Linear Quadratic Optimal Control Research Articles

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.

Suboptimal consensus protocol design for a class of multiagent systems

This article presents a new technique for suboptimal consensus protocol design for a class of multiagent systems. The proposed technique is based upon the extension of newly developed sufficient conditions for suboptimal linear–quadratic optimal control design, which are derived in this paper by an explication of a noniterative solution technique of the infinite-horizon linear quadratic regulation problem in Krotov framework. For suboptimal consensus protocol design, the structural requirements on the overall feedback gain matrix, which are inherently imposed by agents dynamics and their interaction topology, are recast on a specific matrix introduced in a suitably formulated convex optimization problem. As a result, preassigning the identical feedback gain matrices to a network of homogeneous agents, which act on the relative state variables with respect to their neighbors is not required. The suboptimality of the computed control laws is quantified by implicitly deriving an upper bound on the cost in terms of the solution of a convex optimization problem and initial conditions of the agents instead of specifying it a priori. Numerical examples are provided to demonstrate the implementation of proposed approaches and their comparison with existing methods in the literature.

Robust stabilization of LTI negative imaginary systems using the nearest negative imaginary controller

AbstractThis paper considers the problem of robust stabilization of linear time‐invariant systems with respect to unmodelled dynamics and structure uncertainties. To that end, a methodology to find the nearest negative imaginary system for a given non‐negative imaginary system is presented first. Then, this result is employed to construct a near optimal linear quadratic Gaussian controller achieving desired performance measures. The problem is formulated using port‐Hamiltonian method and the required conditions are defined in terms of linear matrix inequalities. The technique is presented using the fast gradient method to solve the problem systematically. The designed controller satisfies a negative imaginary property and guarantees a robust feedback loop. The effectiveness of the approach is demonstrated by a simulation on a numerical example.

Optimized fuzzy fractional-order linear quadratic tracking control for a nonlinear system

In this paper, a novel fuzzy fractional-order Linear Quadratic Tracking (LQT) controller optimized by a Multi-Objective gray Wolf Algorithm (MOGWA) is designed for a 6 Degree-Of-Freedom (DOF) quadcopter flying system. In order to reach this end, the state-space equations would be divided into two parts, one for attitude and altitude control and another for position control. The LQT method is implemented to determine the constant parameters of the designed controller by minimizing a quadratic form related to errors and control efforts. Then, to substantially improve the performance of the linear quadratic tracking scheme, the fractional-order derivatives of the system errors are considered as the control inputs, while the MOGWA is used for optimal determination of the related orders. A fuzzy system that employs triangular and trapezoidal membership functions as well as the center average defuzzifier and singleton fuzzifier, is designed to timely regulate the gains of the controller. The introduced strategy would be applied on the nonlinear quadcopter flying system, and the related diagrams would be revealed to illustrate the efficiency and ability of the introduced fuzzy optimal fractional-order linear quadratic tracking controller.

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.

Data-Driven Active Learning Control for Bridge Cranes

For positioning and anti-swing control of bridge cranes, the active learning control method can reduce the dependence of controller design on the model and the influence of unmodeled dynamics on the controller’s performance. By only using the real-time online input and output data of the bridge crane system, the active learning control method consists of the finite-dimensional approximation of the Koopman operator and the design of an active learning controller based on the linear quadratic optimal tracking control. The effectiveness of the control strategy for positioning and anti-swing of bridge cranes is verified through numerical simulations.

Applications of the Linear Quadratic Regulator Optimal Control With Completely and Partially Observed Markovian Switching

Abstract We study optimal control problems where the dynamic system evolves according to a linear stochastic differential equation with (a) multiplicative and additive white noise, (b) a mixture of white noise, and (c) white noise and colored noise. The drift rate and the diffusion matrix of the linear dynamic system depend on a continuous-time Markov chain with finite state space that is (I) partially observed and (II) completely observed. Using the results of Wonham filter theory, we reduce the partially observed problems to one with complete observation. We solve the control optimal problems explicitly with the help of dynamic programing technique, and three applications are presented to illustrate our theoretical results.

Linear quadratic optimal control of a high-performance loudspeaker

Linear quadratic optimal control of a high-performance loudspeaker

Toward an optimal twisting-sliding mode control of a three-phase PMSM for electric vehicles

This paper deals with an optimal twisting sliding mode controller (OT-SMC) for the operation of a three phase permanent magnet synchronous machine (PMSM) in an electric vehicle (EV). In order to drive these vehicles, optimal performance is needed with robust control against real-time disturbances such as the variable load torque, uncertainties such as the parameters variation and speed variations between medium, low, and high speed as well as good performance characteristics for enhanced drive quality and longer battery time. Several conventional techniques have been applied to PMSM but they suffer from the problem of uncertainties and disturbances due to the PI regulator. A hybrid approach comprising of a robust nonlinear and optimal controller to achieve these objectives is attempted for driving electrical vehicles. This advanced hybrid controller obtained after the merger of sliding mode control (SMC) and a linear quadratic controller (LQR) is found to outperform existing controllers due to their superb performance characteristics. Furthermore, SMC is designed based on the exponential reaching law for the twisting sliding mode control (T-SMC) in order to ensure stability of the system while reducing the chattering, accelerating the rate of convergence with higher accuracy of the control performance, and the LQR is developed using the steady-state error method (N-LQR) in order to obtaining better performance characteristics. In addition, the hybridization between a twisting SMC and an optimal LQR is characterized by stabilizing and minimizing the oscillations in the permanent regime thus optimizing the system’s performance. Extensive simulation results illustrate the effectiveness and validity of the proposed control for achieving the highest performance of the PMSM.

