Computation of the constrained infinite time linear quadratic regulator

This paper presents an efficient algorithm for computing the solution to the constrained infinite-time, linear quadratic regulator (CLQR) problem for discrete time systems. The algorithm combines multi-parametric quadratic programming with reachability analysis to obtain the optimal piecewise affine (PWA) feedback law. The algorithm reduces the time necessary to compute the PWA solution for the CLQR when compared to other approaches. It also determines the minimal finite horizon , such that the constrained finite horizon LQR problem equals the CLQR problem for a compact set of states . The on-line computational effort for the implementation of the CLQR can be significantly reduced as well, either by evaluating the PWA solution or by solving the finite dimensional quadratic program associated with the CLQR for a horizon of .     

Sliding surface design for discrete VSS using LQR technique with a preset real eigenvalue

This paper presents an LQR-based sliding surface design procedure that allows one to specify a desired weighting matrix and a desired real eigenvalue which is important in discrete VSS. The procedure computes a weighting matrix that simultaneously stays “closest” to the desired one and yields the desired eigenvalue. Sliding surface is then determined from the LQR gain matrix and the desired eigenvalue. It is shown that feasible weighting matrices almost always exist and are not unique. Moreover, the underlying constrained optimization problem is convex and least-squares method may be used to simplify or solve the problem. The paper ends with two numerical examples that illustrate the design procedure.     

Move blocking strategies in receding horizon control

In order to deal with the computational burden of optimal control, it is common practice to reduce the degrees of freedom by fixing the input or its derivatives to be constant over several time-steps. This policy is referred to as ‘move blocking’. This paper will address two issues. First, a survey of various move blocking strategies is presented and the shortcomings of these blocking policies, such as the lack of stability and constraint satisfaction guarantees, will be illustrated. Second, a novel move blocking scheme, ‘Moving Window Blocking’ (MWB), will be presented. In MWB, the blocking strategy is time-dependent such that the scheme yields stability and feasibility guarantees for the closed-loop system. Finally, the results of a large case study that illustrate the advantages and drawbacks of the various control strategies discussed in this paper and the implementation of the MWB scheme on a mechanical system are presented.     

On stochastic Riccati equations for the stochastic LQR problem

This paper studies an approximation of stochastic Riccati equations for stochastic LQR problems some of which may be even with indefinite control weight costs.     

电子横移伺服LQR最优控制器设计

为了提高系统的稳态精度和动态性能,将LQR最优控制算法应用于高速经编机电子横移伺服控制系统.建立了伺服系统状态空间模型,引入了一类具有指数衰减度的二次型函数作为系统性能指标.仿真结果表明,所提出的电子横移伺服LQR最优控制算法易于工程实现,保证了系统的稳定性,具有良好的频响特性,改善了系统的动态特性,实现了无静差高精度控制.     

On the infinite-horizon LQ tracker

The infinite-horizon tracking problem is considered in the linear-quadratic optimal control framework. Computationally, one term in the control signal is a constant gain, stabilizing, full-state feedback found by solving an algebraic Riccati equation; the second term involves a steady-state function v_ss(t) that solves an auxiliary, forced differential equation with unknown initial condition. In this article, a linear system of algebraic equations is derived to determine v_ss(0). Numerical examples are included to illustrate the result.     

无穷时域的连续时间广义预测控制

将无穷时域的性能指标引入连续时间的广义预测控制,通过施加新的终端等式约束,把包含无穷时域性能指标的优化问题转化成可解的二次规划问题.利用后退时域性能指标的单调性,给出了保证连续时间广义预测控制闭环稳定性的条件.仿真例子验证了该算法的有效性.     

Linear programming and model predictive control

The practicality of model predictive control (MPC) is partially limited by the ability to solve optimization problems in real time. This requirement limits the viability of MPC as a control strategy for large scale processes. One strategy for improving the computational performance is to formulate MPC using a linear program. While the linear programming formulation seems appealing from a numerical standpoint, the controller does not necessarily yield good closed-loop performance. In this work, we explore MPC with an l₁ performance criterion. We demonstrate how the non-smoothness of the objective function may yield either dead-beat or idle control performance.     

Fast algorithms for generalized predictive control

Fast algorithms for generalized predictive control (GPC) are derived by adopting an approach whereby dynamic programming and a polynomial formulation are jointly exploited. They consist of a set of coupled linear polynomial recursions by which the dynamic output feedback GPC law is recursively computed wwith only O(Nn) computations for an n-th order plant and N-steps prediction horizon.     

Nonlinear sampled-data output feedback receding horizon control

In this paper, we present an output feedback receding horizon control for a class of SISO nonlinear systems. A globally stabilizing state feedback receding horizon control scheme is combined with a discretized high gain observer. This is motivated by the fact that measurable system’s outputs are only available at specific sampling intervals. Our result follows from the application of a separation principle applicable to a class of sampled-data nonlinear systems. It is shown that the output-feedback scheme recovers the performance (rate of convergence) achieved under state feedback receding horizon control for a sufficiently large observer gain and sampling frequency.     

