Stability of sampled-data piecewise affine systems: A time-delay approach

This paper addresses the stability analysis of sampled-data piecewise-affine (PWA) systems consisting of a continuous-time plant in feedback connection with a discrete-time emulation of a continuous-time state feedback controller. The sampled-data system is first considered as a continuous-time system with a variable time delay. Conditions under which the trajectories of the sampled-data closed-loop system will converge to an attracting invariant set are then presented. It is also shown that when the sampling period converges to zero, these conditions coincide with sufficient conditions for non-fragility of the stabilizing continuous-time PWA state feedback controller. The results are successfully applied to a helicopter example.     

A mathematical framework for the control of piecewise-affine models of gene networks

This article introduces results on the control of gene networks, in the context of piecewise-affine models. We propose an extension of this well-documented class of models, where some input variables can affect the main terms of the equations, with a special focus on the case of affine dependence on inputs. Some generic control problems are proposed, which are qualitative, respecting the coarse-grained nature of piecewise-affine models. Piecewise constant feedback laws that solve these control problems are characterized in terms of affine inequalities, and can even be computed explicitly for a subclass of inputs. The latter is characterized by the condition that each state variable of the system is affected by at most one input variable. These general feedback laws are then applied to a two-dimensional example, consisting in two genes inhibiting each other. This example has been observed in real biological systems, and is known to present a bistable switch for some parameter values. Here, the parameters can be controlled, allowing to express feedback laws leading to various behaviours of this system, including bi-stability as well as situations involving a unique global equilibrium.     

PWL and PWA ℋ ∞ controller synthesis for uncertain PWA slab systems: LMI approach

This article presents new piecewise-affine (PWA) and piecewise-linear (PWL) ?_∞ controller synthesis methods for uncertain PWA slab systems. The synthesis problems are formulated as convex programs subject to a set of Linear Matrix Inequalities (LMIs), which can then be solved efficiently using available software. No other synthesis methods available in the literature can design a PWA controller for an uncertain PWA system using convex optimisation without constraining the controller gain matrix. Therefore this is the main contribution of the article. The proposed controller synthesis methodologies are applied to a circuit example.     

Piecewise-affine Lyapunov functions for discrete-time linear systems with saturating controls

This paper is concerned with piecewise-affine (PWA) functions as Lyapunov function candidates for stability analysis of time-invariant discrete-time linear systems with saturating closed-loop control inputs. Using a PWA model of saturating closed-loop system, new necessary and sufficient conditions for a PWA function be a Lyapunov function are presented. Based on linear programming formulation of these conditions, an effective algorithm is proposed for construction of such Lyapunov functions for estimation of the region of local asymptotic stability. Compared to piecewise-linear functions, like Minkowski functions, PWA functions are more adequate to capture the dynamical effects of saturation nonlinearities, giving strictly less conservative results. The complexity of the proposed approach is polynomial in state dimension and exponential in saturating control dimension, being hence appropriate for problems with large state dimension but with few saturating inputs.     

Stability analysis of piecewise-affine systems using controlled invariant sets

This paper presents new stability conditions for closed-loop piecewise-affine (PWA) systems. The result is based on controlled invariant sets for PWA systems, which are defined by extending the notion of semi-ellipsoidal invariant sets for constrained linear systems reported in previous research. The paper shows that by proper use of the control input, concatenations of semi-ellipsoidal sets can be made invariant for the trajectories of PWA systems. Furthermore, based on these controlled invariant sets, the paper presents a result for stability of a closed-loop PWA system which is less conservative than existing approaches in the literature. In this result, it is shown that a PWA system is stable inside the intersection of any level set for a local Lyapunov function and the design set where the function is defined, provided the flow points inwards at the boundaries of the intersection. This result is less conservative than previous approaches and it enables the designer to have an estimation of a much larger region of exponential stability then it would be possible using previous results. A numerical example is presented, in which it is made clear by comparison with previous approaches that the estimated region of stability can be made significantly larger using the new stability conditions developed in this paper.     

智能微电网分布式云储能系统安全稳定控制

针对目前智能微电网分布式云储能系统运行架设成本较高且难寻行之有效整体控制规律的问题,引入一种基于观测器的 控制器设计方法,用以处理数学模型体现为具有范数有界时变参数不确定性的一类分段式储能系统。通过采用广义分段仿射Lyapunov函数、投影定理以及几个基本引理,提出了对于由所设计控制器构成的闭环系统基于观测器且满足鲁棒 性能指标的反馈控制器设计方法。通过求解一组包含参变量的LMIs,可以得到保证广义分段仿射系统具有 性能的反馈控制器增益和观测器增益,数值仿真得到干扰抑制度 =6.5896,通过将云储能供电系统与传统电力系统进行比较,在证明所提设计方法有效性的同时也验证了该方法的卓越性。     

Stability of sampled-data piecewise-affine systems under state feedback

This paper addresses stability of sampled-data piecewise-affine (PWA) systems consisting of a continuous-time plant and a discrete-time emulation of a continuous-time state feedback controller. The paper presents conditions under which the trajectories of the sampled-data closed-loop system will exponentially converge to a neighborhood of the origin. Moreover, the size of this neighborhood will be related to bounds on perturbation parameters related to the sampling procedure, in particular, related to the sampling period. Finally, it will be shown that when the sampling period converges to zero the performance of the stabilizing continuous-time PWA state feedback controller can be recovered by the emulated controller.     

