Discrete-time Extremum Seeking Research Articles

Extremum Seeking Control for Stiffness Auto-Tuning of a Quasi-Passive Ankle Exoskeleton

Recently, it has been shown that light-weight, passive, ankle exoskeletons with spring-based energy store-and-release mechanisms can reduce the muscular effort of human walking. The stiffness of the spring in such a device must be properly tuned in order to minimize the muscular effort. However, this muscular effort changes for different locomotion conditions (e.g., walking speed), causing the optimal spring stiffness to vary as well. Existing passive exoskeletons have a fixed stiffness during operation, preventing it from responding to changes in walking conditions. Thus, there is a need of a device and auto-tuning algorithm that minimizes the muscular effort across different walking conditions, while preserving the advantages of passive exoskeletons. In this letter, we developed a quasi-passive ankle exoskeleton with a variable stiffness mechanism capable of self-tuning. As the relationship between the muscular effort and the optimal spring stiffness across different walking speeds is not known a priori, a model-free, discrete-time extremum seeking control (ESC) algorithm was implemented for real-time optimization of spring stiffness. Experiments with an able-bodied subject demonstrate that as the walking speed of the user changes, ESC automatically tunes the torsional stiffness about the ankle joint. The average RMS EMG readings of tibialis anterior and soleus muscles at slow walking speed decreased by 26.48% and 7.42%, respectively.

Extremum Seeking Algorithms based on Non-Commutative Maps

In this work, discrete-time extremum seeking algorithms for unconstrained optimization problems are developed. A general class of non-commutative maps and one- and two point function evaluation polices are presented to approximate a gradient-descent algorithm, suitable for extremum seeking problems. Moreover, adaptive step size rules are discussed to achieve faster convergence and vanishing steady state oscillations.

Multi-agent source seeking via discrete-time extremum seeking control

Recent developments in extremum seeking theory have established a general framework for the methodology, although the specific implementations, particularly in the context of multi-agent systems, have not been demonstrated. In this work, a group of sensor-enabled vehicles is used in the context of the extremum seeking problem using both local and global optimisation algorithms to locate the extremum of an unknown scalar field distribution. For the former, the extremum seeker exploits estimates of gradients of the field from local dithering sensor measurements collected by the mobile agents. It is assumed that a distributed coordination which ensures uniform asymptotic stability with respect to a prescribed formation of the agents is employed. An inherent advantage of the frameworks is 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 field sampling noise. Subsequently, global extremum seeking with multiple agents is investigated and shown to give rise to robust practical convergence whose speed can be improved via computational parallelism. Nonconvex field distributions with local extrema can be accommodated within this global framework.

Finite-horizon LQ control for unknown discrete-time linear systems via extremum seeking

We present a non-model based approach for asymptotic, locally exponentially stable attainment of the optimal open-loop control sequence for unknown, discrete-time linear systems with a scalar input, where not even the dimension of the system is known. This control sequence minimizes the finite-time horizon cost function, which is quadratic in the measured output and in the input. We make no assumptions on the stability of the unknown system, but we do assume that the system is reachable. The proposed algorithm employs the multi-variable discrete-time extremum seeking approach to minimize the cost function, extending results established for the scalar discrete-time extremum seeking method. Simulation results show that the Hessian's condition number, used as a measure of the optimization problem's level of difficulty, increases with both the system's level of instability and the length of the finite horizon for a scalar system. Thus, we suggest solving well-conditioned, shorter time horizon optimal control problems to obtain good initial control estimates for longer time horizon problems if possible. We also show that the algorithm accommodates input constraints by employing the projection operator and introduce a Newton-based discrete-time extremum seeking controller, which removes the convergence rate's dependence on the unknown Hessian.

Extremum Seeking Control of a Variable Valve timing Engine

In this paper, a discrete-time extremum seeking control algorithm using sliding mode iS proposed for a variable valve timing (VVT) engine control system. With the extremum seeking controller, the control inputs of the VVT engine, including the intake and the exhaust valves open and close timings, the fuel-air ratio, and the spark ignition timing are determined by the sliding mode control rules so that a cost function, which is a combination of the fuel consumption and the NOx emissions is minimized while the output torque tracks a desired signal.