Robustness of H/sup ∞/ controllers to nonlinear perturbations

We study the robustness of minimax controllers, originally designed for nominal linear or nonlinear systems, to unknown static nonlinear perturbations in the state dynamics, measurement equation, and performance index. When the nominal system is linear, we consider both perfect state measurements and general imperfect state measurements; in the case of nominally nonlinear systems, we consider perfect state measurements only. Using a differential game theoretic approach, we show for the former class that, as the perturbation parameter (say, e>0) approaches zero, the optimal disturbance attenuation level for the overall system converges to the optimal disturbance attenuation level for the nominal system if the nonlinear structural uncertainties satisfy certain prescribed growth conditions. We also show that anH∞-controller, designed based on a chosen performance level for the nominal linear system, achieves the same performance level when the parameter e is smaller than a computable threshold, except for the finite-horizon imperfect state measurements case. For that case, we show that the design of the nominal controller must be based on a decreased confidence level of the initial data, and a controller thus designed again achieves a desired performance level in the face of nonlinear perturbations satisfying a computable norm bound. In the case of nominally nonlinear systems, and assuming that the nominal system is solvable, we obtain sufficient conditions such that the nominal controller achieves a desired performance in the face of perturbations satisfying computable norm bounds. In this way, we provide a characterization of the class of uncertainties that are tolerable for a controller designed based on the nominal system. The paper also presents two numerical examples; in one of these, the nominal system is linear; in the other one, it is nonlinear.

We study time-scale separation and robust controller design for a class of singularly perturbed nonlinear systems under perfect state measurements. The system dynamics are taken to be jointly linear in the fast state variables, control and disturbance inputs, but nonlinear in the slow state variables. Since global timescale separation may not always be possible for nonlinear singularly perturbed systems, we restrict our attention here to some closed subset of the state space, on which a timescale separation holds for sufficiently small values of the singular perturbation parameter. We construct a slow controller and a composite controller based on the solutions of particular slow and fast games obtained using time-scale separation. For the class of systems for which the slow controller can be selected to be robust with respect to small regular structural perturbations on the slow subsystem, we show under some growth conditions that the composite controller can achieve any desired level of performance that is larger than the maximum of the performance levels for the slow and fast subsystems,. A slow controller, however, is not generally as robust as the composite controller; but, still under some conditions which are delineated in the paper, the fast dynamics can be totally ignored. The paper also presents a numerical example to illustrate the theoretical results.

H¿-optimal control for singularly perturbed systems¿ Part I: Perfect state measurements

Presents techniques for controlling the motion of an underactuated vehicle when disturbances are present and only imperfect state measurements are available for feedback. A state feedback controller is developed and then it is converted to an imperfect state measurement feedback controller. The state feedback tracking control law uses an H/sup /spl infin//-optimal design and produces a locally exponentially stable closed-loop system. The imperfect state measurement feedback controller combines the state feedback control law with an H/sup /spl infin//-filter to estimate the states and achieves a modified form of disturbance attenuation. The state estimator exploits a unique structure in the nonlinear equations of motion to develop a direct solution. MATLAB simulations illustrate both control algorithms for an underactuated ship model.

This correspondence is concerned with asymptotic properties on the codeword length of a fixed-to-variable length code (FV code) for a general source {X/sup n/}/sub n=1//sup /spl infin// with a finite or countably infinite alphabet. Suppose that for each n /spl ges/ 1 X/sup n/ is encoded to a binary codeword /spl phi//sub n/(X/sup n/) of length l(/spl phi//sub n/(X/sup n/)). Letting /spl epsiv//sub n/ denote the decoding error probability, we consider the following two criteria on FV codes: i) /spl epsiv//sub n/ = 0 for all n /spl ges/ 1 and ii) lim sup/sub n/spl rarr//spl infin///spl epsiv//sub n/ /spl les/ /spl epsiv/ for an arbitrarily given /spl epsiv/ /spl isin/ [0,1). Under criterion i), we show that, if X/sup n/ is encoded by an arbitrary prefix-free FV code asymptotically achieving the entropy, 1/nl(/spl phi//sub n/(X/sup n/)) - 1/nlog/sub 2/ 1/PX/sup n/(X/sup n/) /spl rarr/ 0 in probability as n /spl rarr/ /spl infin/ under a certain condition, where P/sub X//sup n/ denotes the probability distribution of X/sup n/. Under criterion ii), we first determine the minimum rate achieved by FV codes. Next, we show that 1/nl(/spl phi//sub n/(X/sup n/)) of an arbitrary FV code achieving the minimum rate in a certain sense has a property similar to the lossless case.

