Showing papers on "Lyapunov equation published in 1984"
TL;DR: This paper collects the bounds that have been presented up to now and summarizes them in an unified form to prove particularly convenient for those wishing to get a ready estimate of the solution while solving the equations numerically or to develop theoretical results that rely on these bounds.
Abstract: In recent years, several bounds have been reported for different measures of the ‘ extent’ or ‘ size ’ of the solution of the algebraic matrix equations arising in control theory, such as the Riccati equation and the Lyapunov equation. This paper collects the bounds that have been presented up to now and summarizes them in an unified form. This will prove particularly convenient for those wishing to get a ready estimate of the solution while solving the equations numerically or to develop theoretical results that rely on these bounds.
116 citations
TL;DR: In this article, the Lyapunov functional introduced by Newman can be used to prove the stability of the Barenblatt-Pattle solutions of the porous medium equation.
Abstract: This article continues the analysis of the preceding article, showing that the Lyapunov functional introduced by Newman can be used to prove the stability of the Barenblatt–Pattle solutions of the porous medium equation.
49 citations
TL;DR: In this article, a method for calculating the leading Lyapunov exponent directly from experimental data for systems having a strange attractor with dimensionality near 2 is presented. But the method is exact only for one-dimensional maps and gives good results for systems that have approximate onedimensional maps associated with them even in the presence of some noise.
Abstract: A new method is presented for calculating the leading Lyapunov exponent directly from experimental data for systems having a strange attractor with dimensionality near 2. The method is exact for one-dimensional maps and gives good results for systems that have approximate one-dimensional maps associated with them even in the presence of some noise. Numerical examples are given.
39 citations
01 Dec 1984
TL;DR: The work shows that iterative ARE solutions offer a viable alternative to the Shur eigenvector approach that is a generally accepted reference and demonstrates numerical robustness, accurate results, rapid (super linear) convergence, algorithmic simplicity, and modest storage requirements.
Abstract: Roberts matrix sign function solution to the ARE is defined so as to speed convergence and reduce storage requirements; our work extends ideas proposed by R. Byers, ref(8). Features of the sign function presented here are: (a) our formulation of the Roberts-Byers algorithm recurses on the symmetric transformed Hamiltonian, which reduces storage requirements, (b) the symmetric indefinite matrix inversion required by the algorithm is carried out using LINPACK (and our excellent numerical results reflect the wisdom of this choice); corrections to the computed (approximate) solution are obtained by applying the same algorithms to the translated problem (which improves upon the linear Lyapunov equation correction that has been used), and (d) simple (but somewhat ad hoc) convergence criteria are proposed to reduce computation. The algorithm described in this work has been tested on a variety of continuous time ARE test problems, and the results have been very satisfactory. Tests on numerically ill-conditioned problems produced results of comparable accuracy with those obtained by the Shur vector RICPACK method. Our sign function iterative ARE solution demonstrates numerical robustness, accurate results, rapid (super linear) convergence, algorithmic simplicity, and modest storage requirements. Our work shows that iterative ARE solutions offer a viable alternative to the Shur eigenvector approach that is a generally accepted reference.
33 citations
Book•
01 Jan 1984TL;DR: This chapter discussesrete models in Systems Engineering, a branch of engineering that focuses on the design of systems with low levels of control, and some of the approaches used, such as the tried and tested Two-Stage Control Design.
