Linear Quadratic Optimal Control Problem Research Articles

Linear quadratic optimal control turnpike in finite and infinite dimension: Two-term expansion of the value function

In this paper, we consider a linear quadratic (LQ) optimal control problem in both finite and infinite dimensions. We derive an asymptotic expansion of the value function as the fixed time horizon T tends to infinity. The leading term in this expansion, proportional to T, corresponds to the optimal value attained through the classical turnpike theory in the associated static problem. The remaining terms are associated with optimal stabilization problems towards the turnpike.

Decentralized stochastic linear-quadratic optimal control with risk constraint and partial observation

This paper addresses a risk-constrained decentralized stochastic linear-quadratic optimal control problem with one remote controller and one local controller, where the risk constraint is posed on the cumulative state weighted variance in order to reduce the oscillation of system trajectory. In this model, local controller can only partially observe the system’s state, and sends the estimate of state to remote controller through an unreliable channel, whereas the channel from remote controller to local controller is perfect; here, the unreliable channel means the Bernoulli channel, that is, when data is transmitted on this channel, there is a certain probability of failure. For the considered constrained optimization problem with Bernoulli channel, we first punish the risk constraint into cost function through Lagrange multiplier method, and the resulting augmented cost function will include a quadratic mean-field term of state. In the sequel, for any but fixed multiplier, explicit solutions to finite-horizon and infinite-horizon mean-field decentralized linear-quadratic problems are derived together with necessary and sufficient condition on the mean-square stability of optimal system. Then, approach to find the optimal Lagrange multiplier is presented based on bisection method. Finally, two numerical examples are given to show the efficiency of the obtained results.

Optimal control for both forward and backward discrete-time systems

Forward discrete-time systems use past information to update the current state, while backward discrete-time systems use future information to update the current state. This study focuses on optimal control problems within the context of forward and backward discrete-time systems. We begin by investigating a general optimal control problem for both forward and backward discrete-time systems. Leveraging the inherent properties of these systems and the Bellman optimality principle, we derive recursive equations as a means to solve such optimal control problems. Using these recursive equations, we obtain analytical expressions for both the optimal controls and optimal values of bang–bang and linear quadratic optimal control problems. Finally, we present a numerical example and an industrial wastewater treatment problem to illustrate and demonstrate our findings.

Pointwise Error Estimates of Numerical Solutions to Linear Quadratic Optimal Control Problems

An auxiliary optimal control problem is formulated that provides with its unique solution, a continuous representation of the global error of a numerical approximation to the solution of a linear quadratic optimal control problem. The resulting error functions are characterized as the unique solutions to an optimality system that is reformulated as a boundary value problem. With this formulation, reliable pointwise error estimates can be generated utilizing well-established techniques of defect control. A novel algorithm based on defect correction and defect control is presented that generates pointwise approximations to the global error of numerical optimal control solutions on a uniform grid. It is proven and numerically validated that this algorithm can generate pointwise estimates that approximate the true global error with a prescribed accuracy.

Turnpike Properties for Mean-Field Linear-Quadratic Optimal Control Problems

Turnpike Properties for Mean-Field Linear-Quadratic Optimal Control Problems

Forward–backward stochastic evolution equations in infinite dimensions and application to LQ optimal control problems

This paper focuses on the study of forward–backward stochastic evolution equations (FBSEEs), which are a class of nonlinear fully coupled forward–backward stochastic differential equations (FBSDEs), in infinite dimensions. Drawing inspiration from various linear-quadratic (LQ) optimal control problems, we apply a set of domination-monotonicity conditions that are more relaxed compared to general conditions. Within this framework, we employ the method of continuation to establish the unique solvability result and provide a pair of solution estimates. Notably, to address the challenges posed by the infinite-dimensional setting, we introduce a class of approximating equations, as Itô’s formula is not directly applicable. Conversely, these results find application in various LQ problems, where the stochastic Hamiltonian systems precisely correspond to the FBSEEs satisfying the aforementioned domination-monotonicity conditions. Consequently, by solving the corresponding stochastic Hamiltonian systems, we can obtain explicit expressions for the optimal controls.

