A stable state-space realization in the formulation of H∞norm computation

In the two block H^inftyoptimization problem, usually we are given the state-space realizations of the proper rational matricesR_{1}(s)andR_{2}(s)whose poles are all the open right-half plane. Two problems are studied in the note. The first is the evaluation ofphi(s)R_{1}(s)ats = s_{k}, k = 1, 2, ..., n, wherephi(s)is an inner function whose zeros{s_{k}, k = 1, 2, ..., n }are the poles ofR_{1}(s). This evaluation is essential if Chang and Pearson's method is used for computing the optimal H^inftynorm. The problem is solved in state space via the solutions of Lyapunov equations. Neither polynomial matrix manipulations nor numerical pole-zero cancellations are involved in the evaluation. The second problem is to find a stable state-space realization ofS(s) = U(s)R_{2}(s)whereU(s)is an inner matrix. This problem arises in the spectral factorization ofgamma^{2} - R_{2}^{ast}R_{2}. Doyle and Chu had a method for constructing stableS(s)based on a minimal realization ofR_{2}(s). An alternate method is proposed. The alternate method does not require a minimal realization ofR_{2}(s)and only a Lyapunov equation is involved.     

A differential representation for multivariable linear systems with disturbances

It is shown how to compute a differential representation for a multivariable linear system with disturbancesdot{x}(t)=Ax(t)+Bu(t)+ W_{x}w(t)y(t)=Cx(t)+ Eu(t)+ W_{y}w(t). Explicit formulas forM_{y}(D)andM_{z}(D)in a differential equivalent representationP(D)z(t)=u(t)+M_{z}(D)w(t)y(t)=R(D)z(t)+M_{y}(D)w(t)are presented in this paper.     

The time-optimal design of linear dynamic systems

The following general time-optimal design problem is studied: determine a real constant square matrix,A, subject to specified constraints, to minimize the transit time between specified endpoints while satisfying the vector differential equationdot{x}(t) = Ax(t). Two specific kinds of constraints onAare considered: 1) where the individual elements ofAare free but the matrix as a whole must satisfyQ(A) leq hat{Q}whereQ(A)is a specified homogeneous function of the elements ofAandhat{Q}is a given upper bound and 2) where the elements ofAare individually bounded. Theoretical results show that both problems can be solved by first solving a related minimum cost fixed time problem. The latter problem is solved iteratively by using the generalized Newton-Raphson method for two point boundary value problems to provide a set of linear equations at each iteration. The cost functionQis then minimized subject to these equations using appropriate optimization techniques.     

On the span reachability of polynomial state-affine systems

The purpose of this report is to derive an explicit condition for the span reachability of a discrete polynomial state-affine system described byx(k+1)=(A_{0} +Sigmamin{i=1}max{r}u^{i}(k)A_{i})x(k)+ summin{i=1}max{r} u^{i}(k)B_{i}, (k=0,1,...)(1) whereris a positive integer,x in R^{n}, u in R^{1},u^{i}denotes the ith power ofu, and Aiand Biare matrices of appropriate dimensions. In order to define input sequences which can construct reachable state vectors from the origin to span the whole state space, a generalized type of the Vandermonde's matrix is newly defined and utilized fully. Although the algebraic structure of (1) is more complicated than discrete bilinear systems, the result turns out to be quite analogous to each other.     

Sufficient conditions for function space controllability and feedback stabilizability of linear retarded systems

New sufficient conditions for function space controllability and hence feedback stabilizability of linear retarded systems are presented. These conditions were obtained by treating the retarded systems as a special case of an abstract equation in Hilbert spaceR^{n}times L_{2}([- h, 0], R^{n})(denoted asM_{2}). For systems of typecdot{x}(t)=A_{0}x(t)+A_{1}x(t-h)+Bu(t), it is shown that most of controllability properties are described by a certain polynomial matrixP(lambda), whose columns can be generated by an algorithm comparingA_{0}^{i}B,A_{0}^{i} Band mixed powers of A0and A1multiplied byB.It is shown that the M2-approximate controllability of the system is guaranteed by certain triangularity properties ofP(lambda). By using the Luenberger canonical form, it is shown that the system is M2-approximately controllable if the pair(A_{1},B)is controllable and if each of the spaces spanned by columns of[B,A_{1}B,... ,A_{1}^{j}B], j=O...n-1, is invariant under transformation A0. Other conditions of this type are also given. Since the M2-approximate controllability implies controllability of all the eigenmodes of the system, the feedback stabilizability with an arbitrary exponential decay rate is guaranteed under hypotheses leading to M2-approximate controllability. Some examples are given.     

