Showing papers on "Lyapunov equation published in 1979"
TL;DR: In this article, a class of dynamical systems in the presence of uncertainty is formulated by contingent differential equations, and the uncertainty is deterministic; the only assumption is that its value belongs to a known compact set.
Abstract: A class of dynamical systems in the presence of uncertainty is formulated by contingent differential equations. Asymptotic stability (in the sense of Lyapunov) is then developed via generalized dynamical systems (GDS's). The uncertainty is deterministic; the only assumption is that its value belongs to a known compact set. Application to variable structure and model reference control systems are discussed.
594 citations
TL;DR: In this paper, a simple constructive procedure of Berkhout, based on the backwards Levinson algorithm is discussed and an application of the result in stochastic control is mentioned, where the solution to the discrete-time Lyapunov matrix equation in controllable canonical form is shown to be the inverse of the Schur-Cohn matrix.
Abstract: The solution to the discrete-time Lyapunov matrix equation in controllable canonical form is shown to be the inverse of the Schur-Cohn matrix. A simple constructive procedure of Berkhout, based on the backwards Levinson algorithm is discussed and an application of the result in stochastic control is mentioned.
41 citations
TL;DR: A new lower bound on the quadratic cost of linear regulators is derived using some recently established results on norm bounds for the algebraic matrix Riccati and Lyapunov equations.
Abstract: A new lower bound on the quadratic cost of linear regulators is derived using some recently established results on norm bounds for the algebraic matrix Riccati and Lyapunov equations. The bound thus obtained is very attractive computationally and is independent of the initial conditions.
20 citations
TL;DR: It is shown that it is always possible to achieve a lower bound, while the upper bound can be obtained for a specified class of Q matrices which is given.
Abstract: In this paper, new bounds are given for the matrix solution of the Lyapunov equation A'P+ PA+Q=0 . It is shown that it is always possible to achieve a lower bound, while the upper bound can be obtained for a specified class of Q matrices which is given. The results are compared to those given in [1] through some examples.
17 citations
01 Jan 1979
11 citations
01 Dec 1979
TL;DR: In this article, a representation of the solution matrix P to the Lyapunov matrix equation PF + F'P = -LL' is derived, which is a proper subset of the generalized positive real matrices defined by Anderson and Moore.
Abstract: In this paper, a representation of the solution matrix P to the Lyapunov matrix equation PF + F'P = -LL' is derived. We consider the class of m×m matrices Z(?) of real rational functions of a complex variable s, bounded at s = ?, with Z (jw) + Z' (-j?) equal to a nonnegative definite Hermitian matrix for all real ?, and with ? + µ ? 0 for all poles, not necessarily distinct, of Z(s). This last condition is imposed because (1) has a unique solution if and only if ? + µ ? 0 for all eigenvalues of the matrix F. This means that the class {z(s)} is a proper subset of the generalized positive real matrices defined by Anderson and Moore.
8 citations
TL;DR: A,Q∈Rn×n(Cn×N) as mentioned in this paper, A,Q ∈Rnn×n, Qは 正 定 対称 行 列, Aは ǫ(A)<1)
Abstract: た だ し, A,Q∈Rn×n, Qは 正 定 対称 行 列, Aは 正 則 か つ|λi(A)|<1と す る. の 解 行 列Pの 適 当 な 尺度 に よ る “木 き さ”の 限 界 を, Q,Aな ど の “大 き さ” に よ り評 価 した もの で あ る. 連続 形 の リア プ ノ フ方程 式 に関 し て は, 同 様 の 趣 旨 の 報 告 がす で に い くつ か公 表 さ れて い る1)~5). 本 報 告 の 目 的 は, 離 散形 方 程 式 に対 して も連 続 形 方 程 式 に 対 す る結 果 と並 行 した 評 価 式 を得 て お くこ とに あ る. 以 下 で は, P,Qの 固 有 根 を順 に そ れ ぞ れ, 0<α1≦ ...≦ αn, 0<β1≦... ≦βnと し, 行 列X∈Rn×n(Cn×n)に 対 し て ||X||.で,・=1,2,∞な どの 適 当 な 行 列 ノ ル ム を 示 し, μ(X)を||X||. よ り 導 か れ た 行 列 メ ジ ャ ー7)と して u.[X]を 次 式 の よ うに定 め る. μ.[X]=max{-μ.(-X),-μ(X)} ま た, 行 列Aの ス ペ ク トル 半 径 を ρ(A)で 表 わ せ ば, ρ(A)<1で あ る.
7 citations
TL;DR: In this paper, the stability of difference equations, which represent discrete-time motions, is studied on general metric spaces and an analogous theorem to the equivalence between the asymptotic stability of invariant sets and the existence of Lyapunov functions for continuous time motions is proved.
Abstract: The stability of difference equations, which represent discrete-time motions, is studied on general metric spaces. An analogous theorem to the equivalence between the asymptotic stability of invariant sets and the existence of Lyapunov functions for continuous-time motions is proved. One consequence of the results is the reduction of the asymptotic stability to an invariance condition.
6 citations
Journal Article•
4 citations
TL;DR: In this article, a treatment of positive semidefinite solutions to the Lyapunov equation (1) in a complex Hilbert space S which in general is infinite dimensional is presented.
3 citations
TL;DR: The discrete-time Lyapunov matrix equation A ′ Q A - Q = - R is considered and fundamental inequalities declaring some extremal properties of the solution are proposed.
