Steady-state Riccati Equation Research Articles

Optimal production control and marketing plan in two-machine unreliable flexible manufacturing systems

In this paper, we have developed a production planning and marketing model in unreliable flexible manufacturing systems with inconstant demand rate that its rate depends on the level of advertisement on that product. The proposed model is more realistic and useful from a practical point of view. The flexible manufacturing system is composed of two machines that produce a single product. Markovian models frequently have been used in modeling a wide variety of real-world systems under uncertainties. Therefore, in this paper, the inventory balance equation is represented by a continuous-time model with Markov jump process to take into account machines breakdown. The objective is to minimize the expected total cost of the firm over an infinite time horizon. While the total cost consists of the cost of the product surplus, the cost of the production, and the cost of the advertisement. In the process of finding a solution to the problem, we first characterize an optimal control by a class of linear stochastic system where some parameter values are subject to random jump. By defining quadratic cost functions and characterizing the associated limiting optimal control problem, a discrete-time approximation model and an asymptotic optimal control model are developed. It is clear that such a solution exists and can be obtained as a limit of a monotonic sequence with solving the steady-state Riccati equation.

Self-tuning weighted measurement fusion Kalman filtering algorithm

For the multisensor linear discrete system with correlated noises and same measurement matrix, the self-tuning weighted measurement fusion Kalman filtering algorithm is presented when the model parameters and noise variances are all unknown. It can handle the self-tuning fused Kalman filtering, smoothing, and prediction problem and the input white noise deconvolution estimation problem. By the dynamic variance error system analysis (DVESA) method, it is proved that the solution of the self-tuning Riccati equation converges to the solution of the steady-state Riccati equation. Based on the convergence of the self-tuning Riccati equation, the convergence of the proposed self-tuning weighted measurement fusion Kalman estimator is proved. So it has asymptotic global optimality. Applying to the multi-channel autoregressive moving average (ARMA) signal with sensor bias, the corresponding self-tuning weighted measurement fusion Kalman estimator of the signal is also presented, where the estimates of unknown model parameters and noise variances are obtained by the multi-dimension recursive extended least squares (RELS) algorithm, the correlation method and the Gevers–Wouters algorithm with a dead band. One simulation example shows the effectiveness.

Self-tuning measurement fusion white noise deconvolution estimator with correlated noises

For the multisensor linear discrete time-invariant stochastic systems with correlated noises and unknown noise statistics, an on-line noise statistics estimator is presented by using the correlation method. Substituting it into the steady-state Riccati equation, the self-tuning Riccati equation is obtained. Using the Kalman filtering method, based on the self-tuning Riccati equation, a self-tuning weighted measurement fusion white noise deconvolution estimator is presented. By the dynamic error system analysis (DESA) method, it is proved that the self-tuning fusion white noise deconvolution estimator converges to the optimal fusion steady-state white noise deconvolution estimator in a realization, so that it has the asymptotic global optimality. A simulation example for Bernoulli-Gaussian input white noise shows its effectiveness.

Weak and strong stabilization of linear systems

We study the problem of stabilizing a distributed linear system on a subregion of its geometrical domain. We are concerned with two methods: the first approach enables us to characterize a stabilizing control via the steady state Riccati equation, and the second one is based on decomposing the state space into two suitable subspaces and studying the projections of the initial system onto such subspaces. The obtained results are performed through various examples.

Convergence analysis of self-tuning Riccati equation for systems with correlation noises

For linear discrete time-invariant stochastic system with correlated noises, and with unknown state transition matrix and unknown noise statistics, substituting the online consistent estimators of the state transition matrix and noise statistics into steady-state optimal Riccati equation, a new self-tuning Riccati equation is presented. A dynamic variance error system analysis (DVESA) method is presented, which transforms the convergence problem of self-tuning Riccati equation into the stability problem of a time-varying Lyapunov equation. Two decision criterions of the stability for the Lyapunov equation are presented. Using the DVESA method and Kalman filtering stability theory, it proves that with probability 1, the solution of self-tuning Riccati equation converges to the solution of the steady-state optimal Riccati equation or time-varying optimal Riccati equation. The proposed method can be applied to design a new selftuning information fusion Kalman filter and will provide the theoretical basis for solving the convergence problem of self-tuning filters. A numerical simulation example shows the effectiveness of the proposed method.

