Operator Riccati Equation Research Articles

ОБ ОДНОМ ПРИЛОЖЕНИИ ОПЕРАТОРНОГО УРАВНЕНИЯ РИККАТИ

ОБ ОДНОМ ПРИЛОЖЕНИИ ОПЕРАТОРНОГО УРАВНЕНИЯ РИККАТИ

Boundary feedback stabilization of the monodomain equations

Boundary feedback control for a coupled nonlinear PDE-ODE system (in the two and three dimensional cases) is studied. Particular focus is put on the monodomain equations arising in the context of cardiac electrophysiology. Neumann as well as Dirichlet based boundary control laws are obtained by an algebraic operator Riccati equation associated with the linearized system. Local exponential stability of the nonlinear closed loop system is shown by a fixed-point argument. Numerical examples are given for a finite element discretization of the two dimensional monodomain equations.

Optimal linear–quadratic control of coupled parabolic–hyperbolic PDEs

ABSTRACTThis paper focuses on the optimal control design for a system of coupled parabolic–hypebolic partial differential equations by using the infinite-dimensional state-space description and the corresponding operator Riccati equation. Some dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the linear–quadratic (LQ)-optimal control problem. A state LQ-feedback operator is computed by solving the operator Riccati equation, which is converted into a set of algebraic and differential Riccati equations, thanks to the eigenvalues and the eigenvectors of the parabolic operator. 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 nonlinear model. Numerical simulations are performed to show the controller performances.

Mixed H 2/H ∞ control for linear infinite-dimensional systems

In this paper, we consider mixed H2/H∞ control problems for linear infinite-dimensional systems. The first part considers the state feedback control for the H2/H∞ control problems of linear infinite-dimensional systems. The cost horizon can be infinite or finite time. The solutions of the H2/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 H2/H∞ control of linear infinite-dimensional systems with infinite horizon and finite horizon costs. The solutions for the observer-based H2/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.

Finite rank perturbations and solutions to the operator Riccati equation

We consider an off-diagonal self-adjoint finite rank perturbation of a self-adjoint operator in a complex separable Hilbert space $\mathfrak{H}_0 \oplus \mathfrak{H}_1$, where $\mathfrak{H}_1$ is finite dimensional. We describe the singular spectrum of the perturbed operator and establish a connection with solutions to the operator Riccati equation. In particular, we prove existence results for solutions in the case where the whole Hilbert space is finite dimensional.

Positive semigroups and algebraic Riccati equations in Banach spaces

We generalize Wonham’s theorem on solvability of algebraic operator Riccati equations to Banach spaces, namely there is a unique stabilizing solution to $$A^*P+PA-PBB^*P+C^*C=0$$ when (A, B) is exponentially stabilizable and (C, A) is exponentially detectable. The proof is based on a new approach that treats the linear part of the equation as the generator of a positive semigroup on the space of symmetric operators from a Banach space to its dual, and the quadratic part as an order concave map. A direct analog of global Newton’s iteration for concave functions is then used to approximate the solution, the approximations converge in the strong operator topology, and the convergence is monotone. The linearized equations are the well-known Lyapunov equations of the form $$A^*P+PA=-Q$$ , and semigroup stability criterion in terms of them is also generalized.

Output regulation problem for a class of regular hyperbolic systems

This paper investigates the output regulation problem for a class of regular first-order hyperbolic partial differential equation (PDE) systems. A state feedback and an error feedback regulator are considered to force the output of the hyperbolic PDE plant to track a periodic reference trajectory generated by a neutrally stable exosystem. A new explanation is given to extend the results in the literature to solve the regulation problem associated with the first-order hyperbolic PDE systems. Moreover, in order to provide the closed-loop stability condition for the solvability of the regulator problems, the design of stabilising feedback gain and its dual problem design of stabilising output injection gain are considered in this paper. This paper develops an easy method to obtain an adjustable stabilising feedback gain and stabilising output injection gain with the aid of the operator Riccati equation.