Abstract: 1 Discrete Models in Systems Engineering.- 1.1 Introduction.- 1.2 Some Illustrative Examples.- 1.2.1 Direct Digital Control of a Thermal Process.- 1.2.2 An Inventory Holding Problem.- 1.2.3 Measurement and Control of Liquid Level.- 1.2.4 An Aggregate National Econometric Model.- 1.3 Objectives and Outline of the Book.- 1.4 References.- 2 Representation of Discrete Control Systems.- 2.1 Introduction.- 2.2 Transfer Functions.- 2.2.1 Review of Z-Transforms.- 2.2.2 Effect of Pole Locations.- 2.2.3 Stability Analysis.- 2.2.4 Simplification by Continued-Fraction Expansions.- 2.2.5 Examples.- 2.3 Difference Equations.- 2.3.1 The Nature of Solutions.- 2.3.2 The Free Response.- 2.3.3 The Forced Response.- 2.3.4 Examples.- 2.3.5 Relationship to Transfer Functions.- 2.4. Discrete State Equations.- 2.4.1 Introduction.- 2.4.2 Obtaining the State Equations.- A. From Difference Equations.- B. From Transfer Functions.- 2.4.3 Solution Procedure.- 2.4.4 Examples.- 2.5 Modal Decomposition.- 2.5.1 Eigen-Structure.- 2.5.2 System Modes.- 2.5.3 Some Important Properties.- 2.5.4 Examples.- 2.6 Concluding Remarks.- 2.7 Problems.- 2.8 References.- 3 Structural Properties.- 3.1 Introduction.- 3.2 Controllability.- 3.2.1 Basic Definitions.- 3.2.2 Mode-Controllability Structure.- 3.2.3 Modal Analysis of State-Reachability.- 3.2.4 Some Geometrical Aspects.- 3.2.5 Examples.- 3.3 Observability.- 3.3.1 Basic Definitions.- 3.3.2 Principle of Duality.- 3.3.3 Mode-Observability Structure.- 3.3.4 Concept of Detectability.- 3.3.5 Examples.- 3.4. Stability.- 3.4.1 Introduction.- 3.4.2 Definitions of Stability.- 3.4.3 Linear System Stability.- 3.4.4 Lyapunov Analysis.- 3.4.5 Solution and Properties of the Lyapunov Equation.- 3.4.6 Examples.- 3.5 Remarks.- 3.6 Problems.- 3.7 References.- 4 Design of Feedback Systems.- 4.1 Introduction.- 4.2 The Concept of Linear Feedback.- 4.2.1 State Feedback.- 4.2.2 Output Feedback.- 4.2.3 Computational Algorithms.- 4.2.4 Eigen-Structure Assignment.- 4.2.5 Remarks.- 4.2.6 Example.- 4.3 Deadbeat Controllers.- 4.3.1 Preliminaries.- 4.3.2 The Multi-Input Deadbeat Controller.- 4.3.3 Basic Properties.- 4.3.4 Other Approaches.- 4.3.5 Examples.- 4.4 Development of Reduced-Order Models.- 4.4.1 Analysis.- 4.4.2 Two Simplification Schemes.- 4.4.3 Output Modelling Approach.- 4.4.4 Control Design.- 4.4.5 Examples.- 4.5 Control Systems with Slow and Fast Modes.- 4.5.1 Time-Separation Property.- 4.5.2 Fast and Slow Subsystems.- 4.5.3 A Frequency Domain Interpretation.- 4.5.4 Two-Stage Control Design.- 4.5.5 Examples.- 4.6 Concluding Remarks.- 4.7 Problems.- 4.8 References.- 5 Control of Systems with Inaccessible States.- 5.1 Introduction.- 5.2 State Reconstruction Schemes.- 5.2.1 Full-Order State Reconstructors.- 5.2.2 Reduced-Order State Reconstructors.- 5.2.3 Discussion.- 5.2.4 Deadbeat State Reconstructors.- 5.2.5 Examples.- 5.3 Observer-Based Controllers.- 5.3.1 Structure of Closed-Loop Systems.- 5.3.2 The Separation Principle.- 5.3.3 Deadbeat Type Controllers.- 5.3.4 Example.- 5.4 Two-Level Observation Structures.- 5.4.1 Full-Order Local State Reconstructors.- 5.4.2 Modifications to Ensure Overall Asymptotic Reconstruction.- 5.4.3 Examples.- 5.5 Discrete Two-Time-Scale Systems.- 5.5.1 Introduction.- 5.5.2 Two-Stage Observer Design.- 5.5.3 Dynamic State Feedback Control.- 5.5.4 Example.- 5.6 Concluding Remarks.- 5.7 Problems.- 5.8 References.- 6 State and Parameter Estimation.- 6.1 Introduction.- 6.2 Random Variables and Gauss-Markov Processes.- 6.2.1 Basic Concepts of Probability Theory.- 6.2.2 Mathematical Properties of Random Variables.- A. Distribution Functions.- B. Mathematical Expectation.- C. Two Random Variables.- 6.2.3 Stochastic Processes.- A. Definitions and Properties.