Generic nonlinear stabilization of systems with matching algebraic structure

The paper demonstrates that using algebraic methods for the construction of time varying stabilizing controls for general controllable systems which are affine in the control is not only computationally feasible, but delivers generic feedback laws. A single feedback control law can be stabilizing for all systems which have the same algebraic structure and also for systems that can be adequately approximated by this structure. The systems considered are not limited to those whose controllability Lie algebra is nilpotent or even finite dimensional. The stabilizing controls are constructed by the help of an open-loop control problem on an associated Lie group which is posed as a trajectory interception problem in the logarithmic coordinates of flows.     

On the optimal control of bilinear systems and its relation to Lie algebras

The optimal control of bilinear systems is considered and related to the Lie algebra generated by the system matrices. Interesting results obtain when this Lie algebra is nilpotent.     

Motion control of drift-free, left-invariant systems on Lie groups

In this paper we address the constructive controllability problem for drift-free, left-invariant systems on finite-dimensional Lie groups with fewer controls than state dimension. We consider small (ϵ) amplitude, low-frequency, periodically time-varying controls and derive average solutions for system behavior. We show how the pth-order average formula can be used to construct open-loop controls for point-to-point maneuvering of systems which require up to (p-1) iterations of Lie brackets to satisfy the Lie algebra controllability rank condition. In the cases p=2,3, we give algorithms for constructing these controls as a function of structure constants that define the control authority, i.e., the actuator capability, of the system. The algorithms are based on a geometric interpretation of the average formulas and produce sinusoidal controls that solve the constructive controllability problem with O(ϵ^P) accuracy in general (exactly if the Lie algebra is nilpotent). The methodology is applicable to a variety of control problems and is illustrated for the motion control problem of an autonomous underwater vehicle with as few as three control inputs     

A method to obtain the lie group associated with a nilpotent lie algebra

According to Ado and Cartan Theorems, every Lie algebra of finite dimension can be represented as a Lie subalgebra of the Lie algebra associated with the general linear group of matrices. We show in this paper a method to obtain the simply connected Lie group associated with a nilpotent Lie algebra, by using unipotent matrices. Two cases are distinguished, according to the nilpotent Lie algebra is or not filiform.     

自旋量子系统的可控李代数计算

动态系统可控性同是经典和量子控制中研究的一个基本问题,本文研究了单自旋和双自旋量子系统的可控李代数的计算.首先基于量子系统可控的等价性条件,通过单量子系统Hamiltonian算符的李括号运算,给出了与基系数相关的系统可控的充要条件.然后利用Cartan分解方法构造了李代数su(4)的矩阵基,同时根据可控性基本定理提出了Hamiltonian算符多重李括号的计算方法及系统的可控性判据.     

A Counterexample to a Conjecture of Gurvits on Switched Systems

We consider products of matrix exponentials under the assumption that the matrices span a nilpotent Lie algebra. In 1995, Gurvits conjectured that nilpotency implies that these products are, in some sense, simple. More precisely, there exists a uniform bound l such that any product can be represented as a product of no more than I matrix exponentials. This conjecture has important applications in the analysis of linear switched systems, as it is closely related to the problem of reachability using a uniformly bounded number of switches. It is also closely related to the concept of nice reachability for bilinear control systems. The conjecture is trivially true for the case of first-order nilpotency. Gurvits proved the conjecture for the case of second-order nilpotency using the Baker-Campbell-Hausdorff formula. We show that the conjecture is false for the third-order nilpotent case using an explicit counterexample. Yet, the underlying philosophy behind Gurvits' conjecture is valid in the case of third-order nilpotency. Namely, such systems do satisfy the following nice reachability property: any point in the reachable set can be reached using a piecewise constant control with no more than four switches. We show that even this form of finite reachability is no longer true for the case of fifth-order nilpotency.     