Stochastic Linear Quadratic Optimal Control of Speed and Position of Multiple Trains on a Single-Track Line

Stochastic Linear Quadratic Optimal Control of Speed and Position of Multiple Trains on a Single-Track Line

The mean‐field linear quadratic optimal control problem for stochastic systems controlled by impulses

AbstractIn the present note, we state and solve the linear quadratic (LQ) control problem for a class of McKean–Vlasov stochastic differential equations for which the control actions are of impulsive nature. After reformulating the original optimization problem using an orthogonal decomposition of the state variables, we introduce an adequately defined system of two coupled matrix Lyapunov‐type differential equations with jumps. Such equations are central for the definition of the optimal feedback gains corresponding to the closed‐loop representation of the optimal LQ control.

Indefinite linear quadratic optimal control problem for uncertain random discrete-time systems

Indefinite linear quadratic optimal control problem for uncertain random discrete-time systems

Stochastic linear quadratic optimal control problems with expectation-type linear equality constraints on the terminal states

A stochastic linear quadratic (LQ) optimal control problem with an expectation-type linear equality constraint on the terminal state is considered. Under the solvability condition on a stochastic Riccati equation and a surjectivity condition on the linear constraint mapping, the constrained stochastic LQ problem is solved completely by the Lagrangian duality theory. Some equivalent characterizations of the surjectivity condition are discussed by the controllability theory of linear control systems. Especially, the equivalence between the surjectivity condition and a Kalman-type rank condition is proved under proper conditions.

Mean–variance portfolio selection under no-shorting rules: A BSDE approach

This paper revisits the mean–variance portfolio selection problem in continuous-time within the framework of short-selling of stocks is prohibited via backward stochastic differential equation approach. To relax the strong condition in Li et al. (Li et al. 2002), the above issue is formulated as a stochastic recursive optimal linear–quadratic control problem. Due to no-shorting rules (namely, the portfolio taking non-negative values), the well-known “completion of squares” no longer applies directly. To overcome this difficulty, we study the corresponding Hamilton–Jacobi–Bellman (HJB, for short) equation inherently and derive the two groups of Riccati equations. On one hand, the value function constructed via Riccati equations is shown to be a viscosity solution of the HJB equation mentioned before; On the other hand, by means of these Riccati equations and backward semigroup, we are able to get explicitly the efficient frontier and efficient investment strategies for the recursive utility mean–variance portfolio optimization problem.

Research on road tracking and anti-roll game control scheme of commercial vehicle based on Nash equilibrium

In view of the problem that the active suspension conflicts with the road tracking system while resisting roll when the commercial vehicle encounters emergency road conditions in the process of unmanned driving, this article is based on the Nash non-cooperative open-loop feedback game theory, taking the road tracking system, and the active anti-roll system as the participants of the game. An interactive control scheme which can not only accurately steer but also take into account the vertical roll stability is proposed. Firstly, the coupled model of Horizontal pendulum-anti-roll for commercial vehicles is established and extended to the vehicle-road three-degree of freedom closed model. Then, based on the derivation of the linear quadratic optimal control model (LQ), the non-cooperative open-loop Nash game theory is used to obtain the interactive optimal control afferent the road tracking system and the active anti-roll system. Finally, the experimental scheme is verified by hardware in the loop experiment. The results show that the interactive control scheme of road tracking and anti-roll based on Nash non-cooperative game for commercial vehicles allows the vehicle to take the vertical stability into account while turning actively, thus ensuring the safety and stability of the vehicle during driving.

Linear quadratic optimal control for time-delay stochastic system with partial information

ABSTRACT This paper is concerned with the linear quadratic optimal control problem for the time-delay stochastic system with partial information. The main contribution is to derive the equivalent solvability condition and give the explicitly optimal controller under partial information in terms of the Riccati equations. The key is to explicitly solve the forward and backward stochastic difference equations with partial information, which is derived from the stochastic maximum principle.

Stochastic Fokker–Planck Equations for Conditional McKean–Vlasov Jump Diffusions and Applications to Optimal Control

The purpose of this paper is to study optimal control of conditional McKean–Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean–Vlasov jump diffusions, for short). To this end, we first prove a stochastic Fokker–Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate a Hamilton–Jacobi–Bellman equation for the optimal control of conditional McKean–Vlasov jump diffusions. Then we study the situation when the law is absolutely continuous with respect to Lebesgue measure. In that case the Fokker–Planck equation reduces to a stochastic partial differential equation for the Radon–Nikodym derivative of the conditional law. Finally we apply these results to solve explicitly the linear-quadratic optimal control problem of conditional stochastic McKean–Vlasov jump diffusions, and optimal consumption from a cash flow.

Linear quadratic optimal regulation for multiplicative noise systems with special terminal penalty

This paper focuses on the linear quadratic optimal control problem for stochastic systems with a special terminal penalty. The specialty lies in that the terminal penalty is a quadratic term over the conditional expectations of the terminal state. The essential difficulty lies in an additional measurability constraint on the input induced by the conditional expectation term, which is shared by the control problems for the stochastic system with delays, rational expectations problems, control problems involving conditional expectation terms, asymmetric information control and so on. A new idea, dynamic programming combined with orthogonal decomposition, is proposed to deal with this problem. The orthogonal decomposition eliminates the additional measurability constraint on the input. The solvability condition and the optimal controller are obtained by developing generalised Riccati equations. Numerical experiment is provided to support the achieved results.