Improved measures of stability robustness for linear state space models

In this paper, the aspect of "stability robustness" of linear systems is analyzed in the time domain. A bound on the structured perturbation of an asymptotically stable linear system is obtained to maintain stability using a Lyapunov matrix equation solution. The resulting bound is shown to be an improved bound over the ones recently reported in the literature. Also, special cases of the nominal system matrix are considered, for which the bound is given in terms of the nominal matrix, thereby, avoiding the solution of the Lyapunov matrix equation. Examples given include comparison of the proposed approach with the recently reported results.     

Reduced conservatism in stability robustness bounds by state transformation

This note addresses the issue of "conservatism" in the time domain stability robustness bounds obtained by the Lyapunov approach. A state transformation is employed to improve the upper bounds on the linear time-varying perturbation of an asymptotically stable linear time-invariant system for robust stability. This improvement is due to the variance of the conservatism of the Lyapunov approach with respect to the basis of the vector space in which the Lyapunov function is constructed. Improved bounds are obtained, using a transformation, on elemental and vector norms of perturbations (i.e., structured perturbations) as well as on a matrix norm of perturbations (i.e., unstructured perturbations). For the case of a diagonal transformation, an algorithm is proposed to find the "optimal" transformation. Several examples are presented to illustrate the proposed analysis.     

Attraction, stability and robustness for stochastic functional differential equations with infinite delay

This paper establishes the stochastic LaSalle theorem to locate limit sets for stochastic functional differential equations with infinite delay, from which some criteria on attraction, boundedness, stability and robustness are obtained. To illustrate the applications of our results clearly, this paper considers a scalar stochastic integro-differential equation with infinite delay as an example.     

State-space measures for stability robustness

Stability robustness measures for a perturbed linear feedback system are derived based on state-space models of the system. The system may be a continuous-time or discrete-time system. The perturbations are modeled as additive perturbation matrices. Necessary and sufficient conditions for the stability of the perturbed closed-loop system for all perturbations of norm bounded by some positive number are obtained. The destabilizing perturbations of minimal norm are characterized. It is shown by an example that there are cases when the destabilizing perturbations of minimal norm are all complex. The results are expressed in terms of induced operator norms. These are later specialized to the Euclidean norm and expressed in terms of singular values. A simple example is also included to illustrate an application of the results of this note.     

The stability robustness determination of state space models with real unstructured perturbations

This paper considers the robust stability of a linear time-invariant state space model subject to real parameter perturbations. The problem is to find the distance of a given stable matrix from the set of unstable matrices. A new method, based on the properties of the Kronecker sum and two other composite matrices, is developed to study this problem; this new method makes it possible to distinguish real perturbations from complex ones. Although a procedure to find the exact value of the distance is still not available, some explicit lower bounds on the distance are obtained. The bounds are applicable only for the case of real plant perturbations, and are easy to compute numerically; if the matrix is large in size, an iterative procedure is given to compute the bounds. Various examples including a 46th-order spacecraft system are given to illustrate the results obtained. The examples show that the new bounds obtained can have an arbitrary degree of improvement over previously reported ones. This work has been supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A4396.     

On the stabilization and stability robustness against small delaysof some damped wave equations

In this paper we consider a system which can be modeled by two different one-dimensional damped wave equations in a bounded domain, both parameterized by a nonnegative damping constant. We assume that the system is axed at one end and is controlled by a boundary controller at the other end. We consider two problems, namely the stabilization and the stability robustness of the closed-loop system against arbitrary small time delays in the feedback loop. We propose a class of dynamic boundary controllers and show that these controllers solve the stabilization problem when the damping coefficient is nonnegative and the stability robustness problem when the damping coefficient is strictly positive     

Assessment of stability robustness for adaptive controllers

The problem of improving the stability characteristics of discrete-time SISO reduced order adaptive controllers is addressed in this note. A method is proposed to adjust the observer poles of an indirect adaptive scheme in order to ensure stability of the closed-loop system in spite of possible unstructured uncertainties (e.g., process-model order mismatch). Assuming that the identified model transfer function module converges within a previously defined uncertainty range, the frequency response-like compensation method is shown to lead to a stable design, provided that the reduced order estimated model is stable and stably invertible and the system is open-loop stable.     

On the robustness of stability for uncertain time-delay systems

Based on the Lyapunov stability theorem associated with induced norm of matrix and matrix measure techniques, this paper addresses the robust stability analysis for linear time-delay systems with uncertainties. Several new criteria, expressed by concise inequalities, are presented to guarantee the robust stability and stability with a specified decaying rate for the above systems. Both nonlinear norm-bounded and highly structured parametric uncertainties are discussed. Finally, the superiority of the proposed results is demonstrated in the illustrative examples by comparing with those available in the literature     

Intrinsic robustness of global asymptotic stability

Equivalence is shown for discrete time systems between global asymptotic stability and the so-called integral Input-to-State Stability. The latter is a notion of robust stability with respect to exogenous disturbances which informally translates into the statement “no matter what is the initial condition, if the energy of the inputs is small, then the state must eventually be small”.     

Analysis of the local robustness of stability for flows

In this paper the problem of measuring the robustness of stability for a perturbed continuous-time nonlinear system at a singular fixed point is studied. Various stability radii are introduced and their values for the nonlinear system and its linearization are compared. It is shown that they generically coincide. This result may also be used to show generic continuity of linear real stability radii. Some examples are presented showing that it is sometimes necessary to consider the nonlinear system directly, and not simply to rely on the information provided by the linearization.     

