A new vertex result for robustness problems with interval matrix uncertainty

This paper¹ addresses a family of robustness problems in which the system under consideration is affected by interval matrix uncertainty. The main contribution of the paper is a new vertex result that drastically reduces the number of extreme realizations required to check robust feasibility. This vertex result allows one to solve, in a deterministic way and without introducing conservatism, the corresponding robustness problem for small and medium size problems. For example, consider quadratic stability of an autonomous n_x dimensional system. In this case, instead of checking vertices, we show that it suffices to check 2^2n_x specially constructed systems. This solution is still exponential, but this is not surprising because the problem is NP-hard. Finally, vertex extensions to multiaffine interval families and some sufficient conditions (in LMI form) for robust feasibility are presented. Some illustrative examples are also given.     

Worst-case properties of the uniform distribution and randomized algorithms for robustness analysis

In this paper we study aprobabilistic approach which is an alternative to the classical worst-case algorithms for robustness analysis and design of uncertain control systems. That is, we aim to estimate the probability that a control system with uncertain parametersq restricted to a boxQ attains a given level of performance γ. Since this probability depends on the underlying distribution, we address the following question: What is a “reasonable” distribution so that the estimated probability makes sense? To answer this question, we define two worstcase criteria and prove that the uniform distribution is optimal in both cases. In the second part of the paper we turn our attention to a subsequent problem. That is, we estimate the sizes of both the so-called “good” and “bad” sets via sampling. Roughly speaking, the good set contains the parametersq∈Q with a performance level better than or equal to γ while the bad set is the set of parametersq∈Q with a performance level worse than γ. We give bounds on the minimum sample size to attain a good estimate of these sets in a certain probabilistic sense.     

Randomized algorithms for synthesis of switching rules for multimodal systems

In this paper, we consider the design of globally asymptotically stabilizing state-dependent switching rules for multimodal systems, first restricting attention to linear time-invariant (LTI) systems with only two states for the switch, and then generalizing the results to multimodal LTI systems and to nonlinear systems. In all cases, the systems considered do not allow the construction of a single quadratic Lyapunov function and, hence, fall in the class of problems that require multiple Lyapunov functions and thus are nonconvex. To address the challenge of nonconvexity , we introduce probabilistic algorithms, and prove their probability-one convergence under a new notion of convergence. Then, to reduce complexity, we develop modified versions of the algorithm. We also present a class of more general nonconvex problems to which this approach can be applied. The results are illustrated using two- and three-dimensional systems with multiple switch states.     

Bounded error parameter estimation: a sequential analytic centerapproach

A sequential analytic center approach for bounded error parameter estimation is proposed. The authors show that the analytic center minimizes the logarithmic average output error among all the estimates within the membership set and is a maximum likelihood estimator for a class of noise density functions which include parabolic densities and approximations of truncated Gaussian. They also show that the analytic center is easily computable for both offline and online problems with a sequential algorithm. The convergence proof of this sequential algorithm is obtained and, moreover, it is shown that the complexity in terms of the maximum number of Newton iterations is linear in the number of observed data points     

Probabilistic robustness analysis: Explicit bounds for the minimum number of samples

In this paper, we study robustness analysis of control systems affected by bounded uncertainty. Motivated by the difficulty to perform this analysis when the uncertainty enters into the plant coefficients in a nonlinear fashion, we study a probabilistic approach. In this setting, the uncertain parameters q are random variables bounded in a set Q and described by a multivariate density function f(q). We then pose the following question: Given a performance level, what is the probability that this level is attained? The main content of this paper is to derive explicit bounds for the number of samples required to estimate this probability with a certain accuracy and confidence a priori specified. It is shown that the number obtained is inversely proportional to these thresholds and it is much smaller than that of classical results. Finally, we remark that the same approach can be followed to study several problems in a control system context. For example, we can evaluate the worst-case H_∞ norm of the sensitivity function of multi-input multi-output systems or compute μ when the robustness margin is of interest.     

Optimal algorithms theory for robust estimation and prediction

This paper deals with the theory of optimal algorithms for problems which cannot be solved exactly. The theory developed allows for the derivation of new and interesting results in parameter estimation and in time series prediction in situations where no reliable statistical hypothesis can be made on the functions and modeling errors involved, but only a bound on them is known, in particular, the derivation of computationally simple optimal algorithms for these two problems is investigated. The practical effectiveness of the algorithms obtained is illustrated by several numerical examples.     

On the stability of a weighted diamond of real polynomials

Recently, a subset of the robust stability literature concentrated on the so-called diamond of polynomials. In this paper, we study the weighted diamond Q_w defined as

, where the center q^*, the weights w₀, w₁, 3dot , w_n > 0 and the radius r > 0 are given. For this family, we show that an extreme point result holds if and only if a certain ‘interlacing condition’ on the weights is satisfied.     

Research on probabilistic methods for control system design

A novel approach based on probability and randomization has emerged to synergize with the standard deterministic methods for control of systems with uncertainty. The main objective of this paper is to provide a broad perspective on this area of research known as “probabilistic robust control”, and to address in a systematic manner recent advances. The focal point is on design methods, based on the interplay between uncertainty randomization and convex optimization, and on the illustration of specific control applications.     

Extreme point results for robust stabilization of interval plantswith first-order compensators

It has been shown previously that a first-order compensator robustly stabilizes an internal plant family if and only if it stabilizes all of the extreme plants. These extreme plants are obtained by considering all possible combinations for the extreme values of the numerator and denominator coefficients. In this work, the authors prove a stronger result, namely, that it is necessary and sufficient to stabilize only sixteen of the extreme plants. These sixteen plants are generated using the Kharitonov polynomials associated with the numerator and denominator. Furthermore, when additional information about the compensator is specified (sign of the gain and signs and relative magnitudes of the pole and zero), then, in some cases, it is necessary and sufficient to stabilize eight critical plants, while, in other cases, it is necessary and sufficient to stabilize twelve critical plants