- B. Gauss and Markov Processes.- 6.3 Linear Discrete Models with Random Inputs.- 6.3.1 Model Description.- 6.3.2 Some Useful Properties.- 6.3.3 Propagation of Means and Covariances.- 6.3.4 Examples.- 6.4 The Kalman Filter.- 6.4.1 The Estimation Problem.- A. The Filtering Problem.- B. The Smoothing Problem.- C. The Prediction Problem.- 6.4.2 Principal Methods of Obtaining Estimates.- A. Minimum Variance Estimate.- B. Maximum Likelihood Estimate.- C. Maximum A Posteriori Estimate.- 6.4.3 Development of the Kalman Filter Equations.- A. The Optimal Filtering Problem.- B. Solution Procedure.- C. Some Important Properties.- 6.4.4 Examples.- 6.5 Decentralised Computation of the Kalman Fikter.- 6.5.1 Linear Interconnected Dynamical Systems.- 6.5.2 The Basis of the Decentralised Filter Structure.- 6.5.3 The Recursive Equations of the Filter.- 6.5.4 A Computation Comparison.- 6.5.5 Example.- 6.6 Parameter Estimation.- 6.6.1 Least Squares Estimation.- A. Linear Static Models.- B. Standard Least Squares Method and Properties.- C. Application to Parameter Estimation of Dynamic Models.- D. Recursive Least Squares.- E. The Generalised Least Squares Method.- 6.6.2 Two-Level Computational Algorithms.- A. Linear Static Models.- B. A Two-Level Multiple Projection Algorithm.- C. The Recursive Version.- D. Linear Dynamical Models.- E. The Maximum A Posteriori Approach.- F. A Two-Level Structure.- 6.6.3 Examples.- 6.7 Problems.- 6.8 References.- 7 Adaptive Control Systems.- 7.1 Introduction.- 7.2 Basic Concepts of Model Reference Adaptive Systems.- 7.2.1 The Reference Model.- 7.2.2 The Adaptation Mechanism.- 7.2.3 Notations and Some Definitions.- 7.2.4 Design Considerations.- 7.3 Design Techniques.- 7.3.1 Techniques Based on Lyapunov Analysis.- 7.3.2 Techniques Based on Hyperstability and Positivity Concepts.- A. Popov Inequality and Related Results.- B. Systematic Procedure.- C. Parametric Adaptation Schemes.- D. Adaptive Model-Following Schemes.- 7.3.3 Examples.- 7.4 Self-Tuning Regulators.- 7.4.1 Introduction.- 7.4.2 Description of the System.- 7.4.3 Parameter Estimators.- A. The Least Squares Method.- B. The Extended Least Squares Method.- 7.4.4 Control Strategies.- A. Controllers Based on Linear Quadratic Theory.- B. Controllers Based on Minimum Variance Criteria.- 7.4.5 Other Approaches.- A. Pole/Zero Placement Approach.- B. Implicit Identification Approach.- C. State Space Approach.- D. Multivariable Approach.- 7.4.6 Discussion.- 7.4.7 Examples.- 7.5 Concluding Remarks.- 7.6 Problems.- 7.7 References.- 8 Dynamic Optimisation.- 8.1 Introduction.- 8.2 The Dynamic Optimisation Problem.- 8.2.1 Formulation of the Problem.- 8.2.2 Conditions of Optimality.- 8.2.3 The Optimal Return Function.- 8.3 Linear-Quadratic Discrete Regulators.- 8.3.1 Derivation of the Optimal Sequences.- 8.3.2 Steady-State Solution.- 8.3.3 Asymptotic Properties of Optimal Control.- 8.4 Numerical Algorithms for the Discrete Riccati Equation.- 8.4.1 Successive Approximation Methods.- 8.4.2 Hamiltonian Methods.- 8.4.3 Discussion.- 8.4.4 Examples.- 8.5 Hierarchical Optimization Methodology.- 8.5.1 Problem Decomposition.- 8.5.2 Open-Loop Computation Structures.- A. The Goal Coordination Method.- B. The Method of Tamura.- C. The Interaction Prediction Method.- 8.5.3 Closed-Loop Control Structures.- 8.5.4 Examples.- 8.6 Decomposition-Decentralisation Approach.- 8.6.1 Statement of the Problem.- 8.6.2 The Decoupled Subsystems.- 8.6.3 Multi-Controller Structure.- 8.6.4 Examples.- 8.7 Concluding Remarks.- 8.8 Problems.- 8.9 References.
32 citations
TL;DR: In this paper, a stability inequality for nonlinear feedback systems with slope-restricted nonlinearity is derived using Lyapunov's direct method, which is essentially an extension of a recent idea of the author.
