Construction of invariant polytopic sets with specified complexity

The problem of construction of controlled invariant polytopic sets with specified complexity, for linear systems subject to linear state and control constraints, is investigated. First, geometric conditions for the enlargement of a polytopic set by adding a new vertex, in order to produce a polytopic set of specified complexity, are established. Next, conditions for such an enlargement of controlled invariant sets to preserve the controlled invariance property are presented. The established theoretical results are used to develop methods for the construction of admissible controlled invariant sets with specified complexity. Two numerical examples show how these results can be used for the computation of monotonic sequences of admissible controlled invariant sets of specified complexity.     

On invariant sets and closed‐loop boundedness of Lure‐type nonlinear systems by LPV‐embedding

We address the problem of achieving trajectory boundedness and computing ultimate bounds and invariant sets for Lure‐type nonlinear systems with a sector‐bounded nonlinearity. Our first contribution is to compare two systematic methods to compute invariant sets for Lure systems. In the first method, a linear‐like bound is considered for the nonlinearity, and this bound is used to compute an invariant set by regarding the nonlinear system as a linear system with a nonlinear perturbation. In the second method, the sector‐bounded nonlinearity is treated as a time‐varying parameterised linear function with bounded parameter variations, and then invariant sets are computed by embedding the nonlinear system into a convex polytopic linear parameter varying (LPV) system. We show that under some conditions on the system matrices, these approaches give identical invariant sets, the LPV‐embedding method being less conservative in the general case. The second contribution of the paper is to characterise a class of Lure systems, for which an appropriately designed linear state feedback achieves bounded trajectories of the closed‐loop nonlinear system and allows for the computation of an invariant set via a simple, closed‐form expression. The third contribution is to show that, for disturbances that are ‘aligned’ with the control input, arbitrarily small ultimate bounds on the system states can be achieved by assigning the eigenvalues of the linear part of the system with ‘large enough’ negative real part. We illustrate the results via examples of a pendulum system, a Josephson junction circuit and the well‐known Chua circuit. Copyright © 2015 John Wiley & Sons, Ltd.     

Offline Robust Model Predictive Control for Lipschitz Non‐Linear Systems Using Polyhedral Invariant Sets

In this paper the concept of maximal admissible set (MAS) for linear systems with polytopic uncertainty is extended to non‐linear systems composed of a linear constant part followed by a non‐linear term. We characterize the maximal admissible set for the non‐linear system with unstructured uncertainty in the form of polyhedral invariant sets. A computationally efficient state‐feedback RMPC law is derived off‐line for Lipschitz non‐linear systems. The state‐feedback control law is calculated by solving a convex optimization problem within the framework of linear matrix inequalities (LMIs), which leads to guaranteeing closed‐loop robust stability. Most of the computational burdens are moved off‐line. A linear optimization problem is performed to characterize the maximal admissible set, and it is shown that an ellipsoidal invariant set is only an approximation of the true stabilizable region. This method not only remarkably extends the size of the admissible set of initial conditions but also greatly reduces the on‐line computational time. The usefulness and effectiveness of the method proposed here is verified via two simulation examples.     

基于容许集的多胞不确定系统三模RMPC

针对输入和状态受约束的多胞不确定线性系统,提出了基于容许集的扩大吸引域三模鲁棒模型预测控制方法.在多面体不变集离线模型预测控制算法的基础上引入容许集,以多面体不变集序列的并集作为模态1,基于N步容许集的控制容许集作为模态2,并利用离线设计和在线优化的控制策略,设计了三模变终端约束鲁棒模型预测控制算法,以实现系统渐近稳定.该算法不仅降低了在线运算量,而且扩大了吸引域.最后的仿真结果验证了所提出算法的有效性.     

Full‐complexity polytopic robust control invariant sets for uncertain linear discrete‐time systems

This paper presents an algorithm for the computation of full‐complexity polytopic robust control invariant (RCI) sets, and the corresponding linear state‐feedback control law. The proposed scheme can be applied for linear discrete‐time systems subject to additive disturbances and structured norm‐bounded or polytopic uncertainties. Output, initial condition, and performance constraints are considered. Arbitrary complexity of the invariant polytope is allowed to enable less conservative inner/outer approximations to the RCI sets whereas the RCI set is assumed to be symmetric around the origin. The nonlinearities associated with the computation of such an RCI set structure are overcome through the application of Farkas' theorem and a corollary of the elimination lemma to obtain an initial polytopic RCI set, which is guaranteed to exist under certain conditions. A Newton‐like update, which is recursively feasible, is then proposed to yield desirable large/small volume RCI sets.     

