Inverse system design for LPV systems using parameter-dependent Lyapunov functions

This paper addresses the design problem of gain-scheduled inverse systems (GSISs) for linear parameter-varying (LPV) systems, whose state-space matrices are represented as parametrically affine matrices, using parameter-dependent Lyapunov functions (PDLFs), and proposes a method for them via parametrically affine linear matrix inequalities (LMIs). Our method includes robust inverse system (RIS) design as a special case. For RIS design, our method theoretically encompasses the method using constant Lyapunov functions. A design example is included to illustrate our conclusions.     

连续时间多胞线性变参数系统变增益H∞/H2输出反馈控制

针对一类同时具有参数不确定性和外界干扰的非线性系统,提出了一种连续时间多胞线性变参数(LPV)系统变增益H_∞/H_2输出反馈控制方法.首先,对连续时间多胞LPV系统的变增益混合目标(H_∞/H_2指标和区域极点约束)输出反馈控制器综合方法进行了数学描述;其次,引入新的结构化松弛矩阵变量和参数依赖Lyapunov函数,将满足期望性能的混合目标鲁棒动态输出反馈控制问题转化为线性矩阵不等式框架内的有限维凸优化问题,进一步降低了所设计LPV控制器的保守性.最后,以四分之一车辆模型主动悬架系统为研究对象进行仿真,仿真结果验证了本文控制器的有效性.     

Smooth switching LPV controller design for LPV systems

This paper presents a method to design a smooth switching gain-scheduled linear parameter varying (LPV) controller for LPV systems. The moving region of the gain-scheduling variables is divided into a specified number of local subregions as well as subregions for the smooth controller switching, and one gain-scheduled LPV controller is assigned to each of the local subregions. For each switching subregion, a function interpolating two local LPV controllers associated with its neighborhood subregions is designed to satisfy the constraint of smooth transition of controller system matrices. The smooth switching controller design problem amounts to solving a feasibility problem which involves nonlinear matrix inequalities. To find a solution to the feasibility problem, an iterative descent algorithm which solves a series of convex optimization problems is proposed. The usefulness of the proposed controller design method is demonstrated with a control example of a flexible ball-screw drive system.     

Gain-scheduled H∞ control via parameter-dependent Lyapunov functions

Synthesising a gain-scheduled output feedback H_∞ controller via parameter-dependent Lyapunov functions for linear parameter-varying (LPV) plant models involves solving an infinite number of linear matrix inequalities (LMIs). In practice, for affine LPV models, a finite number of LMIs can be achieved using convexifying techniques. This paper proposes an alternative approach to achieve a finite number of LMIs. By simple manipulations on the bounded real lemma inequality, a symmetric matrix polytope inequality can be formed. Hence, the LMIs need only to be evaluated at all vertices of such a symmetric matrix polytope. In addition, a construction technique of the intermediate controller variables is also proposed as an affine matrix-valued function in the polytopic coordinates of the scheduled parameters. Computational results on a numerical example using the approach were compared with those from a multi-convexity approach in order to demonstrate the impacts of the approach on parameter-dependent Lyapunov-based stability and performance analysis. Furthermore, numerical simulation results show the effectiveness of these proposed techniques.     

A design of gain-scheduled control for a linear parameter varying system: an application to flight control

For multi-input–multi-output, multi-parameter, nonlinearly parameter-dependent linear parameter-varying systems, a gain-scheduled control design method using both minimum sensitivity eigenvalue assignment and quadratic stability check is proposed. The designed controller guarantees the stability of the closed-loop system and assigns the eigenvalues of the frozen parameter closed-loop LPV system in the prespecified disjointed regions. Fewer controllers than that of any other method are needed to cover the whole parameter space. The proposed design method is applied to the design of a flight vehicle's back-to-turn (BTT) controller and nonlinear six-degree-of-freedom (6-DOF) simulations are performed to show its usefulness as a gain-scheduled controller design method.     

Filter design for LPV systems using quadratically parameter-dependent Lyapunov functions

This paper considers design problems of robust gain-scheduled H_∞ and H₂ filters for linear parameter-varying (LPV) systems whose state-space matrices are represented as parametrically affine matrices, using quadratically parameter-dependent Lyapunov functions, and proposes methods of filter design via parametrically affine linear matrix inequalities (LMIs). For robust filters, our design methods theoretically encompass those that use constant Lyapunov functions. Several numerical examples are included that demonstrate the effectiveness of gain-scheduled and robust filters using our proposed methods compared with robust filters using existing methods.     

Robust Static Output Feedback H2/H∞ Control Synthesis with Pole Placement Constraints: An LMI Approach

This paper studies the robust static output feedback (SOF) problem considering pole placement constraints for linear systems with polytopic uncertainty as well as linear parameter varying (LPV) systems. New linear matrix inequality (LMI) approaches are proposed for the SOF controller design while the pole placement, H₂, and H_∞ constraints are guaranteed. In addition, the gain-scheduled SOF controller will be designed for LPV systems if system parameters are measured. The proposed methods can be applied to general linear systems without imposing any constraints on system matrices. The performance and effectiveness of the proposed methods are shown using two examples.

