Basis functions design for the approximation of constrained linear quadratic regulator problems encountered in model predictive control

In this paper, the Linear Quadratic Regulator Problem with a positivity constraint on the admissible control set is addressed. Necessary and sufficient conditions for optimality are presented in terms of inner products, projections on closed convex sets, Pontryagin's maximum principle and dynamic programming. The main results are concerned with smoothness of the optimal control and the value function. The maximum principle will be extended to the infinite horizon case. Based on these analytical methods, we propose a numerical algorithm for the computation of the optimal controls for the finite and infinite horizon problem. The numerical methods will be justified by convergence properties between the finite and infinite horizon case on one side and discretized optimal controls and the true optimal control on the other.

In this paper, the Linear Quadratic Regulator Problem with a positively constraint on the admissible control set is addressed. Necessary and sufficient conditions for optimality are presented in terms of inner products, projections on closed convex sets. Pontryagin's maximum principle and dynamic programming. Sufficient and sometimes necessary conditions for the existence of positive stabilizing controls are incorporated. Convergence properties between the finite and infinite horizon case are presented. Besides these analytical methods, we describe briefly a method for the approximation of the optimal controls for the finite and infinite horizon problem.

The sampled-data linear quadratic tracking problem has been transformed into the well known discrete-time linear quadratic tracking problem, just as the sampled-data linear quadratic regulator problem may be transformed into the well known discrete-time linear quadratic regulator problem.

ABSTRACTThis work proposes a new technique to characterize the suboptimality of stabilizing control laws for the infinite‐horizon linear quadratic regulation problem, wherein the norm of the state feedback gain matrix is conditioned to be lesser than a given bound. In contrast to existing techniques where the upper bound on the cost is pre‐specified, this work proposes to compute an upper bound associated with the suboptimal control law(s). To derive the norm‐bounded state feedback gain matrix, sufficient conditions of Krotov global optimal control framework are analyzed for the constrained linear quadratic problem, which typically renders the optimal control problem non‐convex. To address this issue, suitable convex optimization problems are formulated to compute the upper bound independent of initial conditions. The effectiveness of the proposed technique and its comparison with state‐of‐the‐art methods are demonstrated through numerical examples.

In this paper we formulate the problem of dynamically hedging a basket option on multiple underlying stocks and in the presence of proportional transaction costs as a linear quadratic control problem subject to constraints. The linear structure is obtained by sampling over paths of the underlying stocks and linearly parameterizing control actions over basis functions. Two solutions are then proposed. The first involves quadratically penalizing transaction costs in the objective and allows the hedging problem to be solved as a standard unconstrained linear quadratic regulator problem. The second approach uses receding horizon control to solve a quadratic program over a specified prediction horizon, where the cost function utilizes the LQR solution from the first approach. A numerical example illustrates the methodology.

Linear Quadratic Optimal Control of a Wave Equation with Boundary Damping and Pointwise Control Input

This paper investigates and discusses the method of dynamic programming in solving Bolza's cost form of Linear Quadratic Regulator Problems (LQRP). It is the desire of the authors of this paper to experiment numerically the solution of this class of problem using dynamic programming to solve for the optimal controls and the trajectories compared with other numerical methods with a view to further improving the results. The method uses the principle of optimality to reduce mathematically the number of calculations required to determine the optimal control law as well as the corresponding optimal cost functional.

This paper presents a data-driven approach towards time-optimal trajectory generation for Unmanned Aerial Vehicles (UAV's) using a machine-learned trajectory generation mechanism for point-to-point time-optimal trajectories on-the-fly. To train this machine-learned black box trajectory generator off-line, a model-based optimization problem is first constructed for point-to-point time-optimal trajectory generation, with physical constraints on inputs, states, and rates. The formulated optimization problem is then solved off-line for a range of initial and terminal flight states to generate point-to-point data-sets that consist of the optimal state and input trajectories. This information is compressed by parameterizing the input and state trajectories using a set of basis functions. This data is then used to train the neural network-based trajectory planner. The output of the neural network is the basis function coefficient sets for the state and input trajectories (and the total flight time) which can then be used to reconstruct the flight trajectory. Once the neural network is trained, the data-driven on-board trajectory generator is ready to be deployed on the UAV for on-board planning. This approach is demonstrated for two scenarios: (1) the input to the neural network being the initial and terminal flight states and (2) the input to the neural network being initial and terminal flight states as well as physical constraints. To validate the performance of the machine-learned black-box trajectory generator, the root mean squared error between the neural network generated trajectories and the trajectories obtained from solving the optimization problem directly is statistically evaluated. These trajectories are also tested for violation of path constraints (which are not included explicitly in the training or input to the black box planner) by evaluating the mean constraint violation for each path-constrained variable.

