Optimization over state feedback policies for robust control with constraints

This paper is concerned with the optimal control of linear discrete-time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input. It is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence. This implies that a broad class of constrained finite horizon robust and optimal control problems, where the optimization is over affine state feedback policies, can be solved in a computationally efficient fashion using convex optimization methods. This equivalence result is used to design a robust receding horizon control (RHC) state feedback policy such that the closed-loop system is input-to-state stable (ISS) and the constraints are satisfied for all time and all allowable disturbance sequences. The cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance-free state and input sequences. The value of the receding horizon control law can be calculated at each sample instant using a single, tractable and convex quadratic program (QP) if the disturbance set is polytopic, or a tractable second-order cone program (SOCP) if the disturbance set is given by a 2-norm bound.     

Output feedback receding horizon control of constrained systems

This paper considers output feedback control of linear discrete-time systems with convex state and input constraints which are subject to bounded state disturbances and output measurement errors. We show that the non-convex problem of finding a constraint admissible affine output feedback policy over a finite horizon, to be used in conjunction with a fixed linear state observer, can be converted to an equivalent convex problem. When used in the design of a time-varying robust receding horizon control law, we derive conditions under which the resulting closed-loop system is guaranteed to satisfy the system constraints for all time, given an initial state estimate and bound on the state estimation error. When the state estimation error bound matches the minimal robust positively invariant (mRPI) set for the system error dynamics, we show that this control law is time-invariant, but its calculation generally requires solution of an infinite-dimensional optimization problem. Finally, using an invariant outer approximation to the mRPI error set, we develop a time-invariant control law that can be computed by solving a finite-dimensional tractable optimization problem at each time step that guarantees that the closed-loop system satisfies the constraints for all time.     

Receding horizon control of nonlinear systems without differentiability of the optimal value function

In this paper, we show that receding horizon control, when applied to a class of nonlinear systems, has stabilizing properties. Our assumptions are necessarily fairly strong because they guarantee implicity that a smooth stabilizing feedback can be constructed. This is an extension of our previous results which require the finite horizon value function to be continuously differentiable and therefore applies to a more restricted class of system.Continuous differentiability of the value function is replaced by local Lipschitz continuity; Dini derivatives and a generalization of the Invariance Principle are employed to establish stability.     

Numerical optimization method for HJI equations derived from robust receding horizon control schemes and controller design

This paper addresses how to numerically solve the Hamilton-Jacobin-Isaac(HJI)equations derived from the robust receding horizon control schemes.The developed numerical method,the finite dierence scheme with sigmoidal transformation,is a stable and convergent algorithm for HJI equations.A boundary value iteration procedure is developed to increase the calculation accuracy with less time consumption.The obtained value function can be applied to the robust receding horizon controller design of some kind of uncertain nonlinear systems.In the controller design,the finite time horizon is extended into the infinite time horizon and the controller can be implemented in real time.It can avoid the on-line repeated optimization and the dependence on the feasibility of the initial state which are encountered in the traditional robust receding horizon control schemes.     

Robust one-step receding horizon control of discrete-time Markovian jump uncertain systems

This paper proposes a receding horizon control scheme for a set of uncertain discrete-time linear systems with randomly jumping parameters described by a finite-state Markov process whose jumping transition probabilities are assumed to belong to some convex sets. The control scheme for the underlying systems is based on the minimization of the worst-case one-step finite horizon cost with a finite terminal weighting matrix at each time instant. This robust receding horizon control scheme has a more general structure than the existing robust receding horizon control for the underlying systems under the same design parameters. The proposed controller is obtained using semidefinite programming.     

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.     

Robust one-step receding horizon control for constrained systems

This paper proposes a robust receding horizon control scheme for discrete-time uncertain linear systems with input and state constraints. The control scheme is based on the minimization of the worst-case one-step finite horizon cost with a finite terminal weighting matrix. It is shown that the proposed receding horizon control robustly asymptotically stabilizes uncertain constrained systems under some matrix inequality conditions on the terminal weighting matrices. This robust receding horizon control scheme has a larger feasible initial-state set and a more general structure than existing robust receding horizon controls for uncertain constrained systems under the same design parameters. The proposed controller is obtained using semidefinite programming. Copyright © 1999 John Wiley & Sons, Ltd.     