Abstract: The discrete-time Lyapunov matrix equation A ′ Q A - Q = - R is considered. Fundamental inequalities declaring some extremal properties of the solution are proposed. Lower bounds for the minimum and maximum eigenvalues of Q in terms of c A,R⩽ are established. Upper bounds are also attained under some conditions.
31 Aug 1979
TL;DR: In this paper, state feedback regulators are derived which have better robustness characteristics than the standard 60 deg phase margin and 50% gain reduction tolerance of standard linear- quadratic regulators.
Abstract: : State feedback regulators are derived which have better robustness characteristics than the standard 60 deg phase margin and 50% gain reduction tolerance of standard linear- quadratic regulators It is also shown how the Lyapunov equation can be used to design high-integrity regulators for open-loop stable systems (Author)
TL;DR: In this paper, the authors investigated stability, uniform stability and equi-asymptotic stability with respect to the x-components and y-component of a differential equation with time delay.
Abstract: We investigate stability, uniform stability and equi-asymptotic stability with respect to the x-components and y-components of a differential equation with time delay. We also obtain necessary and sufficient conditions for the generalized asymptotic stability of the exponential type with respect to the components which generalizes the work of Corduneanu[3]. We make use of Lyapunov functionals and differential inequalities in our study.
TL;DR: In this article, a sufficient condition of null controllability of a nonlinear system was established using the theory of cones, which is similar to the condition of stability of the nonlinear systems in this paper.
Abstract: Cone-valued Lyapunov functions have been used by several authors to examine the stability of nonlinear systems. However, the general problem of constructing a suitable cone given the nature of nonlinearity seems to be as difficult as that of constructing a suitable Lyapunov function. The construction of a suitable cone, at least in special cases, deserves further investigation. Lakshmikantham [1] has used the theory of cones for stability of nonlinear systems. A more unified theory of cones can be found in the book by Krasnosel'skii [2]. The objective of this paper is to use methods of the theory of cones to establish a sufficient condition of null controllability of \dot{x}(t)= A(t)+ g(t,x,u) .
TL;DR: Using the direct Lyapunov method for distributed systems, the problem of stability of particlelike solutions of the equation of a scalar field with logarithmic nolinearity is solved.
Abstract: Using the direct Lyapunov method for distributed systems, the problem of stability of particlelike solutions of the equation of a scalar field with logarithmic nolinearity is solved. It is shown that nonlinearities of the Heaviside function type, which ensure the existence of exact regular solutions for the Klein-Gordon equation, are not a very fruitful approach because of the mathematical difficulties encountered in studying the stability of such solutions.
TL;DR: The design of structures to a given performance or response index while ensuring certain dynamic stability properties may be based on Lyapunov's second or direct method as mentioned in this paper, where stiffness/damping/mass requirements may be optimally chosen according to an integral square error (or deviation) criterion.
Abstract: The design of structures to a given performance or response index while ensuring certain dynamic stability properties may be based on Lyapunov's second or direct method. For structures subject to initial disturbances and modelled in state equation form, the stiffness/damping/mass requirements may be optimally chosen according to an integral square error (or deviation) criterion.
TL;DR: It is shown that the proposed adaptive algorithms for the discrete adaptive observer based on the Lyapunov's direct method are sufficiently valid where observation noises are present and that they are identical with the adaptive algorithms of a noise-free system.
Abstract: In spite of the large number of designing methods for an adaptive observer proposed by the researchers of stability theorem, the study of an adaptive observer with observation noises has apparently not been published to date. In this paper, a designing method for the discrete adaptive observer based on the Lyapunov's direct method is proposed. It is shown that the proposed adaptive algorithms are sufficiently valid where observation noises are present and that they are identical with the adaptive algorithms of a noise-free system. Futhermore, are investigated the convergence speed and the steady state value of errors between the parameters of the observer and those of the system.
TL;DR: In this article, a new concept of vector Lyapunov functions is introduced as a tool for investigating the dynamic stability behavior of a large scale power system, which is then extended to derive reduced order equivalents of the interconnected power system.
TL;DR: In this article, a method is proposed to enlarge the estimation of the stability region determined by means of Lyapunov's direct method using series expansion, and by putting it closer to the actual stability region.
Abstract: In this paper, a method is proposed for improving the accuracy of the stability criterion by enlarging the estimation of the stability region determined by means of Lyapunov's direct method using series expansion, and by putting it closer to the actual stability region. The Lyapunov function obtained by energy integration is used as the first estimation. The method is applied to a single-machine infinite bus power system, and the stability domain in the δ-ω plane is shown together with the actual stability domain obtained by numerical integration. It is shown that the application of the proposed method results in a considerable improvement in the stability boundary estimation over that given by the original Lyapunov function. Zubov's method, which obtains an accurate stability boundary using the series expansion, is related to the method proposed in this paper. The last section is, therefore, devoted to comparing the two methods and examining the possibility of their application to multi-machine ...
TL;DR: In this article, the concept of p-th-order conditional stability and boundedness for ordinary differential equations in Banach spaces was introduced, and sufficient conditions for conditional stability were given using vector Lyapunov functions.
Abstract: In this paper we introduce the concept of p-th-order conditional stability and boundedness for ordinary differential equations in Banach spaces. Using vector Lyapunov functions and the comparison technique we give sufficient conditions for p-th-order conditional stability and boundedness properties of the abstract differential equation. Our results generalize in many ways various known results of stability and boundedness.