Low-complexity piecewise-affine virtual sensors: theory and design

This paper is focused on the theoretical development and the hardware implementation of low-complexity piecewise-affine direct virtual sensors for the estimation of unmeasured variables of interest of nonlinear systems. The direct virtual sensor is designed directly from measured inputs and outputs of the system and does not require a dynamical model. The proposed approach allows one to design estimators which mitigate the effect of the so-called ‘curse of dimensionality’ of simplicial piecewise-affine functions, and can be therefore applied to relatively high-order systems, enjoying convergence and optimality properties. An automatic toolchain is also presented to generate the VHDL code describing the digital circuit implementing the virtual sensor, starting from the set of measured input and output data. The proposed methodology is applied to generate an FPGA implementation of the virtual sensor for the estimation of vehicle lateral velocity, using a hardware-in-the-loop setting.     

Robust explicit model predictive control via regular piecewise-affine approximation

This paper proposes an explicit model predictive control design approach for regulation of linear time-invariant systems subject to both state and control constraints, in the presence of additive disturbances. The proposed control law is implemented as a piecewise-affine function defined on a regular simplicial partition, and has two main positive features. First, the regularity of the simplicial partition allows one to efficiently implement the control law on digital circuits, thus achieving extremely fast computation times. Moreover, the asymptotic stability (or the convergence to a set including the origin) of the closed-loop system can be enforced a priori, rather than checked a posteriori via Lyapunov analysis.     

Observer-based control of piecewise-affine systems

This paper presents a new synthesis method for both state and dynamic output feedback control of a class of hybrid systems called piecewise-affine (PWA) systems. The synthesis procedure delivers stabilizing controllers that can be proven to give either asymptotic or exponential convergence rates. The synthesis method builds on existing PWA stability analysis tools by transforming the design into a closed-loop analysis problem wherein the controller parameters are unknown. More specifically, the proposed technique formulates the search for a piecewise-quadratic control Lyapunov function and a piecewise-affine control law as an optimization problem subject to linear constraints and a bilinear matrix inequality. The linear constraints in the synthesis guarantee that sliding modes are not generated at the switching. The resulting optimization problem is known to be hard, but suboptimal solutions can be obtained using the three iterative algorithms presented in the paper. The new synthesis technique allows controllers to be designed with a specified structure, such as a combined regulator and observer. The observers in these controllers then enable switching based on state estimates rather than on measured outputs. The overall design approach, including a comparison of the synthesis algorithms and the performance of the resulting controllers, is clearly demonstrated in four simulation examples.     

Delay‐dependent anti‐windup strategy for linear systems with saturating inputs and delayed outputs

This paper addresses the problem of the determination of stability regions for linear systems with delayed outputs and subject to input saturation, through anti‐windup strategies. A method for synthesizing anti‐windup gains aiming at maximizing a region of admissible states, for which the closed‐loop asymptotic stability and the given controlled output constraints are respected, is proposed. Based on the modelling of the closed‐loop system resulting from the controller plus the anti‐windup loop as a linear time‐delay system with a dead‐zone nonlinearity, constructive delay‐dependent stability conditions are formulated by using both quadratic and Lure Lyapunov–Krasovskii functionals. Numerical procedures based on the solution of some convex optimization problems with LMI constraints are proposed for computing the anti‐windup gain that leads to the maximization of an associated stability region. The effectiveness of the proposed technique is illustrated by some numerical examples. Copyright © 2004 John Wiley & Sons, Ltd.     

Piecewise-affine state feedback for piecewise-affine slab systems using convex optimization

This paper shows that Lyapunov-based state feedback controller synthesis for piecewise-affine (PWA) slab systems can be cast as an optimization problem subject to a set of linear matrix inequalities (LMIs) analytically parameterized by a vector. Furthermore, it is shown that continuity of the control inputs at the switchings can be guaranteed by adding equality constraints to the problem without affecting its parameterization structure. Finally, it is shown that piecewise-affine state feedback controller synthesis for piecewise-affine slab systems to maximize the decay rate of a quadratic control Lyapunov function can be cast as a set of quasi-concave optimization problems analytically parameterized by a vector. Before casting the synthesis in the format presented in this paper, Lyapunov-based piecewise-affine state feedback controller synthesis could only be formulated as a bi-convex optimization program, which is very expensive to solve globally. Thus, the fundamental importance of the contributions of the paper relies on the fact that, for the first time, the piecewise-affine state feedback synthesis problem has been formulated as a convex problem with a parameterized set of LMIs that can be relaxed to a finite set of LMIs and solved efficiently to a point near the global optimum using available software. Furthermore, it is shown for the first time that, in some situations, the global can be exactly found by solving only one concave problem.     