Value iteration algorithm for continuous-time linear quadratic stochastic optimal control problems

Value iteration algorithm for continuous-time linear quadratic stochastic optimal control problems

Value iteration for LQR control of unknown stochastic-parameter linear systems

This paper focuses on the linear-quadratic optimal control problem for unknown stochastic-parameter linear systems using reinforcement learning methods. Based on the second moments of random system matrices, a model-based value iteration algorithm is proposed to solve the problem and is proved to be convergent by using the contraction mapping theorem. For the case without knowing any information about the random system matrices, a normalized model-free value iteration algorithm is presented to learn the optimal control law by estimating the data for the next time. The collected data are normalized first in our model-free algorithm to reduce the error of the least squares method. It is proved that our algorithm can obtain an approximate optimal solution. Our algorithm is applicable for any distribution of the random system matrices and does not require an initial mean-square stabilizing control policy. Finally, an example illustrates that our algorithm converges to an approximate optimal control policy and that the normalization step can significantly reduce convergence errors.

Linear quadratic stochastic optimal control with state- and control-dependent noises: A deterministic data approach

This paper investigates an infinite-horizon linear quadratic stochastic optimal control (LQSOC) problem with state- and control-dependent noises, in which two system matrices are unknown. First, a deterministic system is introduced and a relationship among its system trajectory, its control and the matrices to be solved is formulated. Subsequently, based on the deterministic system, an adaptive dynamic programming (ADP) algorithm is established to tackle the LQSOC problem. Then, the convergence analysis is presented by proving the equivalence between the proposed algorithm and an existing policy iteration algorithm. Finally, the effectiveness of the obtained algorithm is verified by a perturbed turbocharged diesel engine optimal control design example.

Lagrangian dual method for solving stochastic linear quadratic optimal control problems with terminal state constraints

A stochastic linear quadratic (LQ) optimal control problem with a pointwise linear equality constraint on the terminal state is considered. A strong Lagrangian duality theorem is proved under a uniform convexity condition on the cost functional and a surjectivity condition on the linear constraint mapping. Based on the Lagrangian duality, two approaches are proposed to solve the constrained stochastic LQ problem. First, a theoretical method is given to construct the closed-form solution by the strong duality. Second, an iterative algorithm, called augmented Lagrangian method (ALM), is proposed. The strong convergence of the iterative sequence generated by ALM is proved. In addition, some sufficient conditions for the surjectivity of the linear constraint mapping are obtained.

Linear quadratic optimal control problems for stochastic evolution equations in infinite horizon

Linear quadratic optimal control problems for stochastic evolution equations in infinite horizon

Value functional and optimal feedback control in linear-quadratic optimal control problem for fractional-order system

In this paper, a finite-horizon optimal control problem involving a dynamical system described by a linear Caputo fractional differential equation and a quadratic cost functional is considered. An explicit formula for the value functional is given, which includes a solution of a certain Fredholm integral equation. A step-by-step feedback control procedure for constructing $ \varepsilon $-optimal controls with any accuracy $ \varepsilon > 0 $ is proposed. The basis for obtaining these results is the study of a solution of the associated Hamilton–Jacobi–Bellman equation with so-called fractional coinvariant derivatives.

Linear-quadratic optimal control problem for mean-field stochastic differential equations with a type of random coefficients

Linear-quadratic optimal control problem for mean-field stochastic differential equations with a type of random coefficients

Accurate solutions of interval linear quadratic regulator optimal control problems with fractional-order derivative

Accurate solutions of interval linear quadratic regulator optimal control problems with fractional-order derivative

Stochastic linear quadratic optimal control problem with terminal state constraints

AbstractThis paper addresses the stochastic linear quadratic (LQ) control problem with first‐ and second‐order moment constraints on the terminal state. The problem is a modified version of the optimal covariance control problem, where the terminal state is steered to a given probability distribution. Studying a multiplicative‐noise stochastic system rather than an additive‐noise system is a salient feature. Unlike the existing ideas in the optimal steering, by using the Lagrange multipliers method and establishing the stochastic maximum principle, our problem is converted into solving forward–backward stochastic difference equations (FBSDEs), which is a special stochastic two‐point boundary‐value problem (TPBVP). We provide the optimal closed‐loop controller and necessary and sufficient solvability conditions by developing a nonhomogeneous relationship between the optimal state and costate in FBSDEs. Finally, numerical examples are given to demonstrate our results.