On the global controllability of perturbed controllable systems

By using a recent theorem of Davison and Kunze [1], it is shown that, if certain conditions hold such that the systemdot{x} = A(x,t)x + B(x,t)uis globally controllable, then the perturbed systemdot{x} = [A(x,t) + epsilontilde{A}(x,t)]x + [B(x,t) + epsilontilde{B}(x,t)u, wheretilde{A}andtilde{B}are bounded, is also globally controllable, provided ε is small enough. In particular, ifdot{x} = A(t)x + B(t)uis controllable, then so is the perturbed systemdot{x} = [A(t) + epsilontilde{A}(x,t)]x + [B(t) + epsilontilde{B}(x,t)]u.     

Transfer function conditions for stabilizability

For the linear time invariant systembigl[matrix {y_{c}(s)cr y_{m}(s)}bigr] = bigl[matrix{G(s)&H(s)cr M(s)&N(s)} bigr] bigl[matrix {u_{c}(s)cr u_{r}(s)}bigr]a necessary and sufficient condition based on the proper rational transfer function matrices{G, H, M, N}is given for the existence of a proper stabilizing controlleru_{c}(s) = C(s)y_{m}(s). The condition states that the McMillan degrees ofM^{+}(s)andbigl[matrix{G^{+}(s)&H^{+}(s)cr M{+}(s)&N{+}(s)}bigr]must be equal, where{G^{+}(s), H^{+}(s), M^{+}(s), N^{+}(s)}represents the unstable components of the partial fraction expansions of{G(s), H(s), M(s), N(s)}.     

On the identification of state-derivative-coupled systems

This short paper treats one aspect of the identification of state-derivative-coupled systems, such asMdot{x}(t) = Ax(t) + Bu(t) + w(t)whereM neq I, andMis invertible. This equation can also be written asdot{x}(t) = F_{1}x(t) + F_{2}u(t) + omega(t). We assume that reduced form parameters (F_{1}, F_{2}) are identifiable and develop a sequence of tests for establishing the identifiability of structural parameters (M, A, B) from (F_{1} F_{2}). The tests are constructive, in that they not only can be used to ascertain the identifiability of (M, A, B); but, if (M, A, B) are not identifiable, can also indicate corrective actions to be taken so that (M, A, B) are identifiable.     

On the structure of maximal(A, B)-Invariant subspaces: A polynomial matrix approach

Given a controllable and observable triple (A, B, C) describing a linear time invariant multivariable system Σ, which gives rise to a full rank transfer function matrixT_{o}(s), the structure of the maximal (A, B)- invariant subspace contained inker Cis investigated using a polynomial matrix approach. Thus, certain connections between the geometric and the polynomial matrix approaches to linear system theory are established.     

On the fuel-optimal singular control of nonlinear second-order systems

Given a body subject to quadratic drag forces so that the positiony(t)and the applied control thrustu(t)are related byddot{y}(t)+adot{y}(t)|dot{y}(t)| = u(t), |u(t)| leq 1, the controlu(t)is found which forces the body to a desired position, and stops it there, and which minimizes the costJ=intliminf{0} limsup{T}{k + |u(t)|}dt. The response timeTis not fixed,k > 0, and|u(t)|is proportional to the rate of flow of fuel. Repeated use of the necessary conditions provided by the Maximum Principle results in the optimum feedback system. It is shown that ifkleq 1, then singular controls exist and they are optimal; ifk > 1, then singular controls are not optimal. Techniques for the construction of the various switch curves are given, and extensions of the results to other nonlinear systems are discussed.     

