Constantino M. Lagoa

The uniform distribution: rigorous justification for its use in robustness analysis

Consider a control system which is operated with admissible values of uncertain parameters which exceed the bounds specified by classical robustness theory. In this case it is important to quantify the tradeoffs between risk of performance degradation and increased tolerance of uncertainty. If a large increase in the uncertainty bound can be established, an acceptably small risk may often be justified. Since robustness problem formulations do not include statistical descriptions of the uncertainty, the question arises whether it is possible to provide such assurances in a “distribution-free” manner. In other words, if ℱ denotes a class of possible probability distributions for the uncertaintyq, we seek some worst-casef* ε ℱ having the following property: The probability of performance satisfaction underf* is smaller than the probability under any otherf ε ℱ. Said another way,f* provides the best possible guarantee. This new framework is illustrated on robust stability problems associated with Kharitonovs theorem and the Edge Theorem. The main results are straightforward to describe: Letp(s, q) denote the uncertain polynomial under consideration and takeP(ω) to be a frequency-dependent convex target set (in the complex plane) for the uncertain valuesp(jω, q). Consistent with value set analysis,P(ω) is assumed to be symmetric with respect to the nominalp(jω, 0). The uncertain parametersqi are taken to be zero-mean independent random variables with known support interval. For each uncertainty, the class ℱ is assumed to consist of density functions which are symmetric and nonincreasing on each side of zero. Then, for fixed frequencyω, the first theorem indicates that the probability thatp(jω, q) is inP(ω) is minimized by the uniform distribution forq. The second theorem, a generalization of the first, indicates that the same result holds uniformly with respect to frequency. Then probabilistic guarantees for robust stability are given in the third theorem. It turns out that in many cases, classical robustness margins can be far exceeded while keeping the risk of instability surprisingly small. Finally, for a much more general class of uncertainty structures, this paper also establishes the fact thatf* can be estimated by a truncated uniform distribution.

A Sparsification Approach to Set Membership Identification of Switched Affine Systems

This paper addresses the problem of robust identification of a class of discrete-time affine hybrid systems, switched affine models, in a set membership framework. Given a finite collection of noisy input/output data and some minimal a priori information about the set of admissible plants, the objective is to identify a suitable set of affine models along with a switching sequence that can explain the available experimental information, while minimizing either the number of switches or subsystems. For the case where it is desired to minimize the number of switches, the key idea of the paper is to reduce this problem to a sparsification form, where the goal is to maximize sparsity of a suitably constructed vector sequence. Our main result shows that in the case of ℓ∞ bounded noise, this sparsification problem can be exactly solved via convex optimization. In the general case where the noise is only known to belong to a convex set N, the problem is generically NP-hard. However, as we show in the paper, efficient convex relaxations can be obtained by exploiting recent results on sparse signal recovery. Similarly, we present both a sparsification formulation and a convex relaxation for the (known to be NP hard) case where it is desired to minimize the number of subsystems. These results are illustrated using two non-trivial problems arising in computer vision applications: video-shot and dynamic texture segmentation.

A sparsification approach to set membership identification of a class of affine hybrid systems

This paper addresses the problem of robust identification of a class of discrete-time affine hybrid systems, switched affine models, in a set membership framework. Given a finite collection of noisy input/output data and some minimal a priori information about the set of admissible plants, the objective is to identify a suitable set of affine models along with a switching sequence that can explain the available experimental information, while optimizing a performance criteria (either minimum number of switches or minimum number of plants). Our main result shows that this problem can be reduced to a sparsification form, where the goal is to maximize sparsity of a given vector sequence. Although in principle this leads to an NP-hard problem, as we show in the paper, efficient convex relaxations can be obtained by exploiting recent results on sparse signal recovery. These results are illustrated using two non-trivial problems arising in computer vision applications: video-shot and dynamic texture segmentation.

Probabilistically Constrained Linear Programs and Risk-Adjusted Controller Design

The focal point of this paper is the probabilistically constrained linear program (PCLP) and how it can be applied to control system design under risk constraints. The PCLP is the counterpart of the classical linear program, where it is assumed that there is random uncertainty in the constraints and, therefore, the deterministic constraints are replaced by probabilistic ones. It is shown that for a wide class of probability density functions, called log-concave symmetric densities, the PCLP is a convex program. An equivalent formulation of the PCLP is also presented which provides insight into numerical implementation. This concept is applied to control system design. It is shown how the results in this paper can be applied to the design of controllers for discrete-time systems to obtain a closed loop system with a well-defined risk of violating the so-called property of superstability. Furthermore, we address the problem of risk-adjusted pole placement.

