Fixed-order Dynamic Compensation Research Articles

Approximate loop transfer recovery method for designing fixed-order compensators

An approach is outlined for designing fixed-order dynamic compensators for multivariable time-invariant linear systems, based on minimizing a linear quadratic performance index. The formulation is done in an output feedback setting that exploits an observer canonical form to represent the compensator dynamics. The formulation also precludes the use of direct feedback of the plant output. The main contribution lies in defining a method for penalizing the states of the plant and of the compensator, and for choosing the distribution on initial conditions so that the loop transfer matrix approximates that of a full-state feedback design. When linear quadratic regulator theory is used to do the full-state feedback design, the approach can result in good gain and phase margin characteristics. Two examples are given to illustrate the effectiveness of the approach. The first treats the problem of pointing a flexible structure, and the second is a helicopter flight control problem using a tenth-order model for the fuselage and rotor dynamics. Both of the examples considered in this paper are for nonsquare plants.

Robust Stability and Performance via Fixed-Order Dynamic Compensation

Two robust control-design problems are considered. The Robust Stabilization Problem involves deterministically modeled, bounded but unknown, time-varying parameter variations, while the Robust Performance Problem includes, in addition, a quadratic performance criterion averaged over stochastic disturbances and maximized over the admissible parameter variations. For both problems the design goal is a fixed-order (i.e., reduced- or full-order) dynamic (strictly proper) feedback compensator. A sufficient condition for solving the Robust Stabilization Problem is given by means of a quadratic Lyapunov function parameterized by the compensator gains. For the Robust Performance Problem the Lyapunov function provides an upper bound for the closed-loop performance. This leads to consideration of the Auxiliary Minimization Problem: Minimize the performance bound over the class of fixed-order controllers subject to the Lyapunov-function constraint. Necessary conditions for optimality in the auxiliary problem thus serve as sufficient conditions for robust stability and performance in the original problem. Two particular bounds are considered for constructing the quadratic Lyapunov function. The first corresponds to a right shift/ multiplicative white noise model, while the second was suggested by recent work of Petersen and Hollot. The main result is an extended version of the optimal projection equations for fixed-order dynamic compensation whose solutions are guaranteed to provide both robust stability and robust performance.

The optimal projection equations with Petersen-Hollot bounds: robust stability and performance via fixed-order dynamic compensation for systems with structured real-valued parameter uncertainty

A feedback control design problem involving structured real-valued plant parameter uncertainties is considered. A quadratic Lyapunov bound suggested by recent work of I.R. Petersen and C.V. Hollot (1986) is utilized in conjunction with the guaranteed cost approach of S.S.L. Chang and T.K.C. Peng (1972) to guarantee robust stability with robust performance bound. Necessary conditions that generalize the optimal projection equations for fixed-order dynamic compensation are used to characterize the controller that minimizes the performance bound. The design equations thus effectively serve as sufficient conditions for synthesizing dynamic output-feedback controllers that provide robust stability and performance.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Optimal nonzero set point regulation via fixed-order dynamic compensation

Standard LQG (linear quadratic Gaussian) theory is generalized to a regulation problem involving specified nonzero setpoints for the state and control variables and nonzero-mean disturbances. For generality, the results are obtained for the problem of fixed-order (i.e. not necessarily full order) dynamic compensation. When the state control, and disturbance offsets are set to zero and the compensator order is set equal to the plant dimension. The standard LQG result is recovered. These results provide the dynamic counterpart for the nonzero set point regulation results obtained by D.S. Bernstein and W.M. Haddad (1987) by means of static controllers. >

Fixed-order dynamic compensation for multivariable linear systems

This paper considers the design of fixed order dynamic compensators for multivariable time invariant linear systems, minimizing a linear quadratic performance cost functional. Attention is given to robustness issues in terms of multivariable frequency domain specifications. An output feedback formulation is adopted by suitably augmenting the system description to include the compensator states. Either a controller or observer canonical form is imposed on the compensator description to reduce the number of free parameters to its minimal number. The internal structure of the compensator is prespecified by assigning a set of ascending feedback invariant indices, thus forming a Brunovsky structure for the nominal compensator.