     

Robust H∞ control for discrete-time polytopic uncertain systems with linear fractional vertices

The robust H∞ control problem for discrete-time uncertain systems is investigated in this paper. The uncertain systems are modelled as a polytopic type with linear fractional uncertainty in the vertices. A new linear matrix inequality (LMI) characterization of the H∞ performance for discrete systems is given by introducing a matrix slack variable which decouples the matrix of a Lyapunov function candidate and the parametric matrices of the system. This feature enables one to derive sufficient conditions for discrete uncertain systems by using parameter-dependent Lyapunov functions with less conservativeness. Based on the result, H∞ performance analysis and controller design are carried out. A numerical example is included to demonstrate the effectiveness of the proposed results.     

Anti-windup controller design using linear parameter-varying control methods

In this paper, we seek to provide a systematic anti-windup control synthesis approach for systems with actuator saturation within a linear parameter-varying (LPV) design framework. The closed-loop induced L2 gain control problem is considered. Different from conventional two-step anti-windup design approaches, the proposed scheme directly utilizes saturation indicator parameters to schedule accordingly the parameter-varying controller. Hence, the synthesis conditions are formulated in terms of linear matrix inequalities (LMIs) that can be solved very efficiently. The resulting gain-scheduled controller is non-linear in general and would lead to graceful performance degradation in the presence of actuator saturation non-linearities and linear performance recovery. An aircraft longitudinal dynamics control problem with two input saturation non-linearities is used to demonstrate the effectiveness of the proposed LPV anti-windup scheme.     

Gain-scheduled ℓ-optimal control for boiler-turbine dynamics with actuator saturation

This paper presents a gain-scheduled approach for boiler-turbine controller design. The objective of this controller design is to achieve tracking performance in the power output and drum pressure while regulating water level deviation. Also, the controller needs to take into account the magnitude and rate saturation constraints on actuators. The nonlinear boiler-turbine dynamics is brought into a linear parameter varying (LPV) form which is a parameter-dependent state-space realization. The LPV form of the boiler-turbine dynamics is characterized by nonlinear dependence on drum pressure, which is naturally the scheduling variable. The controller is designed by utilizing the set-valued method for l¹- optimization, which explicitly addresses state constraints and controller saturations in the design process. The overall gain-scheduled design is augmented by a reference governor to avoid performance degradation in the presence of large tracking commands.     

Gain-scheduling control of LFT systems using parameter-dependent Lyapunov functions

In this paper, we propose a new control design approach for linear fractional transformation (LFT) systems using parameter-dependent Lyapunov functions. Instead of assuming parameter dependency in LFT fashion, we consider general parameter-dependent controllers to achieve better closed-loop performance. Using full-block multipliers, new LPV synthesis conditions have been derived in terms of finite number of linear matrix inequalities (LMIs). Both continuous- and discrete-time cases are discussed. A ship steering example has been used to demonstrate advantages and benefits of the proposed approach.     

Peak-to-peak filtering for periodic piecewise linear polytopic systems

In this paper, the robust peak-to-peak filtering problem for periodic piecewise systems with polytopic uncertainties is investigated. Attention is focused on designing of robust peak-to-peak filters that guarantee the robust asymptotic stability of periodic piecewise filtering error system and satisfy a prescribed peak-to-peak disturbance attenuation level for all admissible uncertainties. A sufficient condition is proposed for robust stability and peak-to-peak performance by employing the constructed time-varying Lyapunov function. Two design approaches based on the Lyapunov function with parameter-dependent matrices and parameter-independent matrices are utilised to solve this problem. An algorithm is proposed to compute the periodic filter parameters governed by non-convex feasibility conditions. Two numerical examples are presented to demonstrate the effectiveness of the proposed techniques.     

Reliable gain-scheduled control of discrete-time systems and its application to CSTR model

This paper is focused on reliable gain-scheduled controller design for a class of discrete-time systems with randomly occurring nonlinearities and actuator fault. Further, the nonlinearity in the system model is assumed to occur randomly according to a Bernoulli distribution with measurable time-varying probability in real time. The main purpose of this paper is to design a gain-scheduled controller by implementing a probability-dependent Lyapunov function and linear matrix inequality (LMI) approach such that the closed-loop discrete-time system is stochastically stable for all admissible randomly occurring nonlinearities. The existence conditions for the reliable controller is formulated in terms of LMI constraints. Finally, the proposed reliable gain-scheduled control scheme is applied on continuously stirred tank reactor model to demonstrate the effectiveness and applicability of the proposed design technique.     

