An Invariance Principle for Nonlinear Hybrid and Impulsive Dynamical Systems
V.-S. Chellaboina, S. P. Bhat and W. M. Haddad
American Control Conference, June 2000
 
 
Abstract-In this paper we develop an invariance principle for dynamical systems possessing left-continuous flows. Specifically, we show that left-continuity of the system trajectories in time for each fixed state point and continuity of the system trajectory in the state for every time in some dense subset of the semi-infinite interval are sufficient for establishing an invariance principle for hybrid and impulsive dynamical systems. As a special case of this result we state and prove new invariant set stability theorems for a class of nonlinear impulsive dynamical systems; namely, state-dependent impulsive dynamical systems. These results provide less conservative stability conditions for impulsive systems as compared to classical results in the literature and allow us to address the stability of limit cycles and periodic orbits of impulsive systems.


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Lyapunov Analysis of Semistability
S. P. Bhat and D. S. Bernstein
American Control Conference, 1999
 
 
Abstract-Semistability is the property whereby the solutions of a system converge to stable equilibrium points determined by the initial conditions. Important applications of this notion of stability include lateral aircraft dynamics and the dynamics of chemical reactions. A notion central to semistability theory is that of convergence in which every solution converges to a limit point that may depend upon the initial condition. We give sufficient conditions for convergence and semistability of nonlinear systems. By way of illustration, we apply these results to study the semistability of linear systems and some nonlinear systems.
 
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Nonegativity, Reducibility and Semistability of Mass Action Kinetics
D. S. Bernstein and S. P. Bhat
IEEE Conference on Decision and Control, 1999
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Average-Preserving Symmetries and Equipartition in Linear
Hamiltonian Systems

Sanjay P. Bhat & Dennis S. Bernstein
IEEE Conference on Decision and Control, 2004

Abstract:This paper analyzes equipartition in linear Hamiltonian systems in a deterministic setting. We consider the group of phase space symmetries of a stable linear Hamiltonian system, and characterize the subgroup of symmetries whose elements preserve the time averages of quadratic functions along the trajectories of the system. As a corollary, we show that if the system has simple eigenvalues, then every symmetry preserves averages of quadratic functions. As an application of our results to linear undamped lumped-parameter systems, we provide a novel proof of the virial theorem using symmetry. We also show that under the assumption of distinct natural frequencies, the time-averaged energies of two identical substructures of a linear undamped structure are equal. Examples are provided to illustrate the results.

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Controllability of Spacecraft Attitude Under Magnetic Actuation

Sanjay P. Bhat & Ajit S. Dham
Proceedings of the IEEE Conference on Decision and Control, 2003

Abstract: In this paper, we apply nonlinear controllability theory to the attitude dynamics of a magnetically actuated spacecraft in a Keplerian orbit in the geomagnetic field. The variation of the geomagnetic field along the orbit makes the dynamical equations time varying in nature. Hence we first present sufficient conditions for accessibility, strong accessibility and controllability of a general time-varying system. We apply these sufficient conditions to show that the attitude dynamics of a spacecraft actuated by three magnetic actuators and subjected to a general  time-varying magnetic field are strongly accessible if the magnetic field and its first two time derivatives are linearly independent at every instant. In addition, if the magnetic field is periodic in time, then the attitude dynamics of the spacecraft are controllable. These results are used to show that the attitude dynamics of a spacecraft actuated by three magnetic actuators in a closed Keplerian orbit in a nonrotating dipole approximation of the geomagnetic field are strongly accessible and controllable if the orbital plane contains neither the geomagnetic equator nor the geomagnetic poles.

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Arc-length-based Lyapunov tests for convergence and stability in systems
having a continuum of equilibria

S. P. Bhat and D. S. Bernstein
Proceedings of the American Control Conference, 2003

Abstract: This paper focuses on the stability analysis of systems having a continuum of equilibria. Two notions that are of particular relevance to such systems are convergence and semistability. Convergence is the property whereby every solution converges to a limit point that may depend on the initial condition. Semistability is the additional requirement that all solutions converge to limit points that are Lyapunov stable. In this paper, we relate convergence and stability to arc length of the orbits. More specifically, we show that a system is convergent if all of its orbits have finite arc length, while an equilibrium is Lyapunov stable if the arc length (considered as a function of the initial condition) is continuous at the equilibrium, and semistable if the arc length is continuous in a neighborhood of the equilibrium. Next we derive arc-length-based Lyapunov results for convergence and stability. These results do not require the Lyapunov function to be positive definite. Instead, these results involve an inequality relating the righthandside of the differential equation and the Lyapunov function derivative. This inequality makes it possible to deduce properties of the arc length function and thus leads to sufficient conditions for convergence and stability. Finally, we give additional assumptions under which the converses of all the main results hold.

