A design procedure for multivariable regulators

A computational algorithm is presented for the design of a compensator for linear multivariable regulators. The compensated system is required to be structurally stable, in the sense of preserving output regulation and internal (loop) stability in the face of small parameter variations or uncertainties. The design technique is based on the internal model principle [1]. It is shown that the distance between the system transmission zeros and eigenvalues of the exogenous dynamics may be taken as a measure of the well-conditioning of the algorithm. The design algorithm is demonstrated by application to a two-area power system.     

On alternative methodologies for the design of robust linear multivariable regulators

This paper presents two synthesis algorithms which embody the two major variants of the numerous methodologies which have been proposed for the design of multivariable linear regulators which exhibit the property of disturbance rejection with or without additional robustness qualities. It is shown that these two procedures generally lead to substantively different compensator structures.     

A staircase model for unknown multivariable systems and design of regulators

Based on a very simple model, a discrete-time integrating regulator can be designed for a linear time-invariant system. The linear system is not restricted to be scalar with a monotone step response, as required in previously published literature. The number of tuning parameters in the controller depends on the number of steps in the staircase model which can be properly chosen so that the closed-loop system is an asymptotically stable system with a specified tracking speed.     

Further results on the observer design problem

In this paper the observer design problem for nonlinear systems is considered. Sufficient Lyapunov-like conditions are presented for the existence of a nonlinear observer. The theory we develop considerably improves and extends the results of our recent work [14].     

Regional pole placement of multivariable systems under controlstructure constraints

Many controller realizations are structurally constrained. Some typical examples are static output feedback, constant gain feedback for multiple operating points of a system, two-controller feedback, and decentralized feedback. A general class of problems of regional pole placement of multivariable systems with such control structure constraints is considered and a unified numerical method is given to solve them. First a problem in this class is converted to a problem of solving a system of equalities and inequalities. This system is then solved by using a modified homotopy method     

On the robustness of LQ regulators

It is shown that in spite of its large gain and phase margins the linear quadratic state feedback regulator may suffer from poor robustness where small changes in the parameters of the system may lead to fast unstable closed-loop modes.     

Synthesis of linear multivariable regulators

This paper generalizes the results obtained by the authors in [1] and presents new results on the construction of proper controllers for linear multivariable regulators. These results may lead to efficient computation methods that do not involve Smith forms or the order relations used in [1] to construct proper controllers. The methods are based on the recently developed theory of Λ-generalized polynomials [2]. We discuss several numerical algorithms and how they can be applied to the regulator problem. We also point out an important computational problem in need of an efficient numerical algorithm.     

Further results on robustness of (possibly discontinuous) sample and hold feedback

We demonstrate that sample and hold state feedback control (possibly discontinuous with respect to the state) is robust when the closed loop system possesses an appropriate Lyapunov function. We first show that if a Lyapunov decrease over sampling periods exists for the nominal system, this decrease can be maintained with some degradation relative to a sufficiently small additive perturbation. We then proceed to catalog several applications of this robustness, e.g., robustness to measurement noise, computational delays, or fast actuator dynamics.     

On the robustness of multivariable linear feedback systems

One of the useful indicators of the robustness of a multivariable linear feedback system is the largest singular value of the nominal closed-loop transfer matrix. It is shown that while comparing different, not necessarily diagonal, closed-loop transfer matrices which have the same diagonal elements, the diagonal closed-loop transfer matrix has the greatest robustness. For plants with not "too" large parameter uncertainty, this result also guarantees the maximization of disturbance rejection, and the minimization of the control signal "power" at the plant's input.     

On improving the robustness of LQ regulators

Soroka and Shaked have shown that a linear quadratic state feedback regulator may suffer from poor robustness where small changes in the parameters of the system can lead to fast unstable closed-loop modes. It is shown there that the stability properties can be improved if possible nonminimum phase behavior is allowed for in modeling the system prior to the controller design.     

