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.     

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)|.     

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.     

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.     

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.     

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.     

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.     

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.     

Stabilizability of the systemẋ(t)=Fx(t)+Gu(t-h)

Stabilizability problem for the systemdot{x}(t)= Fx(t) + Gu(t - h)is considered. For appropriate discrete modelx_{k+1} = Ax_{k} + Bu_{k-1}the feedback controller which has the formu_{k} =Sigmamin{i=0}max{l}F_{i}x_{k-i}is proposed. It is proven that controllability of the pair (A,B) and cyclicity of theAmatrix imply stabilizability. Some extensions and applications are also mentioned.     

A minimum principle for a class of discrete-time stochastic systems

A minimum principle is obtained for discrete-time stochastic systems described by the stochastic difference equationx_{k+1} = A_{k}x_{k} + phi_{k}(u_{k})+w_{k}where{w_{k}, k = 0, ... ,N - }is la sequence of independent random vector variables. The control action ukis constrained to belong to a compact set Uk, and the setphi_{k}(U_{k}), k = 0,..., N - 1is convex. The system is open-loop.     

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.     

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.     

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.     

Exact control of linear systems with multiple controls

It is shown how the linear control systemdot{x} = Ax + BuwhereAandBaren times nandn times mmatrices, respectively, can be driven from an arbitrary initial conditionx(0) = x_{0} in E^{n}to an arbitrary final conditionx(T) = x_{f} in E^{n}by a vectoru = (u_{i})of piecewise constant controlsu_{i}(t)in an arbitrary time periodT. A simple algorithm for findinguis given along with a worked example.     

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.     

Almost Periodic Solution for Shunting Inhibitory Cellular Neural Networks with Time-varying Delays and Variable Coefficients

In this paper, a class of two-dimensional shunting inhibitory cellular neural networks with time-varying delays and variable coefficients as following system

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.     

A synthesis theory for linear time-varying feedback systems with plant uncertainty

Given a feedback system containing a linear, time-varying (LTV) plant with significant plant uncertainty, it is required that the system response to command and disturbance inputs satisfy specified tolerances over the range of plant uncertainty. The synthesis procedure guarantees the latter satisfied, providing that they are of the following form. Leth(t',tau)be the system response att'= t - taudue to a command inputdelta(t - tau), andh_{tau}(s)=int liminf{0}limsup{infty}h(t',tau)e^-{st'}dt'is the Laplace transform ofh(t',tau). There is given a setM_{tau}(omega)={m_{tau}(omega)} , omega in[0, infty), with the requirement that|h_{tau}(jomega)| in M_{tau}(omega), over the range of plant uncertainty. The disturbance response tolerances are of the same form, in response to a disturbance inputdelta (t- tau). The acceptable response setM_{tau}(omega)can depend on τ. The design emerges with a fixed pair of LTV compensation networks and can be considered applicable to time-domain response tolerances, to the extent that a set of bounds on a time function can be translated into an equivalent set on its frequency response. The design procedure utilizes only time-invariant frequency response concepts and is conceptually easy to follow and implement. At any fixed τ, the time-varying system is converted into an equivalent time-invariant one with plant uncertainty, for which an exact solution is available, with "frozen" time-invariant compensation. Schauder's fixed-point theorem is used to prove the equivalence of the two systems. The ensemble over τ of the time-invariant compensation gives the final required LTV compensation. It is proven that the design is stable and nonresonant for all bounded inputs.     

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}.