A Jordan control canonical form for singular systems

A canonical form for controllable singular systems modulo restricted system equivalence is presented. The canonical form respects the decomposition of the singular system into a regular and an impulsive part. The construction is based on Popov's control canonical form and on the Jordan canonical form recently developed by Hinrichsen and Prätzel-Wolters. Continuity properties are analysed and the cellular partition of the orbit space induced by the canonical form is briefly discussed.     

A singular perturbation model of reliability in systems control

In many engineering systems, the intensity of the input noise is small and the state trajectory is mainly due to the deterministic part of the system structure. When this is the case for a white input noise, the mean exit-time from the reliability region may be represented as a solution to a partial differential equation with a small parameter multiplying the highest derivatives. Hence, the evaluation of the mean exit-time from the reliability region reduces to the solution of a singularly perturbed partial differential equation for which an asymptotic solution can be evaluated analytically in a fairly general case. The mean exit-time is then used to measure the reliability of some decentralized control policies for linear stochastic systems.     

Decentralization of large-scale systems: A new canonical form for linear multivariable systems

This note presents a new canonical form for linear multivariable systems. This canonical form casts and partitions the system to the so-called component connection model (CCM). The application of this canonical form to the problem of decentralized stabilization and estimation is discussed.     

Sensitivity models of canonical form delay-differential systems

In the study of adaptive control problems and design of low sensitivity systems, it is often required to evaluate the sensitivity of systems to parameter variations. This correspondence presents a method for obtaining low-order sensitivity models for canonical form single-input linear constant delay differential systems. Total symmetry and complete simultaneity (TSACS) properties of the first-order sensitivity functions are provided. They subsume and reduce to the results of Wilkie and Perkins [7] in the absence of delay.     

An observable canonical form of discrete-time bilinear systems

A canonical form of discrete-time bilinear systems is investigated. A new concept of sequential observability is introduced and it is shown that the discrete bilinear system is sequentially observable if and only if it can be transformed into some canonical form. One of the distinctions between discrete-time and continuous-time bilinear systems is discussed from uniform observability and sequential observability.     

Stability margins of singular perturbation systems via state-feedback control

The gain and phase margins of singular perturbation systems are analysed under unmodelled high-frequency dynamics control, composite control, and the original full-order linear quadratic (LQ) control. The analysis is on the basis that there is a good relation between the minimum singular value of return difference transfer matrix and the stability margins. We begin with the examination of stability margins of subsystems and then show that state-feedback control design of subsystems could preserve gain and phase margins for the original full-order singularly perturbed system if the singular perturbation parameter epsiv; is sufficiently small. The effectiveness of ε on stability margins is formulated and determined. It is found that the effectiveness can be evaluated by a simple method. Two examples are exploited to illustrate the analytic results.     

具有非线性扰动的广义系统的鲁棒H∞控制

研究具有非线性结构扰动广义系统的鲁棒H∞控制和鲁棒H∞保性能控制问题,该不确定性为时间和状态的函数.且满足Lipschitz条件.目的是分别设计系统的鲁棒H∞控制器和鲁棒H∞保性能控制器.应用线性矩阵不等式方法,分别给出了系统的鲁棒H∞控制器和鲁棒H∞保性能控制器存在的充分条件.当这些条件可解时,分别给出了鲁棒H∞控制器和鲁棒H∞保性能控制器的表达式.最后通过一个仿真算例说明了所给出方法的应用.     

Transformation of time-variable multi-input systems to a canonical form

The problem of transforming a multi-input time-variable system to a canonical form is considered in this correspondence. The necessary and sufficient condition for the existence of the nonsingular transformation is stated and proved. The canonical form, when restricted to time-invariant systems, is the same as the one given in [3]. The advantages of the present method over [3] are explained.     

Stabilization of second-order nonholonomic systems in canonical chained form

Stabilization of a class of second-order nonholonomic systems in canonical chained form is investigated in this paper. First, the models of two typical second-order nonholonomic systems, namely, a three-link planar manipulator with the third joint unactuated, and a kinematic redundant manipulator with all joints free and driven by forces/torques imposing on the end-effector, are presented and converted to second-order chained form by transformations of coordinate and input. A discontinuous control law is then proposed to stabilize all states of the system to the desired equilibrium point exponentially. Computer simulation is given to show the effectiveness of the proposed controller.     

基于奇异摄动理论的非仿射系统的输出调节问题

基于奇异摄动的时标分解理论和动态逆设计的方法,研究了一类非仿射非线性系统的输出调节问题.通过将控制律定义为一个快动态系统的解,得到了指数稳定的误差系统状态方程,从而实现了原系统的输出相对于外系统的参考信号具有零误差的跟踪效果.结果表明在满足标准假设和一组偏微分方程有解的前提下,该输出调节问题是可解的.仿真实例表明了该理论方法的有效性.     