基于多面体终端域的Hammerstein-Wiener非线性预测控制

许多实际系统都可以表示成一种中间为线性动态子系统、输入输出端为非线性静态子系统的Hammerstein-Wiener型非线性模型. 针对输入和输出受约束的Hammerstein-Wiener型非线性系统, 提出一种基于多面体终端域的预测控制综合算法. 离线设计时, 通过构造一系列多面体不变集, 扩大了终端域; 在多面体不变集内, 设计非线性控制律, 减少了常规线性控制律设计的保守性. 在线计算时, 通过求解有限个线性矩阵不等式(Linear matrix inequalities, LMIs)优化问题, 不仅可以满足实时性要求, 而且能够改善控制性能. 仿真结果表明了采用多面体不变集的优越性.     

Computation of maximal safe sets for switching systems

The problem of determining maximal safe sets and hybrid controllers is computationally intractable because of the mathematical generality of hybrid system models. Given the practical and theoretical relevance of the problem, finding implementable procedures that could at least approximate the maximal safe set is important. To this end, we begin by restricting our attention to a special class of hybrid systems: switching systems. We exploit the structural properties of the graph describing the discrete part of a switching system to develop an efficient procedure for the computation of the safe set. This procedure requires the computation of a maximal controlled invariant set. We then restrict our attention to linear discrete-time systems for which there is a wealth of results available in the literature for the determination of maximal controlled invariant sets. However, even for this class of systems, the computation may not converge in a finite number of steps. We then propose to compute inner approximations that are controlled invariant and for which a procedure that terminates in a finite number of steps can be obtained. A tight bound on the error can be given by comparing the inner approximation with the classical outer approximation of the maximal controlled invariant set. Our procedure is applied to the idle-speed regulation problem in engine control to demonstrate its efficiency.     

Robust invariant sets for constrained storage systems

Several items are produced and stored into n buffers in order to supply an external demand without interruptions. We consider the classical problem of determining control laws and smallest buffer levels guaranteeing that an unknown bounded demand is always satisfied. A simple model with n decoupled integrators and n additive bounded disturbances is employed. The coupling arises from bounds on the total production capacity and on the total demand. Invariant set theory for linear and switched linear systems is exploited to compute robust positive invariant sets and controlled robust invariant sets for two commonly adopted scheduling policies. This paper provides the explicit expression of the invariant sets for any arbitrary n.     

一类离散不确定系统的Tube不变集离线鲁棒模型预测控制

针对一类输入和状态受限的离散线性不确定系统,提出了一种基于Tube不变集的离线鲁棒模型预测控制方法.首先针对输入和状态约束线性时不变标准系统,设计了改进的基于多面体不变集的离线模型预测控制算法,并证明了稳定性.其次对于存在未知有界干扰的实际不确定系统,引入了Tube不变集策略,通过设计对应标准模型的最优控制序列和状态轨迹,给出了实际不确定系统的离线Tube不变集控制策略,保证系统状态鲁棒渐近稳定,并收敛于终端干扰不变集.仿真结果验证了该控制方法的有效性.     

基于多面体不变集的变终端约束集RMPC

针对一类状态和输入受约束的多胞不确定线性时变系统,提出了一种基于多面体不变集的变终端约束集鲁棒模型预测控制算法.首先采用基于状态反馈增益的多面体不变集计算方法,给出了一种新的控制不变集序列构造方法,然后以控制不变集序列的并集作为终端约束集,结合在线优化和增益切换,实施变终端约束集双模鲁棒预测控制.该算法不仅有效地扩大了...     

Receding horizon output feedback control for constrained uncertain systems using periodic invariance

In this article, we consider a receding horizon output feedback control (RHOC) method for linear discrete-time systems with polytopic model uncertainties and input constraints. First, we derive a set of estimator gains and then we obtain, on the basis of the periodic invariance, a series of state feedback gains stabilising the augmented output feedback system with these estimator gains. These procedures are formulated as linear matrix inequalities. An RHOC strategy is proposed based on these state feedback and state estimator gains in conjunction with their corresponding periodically invariant sets. The proposed RHOC strategy enhances the performance in comparison with the case in which static periodic gains are used, and increases the size of the stabilisable region by introducing a degree of freedom to steer the augmented state into periodically invariant sets.     

Optimized robust control invariance for linear discrete-time systems: Theoretical foundations

This paper introduces the concept of optimized robust control invariance for discrete-time linear time-invariant systems subject to additive and bounded state disturbances. A novel characterization of two families of robust control invariant sets is given. The existence of a constraint admissible member of these families can be checked by solving a single and tractable convex programming problem in the generic linear-convex case and a standard linear/quadratic program when the constraints are polyhedral or polytopic. The solution of the same optimization problem yields the corresponding feedback control law that is, in general, set-valued. A procedure for selection of a point-valued, nonlinear control law is provided.     