The optimal projection equations for static and dynamic output feedback: The singular case

Oblique projections have been shown to arise naturally in both static and dynamic optimal design problems. For static controllers an oblique projection was inherent in the early work of Levine and Athans, while for dynamic controllers an oblique projection was developed by Hyland and Bernstein. This note is motivated by the following natural question: What is the relationship between the oblique projection arising in optimal static output feedback and the oblique projection arising in optimal fixed-order dynamic compensation? We show that in nonstrictly proper optimal output feedback there are, indeed, three distinct oblique projections corresponding to singular measurement noise, singular control weighting, and reduced compensator order. Moreover, we unify the Levine-Athans and Hyland-Bernstein approaches by rederiving the optimal projection equations for combined static/dynamic (nonstrictly proper) output feedback in a form which clearly illustrates the role of the three projections in characterizing the optimal feedback gains. Even when the dynamic component of the nonstrictly proper controller is of full order, the controller is characterized by four matrix equations which generalize the standard LQG result.

Optimal projection equations for discrete-time fixed-order dynamic compensation of linear systems with multiplicative white noise

The optimal projection equations for discrete-time reduced-order dynamic compensation are generalized to include the effects of state-, control- and measurement-dependent noise. In addition, the discrete-time static output feedback problem with multiplicative disturbances is considered. For both problems, the design equations are presented in a concise, unified manner to facilitate their accessibility for developing numerical algorithms for practical applications.

The Optimal Projection Equations for Finite-Dimensional Fixed-Order Dynamic Compensation of Infinite-Dimensional Systems

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 ofa 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 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 finite-dimensional fixed-orderdynamic compensator is characterized byfour operator equations (two modified Riccati equations and two modified Lyapunov equations) coupled by an oblique whose rank is precisely equal to the order ofthe compensatorand which determines the 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 ofmodified 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.

The optimal projection equations for reduced-order state estimation

First-order necessary conditions for optimal, steady-state, reduced-order state estimation for a linear, time-invariant plant in the presence of correlated disturbance and nonsingular measurement noise are derived in a new and highly simplified form. In contrast to the lone matrix Riccati equation arising in the full-order (Kalman filter) case, the optimal steady-state reduced-order estimator is characterized by three matrix equations (one modified Riccati equation and two modified Lyapunov equations) coupled by a projection whose rank is precisely equal to the order of the estimator and which determines the optimal estimator gains. This coupling is a graphic reminder of the suboptimality of proposed approaches involving either model reduction followed by "full-order" estimator design or full-order estimator design followed by estimator-reduction techniques. The results given here complement recently obtained results which characterize the optimal reduced-order model by means of a pair of coupled modified Lyapunov equations [7] and the optimal fixed-order dynamic compensator by means of a coupled system of two modified Riceati equations and two modified Lyapunov equations [6].

The optimal projection equations for fixed-order dynamic compensation

First-order necessary conditions for quadratically optimal, steady-state,fixed-order dynamic compensation of a linear, time-invariant plant in the presence of disturbance and observation noise are derived in a new and highly simplified form. In contrast to the pair of matrix Riccati equations for the full-order LQG case, the optimal steady-state fixed-order dynamic compensator is characterized by four matrix equations (two modified Riccati equations and two modified Lyapunov equations) coupled by a projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains. The coupling represents a graphic portrayal of the demise of the classical separation principle for the reduced-order controller case.

On pole assignment by output feedback

A new result of the problem of pole assignment by output feedback is presented. A simple condition is given under which min (n, m + r − 1) closed-loop poles can be assigned arbitrarily by a gain output feedback, where n, mand r are the dimensions of the state, output and input vectors, respectively. Under the same condition, a new lower bound for the number of poles assignable by a fixed-order dynamic compensator is given which improves considerably on the results of Breach and Pearson (1970) and Ahmari and Vacroux (1973). A design algorithm and illustrative examples are also given.

Design of optimal constrained dynamic compensators for non-stationary linear stochastic systems

This paper deals with the design of fixed-order dynamic compensators for non-stationary linear stochastic systems with noisy observations, where the observation noise need not necessarily be white. An integral quadratic performance index defined over a finite time interval is employed and this yields a matrix variational problem in the compensator parameters. It is shown how the optimal, possibly time-varying, compensator parameters rnay be determined by direct solution of this variational problem using a conjugate-gradient technique. Consideration is also given to finding suboptimal compensators that are simpler to implement. In particular, an algorithm is proposed for designing compensators having gains that are constrained to be piecewise-constant functions of time, with provision for optimally choosing the instants at which gains changes occur. An illustrative numerical example is included.

Design of optimal constrained dynamic compensators for linear stationary stochastic servomechanisms

Techniques are developed for designing fixed-order time-invariant dynamic compensators for linear stochastic multivariable servomechanisms subject to plant disturbances and measurement noise. A quadratic performance criterion is employed and the resulting parameter optimization problem is treated using a state-variable formulation. Computable expressions are derived for the cost and its gradient with respect to the compensator parameters. The optimal parameters are then determined using a gradient-type minimization algorithm. An illustrative numerical example is presented.