On finite volterra series which admit hamiltonian realizations

In this paper we give necessary and sufficient conditions for a stationary finite Volterra series to have a linear (in the controls) analytic realization, which at the same time has a Hamiltonian structure. The result generalizes that for linear systems, where the condition is that the impulse response should be an odd function, and is expressed as a particular symmetry condition on the Volterra kernals. The relation between this problem and that of the inverse problem in Newtonian mechanics is explored. The finiteness of the Volterra series implies a nilpotence condition on a certain Lie algebra defined by the realization. The additional requirement that the system be Hamiltonian adds further structure to the classical representation of a nilpotent Lie algebra by lower triangular matrices.Work supported in part by the Office of Naval Research, Grant No. N00014-75-C-0648 while the author was on leave in the Division of Applied Sciences, Harvard University, Cambridge, Mass. U.S.A.Work supported in part by S.E.R.C. Grant No. GR/B/9116.7.     

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.     

Decomposition and stability of linear systems with multiplicative noise

This paper concerns a representation of solutions and the stability of linear systems with multiplicative white noise, which is described by a vector Ito stochastic differential equation. The solution can be represented as a finite product of exponential matrices if Lie algebra generated by system matrices is solvable. If Lie algebra is not solvable, it is shown by the decomposition principle of Lie algebra that the problem of solving an equation can be reduced to the problem of solving a set of equations, whose corresponding Lie algebra is simple. Noting the structure of the sample solution, we present a technique of obtaining asymptotic stability conditions of sample solutions w.p.1, in the pth-order moment and in the pth-mean moment. The necessary and/or sufficient conditions of stability in some stochastic sense are obtained under certain conditions.     

Logarithmic control,trajectory tracking,and formation for nonholonomic vehicles on Lie group SE(2)

This paper studies the problem of set-point stabilisation, trajectory tracking, and formation tracking for unicycle-type vehicles by taking advantage of the exponential coordinates and the other mathematical tools provided by the Lie group setting of the special Euclidean group SE(2). Motivated by recent developments in geometric control approaches for holonomic systems, we first study the stabilisation problem for nonholonomic constrained kinematic systems by improving the logarithmic stabilising control laws in such a way that they can satisfy nonholonomic constraints too. We then extend the control design to the problem of trajectory tracking by proposing auxiliary systems and investigating the conditions for which the adjoint map in the Lie algebra preserves the velocity constraint. This leads to a global control law which is valid for general time-varying as well as non-smooth trajectories. The tracking control law is also applied to the problem of leader-follower formation tracking by constructing nonholonomic virtual leaders in the desired formation. Finally, the kinematic control law is translated to a differential drive robot using the backstepping technique. To demonstrate the effectiveness of the controllers, the numerical results are presented and the performances are compared with the other three controllers in the literature.     

Optimal filters for bilinear systems with nilpotent Lie algebras

We consider a bilinear signal process driven by a Gauss-Markov process which is observed in additive, white, Gaussian noise. An exact stochastic differential equation for the least squares filter is derived when the Lie algebra associated with the signal process is nilpotent. It is shown that the filter is also bilinear and moreover that it satisfies an analogous nilpotency condition. Finally, some special cases and an example are discussed, indicating ways of reducing the filter dimensionality.     

The Lie algebra of a nonlinear dynamical system and its application to control

Using a recently introduced Lie algebra associated with a nonlinear system, several new results in dynamical systems and control theory are obtained. In particular, we show that the solutions carry the structure of the associated Lie group. A number of stability and boundedness results are given and a generalization of classical modal control is developed.     

Nonlinear filtering problems with finite-dimensional matrix estimation algebras

Matrix estimation algebras are Lie algebras that arise naturally in the nonlinear filtering problem for systems with random structure. These are systems that have finitely many modes of operation, where transitions from one mode to another are governed by a finite-state Markov process. Necessary conditions for finite dimensionality of matrix estimation algebras are given. Finite-dimensional filter realizations are constructed for some of the cases for which the matrix estimation algebra is finite dimensional.     