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Linear Output-Reversible Systems

D. S. Bernstein and S. P. Bhat
Proceedings of the American Control Conference, 2003

Abstract: The arrow of time remains one of physics' most puzzling questions. Some physical processes (such as a planet orbiting the sun) have no preferred direction in time, while others (such as a melting snowman) do. In this paper we define the notion of output reversibility, which concerns the existence of an initial condition with the property that the resulting output is the time-reversed image of a given output on a specified finite time interval. Our main result is a spectral symmetry condition that provides a complete characterization of single-input, single-output, output-reversible systems. As special cases, the class of output-reversible systems includes rigid body and Hamiltonian systems.

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Nontangency-Based Lyapunov Tests for Stability and Convergence

S. P. Bhat and D. S. Bernstein
Proceedings of the American Control Conference, 2001

Abstract: We give new results for Lyapunov and asymptotic stability of nonlinear systems. In addition, we also give results for convergence and semistability. Convergence is the property whereby every trajectory of a system converges to a limit point that may depend upon the initial condition. Semistability is the additional requirement that the limit points of the trajectories of a convergent system also be Lyapunov stable. Our results do not make assumptions of sign definiteness on the Lyapunov function. Instead, our results use a novel nontangency condition between the system dynamics and the level sets of the Lyapunov function or its derivative. Using this nontangency condition, we extend previously known Lyapunov stability and asymptotic stability results involving semidefinite Lyapunov functions.

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A Nontangency-Based Sufficient Condition for Boundedness of Orbits

S. P. Bhat
Proceedings of the IEEE Conference on Decision and Control, 2003

Abstract: We give a Lyapunov test for the boundedness of orbits of a dynamical system. Unlike previous sufficient conditions for boundedness, our test does not require the Lyapunov function to be proper or weakly proper. Instead, our result uses a nontangency condition between the dynamics and the zero level set of the Lyapunov function.

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Output-Feedback Semiglobal Stabilization of Stall Dynamics for Eliminating
Hysteresis and Surge in Axial-Flow Compressors

N. Chaturvedi and S. P. Bhat
Proceedings of the American Control Conference, 2003

Abstract: This paper deals with the use of feedback control to eliminate the problems of hysteresis and surge associated with axial-flow compressors. We present a dynamic feed-back controller that semiglobally stabilizes every rotating stall equilibrium, and a range of axisymmetric equilibria of the Moore-Greitzer model for axial-flow compressors. The dynamic controller combines a two-state-feedback back-stepping controller from the literature with a nonlinear high-gain observer that estimates the mass flow through the compressor from measurements of the pressure rise across it. Given an equilibrium and a compact inner bound on the domain of attraction, we use Lyapunov techniques to compute an explicit lower bound on the observer gain such that the specified equilibrium is asymptotically stable for the closed-loop system with a domain of attraction that contains the specified inner bound. Simulation results are used to demonstrate that the closed-loop compressor does not exhibit hysteresis and surge oscillations even in response to large and sudden changes in the throttle setting.

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Energy equipartition and the emergence of damping in lossless systems

D. S. Bernstein and S. P. Bhat
 Proceedings of the IEEE Conference on Decision and Control, 2002

Abstract: Deterministic linear systems techniques were used to analyze the vibrational energy of systems of undamped coupled oscillators with identical coupling. First, a single undamped oscillator was considered, and it was shown that the time-averaged potential energy and the time-averaged kinetic energy converge to the same value. Next a collection of n identical undamped oscillators with lossness coupling was considered. Implications of the results for the emergence of damping in lossless systems were outlined.

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Hysteretic Systems and Step-Convergent Semistability

S. L. Lacy, D. S. Bernstein and S. P. Bhat
Proceedings of the American Control Conference, 2000

Abstract: Hysteresis is usually characterized as a memory-dependent relationship between inputs and outputs. While various operator models have been proposed, it is often convenient for engineering applications to approximate hysteretic behavior by means of finite-dimensional differential models. In the present paper we show that step-convergent semistable systems (that is, semistable systems with convergent step response) give rise to multiple-valued maps under quasi-static operation. By providing a connection between semistability and hysteresis, our goal is to provide a class of differential models for representing hysteretic behavior.


