Quadratic Optimal Control Problem Research Articles

Uncertain random problem for multistage switched systems

<abstract><p>Optimal control problems for switched systems how best to switch between different subsystems. In this paper, two kinds of linear quadratic optimal control problems for multistage switched systems composing of both randomness and uncertainty are studied. Chance theory brings us a useful tool to deal with this indeterminacy. Based on chance theory and Bellman's principle, the analytical expressions are derived for calculating both the optimal control input and the optimal switching control law. Optimal control is implemented by genetic algorithm instead of enumerating all the elements of a series of sets whose size grows exponentially. Finally, the results of numerical examples are provided to illustrate the effectiveness of the proposed method.</p></abstract>

A Value Iteration Approach to Adaptive Optimal Control of Linear Time-Delay Systems

This paper studies the adaptive optimal control for linear time-delay systems described by delay differential equations (DDEs). A key strategy is to exploit the value iteration (VI) approach to solve the linear quadratic optimal control problem for time-delay systems. However, previous learning-based control methods are all exclusively devoted to discrete-time time-delay systems. In this article, we aim to fill in the gap by developing a learning-based VI approach to solve the infinite-dimensional algebraic Riccati equation (ARE) for continuous-time time-delay systems. One nice feature of the proposed VI approach is that an initial admissible controller is not required to start the algorithm. The efficacy of the proposed methodology is demonstrated by the example of autonomous driving.

The Numerical Solution for Quadratic Optimal Control Problems by Using Chebyshev and Legendre Polynomials

الغرض من هذا البحث هو حل مشاكل التحكم الأمثل من الدرجة الثانية (QOCP) عدديًا بمساعدة متعددة حدود Chebyshev و Legendre كوظائف أساسية لإيجاد حل للتحكم الأمثل (QOC) تقريبًا. سنشرح خوارزميات الحل بالأمثلة ونستخدم برنامج Mathcad للوصول إلى النتيجة الدقيقة.مقدمةتتمثل مشكلة التحكم المثلى في العثور على عنصر تحكم u ^ * (t) يقلل من مؤشر أداء معين مع تلبية معادلات حالة النظام والقيود. [1]نستخدم طرق التقريب لحل مشكلة التحكم الأمثل اعتمادًا على متعدد حدود Chebyshev في المرة الأولى ومتعدد حدود Legendre ، وبعد ذلك سنقترب من هذه الحلول ذات الوقت الخطي المستمر. للوصول إلى الحلول التقريبية ، نستخدم المعادلات التفاضلية الخطية متعددة المدى لـ u (t) و x (t) لكل من متعدد حدود Chebyshev و Legendre ونجعل شروط هذه المعادلات كمصفوفة مربعة للعثور على هذه القيم بواسطة نظام المصفوفات.عندما نستخدم هذه كثيرات الحدود في حلول تقريبية ، تم تقييم النتائج باستخدام الفهرس مع n = 5.سنشرح هذه الخوارزميات من خلال أخذ بعض الأمثلة لمشاكل التحكم التربيعي.تم تحديد المشكلة التربيعية الخطية على النحو التالي ؛قلل من الوقت المستمر التربيعيدالة التكلفة J = ∫_ (t_0) ^ (t_f) ▒ 〖(x ^ T Qx + u ^ T Ru) □ (24 & dt)〗 ... (1)مع مراعاة معادلات حالة النظام الخطي ؛س ̇ (t) = Dx (t) + Eu (t) ، ... (2)حيث الشرط الأولي x (0) = x_0 والمصفوفات (D و E و Q و R) هي

Open-Loop Decentralized LQ Control Problem With Multiplicative Noise

This article considers the decentralized linear–quadratic (LQ) optimal control problem for the multiplicative-noise stochastic systems in discrete time with asymmetric information. There exist two controllers in the system, and each controller has its own information in general structure. The adopted information structure is adapted open loop for both controllers. The main contribution is to derive the necessary and sufficient condition for the unique solvability of the LQ control problem with a general asymmetric information structure and present the explicitly optimal controllers in terms of the proposed Riccati equations. The key is to explicitly solve the forward and backward stochastic difference equations, which are derived from the maximum principle.

