LQ Control Problem Research Articles

Innerization and Singular LQ Control Problem for Linear Discrete-Time Systems

In this paper, we discuss the LQ problem for discrete-time linear systems with singular weight. The interactor with all-pass property is used in order to make a given strictly proper system be proper. In our approach, the type of Riccati equation to be solved is different from the result for a square transfer function matrix case, where the input weight equals to zero. The relation between our approach and the singular weight case is discussed. Unlike the square case, the explicit solution to the problem cannot be obtained and a recursive method to solve the Riccati equation is shown. An application to the finite settling-time control is also discussed.

On LQ Control Problem for Linear Discrete-Time Systems with Singular Weightings

On LQ Control Problem for Linear Discrete-Time Systems with Singular Weightings

Explicit Solution to the FBSDEs With Asymmetric Information and Application

This brief is focused on the linear forward-backward stochastic differential equations with asymmetric information and multiplicative noises. The main contribution is that we give the explicit solution in terms of Riccati equations. The key lies in establishing the relationship among the forward process and the backward processes. As an application, we solved a kind of stochastic LQ control problems under the asymmetric information by using the derived explicit solution.

The turnpike property and the longtime behavior of the Hamilton–Jacobi–Bellman equation for finite-dimensional LQ control problems

We analyze the consequences that the so-called turnpike property has on the longtime behavior of the value function corresponding to a finite-dimensional linear-quadratic optimal control problem with general terminal cost and constrained controls. We prove that, when the time horizon T tends to infinity, the value function asymptotically behaves as $$W(x) + c\, T + \lambda $$ , and we provide a control interpretation of each of these three terms, making clear the link with the turnpike property. As a by-product, we obtain the longtime behavior of the solution to the associated Hamilton–Jacobi–Bellman equation in a case where the Hamiltonian is not coercive in the momentum variable. As a result of independent interest, we showed that linear-quadratic optimal control problems with constrained control enjoy a turnpike property, also particularly when the steady optimum may saturate the control constraints.

Linear quadratic stochastic optimal control problems with operator coefficients: open-loop solutions

An optimal control problem is considered for linear stochastic differential equations with quadratic cost functional. The coefficients of the state equation and the weights in the cost functional are bounded operators on the spaces of square integrable random variables. The main motivation of our study is linear quadratic (LQ, for short) optimal control problems for mean-field stochastic differential equations. Open-loop solvability of the problem is characterized as the solvability of a system of linear coupled forward-backward stochastic differential equations (FBSDE, for short) with operator coefficients, together with a convexity condition for the cost functional. Under proper conditions, the well-posedness of such an FBSDE, which leads to the existence of an open-loop optimal control, is established. Finally, as applications of our main results, a general mean-field LQ control problem and a concrete mean-variance portfolio selection problem in the open-loop case are solved.

Solutions of differential–algebraic equations as outputs of LTI systems: Application to LQ control problems

In this paper we propose a unified geometric framework for representing all solutions of a Linear Time Invariant Differential–Algebraic Equation (DAE-LTI) as outputs of classical Linear Time Invariant Ordinary Differential Equations (ODE-LTI). The proposed framework is then used to solve an LQ optimal control problem for DAE-LTIs with rectangular matrices.

LQ control based actuator failure compensation

Summary In this paper, LQ control based actuator failure compensation schemes are developed for possibly nonminimum phase overactuated systems with actuator failures. The actuator failure is considered as an unmatched input disturbance signal, and a new disturbance rejection problem is formulated as an LQ control problem using a new cost function to enhance output regulation performance in the presence of actuator failures disturbances. Then the failure compensation design is developed to ensure the maximum reduction of the failure disturbance effect on the system output. New controller structures and design conditions are derived through dynamic programming for compensation of actuator failures, including a finite-time horizon control design and an infinite-time horizon control design, and their observer-based counterparts. Closed-loop stability and output regulation performance are analyzed in details. Simulation results are presented to verify the desired control system performance in the presence of actuator failures, as compared with a traditional design. Moreover, the new LQ based failure compensation schemes are derived in a parametrized framework suitable for adaptive LQ designs to deal with system parameter and failure uncertainties. Copyright © 2015 John Wiley & Sons, Ltd.

Boundary optimal control of coupled parabolic PDE-ODE systems

This paper deals with boundary optimal control problem for coupled parabolic PDEODE systems. The problem is studied using infinite-dimensional state space representation of the coupled PDE-ODE system. Linearization of the non-linear system is established around a steady state profile. Using some state transformations, the linearized system is formulated as a well-posed infinite-dimensional system with bounded input and output operators. It has been shown that the resulting system is a Riesz Spectral system. The LQ-control problem is studied on the basis of the solution of the corresponding eigenvalues problem. The results have been applied to the case study of catalytic cracking reactor with catalyst deactivation. Numerical simulations are performed to illustrate the performances of the developed controller.

Study on Stochastic Linear Quadratic Optimal Control with Quadratic and Mixed Terminal State Constraints

This paper studies the indefinite stochastic LQ control problem with quadratic and mixed terminal state equality constraints, which can be transformed into a mathematical programming problem. By means of the Lagrangian multiplier theorem and Riesz representation theorem, the main result given in this paper is the necessary condition for indefinite stochastic LQ control with quadratic and mixed terminal equality constraints. The result shows that the different terminal state constraints will cause the endpoint condition of the differential Riccati equation to be changed. It coincides with the indefinite stochastic LQ problem with linear terminal state constraint, so the result given in this paper can be viewed as the extension of the indefinite stochastic LQ problem with the linear terminal state equality constraint. In order to guarantee the existence and the uniqueness of the linear feedback control, a sufficient condition is also presented in the paper. A numerical example is presented at the end of the paper.

