The optimal projection equations for finite-dimensional fixed-order dynamic compensation of infinite-dimensional systems

• Published in: SIAM journal on control and optimization 1986, Vol.24 (1), p.122-151
• Main Authors: BERNSTEIN, D. S, HYLAND, D. C
• Format: Article
• Language: English
• Subjects: Applied sciences; Compensation; Computer science; control theory; systems; Control system synthesis; Control theory. Systems; Controllers; Design specifications; Exact sciences and technology; Hilbert space
• ISSN: 0363-0129; 1095-7138
• DOI: 10.1137/0324006

One of the major difficulties in designing implementable finite-dimensional controllers for distributed parameter systems is that such systems are inherently infinite dimensional while controller dimension is severely constrained by on-line computing capability. While some approaches to this problem initially seek a correspondingly infinite-dimensional control law whose finite-dimensional approximation may be of impractically high order, the usual engineering approach involves first approximating the distributed parameter system with a high-order discretized model followed by design of a relatively low-order dynamic controller. Among the numerous approaches suggested for the latter step are model/ controller reduction techniques used in conjunction with the standard LQG result. An alternative approach, developed in [36], relies upon the discovery in [31] that the necessary conditions for optimal fixed-order dynamic compensation can be transformed into a set of equations possessing remarkable structural coherence. The present paper generalizes this result to apply directly to the distributed parameter system itself. In contrast to the pair of operator Riccati equations for the "full-order" LQG case, the optimal finite-dimensional fixed-order dynamic compensator is characterized by four operator equations (two modified Riccati equations and two modified Lyapunov equations) coupled by an oblique projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains. This "optimal projection" is obtained by a full-rank factorization of the product of the finite-rank nonnegative-definite Hilbert-space operators which satisfy the pair of modified Lyapunov equations. The coupling represents a graphic portrayal of the demise of the classical separation principle for the finite-dimensional reduced-order controller case. The results obtained apply to a semigroup formulation in Hilbert space and thus are applicable to control problems involving a broad range of specific partial and functional differential equations.