An “Indefinite Realization” Algorithm via Riccati Difference Equation

This paper studies a realization algorithm for modeling dynamical systems subject to a bounded error. An iterative algorithm is developed based on “indefinite realization”, which is a generalization of stochastic realization. A Riccati difference equation for indefinite realization is derived, and it is shown that the Riccati recursion converges to the stabilizing solution to the steady state Riccati equation under certain assumptions. Numerical simulation results are also included.

A numerical method to approximate optimal production and maintenance plan in a flexible manufacturing system

Abstract The simultaneous planning of the production and the maintenance in a flexible manufacturing system is considered in this paper. The manufacturing system is composed of one machine that produces a single product. There is a preventive maintenance plan to reduce the failure rate of the machine. This paper is different from the previous researches in this area in two separate ways. First, the failure rate of the machine is supposed to be a function of its age. Second, we assume that the demand of the manufacturing product is time dependent and its rate depends on the level of advertisement on that product. The objective is to maximize the expected discounted total profit of the firm over an infinite time horizon. In the process of finding a solution to the problem, we first characterize an optimal control by introducing a set of Hamilton–Jacobi–Bellman partial differential equations. Then we realize that under practical assumptions, this set of equations can not be solved analytically. Thus to find a suboptimal control, we approximate the original stochastic optimal control model by a discrete-time deterministic optimal control problem. Then proposing a numerical method to solve the steady state Riccati equation, we approximate a suboptimal solution to the problem.

An alternate calculation of the discrete-time Kalman filter gain and Riccati equation solution

We describe an algorithm to calculate the steady-state Kalman filter gain and Riccati equation solution for a discrete-time Kalman filter. Our algorithm makes use of an approximate autoregressive model for the one-step predictor and only requires the solutions to linear equations. All of the nonlinear calculations can be made explicitly.

Stochastic Regulator Theory for a Class of Abstract Wave Equations

A class of steady-state stochastic regulator problems for abstract wave equations in a Hilbert space—of relevance to the problem of feedback control of large space structures using co-located controls/sensors—is studied. Both the control operator, as well as the observation operator, are finite-dimensional. As a result, the usual condition of exponential stabilizability invoked for existence of solutions to the steady-state Riccati equations is not valid. Fortunately, for the problems considered it turns out that strong stabilizability suffices. In particular, a closed form expression is obtained for the minimal (asymptotic) performance criterion as the control effort is allowed to grow without bound.

Compensator design for stability enhancement with collocated controllers

A continuum model rather than a finite element model is used. The optimal compensator design is formulated as a stochastic regulator problem and is shown to be solvable by the general infinite-dimensional theory developed by the author despite the lack of exponential stabilizability. Infinite-dimensional steady-state Riccati equations characterizing the feedback control gain and the Kalman filter gain operators can be solved explicitly. The associated performance indexes including the mean-square control-effort are calculated in closed form. As a first approximation, the compensator transfer function can be realized as a bank of bandpass filters in parallel centered at the undamped mode frequencies. Numerical calculations for the gain and bandwidths for a typical configuration are presented. The performance of the compensator is evaluated when in fact in the true model there is no actuator noise. The theoretical problem involved is to show that the infinite-dimensional stochastic process is asymptotically stationary. It is possible to calculate the steady-state covariance in closed form and thereby calculate performance indexes of interest explicitly, facilitating the choice of optimal design parameters. >

Compensator Design for Stability Enhancement with Co-Located Controllers

In this paper we present a stochastic-optimization based approach to the design of compensators for stability enhancement applicable to flexible multibody systems with co-located rate sensros/actuators--for example a truss with offset antennas at one or both ends. A continuum model--rather than a finite ele-ment model--is used, and an essential step in the Compensator Design theory is the formulation of the problem as a wave equation in a Hilbert space. In particular using white noise theory as opposed to the Wiener Process theory the fact that steady state covariances are not nuclear poses no difficulty. The optimal Compensator Design problem is formulated as a stochastic regulator problem. In particular it turns out that we can solve explicitly the infinite dimensional steady state Riccati equations characterizing the feedback control gain and the Kaiman filter gain operators. We can also calculate in closed form the associated performance indices including the "mean square" control effort. We show that, as a first approximation, the Compensator Transfer Function can be realized as a bank of band-pass filters in parallel centered at the undamped mode frequencies. Numerical calculations for the gains and bandwidths for the COFS configuration are presented. We also evaluate the performance of the compensator when in fact in the truth model there is no acutator noise. The theoretical problem involved here is to show that the infinite dimensional stochastic process is asymptotically stationary. We are able to calculate the steady state covariance in closed form and thereby calculate performance indices of interest explicitly, facilitating the choice of optimal design paramters.

