Infinite-dimensional LQ optimal control of a dimethyl ether (DME) catalytic distillation column

This contribution addresses the development of a linear quadratic (LQ) regulator in order to control the concentration profiles along a catalytic distillation column, which is modelled by a set of coupled hyperbolic partial differential and algebraic equations (PDAEs). The proposed method is based on an infinite-dimensional state-space representation of the PDAE system which is generated by a transport operator. The presence of the algebraic equations, makes the velocity matrix in the transport operator, spatially varying, non-diagonal, and not necessarily negative through of the domain. The optimal control problem is treated using operator Riccati equation (ORE) approach. The existence and uniqueness of the non-negative solution to the ORE are shown and the ORE is converted into a matrix Riccati differential equation which allows the use of a numerical scheme to solve the control problem. The result is then extended to design an optimal proportional plus integral controller which can reject the effect of load losses. The performance of the designed control policy is assessed through a numerical study.     

Convolution algebra representation of systems described by linear hyperbolic partial differential equations

The representation of the input-output operator in convolution algebra B(σ₀) is obtained for distributed parameter systems described by linear hyperbolic partial differential equations. Three kinds of system are considered depending on the kind of coefficient matrix corresponding to the space variable derivative, and their properties are studied. Necessary and sufficient conditions for stability are obtained in terms of factorization of the transition matrix. The obtained results allow the use of modern algebraic methods for analysis of such systems.     

Null controllability of an infinite dimensional SDE with state- and control-dependent noise

We consider a controlled stochastic linear differential equation with state- and control-dependent noise in a Hilbert space H. We investigate the relation between the null controllability of the equation and the existence of the solution of “singular” Riccati operator equations. Moreover, for a fixed interval of time, the null controllability is characterized in terms of the dual state. Examples of stochastic PDEs are also considered.     

Optimal control of coupled parabolic–hyperbolic non-autonomous PDEs: infinite-dimensional state-space approach

This paper deals with the design of an optimal state-feedback linear-quadratic (LQ) controller for a system of coupled parabolic–hypebolic non-autonomous partial differential equations (PDEs). The infinite-dimensional state space representation and the corresponding operator Riccati differential equation are used to solve the control problem. Dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the LQ-optimal control problem and also to guarantee the exponential stability of the closed-loop system. Thanks to the eigenvalues and eigenfunctions of the parabolic operator and also the fact that the hyperbolic-associated operator Riccati differential equation can be converted to a scalar Riccati PDE, an algorithm to solve the LQ control problem has been presented. The results are applied to a non-isothermal packed-bed catalytic reactor. The LQ optimal controller designed in the early portion of the paper is implemented for the original non-linear model. Numerical simulations are performed to show the controller performances.     

Infinite horizon linear quadratic stochastic Nash differential games of Markov jump linear systems with its application

In this paper, we deal with the problem of stochastic Nash differential games of Markov jump linear systems governed by Itô-type equation. Combining the stochastic stabilizability with the stochastic systems, a necessary and sufficient condition for the existence of the Nash strategy is presented by means of a set of cross-coupled stochastic algebraic Riccati equations. Moreover, the stochastic H₂/H_∞ control for stochastic Markov jump linear systems is discussed as an immediate application and an illustrative example is presented.     

Stochastic H 2 /H ∞-control for a dynamical system with internal noises multiplicative with respect to state, control, and external disturbance

We consider an optimal control problem for a dynamical system under the influence of disturbances of both deterministic and stochastic nature. The system is defined on a finite time interval, and its diffusion coefficient depends on the control signal. The controller in the feedback circuit is assumed to be static, nonstationary, linear in the state vector, and satisfying the condition ‖L‖_∞ < γ that bounds the norm of operator L: v ?z for the transition of external disturbance to the controllable output signal. Solving the optimization H ₂/H _∞-control problem, we get three matrix functions satisfying a system of two differential equations of Riccati type and one matrix algebraic equation. In the special case of a stochastic system whose diffusion coefficient does not depend on the control signal, the system is reduced to two related Riccati equations.     