Closed-Loop Solvability of Stochastic Linear-Quadratic Optimal Control Problems with Poisson Jumps

The stochastic linear–quadratic optimal control problem with Poisson jumps is addressed in this paper. The coefficients in the state equation and the weighting matrices in the cost functional are all deterministic but are allowed to be indefinite. The notion of closed-loop strategies is introduced, and the sufficient and necessary conditions for the closed-loop solvability are given. The optimal closed-loop strategy is characterized by a Riccati integral–differential equation and a backward stochastic differential equation with Poisson jumps. A simple example is given to demonstrate the effectiveness of the main result.

Preconditioners for robust optimal control problems under uncertainty

AbstractThe discretization of robust quadratic optimal control problems under uncertainty using the finite element method and the stochastic collocation method leads to large saddle‐point systems, which are fully coupled across the random realizations. Despite its relevance for numerous engineering problems, the solution of such systems is notoriously challenging. In this manuscript, we study efficient preconditioners for all‐at‐once approaches using both an algebraic and an operator preconditioning framework. We show in particular that for values of the regularization parameter not too small, the saddle‐point system can be efficiently solved by preconditioning in parallel all the state and adjoint equations. For small values of the regularization parameter, robustness can be recovered by the additional solution of a small linear system, which however couples all realizations. A mean approximation and a Chebyshev semi‐iterative method are proposed to solve this reduced system. We consider a random elliptic partial differential equation whose diffusion coefficient is modeled as an almost surely continuous and positive random field, though not necessarily uniformly bounded and coercive. We further provide estimates of the dependence of the spectrum of the preconditioned system matrix on the statistical properties of the random field and on the discretization of the probability space. Such estimates involve either the first or second moment of the random variables and , where is the spatial domain. The theoretical results are confirmed by numerical experiments, and implementation details are further addressed.

Linear Quadratic Optimal Control Problem for Linear Stochastic Generalized System in Hilbert Spaces

A finite-horizon linear stochastic quadratic optimal control problem is investigated by the GE-evolution operator in the sense of the mild solution in Hilbert spaces. We assume that the coefficient operator of the differential term is a bounded linear operator and that the state and input operators are time-varying in the dynamic equation of the problem. Optimal state feedback along with the well-posedness of the generalized Riccati equation is obtained for the finite-horizon case. The results are also applicable to the linear quadratic optimal control problem of ordinary time-varying linear stochastic systems.

Linear–Quadratic Optimal Control for Discrete-Time Mean-Field Systems With Input Delay

The linear–quadratic (LQ) optimal control and stabilization problems for mean-field systems with input delay (MFSID) are investigated in this article. The necessary and sufficient solvability conditions for LQ control of MFSID are first given in terms of a convexity condition and the solvability of equilibrium conditions. Consequently, by solving the associated mean-field forward and backward stochastic difference equations, the optimal control is derived in terms of the solution of a modified Riccati equation. Furthermore, for the infinite-horizon case, the stabilization problem for MFSID is studied, and the necessary and sufficient stabilizability conditions are obtained. We show that MFSID can be mean square stabilizable if and only if a modified algebraic Riccati equation admits a unique positive-definite solution.

Discrete LQR and ILQR methods based on high order Runge-Kutta discretizations

In this paper, discrete linear quadratic regulator (DLQR) and iterative linear quadratic regulator (ILQR) methods based on high-order Runge-Kutta (RK) discretizations are proposed for solving linear and nonlinear quadratic optimal control problems respectively. As discovered in Hager (2000) [11], direct approaches with RK discretization are equivalent with indirect approaches based on symplectic partitioned Runge-Kutta (SPRK) integration. In this paper, we will reconstruct this equivalence by the analogue of continuous and discrete dynamic programming. Then, based on the equivalence, we discuss the issue that the internal-stage controls produced by direct approaches may have lower order accuracy than the RK method used. We propose order conditions for internal-stage controls and then demonstrate that third or fourth order explicit RK methods cannot avoid the order reduction phenomenon. To overcome this obstacle, we calculate node control instead of internal-stage controls in DLQR and ILQR methods. Another advantage of our methods is high computational efficiency which comes from the usage of feedback technique. A simple numerical example will illustrate the validity of ILQR in solving nonlinear optimal control problems with high-order RK discretizations. In this paper, we also demonstrate that ILQR is essentially a quasi-Newton method with linear convergence rate.