Two singular value inequalities and their implications in H∞approach to control system design

In this note we prove that ifAandBare both nonnegative definite Hermitian matrices andA - Bis also nonnegative definite, then the singular values of A and B satisfy the inequalitiessigma_{i}(A)geq sigma_{i}(B), wherebar{sigma}(cdot) = sigma_{1}(cdot) geq sigma_{2}(cdot) geq '" geq sigma_{m}(cdot) = underbar{sigma}(.)denote the singular values of a matrix. A consequence of this property is that, in a nonsquare H^{infty} optimization problem, ifsup_{omega} bar{sigma}[Z(jsigma)] {underline{underline Delta}} sup_{omega} bar{sigma}[x(jomega)^{T}/ Y(jomega)^{T}]^{T} = lambda, then the singular values ofXandYsatisfy the inequalitylambda^{2} geq max_{i} sup_{omega} [sigma_{i}^{2}(X) + sigma_{m-i-1}^{2}(Y)]wheremis the number of columns of the matrixZ.     

Euclidean controllability of linear delay systems with limited controls

For linear delay systems with limited controls the notion of a proper control system is introduced. If the uncontrolled systemdot{x}(t)= Ax(t)+ Bx(t-h)is uniformly asymptotically stable and the control equationdot{x}(t)=Ax(t)+Bx(t-)+Cu(t)is proper, then the control system is Euclidean null-controllable. We note that controllability is equivalent to the system being proper for delay systems with unlimited power.     

Euclidean null-controllability of linear systems with distributed delays in control

J. Klamka derives a necessary and sufficient condition for the relative controllability of linear systems with distributed delays in control. In this paper the author discusses the Euclidean null-controllability of system described by the equationdot{x}(t) = A(t)x(t) + intliminf{-h}limsup{0} dH(t,s)u(t+s)satisfied almost everywhere on [t_{0},infty). If the systemdot{x}(t)=A(t)x(t)is uniformly asymptotically stable and the controlled systemdot{x}(t) = A(t)x(t) + intliminf{-h}limsup{0} dH(t,s)u(t+s)is proper then the system is null-controllable.     

Stabilization of uncertain systems via linear control

This note considers the problem of stabilizing a linear dynamical system (Σ) whose state equation includes a time-varying uncertain parameter vectorq(cdot). Given the dynamicsdot{x}(t)=A(q(t))x(t)+ B(q(t))u(t)and a bounding setQfor the valuesq(t), the objective is to choose a control lawu(t)=p(x(t))guaranteeing uniform asymptotic stability for all admissible variations ofq(cdot). Our results differ from previous work in one fundamental way; that is, we show that when working with linear controllers, it is possible to dispense with all assumptions onB(cdot)which have been made by previous authors (e.g., see [1]-[9]). This elimination of hypotheses onB(cdot)is accomplished roughly as follows: the system(Sigma) {underline {underline Delta}} (A(q), B(q))is shown to be equivalent to another system(Sigma^{+}) {underline {underline Delta}} (A^{+}(q), B^{+})as far as stabilization is concerned. SinceB^{+}is a constant matrix (independent ofq), the desired result is readily obtained.     

Disturbance decoupling problems via dynamic output feedback

This paper considers the disturbance decoupling problems, with or without internal stability and pole placement, via dynamic output feedback using polynomial and rational matrix techniques. We show that in all three problems considered, the central solvability condition can be expressed as a two-sided matching problemA = BXC, whereA, B, andCare the polynomial system matrices of certain natural subsystems of the system model andXis to be determined over various subrings of the rational functions. This matching problem can in turn be reduced to certain appropriate zero-cancellation conditions on the polynomial system matricesA, B, andC.     

Time-, fuel-, and energy-optimal control of nonlinear norm-invariant systems

Nonlinear systems of the formdot{X}(t)=g[x(t);t]+u(t), wherex(t), u(t), andg[x(t); t]arenvectors, are examined in this paper. It is shown that ifparellelx(t)parellel = sqrt{x_{1}^{2}(t) + ... + x_{n}^{2}(t)}is constant along trajectories of the homogeneous systemdot{X}(t)=g[x(t); t]and if the controlu(t)is constrained to lie within a sphere of radiusM, i.e.,parellelu(t}parellel leq M, for allt, then the controlu^{ast}(t)= - Mx(t} /parellelx(t)parelleldrives any initial statexito 0 in minimum time and with minimum fuel, where the consumed fuel is measured byint liminf{0} limsup{T}parellel u(t) parelleldt. Moreover, for a given response timeT, the controlutilde(t) = -parellelxiparellel x(t)/T parellel x(t) parelleldrivesxito 0 and minimizes the energy measured byfrac{1}{2}int liminf{0} limsup{T}parellelu(t)parellel^{2}dt. The theory is applied to the problem of reducing the angular velocities of a tumbling asymmetrical space body to zero.     