Solvability conditions and design for synchronization of discrete‐time multiagent systems

This paper provides solvability conditions for state synchronization with homogeneous discrete‐time multiagent systems with a directed and weighted communication network under partial‐ or full‐state coupling. Our solvability conditions reveal that the synchronization problem is solvable for all possible, a priori given, set of graphs associated with a communication network only under the condition that the agents are at most weakly unstable (ie, agents have all eigenvalues in the closed unit disc). However, if an upper bound on the eigenvalues inside the unit disc of the row stochastic matrices associated with any graph in a given set of graphs is known, then one can achieve synchronization for a class of unstable agents. We provide protocol design for at most weakly unstable agents based on a direct eigenstructure assignment method and a standard H₂ discrete‐time algebraic Riccati equation. We also provide protocol design for strictly unstable agents (ie, agents have at least one eigenvalue outside the unit disc) based on the standard H₂ discrete‐time algebraic Riccati equation.     

A representation of all solutions of the control algebraic Riccati equation for infinite-dimensional systems

We obtain a representation of all self-adjoint solutions of the control algebraic Riccati equation associated to the infinite-dimensional state linear system Σ(A,B,C) under the following assumptions: A generates a C ₀-group, the system is output stabilizable, strongly detectable and the dual Riccati equation has an invertible self-adjoint non-negative solution.     

The standard H∞ problem in systems with a singleinput delay

The standard H_∞ problem is solved for LTI systems with a single, pure input lag. The solution is based on state-space analysis, mixing a finite-dimensional and an abstract evolution model. Utilizing the relatively simple structure of these distributed systems, the associated operator Riccati equations are reduced to a combination of two algebraic Riccati equations and one differential Riccati equation over the delay interval. The results easily extend to finite time and time-varying problems where the algebraic Riccati equations are substituted by differential Riccati equations over the process time duration     

Delta 域Riccati方程研究：连续与离散的统一方法

基于Ｄｅｌｔａ算子研究连续Ｒｉｃｃａｔｉ方程和离散Ｒｉｃｃａｔｉ方程的统一形式,得到Ｄｅｌｔａ域Ｒｉｃｃａｔｉ方程解的定界估计,本文结果与现有的结果相比,具有较小的保守性,在极限情形下可分别得到连续和离散Ｒｉｃｃａｔｉ方程的相关结论。     

On the infinite-horizon LQ tracker

The infinite-horizon tracking problem is considered in the linear-quadratic optimal control framework. Computationally, one term in the control signal is a constant gain, stabilizing, full-state feedback found by solving an algebraic Riccati equation; the second term involves a steady-state function v_ss(t) that solves an auxiliary, forced differential equation with unknown initial condition. In this article, a linear system of algebraic equations is derived to determine v_ss(0). Numerical examples are included to illustrate the result.     

On closed form solutions for the differential matrix Riccati equation problem

A new formulation of the differential matrix Riccati equation is presented and a closed analytical solution is obtained under the hypothesis that certain commutativity conditions are fulfilled on a transformed space. The formulation generalizes the results of [1] on algebraic equations to differential matrix Riccati equations. To illustrate the usefulness of the method, a closed analytical solution of the differential matrix Riccati equation is obtained inR^{2 times 2}.     

Maximal and Stabilizing Hermitian Solutions for Discrete-Time Coupled Algebraic Riccati Equations

Discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control and H _∞-control of Markovian jump linear systems are considered. First, the equations that arise from the quadratic optimal control problem are studied. The matrix cost is only assumed to be hermitian. Conditions for the existence of the maximal hermitian solution are derived in terms of the concept of mean square stabilizability and a convex set not being empty. A connection with convex optimization is established, leading to a numerical algorithm. A necessary and sufficient condition for the existence of a stabilizing solution (in the mean square sense) is derived. Sufficient conditions in terms of the usual observability and detectability tests for linear systems are also obtained. Finally, the coupled algebraic Riccati equations that arise from the H _∞-control of discrete-time Markovian jump linear systems are analyzed. An algorithm for deriving a stabilizing solution, if it exists, is obtained. These results generalize and unify several previous ones presented in the literature of discrete-time coupled Riccati equations of Markovian jump linear systems. Date received: November 14, 1996. Date revised: January 12, 1999.     