In this paper, the robust stability problem for a class of uncertain two-time-scale neutral systems subject to unstructured perturbations is investigated. The properties of H/sub /spl infin//-norm are used throughout this paper to derive the robust stability criterion. A frequency-domain delay-dependent criterion for the asymptotic stability of the slow neutral subsystem and the fast subsystem of the nominal two-time-scale neutral system is first presented. Under the condition that the slow neutral and fast subsystems of the nominal system are both asymptotically stable, we then propose the allowable bounds for the system uncertainties such that the slow neutral subsystem and the fast subsystem of the original uncertain two-time scale neutral system are both asymptotically stable. Finally, a stability bound /spl epsi//sup */ of the small parameter /spl epsi/ is given such that the original uncertain two-time scale neutral system is asymptotically stable for any /spl epsi/ /spl isin/ (0, /spl epsi//sup */).

Presents an algorithm for minimizing the search effort of a mobile robot navigating in an unknown environment. First we describe a strategy for navigating a cylinder-shaped mobile robot in an area with unknown obstacles using range data. The strategy searches a graph which is constructed incrementally during the navigation process. Then we present a new algorithm for searching the graph which attempts to minimize the search effort, measured by the length of the path traveled by the robot. The algorithm, called A/sub /spl epsiv//*-DFS, combines features of the classical A* and DFS graph-search algorithms, and generates /spl epsiv/-optimal paths. Moreover, A/sub /spl epsiv//*-DFS is general and can be used by any online search strategy. The algorithm uses E as a parameter, where /spl epsiv/=0 corresponds to A* while /spl epsiv/=/spl infin/ corresponds to DFS. We study the performance of A/sub /spl epsiv//*-DFS for various values of /spl epsiv/ on simulated environments, and indicate how to best choose /spl epsiv/ for a given class of environments.

In this paper, we present robust adaptive controller design for SISO linear systems with zero relative degree under noisy output measurements. We formulate the robust adaptive control problem as a nonlinear H/sup /spl infin// optimal control problem under imperfect state measurements, and then solve it using game theory. By using the a priori knowledge of the parameter vector, we apply a soft projection algorithm, which guarantees the robustness property of the closed-loop system without any persistency of excitation assumption on the reference signal. Due to our formulation in state space, we allow the true system to be uncontrollable, as long as the uncontrollable part is stable in the sense of Lyapunov, and the uncontrollable modes on the jw-axis are uncontrollable from the exogenous disturbance input. This assumption allows the adaptive controller to asymptotically cancel out, at the output, the effect of exogenous sinusoidal disturbance, inputs with unknown magnitude, phase, and frequency. These strong robustness properties are illustrated by a numerical example.

In this paper the H/sub /spl infin//-optimal control and risk-sensitive control of linear singularly perturbed, discrete-time systems are considered. It is shown that the Riccati equation associated with the solution of the H/sub /spl infin//-optimal control problem, can be approximated by an outer series solution, and a boundary-layer correction series solution. As the singular perturbation parameter /spl epsiv/ tends to zero, it is shown that the full-order Riccati equation can be approximated by the zeroth-order outer-series solution. It is also recognized that the outer-series solution and the boundary-layer solution, are equivalent to the resulting asymptotic series expansion in powers of /spl epsiv/, of the Riccati equation associated with a certain risk-sensitive control problem. When a two-time-scale decomposition is considered, it is shown that an approximate composite control can be formed, as the sum of the slow and fast subsystems. Unlike the asymptotic expansion approach, it is shown that the original system can be approximated by the maximum of the performance associated with the slow and fast subsystems. Finally, it is shown that the slow and fast subsystems, are equivalent to certain slow and fast subsystems, respectively, resulting from applying a two-time-scale decomposition to a certain risk-sensitive control problem. >

The H/sub /spl infin// filtering Riccati equation, associated with filtering problem, of standard singularly perturbed systems is completely and exactly decomposed into two reduced-order H/sub /spl infin// filtering Riccati equations corresponding to the slow and fast time scales. The pure-slow and pure-fast H/sub /spl infin// filtering Riccati equations are nonsymmetric ones; however, their O(/spl epsiv/) perturbations are symmetric. It will be shown that the Newton method is very efficient for solving the obtained nonsymmetric H/sub /spl infin// filtering Riccati equations. This method is suitable for parallel computations. Due to complete and exact decomposition of the Riccati equation, this procedure might produce new insight into the two-time scale H/sub /spl infin// output feedback problem.