Abstract: Using Lyapunov's direct method, a stability inequality in frequency domain is derived for nonlinear feedback systems with slope-restricted nonlinearity. In the present approach, a transformed system that involves the slope of the nonlinearity is considered, thus leading to a stability inequality that incorporates the slope information (but not the sector information) of the nonlinearity. The approach is essentially an extension of a recent idea of the author.
31 citations
TL;DR: In this paper, it was shown that the system is asymptotically stable if and only if the associated Lyapunov differential equation admits a periodic solution positive definite at each time instance.
30 citations
27 citations
24 citations
TL;DR: In this article, sufficient conditions for the nonexistence of zero input limit cycles in two-dimensional systems which contain a finite number of memoryless nonlinearities are presented, and two different conditions are given.
Abstract: Sufficient conditions for the nonexistence of zero input limit cycles in two-dimensional systems which contain a finite number of memoryless nonlinearities are presented. Two different conditions are given. The first one is based on the frequency domain representation of the linear part of the discrete system, and it is shown to be more general than the one already presented in [1]. The second condition is formulated using the state space representation and is based on the properties of quasidominant matrices and on partial results on the 2-D Lyapunov equation.
24 citations
TL;DR: In this paper, it was shown that whenever there is a Markov solution x with Lyapunov number X then there is another Markov solutions with a stationary angle (or equivalently: an invariant measure for the transition probabilities of (s, A)) with the same Lyapinov number.
Abstract: All nontrivial solutions of x = A(t)x grow exponentially with rate X(x,w)e{A1,...,Xr}, A a (strictly) stationary matrix process. Projecting x to the unit sphere one obtains for each of the Lyapunov exponents Xt a solution xt with stationary angle st. Now if A is a Markov process one can restrict oneself to Markov solutions, i.e., (x, A) shall be a (joint) Markov process (wich is a restriction on the inital conditions). We prove that whenever there is a Markov solution x with Lyapunov number X then there is another Markov solution with a stationary angle (or equivalently: an invariant measure for the transition probabilities of (s, A)) with the same Lyapunov number. This has some consequences, e.g., for the uniqueness of the Lyapunov numbers
TL;DR: In this paper, the determinants of the positive definite solutions of the discrete algebraic Riccati and Lyapunov matrix equations are presented, and lower bounds for the product of the eigenvalues of the matrix solutions are given.
Abstract: Inequalities which are satisfied by the determinants of the positive definite solutions of the discrete algebraic Riccati and Lyapunov matrix equations are presented. The results give lower bounds for the product of the eigenvalues of the matrix solutions. Also for a discrete Lyapunov equation, an algorithm is presented to determine under what conditions a positive diagonal solution will exist. If all the conditions are satisfied, the algorithm also provides such a diagonal solution.
TL;DR: In this paper, a Lyapunov sufficiency condition was developed for steepest descent control to yield global asymptotic stability of the origin in constrained linear systems, where optimal-aim control produced instability.
Abstract: A Lyapunov sufficiency condition is developed for steepest descent controls to yield global asymptotic stability of the origin in constrained linear systems. The results are applied to a previous counterexample for "optimal-aim" control. Using the more flexible Lyapunov steepest descent approach, global asymptotic stability is achieved, where optimal-aim control produced instability.
TL;DR: It is shown that the new bound is less conservative than those currently available, especially in the case of a high-dimensional Lyapunov equation.
TL;DR: In this paper, the authors proved that the Cauchy solutions of the discrete Schrodinger equation with the potential qn =g cos(2πnθ+φ) grow exponentially for every irrational θ, g>1 and almost every φe[0,2π).
Abstract: This paper contains the rigorous proof of the formulated by Andre and Aubry following statement: the Cauchy solutions of the discrete Schrodinger equation with the potential qn=g cos(2πnθ+φ) grow exponentially for every irrational θ, g>1 and almost every φe[0,2π). According to known this fact implies the absence of the absolutely continuous component of the spectrum for the corresponding operator.
TL;DR: In this paper, real indecomposable quasi-Jacobi matrices of class D were characterized, i.e., those which satisfy the Lyapunov equation PA + A′P = −Q with P diagonal and both P and Q positive definite.
01 Dec 1984
TL;DR: In this article, a new approach based on singular perturbation and time scale decomposition is proposed to analyze the stability of large scale power systems using reduced order models (e.g. one machine-infinite bus equivalent) or decomposition.
