The Discrete Time $H_\infty $ Control Problem with Measurement Feedback

Rotational Motion Control by Feedback with Minimum L1-Norm

This paper presents an algorithm for the iterative refinement of matrix solutions for the algebraic Riccati equation in both the continuous and discrete-time cases. It is based on Newton's method for root-finding and extended using the Fréchet derivative framework. The improvement brought by the paper is in considering the complete structure of the algebraic Riccati equations, which typically consider the input-state dependency matrix to be zero. This approach is reasonable for computing linear quadratic regulators and estimators, but must be treated in the field of Robust Control, as the algebraic Riccati equations which arise in H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> /H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> synthesis methods have the complete structure. Numerical results are also presented and it is shown that the method brings important improvements, in particular for ill-conditioned problems, i.e. when the system controllability is ill-conditioned.

New results in the Lyapunov-type algorithm for algebraic MCV Riccati equations

This paper investigates the problem of Hankel norm model reduction for linear systems with time-varying delay in the state. For a given stable system, our attention is focused on the construction of reduced-order model, which guarantees the corresponding error system to be asymptotically stable and has a specified Hankel norm error performance. Two different approaches are proposed to solve this problem. One casts the model reduction into a convex optimization problem by using a linearization procedure, and the other is based on the cone complementarity linearization idea, which casts the model reduction into a sequential minimization problem subject to linear matrix inequality constraints. Both continuous and discrete time cases are considered. A numerical example is provided to show the effectiveness of the proposed theory.

In this study, we define new solution bounds for the infinite-dimensional algebraic Riccati equation. We first suggest an upper bound solution in the continuous case. After that, we develop upper and lower bounds for the algebraic Riccati equation solution in discrete time. We provide a robust control application.

Employing the algebraic Riccati equation for a parametrization of the solutions of the finite-horizon LQ problem: the discrete-time case

A common quadratic Lyapunov function (CQLF) guarantees the asymptotic stability of a set of systems. A complete characterization of the set of systems with such a property have been unsuccessful (except for second-order systems). Thus, for both the continuous-time and discrete-time cases, several subsets of linear systems which have a CQLF are known. Some results indicate that there is a parallelism between the continuous-time case and the discrete-time case. In this paper, we show a new subclass for continuous-time systems which have a CQLF by using a property of M-matrices. We also show the discrete-time counterpart of the above new subclass. Next, it is shown that the whole class of continuous-time linear systems having a CQLF is connected directly with its discrete-time counterpart by using a bilinear transformation. For some known subclasses of systems having a CQLF, the transformation gives a one-to-one correspondence between the continuous-time and discrete-time cases. We further show relationships among the obtained results and other, known results.

The paper deals with the implementation of optimal controller design for linear dynamic systems. The implementation is carried out with the use of quadratic cost function for both cases of continuous and discrete systems. The mathematical optimal control theory is applied to dynamical systems with considering the form of either vector differential equations in continuous time case, or difference equations in discrete time- or sampled data-case. The bulk of the work provides design techniques for synthesizing the optimal control structure by applying the Pontryagin's Maxi mum principle to linear systems with using Matrix Riccati Equation. The work is focused upon regulator problems; that is problems where the goal is to maintain the system states at zero level. The implemented techniques are tested by the application on fourth order practical example represents the linearized control model of two-reach river pollution System. Also comparative study between the continuous and discrete optimal regulator problems will be presented and discussed from a numerical point-of-view.

Algebraic Riccati Equation (ARE) solutions play an important role in many optimal/suboptimal control problems. However, a key assumption in formulation and solution of the ARE is a certain ‘regularity condition’ on the feedthrough term D of the system. For example, formulation of the ARE requires nonsingularity of \(D+D^T\) in positive real dissipative systems and, in the case of bounded real dissipative systems, one requires nonsingularity of \(I-D^T D\). Note that for lossless systems \(D+D^T=0\), while for all-pass systems \(I-D^TD=0\); this rules out the formulation of the ARE. Noting that the ARE solutions are also extremal “storage functions” for dissipative systems, one can speak of storage function for the lossless case too. This contributed chapter formulates new properties of the ARE solution; we then show that this property is satisfied by the storage function for the lossless case too. The formulation of this property is via the set of trajectories of minimal dissipation. We show that the states in a first-order representation of this set satisfy static relations that are closely linked to ARE solutions; this property holds for the lossless case too. Using this property, we propose an algorithm to compute the storage function for the lossless case.

In Part I, a single-channel service system consisting of a service facility and queue, in which the possible times of arrival of units and the possible service times are discrete, is analyzed. A method of calculating the moments of the total service time of units in the system is developed. This total service time is related to the delay caused by the system. When arrivals at different times are assumed to be independent, the “values” of the resulting Markov process can be calculated. These values lead to information about the transient behavior, autocorrelation function, expected first passage time, and expected extra delay that arises if another unit is inserted into the system. An expression for the geometric transform or moment generating function of the probability distribution of the total service time of units in the system is determined. The results are derived for arbitrary arrival and service time distributions. In Part II, the continuous time analog of the system of Part I is considered and analogous results are determined.

If $(\mathcal{A},\mathcal{B})$ is stabilizable, one pretty well knows algebraic conditions for the solvability and for the existence of largest solutions of the algebraic Riccati equation and inequality. \[ \mathcal{A}^ * \mathcal{X} + \mathcal{X}\mathcal{A} - \mathcal{X}\mathcal{B}\mathcal{B}^ * \mathcal{X} + \mathcal{Q} = 0\quad{\text{and}}\quad \mathcal{A}^ * \mathcal{X} + \mathcal{X}\mathcal{A} - \mathcal{X}\mathcal{B}\mathcal{B}^ * \mathcal{X} + \mathcal{Q} \geq 0,\] which leads to immediate existence results for positive definite solutions. In this paper we work out how far these properties may be generalized if $(\mathcal{A},\mathcal{B})$ could have uncontrollable modes on the imaginary axis. Since the relations of the equation and inequality are not as tight any more, we provide separate conditions for the existence of Hermitian or positive definite solutions and give a detailed discussion how to verify them. As auxiliary steps we discuss various new aspects for the corresponding Lyapunov equation/inequality and a complete solvability test for the quadratic equation \[ X^ * RX + SX + (SX)^ * + T = 0 \] with Hermitian R and T. Finally, we briefly point out the consequences of our results for the general state-feedback $H_\infty $-control problem at optimality.

We offer sufficient and realistic criteria for the existence of positive periodic solutions for a class of non-autonomous scalar equations with deviating arguments. Both the continuous time case and the discrete time case are discussed. Our approach is based on the theory of coincidence degree and the related continuation theorem. To show the usefulness of the results obtained, we include applications to several famous population models.

On the minimum rate of impatience for complicated optimal growth paths

This book provides a careful treatment of the theory of algebraic Riccati equations. It consists of four parts: the first part is a comprehensive account of necessary background material in matrix theory including careful accounts of recent developments involving indefinite scalar products and rational matrix functions. The second and third parts form the core of the book and concern the solutions of algebraic Riccati equations arising from continuous and discrete systems. The geometric theory and iterative analysis are both developed in detail. The last part of the book is an exciting collection of eight problem areas in which algebraic Riccati equations play a crucial role. These applications range from introductions to the classical linear quadratic regulator problems and the discrete Kalman filter to modern developments in HD*W*w control and total least squares methods.

Theoretical Study of the Nonlinear Control Algorithms with Continuous and Discrete-Time State Dependent Riccati Equation