Invariance of the strict Hurwitz property for polynomials with perturbed coefficients

Given a strictly Hurwitz polynomialf(lambda) = lambda^{n} + a_{n-1} lambda^{n-1} + a_{n-2}lambda^{n-2}+...+ a_{1}lambda + a_{0}, it is of interest to know how much the coefficients aican be perturbed while simultaneously preserving the strict Hurwitz property. For systems withn leq 4, maximal intervals of the aiare given in a recent paper by Guiver and Bose [1]. In this note, a theorem of Kharitonov is exploited to obtain a general result for polynomials of any degree.     

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.     

A symplectic QR like algorithm for the solution of the real algebraic Riccati equation

A method is presented to solve the real algebraic Riccati equation -XNX + XA + A^{T}X + K = 0, whereK = K^{T}andN = N^{T}. The solution for the corresponding eigenvalue problemMx = lambda x, whereMis a Hamiltonian matrix, is computed by an algorithm similar to the QR algorithm. Special symplectic matrices are used for the transformation ofMsuch that the Hamiltonian form is preserved during the computations.     

On the minimum order of a robust servocompensator

In recent papers [1]-[4], Davison et al. have given the characterization of a minimal-order robust error-driven servocompensator which achieves asymptotic tracking and disturbance rejection. In this note, we establish this minimality property by frequency domain methods. We show that any right coprime factorization of the controller, sayN_{r}(S)D_{r}(S)^{-1}, must have all the elements ofD_{r}(s)divisible byPhi(s), the minimal polynomial of the tracking and disturbance signal generator. Hence, its order must be at leastn_{0}.d(phi)(n0=number of outputs,d(Phi)= degree ofPhi).     

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.     

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.     

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.     

An improved frequency time domain stability criterion for autonomous continuous systems

A sufficient condtion given for the asymptotic stability of a system having a single monotonic nonlinearity with slope confined to[0, k_{2}]and a transfer functionG(jomega), isRe(1 + X(jomega) + Y(jomega) + alphajomega)(G(jomega) + 1/k_{2}) geq 0wherealpha>0 , x(t)leq 0fort leq 0andx(t)=0fort>0 , y(t)leq0fort>0andy(t) = 0fort < 0, andintmin{-infty}max{infty}(| x(t)| + | y(t) | )dt < 1. The improvement consists of the addition of theX(jomega)term which corresponds to a nonzero time function fort<0, resulting inZ(jomega)multipliers whose phase angle is capable of varying from +90° to -90° any desired number of times. As is shown by examples, the new criterion gives better results than existing criteria. Also developed is an improved criterion for an odd monotonic nonlinearity.     

A homotopy method for eigenvalue assignment using decentralized state feedback

This paper addresses the following problem. Given an interconnected systemMcomposed ofNsubsystems of the formA_{i} + B_{i}K_{i},i = 1,..., N , (A_{i}, B_{i}), a controllable pair, and where the off diagonal blocks ofMlie in the image of the appropriate Bi, then is it possible to arbitrarily assign the characteristic polynomial ofMby a suitable selection of the characteristic polynomials ofA_{i} + B_{i}K_{i}? Moreover, is it possible to compute the appropriate characteristic polynomials of theA_{i} + B_{i}K_{i}(or equivalently construct the Ki) needed to do so? The first question is answered by constructing a mappingF: R^{n} rightarrow R^{n}which maps a prescribed set ofnof the feedback gains (elements ofK_{i}, i=1,...,N) to thencoefficients of the characteristic polynomial ofM. The question then becomes, given ap in R^{n}, doesF(x) = phave a solution? The answer is found by constructing a homotopyH: R^{n}x[O.1] rightarrow R^{n}whereH(x,1)= F(x)andH(x,0)is some "trivial" function. Degree theory is then applied to guarantee that there exists anx(t)such thatH(x(t), t) = pfor alltin [0,1]. The parameterized Sard's theorem is then utilized to prove that (with probability 1)x(t)is a "smooth" curve, and hence can be followed numerically fromx(0)tox(1)by the solution of a differential equation (Davidenko's method).     

