L1 analysis and design of sampled-data systems

The focus of this work is L₁-optimal control of sampled-data systems. A converging approximation procedure is derived to compute the L_∞-induced norm of closed-loop finite-dimensional linear time-invariant (LTI) sampled-data systems. An approximation method is developed to synthesize L₁-optimal sampled-data regulators. Finally, an example is provided that illustrates the L₁ analysis and design techniques presented     

Optimal L∞ bounds for disturbancerobustness

Robust disturbance rejection bounds obtained recently by Zhu et al. are tight with respect to the disturbance set that has a specified L₂ norm bound. It is shown that the bounds are also tight with respect to a weighted L₂ disturbance set that has a specified bound on its outer product by choosing a special weight matrix in the L₂ norm. The matrix can computed by numerical iteration     

Rational approximation of L1 optimalcontrollers for SISO systems

The L₁ optimal control problem with rational controllers for continuous-time systems is considered in which it is shown that the optimal L₁ performance index with rational controllers is equal to that of irrational controllers. A sequence of rational controllers that approximates the optimal index is constructed. Convergence properties of such a sequence are studied. That the corresponding sequence of objective transfer functions is shown to converge in weak-* topology in BV(R₊) in the time domain and uniformly in a wider sense in the frequency domain     

The robust H2 control problem: a worst-casedesign

The worst-case effect of a disturbance system on the H ₂ norm of the system is analyzed. An explicit expression is given for the worst-case H₂ norm when the disturbance system is allowed to vary over all nonlinear, time-varying and possibly noncausal systems with bounded L₂-induced operator norm. An upper bound for this measure, which is equal to the worst-case H₂ norm if the exogeneous input is scalar, is defined. Some further analysis of this upper bound is done, and a method to design controllers which minimize this upper bound over all robustly stabilizing controllers is given. The latter is done by relating this upper bound to a parameterized version of the auxiliary cost function studied in the literature     

H∞ optimal control as a weightedWiener-Hopf problem

It is shown that H_∞ optimization is equivalent to weighted H₂ optimization in the sense that the solution of the latter problem also solves the former. The weighting rational matrix that achieves this equivalence is explicitly computed in terms of a state-space realization. The authors do not suggest transforming H_∞ optimization problems to H₂ optimization problems as a computational approach. Rather, their results reveal an interesting connection between H_∞ and H₂ optimization problems which is expected to offer additional insight. For example, H₂ optimal controllers are known to have an optimal observer-full state feedback structure. The result obtained shows that the minimum entropy solution of H_∞ optimal control problems can be obtained as an H₂ optimal solution. Therefore, it can be expected that the corresponding H_∞ optimal controller has an optimal observer-full state feedback structure     

Decomposition of 2-D linear systems into 1-D systems in thefrequency domain

A method is presented for the decomposition of the frequency domain of 2-D linear systems into two equivalent 1-D systems having dynamics in different directions and connected by a feedback system. It is shown that under some assumptions the decomposition problem can be reduced to finding a realizable solution to the matrix polynomial equation X(z₁)P(z₂ )+Q(z₁)Y(z₂ )=D(z₁, z₂). A procedure for finding a realizable solution X(z₁ ), Y(z₂) to the equation is given     

M-dimensional Cayley-Hamilton theorem

The theorem states that every block square matrix satisfies its own m-D (m-dimensional, m⩾1) matrix characteristic polynomial. The exact statement and a simple proof of this theorem are given. The theorem refers to a matrix A subdivided into m blocks, and hence having dimension at least m. The conclusion is that every square matrix A with dimension M satisfies several m-D characteristic matrix polynomials with degrees N₁ . . ., N _m, such that N₁+ . . . +N_m ⩽M     

A linear programming approach for the weighted graph matchingproblem

A linear programming (LP) approach is proposed for the weighted graph matching problem. A linear program is obtained by formulating the graph matching problem in L₁ norm and then transforming the resulting quadratic optimization problem to a linear one. The linear program is solved using a simplex-based algorithm. Then, approximate 0-1 integer solutions are obtained by applying the Hungarian method on the real solutions of the linear program. The complexity of the proposed algorithm is polynomial time, and it is O(n ⁶L) for matching graphs of size n. The developed algorithm is compared to two other algorithms. One is based on an eigendecomposition approach and the other on a symmetric polynomial transform. Experimental results showed that the LP approach is superior in matching graphs than both other methods     

