William H. Moase

Newton-Like Extremum-Seeking for the Control of Thermoacoustic Instability

In practice, the convergence rate and stability of perturbation based extremum-seeking schemes can be very sensitive to the curvature of the plant map. An example of this can be seen in the use of extremum-seeking to reduce the amplitude of thermoacoustic oscillations in premixed, gas-turbine combustors. This sensitivity to the plant map curvature arises from the use of a gradient descent adaptation algorithm. Such extremum-seeking schemes may need to be conservatively tuned in order to maintain stability over a wide range of operating conditions, resulting in slower optimization than could be achieved for a fixed operating condition. This can severely reduce the effectiveness of perturbation based extremum-seeking schemes in some applications. In this paper, a sinusoidally perturbed extremum-seeking scheme using a Newton-like step is developed. Non-local stability results for the scheme are formulated using a Lyapunov analysis. A local analysis of the scheme is given to investigate the influence of plant dynamics and to show that the local rate of convergence is independent of the plant map curvature. The benefit of this plant map curvature independence is then experimentally demonstrated in minimizing the thermoacoustic oscillations in a model premixed combustor.

A unifying approach to extremum seeking: Adaptive schemes based on estimation of derivatives

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.

The forced response of choked nozzles and supersonic diffusers

The response of choked nozzles and supersonic diffusers to one-dimensional flow perturbations is investigated. Following previous arguments in the literature, small flow perturbations in a duct of spatially linear steady velocity distribution are determined by solution of a hyper-geometric differential equation. A set of boundary conditions is then developed that extends the existing work to a nozzle of arbitrary geometry. This analysis accommodates the motion of a plane shock wave and makes no assumption about the nozzle compactness. Numerical simulations of the unsteady, quasi-one-dimensional Euler equations are performed to validate this analysis and also to indicate the conditions under which the perturbations remain approximately linear. The nonlinear response of compact choked nozzles and supersonic diffusers is also investigated. Simple analyses are performed to determine the reflected and transmitted waveforms, as well as conditions for unchoke, ‘over-choke’ and unstart. This analysis is also supported with results from numerical simulations of the Euler equations.

Newton-like extremum-seeking part I: Theory

In practice, the convergence rate and stability of perturbation based extremum-seeking (ES) schemes can be very sensitive to the curvature of the plant map. This sensitivity arises from the use of a gradient descent adaptation algorithm. Such ES schemes may need to be conservatively tuned in order to maintain stability over a wide range of operating conditions, resulting in slower optimisation than could be achieved for a fixed operating condition. This can severely reduce the effectiveness of perturbation based ES schemes in some applications. It is proposed that by using a Newton-like step instead of a more typical gradient descent adaptation law, then the behaviour of the ES scheme near an extremum will be independent of the plant map curvature. In this paper, such a Newton-like ES scheme is developed and its stability and convergence properties are explored.

Semi-Global Stability Analysis of Observer-Based Extremum-Seeking for Hammerstein Plants

This paper considers semi-global stability of an observer-based extremum-seeking (ES) scheme acting on a Hammerstein plant. Unlike previous semi-global analyses, no restriction is placed on the speed of the ES dynamics relative to the plant dynamics. For any set of frequencies of a probing sinusoidal “dither,” it is shown how the plant output can be made to converge to an arbitrarily small neighborhood of its minimum from an arbitrarily large set of initial conditions. Furthermore, in the absence of noise, it is shown how increasing the dither frequency can allow arbitrarily fast convergence of the plant input to a small neighborhood of its optimum. Practical application of the results requires no a priori knowledge of the plant nonlinearity; however, in order to design the ES scheme to operate over a wide range of dither frequencies, some knowledge of the plant dynamics is required. Simulation examples are used to demonstrate these results and to highlight practical considerations when using a high-frequency dither.

A unifying framework for analysis and design of extremum seeking controllers

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.

Dither Re-Use in Nash Equilibrium Seeking

Nash equilibrium seeking (NES) scheme consists of a number of decentralized extremum-seeking (ES) “agents”, each controlling an input such that an associated (selfish) cost is regulated to its steady-state Nash equilibrium. A non-local stability result for the NES scheme is provided which allows two agents to use the same dither signal if the effect of each agent on the others steady-state cost function is sufficiently weak. Its application to plants with quadratic cost functions is presented as an example. It is then demonstrated in simulation that, by reducing the number of distinct dither signals, the proposed scheme still has acceptable convergence properties, while design effort is reduced.

Fast extremum seeking on Hammerstein plants

Partial plant knowledge may be used to develop model-based extremum seekers; however, existing results rely on a type of time-scale separation which leads to slow optimization relative to the plant dynamics. In this work, a fast model-based extremum seeking scheme is proposed for a Hammerstein plant, and semi-global stability results are provided. The structure of a Hammerstein plant is used to advantage in designing filters that enable the extremum seeker to act on a faster time-scale than the plant dynamics. This leads to fast convergence while maintaining semi-global stability.

Fast model-based extremum seeking on Hammerstein plants

Partial plant knowledge may be used to develop model-based extremum seekers, however existing results rely on a type of time-scale separation which leads to slow optimization relative to the plant dynamics. In this work, a fast model-based extremum seeking scheme is proposed for a Hammerstein plant, and semi-global stability results are provided. Simulation results are used to validate the theoretical results.

Continuity and monotonicity of the MPC value function with respect to sampling time and prediction horizon

The digital implementation of model predictive control (MPC) is fundamentally governed by two design parameters; sampling time and prediction horizon. Knowledge of the properties of the value function with respect to the parameters can be used for developing optimization tools to find optimal system designs. In particular, these properties are continuity and monotonicity. This paper presents analytical results to reveal the smoothness properties of the MPC value function in open- and closed-loop for constrained linear systems. Continuity of the value function and its differentiability for a given number of prediction steps are proven mathematically and confirmed with numerical results. Non-monotonicity is shown from the ensuing numerical investigation. It is shown that increasing sampling rate and/or prediction horizon does not always lead to an improved closed-loop performance, particularly at faster sampling rates.