Robust stability with time-varying structured uncertainty

Considers the problem of assessing robust stability in the presence of block-diagonally structured time-varying dynamic uncertainty. It is shown that robust stability holds only if there exist constant scalings which lead to a small gain condition. The notion of stability here is finite-gain stability over finite-energy signals. In sharp contrast to the case of time-invariant dynamic uncertainty, this result is not limited by the number uncertainty blocks. These results parallel previous results regarding finite-gain stability over persistent bounded signals     

Robust stability with structured real parameter perturbations

This paper considers the problem of robust stabilization of a linear time-invariant system subject to variations of a real parameter vector. For a given controller the radius of the largest stability hypersphere in this parameter space is calculated. This radius is a measure of the stability margin of the closed-loop system. The results developed are applicable to all systems where the closed-loop characteristic polynomial coefficients are linear functions of the parameters of interest. In particular, this always occurs for single-input (multioutput) or single-output (multiinput) systems where the transfer function coefficients are linear or affine functions of the parameters. Many problems with transfer function coefficients which are nonlinear functions of physical parameters can be cast into this mathematical framework by suitable weighting and redefinition of functions of physical parameters as new parameters. The largest stability hyperellipsoid for the case of weighted perturbations and a stability polytope in parameter space are also determined. Based on these calculations a design procedure is proposed to robustify a given stabilizing controller. This algorithm iteratively enlarges the stability hypersphere or hyperellipsoid in parameter space and can be used to design a controller Io stabilize a plant subject to given ranges of parameter excursions. These results are illustrated by an example.     

Robust stability under structured and unstructured perturbations

The problem of robust stability for linear time-invariant single-output control systems subject to both structured (parametric) and unstructured (H_∞) perturbations is studied. A generalization of the small gain theorem which yields necessary and sufficient conditions for robust stability of a linear time-invariant dynamic system under perturbations of mixed type is presented. The solution involves calculating the H_∞ -norm of a finite number of extremal plants. The problem of calculating the exact structured and unstructured stability margins is then constructively solved. A feedback control system containing a linear time-invariant plant which is subject to both structured and unstructured perturbations is considered. The case where the system to be controlled is interval is treated, and a nonconservative, easily verifiable necessary and sufficient condition for robust stability is given. The solution is based on the extremal of a finite number of line segments in the plant parameter property of a finite number of line segments in the plant parameter space along which the points closest to instability are encountered     

Robust stability for linear discrete-time systems with structured perturbations

The robust stability problems of linear discrete-time systems with structured perturbations are considered. The necessary and sufficient condition of interval stability is given and the maximal bound of robust stability is presented for systems with a non-negative or non-positive nominal system matrix. Further, it is shown that the given necessary and sufficient condition and the given maximal robust bound are still valid for time-varying perturbations and nonlinear perturbations.     

Robust pole assignment in systems subject to structured perturbations

The problem of robust pole assignment by feedback in a linear, multivariable, time-invariant system which is subject to structured perturbations is investigated. A measure of robustness, or sensitivity, of the poles to a given class of perturbations is derived, and a reliable and efficient computational algorithm is presented for constructing a feedback which assigns the prescribed poles and optimizes the robustness measure.     

Robust stabilization against non-linear time-varying uncertainties

A simple robust stability criterion for a linear multivariable system is developed to guarantee the stability of the closed-loop system subject to non-linear time-varying uncertainties. The generally parameterized stabilizing controller advanced by Youla et al. is combined with the Schur function (class S) to synthesize the robust stabilizer of feedback systems subject to these uncertainties. Both additively and multiplicatively non-linear uncertainties are considered in this paper. Two examples are given to demonstrate the validity of our results.     

Robust performance analysis for linear continuous/discrete time systems with linear dependent and time-varying perturbations

In this paper, both continuous and discrete-time perturbed systems subject to linear dependent and time-varying perturbations are considered. The time domain performance which related to the convergence rate of the initial states is defined based on the continuous/discrete-time Lyapunov functions. The matrix measure is used to derive the robust performance of a perturbed system.     

Robust detection filter design in the presence of time-varying system perturbations

This paper discusses the problem of designing detection filters for a class of linear time-varying systems. The basic contribution of the paper is that by, applying a time-invariant equivalent representation of the original time-varying system, it is possible to construct a detection filter such that the solution of the design problem can be solved using algebraic methods and geometric concepts similar to the case of time invariant situations.     