Computation of stability robustness bounds for state-space modelswith structured uncertainty

This paper makes two contributions related to computing a bound on the size of structured real parameter perturbations under which a nominally stable matrix remains stable. First, a more efficient method is presented for computing existing bounds, requiring less time and storage than other methods. The second contribution is a means of computing tighter bounds, although at a higher computational cost     

2-D stability test via 1-D stability robustness analysis

A new two-dimensional stability test is proposed, based on the stability robustness analysis of a relevant 1-D system family. A linear algebra type algorithm for numerical implementation of the test is also described. Two examples then follow, to illustrate the use of the algorithm and the feasibility of the proposed test.     

Pole assignment with eigenvalue and stability robustness

This paper presents a computation method for pole assignment with eigenvalue and stability robustness. The robustness measure is constructed to balance the tradeoff between an eigenvalue sensitivity measure and a stability robustness measure, both defined in terms of the non-differentiable spectral norm. It is established that the robustness measure can be minimized asymptotically via a sequence of smooth unconstrained minimizations involving some auxiliary objective functions. Moreover, the sequence of minimizers converges to the set of minimizers of the robustness measure. A numerical algorithm with analytical formulae of the gradient of the auxiliary objective functions is provided. Through a numerical example and comparing with other methods, the idea of using a combined robustness measure in the design is illustrated and the effectiveness of the computation is demonstrated.     

Remote two-qubit state creation and its robustness

We consider the problem of remote two-qubit state creation using the two-qubit excitation pure initial state of the sender. The communication line is based on the optimized boundary-controlled chain with two pairs of properly adjusted coupling constants. We show that the communication line can be characterized by a set of parameters independent of the initial state of the sender. These parameters are permanent attributes of a communication line and can be either calculated theoretically or measured in experiment. In particular, they determine the creatable subregion of the receiver’s state space. The creation of a particular state within the creatable region is achieved by a proper choice of the independent parameters of the sender’s initial state (control parameters) and reduces to the solvability of a certain system of algebraic equations. The creation of the two-qubit Werner state is considered as an example. We also study the effects of imperfections of the chain on the state creation.     

Iterative algorithm for improved measures of stability robustness for linear state-space models

The problem of robust stability of linear lime-invariant systems in state-space models is considered. An iterative algorithm based on frequency domain approach is proposed which leads to new stability robustness measures. The case of structural perturbations is considered and the new bounds are shown to be a significant improvement over recent ones reported. In addition, it is shown that the directional information on structured perturbations can easily be incorporated in the new robustness criterion. Several illustrative examples are worked out.     

The graph metric for unstable plants and robustness estimates for feedback stability

In this paper, a "graph metric" is defined that provides a measure of the distance between unstable multivariable plants. The graph metric induces a "graph topology" on unstable plants, which is the weakest possible topology in which feedback stability is robust. Using the graph metric, it is possible to derive estimates for the robustness of feedback stability without assuming that the perturbed and unperturbed plants have the same number of RHP poles. If the perturbed and unperturbed systems have the same RHP poles, then it is possible to obtain necessary and sufficient conditions for robustness with respect to a given class of perturbations. As an application of these results, the design of stabilizing controllers for unstable singularly perturbed systems is studied. Finally, the relationship of the graph metric to the "gap metric" introduced by Zames and El-Sakkary is studied in detail. In particular, it is shown that the robustness results of Zames and El-Sakkary do not enable one to conclude the causality, of the perturbed system, whereas the present results do.     

Maximal polytopic stability domains in parameter space for uncertain systems

Some results on the stability and pole location of families of uncertain systems with characteristic polynomial coefficients affected by structured perturbations are presented. The case in which the coefficients are affine in a set of system physical parameters is considered, assuming that the parameters are subject to polytopic perturbations of a given structure. For these cases, which often arise in system stability robustness analysis, as for example in the robustification of controllers for multivariable closed-loop systems, a procedure is given to estimate maximal stability polytopic regions in the space of parameters. This result is also used to derive simple necessary and sufficient conditions for the discrete-time stability of convex combinations of two polynomials.     

Necessary and sufficient condition for stability robustness under known additive perturbation

A necessary and sufficient condition is derived for the stability robustness of a unity feedback closed-loop system involving a strictly proper plant P₀ and a stabilizing compensator C₀ under the assumption that the plant P₀ is perturbed to P₀ + ΔP₀ where ΔP₀ is a known strictly proper rational matrix.     

Randomized algorithms for stability and robustness analysis of high-speed communication networks

This paper initiates a study toward developing and applying randomized algorithms for stability of high-speed communication networks. The focus is on congestion and delay-based flow controllers for sources, which are "utility maximizers" for individual users. First, we introduce a nonlinear algorithm for such source flow controllers, which uses as feedback aggregate congestion and delay information from bottleneck nodes of the network, and depends on a number of parameters, among which are link capacities, user preference for utility, and pricing. We then linearize this nonlinear model around its unique equilibrium point and perform a robustness analysis for a special symmetric case with a single bottleneck node. The "symmetry" here captures the scenario when certain utility and pricing parameters are the same across all active users, for which we derive closed-form necessary and sufficient conditions for stability and robustness under parameter variations. In addition, the ranges of values for the utility and pricing parameters for which stability is guaranteed are computed exactly. These results also admit counterparts for the case when the pricing parameters vary across users, but the utility parameter values are still the same. In the general nonsymmetric case, when closed-form derivation is not possible, we construct specific randomized algorithms which provide a probabilistic estimate of the local stability of the network. In particular, we use Monte Carlo as well as quasi-Monte Carlo techniques for the linearized model. The results obtained provide a complete analysis of congestion control algorithms for internet style networks with a single bottleneck node as well as for networks with general random topologies.