Relationship between polynomial and state-space solutions of the optimal regulator problem

A relationship is established between the state-space and polynomial matrix solutions of the LQG state-feedback optimal regulator problem. The solution of polynomial matrix Diophantine equations is related to the solution of the steady-state Riccati equation.

Strong stability of quasi-affine transforms of contraction semigroups and the steady-state riccati equation

This paper studies strong stability of the class of strongly continuous Hilbert space semigroups which are quasi-affine transforms of contraction semigroups. Sufficient conditions for such a semigroup to be approximately strong stable are given. Applications to the stabilizability problem of Hilbert space semigroups, using a feedback involving a solution of the steady-state Riccati equation, are made. Our key tool is the generalization of the LaSalle invariance principle by Hale.

Stability of an exponentially stabilizable system

Let A be the generator of a C 0 semigroup T(t), t \geq 0 , and denote by S(t), t \geq 0 , the semigroup generated by A - K , where K is a bounded linear operator on a Hilbert space. In this note we find necessary and sufficient conditions for the original semigroup T(t), t \geq 0 , to be exponentially stable, given that the semigroup S(t), t \geq 0 , is exponentially stable. Applications to feedback stabilization via a steady-state Riccati equation will then be made.

Strong stabilizability and the steady state Riccati equation

We explore strong stabilizability (as opposed to exponential stabilizability) with the aid of the steady state Riccati equation. We show that the latter can have at most one strongly stable solution and obtain some sufficient conditions for existence. We also indicate an application to steady state Kalman filtering where the observation operator is compact so that we may not have exponential stability.

The direct solution of a steady-state Riccati equation arising in optimum boundary control

An algebraic Riccati matrix equation is solved directly, i.e. disregarding the associated problem of optimum boundary control of a one-dimensional diffusion system. A countable set of exact solutions is found and explicitly specified. From this set the stabilizing solution is selected.

Second-order convergent algorithms for the steady-state Riccati equation†

Second-order convergent algorithms for the steady-state Riccati equation†

A square root of a matrix approach to obtain the solution to a steady-state matrix Riccati equation

By utilizing the square root of a matrix approach, a method of generating the solution to a steady-state matrix Riccati type equation (under certain restrictions) is presented. This approach not only yields a closed form expression for the Riccati solution, but also converts the original Riccati equation into other equations which may have numerical or computational advantages. An example is worked out for a second-order case.

Specific optimal control of the linear regulator using a minimal order observer

Abstract The problem of designing an optimal (n — m)-order Luenberger observer for an nth-order time-invariant linear system with m (m <n) outputs is considered. The performance functional for this problem is the expected value of the well-known quadratic regulator measure. It is assumed that only the mean and covariance of the initial system state are known. The observer structure is fixed but the parameter values, including the feedback gain, are left free to be specified via the specific optimal control problem. Necessary conditions for optimality are derived and a design algorithm is obtained. A separation theorem typo result is shown to apply to this problem and consequently the core of the design algorithm requires only the solution of two steady-state Riccati equations, one of order n and one of order (n — m). A simple example is presented to illustrate the procedure

A new computational solution of the linear optimal regulator problem

A new method is presented for the numerical deterruination of the solution of the steady-state matrix Riccati equation. The equation is converted to a canonical form corresponding to Luenberger's canonical representation for controllable multivariable systems. Three special matrices closely associated with Luenberger's canonical form are defined and two related lemmas are established. These results are used to obtain concise expressions for the eigenvectors of the Hamiltonian matrix associated with the canonical Riccati equation in terms of the solutions of a much simpler reduced Hamiltonian system. Using a theorem due to Potter the solution of the Riccati equation is written in terms of the concise eigenvector expressions. The method is particularly well suited to problems in which the ratio of system states to system inputs is large and it can lead to a 26 to 1 reduction in the computational effort required to solve the Riccati equation.