Mechanical Engineering - Research Publications
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ItemAveraging for nonlinear systems on Riemannian manifolds(IEEE, 2013)This paper provides a derivation of the averaging methods for nonlinear time-varying dynamical systems defined on Riemannian manifolds. We extend the results on ℝ n to Riemannian manifolds by employing the language of differential geometry.
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ItemNew Stability Criteria for Switched Time-Varying Systems: Output-Persistently Exciting Conditions(IEEE, 2011-01-01)This paper proposes three tools to facilitate the verification of the output-persistently exciting (OPE) condition and simultaneously, provides new asymptotic stability criteria for uniformly globally stable switched systems. By introducing some related reference systems, the OPE condition of the original system can be reduced or simplified. Both the ideas of classic LaSalle invariance principle and nested Matrosov theorem are used to generate such reference systems. The effectiveness and flexibility of the proposed methods are demonstrated by two applications. From these applications, it can be seen that the flexibility of the proposed method produces a novel set of tools for checking uniform asymptotic stability of switched time-varying systems.
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ItemExtremum seeking control for nonlinear systems on compact Riemannian manifolds(IEEE Press, 2014)This paper formulates the extremum seeking control problem for nonlinear dynamical systems which evolve on Riemannian manifolds and presents stability results for a class of numerical algorithms defined in this context. The results are obtained based upon an extension of extremum seeking algorithms in Euclidean spaces and a generalization of Lyapunov stability theory for dynamical systems defined on Rimannian manifolds. We employ local properties of Lyapunov functions to extend the singular perturbation analysis on Riemannian manifolds. Consequently, the results of the singular perturbation on manifolds are used to obtain the convergence of extremum seeking algorithms for dynamical systems on Riemannian manifolds.
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ItemExtremum Seeking From 1922 To 2010(IEEE, 2010-01-01)Extremum seeking is a form of adaptive control where the steady-state input-output characteristic is optimized, without requiring any explicit knowledge about this input-output characteristic other than that it exists and that it has an extremum. Because extremum seeking is model free, it has proven to be both robust and effective in many different application domains. Equally being model free, there are clear limitations to what can be achieved. Perhaps paradoxically, although being model free, extremum seeking is a gradient based optimization technique. Extremum seeking relies on an appropriate exploration of the process to be optimized to provide the user with an approximate gradient, and hence the means to locate an extremum. These observations are elucidated in the paper. Using averaging and time-scale separation ideas more generally, the main behavioral characteristics of the simplest (model free) extremum seeking algorithm are established.
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ItemCloseness of solutions and averaging for nonlinear systems on Riemannian manifolds(IEEE, 2013)An averaging result for periodic dynamical systems evolving on Euclidean spaces is extended to those evolving on (differentiable) Riemannian manifolds. Using standard tools from differential geometry, a perturbation result for time-varying dynamical systems is developed that measures closeness of trajectories via a suitable metric on a finite time horizon. This perturbation result is then extended to bound excursions in the trajectories of periodic dynamical systems from those of their respective averages, on an infinite time horizon, yielding the specified averaging result. Some simple examples further illustrating this result are also presented.
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ItemAn improved Matrosov theorem for hybrid time-varying systems: A behavior approach(IEEE, 2011-08-29)This paper concerns the stability of hybrid time-varying systems. A behavior approach is used to transfer the functional space on hybrid time domains into a functional space on continuous-time domains. Then a new stability criterion is derived for the transferred continuous-time functional space to derive a nested Matrosov theorem for hybrid time-varying systems. The proposed Matrosov theorem does not require that the equilibrium set is compact, which indicates an extension of current results in literature. The obtained results have also a potential to be used in stability analysis for other types of dynamic systems such as time-delay systems.
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ItemA UNIFYING FRAMEWORK FOR ANALYSIS AND DESIGN OF EXTREMUM SEEKING CONTROLLERS(IEEE, 2012-01-01)We summarize a unifying design approach to continuous-time extremum seeking that was recently reported by the authors. This approach is based on a feedback control paradigm that was to the best of our knowledge explicitly summarized for the first time in this form in our recent work. This paradigm covers some existing extremum seeking schemes, provides a direct link to off-line optimization and can be used as a unifying framework for design of novel extremum seeking schemes. Moreover, we show that other extremum seeking problem formulations can be interpreted using this unifying viewpoint. We believe that this unifying view will be invaluable to systematically design and analyze extremum seeking controllers in various settings.
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ItemA Unifying Approach to Extremum Seeking: Adaptive Schemes Based on Estimation of Derivatives(IEEE, 2010-01-01)A unifying, prescriptive framework is presented for the design of a family of adaptive extremum seeking controllers. It is shown how extremum seeking can be achieved by combining an arbitrary continuous optimization method (such as gradient descent or continuous Newton) with an estimator for the derivatives of the unknown steady-state reference-to-output map. A tuning strategy is presented for the controller parameters that ensures non-local convergence of all trajectories to the vicinity of the extremum. It is shown that this tuning strategy leads to multiple time scales in the closed-loop dynamics, and that the slowest time scale dynamics approximate the chosen continuous optimization method. Results are given for both static and dynamic plants. For simplicity, only single-input-single-output (SISO) plants are considered.
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ItemTrajectory-based proofs for sampled-data extremum seeking control(IEEE, 2013)Extremum seeking of nonlinear systems based on a sampled-data control law is revisited. It is established that under some generic assumptions, semi-global practical asymptotically stable convergence to an extremum can be achieved. To this end, trajectory-based arguments are employed, by contrast with Lyapunov-function-type approaches in the existing literature. The proof is simpler and more straightforward; it is based on assumptions that are in general easier to verify. The proposed extremum seeking framework may encompass more general optimisation algorithms, such as those which do not admit a state-update realisation and/or Lyapunov functions. Multi-unit extremum seeking is also investigated within the context of accelerating the speed of convergence.
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ItemOn a Shubert Algorithm-Based Global Extremum Seeking Scheme(IEEE, 2012)This paper adapts the so-called Shubert algorithm for Extremum Seeking Control (ESC) to seek the global extremum (in presence of local extrema) of general dynamic plants. Different from derivative based methods that are widely used in ESC, the Shubert algorithm is a good representative of sampling optimization methods. With knowledge of the Lipschitz constant of an unknown static mapping, this deterministic algorithm seeks the global extremum. By introducing “waiting time” the proposed Shubert algorithm-based global extremum seeking guarantees the semi-global practical convergence (in the initial states) to the global extremum if compact sets of inputs are considered. Several numerical examples demonstrate how proposed method may be successfully deployed.