Gain-scheduled control via filtered scheduling parameters

Gain-scheduled control via LPV system models enjoys LMI-based synthesis methods and in particular parameter-dependent Lyapunov matrices have been employed to successfully reduce conservatism. Those controllers derived via parameter-dependent Lyapunov matrices, however, end up with depending on derivatives of scheduling parameters. Though this can be avoided by approximating derivatives or restricting Lyapunov matrices to be partly constant, the former loses guarantee of performance and stability and the latter can cause conservatism. This paper proposes a synthesis method of gain-scheduled controllers that depend on filtered scheduling parameters, instead of derivatives, with a concrete guarantee of a performance level. Moreover, it is shown that the performance level of conventional derivative-dependent gain-scheduled controllers is recovered with arbitrarily small errors.     

Special decentralized control problems in discrete-time interconnected systems composed of two subsystems

In this paper, some special decentralized control problems are addressed for discrete-time interconnected systems. First it is pointed out that some subsystems must be unstable to ensure stability of the overall system in some special cases. Then a special kind of decentralized control problem is studied. This kind of problem can be viewed as harmonic control among independent subsystems. Research results show that two unstable systems can generate a stable system through some effective cooperations. In addition, a decentralized controller design method based on linear matrix inequality is also given by using parameter-dependent Lyapunov function method developed for the study of robust stability. A special linear star coupled dynamical network is also considered. The central subsystem must be unstable to stabilize the whole network under a special coupling. Several examples are given to illustrate the results.     

压水堆功率跟踪自适应保性能控制器设计

针对压水堆动态模型的高度非线性和不确定性特点,本文提出一种自适应保性能跟踪控制器(adaptive guaranteed cost control,AGCC)设计方法.首先以堆芯的点堆方程为基础,引入功率跟踪误差的积分项,构造反应堆的增广状态空间模型,再结合线性参数变化(linear parameter varying,LPV)理论,建立了堆芯系统的多胞LPV模型.该控制器的控制输入由状态反馈控制和不确定性补偿组成,结合保性能控制理论和多胞模型理论,求解线性矩阵不等式得到变增益状态反馈矩阵,确保闭环系统全局渐近稳定;利用李亚普诺夫稳定理论得到不确定性参数的自适应律,实现对系统不确定性的动态补偿.仿真结果表明,该控制器不仅对系统不确定项具有自适应性,而且有较好的负荷跟踪性能.     

Switching LPV control designs using multiple parameter-dependent Lyapunov functions

In this paper we study the switching control of linear parameter-varying (LPV) systems using multiple parameter-dependent Lyapunov functions to improve performance and enhance control design flexibility. A family of LPV controllers is designed, each suitable for a specific parameter subregion. They are switched so that the closed-loop system remains stable and its performance is optimized. Two switching logics, hysteresis switching and switching with average dwell time, are examined. The control synthesis conditions for both switching logics are formulated as matrix optimization problems, which are generally non-convex but can be convexified under some simplifying assumptions. The hysteresis switching LPV control scheme is then applied to an active magnetic bearing problem.     

LPV decoupling for control of multivariable systems

This article investigates methods for decoupling multivariable linear parameter varying (LPV) systems and proposes a new interaction measure for decoupled proportional-integral (PI) feedback control design in LPV systems. The proposed approach seeks to benefit the multivariable control of multi-input multi-output (MIMO) systems with variable operating conditions, variable parameters or nonlinear behaviour. This method can improve the tracking performance and reduce the operating conditions variability of such systems with significant coupling in the system dynamics. We design MIMO decoupling feedback LPV controllers to address loop interaction effects. The proposed method uses a parameter-dependent static inversion or SVD decomposition of the system to minimise the effects of the off-diagonal terms in the MIMO system transfer function matrix. A new parameter-dependent interaction measure is introduced based on the SVD decomposition and static inversion which is subsequently utilised for tuning multi-loop PI controller gains. Numerical examples are presented to illustrate the validity of the proposed LPV decoupling methods, as well as the use of the proposed interaction measures for a decoupled multi-loop PI control design.     

Observer-based H∞ resilient control for a class of switched LPV systems and its application

This paper deals with the issue of observer-based H_∞ resilient control for a class of switched linear parameter-varying (LPV) systems by utilising a multiple parameter-dependent Lyapunov functions method. First, attention is focused upon the design of a resilient observer, an observer-based resilient controller and a parameter and estimate state-dependent switching signal, which can stabilise and achieve the disturbance attenuation for the given systems. Then, a solvability condition of the H_∞ resilient control problem is given in terms of matrix inequality for the switched LPV systems. This condition allows the H_∞ resilient control problem for each individual subsystem to be unsolvable. The observer, controller, and switching signal are explicitly computed by solving linear matrix inequalities (LMIs). Finally, the effectiveness of the proposed control scheme is illustrated by its application to a turbofan engine, which can hardly be handled by the existing approaches.