Chris Manzie

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.

Extremum Seeking With Stochastic Perturbations

Extremum seeking (ES) using deterministic periodic perturbations has been an effective method for non-model based real time optimization when only limited plant knowledge is available. However, periodicity can naturally lead to predictability which is undesirable in some tracking applications and unrepresentative of biological optimization processes such as bacterial chemotaxis. With this in mind, it is useful to investigate employing stochastic perturbations in the context of a typical ES architecture, and to compare the approach with existing stochastic optimization techniques. In this work, we show that convergence towards the extremum of a static map can be guaranteed with a stochastic ES algorithm, and quantify the behavior of a system with Gaussian-distributed perturbations at the extremum in terms of the ES constants and map parameters. We then examine the closed loop system when actuator dynamics are included, as the separation of time scales between the perturbation signal and plant dynamics recommended in periodic ES schemes cannot be guaranteed with stochastic perturbations. Consequently, we investigate how actuator dynamics influence the allowable range of ES parameters and necessitate changes in the closed loop structure. Finally simulation results are presented to demonstrate convergence and to validate predicted behavior about the extremum. For the sake of analogy with the classical methods of stochastic approximation, stochastic ES in this technical note is pursued in discrete time.

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.

Effects of moving the center's in an RBF network

In radial basis function (RBF) networks, placement of centers is said to have a significant effect on the performance of the network. Supervised learning of center locations in some applications show that they are superior to the networks whose centers are located using unsupervised methods. But such networks can take the same training time as that of sigmoid networks. The increased time needed for supervised learning offsets the training time of regular RBF networks. One way to overcome this may be to train the network with a set of centers selected by unsupervised methods and then to fine tune the locations of centers. This can be done by first evaluating whether moving the centers would decrease the error and then, depending on the required level of accuracy, changing the center locations. This paper provides new results on bounds for the gradient and Hessian of the error considered first as a function of the independent set of parameters, namely the centers, widths, and weights; and then as a function of centers and widths where the linear weights are now functions of the basis function parameters for networks of fixed size. Moreover, bounds for the Hessian are also provided along a line beginning at the initial set of parameters. Using these bounds, it is possible to estimate how much one can reduce the error by changing the centers. Further to that, a step size can be specified to achieve a guaranteed, amount of reduction in error.

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.

Model Predictive Control of a Fuel Injection System with a Radial Basis Function Network Observer

This paper proposes a new Model Predictive Control scheme incorporating a Radial Basis Function Network Observer for the fuel injection problem. Two new contributions are presented here. First a Radial Basis Function Network is used as an observer for the air system. This allows for gradual adaptation of the observer, ensuring the control scheme is capable of maintaining good performance under changing engine conditions brought about by engine wear, variations between individual engines, and other similar factors. The other major contribution is the use of model predictive control algorithms to compensate for the fuel pooling effect on the intake manifold walls. Two model predictive control algorithms are presented which enforce input, and input and state constraints. In this way stability under the constraints is guaranteed. A comparison between the two constrained MPC algorithms is qualitatively presented, and conclusions are drawn about the necessity of constraints for the fuel injection problem. Simulation results are presented that demonstrate the effectiveness of the control scheme, and the proposed control approach is validated on a four-cylinder spark ignition engine.

Unified frameworks for sampled-data extremum seeking control: Global optimisation and multi-unit systems

Two frameworks are proposed for extremum seeking of general nonlinear plants based on a sampled-data control law, within which a broad class of nonlinear programming methods is accommodated. It is established that under some generic assumptions, semi-global practical convergence to a global extremum can be achieved. In the case where the extremum seeking algorithm satisfies a stronger asymptotic stability property, the converging sequence is also shown to be stable using a trajectory-based proof, as opposed to a Lyapunov-function-type approach. The former is more straightforward and insightful. This allows for more general optimisation algorithms than considered in existing literature, such as those which do not admit a state-update realisation and/or Lyapunov functions. Lying at the heart of the analysis throughout is robustness of the optimisation algorithms to additive perturbations of the objective function. Multi-unit extremum seeking is also investigated with the objective of accelerating the speed of convergence.

Model predictive control of velocity and torque split in a parallel hybrid vehicle

Fuel economy of parallel hybrid electric vehicles is affected by both the torque split ratio and the vehicle velocity. To optimally schedule both variables, information about the surrounding traffic is necessary, but may be made available through telemetry. Consequently, in this paper, a nonlinear model predictive control algorithm is proposed for the vehicle control system to maximise fuel economy while satisfying constraints on battery state of charge, relative position and vehicle performance. Different scenarios are considered including allowing and disallowing overtaking; various hard and soft constraints; and computational aspects of the solution. The optimal control signal vector was found to be characterised by smooth changes in velocity and increases in the motor to engine power ratio as the vehicle accelerates. It was found that using feedforward information about traffic flow in the range of five to fifteen seconds has the potential for significant fuel savings over two urban drive cycles.

A Framework for Extremum Seeking Control of Systems With Parameter Uncertainties

Traditionally, the design of extremum seeking algorithm treats the system as essentially a black-box, which for many applications means disregarding known information about the model structure. In contrast to this approach, there have been recent examples where a known plant structure with uncertain parameters has been used in the online optimization of plant operation. However, the results for these approaches have been restricted to specific classes of plants and optimization algorithms. This paper seeks to provide general results and a framework for the design of extremum seekers applied to systems with parameter uncertainties. General conditions for an optimization method and a parameter estimator are presented so that their combination guarantees convergence of the extremum seeker for both static and dynamic plants. Tuning guidelines for the closed loop scheme are also presented. The generality and flexibility of the proposed framework is demonstrated through a number of parameter estimators and optimization algorithms that can be combined to obtain extremum seeking. Examples of anti-lock braking and model reference adaptive control are used to illustrate the effectiveness of the proposed framework.

Gaussian networks for fuel injection control

Abstract This paper proposes a radial basis function (RBF) based approach for the fuel injection control problem. In the past, neural controllers for this problem have centred on using a cerebellar model articulation controller (CMAC) type network with some success. The current production engine control units also use look-up tables in their fuel injection controllers, and if adaptation is permitted to these look-up tables the overall effect closely mimics the CMAC network. Here it is shown that an RBF network with significantly fewer nodes than a CMAC network is capable of delivering superior control performance on a mean value engine model simulation. The proposed approach requires no a priori knowledge of the engine systems, and on-line learning is achieved using gradient descent updates. The RBF network is then implemented on a four-cylinder engine and, after a minor modification, outperforms a production engine control unit.