Singular PDEs and the single-step formulation of feedback linearization with pole placement

The present work proposes a new formulation and approach to the problem of feedback linearization with pole placement. The problem under consideration is not treated within the context of geometric exact feedback linearization, where restrictive conditions arise from a two-step design method (transformation of the original nonlinear system into a linear one in controllable canonical form with an external reference input, and the subsequent employment of linear pole-placement techniques). In the present work, the problem is formulated in a single step, using tools from singular PDE theory. In particular, the mathematical formulation of the problem is realized via a system of first-order quasi-linear singular PDEs and a rather general set of necessary and sufficient conditions for solvability is derived, by using Lyapunov's auxiliary theorem. The solution to the system of singular PDEs is locally analytic and this enables the development of a series solution method, that is easily programmable with the aid of a symbolic software package. Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback law computed through the solution of the system of singular PDEs, both feedback linearization and pole-placement design objectives may be accomplished in a single step, effectively overcoming the restrictions of the other approaches by bypassing the intermediate step of transforming the original system into a linear controllable one with an external reference input.     

Current Estimation and Path Following for an Autonomous Underwater Vehicle (AUV) by Using a High-gain Observer Based on an AUV Dynamic Model

A path following problem for autonomous underwater vehicles (AUVs) under a nonuniform current is presented in this paper. A dynamic model of an AUV in a nonuniform flow was adopted to develop a high-gain observer (HGO) for estimation of the three-dimensional current velocities along AUV trajectories. The HGO was chosen as a nonlinear estimation algorithm, and the observer gain was computed by solving a Linear Matrix Inequality (LMI) which represented the estimation error dynamics. The current velocities were determined by calculating the differences between the measured absolute velocities of the vehicle and the estimated relative velocities of the vehicle estimated by the observer. The estimation error means of the HGO using the LMI have smaller values than the state observer with a gain matrix determined by the pole-placement approach. For the path following study, the desired curved path was represented by using a Serret-Frenet frame which propagated along the curve. The path-following system includes a guidance law, an update law and a proportional and integral controller. Two cases of numerical simulations were conducted to verify the performance of the path following system combined with HGO for current compensation, and the results of both cases have shown that the AUV reached and converged to the desired path.

     

基于自适应观测器的无速度传感器感应电机控制

针对采用极点配置的自适应速度观测器存在不稳定区域的问题,建立了全阶自适应状态观测器并给出了观测器的速度辨识律.应用Lyapunov稳定性理论,观测器的增益借助于MATLAB LMI工具箱求解两个双线性矩阵不等式得到.在MATLAB 6.5/SIMULINK环境下,建立了无速度传感器感应电机直接转矩控制的仿真实验平台,给出了无速度传感器直接转矩控制的仿真结果.仿真结果表明本文给出的自适应观测器在全速范围内具有很好的稳态性能,并具有很好的鲁棒性.同时,在以TMS320F240为核心的感应电机直接转矩控制系统上进行了速度辨识实验,实验结果验证了方案的有效性.     

LMI optimization approach to robust decentralized controller design

A new approach for design of robust decentralized controllers for continuous linear time‐invariant systems is proposed using linear matrix inequalities (LMIs). The proposed method is based on closed‐loop diagonal dominance. Sufficient conditions for closed‐loop stability and closed‐loop block‐diagonal dominance are obtained. Satisfying the obtained conditions is formulated as an optimization problem with a system of LMI constraints. By adding an extra LMI constraint to the system of LMI constraints in the optimization problem, the robust control is addressed as well. Accordingly, the decentralized robust control problem for a multivariable system is reduced to an optimization problem for a system of LMI constraints to be feasible. An example is given to show the effectiveness of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

A pole-placement design approach for systems with multipleoperating conditions

The author proposes design procedures based on state-space pole-placement techniques for systems with multiple operating conditions. This is the so-called simultaneous pole-placement problem. First, the full state feedback problem is studied, in which a nonlinear local pole-placement solution is proposed. The design condition is formulated in terms of the rank condition of a multimode controllability matrix. Then, the output feedback problem is approached using a multimodel controller design, which is an extension of the observer design to multimode systems. The design is decomposed into separated global pole-placement subproblems and a local pole-placement subproblem. For a system with some operating conditions having modes on the j ω-axis, but no modes at the origin in the open right-half of the complex plane, stabilizability and detectability conditions for the design of an asymptotically stabilizing control are established, without any restriction on the number of inputs or outputs. Relations of this approach to other simultaneous control design approaches are pointed out     

一类随机混杂系统的鲁棒方差控制

对一类结构参数不完全已知的Markov跳变参数系统, 研究使得闭环系统的稳态状态方差小于某个给定的上界, 同时满足一定H∞性能的状态反馈鲁棒方差控制器设计问题. 运用线性矩阵不等式(Linear matrix inequality, LMI)方法, 对系统进行了方差分析, 给出并证明了控制器存在的条件, 进而用一组线性矩阵不等式的可行解给出了控制器的一个参数化表示. 通过建立一个具有LMI约束的凸优化问题, 给出了最小方差鲁棒控制器的设计方法. 最后仿真结果表明了该方法的有效性.     