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Topological Obstruction to Continuous Global Stabilization of
Rotational Motion and the Unwinding Phenomenon

S. P. Bhat and D. S. Bernstein
Proceedings of the American Control Conference, 1998.
Abstract:We show that a continuous dynamical system on a state space that has the structure of a vector bundle on a compact manifold possesses no globally asymptotically stable equilibrium. This result is directly applicable to mechanical systems having rotational degrees of freedom. In particular, the result applies to the attitude motion of a rigid body. In light of this result, we explain how attitude stabilizing controllers appearing in the literature lead to unwinding instead of global asymptotic stability.


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Finite-Time Stability of Homogeneous Systems

S. P. Bhat and D. S. Bernstein
 Proceedings of the American Control Conference, 1997

Abstract: This paper examines finite-time stability of homogeneous systems. The main result is that a homogeneous system is finite-time stable if and only if it is asymptotically stable and has a negative degree of homogeneity.


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Example of Indeterminacy in Classical Dynamics

S. P. Bhat and D. S. Bernstein
Proceedings of the American Control Conference, 1997

Abstract: The case of a particle moving along a nonsmooth constraint under the action of uniform gravity is presented as an example of indeterminacy in a classical situation. The indeterminacy arises from certain initial conditions having non-unique solutions and is due to a failure of the Lipschitz condition at the corresponding points in the phase space of the equation of motion.

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Continuous, Finite-Time Stabilization of the Translational and
Rotational Double Integrators

S. P. Bhat and D. S. Bernstein
IEEE Conference on Control Applications,1996

Abstract: A class of bounded, continuous, time-invariant, finite-time stabilizing feedback laws is given for the double integrator. These controllers are modified to obtain finite-time stabilizing feedbacks for the rotational double integrator that do not exhibit 'unwinding'.

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Second-Order Systems with Singular Mass Matrix and an
Extension of Guyan Reduction

S. P. Bhat and D. S. Bernstein
 Proc. 1995 Design Engineering Technical Conferences,1995


Abstract: The set of consistent initial conditions for a second-order system with singular mass matrix is obtained. In general, such a system can be decomposed (i.e., partitioned) into three coupled subsystems of which the first is algebraic, the second is a regular system of first-order differential equations, and the third is a regular system of second-order differential equations. Under specialized conditions, these subsystems are decoupled. This result provides an extension of Guyan reduction to include viscous damping.
Adaptive Virtual Autobalancing for a Magnetic Rotor with
Unknown Mass Imbalance, II. Dynamic Balancing

K.-Y. Lum, S. P. Bhat, V. T. Coppola, and D. S. Bernstein
  Proc. 1995 Design Engineering Technical Conferences,  1995

Abstract: In Lum et al. (1995), an adaptive control algorithm for the stabilization of a rigid, statically unbalanced rotor moving in the plane was proposed. The control strategy consisted in emulating a mechanical autobalancer using magnetic actuation so as to directly cancel the effects of static mass imbalance. In this present paper, this strategy is extended to the case of a rigid, dynamically unbalanced rotor in six degree-of-freedom motion. The state equations of the controller are based on the equations of motion of a multiple-plane autobalancer, and the control forces partially emulates the interaction between rotor and autobalancer. It is shown in simulation that the adaptive virtual autobalancing control can achieve stabilization of rotor motion as well as adaptation to changes in imbalance.

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Adaptive Virtual Autobalancing for a Magnetic Rotor with
Unknown Mass Imbalance, I. Static Balancing

K.-Y. Lum, S. P. Bhat, V. T. Coppola, and D. S. Bernstein
  Proc. 1995 Design Engineering Technical Conferences, 1995

Abstract: An adaptive control scheme is proposed for stabilizing a planar rotor mounted on a magnetic bearing. The control strategy involves the concept of virtual autobalancing, where the control algorithm emulates the dynamics of a mechanical autobalancer by applying forces that are equivalent to the action of the autobalancer on the rotor. Equations of motion for a planar, torque-free, elastically suspended rotor equipped with an autobalance are derived. Based on these equations, an adaptive controller for the magnetic rotor is formulated. The results are demonstrated in simulation.

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Lyapunov Analysis of Finite-Time Differential Equations

S. P. Bhat and D. S. Bernstein
 Proceedings of the American Control Conference,1995

Abstract: Necessary and sufficient conditions in terms of Lyapunov functions are derived for the finite-time stability of equilibria of systems of differential equations with continuous but non-Lipschitzian right hand sides.

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Lyapunov Stability, Semistability and Asymptotic Stability of
Matrix Second-Order Systems

D. S. Bernstein and S. P. Bhat
  Proceedings of the American Control Conference,1994

Abstract: A self-contained, unified and extended treatment of the stability of matrix second-order systems is presented. The results obtained encompass numerous results from prior literature in addition to new ones. Specifically, in addition to obtaining necessary and sufficient conditions for Lyapunov and asymtotic stability, the case of semistability is considered, a concept first introduced in a previous study on single perturbation on nonlinear systems.Semistability is of particular interest in the analysis of vibrating systems in that it represents the case of 'damped rigid body modes', i.e., systems that eventually come to rest, although not necessarily at a specified equilibrium point. This paper presents the first treatment of semistability for matrix second-order systems.