Abstract: Stability of large scale power systems using direct methods has been investigated either through reduced order models (e.g. one machine-infinite bus equivalent) or by decomposition. The latter method employs artificial mathematical methods for decomposition. In either method the physical picture gets lost and the analysis has to be repated for every disturbance. In this paper we propose a new approach based on singular perturbation and time scale decomposition. The system Lyapunov function gets split into a "slow" Lyapunov function and a number of "fast" Lyapunov functions each for a slowly coherent area. The weighted sum of these Lyapunov functions gets improved in quality as higher order corrections are taken into account. The decomposition is invariant with respect to the disturbance and thus offers a new approach to stability analysis of large scale power systems.
TL;DR: In this article, a method of extending this algorithm to narrowband filters is presented, which uses a transformation which was also presented by Mullis and Roberts, and improves the numerical accuracy of the solution of the Lyapunov equation.
Abstract: The quantization noise generated by a digital filter can be analyzed via the solution of the Lyapunov equation K = AKA^{T} + bb^{T} corresponding to the filter's state variable description. A method given by Mullis and Roberts [1] for the solution of the Lyapunov equation is noted for its speed of implementation, but its applicability is limited to filters with wide bandwidths because of numerical accuracy. A method of extending this algorithm to narrowband filters is presented. It uses a transformation which was also presented by Mullis and Roberts. The transformation distorts the frequency response of the filters and improves the numerical accuracy of the solution of the Lyapunov equation. The development given here is restricted to low-pass or high-pass filters, but it can be extended to bandpass filters.
TL;DR: In this paper, a differentiator-free Lyapunov MRAC scheme with a new error model is suggested, which does not require (2n or 4n) additional filters and generates auxiliary signals, even when the plant relative degree n* ≧ 2; instead, it uses positive feedback in an inner loop.
01 Jan 1984
TL;DR: The main purpose of as discussed by the authors is to stress the importance of Lyapunov exponents for the study of nonlinear deterministic and stochastic systems, in particular, the stabilizing and destabilizing effect of noise can be studied via perturbation theory.
Abstract: The main purpose of this paper is to stress the importance of Lyapunov exponents for the study of nonlinear deterministic and stochastic systems. After some introductory examples we present basic results of Lyapunov exponents for stochastic parameter-excited systems. This includes a formula for the biggest Lyapunov exponent (which determines the stability of the system) from which various quantitative conclusions can be drawn. In particular, the stabilizing and destabilizing effect of noise can be studied via perturbation theory.
TL;DR: This letter shows how the method can be extended to include the contributions from the quantization which occurs at the inputs to time delays, at the "storage nodes" of the filter.
Abstract: The calculation of digital filter quantization noise can be facilitated through the solution of the Lyapunov equation AKA^{T} + bb^{T}=K [1]. This method, however, only includes the contributions from the quantization which occurs at the inputs to time delays, that is, at the "storage nodes" of the filter. This letter shows how the method can be extended to include the contributions at the other nodes, the "signal nodes."
TL;DR: In this paper, the relationship between asymptotic stability, the zeros of the characteristic polynomial and the 2-D Lyapunov equation is investigated for 2D discrete systems.
TL;DR: In this paper, the authors describe procedures for computing the matrix polynomial defining a vector backward autoregressive recursion from a vector forward autoregression recursion, which do not involve a solution of a matrix Lyapunov equation.
Abstract: The aim of this paper is to describe procedures for computing the matrix polynomial defining a vector backward autoregressive recursion from the matrix polynomial defining a vector forward autoregressive recursion. Direct procedures for computing the backward polynomial which do not involve a solution of a matrix Lyapunov equation are described. A novel interpretation is also included of a known procedure which involves the computation of covariance data via the matrix Lyapunov equation. This procedure depends on a standard result connecting forward and reverse time state-space models. A comparison involving operation counts is given of the algorithms.
TL;DR: In this article, a definition of the stability region is given by extending the properties of Lyapunov's definition of sets of sizable measure, and constructive theorems on estimates of regions of stability and attraction are obtained by using certain developments of the second method for a wide class of autonomous and non-autonomous systems that satisfy both the Lipschitz and discontinuous conditions.
01 May 1984
TL;DR: In this article, the authors established the comparison principle of the discrete systems and studied the decomposition problem in the theory of stability, and showed that the properties of an n-order large system can be obtained from an r-order comparison system (1≤r≤n).
TL;DR: A generalization of Lyapunov's second method, formulated by J. Kato, is introduced and with these mathematical means the stability of the controller designed in this paper can be proved.