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The construction of optimal linear and nonlinear regulators. (English) Zbl 0778.49024

Systems, models and feedback: theory and applications, Proc. US-Italy Workshop in Honor of Prof. Antonio Ruberti, Capri/Italy 1992, Prog. Syst. Control Theory 12, 301-322 (1992).
Summary: [For the entire collection see Zbl 0745.00051.]
The regulator servomechanism problem is to design a feedforward and feedback control law to make the output of a given system called the plant, track a signal from a given class. The class of signals to be tracked are described as the output of a second system called the signal generator (or exosystem). The control law consists of feedforward terms involving the state of signal generator, feedback terms involving the state of the plant and mixed terms involving both.
B. A. Francis [SIAM J. Control Optimization 15, 486-505 (1977)] posed and solved the linear regulator problem and A. Isidori and C. I. Byrnes [IEEE Trans. Autom. Control 35, No. 2, 131-140 (1990; Zbl 0704.93034)] generalized this to the nonlinear case. The former showed that the linear regulator problem is solvable only if a certain system of linear equations is solvable. The latter showed that the nonlinear regulator problem is solvable only if a certain system of first order partial differential equations is solvable. J. Huang and W. J. Rugh [ibid. 37, No. 7, 1009-1013 (1992; Zbl 0767.93042); ibid. 37, No. 9, 1395-1398 (1992; Zbl 0767.93034)] did a formal (term by term) analysis of these PDEs and gave sufficient conditions for its solvability. The degree one terms of the Isidori and Byrnes PDE yield the linear equations of Francis, so we refer to the system as the Francis- Byrnes-Isidori (FBI) equations.
We show necessary and sufficient conditions for the term by term solvability of the FBI equations when either the signal generator has a semisimple pole structure or the plant has a semisimple zero structure. We present a proof of Huang-Rugh sufficient conditions.
We also give optimal methods for constructing the nonlinear regulator based on term by term analysis of the Hamilton-Jacobi Bellman HJB equations in the spirit of Al’brecht. This optimal approach to constructing a regulator may be novel even in the linear case.

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
93C15 Control/observation systems governed by ordinary differential equations
93D15 Stabilization of systems by feedback
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