Linear Quadratic Regulator of Discrete-Time Switched Linear Systems

Data-driven Linear Quadratic Regulation via Semidefinite Programming

We use the dynamic programming and approximation techniques to prove the existence of an optimal synthesis for a class of linear discrete time systems with quadratic costs and a fixed target point, in which the optimal control law is given in the feedback form of time and state. Our result establishes an analogy in synthesizing fixed target point linear quadratic regulator problems between linear continuous time systems and this class of linear discrete time systems.

This paper presents a new control approach for piezoelectric switching shunt damping. Recently, semi-active controllers have been used to switch piezoelectric materials in order to damp vibration. These switching shunt circuits allow a small implementation and require only little power supply. However, the control laws to switch these shunts are derived heuristically and therefore it remains unclear, if a better control law for a given shunt topology exists. We present a new control approach based on the Hybrid System Framework. This allows the modelling of the switched composite system as a hybrid system. Once the hybrid system description is obtained, a receding horizon optimal control problem can be solved in order to get the optimal switching sequence. As the computation time to solve this optimisation problem is too long for real-time applications, we will show that the problem can be solved off-line and the solution stored in a look-up table. This allows a real-time implementation of the switch controller. Moreover, control rules can be derived from this look-up table, and we will demonstrate that in some situations the controllers proposed in previous papers generate near optimal switching. In this paper, we will investigate several shunt topologies with switches and compare the performance between the heuristically derived control laws and the optimal new control laws. Simulations and experiments show the improvement with the new controllers. This is very promising, since this new control approach can be applied for more complex shunt circuits with many switches, where the derivation of a switching law would be very difficult.

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.

ABSTRACTFor unknown continuous‐time nonlinear systems with multiple equilibrium points, this article proposes a robust adaptive optimal tracking switching controller. The controller integrates multiple nonlinear approximate optimal tracking controllers, an optimal switching sequence, and a robust compensator. First, multiple nonlinear neural network models are established to construct a controller design model, whose unknown parameters are online estimated by resorting to Lyapunov stability theory. Then, at every instant, the optimal switching sequence and approximate optimal tracking controller are obtained by using embedding transformation and approximate dynamic programming technology. After that, according to the error between the nonlinear system and the given input, a robust compensator is designed. Finally, stability and convergence of the control system are verified, and the simulations are conducted to validate the effectiveness and superiority.

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

This paper presents an optimal voltage control algorithm developed to increase the efficiency of operation of power systems and minimizing the reaction time for compensating voltage deviations due to load fluctuations. Because of the many variables of influence in the optimal voltage control (e.g. line power flow limits, bus voltage limits, etc.), the mathematical formulation is a constrained optimization problem with continuous and discrete variables, known as Optimal Power Flow (OPF), which is non-linear and non-convex. We propose a methodology to convexify the OPF problem based on Semidefinite Programming (SDP), with a novel formulation to include discrete decision variables such as shunt elements and tap changers positions as discrete variables. Finally, the SDP formulation of the case of a DC network is included.

Multi-parametric programming based algorithms for the global solution of bi-level mixed-integer linear and quadratic programming problems

This paper considers an approach to perform perturbation analysis of linear quadratic regulator (LQR) control problem for continuous-time descriptor systems. The investigated control problem is based on solving LMIs (Linear Matrix Inequalities) and applying Lyapunov functions. The paper is concerned with obtaining linear perturbation bounds for the continuous-time LQR control problem for descriptor systems. The computed perturbation bounds can be used to study the effect of perturbations in system and controller on feasibility and performance of the considered control problem. A numerical example is also presented in the paper.

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.

In this article, the estimation problem in a class of nonlinear fractional-order systems is solved by a stable estimator that generalizes the classical exponential observers, establishing a new class called Mittag–Leffler observers. The solution to the linear quadratic regulator problem is proposed to design optimal control laws for fractional-order linear systems and its applications. All the results are based on the Caputo derivative for commensurate fractional-order systems. Numerical simulations validate the proposed theory.

Optimal D-stable control for discrete multiple time-delay systems with parametric uncertainties

In traditional system designs for feedback structural control systems, the structure is assumed to be linear time invariant (LTI). However, extreme load events on the structure can damage the structure leading to violation of the time invariance assumption. This study proposes an online adaptive control strategy that tracks the time varying characteristics of a structural system with the optimal control law recursively derived for the system. Continuous system identification executed concurrently with derivation of optimal control laws is possible using the low-power, dual-core Martlet wireless node developed at the University of Michigan. The wireless platform executes online system identification recursively on one processor pipeline while a control law based on linear quadratic regulation (LQR) is derived and implemented concurrently on the second pipeline. By parallelizing the implementation of the LQR control law and system identification algorithm, the wireless sensing network is shown to be capable of learning and adapting to system changes in real-time. Simulation and experiments on a 4 story benchmark shear structure is utilized to show the validity and scalability of the proposed adaptive feedback control approach.