Slow Decay and Turnpike for Infinite-Horizon Hyperbolic Linear Quadratic Problems

This paper is devoted to analyzing the explicit slow decay rate and turnpike in infinite-horizon linear quadratic optimal control problems for hyperbolic systems. Under suitable weak observability or controllability conditions, lower and upper bounds of the corresponding algebraic Riccati operator are proved. Then based on these two bounds, the explicit slow decay rate of the closed-loop system with Riccati-based optimal feedback control is obtained. The averaged turnpike property for this problem is also further discussed. We then apply these results to LQ optimal control problems constrained to networks of one-dimensional wave equations and also some multidimensional ones with local controls which lack a geometric control condition.

Indefinite linear quadratic optimal control for discrete time‐varying linear rectangular descriptor systems

AbstractIn this article, the indefinite linear quadratic (LQ) optimal control problem for discrete time‐varying rectangular descriptor systems is investigated, in which the indefinite weight matrices are considered in the quadratic cost function. With a suite of equivalent transformations, the indefinite LQ problem for discrete time‐varying rectangular descriptor systems can be substituted by an equivalent indefinite LQ problem for standard state space systems. Then, the transformed equivalent indefinite LQ problem is solved by establishing a dual Krein space model, and the optimal control and optimal cost value are obtained by invoking the Kalman filter of Krein space. The form of state feedback optimal control corresponding to the original system is given. Next, to guarantee the dynamic part for the resulting optimal closed‐loop system has a unique solution, some necessary and sufficient conditions are proposed. Finally, a numerical example is carried out to demonstrate the availability of established results.

Decentralized control for multi-sensors networked systems with different transmission delays and packet dropouts

In this paper, the linear quadratic (LQ) optimal decentralized control and stabilization problems are investigated for multi-sensors networked control systems (MSNCSs) with multiple controllers of different information structure. Specifically, for a MSNCS, in view of the packet dropouts and the transmission delays, each controller may access different information sets. To begin with, the sufficient and necessary solvability conditions for the LQ decentralized control problems are developed. Consequently, for the purpose of deriving the optimal decentralized control strategy, an innovative orthogonal decomposition method is proposed to decouple the forward and backward stochastic difference equations (FBSDEs) from the maximum principle. In the following, we show that the optimal decentralized controller can be calculated according to a set of Riccati-type equations. Finally, a stabilizing controller is derived for the stabilization problem.

Numerical Methods for Solving Linear Time Varying Quadratic Optimal Control Problems

In this article, we discussed linear time varying optimal control problems with quadratic performance index, and approximated control variable, state variable and performance index. There are many different numerical processes for approximating of linear time varying quadratic optimal control problem. Two of those are explained here the Legendre scaling function and Chebyshev scaling function methods using the MATLAB. The methods will be explained, and then applied to test problems to see how they perform. The results show that the Legendre scaling function was the best of the Chebyshev scaling function methods being a computation. Hence, the Legendre scaling function method is more suitable for solving the linear time varying quadratic optimal control systems.

Optimal control of fractional Sturm–Liouville wave equations on a star graph

ABSTRACT In the present paper, we are concerned with a fractional wave equation of Sturm–Liouville type in a general star graph. We first give several existence, uniqueness and regularity results of weak solutions for the one-dimensional case using the spectral theory; we prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal control via the Euler–Lagrange first-order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary conditions and controls of the velocity. We show the existence and uniqueness of minimizers, and by using the first-order optimality conditions with the Lagrange multipliers, we are able to characterize the optimal controls.

Distributed optimal control problems driven by space-time fractional parabolic equations

Abstract We study distributed optimal control problems, governed by space-time fractional parabolic equations (STFPEs) involving time-fractional Caputo derivatives and spatial fractional derivatives of Sturm-Liouville type. We first prove existence and uniqueness of solutions of STFPEs on an open bounded interval and study their regularity. Then we show existence and uniqueness of solutions to a quadratic distributed optimal control problem. We derive an adjoint problem using the right-Caputo derivative in time and provide optimality conditions for the control problem. Moreover, we propose a finite difference scheme to find the approximate solution of the considered optimal control problem. In the proposed scheme, the well-known L1 method has been used to approximate the time-fractional Caputo derivative, while the spatial derivative is approximated using the Grünwald-Letnikov formula. Finally, we demonstrate the accuracy and the performance of the proposed difference scheme via examples.

