Robust Control of Uncertain Time -Delay Systems.

Time-delay systems are common in industries. Direct analysis and synthesis of control systems with time delays are complicated and approximation methods such as Pade approximation are usually applied. However, the issues of control system robustness with respect to model uncertainties and approximation errors have not been sufficiently addressed. This dissertation focus on robustness of time-delay systems, especially robustness with respect to time delays, which has been discussed extensively using Lyapunov second method. We propose two methods in this dissertation to reformulate the problems into standard mu or Hinfinity problems. The first method involves representing the systems in linear functional transformation (LFT) framework and approximating delays by rational transfer functions. The approximation errors are then treated as uncertainties. We show that all the well-known techniques of Hinfinity control theory can be applied to this framework. Consequently, controller design becomes a routine process. We also show that the conventional Lyapunov method is a special case in our proposed framework and our proposed method offers less conservative results. In the second method, we treat uncertain delays as uncertainties with restricted phase angles and extend structured singular value to include phase information. We show that the extended small-mu theorem can be applied to analyze stability and performance of uncertain delay systems with many other type of uncertainties, such as plant model uncertainties and parametric uncertainties. Finally, we generalize the above techniques to linear systems with feedback connected nonlinear elements. Both time invariant and time-varying nonlinearities are discussed by incorporating circle/Popov criterion with small-mu theorem

Synchronization of coupled neural oscillators with heterogeneous delays

We investigate the effects of heterogeneous delays in the coupling of two excitable neural systems. Depending upon the coupling strengths and the time delays in the mutual and self-coupling, the compound system exhibits different types of synchronized oscillations of variable period. We analyze this synchronization based on the interplay of the different time delays and support the numerical results by analytical findings. In addition, we elaborate on bursting-like dynamics with two competing timescales on the basis of the autocorrelation function.Comment: 18 pages, 14 figure

Amplitude bounds for biochemical oscillators

We present a practical method to obtain bounds for the oscillation minima and maxima of large classes of biochemical oscillator models that generate oscillations through a negative feedback. These bounds depend on the feedback nonlinearity and are independent of explicit or effective feedback delays. For specific systems, we obtain explicit analytical expressions for the bounds and demonstrate their effectiveness in comparison with numerical simulations.Comment: 6 pages, 4 figure

Galerkin approximations for the optimal control of nonlinear delay differential equations

Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042

Trapped Modes in Linear Quantum Stochastic Networks with Delays

Networks of open quantum systems with feedback have become an active area of research for applications such as quantum control, quantum communication and coherent information processing. A canonical formalism for the interconnection of open quantum systems using quantum stochastic differential equations (QSDEs) has been developed by Gough, James and co-workers and has been used to develop practical modeling approaches for complex quantum optical, microwave and optomechanical circuits/networks. In this paper we fill a significant gap in existing methodology by showing how trapped modes resulting from feedback via coupled channels with finite propagation delays can be identified systematically in a given passive linear network. Our method is based on the Blaschke-Potapov multiplicative factorization theorem for inner matrix-valued functions, which has been applied in the past to analog electronic networks. Our results provide a basis for extending the Quantum Hardware Description Language (QHDL) framework for automated quantum network model construction (Tezak \textit{et al.} in Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 370(1979):5270-5290, to efficiently treat scenarios in which each interconnection of components has an associated signal propagation time delay

The SLH framework for modeling quantum input-output networks

Many emerging quantum technologies demand precise engineering and control over networks consisting of quantum mechanical degrees of freedom connected by propagating electromagnetic fields, or quantum input-output networks. Here we review recent progress in theory and experiment related to such quantum input-output networks, with a focus on the SLH framework, a powerful modeling framework for networked quantum systems that is naturally endowed with properties such as modularity and hierarchy. We begin by explaining the physical approximations required to represent any individual node of a network, eg. atoms in cavity or a mechanical oscillator, and its coupling to quantum fields by an operator triple $(S,L,H)$. Then we explain how these nodes can be composed into a network with arbitrary connectivity, including coherent feedback channels, using algebraic rules, and how to derive the dynamics of network components and output fields. The second part of the review discusses several extensions to the basic SLH framework that expand its modeling capabilities, and the prospects for modeling integrated implementations of quantum input-output networks. In addition to summarizing major results and recent literature, we discuss the potential applications and limitations of the SLH framework and quantum input-output networks, with the intention of providing context to a reader unfamiliar with the field.Comment: 60 pages, 14 figures. We are still interested in receiving correction