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.     

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.     

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.     

A note on the matrix equation A' LA-L = -K

Matrices WiandN_{i},which result from application of Krylov's algorithm to the matricesAandBrelated byA = (B + I)(B - I)^{-1},are shown to be row equivalent, i.e.,N_{i} = M_{i}W_{1}. This result is applied to solution of the Lyapunov matrix equation for discrete-time systems,A'LA - L = -K.     

An L∞error bound for the phase approximation problem

Given a random process spectral factorw(cdot), the phase approximation algorithm, initiated by Jonckheere and Helton [1], constructs a reduced-order spectral factorhat{w}(cdot)such thatparallel w/w^{ast}-hat{w}/ hat{w}^{ast}parallelis small in the Hankel-norm sense. In this note, we derive theL^{infty}error boundparallel w/w^{ast} - hat{w}/hat{w}^{ast}parallel_{infty} leq 4(sigma_{k+1} + ... +sigma_{N}), where the σ's are the canonical correlation coefficients. A similar result holds in the multivariable case.     

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.     

The stability of an nth-order nonlinear time-varying differential system

The stability of a system described by annth order differential equationy^{(n)} + a_{n-1}y^{(n-1)} + . . . + a_{1}y + a_{0} = 0wherea_{i}=a_{i}(t, y, dot{y}, . . . , y^{(n-1)}), i=0, 1, . . . , n - 1, is considered. It is shown that if the roots of the characteristic equation of the system are always contained in a circle on the complex plane with center(-z, 0), z > 0, and radius Ω such thatfrac{z}{Omega} > 1 + nC_{[n/2]}where[n/2]= nearest integergeq n/2andnC_{m} = n!/m!(n-m)!, wherenandmare integers, then the system is uniformly asymptotically stable in the sense of Liapunov.     

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.     

An algorithm to solve matrix equationsPH^{T} = GandP= φPφT+ γγT

An iterative algorithm is presented for the numerical solution of matrix equationsPH^{T} = GandP = PhiPPhi^{T} + GammaGamma^{T}, whereP geq 0andG, H, andPhiare given. These equations arise in various identification, network synthesis, and stability analysis problems.     

Direct solution method forA_{1}E +EA_{2}=-D

A direct method-a method without truncation or convergence errors-for the solution ofA_{1} E + EA_{2} = - D, whereA_{1} in R^{n_{1} times n_{1}}andA_{2} in R^{n_{2} times n_{2}}, is described. The only assumption on the matrices A1and A2is that the spectra of A1and-A_{2}be disjoint. The method requires storage for order2n_{1}^{2}+3n_{1}n_{2}+2n_{2}^{2}variables and requires ordern{1}(n_{1}^{2}+ n_{1}n_{2} + n_{2}^{2})n_{2}multiplications and divisions.     

Cancellations in multivariable continuous-time and discrete-time feedback systems treated by greatest common divisor extraction

This correspondence considers a multivariable system with proper rational matrix transfer functions G0and Gfin the forward and feedback branches, respectively. It develops a strictly algebraic procedure to obtain polynomials whose zeros are the poles of the matrix transfer functions from input to output (Hy), and from input to error (He). G0and Gfare given in the polynomial matrix factored formN_{0}D_{0}^{-1}andD_{f}^{-1}N_{f}. The role of the assumption det [I + G_{f}(infty)G_{0}(infty)] neq 0and the relation between the zeros of det [I + G_{f}G_{0}] and the poles of Hyand Heare indicated. The implications for stability analysis of continuous-time as well as discrete-time systems are stressed.     

Finite series solutions for the transition matrix and the covariance of a time-invariant system

The transition matrixvarphicorresponding to then-dimensional matrixAcan be represented byvarphi(t) = g_{1}(t)I + g_{2}(t)A + ... + g_{n}(t)A^{n-1}, where the vectorg^{T} = (g_{1}, ... , g_{n})is generated fromdot{g}^{T} = g^{T}A_{c}, g^{T}(0) = (1, 0, ... , 0)and Acis the companion matrix toA. The result is applied to the covariance differential equationdot{C} = AC + CA^{T} + Qand its solution is written as a finite series. The equations are presented in a form amenable for implementation on a digital computer.     

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.     

Improving the predictions of the circle criterion by combining quadratic forms

By using a Lyapunov function which consists of different quadratic forms in various sectors of the (u, (du/dtau)) plane, the prediction of the circle criterion that the null solution of(d^{2}u/dtau^{2}) + 2(du/dtau) + f(tau, u, (du/dtau))cdotp u = 0is asymptotically stable for0 leq alpha < f(cdotp) < beta, withbeta = (sqrt{alpha} + 2)^{2}, is improved tobeta = [{frac{(sqrt{alpha} + 1)^{2} + 1 + sqrt{(sqrt{alpha} + 1)^{4} + 2 (sqrt{alpha} + 1)^{2} + 5}}{2}}^{frac{1}{2}} + 1 ]^{2}.     

Superposition laws for solutions of differential matrix Riccati equations arising in control theory

Representation formulas are given for the general solution of theN times Nmatrix Riccati equationdot{W} = A + WB + CW + WDWusingnknown solutions, withn = 1, ..., 5(n-representations). The 5- representation is a superposition formula, in that it expresses the general solution explicitly as a function of five particular solutions and N²arbitrary constants (N geq 2), using no further information. The representation formulas can be used in numerical calculations. The 4- and 5- representations are specially useful when a solutionW(t)has a singularity for some finitet = t_{0}. They also clarify the properties of the solution space: the matrix elements ofW(t)are meromorphic functions ofthaving simple poles as the only possible singularities. The relation between the representation formulas and previously known results is discussed.     

Minimum-energy control of a second-order nonlinear system

This paper establishes the bounded control functionu(t)which minimizes the total energy expended by a submerged vehicle (for propulsion and hotel load) in a rectilinear translation with arbitrary initial velocity, arbitrary displacement, and zero final velocity. The motion of the vehicle is determined by the nonlinear differential equationddot{x}+adot{x}|dot{x}| = u, a > 0. The performance index to be minimized is given byS =int_{0}^{T}(k+udot{x})dt, withTopen andk > 0.The analysis is accomplished with the use of the Pontryagin maximum principle. It is established that singular controls can result whenk leq 2 sqrt{U^{3}/a}.Uis the maximum value of|u(t)|.     

Maximum-likelihood estimation of a process with random transitions (failures)

A process with random transitions is represented by the difference equationx_{n} = x_{n-1}+ u_{n}where unis a nonlinear function of a Gaussian sequence w_{n}. The nonlinear function has a threshold such thatu_{n} =0for|w_{n}| leq W. This results in a finite probability of no failure at every step. Maximum likelihood estimation of the sequenceX_{n}={x_{0},...,x_{n}}given a sequence of observationsY_{n} = { y_{1},...,y_{n} }gives rise to a two-point boundary value (TPBV) problem, the solution of which is suggested by the analogy with a nonlinear electrical ladder network. Examples comparing the nonlinear filter that gives an approximate solution of the TPBV problem with a linear recursive filter are given, and show the advantages of the former. Directions for further investigation of the method are indicated.     

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.