An infinite ε-bound stabilization design for a class of singularly perturbed systems

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 */).

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.

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.

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.

An LMI-Based Approach to Static Output Feedback Stabilization of T—S Fuzzy Systems

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. >

We study the robustness of H/sup /spl infin// controllers to unknown static nonlinear perturbations in the state dynamics, the measurement equation, and the performance index. When the nominal system is linear, we consider both perfect and general imperfect state measurements, and 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, /spl epsiv/>0) approaches zero, the optimal disturbance attenuation level for the overall system converges to a value that is bounded above by the optimal disturbance attenuation level for the nominal system if the nonlinear structural uncertainties satisfy prescribed growth conditions. In particular, the overall optimal disturbance attenuation level converges to the nominal optimal disturbance attenuation level under perfect state measurements. We also show that the H/sup /spl infin//-optimal controller designed for a chosen performance level for the nominal linear system achieves the same performance level when the parameter |/spl epsiv/| 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 is tolerable for a controller designed based on the nominal system. >

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).

Complementary sequences (CS) have peak-to-average power ratio (PAR) /spl les/ 2 under the one-dimensional continuous discrete Fourier transform (DFT/sub 1//sup /spl infin//). Davis and Jedwab (see IEEE Trans. Inform. Theory, vol.45, no.7, p.2397-2417, 1999) constructed binary CS (DJ set) for lengths 2/sup n/ described by s = 2/sup -n/2/ (-1)/sup p(x)/, p(x) = /spl Sigma//sub j=0//sup L-2/x/sub /spl pi/(j)/x/sub /spl pi/(j+1)/+c/sub j/x/sub j/+k, c/sub j/, k /spl isin/ Z/sub 2/. Hamming distance, D, between sequences in this set satisfies D /spl ges/ 2/sup n-2/. However the rate of the DJ set vanishes for n /spl rarr/ /spl infin/, and higher rates are possible for PAR /spl les/ O(n) and D large. We present such a construction which generalises the DJ set. These codesets have PAR /spl les/ 2/sup t/ under all linear unimodular unitary transforms (LUUTs), including all one and multi-dimensional continuous DFTs, and D /spl ges/ 2/sup n-d/ where d is the maximum algebraic degree of the chosen subset of the complete set.

Scheduling schemes for an integrated flight and propulsion control system

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.

Studies the singular nonlinear state feedback H/sub /spl infin// problem for affine nonlinear systems. The authors' approach yields a sufficient and, under an extra assumption, also necessary condition for a singular nonlinear system to construct a state feedback which leads to an L/sub 2/ gain less then a certain bound /spl gamma/ for the closed loop system. This condition is in terms of a set of Hamilton Jacobi inequalities parameterised by a parameter /spl epsiv/.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

In previous papers, the Fourier transform (FT) has been generalized into the fractional Fourier transform (FRFT), the linear canonical transform (LCT), and the simplified fractional Fourier transform (SFRFT). Because the cosine, sine, and Hartley transforms are very similar to the FT, it is reasonable to think they can also be generalized by the similar way. We introduce several new transforms. They are all the generalization of the cosine, sine, or Hartley transform. We first derive the fractional cosine, sine, and Hartley transforms (FRCT/FRST/FRHT). They are analogous to the FRFT. Then, we derive the canonical cosine and sine transforms (CCT/CST). They are analogous to the LCT. We also derive the simplified fractional cosine, sine, and Hartley transforms (SFRCT/SFRST/SFRHT). They are analogous to the SFRFT and have the advantage of real-input-real-output. We also discuss the properties, digital implementation, and applications (e.g., the applications for filter design and space-variant pattern recognition) of these transforms. The transforms introduced in this paper are very efficient for digital implementation. We can just use one half or one fourth of the real multiplications required for the FRFT and LCT to implement them. When we want to process even, odd, or pure real/imaginary functions, we can use these transforms instead of the FRFT and LCT. Besides, we also show that the FRCT/FRST, CCT/CST, and SFRCT/SFRST are also useful for the one-sided (t /spl isin/ [0, /spl infin/]) signal processing.