Radially truncated uniform distributions for probabilistic robustness of control systems

An approach to probabilistic robustness is developed in the context of unstructured uncertainty. For a control system with a bound on the uncertain quantities of interest, the probabilistic robustness margin R/sub max/(/spl epsiv/) describes the radius of tolerable uncertainty as a function of the risk level 0/spl les//spl epsiv//spl les/1. In addition, associated with the performance risk probability p=/spl epsiv/, the computed radius R/sub max/(/spl epsiv/) is guaranteed for a large class of radially symmetric nonincreasing density functions. In other words, the results are distribution free in the sense that the user does not need to have a detailed description of the statistics of the uncertainty other than a radial bound.

Distributionally Robust Monte Carlo Simulation: A Tutorial Survey

Abstract Whereas the use of traditional Monte Carlo simulation requires probability distributions for the uncertain parameters entering the system, distributionally robust Monte Carlo simulation does not. According to this new theory, instead of carrying out simulations using some rather arbitrary probability distribution such as Gaussian for the uncertain parameters, we provide a rather different prescription based on distributional robustness considerations. Motivated by manufacturing considerations, a class of distributions ℱ is specified and the results of the simulation hold for all f ∈ ℱ. This new method of Monte Carlo simulation was developed with the robustician in mind in that we begin only with bounds on the uncertain parameters and no a priori probability distribution is assumed.

Robust identification of switched affine systems via moments-based convex optimization

This paper addresses the problem of robust identification of a class of discrete-time affine hybrid systems, switched affine models, in a set membership framework. Given a finite collection of noisy input/output data and a bound on the number of subsystems, the objective is to identify a suitable set of affine models along with a switching sequence that can explain the available experimental information. Our method builds upon an algebraic procedure proposed by Vidal et al. for noise free measurements. In the presence of norm bounded noise, this algebraic procedure leads to a very challenging nonconvex polynomial optimization problem. Our main result shows that this problem can be reduced to minimizing the rank of a matrix whose entries are affine in the optimization variables, subject to a convex constraint imposing that these variables are the moments of an (unknown) probability distribution function with finite support. Appealing to well known convex relaxations of rank leads to an overall semi-definite optimization problem that can be efficiently solved. These results are illustrated with two examples showing substantially improved identification performance in the presence of noise.

End-to-end optimal algorithms for integrated QoS, traffic engineering, and failure recovery

This paper addresses the problem of optimal quality of service (QoS), traffic engineering (TE) and failure recovery (FR) in computer networks by introducing novel algorithms that only use source inferrable information. More precisely, optimal data rate adaptation and load balancing laws are provided which are applicable to networks where multiple paths are available and multiple classes of service (CoS) are to be provided. Different types of multiple paths are supported, including point-to-point multiple paths, point-to-multipoint multiple paths, and multicast trees. In particular, it is shown that the algorithms presented only need a minimal amount of information for optimal control, i.e., whether a path is congested or not. Hence, the control laws provided in this paper allow source inferred congestion detection without the need for explicit congestion feedback from the network. The proposed approach is applicable to utility functions of a very general form and endows the network with the important property of robustness with respect to node/link failures; i.e., upon the occurrence of such a failure, the presented control laws reroute traffic away from the inoperative node/link and converge to the optimal allocation for the ldquoreducedrdquo network. The proposed control laws set the foundation for the development of highly scalable feature-rich traffic control protocols at the IP, transport, or higher layers with provable global stability and convergence properties.

Hybrid system identification via sparse polynomial optimization

In this paper, the problem of identifying discrete time affine hybrid systems with measurement noise is considered. Given a finite collection of measurements and a bound on the noise, the objective is to identify a hybrid system with the smallest number of sub-systems that is compatible with the a priori information. While this problem has been addressed in the literature if the input/output data is noise-free or corrupted by process noise, it remains open for the case of measurement noise. To handle this case, we propose a new approach based on recasting the problem into a polynomial optimization form and exploiting its inherent sparse structure to obtain computationally tractable problems. Combining these ideas with a randomized Hit and Run type approach leads to further computational complexity reduction, allowing for solving realistically sized problems. Numerical examples are provided, illustrating the effectiveness of the algorithm and its potential to handle large size problems.

Unconstrained optimal control of regular languages

This paper formulates an unconstrained optimal policy for control of regular languages realized as deterministic finite state automata (DFSA). A signed real measure quantifies the behavior of controlled sublanguages based on a state transition cost matrix and a characteristic vector as reported in an earlier publication. The state-based optimal control policy is obtained by selectively disabling controllable events to maximize the measure of the controlled plant language without any further constraints. Synthesis of the optimal control policy requires at most n iterations, where n is the number of states of the DFSA model. Each iteration solves a set of n simultaneous linear algebraic equations. As such, computational complexity of the control synthesis is polynomial in n.