A framework for robustness analysis of constrained finite receding horizon control

A framework for robustness analysis of input-constrained finite receding horizon control is presented. Under the assumption of quadratic upper bounds on the finite horizon costs, we derive sufficient conditions for robust stability of the standard discrete-time linear-quadratic receding horizon control formulation. This is achieved by recasting conditions for nominal and robust stability as an implication between quadratic forms, lending itself to S-procedure tools which are used to convert robustness questions to tractable convex conditions. Robustness with respect to plant/model mismatch as well as for state measurement error is shown to reduce to the feasibility of linear matrix inequalities. Simple examples demonstrate the approach.     

基于终端不变集的 Markov 跳变系统约束预测控制

针对离散 Markov 跳变系统, 研究带输入输出约束的有限时域预测控制问题. 对于给定预测时域内的每条模态轨迹, 设计控制输入序列, 驱动系统状态到达相应的终端不变集内, 在预测时域外, 则寻求一个虚拟的状态反馈控制器以保证系统的随机稳定性, 在此基础上, 分别给出了以线性矩阵不等式 (LMI) 描述的带输入、输出约束预测控制器的设计方法.     

On Stability of Constrained Receding Horizon Control with Finite Terminal Weighting Matrix

In this paper, a new stabilizing receding horizon control, based on a finite input and state horizon cost with a finite terminal weighting matrix, is proposed for time-varying discrete linear systems with constraints. We propose matrix inequality conditions on the terminal weighting matrix under which closed-loop stability is guaranteed for both cases of unconstrained and constrained systems with input and state constraints. We show that such a terminal weighting matrix can be obtained by solving a linear matrix inequality (LMI). In the case of constrained time-invariant systems, an artificial invariant ellipsoid constraint is introduced in order to relax the conventional terminal equality constraint and to handle constraints. Using the invariant ellipsoid constraints, a feasibility condition of the optimization problem is presented and a region of attraction is characterized for constrained systems with the proposed receding horizon control.     

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 control of discrete-time state-dependent jump linear systems with probabilistic constraints: A receding horizon approach

In this article, we consider a receding horizon control of discrete-time state-dependent jump linear systems, a particular kind of stochastic switching systems, subject to possibly unbounded random disturbances and probabilistic state constraints. Due to the nature of the dynamical system and the constraints, we consider a one-step receding horizon. Using inverse cumulative distribution function, we convert the probabilistic state constraints to deterministic constraints, and obtain a tractable deterministic receding horizon control problem. We consider the receding horizon control law to have a linear state-feedback and an admissible offset term. We ensure mean square boundedness of the state variable via solving linear matrix inequalities off-line, and solve the receding horizon control problem on-line with control offset terms. We illustrate the overall approach applied on a macroeconomic system.     

Receding Horizon Controls for Input-Delayed Systems

This paper presents a receding horizon control (RHC) for an unconstrained input-delayed system. To begin with, we derive a finite horizon optimal control for a quadratic cost function including two final weighting terms. The RHC is easily obtained by changing the initial and final times of the finite horizon optimal control. A linear matrix inequality (LMI) condition on two final weighting matrices is proposed to meet the cost monotonicity, under which the optimal cost on the horizon is monotonically nonincreasing with time and hence the asymptotical stability is guaranteed only if an observability condition is met. It is shown through simulation that the proposed RHC stabilizes the input-delayed system effectively and its performance can be tuned by adjusting weighting matrices with respect to the state and the input.      

Receding horizon time-optimal control for a class of differentially flat systems

A closed-loop, time-optimal path-following control scheme is proposed for a class of constrained differentially flat systems. Within a receding horizon framework, a finite horizon optimisation problem is solved at each sample, using available state feedback and feedforward path information. Irrespective of horizon length, the proposed formulation guarantees exact path-following. Moreover, the requirements under which the proposed algorithm achieves minimum-time path-following are established. Simulations conducted with a rigid X–Y table model confirm the theoretical results.     

鲁棒后退时域控制中HJI方程的数值解法及控制器设计

本文讨论了由不确定非线性系统鲁棒后退时域控制(robust receding horizon control,RRHC)策略导出的Hamilton-Jacobin-Isaac(HJI)方程的求解,提出了一种新的带反曲变换的有限差分算法计算值函数,所提出算法对HJI方程的求解是一种稳定且收敛的算法.同时提出基于边界值迭代的加速过程,加速优化问题的求解,在花费更少计算时间的前提下,提高计算精度.所求得的值函数可直接应用于一类不确定非线性系统鲁棒后退时域控制器的设计,在控制器设计中,传统鲁棒后退时域控制策略中的有限时域被扩展到无限时域,求得的控制器可实时实现,避免对初始点可解性的依赖以及反复在线优化问题.     

