Decentralized control and state estimation of linear time-periodic systems

The main contributions of this article are the design of a decentralized controller and state estimator for linear time-periodic systems with fixed network topologies. The proposed method to tackle both problems consists of reformulating the linear periodic dynamics as a linear time-invariant system by applying a time-lifting technique and designing a discrete-time decentralized controller and state estimator for the time-lifted formulation. The problem of designing the decentralized estimator is formulated as a discrete-time Kalman filter subject to sparsity constraints on the gains. Two different algorithms for the computation of steady-state observer gains are tested and compared. The control problem is posed as a state feedback gain optimization problem over an infinite-horizon quadratic cost, subject to a sparsity constraint on the gains. An equivalent formulation that consists in the optimization of the steady-state solution of a matrix difference equation is presented and an algorithm for the computation of the decentralized gain is detailed. Simulation results for the practical cases of the quadruple-tank process and an extended 40-tank process are presented that illustrate the performance of the proposed solutions, complemented with numerical simulations using the Monte Carlo method.     

Periodic disturbance cancellation with uncertain frequency

This paper presents a new algorithm to cancel periodic disturbances with uncertain frequency. The disturbances are cancelled using an internal model structure with adaptive frequency, in parallel with a stabilizing controller. The time-varying internal model controller's states, in steady state, can be mapped to two time-invariant variables: the magnitude or energy of the internal model and frequency of the disturbance. An additional integral controller then can be used to reduce the difference between the internal model controller (IMC) and disturbance frequency to zero. The stability of the feedback control system with this algorithm and convergence of the algorithm to the correct frequency with exact disturbance cancellation are justified by singular perturbation and averaging theories. The algorithm is locally exponentially stable, rather than asymptotically stable. Simulations demonstrate the performance of the algorithm, the ability of this algorithm to identify the frequency of periodic disturbances and to reject periodic disturbances with uncertain frequency.     

Robust model predictive control with guaranteed setpoint tracking

In this paper a novel robust model predictive control (RMPC) algorithm is proposed, which is guaranteed to stabilize any linear time-varying system in a given convex uncertainty region while respecting state and input constraints. Moreover, unlike most existing RMPC algorithms, the proposed algorithm is guaranteed to remove steady-state offset in the controlled variables for setpoints (possibly) different from the origin when the system is unknown linear time-invariant. The controller uses a dual-mode paradigm (linear control law plus free control moves to reach an appropriate invariant region), and the key step is the design of a robust linear state feedback controller with integral action and the construction of an appropriate polyhedral invariant region in which this controller is guaranteed to satisfy the process constraints. The proposed algorithm is efficient since the on-line implementation only requires one to solve a convex quadratic program with a number of decision variables that scale linearly with the control horizon. The main features of the new control algorithm are illustrated through an example of the temperature control of an open-loop unstable continuous stirred tank reactor.     

Output stabilization via pole placement of discrete-time linearperiodic systems

The output stabilization problem for discrete-time linear periodic systems is solved. Both the state-feedback control law and the state-predictor are based on a suitable time-invariant state-sampled reformulation associated with a periodic system. Preliminary concepts of periodic system theory are briefly recalled. In particular, the structural properties of a linear discrete-time periodic system are properly related to those of a time-invariant system associated with it. By resorting to such a time-invariant reformulation, the output stabilization problem via pole placement is solved. The stabilizing controller is constituted by a control law and an asymptotic state predictor, both of which are shown to be periodic     

Satisfactory control of discrete-time linear periodic systems

In this paper satisfactory control for discrete-time linear periodic systems is studied. Based on a suitable time-invariant state sampled reformulation, periodic state feedback controller has been designed such that desired requirements of steady state covariance, H-infinity rejection bound and regional pole assignment for the periodic system are met simultaneously. By using satisfactory control theory, the problem of satisfactory periodic controller can be transformed into a linear programming problem subject to a set of linear matrix inequalities （LMIs）, and a feasible designing approach is presented via LMI technique. Numeric example validates the obtained conclusion.     

A parametric periodic Lyapunov equation with application in semi-global stabilization of discrete-time periodic systems subject to actuator saturation

This paper is concerned with semi-global stabilization of discrete-time linear periodic systems subject to actuator saturation. Provided that the open loop characteristic multipliers are within the closed unit circle, a low gain feedback design approach is proposed to solve the problem by state feedback. Our approach is based on the solution to a parametric discrete-time periodic Lyapunov equation. The proposed approaches not only generalize the corresponding results for time-invariant systems to periodic systems, but also reveal some important intrinsic properties of this class of periodic matrix equations. A numerical example is worked out to illustrate the effectiveness of the proposed approaches.     

Stability and stabilization of discrete-time periodic linear systems with actuator saturation

This paper is concerned with the problems of stability and stabilization for discrete-time periodic linear systems subject to input saturation. Both local results and global results are obtained. For local stability and stabilization, the so-called periodic invariant set is used to estimate the domain of attraction. The conditions for periodic invariance of an ellipsoid can be expressed as linear matrix inequalities (LMIs) which can be used for both enlarging the domain of attraction with a given controller and synthesizing controllers. The periodic enhancement technique is introduced to reduce the conservatism in the methods. As a by-product, less conservative results for controller analysis and design for discrete-time time-invariant systems with input saturation are obtained. For global stability, by utilizing the special properties of the saturation function, a saturation dependent periodic Lyapunov function is constructed to derive sufficient conditions for guaranteeing the global stability of the system. The corresponding conditions are expressed in the form of LMIs and can be efficiently solved. Several numerical and practical examples are given to illustrate the theoretical results proposed in the paper.     

An LMI-based controller synthesis for periodic trajectories in a class of nonlinear systems

The use of finite-dimensional linear time-invariant controllers for the stabilization of periodic solutions in sinusoidally forced nonlinear systems is investigated. By mixing results concerning absolute stability of nonlinear systems and robustness of linear systems, a linear matrix inequality-based controller synthesis technique is developed. The synthesis algorithm yields the controller maximizing a lower bound of the maximum amplitude of the forcing input, for which the corresponding periodic solutions are guaranteed to be stable. The Duffing oscillator is employed to illustrate the main features of the proposed synthesis technique.     

On the output control of multirate systems subject to arbitraryexogenous signals

The design of multi-input, multi-output (MIMO) digital control systems in which each plant output is sampled at its own rate, whereas the inputs are updated all together, is considered. Specifically, a solution is given to the problem of asymptotically zeroing the system error when the exogenous signals coincide with the free motion of any unstable system. The proposed controller consists of a time-invariant internal model of the exogeneous signals and a periodic stabilizer     

新型混合励磁导向系统的状态反馈控制

提出一种新型混合励磁导向系统,阐述其结构特点与工作机理,利用等效磁路法得出磁力解析表达式,结合牛顿第二定律和绕组电路方程,推导出混合励磁导向系统的运动方程.建立以气隙、速度、电流为状态变量的线性化电压控制模型,设计出状态反馈控制器并进行仿真.仿真结果表明:新型混合励磁系统的结构是合理的,理论分析和建模方法是切实可行的.设计的反馈控制器在无干扰情况下,控制电流保持为零,在干扰条件下通过调整控制电流能够实现导向系统的稳定导向,研究成果为此类混合励磁导向系统的工业应用提供理论价值和设计参考.