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The Nyquist robust sensitivity margin is proposed as a new scalar indicator of robust stability that also provides a meaningful quantitative assessment of the worst sensitivity realized by the uncertain closed loop. After formulating and discussing in detail the underlying optimization problem required for the calculation of the margin, the approach is applied to the characterization of the robust stability of a closed-loop featuring a linear system with an affine uncertainty structure and a parametric uncertainty set described by a real rectangular polytope. The capabilities of the methodology are illustrated through examples, which include an approach for quantifying alternative robustness margins, such as a parametric stability margin. The computational algorithm is systematic and can be carried out with high numerical precision. Copyright © 2005 John Wiley & Sons, Ltd.

The critical direction theory for analyzing the robustness of uncertain feedback systems is extended to include nonlinear elements in the closed loop, making the approach applicable to a wider variety of systems. A redefinition of the critical perturbation radius is introduced, preserving key properties of the original theory. Necessary and sufficient conditions for robust stability are developed, and analytic conditions are derived to determine the Nyquist robust stability margin for systems with circular uncertainty. An example is presented to illustrate the theory.

The standard critical-direction theory quantifies in an effective fashion the robust stability of uncertain systems by restricting the analysis to a single direction on the Nyquist plane. The paper proposes a modification to take into account plants in the uncertainty set that define the worst sensitivity. The Nyquist robust sensitivity margin is proposed as a new scalar indicator of robust stability, and this concept is applied to the analysis of linear systems with a parametric real affine uncertainty structure. The algorithm involved in the computation of the Nyquist robust sensitivity margin is described along with examples that illustrate the advantages of the new concept.

A new l 1 / H ∞ technique for the design of robust predictive regulators that account for time-domain constraints is proposed. The controller robustly stabilizes a system where the open-loop plant model is corrupted by unstructured uncertainty, and the output signal is affected by bounded but possibly persistent exogenous disturbances. Adequate nominal performance in the regulation mode is achieved by optimizing an l 1 objective that minimizes the worst-case amplification of the disturbance peaks. Robust stability is obtained by satisfying classical frequency-domain conditions through an H ∞ problem. The resulting linear predictive regulator design quantifies the attamable limit of performance with respect to the satisfaction of the time-domain constraints.

The problem of asymptotic stabilization of uncertain bilinear systems containing saturating actuators is discussed. Using a state-feedback approach, sufficient conditions are derived to guarantee the stability of the saturating system in the presence of a norm-bounded, possibly nonlinear uncertainty. The time-domain robust stability conditions presented for continuous-time bilinear systems make use of the measure of a matrix and depend on bounds of the relevant matrices and the initial state. The region of attraction for the system is characterized.

A systematic and complete method to design robust predictive controllers for unconstrained linear systems is proposed. The synthesis procedure is based on rigorous theoretical foundations, without resorting to approximations or ad hoc design guidelines, yet it remains a viable tool for practical application. A significant feature is that the robust predictive controller retains the servo performance of a nominal predictive controller designed using conventional methods. In addition, the robust predictive controller can be designed to guarantee perfect steady-state rejection of asymptotically constant disturbances. The robust design method is developed for systems affected by unmodeled dynamics and is based on solving a discrete-time model-matching problem. It is shown that the robust controller can be classified legitimately as a predictive controller because it minimizes the same performance functional as the nominal predictive controller. An illustrative design example is given.

This paper considers the design of a sliding mode control (SMC) to stabilize a class of uncertain homogenous bilinear systems. A new approach based on the concept of integral sliding mode is proposed along with an auxiliary control law to overcome the key problems associated with using the classical SMC for such systems. The design is robust with respect to a family of conically bounded perturbations in the sense that it ensures reachability of the sliding manifold as well as asymptotic closed-loop stability in the presence of any perturbation belonging to the uncertainty family.

A systematic and complete design procedure for robust predictive controllers is proposed. The synthesis method is based on rigorous theoretical foundations, without resorting to approximations or ad hoc design guidelines, yet it remains a viable tool for practical utilization. A significant feature is that the robustified predictive controller retains the servo performance of a nominal predictive controller designed using conventional methods. In addition, the robust predictive controller can be designed to guarantee perfect steady-state rejection of asymptotically constant disturbances. The robust design method is developed for systems affected by unmodeled dynamics, and is based on solving a discrete-time model-matching problem. It is shown that the robustified controller can legitimately be classified as a predictive controller. An illustrative design example is given.

The major principal direction alignment (MPDA) principle, developed in the context of robust control theory, states that a necessary and sufficient condition for the spectral radius of a matrix to equal its maximum singular value is that the major input and the major output principal directions of the matrix be aligned. The MPDA principle emerged from a study of the derivatives of the maximum singular value. An ambiguity that occurs when the maximum singular value is repeated is considered in this paper, together with a modified statement of the major principal direction alignment principle. The new necessary and sufficient condition for the spectral radius of a matrix to equal its maximum singular value is that there exists at least one major input and output principal direction pair of the matrix that is aligned. A rigorous proof is provided for the new necessary and sufficient condition, which makes use of early results on dual norms and dual vectors. An example is presented to illustrate the results.

A formal robustness stability analysis of popular proportional-integral (PI) controller tuning rules for systems approximated by a first-order-plus-time-delay model is proposed. The uncertainty in the process model is represented by multiplicative parametric perturbations in the process gain, process time constant, and process time-delay. The zero-exclusion principle is used to characterize the robustness of the uncertain system in terms of the set of all perturbations that result in stable closed-loops. The robustness results recover the standard gain and phase margin concepts as special cases. In addition, a parametric stability margin is introduced for this class of problems as a generic metric via which alternative PI controller tuning rules may be compared in terms of robustness to simultaneous variations in the all three model parameters. The results of the paper can be applied to several disturbance-rejection and tracking PI tuning rules in widespread use, and permits comparing the tuning rules in terms of their relative robustness. It is shown for example that the integral-square-error tuning rule for disturbance rejection can be destabilized by a 7% simultaneous variation in the system parameters.

