Stability of polynomials under coefficient perturbation

Let the real polynomiala_{0}x^{n} + a_{1}x^{n-1} + ... + a_{n-1}x + a_{n}be stable and let the real numbersb_{k}, c_{k} geq 0, 0 leq k leq n, be given. We present a simple determinant criterion for finding the largestt_{0} geq 0such that the polynomialalpha_{0}x^{n} + alpha_{1}x^{n-1}+ ... +alpha_{n-1}x + alpha_{n}is stable for allalpha_{k} in (a_{k} - b_{k}t_{0}, a_{k} + C_{k}t_{0}) cup {a_{k}}, 0 leq k leq n. Several further observations allow us to reduce the computational cost considerably.     

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.     

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.     

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.     

Stability conditions for annth-order nonlinear time-varying differential system

The stability of a system described by thenth-order differential equationy^(n) + a_{n-1}Y^(n-1) + ... + a_{1}dot{Y} + a_{0}y = 0wherea_{i} = a_{i}(t, y, dot{y}, ... , y^(n-1)),i = 0,1,2, ... , n-1is considered. It is shown that if the instantaneous roots of the characteristic equation of the system are always contained in a circle on the complex plane with center (- z, 0),z > 0and radius ω such thatfrac{z}{Omega} > {{1, n = 1}{sqrt{2n(n-1)}, n geq 2}then the system is uniformly asymptotically stable in the sense of Liapunov.     

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.     

Instability of slowly varying systems

Instability criteria are obtained for systems described bydot{x} = A(t)xwhen the parameters are slowly varying. In particular it is shown that, whenA(t)has eigenvalues in the right-half plane and all eigenvalues are bounded away from the imaginary axis, then ifsup_{t geq 0} parallel dot{A}(t)parallelis sufficiently small, the system has unbounded solutions. Results are also given for systems of the formdot{x} = A(t)x + f(x, t), and the dichotomy of solutions is studied in both the linear and nonlinear cases.     

Optimum control of linear systems with a "Modified" energy constraint

Linear control processes are considered under the following optimization criteria: 1) minimization of the terminal error and 2) minimization of the required time (T) to reach a desirable state. The constraint on the control vector(u(t))is considered to beintliminf{t_{j}} limsup{t_{j+1}} u'(t)Qu(t)dt leq c_{j} j = 1, 2, ... mwhereQis a positive definite matrix, andt_{j} < t represents any given interval contained in the interval0, and cjcan be considered as the available energy during that interval. A condition of optimality has been obtained which can be used analytically. Furthermore, a numerical procedure is developed for determination of the optimum control vector.     

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)|.     

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.     

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.     

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.     

Zeros of f(z) = (az-b)npm (cz-d)n

The zeros off(z) = (az - b)^{n} pm (cz - d)^{n}are found to lie on a circle of radius|(ad - cb)/(|a|^{2} - |c|^{2})|with its center atz = (a^{ast}b - c^{ast}d)/(|a|^{2} - |c|^{2}), wherea, b, c, anddare complex numbers andnis assumed real. When|a| = |c|the locus of the zeros is a straight line perpendicular to the line joining the pointsb/aandb/cand intersecting it atz = 0.5(b/a + d/c). The zeros are found analytically and constructed geometrically.     

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).     

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.     

Application of the describing function technique in a single-loop feedback system with two nonlinearities

A graphic procedure is presented which allows the describing function technique to be extended to a single-loop feedback system with two nonlinearities. The graphic technique is very simple and immediately allows qualitative answers, or quantitative answers subject to the usual errors and restrictions of the describing function technique, to be obtained regarding the presence of limit cycles, regions of stability, instability, etc. The method essentially is as follows. A plot ofG_{1}(jomega) G_{2}(j_omega)in Fig. 1 vs. ω is made, and the point of intersection ofG_{1}(jomega) G_{2}(jomega)with the negative real axis is noted, for example, atG_{1}(jomega^{ast}) G_{2}(jomega^{ast}) =-1/Gamma, Gamma > 0. By plotting |G_{d_{1}}(A1)| vs. A1in the second quadrant, and|G_{d_{2}}(A_{2})|vs. A2in the fourth quadrant, it is possible to plot a curve (relating |G_{d_{1}}| vs. |G_{d_{1}}|) in the first quadrant. If this curve intersects|G_{d_{1}}| |G_{d_{2}}| = Gamma, a limit cycle exists in the system. If no intersection takes place, then no limit cycle exists in the system.     

Sensitivities of stability constraints and their applications

Let the real polynomial(a(s) = a_{0} + a_{1}s + ... + a_{n}s^{n}with the coefficients being known differentiable functionsa_{k}(x)be given and let the constraintsg_{i}(x) > 0determine the strictly Hurwitz property of the polynomiala(s). A simple and efficient method to calculate the derivativespartial g_{i}(x)/partial x_{j}is proposed. Then, the application of the method to the problem of stability of polynomials under coefficient perturbation by gradient optimization is discussed. Also, a theorem characterizing the stability region and the newly introduced regions of nondestabilizing perturbations is given.     

Robust solutions of perturbed linear equations

A perturbed system of linear equalitieslangle a_{i},x rangle = b_{i}, i = 1,2,...,n;a_{i} inA_{i};b_{i},inB_{i};xinX(the sets Aiand the intervals Biprescribed a priori) is said to be robust if a solution vectorx_{0}inXcan be found resulting inlangle a_{i},x_{0}rangle in B_{i}for alla_{i} inA_{i}and alli = 1, 2,...,n. A numerical "test for robustness" is developed. This test is seen to involve 2n parameters at most-even when the solution setXis an infinite-dimensional vector space.     

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.     

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.