A design method of suboptimal LQ control law for mode transition systems

AbstractAn LQ control problem for mode transition systems is considered and a design method of sub-optimal control law, which guarantees the upper/lower bounds of the corresponding cost-functional, is derived. The control law is given by piecewise affine function of the state and each gain is easily obtained based on the solutions of finite-horizon LQ control problems. Furthermore, the achievable performance of the control law is characterized in terms of the initial state. The design method is applied to the start-up control of a hot strip mill tension/looper system and the feature of the control system is investigated.

《第5回》ディスクリプタシステムに対するLQ制御と一般化代数 Riccati 方程式

《第5回》ディスクリプタシステムに対するLQ制御と一般化代数 Riccati 方程式

Verified Numerical Computation in LQ Control Problem

Verified Numerical Computation in LQ Control Problem

Numerical algorithms and existence results on LQ control of descriptor systems with conditions on x(0−) and stability

We present a numerical algorithm, based on using orthogonal matrices, and sufficient existence conditions for an LQ control problem of descriptor systems, which generalise the existing results. The associated Hamiltonian pencil may be singular and may have infinite generalised eigenvalues (IGEs) of multiplicity > 1. Besides on x(0+), we present optimality conditions on x(0−). We present sufficient conditions for stability of the optimal closed-loop descriptor system, as well as a stabilisation algorithm.

LQ Control Problem based on Numerical Computation with Guaranteed Accuracy

This paper presents LQ control problem based on numerical computation with guaranteed accuracy. By using the proposed methods, it is possible to guarantee numerical quality about design of control system. This paper also proposes a problem which finds the numerical optimal controller from a set of solutions solved by verified LQ control problem.

Why “state” feedback?

We study the linear quadratic control problem from a representation-free point of view, and we show that this formulation brings forth two self-contained and original proofs of the optimality of state feedback control laws; these proofs which do not depend on an a priori state-space representation. Moreover, we show an orthogonality property characterizing the set of optimal trajectory of a LQ-control problem.

충돌각 제어 호밍유도법칙의 역최적 문제

본 논문에서는 충돌각 구속조건을 갖는 비행체 호밍 제어 유도법칙에 대한 역최적 문제를 제시한다. 편향비례항법 유도법칙의 이득과 LQ 문제에서의 가중치와의 관계를 규명하고, Riccati 방정식으로부터 제어입력이 LQ 최적제어가 되기 위한 영역을 제시한다. 이를 근거로 종말 구속조건을 만족하는 호밍 유도법칙의 제어이득이 최적제어법칙이 되기 위한 범위를 제안한다. 이론적 해석 결과의 타당성은 3-DOF 모의시험을 통하여 확인한다. An inverse optimal problem for homing guidance with angular constraint is addressed. The gains of BPN(Biased PN) are investigated by duality analysis related to the weighting matrices of the performance index in the LQ control problem. Moreover, the criteria for the existence of optimal gains are derived from the generalized Riccati equation. Based on the conditions we achieve the gain set of BPN to be optimal solution to the LQ problem with terminal constraints. To validate and demonstrate the proposed approach 3-DOF simulations are carried out.

Formula omitted]-optimal rejection with preview in the continuous-time domain

The synthesis of a feedforward unit for H 2 -optimal rejection of previewed signals in continuous time-invariant linear systems is considered. The H 2 -optimal compensator herein devised consists of a finite impulse response system working in connection with a standard dynamic unit. The design strategy is based on a simple interpretation of the H 2 -optimal rejection problem with preview as a compound optimal control problem, i.e. as a problem consisting of a finite-horizon LQ control problem with constrained final state, an infinite-horizon LQ control problem, and the problem of minimizing the global cost functional. An explicit description of the compensator is derived by exploiting some results on the parameterization of the solutions of the associated autonomous Hamiltonian system.

A MODEL PREDICTIVE CONTROL BASED ON INPUT-OUTPUT CHARACTERIZATION OF FINITE-HORIZON LINEAR SYSTEMS

An LQ control problem for constrained discrete-time systems is discussed based on model predictive control with reduced order quadratic programming (QP) problem. By introducing a singular value decomposition for linear systems, where zero-order holder is embedded in the input signals, an approximation method of model predictive control is derived. The feature of resulting control system is illustrated with numerical examples.

GENERAL LQ PROBLEM FOR INFINITE JUMP LINEAR SYSTEMS AND THE MINIMAL SOLUTION OF ALGEBRAIC RICCATI EQUATIONS

The paper addresses the LQ control problem for systems with countable Markov jump parameters, and the associated coupled algebraic Riccati equations. The problem is considered in a general optimization setting in which the solution is not required to be stabilizing in any sense. We show that a necessary and sufficient condition for a solution to the control problem to exist is that the Riccati equations have a nonempty set of solutions, which generalizes previous known results requiring stabilizability as a sufficient condition. We clarify the connection between the minimal solution of the Riccati equation and the control problem, showing that the minimal solution provides the synthesis of the optimal control. The derived results strengthen the relations of the theory of Markov jump systems with the one of linear deterministic systems. An illustrative example is included.

Feedback stabilization of Navier–Stokes equations

One proves that the steady-state solutions to Navier{Stokes equations with internal con- trollers are locally exponentially stabilizable by linear feedback controllers provided by a LQ control problem associated with the linearized equation. yt(x;t) y(x;t )+( y r)y(x;t )= m(x)u(x;t )+ f0(x )+ rp(x;t); (x;t) 2 Q ( ry)(x;t )=0 ; 8 (x;t) 2 Q =(0;1) y =0 ; on = @(0;1) y(x; 0) =y0(x) ;x 2 :