LQG control with an H∞ performance bound:a Riccati equation approach

An LQG (linear quadratic Gaussian) control-design problem involving a constraint on H^∞ disturbance attenuation is considered. The H^∞ performance constraint is embedded within the optimization process by replacing the covariance Lyapunov equation by a Riccati equation whose solution leads to an upper bound on L₂ performance. In contrast to the pair of separated Riccati equations of standard LQG theory, the H^∞-constrained gains are given by a coupled system of three modified Riccati equations. The coupling illustrates the breakdown of the separation principle for the H^∞ -constrained problem. Both full- and reduced-order design problems are considered with an H^∞ attenuation constraint involving both state and control variables. An algorithm is developed for the full-order design problem and illustrative numerical results are given     

A formal analysis of the fault-detecting ability of testing methods

Several relationships between software testing criteria, each induced by a relation between the corresponding multisets of subdomains, are examined. The authors discuss whether for each relation R and each pair of criteria, C₁ and C₂ , R(C₁, C₂) guarantees that C₁ is better at detecting faults than C₂ according to various probabilistic measures of fault-detecting ability. It is shown that the fact that C ₁ subsumes C₂ does not guarantee that C₁ is better at detecting faults. Relations that strengthen the subsumption relation and that have more bearing on fault-detecting ability are introduced     

Filtering and smoothing in an H∞ setting

The problems of filtering and smoothing are considered for linear systems in an H^∞ setting, i.e. the plant and measurement noises have bounded energies (are in L₂), but are otherwise arbitrary. Two distinct situations for the initial condition of the system are considered; the initial condition is assumed known in one case, while in the other the initial condition is not known but the initial condition, the plant, and measurement noise are in some weighted ball of RⁿXL₂. Finite-horizon and infinite-horizon cases are considered. Necessary and sufficient conditions are presented for the existence of estimators (both filters and smoothers) that achieve a prescribed performance bound, and algorithms that result in performance within the bounds are developed. In case of smoothers, the optimal smoother is also presented. The approach uses basic quadratic optimization theory in a time-domain setting, as a consequence of which both linear time-varying and time-invariant systems can be considered with equal ease. (In the smoothing problem, for linear time-varying systems, one considers only the finite-horizon case)     

H2-optimal sampled-data control

The H₂-optimal control of continuous-time linear time-invariant systems by sampled-data controllers is discussed. Two different solutions, state space and operator theoretic, are given. In both cases, the H₂ sampled-data problem is shown to be equivalent to a certain discrete-time H₂ problem. Other topics discussed include input-output stability of sampled-data systems, performance recovery in digital implementation of analog controllers, and sampled-data control of systems with the possibility of multiple-time delays     

Performance evaluation of dynamic sharing of processors intwo-stage parallel processing systems

The performance of job scheduling is studied in a large parallel processing system where a job is modeled as a concatenation of two stages which must be processed in sequence. P_i is the number of processors required by stage P as the total number of processors in the system. A large parallel computing system is considered where Max(P₁, P₂)⩾P≫1 and Max(P₁ , P₂)≫Min(P₁, P₂). For such systems, exact expressions for the mean system delay are obtained for various job models and disciplines. The results show that the priority should be given to jobs working on the stage which requires fewer processors. The large parallel system (i.e. P≫1) condition is then relaxed to obtain the mean system time for two job models when the priority is given to the second stage. Moreover, a scale-up rule is introduced to obtain the approximated delay performance when the system provides more processors than the maximum number of processors required by both stages (i.e. P>Max(P₁, P₂)). An approximation model is given for jobs with more than two stages     

The stability of a family of polynomials can be deduced from afinite number 0(k3) of frequency checks

Let φ(s,a)=φ₀(s,a)+ a₁φ₁(s)+a₂ φ₂(s)+ . . .+a_kφ _k(s)=φ₀(s)-q(s, a) be a family of real polynomials in s, with coefficients that depend linearly on parameters a_i which are confined in a k-dimensional hypercube Ω_a . Let φ₀(s) be stable of degree n and the φ_i(s) polynomials (i⩾1) of degree less than n. A Nyquist argument shows that the family φ(s) is stable if and only if the complex number φ₀(jω) lies outside the set of complex points -q(jω,Ω_a) for every real ω. In a previous paper (Automat. Contr. Conf., Atlanta, GA, 1988) the authors have shown that -q(jω,Ω_a ), the so-called `-q locus', is a 2k convex parpolygon. The regularity of this figure simplifies the stability test. In the present paper they again exploit this shape and show that to test for stability only a finite number of frequency checks need to be done; this number is polynomial in k, 0(k³), and these critical frequencies correspond to the real nonnegative roots of some polynomials     

