Sei Zhen Khong

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.

Multidimensional global extremum seeking via the DIRECT optimisation algorithm

DIRECT is a sample-based global optimisation method for Lipschitz continuous functions defined over compact multidimensional domains. This paper adapts the DIRECT method with a modified termination criterion for global extremum seeking control of multivariable dynamical plants. Finite-time semi-global practical convergence is established based on a periodic sampled-data control law, whose sampling period is a parameter which determines the region and accuracy of convergence. A crucial part of the development is dedicated to a robustness analysis of the DIRECT method against bounded additive perturbations on the objective function. Extremum seeking involving multiple units is also considered within the same context as a means to increase the speed of convergence. Numerical examples of global extremum seeking based on DIRECT are presented at the end.

Robust Stability Analysis for Feedback Interconnections of Time-Varying Linear Systems

Feedback interconnections of causal linear systems are studied in a continuous-time setting. The developments include a linear time-varying (LTV) generalization of Vinnicombes nu-gap metric and an integral-quadratic-constraint-based robust L2-stability theorem for uncertain feedback interconnections of potentially open-loop unstable systems. These main results are established in terms of Toeplitz--Wiener--Hopf and Hankel operators, and the Fredholm index, for a class of causal linear systems with the following attributes: (i) a system graph (i.e., subspace of L2 input-output pairs) admits normalized strong right (i.e., image) and left (i.e., kernel) representations, and (ii) the corresponding Hankel operators are compact. These properties are first verified for stabilizable and detectable LTV state-space models to initially motivate the abstract formulation, and subsequently verified for frequency-domain multiplication by constantly proper Callier--Desoer transfer functions in analysis that confirms consistency of the developments with the time-invariant theory. To conclude, the aforementioned robust stability theorem is applied in an illustrative example concerning the feedback interconnection of distributed-parameter systems over a network with time-varying gains. (Less)

Iterative learning control based on extremum seeking

This paper proposes a non-model based approach to iterative learning control (ILC) via extremum seeking. Single-input-single-output discrete-time nonlinear systems are considered, where the objective is to recursively construct an input such that the corresponding system output tracks a prescribed reference trajectory as closely as possible on finite horizon. The problem is formulated in terms of extremum seeking control, which is amenable to a range of local and global optimisation methods. Contrary to the existing ILC literature, the formulation allows the initial condition of each iteration to be incorporated as an optimisation variable to improve tracking. Sufficient conditions for convergence to the reference trajectory are provided. The main feature of this approach is that it does not rely on knowledge about the systems model to perform iterative learning control, in contrast to most results in the literature.

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.

Extremum seeking of dynamical systems via gradient descent and stochastic approximation methods

This paper examines the use of gradient based methods for extremum seeking control of possibly infinite-dimensional dynamic nonlinear systems with general attractors within a periodic sampled-data framework. First, discrete-time gradient descent method is considered and semi-global practical asymptotic stability with respect to an ultimate bound is shown. Next, under the more complicated setting where the sampled measurements of the plants output are corrupted by an additive noise, three basic stochastic approximation methods are analysed; namely finite-difference, random directions, and simultaneous perturbation. Semi-global convergence to an optimum with probability one is established. A tuning parameter within the sampled-data 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.

On the definiteness of graph Laplacians with negative weights: Geometrical and passivity-based approaches

The positive semidefiniteness of Laplacian matrices corresponding to graphs with negative edge weights is studied. Two alternative proofs to a result by Zelazo and Bürger (Theorem 3.2), which provides upper bounds on the magnitudes of the negative weights in terms of effective resistances within which to ensure definiteness of the Laplacians, are provided. Both proofs are direct and intuitive. The first employs purely geometrical arguments while the second relies on passivity arguments and the laws of physics for electrical circuits. The latter is then used to establish consensus in multi-agent systems with generalized high-order dynamics. A numerical example is given at the end of the paper to highlight the result.

On the convergence of iterative learning control

We derive frequency-domain criteria for the convergence of linear iterative learning control (ILC) on finite-time intervals that are less restrictive than existing ones in the literature. In particular, the former can be used to establish the convergence of ILC in certain cases where the latter are violated. The results cover ILC with non-causal filters and provide insights into the transient behaviors of the algorithm before convergence. We also stipulate some practical rules under which ILC can be applied to a wider range of applications.

Positive systems analysis via integral linear constraints

Closed-loop positivity of feedback interconnections of positive monotone nonlinear systems is investigated. It is shown that an instantaneous gain condition on the open-loop systems which implies feedback well-posedness also guarantees feedback positivity. Furthermore, the notion of integral linear constraints (ILC) is utilised as a tool to characterise uncertainty in positive feedback systems. Robustness analysis of positive linear time-varying and nonlinear feedback systems is studied using ILC, paralleling the well-known results based on integral quadratic constraints.

Aerodynamic Shape Optimization via Global Extremum Seeking

Optimization of aerodynamic shapes using computational fluid dynamics (CFD) approaches has been successfully demonstrated over a number of years; however, the typical optimization approaches employed utilize gradient algorithms that guarantee only the local optimality of the solution. While numerous global optimization techniques exist, they are usually too time consuming in practice. In this brief, a modified global optimization algorithm (DIRECT-L) is introduced and is utilized in the context of sampled-data global extremum seeking. The theoretical framework and conditions under which the convergence to the steady state of the CFD solver can be interpreted as plant dynamics are stated. This method alleviates the computational burden by reducing sampling and requiring only partial convergence of the CFD solver for each iteration of the optimization design process. The approach is demonstrated on a simple example involving drag minimization on a 2-D aerofoil.