Stochastic optimal control and piecewise parameterization and optimization method for inventory control system improvement

In recent decades, stochastic control systems have been widely used in industrial production, biomedicine, aerospace, military strategy and so on. In this paper, an approximate optimal strategy derived from a stochastic linear quadratic (SLQ) optimal control problem is considered, and a piecewise parameterization and optimization (PPAO) method is proposed. Firstly, using the principle of dynamic programming, the control form of SLQ optimal control problem is relevant to a Riccati differential equation. It is well known that the Riccati differential equation is difficult to be solved analytically. Thus, we present a PPAO method for finding an approximate optimal strategy for stochastic control problems. Here, a piecewise parameter control can be obtained by solving first-order differential equations rather than Riccati differential equations. Finally, the inventory control problems with different dimensions are used to justify the feasibility of PPAO method, and the results compared with the original optimal control are given. The results show that parametric control greatly simplifies the control form, and a small model error is ensured.

Linear-quadratic optimal control of second-order parabolic systems

This paper considers the linear-quadratic (LQ) optimal control problem for systems governed by a class of second-order parabolic partial differential equations (PDEs). This problem is significant in many areas of mathematics, control, engineering and so on, and thus has received great attention in the past decades. Different from the previous articles where the operator is applied to present the controller, the main contribution of this paper is to propose the discretisation-then-continuousization method, which is explicit and implementable. The solvability condition of the LQ optimal control problems is given based on a set of differential Riccati equations, and an explicit numerical calculation way of these equations and the design of the optimal controller are provided.

A generalization of the Riccati recursion for equality‐constrained linear quadratic optimal control

AbstractThis paper introduces a generalization of the well‐known Riccati recursion for solving the discrete‐time equality‐constrained linear quadratic optimal control problem. The recursion can be used to compute problem solutions as well as optimal feedback control policies. Unlike other tailored approaches for this problem class, the proposed method does not require restrictive regularity conditions on the problem. This allows its use in nonlinear optimal control problem solvers that use exact Lagrangian Hessian information. We demonstrate that our approach can be implemented in a highly efficient algorithm that scales linearly with the horizon length. Numerical tests show a significant speed‐up of about one order of magnitude with respect to state‐of‐the‐art general‐purpose sparse linear solvers. Based on the proposed approach, faster nonlinear optimal control problem solvers can be developed that are suitable for more complex applications or for implementations on low‐cost or low‐power computational platforms. The implementation of the proposed algorithm is made available as open‐source software.

Robust Solution of the Multi-Model Singular Linear-Quadratic Optimal Control Problem: Regularization Approach

We consider a finite horizon multi-model linear-quadratic optimal control problem. For this problem, we treat the case where the problem’s functional does not contain a control function. The latter means that the problem under consideration is a singular optimal control problem. To solve this problem, we associate it with a new optimal control problem for the same multi-model system. The functional in this new problem is the sum of the original functional and an integral of the square of the Euclidean norm of the vector-valued control with a small positive weighting coefficient. Thus, the new problem is regular. Moreover, it is a multi-model cheap control problem. Using the solvability conditions (Robust Maximum Principle), the solution of this cheap control problem is reduced to the solution of the following three problems: (i) a terminal-value problem for an extended matrix Riccati type differential equation; (ii) an initial-value problem for an extended vector linear differential equation; (iii) a nonlinear optimization (mathematical programming) problem. We analyze an asymptotic behavior of these problems. Using this asymptotic analysis, we design the minimizing sequence of state-feedback controls for the original multi-model singular optimal control problem, and obtain the infimum of the functional of this problem. We illustrate the theoretical results with an academic example.

Linear Quadratic Optimal Control of Discrete-Time Stochastic Systems Driven by Homogeneous Markov Processes

Random terms in many natural and social science systems have distinct Markovian characteristics, such as Markov jump-taking values in a finite or countable set, and Wiener process-taking values in a continuous set. In general, these systems can be seen as Markov-process-driven systems, which can be used to describe more complex phenomena. In this paper, a discrete-time stochastic linear system driven by a homogeneous Markov process is studied, and the corresponding linear quadratic (LQ) optimal control problem for this system is solved. Firstly, the relations between the well-posedness of LQ problems and some linear matrix inequality (LMI) conditions are established. Then, based on the equivalence between the solvability of the generalized difference Riccati equation (GDRE) and the LMI condition, it is proven that the solvability of the GDRE is sufficient and necessary for the well-posedness of the LQ problem. Moreover, the solvability of the GDRE and the feasibility of the LMI condition are established, and it is proven that the LQ problem is attainable through a certain feedback control when any of the four conditions is satisfied, and the optimal feedback control of the LQ problem is given using the properties of homogeneous Markov processes and the smoothness of the conditional expectation. Finally, a practical example is used to illustrate the validity of the theory.