Robust control performance of time-varying human controller models

Previous models for human operator control performance have exhibited stochastic characteristics. Even with a repetitive task and a highly motivated and trained subject, there is clear variability of output (control actions). This variability, termed remnant, has been previously modeled as observation and motor noise associated with the operator. Recently, a new approach was taken: remnant was modeled as a time-varying operator characteristic associated with changes in the relative weighting of various performance measures, together with lack of understanding of the dynamics of the plant being controlled. The present work extends this concept and shows that the performance of time-varying human controller models is very robust with respect to changes in the time variability of the operator model. The results are particularly heartening in view of the inherent difficulties associated with modeling human control performance.     

Robust distributed model predictive control of linear systems with structured time-varying uncertainties

In this work, synthesis of robust distributed model predictive control (MPC) is presented for a class of linear systems subject to structured time-varying uncertainties. By decomposing a global system into smaller dimensional subsystems, a set of distributed MPC controllers, instead of a centralised controller, are designed. To ensure the robust stability of the closed-loop system with respect to model uncertainties, distributed state feedback laws are obtained by solving a min–max optimisation problem. The design of robust distributed MPC is then transformed into solving a minimisation optimisation problem with linear matrix inequality constraints. An iterative online algorithm with adjustable maximum iteration is proposed to coordinate the distributed controllers to achieve a global performance. The simulation results show the effectiveness of the proposed robust distributed MPC algorithm.     

Robust stability bounds on time-varying perturbations for state-space models of linear discrete-time systems

The stability robustness of linear discrete-time systems in the time domain is addressed using the Lyapunov approach. Bounds on linear time-varying perturbations that maintain the stability of an asymptotically stable linear time-invariant discrete-time nominal system are obtained for both structured and unstructured independent perturbations. Bounds are also derived assuming that various elements of the system matrix are perturbed dependently. The result for the structured perturbation case is extended to the stability analysis of interval matrices.     

Robust performance analysis for systems with structured uncertainty

This paper analyses robust performance measures for linear time-invariant systems with norm-bounded time-varying structured uncertainty. We consider two robust performance measures. One is the worst-case peak value of the error signal in response to the disturbance with a known energy. The other is the worst-case energy of the error signal in response to impulsive disturbance. In both cases, the ‘worst case’ is taken over all admissible uncertainties and disturbances. The notion of robust stability is Q-stability, or the scaled H_∞ norm bound. Our main results provide an upper bound for each of the robust performance measures in terms of a positive definite matrix which satisfies a linear matrix inequality (LMI) together with a scaling matrix. Hence, the best bound in this LMI formulation can be computed by convex programming.     

Robust stabilization in an observer-controller feedback system under nonlinear time-varying perturbations or unmodeled dynamics

In this note a new robust criterion is proposed to analyze multivariable observer-controller compensator feedback systems under additive nonlinear perturbations. In order to stabilize these systems, a more general parameterized observer-controller compensator is introduced, and a design algorithm is proposed for robust stabilization against the nonlinear perturbations. Finally, an example is given to illustrate our results.     

Robust performance with fixed and worst-case signals for uncertain time-varying systems

In this paper we focus our attention on the determination of upper bounds of the l^∞ norm of the output of a linear discrete-time dynamic system driven by a step input, in the presence of both persistent unknown, but, l^∞ bounded disturbances and memoryless time-varying model uncertainty. For the same type of systems we also analyze the transient behavior of the step response in terms of its overshoot. The problem is solved in a constructive way by determining appropriate invariant sets contained in a given convex region. Finally, we show how to extend these results to continuous-time systems.     

The time-varying gap and coprime factor perturbations

The connection between the time-varying gap metric and two-block problems is utilized to obtain criteria for robust stabilization of linear, discretetime, time-varying systems. In particular we give a formula for the optimal minimal angle for a stabilizable linear time-varying system and show that it has a maximally stabilizing controller.     

Robust control with structure perturbations

The problem of making a given stabilizing controller robust so that the closed-loop system remains stable for prescribed ranges of variations of a set of physical parameters in the plant. The problem is treated in the state-space and transfer-function domains. In the state-space domain a stability hypersphere is determined in the parameter space using Lyapunov theory. The radius of this hypersphere is iteratively increased by adjusting the controller parameters until the prescribed perturbation ranges are contained in the stability hypersphere. In the transfer-function domain a corresponding stability margin is defined and optimized on the basis of the recently introduced concept of the largest stability hypersphere in the space of coefficients of the closed-loop characteristic polynomial. The design algorithms are illustrated by examples     

Robust stability under additive perturbations

We consider a MIMO linear time-invariant feedback system¹S(P, C)which is assumed to beu-stable. The plantPis subjected to an additive perturbationDelta Pwhich is proper but not necessarily stable. We prove that the perturbed system isu-stable if and only ifDelta P[I + Q.Delta P]^{-1}isu-stable. (HereQ: = C(I + PC)^{-1}.)