<inline-formula> <tex-math notation="TeX">$H^{\infty}$</tex-math></inline-formula> Controller Design for Preview and Delayed Systems

The $H^{\infty}$ control problem of general preview/delayed systems is solved using analytic solutions of the corresponding operator Riccati equations. The solution to the problem can be applied to a broad range of input/output delayed systems and enables the handling of preview/delayed control problems. The solvability condition is characterized by the roots of the transcendental equations and the control law for the general problem is given based on a predictive compensation with an integro-differential observer. Some interpretations of typical control problems are presented based on the solvability condition and the resulting control law.

LQ (optimal) control of hyperbolic PDAEs

The linear quadratic control synthesis for a set of coupled first-order hyperbolic partial differential and algebraic equations is presented by using the infinite-dimensional Hilbert state-space representation of the system and the well-known operator Riccati equation (ORE) method. Solving the algebraic equations and substituting them into the partial differential equations (PDEs) results in a model consisting of a set of pure hyperbolic PDEs. The resulting PDE system involves a hyperbolic operator in which the velocity matrix is spatially varying, non-symmetric, and its eigenvalues are not necessarily negative through of the domain. The C0-semigroup generation property of such an operator is proven and it is shown that the generated C0-semigroup is exponentially stable and, consequently, the ORE has a unique and non-negative solution. Conversion of the ORE into a matrix Riccati differential equation allows the use of a numerical scheme to solve the control problem.

Dichotomous Hamiltonians with unbounded entries and solutions of Riccati equations

An operator Riccati equation from systems theory is considered in the case that all entries of the associated Hamiltonian are unbounded. Using a certain dichotomy property of the Hamiltonian and its symmetry with respect to two different indefinite inner products, we prove the existence of nonnegative and nonpositive solutions of the Riccati equation. Moreover, conditions for the boundedness and uniqueness of these solutions are established.

Multi-photon Rabi model: Generalized parity and its applications

Abstract Quantum multi-photon spin–boson model is considered. We solve an operator Riccati equation associated with that model and present a candidate for a generalized parity operator allowing to transform spin–boson Hamiltonian to a block-diagonal form what indicates an existence of the related symmetry of the model.

Optimal Boundary Control of Reaction–Diffusion Partial Differential Equations via Weak Variations

This paper derives linear quadratic regulator (LQR) results for boundary-controlled parabolic partial differential equations (PDEs) via weak variations. Research on optimal control of PDEs has a rich 40-year history. This body of knowledge relies heavily on operator and semigroup theory. Our research distinguishes itself by deriving existing LQR results from a more accessible set of mathematics, namely weak-variational concepts. Ultimately, the LQR controller is computed from a Riccati PDE that must be derived for each PDE model under consideration. Nonetheless, a Riccati PDE is a significantly simpler object to solve than an operator Riccati equation, which characterizes most existing results. To this end, our research provides an elegant and accessible method for practicing engineers who study physical systems described by PDEs. Simulation examples, closed-loop stability analyses, comparisons to alternative control methods, and extensions to other models are also included.

Linear-quadratic regulator with intermediate points for degenerate equations with unbounded operator

We study a linear-quadratic optimal control problem in a Hilbert space where the state equation is unsolvable with respect to the derivative and contains an unbounded operator. The performance index is the sum of the integral of a quadratic form with respect to the control and the state variable on a finite or infinite time interval and also quadratic forms with respect to the differences between the state variable values at fixed points and given values. The optimal control is presented in the feedback form using the implicit differential operator Riccati equation or the operator equation of the Riccati type under a special symmetry condition for the solution. The minimal value of the minimized functional is calculated. Some illustrative examples are also given.