Distributed receding horizon control for multi-vehicle formation stabilization

We consider the control of interacting subsystems whose dynamics and constraints are decoupled, but whose state vectors are coupled non-separably in a single cost function of a finite horizon optimal control problem. For a given cost structure, we generate distributed optimal control problems for each subsystem and establish that a distributed receding horizon control implementation is stabilizing to a neighborhood of the objective state. The implementation requires synchronous updates and the exchange of the most recent optimal control trajectory between coupled subsystems prior to each update. The key requirements for stability are that each subsystem not deviate too far from the previous open-loop state trajectory, and that the receding horizon updates happen sufficiently fast. The venue of multi-vehicle formation stabilization is used to demonstrate the distributed implementation.     

On the finite time performance of model predictive control

This paper addresses the finite time performance of model predictive control (MPC) for linear-time-invariant (LTI) systems without constraints. The performance of MPC is compared with that of finite horizon optimal control to find out how well model predictive control can perform relative to the optimal performance with the same or different horizons. By exploring the properties of the Riccati difference equation (RDE), an upper and a lower bound of the ratio between the finite time performance of MPC and finite horizon optimal cost are obtained. It is possible to extend the obtained results to more complicated systems such as nonlinear dynamic systems with constraints with appropriate generalizations. Simulation example supports our results.     

LQR Integrated with Fuzzy Control for Aerial Refueling Autopilot

Linear Quadratic Regulator （LQR） is modem linear control that is suitable for multivariable state feedback and is known to yield good performance for linear systems or for nonlinear systems where the nonlinear aspects are presented. The fuzzy control is known to have the ability to deal with nonlinearities without having to use advanced mathematics. The LQR integrated fuzzy control （LQRIFC） simultaneously makes use of the good performance of LQR in the region close to switching curve, and the effectiveness of fuzzy control in region away from switching curve. A new analysis of the fuzzy system behavior presented helps to make possible precise integration of LQR features into fuzzy control. The LQRIFC is verified by simulation to suppress the uncertainty instability more effectively than the LQR besides minimizing the time of the mission proposed.     

Self-triggered linear quadratic control

Self-triggered control is a recently proposed paradigm that abandons the more traditional periodic time-triggered execution of control tasks with the objective of reducing the utilization of communication resources, while still guaranteeing desirable closed-loop behavior. In this paper, we introduce a self-triggered strategy based on performance levels described by a quadratic discounted cost. The classical LQR problem can be recovered as an important special case of the proposed self-triggered strategy. The self-triggered strategy proposed in this paper possesses three important features. Firstly, the control laws and triggering mechanisms are synthesized so that a priori chosen performance levels are guaranteed by design. Secondly, they realize significant reductions in the usage of communication resources. Thirdly, we address the co-design problem of jointly designing the feedback law and the triggering condition. By means of a numerical example, we show the effectiveness of the presented strategy. In particular, for the self-triggered LQR strategy, we show quantitatively that the proposed scheme can outperform conventional periodic time-triggered solutions.     

Receding horizon control applied to optimal mine planning

In this paper we show that the problem of optimal mine planning can be cast in the framework of receding horizon control. Traditional formulations of this problem have cast it in the framework of mixed integer linear programming. In this paper, we present an alternative formulation of the mine planning problem using the “language” of control engineering. We show that this alternative formulation gives rise to new insights which have the potential to lead to improved computational procedures. The advantages are illustrated by an example incorporating many practical features of an actual mine planning problem.     

一种线性二次型调节器闭环稳定的充分条件及在预测控制中的应用

提出了终态松弛的线性二次型Kleinman调节器,给出并证明了该调节器保证系统闭环稳定的一个充分条件.将结果用于预测控制的稳定性分析,得到选择代价函数中加权阵的定量准则.     

Output feedback model predictive control for LPV systems based on quasi-min–max algorithm

This paper proposes a robust output feedback model predictive control (MPC) scheme for linear parameter varying (LPV) systems based on a quasi-min–max algorithm. This approach involves an off-line design of a robust state observer for LPV systems using linear matrix inequality (LMI) and an on-line robust output feedback MPC algorithm using the estimated state. The proposed MPC method for LPV systems is applicable for a variety of systems with constraints and guarantees the robust stability of the output feedback systems. A numerical example for an LPV system subject to input constraints is given to demonstrate its effectiveness.     

An improved approach for constrained robust model predictive control

In this paper, we present a new technique to address constrained robust model predictive control. The main advantage of this new approach with respect to other well-known techniques is the reduced conservativeness. Specifically, the technique described in this paper can be applied to polytopic uncertain systems and is based on the use of several Lyapunov functions each one corresponding to a different vertex of the uncertainty's polytope.     

On a sufficient transversality condition for infinite horizon optimal control problems

We prove that, for a class of optimal control problems with infinite horizon, convergence to zero for the co-state variable when time goes to infinity is a sufficient transversality condition.