A linear quadratic regulator problem where a small parameter μ2multiplies the control cost is considered. As \mu \rightarrow 0 , it exhibits a dynamic behavior with multiple time-scales, whereas for \mu = 0 , it is a singular control problem. At first a multiparameter singularly perturbed model of the given problem is constructed by transferring the singularity that exists in the performance index to the dynamic equations. Then the problem is decomposed into several subproblems of minimal order, each pertaining to only one time-scale. The method developed permits under a single framework the characterization of asymptotic behavior of optimal closed-loop poles, state and control trajectories, performance index, and optimal transfer function as \mu \rightarrow 0 . Moreover, the asymptotic solution brings into focus 1) the nature of singular control and its connection to transmission zeros and 2) the presence of various types of impulses and other higher order distributions in both state and control. The approach unifies and extends significantly several ideas previously somewhat disjoint.

For the continuous infinite horizon time-invariant linear quadratic regulator problem (LQR), in the paper, the optimal state feedback controller based on the estimated state of the observer can be decoupled by the proposed approach which resulting one continuous time algebraic Riccati equation (CARE) for the controller design and one matrix equality equation (MEE) for the observer design. A coupling term related the CARE of the controller is found to be existed in the MEE of the observer. Unlike the separate principle to design the controller and observer separately without any coupling term, the design of the observer should consider the coupling term related to the CARE of the controller. The coupling problem between the controller and the observer usually exists in the linear matrix inequality (LMI) approach, and it is the main problem to be solved. The two-stage scheme is popular in the LMI approach, and the proposed method is similar to it, but adopting equality instead of inequality.

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Model predictive control (MPC) is a multi-objective control technique that can handle system constraints. However, the performance of an MPC controller highly relies on a proper prioritization weight for each objective, which highlights the need for a precise weight tuning technique. In this paper, we propose an analytical tuning technique by matching the MPC controller performance with the performance of a linear quadratic regulator (LQR) controller. The proposed methodology derives the transformation of a LQR weighting matrix with a fixed weighting factor using a discrete algebraic Riccati equation (DARE) and designs an MPC controller using the idea of a discrete time linear quadratic tracking problem (LQT) in the presence of constraints. The proposed methodology ensures optimal performance between unconstrained MPC and LQR controllers and provides a sub-optimal solution while the constraints are active during transient operations. The resulting MPC behaves as the discrete time LQR by selecting an appropriate weighting matrix in the MPC control problem and ensures the asymptotic stability of the system. In this paper, the effectiveness of the proposed technique is investigated in the application of a novel vehicle collision avoidance system that is designed in the form of linear inequality constraints within MPC. The simulation results confirm the potency of the proposed MPC control technique in performing a safe, feasible and collision-free path while respecting the inputs, states and collision avoidance constraints.

Linear quadratic regulator problem with inequality control constraints for flexible space structures

This work relates musculotendon dynamics and neurological controls to a continuum analysis of bone in order to show how the effects of muscular control affect the development of stress and strain in skeletal members. This is achieved by incorporating standard Hill-type models of musculotendon actuators into mathematical models of human motion in which the human body is modeled as an ensemble of articulating segments. The join torques and reaction forces as predicted by this analysis are incorporated into a stress analysis of the segmental links by utilizing an approach in which the equations of motion for the segmental links are derived as a hybrid parameter system. This method accounts for both the rigid body motions of the articulating links and the elastic deformations that represent the continuum effects in the bone. The effect of optimal control strategies upon stress development are also investigated. In particular, neural controls for the musculotendon actuators are derived by formulating a linear quadratic regulator problem in order to investigate control strategies that stabilize the musculoskeletal system about a nominal trajectory. Strategies that are considered correspond to controlling join positions, regulating muscle lengths or tendon stiffness, and full-state feedback control.

In this paper, we propose a method for estimating the algebraic Riccati equation (ARE) with respect to an unknown discrete-time system from the system state and input observation. The inverse optimal control (IOC) problem asks, “What objective function is optimized by a given control system?” The inverse linear quadratic regulator (ILQR) problem is an IOC problem that assumes a linear system and quadratic objective function. The ILQR problem can be solved by solving a linear matrix inequality that contains the ARE. However, the system model is required to obtain the ARE, and it is often unknown in fields in which the IOC problem occurs, for example, biological system analysis. Our method directly estimates the ARE from the observation data without identifying the system. This feature enables us to economize the observation data using prior information about the objective function. We provide a data condition that is sufficient for our method to estimate the ARE. We conducted a numerical experiment to demonstrate that our method can estimate the ARE with less data than system identification if the prior information is sufficient.