Simultaneous controller design for linear time-invariant systems

The use of generalized sampled-data hold functions (GSHF) in the problem of simultaneous controller design for linear time-invariant plants is discussed. This problem can be stated as follows: given plants P₁, P₂, . . ., P_N , find a controller C which achieves not only simultaneous stability, but also simultaneous optimal performance in the N given systems. By this, it is meant that C must optimize an overall cost function reflecting the closed-loop performance of each plant when it is regulated by C. The problem is solved in three aspects: simultaneous stabilization, simultaneous optimal quadratic performance, and simultaneous pole assignment in combination with simultaneous intersampling performance     

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     

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     

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     

Provable improvements on branch testing

This paper compares the fault-detecting ability of several software test data adequacy criteria. It has previously been shown that if C₁ properly covers C₂, then C₁ is guaranteed to be better at detecting faults than C₂, in the following sense: a test suite selected by independent random selection of one test case from each subdomain induced by C₁ is at least as likely to detect a fault as a test suite similarly selected using C₂. In contrast, if C₁ subsumes but does not properly cover C₂, this is not necessarily the case. These results are used to compare a number of criteria, including several that have been proposed as stronger alternatives to branch testing. We compare the relative fault-detecting ability of data flow testing, mutation testing, and the condition-coverage techniques, to branch testing, showing that most of the criteria examined are guaranteed to be better than branch testing according to two probabilistic measures. We also show that there are criteria that can sometimes be poorer at detecting faults than substantially less expensive criteria     

A qualitative bound in robustness of stabilization by statefeedback

It is proved that placing the poles of a linear time-invariant system arbitrarily far to the left of the imaginary axis is not possible if small perturbations in the model coefficients are taken into account. Given a nominal controllable system (A₀, B ₀) with one input and at least two states and an open ball around B₀ (no matter how small), there exists a real number γ and a perturbation B within that ball such that for any feedback matrix K placing the eigenvalues of A ₀+B₀K to the left of Res=γ, there is an eigenvalue of A₀+BK with real part not less than γ     

A note on a classical bound for the moduli of all zeros of apolynomial

Considers the monic polynomial f(z):=z ⁿ+a_n-1z^n-1+. . .+a₀ in the complex variable z with complex coefficients. Under the assumption that the nonleading coefficients of f lie in the disk |z|⩽A the authors give an estimate for the smallest disk |z|⩽R containing all zeros of f. The estimate has a guaranteed precision of a few percent. They proceed similarly to obtain a zero-free disk |z |⩽r     

Adaptive structuring of binary search trees using conditionalrotations

Consider a set A={A₁,A₂ ,. . ., A_n} of records, where each record is identified by a unique key. The records are accessed based on a set of access probabilities S=[s₁,s₂ ,. . ., s_N] and are to be arranged lexicographically using a binary search tree (BST). If S is known a priori, it is well known that an optimal BST may be constructed using A and S. The case when S is not known a priori is considered. A new restructuring heuristic is introduced that requires three extra integer memory locations per record. In this scheme, the restructuring is performed only if it decreases the weighted path length (WPL) of the overall resultant tree. An optimized version of the latter method, which requires only one extra integer field per record has, is presented. Initial simulation results comparing this algorithm with various other static and dynamic schemes indicates that this scheme asymptotically produces trees which are an order of magnitude closer to the optimal one than those produced by many of the other BST schemes reported in the literature     

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     

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     

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     

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     

Bicoprime factorizations of the plant and their relation to right and left-coprime factorizations

In a general algebraic framework, starting with a bicoprime factorization P=N_prD^-1 N_pl, a right-coprime factorization N_p D_p^-1, a left-coprime factorization D^-1_pN_p, and the generalized Bezout identities associated with the pairs (N_p, D_p) and (D˜ _p, N˜_p) are obtained. The set of all H-stabilizing compensators for P in the unity-feedback configuration S(P, C) are expressed in terms of (N_pr, D, N _pt) and the elements of the Bezout identity. The state-space representation P=C(sI-A)^-1B is included as an example     

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     

Solution to Morgan's problem

A necessary and sufficient condition is presented for the solution of the row-by-row decoupling problem (known as Morgan's problem) in the general case, that is, without any restrictive assumption added to the system to the feedback law u=Fx+Gy (G may be noninvertible). This is a structural condition in terms of invariant lists of integers which are easily computable from a given state realization of the system. These integers are the infinite zero orders (Morse's list I₄) and the essential orders of the system, which only depend on the input-output behavior, and Morse's list I₂ of the system, which depends on the choice of a particular state realization     

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     

Efficient algorithms for selection of recovery points in tree taskmodels

Efficient solutions to the problem of optimally selecting recovery points are developed. The solutions are intended for models of computation in which task precedence has a tree structure and a task may fail due to the presence of faults. An algorithm to minimize the expected computation time of the task system under a uniprocessor environment has been developed for the binary tree model. The algorithm has time complexity of O(N₂), where N is the number of tasks, while previously reported procedures have exponential time requirements. The results are generalized for an arbitrary tree model     

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     

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     

Damping ratio of polynomials with perturbed coefficients

Let a family of polynomials be P(s)=t ₀Sⁿ+t₁s ^n-1 . . .+t_n where O<α _j⩽t_j⩽β. Recently, C.B. Soh and C.S. Berger have shown that a necessary and sufficient condition for this equation to have a damping ratio of φ is that the 2ⁿ⁺¹ polynomials in it which have t_k=α_k or t_k=β_k have a damping ratio of φ. The authors derive a more powerful result requiring only eight polynomials to be Hurwitz for the equation to have a damping ratio of φ using Kharitonov's theorem for complex polynomials