A Topological Obstruction to Continuous Global Stabilization of Rotational Motion and the Unwinding Phenomenon
S. P. Bhat and D. S. Bernstein
Systems and Control Letters, 2000
 
 
 
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 obtained using local coordinates lead to unwinding instead of global asymptotic stability.
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 Finite-Time Stability of Continuous, Autonomous Systems
 S. P. Bhat and D. S. Bernstein
SIAM Journal of Control and Optimization, 2003.
 
Abstract- Finite-time stability is defined for equilibria of continuous but non-Lipschitzian autonomous systems. Continuity, Lipschitz continuity and Holder continuity of the settling-time function are studied and illustrated with several examples. Lyapunov and converse Lyapunov results involving scalar differential inequalities are given for finite-time stability. It is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov results can only assure the existence of continuous Lyapunov functions. Finally, the sensitivity of finite-time-stable systems to perturbations is investigated.
 
 
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Abstract- A class of bounded continuous time-invariant finite-time stabilizing feedback laws is given for the double integrator. Lyapunov theory is used to prove finite-time convergence. For the rotational double integrator, these controllers are modified to obtain finite-time-stabilizing feedbacks that avoid "unwinding."
 
 
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Example of Indeterminacy in Classical Dynamics
Sanjay P. Bhat and D.S. Bernstein
International Journal of Theoretical Physics, 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 nonunique solutions and is due to the failure of the Lipschitz condition at the corresponding points in the phase space of the equation of motion.
 
 
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Lyapunov Stability, Semistability, And Asymptotic Stability Of Matrix Second-Order Systems
 
D. S. Bernstein and S. P. Bhat
ASME Journal of Vibrations and Acoustics, 1995
 
Abstract-Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.
 
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Controllability of Nonlinear Time-Varying Systems:
Applications to Spacecraft Attitude Control using Magnetic Actuation

S. P. Bhat
IEEE Transactions on Automatic Control, 2005

Abstract-Nonlinear controllability theory is applied to the time-varying attitude dynamics of a magnetically actuated spacecraft in a Keplerian orbit in the geomagnetic field.  First, sufficient conditions for accessibility, strong accessibility and controllability of a general time-varying system are presented. These conditions involve application of Lie-algebraic rank conditions to the autonomous extended system obtained by augmenting the state of the original time-varying system by the time variable, and require the rank conditions to be checked only on the complement of a finite union of level sets of a finite number of smooth functions. At each point of each level set, it is sufficient to verify escape conditions involving Lie derivatives of the functions defining the level sets along linear combinations over smooth functions of vector  fields in the accessibility algebra. These sufficient conditions are used 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 time derivative 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 does not coincide with the geomagnetic equatorial plane.



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

N. A. Chaturvedi and S. P. Bhat
IEEE Transactions on Control Systems Technology, accepted for publication


Abstract-This paper deals with the use of feedback control to prevent hysteresis and surge in axial-flow compressors. We present a dynamic output-feedback 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-variable-feedback backstepping  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. We use a numerical example  to illustrate how the inner bound on the domain of attraction can be specified so that the closed-loop compressor does not exhibit hysteresis and surge oscillations even in response to changes in the throttle setting that are dictated by large and sudden changes in the desired operating point. Simulation results are used to demonstrate the absence of hysteresis and surge in the closed-loop compressor dynamics.


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Nontangency-Based Lyapunov Tests for Convergence and
 Stability in Systems Having a Continuum of Equilibria

S. P. Bhat and D. S. Bernstein
 SIAM Journal of Control and Optimization, 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 solutionsconverge to limit points that are Lyapunov stable. We give new Lyapunov-function-based results for convergence and semistability of nonlinear systems. These results do not make assumptions of sign definiteness on the Lyapunov function. Instead, our results use a novel condition based on nontangency between the vector field and invariant or negatively invariant subsets of the level or sublevel sets of the Lyapunov function or its derivative and represent extensions of previously known stability results involving semidefinite Lyapunov functions. To illustrate our results we deduce convergence and semistability of the kinetics of the Michaelis--Menten chemical reaction and the closed-loop dynamics of a scalar system under a universal adaptive stabilizing feedback controller.


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Geometric Homogeneity with Applications to Finite-Time Stability

S. P. Bhat and D. S. Bernstein
Mathematics of Control, Signals and Systems, 2005

Abstract: This paper studies properties of homogeneous systems in a geometric, coordinate-free setting. A key contribution of this paper is a result relating regularity properties of a homogeneous function to its degree of homogeneity and the local behavior of the dilation near the origin. This result makes it possible to extend previous results on homogeneous systems to the geometric framework. As an application of our results, we consider finite-time stability of homogeneous systems. The main result that links homogeneity and finite-time stability is that a homogeneous system is finite-time stable if and only if it is asymptotically stable and has a negative degree of homogeneity. We also show that the assumption of homogeneity leads to stronger properties for finite-time stable systems.

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

Sanjay P. Bhat, Dennis S. Bernstein
SIAM Journal of Matrix Analysis and Applications, 1996

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.

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An Invariance Principle for Nonlinear Hybrid and Impulsive Dynamical Systems

V.-S. Chellaboina, S. P. Bhat and W. M. Haddad
 Nonlinear Analysis: Theory, Methods and Applications, 2003
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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