The geometry,controllability, and flatness property of the n-bar system

We propose a kinematic model of a system moving in an (m?+?1)-dimensional euclidean space and consisting of n rigid bars attached successively to each other and subject to the nonholonomic constraints that the instantaneous velocity of the source point of each bar is parallel to that bar. We prove that the associated control system is controllable and feedback equivalent to the m-chained form around any regular configuration. As a consequence, we deduce that the n-bar system is flat and show that the Cartesian position of the source point of the last (from the top) bar is a flat output. The n-bar system is a natural generalisation of the n-trailer system and we provide a comparison of flatness properties of both systems.     

Euler's discretization,dynamic equivalence and linearization of control systems

It is shown that two continuous-time control systems are dynamically feedback equivalent if and only if their Euler's discretizations are h-dynamically feedback equivalent for every discretization step h. In particular, a continuous-time system is dynamically feedback linearizable if and only if its Euler's discretization is h-dynamically feedback linearizable for every h?>?0. The proofs of these results are based on algebraic characterizations of dynamic feedback equivalence for continuous-time and discrete-time systems. Two continuous-time systems are dynamically feedback equivalent if and only if their differential algebras are isomorphic. Similarly, two discrete-time systems are dynamically feedback equivalent if and only if their difference algebras are isomorphic. Differential algebras corresponding to continuous-time systems and difference algebras corresponding to discretizations of those systems form two categories. Discretization induces a covariant functor from one category to the other. This functor may be inverted as the difference algebras are equipped with the whole family of difference operators corresponding to all discretizations steps h.     

Flatness Conservation in the n-trailer System Equipped with a Sliding Kingpin Mechanism

Nonlinear systems, which are differentially flat, have several properties that can be useful on designing effective controllers. In this paper we show that the n-trailer system equipped with a sliding kingpin mechanism is a differentially flat system, like its non-sliding kingpin counter part. The sliding kingpin technique is used to eliminate the undesired deviation of the path of each intermediate vehicle from that of the leading one (off-tracking phenomenon). The linearizing outputs of the flat system are the Cartesian coordinates of the middle of the last semi-trailer's axle. The state space and the kinematic equations of the new modified system are derived and the conditions for flatness are examined. The flatness conservation is also checked relatively to several kinds of dynamic sliding feedback control.     

Orbital feedback linearization for multi‐input control systems

A nonlinear control system is said to be orbital feedback linearizable if there exist an invertible static feedback and a change of time scale (depending on the state) which transform the system into a linear system. We give geometric necessary and sufficient conditions describing multi‐input control‐affine systems that are orbital feedback linearizable out of equilibria and in the case of equal controllability indices. We also describe a construction of the time rescaling needed to orbitally linearize the system. Moreover, we analyze close relations between orbital feedback linearizable control‐affine systems and control‐linear systems that are feedback equivalent to a multi‐chained form comparing geometric structures corresponding to both problems. We illustrate our results by two examples, one being a rigid bar moving in .Copyright © 2014 John Wiley & Sons, Ltd.     

Adaptive control of feedback linearizable systems: a modelling error compensation approach

An adaptive output feedback controller for single input feedback linearizable systems is proposed. The output derivatives are estimated with a state high-gain observer, and the matched uncertainties are handled using a modelling error compensation method. Compared with existing adaptive methods, this approach avoids overparameterizations yielding an (n+1)-order compensator, where n is the system dimension. Semiglobal stability is proven with the aid of existing results on the stability of controlled nonlinear systems with high-gain state observation. Copyright © 1999 John Wiley & Sons, Ltd.     

Feedback linearization of a flexible manipulator near its rigid body manifold

It is shown that a ‘slow’ 2n-dimensional manifold exists in the 4n-dimensional state space of an n-link manipulator with n flexible joints. While the flexible manifold is not linearizable by feedback, its restriction to'the slow manifold is. A method is given for computing a feedback control which achieves an approximate linearization.     

Nonlinearizable single-input control systems do not admit stationary symmetries

The main result of the paper states that almost any analytic single-input control system, which is truly nonlinear, that is not feedback linearizable, with controllable linearization at an equilibrium point, does not admit any symmetry preserving that point. By almost any system, we mean that we exclude a small class of odd systems, that admit just one nontrivial symmetry conjugated to minus identity. The obtained results are based on a recent classification of nonlinear single-input systems under formal feedback. We also describe symmetries of feedback linearizable systems.     