Further study of the linear regulator with disturbances--The case of vector disturbances satisfying a linear differential equation

In a previous paper [1], the conventional optimal linear regulator theory was extended to accommodate the case of external input disturbancesomega(t)which are not directly measurable but which can be assumed to satisfyd^{m+1}omega(t)/dt^{m+1} = 0, i.e., represented asmth-degree polynomials in timetwith unknown coefficients. In this way, the optimal controlleru^{0}(t)was obtained as the sum of: 1) a linear combination of the state variablesx_{i}, i = 1,2,...,n, plus 2) a linear combination of the first(m + 1)time integrals of certain other linear combinations of the state variables. In the present paper, the results obtained in [1] are generalized to accommodate the case of unmeasurable disturbancesomega(t)which are known only to satisfy a givenrhoth-degree linear differential equationD: d^{rho}omega(t)/dt^{rho} + beta_{rho}d^{rho-1}omega(t)/dt^{rho-1}+...+beta_{2}domega/dt + beta_{1}omega=0where the coefficientsbeta_{i}, i = 1,...,rho, are known. By this means, a dynamical feedback controller is derived which will consistently maintain state regulationx(t) approx 0in the face of any and every external disturbance functionomega(t)which satisfies the given differential equationD-even steady-state periodic or unstable functionsomega(t). An essentially different method of deriving this result, based on stabilization theory, is also described, In each cases the results are extended to the case of vector control and vector disturbance.     

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.     

Conditions for a feedback transfer function matrix to be proper

Given rational matrixhat{G}(s)and a constant matrixK, necessary and sufficient conditions are given for the ration matrixhat{H}(s) = hat{G}(s) [I + Khat{G}(s)]^{-1}to be proper.     

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.     

Multivariable feedback, sensitivity, and decentralized control

In this paper the problem of sensitivity, reduction by feedback is studied and related to a problem of decentralized control. A plant will be represented by anN times Nmatrix of frequency responses, which may be unstable or irrational. The object will be to find conditions onP(s)under which a diagonal feedbackF(s)can make the sensitivityparallel{I + P(s)F(s)}^{-1}parallelarbitrarily small over some specified frequency interval [-jomega_{0}, jomega_{0}] without violating a global sensitivity, boundparallel{I+ P(s)F(s)}^{-1}parallel leq M, (Mgeqsome const. >1) forRe(s) geq 0. It will be shown that such a diagonal feedback of the "high gain" type can be constructed wheneverP^{-1}(s)is analytic inRe(s)geq 0, P(s)satisfies an attenuation condition nears = infty, andP(s)approaches diagonal dominance at high frequencies. It will also be shown that these conditions on the plant can be interpreted as conditions for the existence of a decentralized wide-band control scheme.     

A characterization of eAtand a constructive proof of the controllability criterion

A differential equation characterizing the functionsalpha_{i}(t), which arise when e^Atis expressed asalpha_{0}(t)I + ... + alpha_{n-1}(t) A^{n-1}, is derived. It is shown that the set of functions{alpha_{i}(t)}is linearly independent over any nonzero interval. Using this fact, a constructive proof is given for the well-known criterion for a linear time-invariant system to be controllable, namely, rank[B|AB| ... |A^{n-1}B] = n.     

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.     

Global transformations of nonlinear systems

Recent results have established necessary and sufficient conditions for a nonlinear system of the formdot{x}(t) = f(x(t))-u(t)g(x(t)). withf(0) = 0, to be locally equivalent in a neighborhood of the origin in Rⁿto a controllable linear system. We combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system. In doing so we introduce a technique for constructing a transformation under the assumptions that{gldot[fdotg],...,(ad^{n-1}fldotg)}span ann-dimensional space and that{gldot[fldot g],...,(ad^{n-2}fldotg)}is an involutive set.     

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.     

Optimal control of service in tandem queues

Customers arrive in a Poisson stream into a network consisting of twoM/M/1service stations in tandem. The service rateu in [0, a]at station 1 is to be selected as a function of the state (x_{1}, x_{2}) where xiis the number of customers at stationiso as to minimize the expected total discounted or average cost corresponding to the instantaneous costc_{1}x_{1} + c_{2}x_{2}. The optimal policy is of the formu=aoru=0according asx_{1} < S(x_{2}) or x_{1} geq S(X_{2})andSis a switching function. For the case of discounted cost, the optimal process can be nonergodic, but it is ergodic for the case of average cost.     

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)|.