The asymptotic behavior of the /spl epsiv/-entropy of ellipsoids in an n-dimensional Hamming space whose coefficients take only two different values is investigated as n /spl rarr/ /spl infin/. Explicit expressions for the main terms of the asymptotic representation for the /spl epsiv/-entropy of such ellipsoids are obtained under different relations between /spl epsiv/ and parameters defining these ellipsoids.

Given a collection of segments in the plane that intersect pairwise at most k times, regarding the segments as opaque barriers, their upper envelope consists of the portions of the segments visible from point (0,+/spl infin/). We give efficient parallel methods for finding the upper envelope of k-intersecting segments for any integer k/spl ges/0, in the weakest shared memory model, the EREW PRAM. We show that the upper envelope of n k-intersecting segments can be found in 0(log/sup 1+/spl epsiv//n) time using 0(/spl lambda//sub k+1/(n)/log/sup /spl epsiv//n) processors for any /spl epsiv/>0, where /spl lambda//sub k+2/(n)/sup 1/ is the size of the upper envelope. In particular, for line segments we show the following optimal algorithms: the upper envelope of n line segments can be found in O(log n) time using O(n) processors, and if the line segments are nonintersecting and sorted, the envelope can be found in O(log n) time using O(n/log n) processors. We also show that our methods imply a fast sequential result: the upper envelope of n sorted line segments can be found in O(n log log n) time sequentially, which improves the known lowest upper bound O(n log n).

This paper presents two kinds of robust controllers for stabilizing singularly perturbed discrete bilinear systems. The first one is an /spl epsi/-dependent controller that stabilizes the closed-loop system for all /spl epsi//spl isin/(0, /spl epsi//sub 0/*), where /spl epsi//sub 0/* is the prespecified upper bound of the singular perturbation parameter. The second one is an /spl epsi/-independent controller, which is able to stabilize the system in the entire state space for all /spl epsi//spl isin/(0, /spl epsi/*), where /spl epsi/* is the exact upper /spl epsi/-bound. The /spl epsi/* can be calculated by the critical stability criterion once the robust controller is determined. An example is presented to illustrate the proposed schemes.

A novel parameter-dependent anti-windup compensator is developed to improve the performance of a constrained model reference adaptive controller. The combined control structure solves the input saturation and stability problem for inertia varying quadcopters. The control synthesis follows the conventional two-step anti-windup design paradigm where a nominal controller is designed without consideration of the input saturation, and the anti-windup compensator is designed to minimize deviations from nominal performance caused by saturated inputs. To account for varying inertia of the quadcopter during package retrieval/delivery routines, the inertia parameters of the vehicle/package are estimated with an online recursive identification technique. These estimates are used by the model reference adaptive controller to ensure stability of the nominal (unsaturated) system and to schedule the parameter-dependent anti-windup compensator. The performance and stability conditions of the parameter-dependent anti-windup compensator are formulated as a set of parameter-dependent linear matrix inequalities. When solved, the linear matrix inequalities yield a gain-scheduled anti-windup compensator that ensures stability and minimizes the deviation from nominal model reference adaptive control performance when saturation occurs. The effectiveness of the combined control scheme is demonstrated by simulations of an input constrained quadcopter lifting a payload of unknown mass.

We provide some evidence that unique k-SAT is as hard to solve as general k-SAT, where k-SAT denotes the satisfiability problem for k-CNFs and unique k-SAT is the promise version where the given formula has 0 or 1 solutions. Namely, defining for each k/spl ges/1, s/sub k/=inf{/spl delta//spl ges/0|/spl exist/aO(2/sup /spl delta/n/)-time randomized algorithm for k-SAT} and, similarly, /spl sigma//sub k/=inf{/spl delta//spl ges/0|/spl exist/aO(2/sup /spl delta/n/)-time randomized algorithm for Unique k-SAT}, we show that lim/sub k/spl rarr//spl infin//s/sub k/=lim/sub k/spl rarr//spl infin///spl sigma//sub k/. As a corollary, we prove that, if Unique 3-SAT can be solved in time 2/sup /spl epsi/n/ for every /spl epsi/>0, then so can k-SAT for k/spl ges/3. Our main technical result is an isolation lemma for k-CNFs, which shows that a given satisfiable k-CNF can be efficiently probabilistically reduced to a uniquely satisfiable k-CNF, with nontrivial, albeit exponentially small, success probability.