A‐Orbital feedback linearization of multiinput control affine systems

This article deals with transformations of multiinput nonlinear control systems into linear controllable systems. For multiinput control affine systems, the notion of A‐orbital feedback linearizability is introduced which generalizes the notion of orbital feedback linearizability and is based on input‐dependent time scalings. A necessary and sufficient condition for A‐orbital feedback linearizability is derived for multiinput control affine systems. On the basis of this condition, an A‐orbital feedback linearization algorithm is developed. It is revealed that the proposed concept extends the existing approaches to orbital feedback linearization. More precisely, it is proved that if a system is A‐orbitally feedback linearizable in a neighborhood of some point, the dimension of the state is greater than that of the input by at least three, and the time scaling essentially depends on the input, then the system cannot be orbitally feedback linearized around that point.     

Termination detection for dynamically distributed systems with non-first-in-first-out communication

We propose a new algorithm for detecting termination of distributed systems. The algorithm works correctly whether the system is static or dynamic, whether the interprocess communication is synchronous or asynchronous, and whether the communication channels are first-in-first-out or not. The algorithm requires, in the worst case, O(nm) control messages in all, where n is the number of processes in the system and m is the total number of messages transmitted during the operation of the system. After the system terminates, the algorithm is able to detect the termination using only O(n) control messages; it is optimal if the system concerned is static.     

Multi-input control-affine systems static feedback equivalent to a triangular form and their flatness

In this paper, we give a complete geometric characterisation of systems locally static feedback equivalent to a triangular form compatible with the chained form, for m = 1, respectively with the m-chained form, for m ≥ 2, where the number of controls is m + 1. They are x-flat systems. We provide a system of first-order PDE's to be solved in order to find all x-flat outputs, for m = 1, respectively all minimal x-flat outputs, for m ≥ 2. We illustrate our results by examples, in particular, by an application to a mechanical system: the coin rolling without slipping on a moving table.     

Least-squares integration of one-dimensional codistributions with application to approximate feedback linearization

We study the problem of approximating one-dimensional nonintegrable codistributions by integrable ones and apply the resulting approximations to approximate feedback linearization of single-input systems. The approach derived in this paper allows a linearizable nonlinear system to be found that is close to the given system in a least-squares (L ₂) sense. A linearly controllable single-input affine nonlinear system is feedback linearizable if and only if its characteristic distribution is involutive (hence integrable) or, equivalently, any characteristic one-form (a one-form that annihilates the characteristic distribution) is integrable. We study the problem of finding (least-squares approximate) integrating factors that make a fixed characteristic one-form close to being exact in anL ₂ sense. A given one-form can be decomposed into exact and inexact parts using the Hodge decomposition. We derive an upper bound on the size of the inexact part of a scaled characteristic one-form and show that a least-squares integrating factor provides the minimum value for this upper bound. We also consider higher-order approximate integrating factors that scale a nonintegrable one-form in a way that the scaled form is closer to being integrable inL ₂ together with some derivatives and derive similar bounds for the inexact part. This allows a linearizable nonlinear system that is close to the given system in a least-squares (L ₂) sense together with some derivatives to be found. The Sobolev embedding techniques allow us to obtain an upper bound on the uniform (L _∞) distance between the nonlinear system and its linearizable approximation. This research was supported in part by NSF under Grant PYI ECS-9396296, by AFOSR under Grant AFOSR F49620-94-1-0183, and by a grant from the Hughes Aircraft Company.     

A design procedure for pole assignment using output feedback

A new design procedure is presented for assigning closed-loop eigenvalues of multi-variable, linear systems using output feedback control. Subject to certain mild restrictions, the number of poles which can be assigned to arbitrary, distinct locations is mill [m + r ? 1, n], where m, r and n are the dimensions of the output, control and state vectors, respectively. The design procedure provides an extension of the results of Davison and Chatterjee (1971) and Jameson (1970) but allows a significantly larger number of eigenvalues to be assigned. The simplicity of the design procedure is demonstrated by a numerical example.     

Differential flatness of two one-forms in arbitrary number of variables

Given a differentially flat system of ODEs, flat outputs that depend only on original variables but not on their derivatives are called zero-flat outputs and systems possessing such outputs are called zero-flat. In this paper we present a theory of zero-flatness for a system of two one-forms in arbitrary number of variables (t,x¹,…,x^N). Our approach splits the task of finding zero-flat outputs into two parts. First part involves solving for distributions that satisfy a set of algebraic conditions. The second part involves finding an integrable distribution from the solution set of the first part. Typically this part involves solving PDEs. Our results are also applicable in determining if a control affine system in n states and n−2 controls has flat outputs that depend only on states. We illustrate our method by examples.     

