All optimal controls for the singular linear-quadratic problem with stability : related algebraic and geometric results

The present Report is the follow-up of [11, in which the linear-quadratic control problem without stability (also called the free end-point problem) has been considered. Here, we compute all optimal controls for the problem where the state trajectory is required to converge to zero as time goes to infinity. Our approach is based upon the generalized dual structure algor ithm. an extended version of Silverman's dual structure algorithm. The paper will show that this method yields several interesting by-results as well as more insight in the unique and the non-unique parts of the optimal state trajectories.

[1]  A. Jameson,et al.  Singular Perturbations and Singular Arcs-Part I , 1975 .

[2]  J. Willems,et al.  Singular optimal control: A geometric approach , 1986 .

[3]  T. Geerts.  All optimal controls for the singular linear-quadratic problem without stability; a new interpretation of the optimal cost , 1989 .

[4]  L. Silverman,et al.  System structure and singular control , 1983 .

[5]  J. Willems.  Least squares stationary optimal control and the algebraic Riccati equation , 1971 .

[6]  J. Dion,et al.  Structure at infinity of linear multivariable systems a geometric approach , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[7]  M. Hautus.  Stabilization, controllability and observability of linear autonomous systems , 1970 .

[8]  B. Francis.  The optimal linear-quadratic time-invariant regulator with cheap control , 1979 .

[9]  Ton Geerts,et al.  Continuity properties of the cheap control problem without stability , 1989 .

[10]  Johannes Schumacher,et al.  The role of the dissipation matrix in singular optimal control , 1983 .

[11]  John B. Moore,et al.  Generalizations of singular optimal control theory , 1971 .

[12]  Johan Grasman.  Non-uniqueness in singular optimal control : (preprint) , 1979 .

[13]  L. Silverman.  Inversion of multivariable linear systems , 1969 .

[14]  H. Rosenbrock.  The zeros of a system , 1973 .

[15]  Ahw Ton Geerts.  A new method to determine the infinite zero structure of a linear time-invariant system , 1988 .

[16]  Mlj Malo Hautus,et al.  The output-stabilizable subspace and linear optimal control , 1990 .

[17]  J. Pearson.  Linear multivariable control, a geometric approach , 1977 .

[18]  A. Jameson,et al.  Cheap control of the time-invariant regulator , 1975 .

[19]  Ahw Ton Geerts.  The computation of optimal controls for the singular linear-quadratic problem that yield internal stability , 1989 .

[20]  N. Karcanias,et al.  Poles and zeros of linear multivariable systems : a survey of the algebraic, geometric and complex-variable theory , 1976 .

[21]  J. B. Pearson,et al.  Singular Perturbations and Singular Arcs Part II , 1977 .

[22]  J. Schumacher,et al.  A nine-fold canonical decomposition for linear systems , 1984 .