Invariant control sets on flag manifolds

LetG be a semisimple Lie group and letS G be a subsemigroup with nonempty interior. In this paper we study invariant control sets for the action ofS on homogeneous spaces ofG. These sets on the boundary manifolds of the group are characterized in terms of the semisimple elements contained in intS. From this characterization a result on controllability of control systems on semisimple Lie groups is derived. Invariant control sets for the action of S on the boundaries of larger groups¯G withG ¯G are also studied. This latter case includes the action ofS on the projective space and on the flag manifolds.This research was partially supported by CAPES, Grant No. 3578/82-4.     

Controllability of discrete-time control system on the symplectic group

Let x_n+1=exp(A+uB)x_n be a discrete-time control system on with A and B matrices in the symplectic Lie algebra . This paper gives sufficient conditions for the global controllability of the system. These conditions are similar to those of Martin (Math. Control Signal Systems 8 (1995) 279).     

Controllability on classical Lie groups

We consider the problem of accessibility and controllability of certain bilinear systems. These evolve on Lie groups whose Lie algebras are the normal real forms of complex simple Lie algebras. Previous results by other authors were obtained under the assumption that the controlled vector field is strongly regular. Our paper is aimed at weakening this requirement, and involves relating the root structure of elements in a Lie algebra as above to the nodal connection graphs obtained from their standard matrix representations. This is in turn related to a standard irreducibility assumption on the uncontrolled vector field. The abstract results on generation of Lie algebras are of some independent interest. The work of this author was supported in part by a Fulbright grant, while she was visiting the Arizona State University, and by the Centro de Matemática da Universidade de Coimbra/INIC. The work of this author was partially supported by AFOSR Contract No. 85-0224A.     

The reachable set of a linear endogenous switching system

In this work, switching systems are named endogenous when their switching pattern is controllable. Linear endogenous switching systems can be considered as a particular class of bilinear control systems. The key idea is that both types of systems are equivalent to polysystems, i.e. to systems whose flow is piecewise smooth. The reachable set of a linear endogenous switching system can be studied consequently. The main result is that, in general, it has the structure of a semigroup, even when the Lie algebra rank condition is satisfied since the logic inputs cannot reverse the direction of the flow. The adaptation of existing controllability criteria for bilinear systems is straightforward.     

Integral control on Lie groups

In this paper, we extend the popular integral control technique to systems evolving on Lie groups. More explicitly, we provide an alternative definition of “integral action” for proportional(–derivative)-controlled systems whose configuration evolves on a nonlinear space, where configuration errors cannot be simply added up to compute a definite integral. We then prove that the proposed integral control allows to cancel the drift induced by a constant bias in both first order (velocity) and second order (torque) control inputs for fully actuated systems evolving on abstract Lie groups. We illustrate the approach by 3-dimensional motion control applications.     

Controllability of right invariant systems on real simple Lie groups of typeF 4,G 2,C n,andB n *

We deal with controllability of right-invariant systems for some real simple Lie groups ofF ₄,G ₂,C _n , andB _n types. We prove that the so-calledcontrollability rank condition is a necessary and sufficient condition for controllability for an open class of systems. In other papers, analogous results were obtained for Lie groups of the remaining types (i.e.,E ₆,E ₇,E ₈,A _n , andD _n ) using a special property of the root systems of their Lie algebras.     

Formation tracking control for multiple rigid bodies on matrix Lie groups: A system decomposition approach

This paper addresses the finite‐time formation tracking control problem for multiple rigid bodies whose dynamics are defined on the matrix Lie groups, including the special Euclidean group SE(2),SE(3) as specific cases. The reference trajectory in a form of rotational and translational motions is generated offline in advance as the virtual leader for tracking. Moreover, the formation time is specified according to the task, and the desired formation shape is given a priori with respect to the virtual leader. By virtue of the system decomposition approach, intrinsic formation tracking laws are derived for arbitrary initial velocities for the rigid bodies. The tracking controllers are intrinsic meaning that the dynamics and controllers are developed in the body‐fixed frame without a global reference frame. Moreover, based on the geometric convex combination on SE(3), the result is extended to the distributed case where only the neighboring agents' state are used. Two numerical simulations on SE(2) and SE(3) are given respectively to illustrate the validity and robustness of the proposed controllers. Copyright © 2017 John Wiley & Sons, Ltd.     

Optimal control on Lie groups with applications to attitude control

In this paper we solve a class of optimal control problems on Lie groups in the sense that we derive differential equations which the optimal controls must satisfy. These results are applied to the attitude control of a spacecraft modeled as a rigid body. Specifically, we derive control laws (both in open-loop and closed-loop form) to maneuver the spacecraft between two given rotational states in finite time. The laws are such that a cost functional measuring the over-all angular velocity during the spacecraft’s motion is minimized. They do not require recourse to numerical methods and hence can be easily implemented in an on-board attitude control system. After dealing with a three-axis controlled spacecraft we also discuss the case that only torques about two principal axes of an axisymmetric spacecraft can be exerted.     