多点悬浮系统的解耦及控制系统设计

本文主要讨论多点磁悬浮系统建模和解耦问题。在建立多点悬浮模型的基础上,利用矩阵变换技巧,实现系统运动状态的解耦,同时根据实际悬浮系统的稳定性能要求和极点配置理论设计一个闭环控制系统,由反馈矩阵K给出反馈控制。     

Transmission zero assignment using semistate descriptions

The problem of finite/infinite transmission zero assignment is examined by squaring a system from the outputs to the inputs. In particular, this problem is studied in two cases: state-accessible systems and partially-state accessible systems. It is shown that the problem of transmission zero assignment for state-accessible state-variable systems is equivalent to a pole-placement problem with state feedback for a generalized system, which always has a solution. In the case of partially-state-accessible systems, it is shown that the transmission zero assignment problem is equivalent to a pole-placement problem with output feedback for a generalized system. In both cases the block Hessenberg form of the system and the extended lower triangular Hessenberg form are exploited to formulate the problem     

A functional equations approach to nonlinear discrete-time feedback stabilization through pole-placement

The present work proposes a new approach to the nonlinear discrete-time feedback stabilization problem with pole-placement. The problem's formulation is realized through a system of nonlinear functional equations and a rather general set of necessary and sufficient conditions for solvability is derived. Using tools from functional equations theory, one can prove that the solution to the above system of nonlinear functional equations is locally analytic, and an easily programmable series solution method can be developed. Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear discrete-time state feedback control law that are both computed through the solution of the system of nonlinear functional equations, the feedback stabilization with pole-placement design objective can be attained under rather general conditions. The key idea of the proposed single-step design approach is to bypass the intermediate step of transforming the original system into a linear controllable one with an external reference input associated with the classical exact feedback linearization approach. However, since the proposed method does not involve an external reference input, it cannot meet other control objectives such as trajectory tracking and model matching.     

Stability analysis and anti-windup design of switched systems with actuator saturation

The stability analysis and anti-windup design problem is investigated for two linear switched systems with saturating actuators by using the single Lyapunov function approach. Our purpose is to design a switching law and the anti-windup compensation gains such that the maximizing estimation of the domain of attraction is obtained for the closed-loop system in the presence of saturation. Firstly, some sufficient conditions of asymptotic stability are obtained under given anti-windup compensation gains based on the single Lyapunov function method. Then, the anti-windup compensation gains as design variables are presented by solving a convex optimization problem with linear matrix inequality (LMI) constraints. Two numerical examples are given to show the effectiveness of the proposed method.     

不确定线性周期系统的满意滤波

针对线性周期多采样率系统的满意滤波问题,根据满意估计思想,将结构不确定周期系 统满足滤波系统稳定与H∞指标的满意估计问题转化为Riccati矩阵不等式求解问题,无须采用 通常的提升手段而直接运用数值法与内点法进行周期系统多指标设计.利用线性矩阵不等式技 术(LMI)提出了一种基于LMI的满意滤波器设计方法.仿真例子验证了相关的结论.     

Reliable LQ fuzzy control for continuous-time nonlinear systems with actuator faults

This paper deals with the reliable linear quadratic (LQ) fuzzy control problem for continuous-time nonlinear systems with actuator faults. The Takagi-Sugeno (T-S) fuzzy model is employed to represent a nonlinear system. By using multiple Lyapunov functions, an improved linear matrix inequality (LMI) method for the design of reliable LQ fuzzy controllers is investigated, which reduces the conservatism of using a single Lyapunov function. The different upper bounds on the LQ performance cost function for the normal and different actuator fault cases are provided. A suboptimal reliable LQ fuzzy controller is given by means of an LMI optimization procedure, which can not only guarantee the stability of the closed-loop overall fuzzy system for all cases, but also provide an optimized upper bound on a weighted average LQ performance cost function. Finally, numerical simulations on the chaotic Lorenz system are given to illustrate the application of the proposed design method.     