Model‐Free H∞ Control Design for Unknown Continuous‐Time Linear System Using Adaptive Dynamic Programming

In this paper, a new online model‐free adaptive dynamic programming algorithm is developed to solve the H_∞ control problem of the continuous‐time linear system with completely unknown system dynamics. Solving the game algebraic Riccati equation, commonly used in H_∞ state feedback control design, is often referred to as a two‐player differential game where one player tries to minimize the predefined performance index while the other tries to maximize it. Using data generated in real time along the system trajectories, this new method can solve online the game algebraic Riccati equation without requiring the full knowledge of system dynamics. A rigorous proof of convergence of the proposed algorithm is given. Finally, simulation studies on two examples demonstrate the effectiveness of the proposed method.     

A symbolic algorithm for computing recursion operators of nonlinear partial differential equations

A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear partial differential equation (PDE) to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order symmetries, which is a key feature of complete integrability. Completely integrable nonlinear PDEs have a bi-Hamiltonian structure and a Lax pair; they can also be solved with the inverse scattering transform and admit soliton solutions of any order.

A straightforward method for the symbolic computation of polynomial recursion operators of nonlinear PDEs in (1+1) dimensions is presented. Based on conserved densities and generalized symmetries, a candidate recursion operator is built from a linear combination of scaling invariant terms with undetermined coefficients. The candidate recursion operator is substituted into its defining equation and the resulting linear system for the undetermined coefficients is solved.

The method is algorithmic and is implemented in Mathematica. The resulting symbolic package PDERecursionOperator.m can be used to test the complete integrability of polynomial PDEs that can be written as nonlinear evolution equations. With PDERecursionOperator.m, recursion operators were obtained for several well-known nonlinear PDEs from mathematical physics and soliton theory.     

Event-triggered control for semi-global stabilisation of systems with actuator saturation

This paper investigates the problem of event-triggered control for semi-global stabilisation of null controllable systems subject to actuator saturation. First, for a continuous-time system, novel event-triggered low-gain control algorithms based on Riccati equations are proposed to achieve semi-global stabilisation. The algebraic Riccati equation with a low-gain parameter is utilised to design both the event-triggering condition and the linear controller; a minimum inter-event time based on the Riccati ordinary differential equation is set a priori to exclude the Zeno behaviour. In addition, the high–low gain techniques are utilised to extend the semi-global results to event-based global stabilisation. Furthermore, for a discrete-time system, novel low-gain and high–low-gain control algorithms are proposed to achieve event-triggered stabilisation. Numerical examples are provided to illustrate the theoretical results.     

Mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control for linear infinite-dimensional systems

In this paper, we consider mixed H ₂/H _∞ control problems for linear infinite-dimensional systems. The first part considers the state feedback control for the H ₂/H _∞ control problems of linear infinite-dimensional systems. The cost horizon can be infinite or finite time. The solutions of the H ₂/H _∞ control problem for linear infinitedimensional systems are presented in terms of the solutions of the coupled operator Riccati equations and coupled differential operator Riccati equations. The second part addresses the observer-based H ₂/H _∞ control of linear infinite-dimensional systems with infinite horizon and finite horizon costs. The solutions for the observer-based H ₂/H _∞ control problem of linear infinite-dimensional systems are represented in terms of the solutions of coupled operator Riccati equations. The first-order partial differential system examples are presented for illustration. In particular, for these examples, the Riccati equations are represented in terms of the coefficients of first-order partial differential systems.