Electrical and Electronic 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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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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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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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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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.
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ItemOn sampled-data extremum seeking control via stochastic approximation methods(IEEE, 2013-01-01)This note establishes a link between stochastic approximation and extremum seeking of dynamical nonlinear systems. In particular, it is shown that by applying classes of stochastic approximation methods to dynamical systems via periodic sampled-data control, convergence analysis can be performed using standard tools in stochastic approximation. A tuning parameter within this framework is the period of the synchronised sampler and hold device, which is also the waiting time during which the system dynamics settle to within a controllable neighbourhood of the steady-state input-output behaviour. Semiglobal convergence with probability one is demonstrated for three basic classes of stochastic approximation methods: finite-difference, random directions, and simultaneous perturbation. The tradeoff between the speed of convergence and accuracy is also discussed within the context of asymptotic normality of the outputs of these optimisation algorithms.
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ItemOpen Problems in Reset Control(IEEE, 2013-01-01)It is well-known that there are fundamental performance limitations in the design of linear feedback control systems for single-input-single-output (SISO) linear-time-invariant (LTI) plants. These performance limitations sometimes include overshoot and rise time. This paper shows that for some examples of SISO LTI systems, it is possible to find suitable reset controllers that can overcome such performance limitations, though there are still some robust and implementable issues that need to be solved. This naturally leads to the formulation of several open research problems that we specify.
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ItemMulti-agent gradient climbing via extremum seeking control(IFAC - International Federation of Automatic Control, 2014)A unified framework based on discrete-time gradient-based extremum seeking control is proposed to localise an extremum of an unknown scalar field distribution using a group of equipped with sensors. The controller utilises estimates of gradients of the field from local dithering sensor measurements collected by the mobile agents. It is assumed that distributed coordination which ensures uniform asymptotic stability with respect to a prescribed formation of the agents is employed. The framework is useful in that a broad range of nonlinear programming algorithms can be combined with a wide class of cooperative control laws to perform extreme source seeking. Semi-global practical asymptotically stable convergence to local extrema is established in the presence of bounded field sampling noise.
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ItemImproving L₂ Gain Performance of Linear Systems by Reset Control(IFAC - International Federation of Automatic Control, 2014)In this paper, new Lyapunov-based reset rules are constructed to improve C2 gain performance of linear-time-invariant (LTI) systems. By using the hybrid system framework, sufficient conditions for exponential and finite gain C2 stability are presented. It is shown that the C2 gain of the closed loop system with resets can be improved compared with the base system. Numerical example supports our results.