Jinglai Shen

Dynamics and control of a 3D pendulum

New pendulum models are introduced and studied. The pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force and control forces and moments. Several different pendulum models are developed to analyze properties of the uncontrolled pendulum. Symmetry assumptions are shown to lead to the planar 1D pendulum and to the spherical 2D pendulum models as special cases. The case where the rigid body is asymmetric and the center of mass is distinct from the pivot location leads to the 3D pendulum. Rigid pendulum and multi-body pendulum control problems are proposed. The 3D pendulum models provide a rich source of examples for nonlinear dynamics and control, some of which are similar to simpler pendulum models and some of which are completely new.

Linear Complementarity Systems: Zeno States

A linear complementarity system (LCS) is a hybrid dynamical system defined by a linear time-invariant ordinary differential equation coupled with a finite-dimensional linear complementarity problem (LCP). The present paper is the first of several papers whose goal is to study some fundamental issues associated with an LCS. Specifically, this paper addresses the issue of Zeno states and the related issue of finite number of mode switches in such a system. The cornerstone of our study is an expansion of a solution trajectory to the LCS near a given state in terms of an observability degree of the state. On the basis of this expansion and an inductive argument, we establish that an LCS satisfying the P-property has no strongly Zeno states. We next extend the analysis for such an LCS to a broader class of problems and provide sufficient conditions for a given state to be weakly non-Zeno. While related mode-switch results have been proved by Brunovsky and Sussmann for more general hybrid systems, our analysis exploits the special structure of the LCS and yields new results for the latter that are of independent interest and complement those by these two and other authors.

Lyapunov Stability of Complementarity and Extended Systems

A linear complementarity system (LCS) is a piecewise linear dynamical system consisting of a linear time-invariant ordinary differential equation (ODE) parameterized by an algebraic variable that is required to be a solution to a finite-dimensional linear complementarity problem (LCP), whose constant vector is a linear function of the differential variable. Continuing the authors’ recent investigation of the LCS from the combined point of view of system theory and mathematical programming, this paper addresses the important system-theoretic properties of exponential and asymptotic stability for an LCS with a C

Strongly Regular Differential Variational Systems

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Generating Functions of Switched Linear Systems: Analysis, Computation, and Stability Applications

state trajectory. The novelty of our approach lies in our employment of a quadratic Lyapunov function that involves the auxiliary algebraic variable of the LCS; when expressed in the state variable alone, the Lyapunov function is piecewise quadratic, and thus nonsmooth. The nonsmoothness feature invalidates standard stability analysis that is based on smooth Lyapunov functions. In addition to providing sufficient conditions for exponential stability, we establish a generalization of the well-known LaSalle invariance theorem for the asymptotic stability of a smooth dynamical system to the LCS, which is intrinsically a nonsmooth system. Sufficient matrix-theoretic copositivity conditions are introduced to facilitate the verification of the stability properties. Properly specialized, the latter conditions are satisfied by a passive-like LCS and certain hybrid linear systems having common quadratic Lyapunov functions. We provide numerical examples to illustrate the stability results. We also develop an extended local exponential stability theory for nonlinear complementarity systems and differential variational inequalities, based on a new converse theorem for ODEs with B-differentiable right-hand sides. The latter theorem asserts that the existence of a “B-differentiable Lyapunov function” is a necessary and sufficient condition for the exponential stability of an equilibrium of such a differential system.

Stability and Stabilization of Relative Equilibria of Dumbbell Bodies in Central Gravity

A differential variational system is defined by an ordinary differential equation (ODE) parameterized by an algebraic variable that is required to be a solution of a finite-dimensional variational inequality containing the state variable of the system. This paper addresses two system-theoretic topics for such a nontraditional nonsmooth dynamical system; namely, (non-)Zenoness and local observability of a given state satisfying a blanket strong regularity condition. For the former topic, which is of contemporary interest in the study of hybrid systems, we extend the results in our previous paper, where we have studied Zeno states and switching times in a linear complementarity system (LCS). As a special case of the differential variational inequality (DVI), the LCS consists of a linear, time-invariant ODE and a linear complementarity problem. The extension to a nonlinear complementarity system (NCS) with analytic inputs turns out to be non-trivial as we need to use the Lie derivatives of analytic functions in order to arrive at an expansion of the solution trajectory near a given state. Further extension to a differential variational inequality is obtained via its equivalent Karush-Kuhn-Tucker formulation. For the second topic, which is classical in system theory, we use the non-Zenoness result and the recent results in a previous paper pertaining to the B-differentiability of the solution operator of a nonsmooth ODE to obtain a sufficient condition for the short-time local observability of a given strongly regular state of an NCS. Refined sufficient conditions and necessary conditions for local observability of the LCS satisfying the P-property are obtained

On the asymptotics of penalized spline smoothing

In this paper, a unified framework is proposed to study the exponential stability of discrete-time switched linear systems and, more generally, the exponential growth rates of their trajectories under three types of switching rules: arbitrary switching, optimal switching, and random switching. It is shown that the maximum exponential growth rates of system trajectories over all initial states under these three switching rules are completely characterized by the radii of convergence of three suitably defined families of functions called the strong, the weak, and the mean generating functions, respectively. In particular, necessary and sufficient conditions for the exponential stability of the switched linear systems are derived based on these radii of convergence. Various properties of the generating functions are established, and their relations are discussed. Algorithms for computing the generating functions and their radii of convergence are also developed and illustrated through examples.

Equations of motion for the triaxial attitude control testbed

A dumbbell-shaped rigid body can be used to represent certain large spacecraft or asteroids with bimodal mass distributions. Such a dumbbell body is modeled as two identical mass particles connected by a rigid, massless link. Equations of motion for the five degrees of freedom of the dumbbell body in a central gravitational field are obtained. The equations of motion characterize three orbit degrees of freedom, two attitude degrees of freedom, and the coupling between them. The system has a continuous symmetry due to a cyclic variable associated with the angle of right ascension of the dumbbell body. Reduction with respect to this symmetry gives a reduced system with four degrees of freedom. Relative equilibria, corresponding to circular orbits, are obtained from these reduced equations of motion; the stability of these relative equilibria is assessed. It is shown that unstable relative equilibria can be stabilized by suitable attitude feedback control of the dumbbell. Nomenclature er = unit vector along local vertical (radial) direction ex = unit vector along longitudinal axis of dumbbell ey, ez = orthogonal unit vectors spanning plane perpendicular to dumbbell axis eλ

Linear Complementarity Systems with Singleton Properties: Non-Zenoness

Abstract: This paper performs an asymptotic analysis of penalized spline estimators. We compare P -splines and splines with a penalty of the type used with smoothing splines. The asymptotic rates of the supremum norm of the difference between these two estimators over compact subsets of the interior and over the entire interval are established. It is shown that a Pspline and a smoothing spline are asymptotically equivalent provided that the number of knots of the P-spline is large enough, and the two estimators have the same equivalent kernels for both interior points and boundary points.

Mathematical Models for the Triaxial Attitude Control Testbed

The triaxial attitude control testbed has been developed as part of a research program on spacecraft multibody rotational dynamics and control. In this paper, equations of motion are derived and presented in various forms. Actuation mechanisms are incorporated into the models including: moment actuators that are fixed to the triaxial base body, as well as reaction wheel actuators and proof mass actuators that are fixed to the triaxial base body. The models also allow incorporation of unactuated auxiliary bodies that are constrained to move relative to the triaxial base body. The models expose the dynamic coupling between the rotational motion of the triaxial base body, the relative or shape motion of the auxiliary degrees of freedom, and dynamics associated with actuation mechanisms.