On the analysis of nonlinear nilpotent switched systems using the Hall–Sussmann system

We consider an open problem on the stability of nonlinear nilpotent switched systems posed by Daniel Liberzon. Partial solutions to this problem were obtained as corollaries of global nice reachability results for nilpotent control systems. The global structure is crucial in establishing stability. We show that a nice reachability analysis may be reduced to the reachability analysis of a specific canonical system, the nilpotent Hall–Sussmann system. Furthermore, local nice reachability properties for this specific system imply global nice reachability for general nilpotent systems. We derive several new results revealing the elegant Lie-algebraic structure of the nilpotent Hall–Sussmann system.     

切换系统输出反馈镇定的特征结构配置方法

考虑由单输入--单输出线性子系统构成的切换系统, 通过解析方法设计输出反馈控制, 使闭环子系统具有公共二次Lyapunov函数. 基于可解Lie代数条件及广义Sylvester方程的通解, 利用特征结构配置方法建立了反馈控制的存在性准则, 并由此给出了控制器的设计形式. 仿真算例实现了设计过程并验证了方法的有效性.     

Nonlinear systems possessing linear symmetry

This paper tackles linear symmetries of control systems. Precisely, the symmetry of affine nonlinear systems under the action of a sub‐group of general linear group GL(n,?). First of all, the structure of state space (briefly, ss) symmetry group and its Lie algebra for a given system is investigated. Secondly, the structure of systems, which are ss‐symmetric under rotations, is revealed. Thirdly, a complete classification of ss‐symmetric planar systems is presented. It is shown that for planar systems there are only four classes of systems which are ss‐symmetric with respect to four linear groups. Fourthly, a set of algebraic equations are presented, whose solutions provide the Lie algebra of the largest connected ss‐symmetry group. Finally, some controllability properties of systems with ss‐symmetry group are studied. As an auxiliary tool for computation, the concept and some properties of semi‐tensor product of matrices are included. Copyright © 2006 John Wiley & Sons, Ltd.     

Stability of switched systems: a Lie-algebraic condition

We present a sufficient condition for asymptotic stability of a switched linear system in terms of the Lie algebra generated by the individual matrices. Namely, if this Lie algebra is solvable, then the switched system is exponentially stable for arbitrary switching. In fact, we show that any family of linear systems satisfying this condition possesses a quadratic common Lyapunov function. We also discuss the implications of this result for switched nonlinear systems.     

Accessibility of bilinear networks of systems: control by interconnections

We analyze controllability properties for a class of bilinear interconnected systems, consisting of networks of linear systems, where the coupling parameters act as control variables. We characterize the system Lie algebra of the resulting bilinear control systems. Necessary and sufficient conditions for accessibility are derived in terms of the underlying interconnection graph. Our results generalize earlier work by Brockett on controllability of bilinear output feedback systems, as well as recent work of Costello and Egerstedt on the control of information-exchange networks for distributed computing.     

Approximation of almost solvable bilinear systems

The complexity of the Wei-Norman integration for a bilinear system can be reduced by approximating the associated Lie algebra by one of lower dimension. An algorithm is given for approximating a class of bilinear systems in this way in the case their Lie algebras are almost solvable.     

Exponential stabilization of driftless nonlinear control systemsusing homogeneous feedback

This paper focuses on the problem of exponential stabilization of controllable, driftless systems using time-varying, homogeneous feedback. The analysis is performed with respect to a homogeneous norm in a nonstandard dilation that is compatible with the algebraic structure of the control Lie algebra. It can be shown that any continuous, time-varying controller that achieves exponential stability relative to the Euclidean norm is necessarily non-Lipschitz. Despite these restrictions, we provide a set of constructive, sufficient conditions for extending smooth, asymptotic stabilizers to homogeneous, exponential stabilizers. The modified feedbacks are everywhere continuous, smooth away from the origin, and can be extended to a large class of systems with torque inputs. The feedback laws are applied to an experimental mobile robot and show significant improvement in convergence rate over smooth stabilizers