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.     

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.     

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.     

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.     

Optimal observations for minimum variance filtering

Consider optimal filtering for a linear, discrete-time, dynamical system with scaler state xkand observation variance nk1at time tk. Then the error varianceP(k|j)corresponding to the minimum-variance estimate of xkgiven the observations through tj,k geq j, is a convex function of (n1. . .,nj) on the nonnegative orthant of Rⁱ.     

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).     

FFT calculation of a determinantal polynomial

An algorithm to find the coefficients of thes-polynomialD(s) = |H(s)|is obtained, whereH(s)is an arbitrarys-polynomial square matrix. The algorithm, based on the fast Fourier transform (FFT), is of an order of magnitude faster than existing methods.     

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)|.     

Remarks on the L/sup p/-input converging-state property

Let X /spl sub/ /spl Ropf//sup N/ and consider a system x/spl dot/ = f(x,u), f : X /spl times/ /spl Ropf//sup M/ /spl rarr/ /spl Ropf//sup N/, with the property that the associated autonomous system x/spl dot/ = f (x,0) has an asymptotically stable compactum C with region of attraction A. Assume that x is a solution of the former, defined on [0,/spl infin/), corresponding to an input function u. Assume further that, for each compact K /spl sub/ X, there exists k > 0 such that |f(z,v) - f(z,0)| /spl les/ k|v| for all (z,v) /spl isin/ /spl times/ /spl Ropf//sup M/. A simple proof is given of the following L/sup p/-input converging-state property: if u /spl isin/ L/sup p/ for some p /spl isin/ [1,/spl infin/) and x has an /spl omega/-limit point in A, then x approaches C.     

Singular systems: A new approach in the time domain

A new approach in the time domain is developed for the study of singular linear systems of the formEdot{x} = Ax + Bu, y = CxwithEsingular. Central to the approach is the fundamental triple((alpha E - A)^{-1}E, (alphaE - A)^{-1}B, C)with α a real number satisfying det(alpha E - A) neq 0. Controllability and observability matrices are expressed in terms of the fundamental triple. New tests for impulse controllability and impulse observability are also established. Feedback control problems including pole placement, decoupling, and disturbance localization are studied by use of a modified proportional and derivative feedback of the state in the form ofu = F(alpha x - dot{x})+ v.     

Filtering in discrete-time systems with state and observation noises correlated on a finite time interval

The aim of this note is to give recurrence equations for the state estimate and the error covariance of a linear discrete-time system whose state and observation noises arc correlated on a finite time interval. More precisely, we give the recurrence state estimation equations for the systemx(k + 1) = A (k)x(k) + C(k)w(k)(1)y(k) = B(k)x(k) + upsilon(k), k = 0, 1, 2, ..., (2) under the assumptions that noises w(k) and w(l) are uncorrelated for|k - l| > q, whilew(k)andupsilon(l)are uncorrelated forl - k < -sorl - k > t, whereq, s, tare known nonnegative integers.     

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 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.     

A counterexample on continuous coprime factors

In Georgiou and Smith (1992), the following question was raised: Consider a linear, shift-invariant system on L₂[0, ∞). Let the graph of the system have Fourier transform (MN)H² (i.e., the system has a transfer function P=N/M) where M, N are elements of C_A={f∈H^∞: f is continuous on the compactified right-half plane}. Is it possible to normalize M and N (i.e., to ensure |M|²+|N|²=1) in C_A? The author shows by example that this is not always possible     

Determination of the roots and of the global extremum of a lipschitz function

Conclusion The obvious deficiency of the method (1.3), (1.9) is the possible difficulty of the operation . In connection with this one can note that all the above given statements remain valid if the number is replaced by some positive lower bound of |f(t _k ,x)| on .In computational methods, the presence of the Lipschitz constant is considered as a deficiency. In connection with this we can note that the Lipschitz constant L can be replaced by any of its upper estimates. For example, for a differentiable function f(z) one can take .Translated from Kibernetika, No. 2, pp. 71–74, March–April, 1987.     

Finding Global Minima of Maximum Functions by Using Exclusion Functions without Derivatives

The first part of the paper introduces a new exclusion function without derivatives and describes its fundamental characteristics. The second part uses the new notion for finding global minimum of a real function of few variables g : [a, b] Rm R, g(x) = max{g1(x),...,gn(x)}, x [a, b], where g1(x),...,gn(x) are special multivariant real functions. The structure of the Maple program and some numerical examples are also presented in this part.     

Continuous-time optimal control theory for cost functionals including discrete state penalty terms

In this short paper, continuous-time optimal control problems are treated, where the cost functional includes additional discrete state penalty termsK_{j}(x(t_{j}))at specified timest_{j}(j = 1,..., N - 1)in the interior of the time interval[t_{0}, t_{N}]of the problem. Necessary conditions for a control to be optimal are given in the form of a minimum principle. The costate is necessarily discontinuous at the timest_{j}(j = 1,..., N - 1). Consequently, the optimal control is discontinuous at tj, in general. A numerical example is given.     

Vague故障诊断方法在振动故障诊断中的应用

提出模糊数据[hij]化成Vague数据[hij]的转化公式：[Ki(hj)=hij=[tij,1-fij]=(hij)2,(hij)1/2],以及Vague集[H]和[G]之间的相似度量公式：[Mm(H,G)=1ni=1n3-f(m)hi-f(m)gi-c(m)hi-c(m)gi-d(m)hi-d(m)gi3+f(m)hi-f(m)gi+c(m)hi-c(m)gi+d(m)hi-d(m)gi]。应用Vague故障诊断方法,进行汽轮发电机组的振动故障诊断,其效果是理想的。     

The stable regulator problem and its inverse

This paper provides a detailed survey and extension of certain properties of the stable regulator problem: determinemininf{u} intmin{0}max{infty} x'Qx + u'u dtsubject todot{x} = Fx + Gu; x(0) = x_{0}; liminf{t rightarrow infty} x(t) = 0whereQis not necessarily sign definite. First, equivalence conditions recently given by Willems for the existence of the minimum are extended to include statements in terms of the Hamiltonian matrix and spectral factorization. This provides a precise relation between the time-domain and frequency-domain solutions to the problems. Second, the inverse problem of whether a given feedbacku = -Kxis optimal for someQis easily resolved, as is the redundancy problem of distinct Q1and Q2, resulting in the same optimal control.