A singular perturbation canonical form of invertible systems: determination of multivariate root-loci

A new canonical form of multivariable invertible systems is given. This form is suitable for the singular perturbation analysis of such singular problems as multi-variable asymptotic root-loci under high-gain feedback, cheap control, state variable estimation with weak measurement noise, etc. Appropriate selection and decomposition of state variables into slow and fast changing groups is a key feature that loads to well-defined singular perturbation models. The selection of state variables is accomplished without explicitly calculating any eigenvalues of the given system but by knowing the direct relationship of the output and its differential coefficients with the input. The use of the canonical form to characterize the multivariable root-loci under high-gain feedback is demonstrated.     

Synthesis of control policies for manufacturing systems using the control canonical form

In this paper, multivariable systems theory is used to synthesize control policies for a manufacturing system which is modelled as a linear discrete-time multivariable dynamical system. The state variables of the system are the rates-of-flow of parts at the various work stations, the buffer inventories, anil the finished product inventories. The control inputs are the man-hours per week required for the various processes involved in the manufacturing system, anil the disturbance inputs are the demands for semi-finished and finished parts. The behaviour of the controlled manufacturing system is illustrated by the presentation of the results of computer simulation studies showing typical transient and steady-state response characteristics,     

Simultaneous stabilization for a collection of nonlinear control systems in canonical form

A controller design method is provided to simultaneously stabilize a collection of nonlinear control systems in canonical form. It is shown that, under a mild assumption, any collection of nonlinear systems in canonical form can be simultaneously stabilized by one continuous state feedback controller. A constructive universal formula is presented explicitly. An illustrative example is given to demonstrate the validity of the method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

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.     

An algorithm for calculation of the Jordan canonical form of a matrix

Jordan canonical forms are used extensively in the literature on control systems. However, very few methods are available to compute them numerically. Most numerical methods compute a set of basis vectors in terms of which the given matrix is diagonalized when such a change of basis is possible. Here, a simple and efficient method is suggested for computing the Jordan canonical form and the corresponding transformation matrix. The method is based on the definition of a generalized eigenvector, and a natural extension of Gauss elimination techniques.     

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.     

Decentralized control and disturbance attenuation for large-scale nonlinear systems in generalized output-feedback canonical form

A global decentralized robust adaptive output-feedback dynamic compensator is proposed for stabilization, tracking, and disturbance attenuation of the decentralized generalized output-feedback canonical form. This represents the largest class for which decentralized robust adaptive output-feedback tracking and disturbance attenuation results are currently available. The system is allowed to contain unknown parameters multiplying output-dependent nonlinearities, and, also, unknown nonlinearities satisfying certain bounds. Under the assumption that a constant matrix can be found for each subsystem to achieve a certain property, it is shown that reduced-order observers and backstepping controllers can be designed to achieve practical stabilization of the tracking error in each subsystem in the presence of bounded disturbance inputs. Sufficient conditions under which asymptotic tracking and stabilization can be achieved are also obtained. Signal gains from disturbance inputs to tracking errors are presented in the input-to-output-practical-stability and integral-input-to-output-practical-stability frameworks. A particular case in which the standard -gain disturbance attenuation is achieved is also provided.     

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.     

Feedback control for non-linear singular systems

An invertible non-linear map is developed, which transforms a non-linear singular system into a regular one and preserves local properties under the application of specific feedback control laws. The presented algorithm is easily implementable on a digital computer, since it is of closed form. An application of the above algorithm is presented on an autonomous vehicle.     

Learning from NN output feedback control of nonlinear systems in Brunovsky canonical form

In this paper, we investigate the learning issue in the adaptive neural network (NN) output feedback control of nonlinear systems in Brunovsky canonical form with unknown affine term. With only output measurements, a high-gain observer (HGO) is employed to estimate the derivatives of the system output which may be associated with the generation of peaking phenomenon. The adverse effect of peaking on learning and its elimination strategies are analyzed. When the gain of HGO is chosen too high, it may cause the failure of learning from the unknown closed-loop system dynamics. Hence, the gain of HGO is not chosen too high to relieve peaking and guarantee the accuracy of the estimated system states. Then, learning from the unknown closed-loop system dynamics can be achieved. When repeating the same or similar control tasks, a neural learning controller is presented which can effectively recall and reuse the learned knowledge to guarantee the output tracking performance. Finally, simulation results demonstrate the effectiveness of the proposed scheme.     

Controllability canonical (phase-variable) form for non-linear time-variable systems

A controllability canonical form for non-linear time-variable systems with a control variable appearing linearly is introduced. The global controllability of this nonlinear phase-variable form is given independently of the non-linearities. A transformation to the controllability canonical form requires the regularity of a matrix which can be considered the controllability matrix of a non-linear time-variable system. A problem which still remains unsolved is the actual computation of the non-linear controllability canonical form for a given system.     

An approach to a singular optimal control problem for non-linear systems using differential form theory

By applying differential form theory, we consider the singular control problem for non-linear systems with control variables appearing linearly in both the system dynamics and the performance index. First, we derive necessary conditions of singular optimality for a single-input system, including the relation to the Euler-Poisson equation and to the generalized Legendre-Clebsch condition. Defining the degree of singularity, we develop necessary conditions satisfied by the singular trajectory embedded in a reduced space. For a time-invariant system, we clarify the relation between the dynamic and the related static optimality. Second, we derive necessary conditions for singular optimality for a multi-input system where the dimension of the control vector is equal to that of the state space. We show that the Shima-Sawaragi condition for the optimality of boundary controls and the generalized Legendre-Clebsch condition are obtained from these conditions. The results are also applied to the analysis of a time-invariant system.     

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.     

线性广义系统的迭代学习控制

针对线性时不变广义系统的迭代学习控制问题．利用时间加权范数性质．通过Frobenius范数给出广义系统在D型和PD型闭环学习律作用下系统的实际输出轨迹逐渐逼近理想输出轨迹的充分条件．并指出在D型闭环学习律的基础上加上P型闭环学习律不影响控制系统的收敛性．但可以改变系统的性能．仿真算例说明了该方法的有效性．     

不确定广义系统鲁棒镇定控制器的设计

讨论了线性定常广义系统的稳定性, 给出了这种系统H_∞ 范数形式的稳定性判据, 在此基础上, 利用线性矩阵不等式 (LMI)方法讨论了含有不确定参数的线性广义控制系统的鲁棒镇定问题, 并相应地给出了鲁棒镇定控制器的设计.     

A canonical expansion for nonlinear systems

The importance of differential geometry, in particular, Lie brackets of vector fields, in the study of nonlinear systems is well established. Under very mild assumptions, we show that a real-analytic nonlinear system has an expansion in which the coefficients are computed in terms of Lie brackets. This expansion occurs in a special coordinate system. We also explain the concept of a pure feedback system. For control design involving a nonlinear system, one approach is to put the system in its canonical expansion and approximate by that part having only feedback paths.     

A new canonical form for an adaptive observer

A new canonical form for an adaptive observer is presented which estimates the state and identifies the parameters of an unknownnth order linear time-invariant system. This is the simplest adaptive observer presented so far in the literature. The adaptive scheme is shown to be globally asymptotically stable, thus guaranteeing the convergence of the identification process.