时变不确定奇异时滞系统的时滞相关H∞控制

研究一类含有时变时滞的不确定奇异系统的H∞控制问题,采用Lyapunov泛函方法和线性矩阵不等式（LMI）工具,给出闭环系统正则、渐近稳定且具有范数界的时滞相关充分条件;通过求解相应的线性矩阵不等式,得到奇异系统的H∞控制律。与已有结论相比,该方法无需模型变换。数值算例说明该方法的有效性。     

一类Takagi-Sugeno模糊离散广义系统的稳定性判据

利用Lyapunov方法,研究了一类Takagi-Sugeno(T-S)模糊离散广义系统的稳定性问题．给出了该系统一致正则、因果和稳定的充分条件,并把此条件用线性矩阵不等式(LMI)表示．通过矩阵分解把广义系统的非严格矩阵不等式约束转化成严格的矩阵不等式约束,从而可以用LMI工具箱一次判定系统的稳定性,算例说明所给方法使用方便．     

基于T-S模型的非线性时滞系统非脆弱保性能模糊控制

针对一类用T-S模糊模型描述的非线性时滞系统,采用状态反馈的并行分布补偿方法,研究其保守性较小的非脆弱保性能模糊控制问题,使闭环系统在控制器存在加性摄动的情况下,其闭环性能指标值低于确定的上界.利用线性矩阵不等式处理方法,导出了非脆弱保性能模糊控制律的存在条件,通过建立和求解一个线性矩阵不等式问题,给出了非脆弱保性能模糊控制律的设计方法.仿真结果表明了所提出方法的有效性.     

Constrained pole-placement using transformation and LQ-design

A new procedure for constrained pole-placement is presented. The poles of the closed loop system are placed within a circle in the left half plane. The design procedure is noniterative and the solution is obtained via linear quadratic (LQ) design for a transformed system. The circle can be chosen in order to obtain good transient properties of the closed loop system. The method combines the properties of pole-placement and LQ-design.     

基于线性矩阵不等式的分散鲁棒跟踪控制器设计

应用线性矩阵不等式（ＬＭＩ）方法研究不确定性关联大系统的分散鲁棒输出跟踪控制问题。系统中不确定项具有数值界,可不满足匹配条件。基于不确定基项的表达形式,给出了存在分散控制鲁棒跟踪控制器的ＬＭＩ条件。在此基础上,通过建立求解受ＬＭＩｓ约束的凸优化问题,提出了具有较小反馈增益ＬＭＩ设计方法,使受控系统渐近跟踪给定的参考输入。ＬＭＩ方法求解简单,最后用示例说明了该方法的应用。     

Robust controllability assessment via semi-definite programming

This paper presents a new framework for systematic assessment of the controllability of uncertain linear time-invariant (LTI) systems. The objective is to evaluate controllability of an uncertain system with norm-bounded perturbation over the entire uncertain region. The method is based on a singular-value minimization problem, over the entire complex plane. To solve the problem, first, a necessary and sufficient condition is proposed to avert the difficulties of griding over the complex plane, and to verify the controllability of directed and undirected networks in a single step. Secondly, the results are utilized to formulate the problem as two Lyapunov-based linear matrix inequalities (LMIs) for undirected networks, and four LMIs, for directed ones. The proposed approach is subsequently extended to evaluate the maximum guaranteed distance to uncontrollability of a system through a quasi-convex optimization problem whose solution is guaranteed to be globally optimal. By duality, analogous results are established for robust observability of directed and undirected networks. The proposed framework is implemented efficiently using LMI tools, and provides a fast and reliable tool for the assessment of robust controllability (and by duality, robust observability). It is also extendable to robust control system design problems such as control node selection for uncertain systems.     

New Approach to Robust$ cal D$-Stability Analysis of Linear Time-Invariant Systems With Polytope-Bounded Uncertainty

This note presents a new approach to robust$ cal D$-stability analysis of linear time-invariant systems with polytope-bounded uncertainty. The proposed approach combines sufficient conditions for robust$ cal D$-stability in terms of feasibility problems with linear matrix inequalities (LMI) constraints and a new polytope partition technique. If the initial polytope does not attain the robust$ cal D$-stability sufficient condition, the polytope is successively subdivided until all subpolytopes attain the sufficient condition, in the case of robustly$ cal D$-stable uncertain system, or it is found a subpolytope vertex that does not attain the regional pole-placement constraints, in the case of an uncertain system that is not robustly$ cal D$-stable. It is presented a new general format polytope partition technique that allows the implementation of the proposed approach. The efficiency of the proposed approach is verified by means of illustrative examples and three different LMI-based analysis formulations.