Optimal input sets for steering quantized systems

Limited capacity of communication channels has strongly pushed the analysis of control systems subject to a quantized input set. Quantized control system of type x ⁺ = x + u, where the u takes values in a set of 2m + 1 integer numbers, symmetric with respect to 0 arise in some fundamental situations, e.g., flat, nilpotent, and linear systems with quantized feedback. In this paper we consider this special type of systems and analyze the reachable set after K steps. We find explicit expressions, for each K and for each m, of m input values such that the reachable set after K steps is as large as possible.     

Dynamic anti-windup scheme for feedback linearizable nonlinear control systems with saturating inputs

This paper proposes a dynamic compensation scheme for input-constrained feedback linearizable nonlinear systems to cope with the windup phenomenon. Given a dynamic feedback linearizing controller designed without considering its input constraint, an additional dynamic compensator is proposed to account for the constraint. This dynamic anti-windup is based on the minimization of a reasonable performance index. The proposed strategy is a nonlinear extended version of [Park, J.-K., & Choi, C.-H. (1995). Dynamic compensation method for multivariable control systems with saturating actuators. IEEE Transactions on Automatic Control, 40(9), 1635–1640] with simplified derivation of an optimization solution under relaxed assumptions. The parameter matrices and structure of the solution are explicitly decided by mathematical optimization for infinite horizon without tuning of design parameters unlike previous schemes. During input saturation, the role of the anti-windup scheme with the proposed dynamic feedback compensator is to maintain the controller states to be exactly the same as those without input saturation. The local asymptotic stability and the total stability of the resulting systems are proved. The usefulness of the proposed design method is illustrated by comparative simulations for a constrained control system.     

非线性控制系统的拓扑结构

本文的目的是描述与分析非线性控制系统的拓扑结构.文中首先定义可线性化系统,然 后引入Whitney拓扑,并用此定义非线性控制系统族上的拓扑.在这个拓扑下,证明了除单 输入二维系统外,在一般情况下能线性化的系统是一个零测集.最后讨论一般反馈线性化问 题.对输入数仅比系统维数少一的情况给出了充要条件,并证明了在此情况下几乎所有的系 统均可线性化.     

Minimizing migrations in fair multiprocessor scheduling of persistent tasks

Suppose that we are given n persistent tasks (jobs) that need to be executed in an equitable way on m processors (machines). Each machine is capable of performing one unit of work in each integral time unit and each job may be executed on at most one machine at a time. The schedule needs to specify which job is to be executed on each machine in each time window. The goal is to find a schedule that minimizes job migrations between machines while guaranteeing a fair schedule. We measure the fairness by the drift d defined as the maximum difference between the execution times accumulated by any two jobs. As jobs are persistent we measure the quality of the schedule by the ratio of the number of migrations to time windows. We show a tradeoff between the drift and the number of migrations. Let n = qm + r with 0 < r < m (the problem is trivial for n ≤ m and for r = 0). For any d ≥ 1, we show a schedule that achieves a migration ratio less than r(m − r)/(n(q(d − 1)) + ∊ > 0; namely, it asymptotically requires r(m − r) job migrations every n(q(d − 1) + 1) time windows. We show how to implement the schedule efficiently. We prove that our algorithm is almost optimal by proving a lower bound of r(m − r)/(nqd) on the migration ratio. We also give a more complicated schedule that matches the lower bound for a special case when 2q ≤ d and m = 2r. Our algorithms can be extended to the dynamic case in which jobs enter and leave the system over time.     

双线性系统反馈线性化鲁棒控制的新方法

研究了单输入的双线性系统的精确线性化及其鲁棒控制器设计问题,避免了用微分 几何的李导数、李括号的复杂运算,只用纯代数的方法给出了一般单输入双线性系统精确线 性化的充分条件.同时,在非线性不确定的条件下,讨论了双线性系统精确线性化的鲁棒控制 器的设计问题.     

On pole assignment of linear systems by gain output feedback

In this paper, the Geometric Approach is used to derive in a straightforward way a sufficient condition for pole assignability by gain output feedback. This result leads to a pole assignment procedure which reduces to solving a system of min (n - m, n - p) polynomial equations where n is the number of states, m the number of inputs, p the number of outputs. In the case where m + p > n, this system clearly appears to be linear. The degrees of freedom related to the pole assignment problem are expressed in terms of (right or left) eigenvectors.