Motion on submanifolds of noninvariant holonomic constraints for a kinematic control system evolving on a matrix Lie group

For a control system on a matrix Lie group with one or more configuration constraints that are not left/right invariant, finding the combinations of (kinematic) control inputs satisfying the motion constraints is not a trivial problem. Two methods, one coordinate-dependent and the other coordinate-free are suggested. The first is based on the Wei–Norman formula; the second on the calculation of the annihilator of the coadjoint action of the constraint one-form at each point of the group manifold. The results are applied to a control system on SE(3) with a holonomic inertial constraint involving the noncommutative part in a nontrivial way. The difference in terms of compactness of the result between the two methods is considerable.     

Robust formation control of multiagent systems on the Lie group SE(3)

This paper addresses the robust formation control problem of multiple rigid bodies whose kinematics and dynamics evolve on the Lie group SE(3). First, it is assumed that all followers have access to the state information of a virtual leader. Then, a novel adaptive super‐twisting sliding mode control with an intrinsic proportional‐integral‐derivative sliding surface is proposed for the formation control problem of multiagent system using a virtual structure (VS) approach. The advantages of this control scheme are twofold: elimination of the chattering phenomenon without affecting the control performance and no requirement of prior knowledge about the upper bound of uncertainty/disturbance due to adaptive‐tuning law. Since the VS method is suffering from the disadvantages of centralized control, in the second step, considering a network as an undirected connected graph, we assume that only a few agents have access to the state information of the leader. Afterward, using the gradient of modified error function, a distributed adaptive velocity‐free consensus‐based formation control law is proposed where reduced‐order observers are introduced to remove the requirements of velocity measurements. Furthermore, to relax the requirement that all agents have access to the states of the leader, a distributed finite‐time super‐twisting sliding mode estimator is proposed to obtain an accurate estimation of the leader's states in a finite time for each agent. In both steps, the proposed control schemes are directly developed on the Lie group SE(3) to avoid singularity and ambiguities associated with the attitude representations. Numerical simulation results illustrated the effectiveness of the proposed control schemes.     

Closed loop parameter identifiability and adaptive control of a linear stochastic system

We show that if the parameters of a linear stochastic control system are identifiable using a persistently existing input then the same systems remains parameter identifiable if the system operates under closed loop in a certain way. We assume that the controller itself depends on the test value of the system parameters, and as usual the control signal is dithered. The estimation problem is formulated as the problem of minimizing an appropriate asymptotic cost function. It is shown that a suitable modification of the gradient of the cost function converts our problem into another problem for which Ljung's scheme can be applied. Thus the theorem provides a general method for the solution of the local adaptive control problem.     

Global controllability of linear autonomous systems: A geometric consideration

The global controllability in finite time of a linear autonomous system with restrained controls is investigated. Necessary and sufficient conditions are obtained by an approach based on the consideration of geometric properties of the system.     

An estimation problem in compact Lie groups

In this paper an estimation problem is formulated and solved where the process evolves in a compact, semisimple Lie group. The problem is to find explicitly the conditional probability density of a random initial condition of a Brownian motion in the Lie group given a vector-valued function of the group-valued Brownian motion at some fixed positive time. This function is constant on orbits of the Lie group formed by conjugation. The conditional probability density function has a simple, explicit form using some orbit properties of the Lie group. Many physical problems can be modelled as evolving on compact, semisimple Lie groups, e.g. the motion of a rigid body with a fixed point.     

Approximation of a switched linear system by a nonswitched homogeneous polynomial system

1 Introduction A switched system is a hybrid dynamical system con- sisting of a family of continuous-time subsystems and a rule that describes how the subsystems switch among them. Many such systems encountered in actual practice exhibit switching among several subsystems which depend on vari- ous physical phenomena. Switching among different system structures is an essential feature of many engineering appli- cations such as power systems as investigated in [1]. Life in the Playa wetlands an…     

Approximation of a switched linear system by a nonswitched homogeneous polynomial system

It is shown that given a switched linear control system which satisfy the controllability rank condition, it is possible to construct a non-switched homogenous polynomial control system in such a way that the trajectories of the latter may be arbitrarily well approximated norm on finite time intervals.     

Discrete-time linear periodic systems: A note on the reachability and controllability interval length

Discrete-time linear systems with periodic coefficients of period T are considered in this paper. The reachability and controllability indices at time t for periodic systems, μ_rt and μ_ct, are defined. It is shown that the reachability [controllability] subspace at time t does coincide with the reachability [controllability] subspace over the interval (t − μ_rt,T, t) [(t, t + μ_ct.,T)] and properly includes the reachability [controllability] subspace over the interval (t −(μ_rt−1) T, t) [(t, t +(μ_cr−1)T)].     

线性控制系统的生存域

利用非光滑分析工具,讨论了线性控制系统的生存域问题．如果一个集合是生存域,则此集合沿着系统系数矩阵具有非负特征值的特征向量方向移动得到的集合仍然是一个生存域,从而可以得到线性系统的一种新的既非椭球也非多面体的生存域．所提出的方法只需计算系统导数矩阵的特征值,简便易行.     

李群平均方法在单轮姿态和运动控制中的应用

针对一个单轮系统,采用欧拉群描述其位形空间,对于Wei-Norman方程中不能精确求解的参数,采用平均方法对其进行近似计算,从而使被控系统在满足李代数可控性秩条件下,在李群上的平均轨迹能以一定的精度逼近真实轨迹.通过对单轮系统的仿真实验与结果分析,证明了所用方法的有效性,并指出该方法更适合驱动一类量子力学系统的状态.