Boundary optimal (LQ) control of coupled hyperbolic PDEs and ODEs

This contribution addresses the development of a linear quadratic (LQ) regulator for a set of hyperbolic PDEs coupled with a set of ODEs through the boundary. The approach is based on an infinite-dimensional Hilbert state-space description of the system and the well-known operator Riccati equation (ORE). In order to solve the optimal control problem, the ORE is converted to a set of matrix Riccati equations. The feedback operator is found by solving the resulting matrix Riccati equations. The performance of the designed control policy is assessed by applying it to a system of interconnected continuous stirred tank reactor (CSTR) and a plug flow reactor (PFR) through a numerical simulation.

Sub-Optimal Control for Nonlinear Heat Equations

This paper presents a successive approximation approach (SAA) designing optimal controllers for a class of nonlinear heat equations with a quadratic performance index. By using the SAA, the optimal control problem for nonlinear heat equations is transformed into a sequence of nonhomogeneous linear differential Riccati operator equations. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoin vector sequence. By using the finite-step iteration of the nonlinear compensation sequence, we can obtain a suboptimal control law.

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.

Optimal LQ-Control of a PDAE Model of a Catalytic Distillation Process

In this contribution a linear quadratic (LQ) control design for a partial differential and algebraic equation (PDAE) system which represents a catalytic distillation process, is presented. The model involves a set of coupled partial differential equations (PDEs), ordinary differential equations (ODEs), and algebraic equations (AEs). The design is based on an infinite-dimensional state-space representation of the system in a Hilbert space and the well-known operator Riccati equation (ORE) method. The underlying PDE-ODE-AE system is converted to one containing coupled PDEs-ODEs, in which the PDE part involves a hyperbolic operator with a space-varying and non-symmetric velocity matrix whose eigenvalues are not necessarily negative through of the domain. Moreover, in the resulting PDE-ODE system, the control variable acts through the ODE part on the boundaries of the PDE section. Using the boundary control transformation method, the PDE-ODE system is represented in an infinite-dimensional state-space with a homogeneous boundary condition. The stabilizability and the detectability properties of the resulting system are explored, which provide guarantees of the existence and uniqueness of the solution to the resulting ORE. The ORE is solved by converting it to a set of equivalent matrix Riccati equations. The designed LQ controller is implemented on the process and its performance is evaluated.

Notes on the Riccati operator equation in open quantum systems

A recent problem [B. Gardas, J. Math. Phys. 52, 042104 (2011)] concerning an antilinear solution of the Riccati equation is solved. We also exemplify that a simplification of the Riccati equation, even under reasonable assumptions, can lead to a not equivalent equation.

LQR control of an infinite dimensional time-varying CSTR-PFR system

AbstractThis contribution addresses the development of a Linear Quadratic Regulator (LQR) for a set of time-varying hyperbolic PDEs coupled with a set of time-varying ODEs through the boundary. The approach is based on an infinite-dimensional Hilbert state-space description of the system and the well-known operator Riccati equation (ORE). In order to solve the optimal control problem, the ORE is converted to a set of equivalent differential and algebraic matrix Riccati equations. The feedback operator can then be found by solving the resulting matrix Riccati equations. The performance of the designed control policy is assessed by applying it to a system of interconnecting continuous stirred tank reactor (CSTR) and a plug flow reactor (PFR) through a numerical simulation.

Operator Stieltjes integrals with respect to a spectral measure and solutions of some operator equations

We introduce the concept of Stieltjes integral of an operator-valued function with respect to the spectral measure associated with a normal operator. We give sufficient conditions for the existence of this integral and find bounds on its norm. The results obtained are applied to the Sylvester and Riccati operator equations. Assuming that the entry C is a normal operator, the spectrum of the entry A is separated from the spectrum of C, and D is a bounded operator, we obtain a representation for the strong solution X to the Sylvester equation XA-CX=D in the form of an operator Stieltjes integral with respect to the spectral measure of C. By using this result we then establish sufficient conditions for the existence of a strong solution to the operator Riccati equation YA-CY+YBY=D where B is another bounded operator.