Two properties of l1-optimal controllers

The solution of the l₁ sensitivity minimization problem is shown to have two properties which contrast markedly with properties of the solutions to the better-known H∞ sensitivity minimization problem or the LQG (linear quadratic Gaussian) problem is given of a one-parameter family of first-order plants where the order of the l₁-optimal compensator can be arbitrarily large, and thus it is impossible to bound the order of an l₁-optimal compensator in terms of the order of the plant. A plant is considered which has a continuous one-parameter family of l₁-optimal compensators, and thus l₁-optimal compensators need not be unique. The author's two examples are considered to answer two questions left open by M.A. Daleh and J.B. Pearson (ibid., vol.32, p.314-23, 1987)     

On Morse's new adaptive controller: parameter convergence andtransient performance

Some transient and asymptotic performance properties are established for the model reference adaptive controller proposed by A.S. Morse (Proc. US-Italy Joint Seminar Syst., Models Feedback 1992). It is proved that the L₂ norm of the tracking error is uniformly bounded by the initial parameter estimation error. Further, if the initial conditions are sufficiently small, it is shown that the L_∞ norm of the tracking error is uniformly bounded by the L_m2 norm of the reference signal. These transient bounds are independent of the signals richness and the adaptation gain, making them arguably the strongest transient results available in the literature. Second, excitation conditions for exponential stability, a property which is well known to insure some local performance measures, are given. To this end, it is shown that, modulo a signal dependent time scale change, Morse's estimator is equivalent to a normalized gradient plus filtering identifier     

Distributed detection with consulting sensors and communicationcost

The problem of distributed detection with consulting sensors in the presence of communication cost associated with any exchange of information (consultation) between sensors is considered. The system considered has two sensors, S1 and S2; S1 is the primary sensor responsible for the final decision u₀ , and S2 is a consulting sensor capable of relaying its decision u₂ to S1 when requested by S 1. The final decision u₀ is either based on the raw data available to S1 only, or, under certain request conditions, also takes into account the decision u₂ of sensor S2. Random and nonrandom request schemes are analyzed and numerical results are presented and compared for Gaussian and slow-fading Rayleigh channels. For each decision-making scheme, an associated optimization problem is formulated whose solution is shown to satisfy certain set design criteria that the authors consider essential for sensor fusion     

Mixed H2/H∞ control:a convex optimization approach

The problem of finding an internally stabilizing controller that minimizes a mixed H₂/H_∞ performance measure subject to an inequality constraint on the H_∞ norm of another closed-loop transfer function is considered. This problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint. Both the state-feedback and output-feedback problems are considered. It is shown that in the state-feedback case one can come arbitrarily close to the optimal (even over full information controllers) mixed H₂/H_∞ performance measure using constant gain state feedback. Moreover, the state-feedback problem can be converted into a convex optimization problem over a bounded subset of (n×n and n ×q, where n and q are, respectively, the state and input dimensions) real matrices. Using the central H_∞ estimator, it is shown that the output feedback problem can be reduced to a state-feedback problem. In this case, the dimension of the resulting controller does not exceed the dimension of the generalized plant     

An H∞ approach to feedback design withtwo objective functions

The problem of tightly bounding and shaping the frequency responses of two objective functions T_i(s)( i=1,2) associated with a closed-loop system is considered. It is proposed that an effective way of doing this is to minimize (or bound) the function max {∥T₁(s)∥ _∞, ∥T₂(s)∥_∞} subject to internal stability of the closed-loop system. The problem is formulated as an H^∞ control problem, and an iterative solution is given     

Damping margins of polynomials with perturbed coefficients

Considers the polynomial P(s)=t₀ Sⁿ+t₁ S^n-1 +···+t_n where 0<a _j⩽t_j⩽b_j. Recently, V.L. Kharitonov (1978) derived a necessary and sufficient condition for this polynomial to have only zeros in the open left-half plane. Two lemmas are derived to investigate the existence of theorems similar to the theorem of Kharitonov. Using these lemmas, the theorem of Kharitonov is generalized for P(s) to have only zeros within a sector in the complex plane. The aperiodic case is also considered