Indefinite linear quadratic optimal control for continuous-time rectangular descriptor Markov jump systems: infinite-time case

The indefinite linear quadratic (ILQ) optimal control problem has many important applications in financial, economic systems, etc., which has been widely researched in stochastic systems and descriptor systems, etc. However, for ILQ problem of rectangular descriptor Markov jump systems (DMJSs), because there are impulses simultaneously in descriptor subsystems and at the switching time, it is theoretically difficult to research and there is no result yet. This paper discusses the ILQ optimal control problem for continuous-time linear rectangular DMJSs. Firstly, under some rank conditions and inequality conditions, the ILQ problem for rectangular DMJSs can be equivalently transformed into standard LQ problem for Markov jump systems (MJSs) by using elementary linear algebra method. Then based on the LQ theory of MJSs, the solvable sufficient condition of the ILQ problem for rectangular DMJSs and the non-negative optimal cost value are obtained. The optimal control can be synthesised as state feedback, and the resulting optimal closed-loop system has the stochastically stable solution. In addition, with some rank inequality assumptions, the differential subsystem of the resulting optimal closed-loop system can be ensured to have a unique solution. Finally, two numerical examples are provided to illustrate the effectiveness of the methods proposed in this paper.

Optimal solution of the liquidation problem under execution and price impact risks

We consider an investor that trades continuously and wants to liquidate an initial asset position within a prescribed time interval. As a consequence of his trading activity, during the execution of the liquidation order, the investor has no guarantees that the placed order is executed immediately; it may go unfilled, partially filled or filled in excess. The uncertainty in the execution affects the trading activity of the investor and the asset share price dynamics generating additional sources of noise: the execution risk and the price impact risk, respectively. Assuming the two sources of noise correlated and driven by the cumulative effect of the investor trading strategy, we study the problem of finding the optimal liquidation strategy adopted by the investor in order to maximize the expected revenue resulting from the liquidation. The mathematical model of the liquidation problem presented here extends the model of Almgren and Chriss [Optimal execution of portfolio transactions. J. Risk, 2000, 3(2), 5–39] to include execution and price impact risks. The liquidation problem is modeled as a linear quadratic stochastic optimal control problem with the finite horizon and, under some assumptions about the functional form for the magnitude of execution and price impact risks, is solved explicitly. The derived solution coincides with the optimal trading strategy obtained in the absence of execution uncertainty for an asset price with a modified growth rate. This suggests that the uncertainty in the execution modifies the directional view of the investor about the future growth rate of the asset price.

LQ control of Itô stochastic system with asymmetric information

This paper is concerned with the linear quadratic (LQ) optimal control problem for the stochastic system under asymmetric information structure. The system contains two controllers and multiplicative noises where each controller has its own information generated by part of the noises. The main contribution is to give the explicitly optimal controllers via Riccati equations. Through the stochastic maximum principle, the solvability of the optimal control problem is reduced to that of corresponding forward and backward stochastic differential equations (FBSDEs). The innovation of this paper is to establish the relationship between forward and backward processes under the general asymmetric information.

Well-posed control problems related to second-order differential inclusions

<p style='text-indent:20px;'>The paper deals with quadratic optimal control problems, we study the equivalence between well-posed problems and affinity on the control for a second-order differential inclusions with two-points conditions, governed by a maximal monotone operator in a finite dimensional space.</p>

A Sequential Convex Programming Approach to Solving Quadratic Programs and Optimal Control Problems With Linear Complementarity Constraints

Mathematical programs with complementarity constraints are notoriously difficult to solve due to their nonconvexity and lack of constraint qualifications in every feasible point. This letter focuses on the subclass of quadratic programs with linear complementarity constraints. A novel approach to solving a penalty reformulation using sequential convex programming and a homotopy on the penalty parameter is introduced. Linearizing the necessarily nonconvex penalty function yields convex quadratic subproblems, which have a constant Hessian matrix throughout all iterates. This allows solution computation with a single KKT matrix factorization. Furthermore, a globalization scheme is introduced in which the underlying merit function is minimized analytically, and guarantee of descent is provided at each iterate. The algorithmic features and possible computational speedups are illustrated in a numerical experiment.