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 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.     

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.     

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.     

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.     

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.     

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.     

On the stability properties of polynomials with perturbed coefficients

Given a polynomialP_{c}(S) = S^{n} + t_{1}S^{n-1} + ... t_{n} = 0which is Hurwitz orP_{d}(Z) = Z^{n} + t_{1}Z^{n-1} + ... t_{n} = 0which has zeros only within or on the unit circle, it is of interest to know how much the coefficients tjcan be perturbed while preserving the stability properties. In this note, a method is presented for obtaining the largest hypersphere centered att^{T} = [t_{1} ... t_{n}]containing only polynomials which are stable.     

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.     

Comments on "Exact control of linear systems with multiple controls"

Given a linear systemdot{x} = Ax + Bu, whereAandBaren times nandn times mmatrices, withm leq nandBis of full rank, Farlow's theorem in the paper gives a necessary condition for the existence of coefficients fornmrectangular pulse functions that constitute the control vector, so that the state of the system is driven to zero in a finite given time. It is shown in this note that rank adj(A - s_{i}I), whereAhasndistinct eigenvaluess_{i}, i = 1, 2,..., n, is one. As a result, there are onlynlinearly independent equations in Farlow's condition for thenmcoefficients and as such onlynof the coefficients are determined uniquely. The othern(m -1)coefficients can be chosen to achieve other desirable characteristics of the system response.     

A new sufficient condition for generic pole assignment by output feedback

We show that for strictly proper systems withmoutputs,linputs (m geq l), McMillan degreen, and controllability indexeslambda_{1} geq lambda_{2} geq ...geq lambda_{1} >0, one can in the generic case arbitrarily assign min((q + 1)m + q + b(l - 1), n + q)closed-loop poles with a proper orderqcompensator. This represents an improvement over results reported earlier in the literature.     

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.     

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.     

Consistent estimation of system order

We consider a parameterized family{S_{alpha}, alpha in a}, a subset R^{infty}, of systems or sources having stochastic outputs{x_{n}}that are partially described by a statistic (e.g, correlation function)sigma_{alpha}(tau). If we representalpha=(alpha_{1},alpha_{2}...,alpha_{n}...), then by the system order M_{alpha} we mean the indexnof the last no nonzero term in the expansinn of α. Our objective is to generate a sequence{hat{M}_{n}(x_{1},... ,X_{n})}of estimates of theM_{alpha}^{0}that converge to it at least in probability. We provide conditions ensuring the existence of such a statistically consistent sequence of estimators, as wen as improved conditions yielding convergence in mean-square and with probability one. We establish existence by providing a method for constructing a family of consistent estimators of system order. We then apply our method to estimate the order of a scalar moving averages process and the order of a scalar autoregressive process. Our present results are primarily Of a theoretical nature, as we lack the efficiency and simulation studies desirable in support of a practical estimator of system order.     

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.     

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.     

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 order reduction of linear function observers

This note analyzes a new algorithm presented in [1] for designing a linear function observer with a minimum number of arbitrary poles. It shows that the maximum order of the observer of [1] isnu + ... + nu_{p} - pforp leq qandn - qforp geq q, instead ofp(nu_{1} - 1)as suggested in [1], wherenu{i}, i = 1top, are the descending ordered observability indexes of system (A, C) andn,p, andqare the order of the system, the number of the functions, and the number of the system outputs, respectively. This note also shows the significance of this result. For presentational purposes, only a special case of [1] is considered here. However, the technical properties as proved in this note are general.     

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.     

Generalization of Leverrier's algorithm to polynomial matrices of arbitrary degree

Leverrier's algorithm to find the determinant of (sI_{r}-A) is generalized to the polynomial matrices of arbitrary degree which are in the formI_{r}s^{n} + H_{n-1}s^{n-1} + ... + H_{1}s + H_{0}.