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Controllability of Spacecraft Attitude using Control Moment Gyroscopes

S. P. Bhat and P. K. Tiwari
  Proceedings of the American Control Conference, June 2006
Abstract:This paper describes an application of nonlinear controllability theory to the problem of spacecraft attitude control using control moment gyroscopes (CMGs). Nonlinear controllability theory is used to show that a spacecraft carrying one or more CMGs is controllable on every angular momentum level set in spite of the presence of singular CMG configurations, that is, given any two states having the same angular momentum, any one of them can be reached from the other using suitably chosen motions of the CMG gimbals. This result is used to obtain sufficient conditions on the momentum volume of the CMG array that guarantee the existence of gimbal motions which steer the spacecraft to a desired spin state or rest attitude.


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Optimal Planar Turns Under Acceleration Constraints

V. Aneesh and S. P. Bhat
  IEEE Conference on Decision and Control, December 2006.
Abstract: This paper considers the problem of finding optimal trajectories for a particle moving in a two-dimensional plane from a given initial position
and velocity to a specified terminal heading under a magnitude constraint on the acceleration. The cost functional to be minimized is
the integral over time of a general non-negative power of the particle's speed. Special cases of such a cost functional include travel time and
path length.  Unlike previous work on related problems, variations in the magnitude of the velocity vector are allowed. Pontryagin's maximum
principle is used to show that the optimal trajectories possess a special property whereby the vector that divides the angle between the velocity
and acceleration vectors in a specific ratio, which depends on the cost functional, is a constant. This property is used to obtain the optimal acceleration vector and the parametric equations of the corresponding optimal paths. Solutions of the time-optimal and the length-optimal problems are obtained as special cases.


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Planar Length-Optimal Paths Under Acceleration Constraints

V. Aneesh and S. P. Bhat
  Proceedings of the American Control Conference, June 2006

Abstract: This paper considers the problem of finding minimum length trajectories for a vehicle moving in a two-dimensional plane from a given initial position and velocity to a specified terminal heading under a magnitude constraint on the acceleration. Unlike previous work on related problems, variations in the magnitude of the velocity vector are allowed.  The Pontryagin's maximum principle is used to show that the length-optimal paths possess a special property whereby the angle bisector between the acceleration and velocity vectors is a constant. This property is used to obtain the optimal acceleration vector and to show that the length-optimal paths are straight line segments or arcs of alysoids. A numerical example is presented and the solutions of the length-optimal problem are compared with those of the corresponding time- optimal problem.

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 Time-Optimal Attitude Reorientation at Constant Angular Velocity Magnitude with Bounded Angular Acceleration

  M. Modgalya and S. P. Bhat
  IEEE Conference on Decision and Control, December 2006.

Abstract: This paper considers the problem of steering the orientation of an inertially symmetric rigid body of unit moment of inertia from an initial attitude and nonzero angular velocity to a specified terminal attitude in minimum time under an upper limit on the magnitude of angular acceleration with the magnitude of the angular velocity constrained to remain constant. Optimal control theory is used to show that singular optimal arcs are uniform eigenaxis rotations in which the body rotates at a uniform
rate about a body-fixed axis, while nonsingular arcs are coning motions in which the body angular velocity vector rotates at a uniform rate about a body-fixed axis. Symmetries of the problem are exploited to further show that  every optimal trajectory consists of at most one coning motion followed either by  one uniform eigenaxis rotation or several coning motions of equal duration.

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Semi-global Practical Stability of Periodic Time-Varying Systems via Averaging: A Lyapunov Approach 

  S. P. Bhat and R. V. Cowlagi
  IEEE Conference on Decision and Control, December 2006.

Abstract: This paper considers semi-global practical stability of a general time-varying, parameter dependent nonlinear system. A Lyapunov result for uniform semi-global practical stability of such a system is given. This sufficient condition is applied to a periodically time-varying system in the standard averaging form to
obtain a sufficient condition on the averaged system for the time-varying system to be uniformly semi-globally practically stable. The sufficient condition requires the existence of a Lyapunov function that guarantees the semi-global practical stability of the averaged system, and is thus weaker than previous averaging based stability
results which require the equilibrium of the averaged system to be exponentially or asymptotically stable. The proof is based on Lyapunov techniques, and does not depend on classical averaging results.


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