A sliding mode control scheme has been designed to ensure the asymptotic stability of a linear system with delay in both the input and state variables. A key step is the use of a transformation that eliminates the input delay. The resulting state-delay system is controlled through a sliding mode approach, where the equivalent control term uses state feedback and the discontinuous control term involves the sign and value of the switching function. It is shown that the sliding mode is reached in finite time. Stability conditions are derived using a Lyapunov technique combined with a Razumikhin approach to bound the difference between the original and the transformed states. The result is a set of sufficient conditions for asymptotic stability. The theoretical developments are illustrated via an example.

The robustness of predictive controllers is studied for systems with real parametric uncertainties that belong to an ellipsoidal or hyperbox domain. A parametric stability margin is proposed as a quantitative measure of robust stability. The parametric stability margin for ellipsoidal uncertainties is calculated through the solution of a linear-quadratic program, and for hyperbox uncertainties it is calculated via a linear program. A robust controller design methodology is proposed that involves determining the parametric stability margin over a range of values for the predictive controller tuning parameters. The controller robustification approach is presented via illustrative examples.

The critical direction theory for analyzing the robustness of uncertain feedback systems is extended to include nonlinear elements in the closed loop, making the approach applicable to a wider class of systems. A redefinition of the critical perturbation radius is introduced, preserving key properties of the original theory. Necessary and sufficient conditions for robust stability are developed, and the case of real affine parametric uncertainties is treated in detail. An example is presented to illustrate the theory.

This paper takes advantage of the critical direction theory to select a meaningful static weight for the H/sub /spl infin// design of robust controller for parametrically uncertain plants. The mechanisms leading to conservatism in an earlier approach utilizing a constant overbounding uncertainty bound are analyzed first; then, an alternative less conservative uncertainty bound is presented. Finally, an example is used to illustrate that the proposed approach can substantially increase the stabilizable uncertainty level.

By transforming the affine parametric uncertain system problem into a linear feasibility problem, a new and simple approach to the problem of finding robust stability margins for SISO systems with real affine parametric uncertainties is proposed. The approach recovers previous results using real-p analysis given in the literature. A comprehensive example shows that these approaches are equivalent in solving for the parametric stability margin for the affine uncertainty problem. The results in this paper demonstrate that the stability radius approach may be promising in solving more general real structured uncertainty problems with general value-sets.

The critical direction theory is proposed as a new tool for analyzing the robustness of uncertain systems. The analysis is done using Nyquist arguments involving eigenvalues instead of singular values, and yields exact necessary and sufficient conditions for robust stability. The concept of a critical line on the Nyquist plane is defined and used to calculate the Nyquist robust stability margin. The new approach gives alternatives to computing exact stability margins in some cases of highly directional uncertainty templates where other methods are not applicable. An overview of major robust stability results reported in the literature is given for the purpose of comparison with the perspective of the new paradigm.

A technique is proposed for synthesizing pole-placement controllers capable of assigning the closed-loop poles of an family of plants inside a specified design region. The design is applicable to plants characterized by uncertain parameters that are known to lie inside an ellipsoid. Ellipsoidal parametric uncertainty descriptions are particularly useful because they arise naturally in the context of parameter estimation, and hence have the advantage of being quantifiable using experimental input-output data. The control-design problem is posed as a minimax program that is shown to be convex, and that can be solved using numerical techniques with global convergence guarantees.

A systematic method of designing robust predictive controllers for systems with parametric ellipsoidal uncertainty is proposed. Ellipsoidal uncertainty descriptions arise in many engineering applications and are relevant to predictive control operations where model parameters are often found by fitting experimental data. A significant feature is that the robust predictive controller retains the servo performance of a nominal predictive controller designed using conventional methods. The synthesis procedure involves solving a quasi-convex optimization problem that has analytic expressions for the gradients. The optimization problem is based on rigorous theoretical foundations for robust stability, and convergence to the global solution is guaranteed. An illustrative design example is given.

A static state-feedback controller is designed for a system that includes an uncertain input delay and a conically-bounded state-dependent perturbation. The ultimate goal is to robustly stabilize the system with respect to all allowed delays and perturbations. In addition, the design seeks to satisfy a 2-norm constraint in the control input. The technique is based on satisfying sufficient conditions expressed as linear matrix inequalities (LMI). A Lyapunov-Razumikhin approach is employed in the stability analysis to produce one set of LMI, while the input constraint introduces another group of LMI. The method yields an analytical expression for a state-feedback gain matrix that stabilizes the closed-loop system for all uncertainties and satisfies the input constraints.

A simple approach to the problem of finding robust stability margins for SISO systems with complex parametric uncertainties is proposed. The technique is based on the critical direction theory, and is an alternative to existing methods such as those based on structured singular value formulations. The uncertainties considered belong to highly structured domains satisfying a radial convexity condition. Explicit and exact stability conditions are derived using intuitive geometric arguments. The approach recovers previous results given in the literature for the special case of disk-bounded affine complex uncertainties. Examples are given for rectangular and elliptical uncertainty domains.

A sliding mode control design is proposed for a class of uncertain input-delay systems based on transformation that converts the original system into a delay-free form. It is shown that asymptotic stability of the transformed system guarantees the asymptotic stability of the original system. The reaching condition and the asymptotic stability are assured through a proper choice of the switching function and the controller parameters. The advantages over state feedback are discussed along with an approach for reducing the problem of chattering.