Robust receding horizon control of constrained nonlinear systems

We present a method for the construction of a robust dual-mode, receding horizon controller which can be employed for a wide class of nonlinear systems with state and control constraints and model error. The controller is dual-mode. In a neighborhood of the origin, the control action is generated by a linear feedback controller designed for the linearized system. Outside this neighborhood, receding horizon control is employed. Existing receding horizon controllers for nonlinear, continuous time systems, which are guaranteed to stabilize the nonlinear system to which they are applied, require the exact solution, at every instant, of an optimal control problem with terminal equality constraints. These requirements are considerably relaxed in the dual-mode receding horizon controller presented in this paper. Stability is achieved by imposing a terminal inequality, rather than an equality, constraint. Only approximate minimization is required. A variable time horizon is permitted. Robustness is achieved by employing conservative state and stability constraint sets, thereby permitting a margin of error. The resultant dual-mode controller requires considerably less online computation than existing receding horizon controllers for nonlinear, constrained systems     

RECEDING HORIZON H∞ CONTROL FOR SYSTEMS WITH A STATE‐DELAY

This paper proposes the receding horizon H_∞ control (RHHC) for linear systems with a state‐delay. We first proposes a new cost function for a finite horizon dynamic game problem. The proposed cost function includes two terminal weighting terms, each of which is parameterized by a positive definite matrix, called a terminal weighting matrix. Secondly, we derive the RHHC from the solution to the finite dynamic game problem. Thirdly, we propose an LMI condition under which the saddle point value satisfies the nonincreasing monotonicity. Finally, we show the asymptotic stability and H_∞‐norm boundedness of the closed‐loop system controlled by the proposed RHHC. Through a numerical example, we show that the proposed RHHC is stabilizing and satisfies the infinite horizon H_∞‐norm bound.     

Finite-time behavior of inner systems

In this paper, we investigate how nonminimum phase characteristics of a dynamical system affect its controllability and tracking properties. For the class of linear time-invariant dynamical systems, these characteristics are determined by transmission zeros of the inner factor of the system transfer function. The relation between nonminimum phase zeros and Hankel singular values of inner systems is studied and it is shown how the singular value structure of a suitably defined operator provides relevant insight about system invertibility and achievable tracking performance. The results are used to solve various tracking problems both on finite as well as on infinite time horizons. A typical receding horizon control scheme is considered and new conditions are derived to guarantee stabilizability of a receding horizon controller.     

Sliding mode prediction based control algorithm for discrete-time non-linear uncertain coupled systems

By introducing predictive control strategy into the design of sliding mode control (SMC), a novel SMC algorithm for a class of discrete-time non-linear uncertain coupled systems is presented in this paper. To enlighten by the recursive sliding mode approach, a special sliding mode prediction model (SMPM) is created at first. Then taking model mismatch into consideration, the error between the output of SMPM and the practical sliding mode value is used to make feedback correction for SMPM. Applying receding horizon optimization, the desired sliding mode control law which is a non-switching type, is obtained subsequently. The reachability of sliding mode is achieved by making predictive value of sliding mode track the expected sliding mode reference value. Due to feedback correction and receding horizon optimization, the influence of uncertainties can be compensated in time, strong robustness to matched or unmatched uncertainties is possessed. Theoretical analysis proves the closed-loop system is robustly stable, without requiring the known boundaries of uncertainties. Simulation results of a numerical example and a rotational inverted pendulum illustrate the validity of the proposed algorithm.     

Receding horizon guidance laws for constrained missiles with autopilot lags

In this paper, a new stabilizing receding horizon control (RHC) scheme is proposed for linear continuous time-invariant systems with input and state constraints. The control scheme is based on the minimization of the finite horizon cost with a finite terminal weighting matrix subject to constraints. A new algorithm is suggested to implement the RHC scheme for a constrained receding horizon guidance law (CRHG). The proposed CRHG, which does not use information on the time-to-go, is obtained by using linear matrix inequality (LMI) optimization. Through simulation examples, the performance of the proposed CRHG is illustrated.