Deadbeat control: A special inverse eigenvalue problem

In this paper we give a numerical method to construct a rankm correctionBF (where then ×m matrixB is known and them ×n matrixF is to be found) to an ×n matrixA, in order to put all the eigenvalues ofA +BF at zero. This problem is known in the control literature as “deadbeat control”. Our method constructs, in a recursive manner, a unitary transformation yielding a coordinate system in which the matrixF is computed by merely solving a set of linear equations. Moreover, in this coordinate system one easily constructs the minimum norm solution to the problem. The coordinate system is related to the Krylov sequenceA−1B,A−2B,A−3B, .... Partial results of numerical stability are also obtained.

[1]  Daniel Boley,et al.  Computing the controllability - observability decomposition of a linear time-invariant dynamic system, a numerical approach , 1981 .

[2]  C. Mullis,et al.  Time optimal discrete regulator gains , 1972 .

[3]  A. Varga.  A Schur method for pole assignment , 1981 .

[4]  Mihail M. Konstantinov,et al.  Synthesis of linear systems with desired equivalent form , 1980 .

[5]  W. Wonham.  Linear Multivariable Control: A Geometric Approach , 1974 .

[6]  P. Dooren.  The generalized eigenstructure problem in linear system theory , 1981 .

[7]  C. Paige,et al.  An algorithm for pole assignment of time invariant multi-input linear systems , 1982, 1982 21st IEEE Conference on Decision and Control.

[8]  R. V. Patel,et al.  Computation of minimal-order state-space realizations and observability indices using orthogonal transformations , 1981 .

[9]  F Rikus Eising,et al.  Pole assignment, a new proof and algorithm , 1982 .

[10]  G. Franklin,et al.  Deadbeat control and tracking of discrete-time systems , 1982 .

[11]  Hajime Akashi,et al.  On time optimal control of linear discrete-time systems by geometric approach , 1978 .

[12]  A. Varga.  Numerically stable algorithm for standard controllability form determination , 1981 .

[13]  Control of Linear Systems Via the Serial Canonical Form , 1979 .

[14]  Frank L. Lewis.  A general Riccati equation solution to the deadbeat control problem , 1982 .

[15]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[16]  P. Dooren.  The Computation of Kronecker's Canonical Form of a Singular Pencil , 1979 .

[17]  Huibert Kwakernaak,et al.  Linear Optimal Control Systems , 1972 .

[18]  C. Paige,et al.  An algorithm for pole assignment of time invariant linear systems , 1982 .

[19]  L. Silverman,et al.  Stable extraction of Kronecker structure of pencils , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[20]  G. Stewart.  Introduction to matrix computations , 1973 .