Stochastic approximation type methods for constrained systems: Algorithms and numerical results

A stochastic version of the standard nonlinear programming problem is considered. A functionf(x)is observed in the presence of noise, and we seek to minimizef(x)forx in C = {x:q^{i}(x) leq 0}, whereq^{i}(x)are constraints. Numerous practical examples exist. Algorithms are discussed for selecting a sequence Xnwhich converges wp 1 to a point where a necessary condition for optimality holds. The algorithms use, of course, noise-corrupted observations on thef(x). Numerical results are presented. They indicate that the approach is quite versatile, and can be a useful tool for systematic Monte-Carlo optimization of constrained systems, a much-neglected area. However, many practical problems remain to be resolved, e.g., investigation of efficient one-dimensional search methods and of the tradeoffs between the effort spent per search cycle and the number of search cycles.     

Componentwise asymptotic stability of linear constant dynamical systems

This note deals with a special type of asymptotic stability, namely componentwise asymptotic stability with respect to the vectorgamma(t)(CWASγ) of systemS: dot{x} = Ax + Bu, t geq 0, wheregamma(t) > 0(componentwise inequality) andgamma(t) rightarrow 0ast rightarrow + infty.Sis CWASγ if for eacht_{0} geq 0and for each|x(t_{0})| leq gamma (t_{0}) (|x (t_{0})|with the components|x_{i}(t_{0})|the free response ofSsatisfies|x(t)| leq gamma (t)for eacht geq t_{0}. Forgamma(t){underline { underline delta} } alphae^{-beta t}, t geq 0, withalpha > 0andbeta > 0(scalar), the CWEAS (E= exponential) may be defined.Sis CWAS γ (CWEAS) if and only ifdot{gamma}(t) geq bar{A}gamma(t), t geq 0 (bar{A}alpha < 0); A {underline { underline delta} } (a_{ij})andbar{A}has the elements aijand|a_{ij}|, i neq j. These results may be used in order to evaluate in a more detailed manner the dynamical behavior ofSas well as to stabilizeScomponentwise by a suitable linear state feedback.     

Stability of sampled-data composite systems with many nonlinearities

A sampled-data composite system given by a set of vector difference equationsx_{i}(tau + 1) - x_{i}(tau) = sum min{j = 1} max{n} A_{ij} f_{j}[x_{j}(tau)], i = 1 ..., nis dealt with. The system given byx_{i}(tau + 1) - x_{i}(tau) = A_{ij} f_{i}[x_{i}(tau)]is referred to as theith isolated subsystem. It is shown that the composite system is asymptotically stable in the large if the fisatisfy certain conditions and the leading principal minors of the determinant|b_{ij}|, i,j = 1, ..., n,are all positive. Here, the diagonal element biiis a positive number such that|x_{i}(tau + 1)| - |x_{i}(tau) | leq - b_{ij}| f_{i}[x_{i}(tau)]|holds with regard to the motion of theith isolated subsystem, and the nondiagonal elementb_{ij} , i neq j, is the minus of|A_{ij}|, which is defined as the maximum of|A_{ij}x_{j}|, for|x_{j}| = 1. Some extensions of this result are also given. Composite relay controlled systems are studied as examples.     

Finite state stochastic games: Existence theorems and computational procedures

Let{X_{n}}be a Markov process with finite state space and transition probabilitiesp_{ij}(u_{i}, v_{i})depending on uiandv_{i}.State 0 is the capture state (where the game ends;p_{oi} equiv delta_{oi});u = {u_{i}}andv = {v_{i}}are the pursuer and evader strategies, respectively, and are to be chosen so that capture is advanced or delayed and the costC_{i^{u,v}} = E[Sum_{0}^{infty} k (u(X_{n}), v(X_{n}), X_{n}) | X_{0} = i]is minimaxed (or maximined), wherek(alpha, beta, 0) equiv 0. The existence of a saddle point and optimal strategy pair or e-optimal strategy pair is considered under several conditions. Recursive schemes for computing the optimal or ε-optimal pairs are given.