DC/DC变换器的PWA模型及预测控制

利用υ步离散法,得到变换器输入控制变量与状态变量之间的直接映射关系,基于混杂系统理论分析系统的动态方程,建立其分段仿射模型。在此模型的基础上,结合非线性预测控制算法,通过模型预测系统的输出,利用反馈校正误差,给出二次型性能指标的优化计算方法,并由此设计预测控制器。最后,以Buck功率变换器为研究对象,通过与峰值电流控制算法的仿真结果进行比较,验证模型的正确性以及控制器设计的有效性。     

Dynamic output feedback H ∞ control for affine fuzzy systems

This article investigates the problem of designing H _∞ dynamic output feedback controllers for nonlinear systems, which are described by affine fuzzy models. The system outputs have been chosen as premise variables, which can guarantee that the plant and the controller always switch to the same region. By using a piecewise Lyapunov function and adding slack matrix variables, a piecewise-affine dynamic output feedback controller design method is obtained in the formulation of linear matrix inequalities (LMIs), which can be efficiently solved numerically. In contrast to the existing work, the proposed approach needs less LMI constraints and leads to less conservatism. Finally, numerical examples illustrate the effectiveness of the new result.     

一种固定时间收敛模型参考终端滑模控制方法

针对一类具有模型不确定性和外部扰动的时变非线性系统, 基于模型参考控制方法, 设计了具有固定时间收敛特性的终端滑模控制器. 首先, 提出一种带有输入饱和限幅和补偿信号滤波的模型参考控制结构; 然后针对广义误差信号, 采用新型终端滑模面设计了补偿控制器, 较好地平衡靠近和远离平衡点的收敛速度. 基于李雅普诺夫方法证明了闭环系统的稳定性和固定时间收敛特性, 并给出了收敛时间上界. 最后将该方法应用到含有极限环的非线性系统跟踪控制中, 仿真结果验证了该方法的有效性.     

Switched model predictive control for performance enhancement

This article presents a switched model predictive control (MPC) algorithm for non-linear discrete time systems where the weights on the state and control variables in the cost function to be minimised depend on the current value of the state. In so doing, with a reduced computational burden one can easily include, in the problem formulation, a number of control specifications, such as the requirement to avoid critical regions in the state space or to reduce as much as possible the use of some actuators in other zones of the state space. The proposed MPC method has stability properties and is applied for control of a thermal system. The reported simulation results witness its favourable characteristics with respect to a standard MPC implementation.     

Stability analysis for networked control systems based on event-time-driven mode

This article studies the problem of stability analysis for a class of networked control systems (NCS), whose control gain is assumed to be known. A switched delay system model is obtained based on the event-time-driven scheme. To solve the stability problem of the new NCS model, a new approach based on the Lyapunov functional exponential estimation method is introduced. By using this method, sufficient conditions are developed to guarantee exponential stability of the considered system. The results obtained may be less conservative than the existing ones from the perspective of maximum allowable transfer interval because the unavailable time is considered in this article. An example is given to show the effectiveness of the proposed method.     

基于未知动态观测器双惯量伺服系统低频主动谐振抑制

针对双惯量伺服系统存在的机械谐振, 本文提出了一种基于扰动观测器的滑模主动低频谐振控制方法. 利 用可测量的电机端的位置信号, 对其进行滤波操作得到位置的滤波变量, 根据不变流形原理设计未知扰动观测器, 实现对传递力矩的在线估计, 并将估计值融入到控制器设计中对其进行补偿. 为了消除传统滑模存在颤振现象, 基 于正切函数提出了一种新颖滑模面设计方法. 在此基础上, 设计了滑模控制器实现了对期望信号的准确跟踪, 并利 用李雅普诺夫稳定性理论证明了闭环系统稳定性. 最后通过实验对比, 验证了本文提出的控制策略有效性, 此外该 算法建构简单不需要计算电机加速度的信息, 利于在实际中应用.     

不确定非仿射系统的引入动态逆的自抗扰控制器设计

针对一类不确定非仿射严反馈非线性系统,提出一种引入动态逆的线性自抗扰控制器设计方法.首先,利用微分同胚映射将严反馈非线性系统变换为积分串联型系统,然后针对积分串联型系统设计线性自抗扰控制器.提出的线性自抗扰控制器将闭环系统划分为3个时间尺度,其中线性扩张状态观测器位于最快的时间尺度上,用来估计系统的状态和总和扰动,动态逆位于次快的时间尺度上用以求解非仿射情况下的控制律,系统动态位于最慢的时间尺度上.利用奇异摄动理论分析了闭环系统的稳定性和性能.提出的自抗扰控制设计方法同样适用于控制增益不确定的仿射非线性系统.仿真和实验结果验证了提出的线性自抗扰控制器的可行性.