Stability robustness of linear quadratic regulators

In quadratic optimal control theory, the multivariable linear quadratic regulator is guaranteed to have excellent stability margins if the weight on the control inputs is diagonal. However, for the non‐diagonal case, it may suffer from poor robustness. In this paper, these robustness properties are studied in relation to weight selection. For general weighting matrices, a new lower bound on the minimum singular value of the return difference is presented. New guaranteed stability margins are also presented. This gives a formal mathematical basis for guidelines for weight selection. Copyright © 2015 John Wiley & Sons, Ltd.     

Frequency domain synthesis of multivariable linear regulators

This paper considers a general class of multivariable linear regulators and determines conditions under which there exist proper controllers such that output regulation is achieved with internal stability.     

Maximizing the configuration robustness for parallel multi-purpose machines under setup cost constraints

This paper focuses on the configuration of a parallel multi-purpose machines workshop. An admissible configuration must be chosen in order to ensure that a load-balanced production plan meeting the demand exists. Moreover, the demand is strongly subject to uncertainties. That is the reason why the configuration must exhibit robustness properties: the load-balancing performance must be guaranteed with regard to a given range of uncertainties. A branch-and-bound approach has been developed and implemented to determine a cost-constrained configuration that maximizes a robustness level. Computational results are reported for both academic and industrial-scale instances. More than 80% of the academic instances are solved to optimality by the proposed method. Moreover, this method appears to be a good heuristic for industrial-scale instances.     

Design of linear multivariable continuous-time output-feedback regulators

In this paper, the method of entire eigenstructure assignment (Kimura 1975, Moore 1976, Porter and D'Azzo 1917) is applied to the design of linear multivariable continuous-time output-feedback regulators. It is shown that, in the case of self-conjugate distinct eigenvalue spectra, the closed-loop eigenstructure assignable by output feedback is constrained by the requirement that the eigenvectors and reciprocal eigenvectors lie in well-defined subspaces. The method is illustrated by designing an output-feedback regulator for a third-order continuous-time system.     

Robust hyperplane design in multivariable variable structure control systems

The problem of designing a stable sliding mode giving robust performance of a variable structure control system is studied. A set of specified eigenvalues is assigned to the closed-loop feedback system during the sliding mode. The design philosophy seeks to build on the known robustness and invariance properties associated with the use of discontinuous (or continuous) non-linear controls, by assigning the eigenvectors associated with the sliding-mode eigenvalues in a manner that leads to a robust control scheme. A previously developed canonical form for the hyperplane design problem is again employed. The design technique centres on a recently published eigenvalue assignment algorithm, and is illustrated by the inclusion of an example.     

On the robustness of LQ regulators for discrete-time systems

A sufficient condition for quadratic stabilizability of uncertain linear discrete-time systems using LQ regulators is presented. The true system is represented by a nominal model plus additive terms representing the most common types of uncertainties in the state and input matrices. Sufficient robustness bounds as well as a robustification procedure for discrete-time LQ regulators are presented and the conservativeness of the proposed condition is discussed     

On the robustness of multivariable linear feedback systems in state-space representation

A singular values-based approach to specify the robustness of a multivariable linear feedback system in state-space representation is investigated. The robustness measure which is considered is the largest spectral norm of an additive uncertainty in the closed-loop system matrix, for which stability is guaranteed. It is shown that under the constraint of prescribed pole placement, a lower bound for the robustness measure is maximized, when the Frobenius norm of the closed-loop system matrix is minimized.     

Linear control design under phase constraints

A new design method is proposed for linear stabilizing controllers which satisfy the given constraints on the phase variables and control input and minimize the upper bound on the maximum deviation of the plant output. The method is based on the apparatus of linear matrix inequalities; the theory is exemplified via the linear shock absorber design.     

The asymptotic behavior of the root-loci of multivariable optimal regulators

The loci of the closed-loop poles of the multivariable, time-invariant, linear optimal regulator are shown to group into the left half-plane part of several Butterworth configurations as the weight on the input in the criterion approaches zero. It is proved that these configurations are of even order and that they are always centered at the origin. The number of configurations of any even order, their radii, and the angle of their corresponding asymptotes are expressed in terms of the criterion and the system constant matrices.