Analysis and design for discrete-time linear systems subject to actuator saturation

We present a method to estimate the domain of attraction for a discrete-time linear system under a saturated linear feedback. A simple condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant. Moreover, the condition can be expressed as linear matrix inequalities (LMIs) in terms of all the varying parameters and hence can easily be used for controller synthesis. The following surprising result is revealed for systems with single input: suppose that an ellipsoid is made invariant with a linear feedback, then it is invariant under the saturated linear feedback if and only if there exists a saturated (nonlinear) feedback which makes the ellipsoid invariant. Finally, the set invariance condition is extended to determine invariant sets for systems with persistent disturbances. LMI based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements.     

一种改进的鲁棒约束预测控制器的综合设计方法

针对多包描述的不确定系统,提出一种新的鲁棒约束预测控制器.高线设计多包系统worst-case情况下性能最优的不变集,在线求解多包系统无穷时域性能指标的rain-max优化问题.设计方法采用了时变的终端约束集,扩大了初始可行域,并能获得较优的控制性能.仿真结果验证了该方法的有效性.     

Control synthesis for invariant sets of families of linear and nonlinear discrete systems with bounded disturbances

Two sets of controlled linear and nonlinear discrete-time systems under bounded disturbances are analyzed. The control synthesis problem for these two sets of systems, including systems with nonlinear functions of a certain class, is constructively solved. The optimal control minimizes the radius of the invariant set. An illustrative example is presented.     

Constrained MPC for uncertain linear systems with ellipsoidal target sets

Robustly feasible invariant sets provide a way of identifying stabilizable regions for uncertain/time-varying linear systems with input constraints under fixed state feedback control laws. With the introduction of extra degrees of freedom in the form of perturbed control laws, these stabilizable regions can be enlarged. This was done in Lee and Kouvaritakis (Automatica 36 (2000) 1497–1504) in conjunction with polyhedral invariant sets and the aim here is to extend this work using ellipsoidal target sets. We also extend the analysis to take into account both polytopic and unstructured bounded disturbances, as well as unstructured uncertainties.     

MPC of constrained discrete-time linear periodic systems — A framework for asynchronous control: Strong feasibility, stability and optimality via periodic invariance

State-feedback model predictive control (MPC) of discrete-time linear periodic systems with time-dependent state and input dimensions is considered. The states and inputs are subject to periodically time-dependent, hard, convex, polyhedral constraints. First, periodic controlled and positively invariant sets are characterized, and a method to determine the maximum periodic controlled and positively invariant sets is derived. The proposed periodic controlled invariant sets are then employed in the design of least-restrictive strongly feasible reference-tracking MPC problems. The proposed periodic positively invariant sets are employed in combination with well-known results on optimal unconstrained periodic linear-quadratic regulation (LQR) to yield constrained periodic LQR control laws that are stabilizing and optimal. One motivation for systems with time-dependent dimensions is efficient control law synthesis for discrete-time systems with asynchronous inputs, for which a novel modeling framework resulting in low dimensional models is proposed. The presented methods are applied to a multirate nano-positioning system.     

On the computation of convex robust control invariant sets for nonlinear systems

In this paper we provide a method to compute robust control invariant sets for nonlinear discrete-time systems. A simple criterion to evaluate if a convex set in state space is a robust control invariant set for a nonlinear uncertain system is presented. The criterion is employed to design an algorithm for computing a polytopic robust control invariant set. The method is based on the properties of DC functions, i.e. functions which can be expressed as the difference of two convex functions. Since the elements of a wide class of nonlinear functions have DC representation or, at least, admit an arbitrarily close approximation, the method is quite general. The algorithm requires relatively low computational resources.     

基于容许集的变终端约束MPC控制

针对一类输入和状态受约束的离散线性系统,提出一种基于Ⅳ步容许集的变终端约束集模型预测控制方法.首先给出多面体不变集序列作为终端约束集的离线模型预测控制算法,扩大了终端约束集.为进一步扩大初始状态可镇定区域,引入N步容许集,设计了基于容许集的变终端约束集模型预测控制方法.该算法采用离线设计、在线优化方法,实现了系统渐近稳定,不仅降低了在线运算量,而且扩大了初始状态可镇定区域.仿真结果表明了算法的有效性.     

多胞约束LPV系统多面体不变集RMPC算法

针对一类输入和输出受约束且具有多胞结构的离散LPV 系统, 提出一种基于多面体不变集的鲁棒模型预测控制(RMPC) 算法. 选取一系列收敛于原点的离散状态点, 计算每个状态的反馈控制率, 构建相应的多面体不变集. 在每一个采样时刻, 确定包含当前状态的最小多面体不变集, 通过计算与相邻两个多面体不变集的位置关系, 执行连续的状态反馈控制率. 仿真结果表明, 相比椭圆不变集离线RMPC算法, 所提出算法扩大了系统的稳定区域, 取得了保守性较小的结果.