–777

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Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 262 - 283
ISSN (online): 1095-7138

History

Submitted: 19 April 1989
Accepted: 13 December 1990
Published online: 14 July 2006

Keywords

  1. ℋ∞-optimal control
  2. game theory
  3. indefinite Riccati equations
  4. four-block general distance problems
  5. worst-case design

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