Robustness of dynamic output feedback control for singular perturbation systems

The problem of the robustness of dynamic output feedback control for singular perturbation systems is investigated. The solution to this problem is reduced to the simultaneous design of static output feedback controllers for the fast subsystem and the so-called auxiliary system of the slow subsystem. Some conditions are proposed to ensure the robustness of the actual closed-loop system. The exact upper bound of the parasitic parameter for the controlled system is also determined. Finally, an actual model which failed in dynamic output feedback control in [4] is reexamined successfully here.     

Output regulation of nonaffine nonlinear systems using singular perturbation theory

A control synthesis method for output regulation based on singular perturbation theory combined with inverting design is considered for a class of nonaffine nonlinear systems. The resulting control signal is defined as a solution to ＂fast＂ dynamics which inverts a series error model, whose state is exponentially stable. It is shown that, under sufficient conditions being consistent with the assumptions of singular perturbation theory, this problem is solvable with （ε） tracking error if and only if a set of first-order nonlinear partial differential equations are solvable. The control law can be easily constructed and the simulations show the feasibility and effectiveness of the controller.     

Generalized quadratic stability for continuous-time singular systems with nonlinear perturbation

This note considers the generalized quadratic stability problem for continuous-time singular system with nonlinear perturbation. The perturbation is a function of time and system state and satisfies a Lipschitz constraint. In this work, a sufficient condition for the existence and uniqueness of solution to the singular system is firstly presented. Then by using S-procedure and matrix inequality approach, a necessary and sufficient condition is presented in terms of linear matrix inequality, under which the maximal perturbation bound is obtained to guarantee the generalized quadratic stability of the system. That is, the system remains exponential stable and the nominal system is regular and impulse free. Furthermore, robust stability for nonsingular systems with perturbation can be obtained as a special case. Finally, the effectiveness of the developed approach for both singular and nonsingular systems is illustrated by numerical examples.     

Equivalence of nonlinear systems to triangular form: the singular case

The problem of state equivalence of a given nonlinear system to a triangular form is considered here. The solution of this problem has been known for the regular case, i.e. when there exists a certain nested sequence of regular and involutive distributions. It is also known that in this case the corresponding system is linearizable using a smooth coordinate change and static state feedback. This paper deals with the singular case, i.e. when the nested sequence of involutive distributions of the system contains singular distributions. Special attention is paid to the so-called bijective triangular form. Geometric, coordinates-free criteria for the solution of the above problem as well as constructive, verifiable procedures are given. These results are used to obtain a result in the nonsmooth stabilization problem.     

Observing non-linear time-variable systems through a canonical form observer

Bestle and Zeitz have introduced a non-linear time-variable observer canonical form which permits, through a non-linear state transformation, the use of the eigenvalue assignment method to design observers for non-linear time-variable dynamic systems. In this paper, a necessary and sufficient condition under which the transformation exists is given. It is also proved that the transformation could be determined by integrating single variable functions instead of solving a set of partial differential equations as shown by Bestle and Zeitz. Examples are included to show how to do the transformation.     

线性不确定奇异摄动系统的稳定控制鲁棒界

对于具有不确定量的奇异摄动系统，利用Lyapunov稳定性理论研究其稳定鲁棒控制问题。在一定条件下得到了任一理想奇异摄动系统的稳定鲁棒控制，也是不确定奇异摄动系统具有一定鲁棒界的稳定控制。给出了闭环奇异摄动系统渐近稳定的条件，讨论了鲁棒界及其变化范围。     

随机非线性扰动下广义马氏跳变系统的控制器设计

针对实际系统中非线性扰动以一定概率存在的情况, 讨论广义马氏跳变系统的控制问题. 首先, 运用Bernoulli 变量描述系统中非线性扰动的存在与否, 建立相应的数学模型, 得到在上述一般条件下系统的随机容许性与解的存在唯一性条件. 然后, 以LMI 形式给出模态依赖和模态独立控制器的存在条件. 与现有结果相比, 所得结果考虑了非线性扰动的存在概率, 具有较小的保守性. 最后, 通过数值例子验证了所得结果的有效性和优越性.

     

Deadbeat control of discrete-time systems with singular state matrix using canonical decomposition

The problem of deadbeat control of controllable discrete-time systems with singular state matrix is considered. Our approach is based on the representation of the system obtained by isolating the reachable part of the system and the controllable but unreachable part using the Kalman structure theorem. It is shown that, eigenvalue assignment for controllable but unreachable single-input systems does not lead to a unique control law. In fact, the extra degrees of freedom available can be used to shape the transient response characteristics of the system.