Parametric eigenvalue assignment by constant output feedback – a cascaded approach

Pole placement by parametric output feedback

This paper proposes a simple necessary condition for the stabilisation of an inherently unstable linear system with constant output feedback. This is a sign criterion obtained directly from the linear constraints on the attainable closed-loop characteristic polynomials when constant output feedback is employed. The condition is first established for single-input systems and then extended to multi-input systems, where the output-feedback matrices are restricted to have unity rank. Stabilisation with unconstrained output-feedback matrices is also discussed. Illustrative examples on stabilisation are given.

In this paper, the pole placement problem by constant output feedback for linear time invariant systems is investigated. In particular, is proven that, for the class of linear time invariant systems with m inputs, p outputs, and McMillan degree n, the condition (m + p) > n is necessary for the solution of the arbitrary pole placement problem by constant output feedback.

The paper presents the design of piecewise constant periodic output feedback control for a discrete-time singularly perturbed system resulting from the discretization of a continuous-time standard singularly perturbed system. By a suitable linear transformation of state variables, the given continuous-time singularly perturbed model is converted into a block triangular form in which the fast subsystem is decoupled. The discrete-time model corresponding to the transformed model also exhibits a two time scale property if sampling period is larger than the parameter E. Now an output injection matrix is found that stabilizes the slow subsystem. The periodic output feedback gain is then calculated only for the slow subsystem and the same for the fast subsystem is set equal to zero. Finally the periodic output feedback gain for the composite system is obtained using the periodic output feedback gains computed separately for the slow and fast subsystems. An approach has been suggested whereby the determination of periodic output feedback gain for the slow subsystem can be converted into an optimization problem. By minimization of the suggested performance index the closed loop system behavior is improved. The method has been applied to a large pressurized heavy water reactor (PHWR) for control of xenon-induced spatial oscillations. A particular grouping of state variables has been suggested for obtaining the model in standard singularly perturbed form. The periodic output feedback gain is then calculated. The efficacy of control has been demonstrated by simulation of transient behavior of the nonlinear model of the PHWR.

A comprehensive use of eigenstructure assignment design methodology is presented. The implementation of this technique, in the context of advanced flight controls, is shown to be broad and flexible enough to accommodate a host of practical situations such as: constant output feedback, constrained output feedback, or output feedback with dynamic compensation for both continuous and discrete systems. In addition, the design of robust closed loop control systems is reviewed. Numerical examples corresponding to each case are presented in the context of flight controls.

Gradient matrices are formulated for the linear multivariable system with output feedback. The incremental motion of the poles and zeros and the coefficients of the numerator and denominator transfer function polynomials is determined as a function of the elements of an output feedback gain matrix. Also considered is the incremental movement of zeros and the coefficients of the numerator transfer function polynomials with respect to the elements of an input gain matrix. In addition, the characteristic polynomial gradient results are used to study the bounds on the number of independently assignable poles for linear multivariable systems with constant output feedback. The investigation is extended to estimate the minimum order dynamic output feedback compensator necessary to establish full polo placement for the expanded order system. Examples are included to illustrate the methods involved and to make comparisons with previous results.

In this paper, it is shown that by exploiting the explicit para- metric state feedback solution, it is feasible to obtain the ultimate solution to minimum sensitivity problem. A numerical algorithm for construction of a robust state feedback in eigenvalue assignment problem for a control- lable linear system is presented. By using a generalized parametric vector companion form, the problem of eigenvalue assignment with minimum sensitivity is re-formulated as an unconstrained minimization problem. The derived explicit expressions of the solutions allow minimization of the sensitivity problem by using a powerful search technique.

Canonical forms are developed for a class of linear hyperbolic systems. They are then applied to solve the problem of eigenvalue assignment by distributed feedback and boundary control. The duality of this problem is demonstrated to one of eigenvalue assignment by boundary feedback of an adjoint system subject to distributed control. For both systems it is shown that by feedback, the set $\{ \rho _j \} $, $j \in \mathbb{Z}$, can be assigned as eigenvalues of the closed loop system, subject to an asymptotic condition on the set $\{ \rho _j \} $. The feedback control is explicitly characterized. Analogous results are obtained for the problem of eigenvalue assignment by distributed feedback and distributed control.

A General Design Approach for Optimal Output Feedback Control

This paper presents as its main result a new and efficient method for the design of optimal dynamic compensators for linear multivariable control systems. The discussion starts with a fresh look into the design of optimal constant gain output feedback. In this context, three new ideas are introduced that prove their usefulness particularly in the dynamic compensator case: (1) deterministic measurement disturbances are included in the formulation of the underlying optimization problem, (2) the relationship between a deterministic and a stochastic optimal control problem is shown, (3) by proper modifications of the quadratic cost function, the effort involved in the numerical solution of the optimal output feedback problem is considerably reduced. Furthermore, a canonical form is introduced for the dynamic compensator and an algorithm is developed for the numerical computation of its free parameters. It is based on quasi-newtonian gradient methods and an efficient line search procedure. Some features of the new design method are demonstrated by means of a non-trivial benchmark taken from the literature.

This paper solves a class of optimally scaled /spl Hscr//sub /spl infin// control problems with constant scalings and output feedback. This class of problems is characterized by a special rank-one assumption on the transfer matrix of the nominal plant. It is shown that, under this special assumption, the optimally scaled /spl Hscr//sub /spl infin// problem can be reduced to a convex optimization problem involving linear matrix inequalities, which can be solved numerically. We illustrate the results with an engineering problem; we use our method to find a controller that meets flying quality requirements for an F-16 aircraft despite uncertainty in the location of the center of gravity.

In this paper, the problem of control design for discrete two-time-scale systems using piecewise constant periodic output feedback is considered. The discrete two-time-scale system is decomposed into a slow and fast subsystem and then a composite controller is designed which provides an output injection gain. The output injection gain has been used to compute periodic output feedback. The method is illustrated by a numerical example.

Theorems are given to demonstrate that the tracking properties of reduced-order models obtained by Padé-type approximation methods are unchanged under constant (and dynamic) state or output feedback. Hence it follows that an optimal feedback law for the reduced-order model may be used for suboptimal control of the high-order system, with similar tracking of the output as for the open-loop case.

This paper describes an applications oriented approach to the generation of optimal output feedback gains for linear time-invariant systems which is dependent of the open loop stability.

A solution is presented for the problem of diagonalization (row-by-row decoupling). The problem is solved using a constant precompensator and a dynamic output feedback compensator of a p*m linear time-invariant system. The solvability condition is compact and concerns the dimension of a single subspace defined via the concepts of essential rows and static kernels associated with the transfer matrix. A characterization of the set of all solutions to the problem is also given. In solving this dynamic feedback problem, a complete solution to its state-feedback counterpart, namely, the restricted state-feedback problem of diagonalization, is also presented. >