Systems control theory applied to natural and synthetic musical sounds

Systems control theory is a far developped field which helps to study stability, estimation and control of dynamical systems. The physical behaviour of musical instruments, once described by dynamical systems, can then be controlled and numerically simulated for many purposes. The aim of this paper is twofold: first, to provide the theoretical background on linear system theory, both in continuous and discrete time, mainly in the case of a finite number of degrees of freedom ; second, to give illustrative examples on wind instruments, such as the vocal tract represented as a waveguide, and a sliding flute

Duality and H∞H_{\infty}-Optimal Control Of Coupled ODE-PDE Systems

In this paper, we present a convex formulation of $H_{\infty}$-optimal control problem for coupled linear ODE-PDE systems with one spatial dimension. First, we reformulate the coupled ODE-PDE system as a Partial Integral Equation (PIE) system and show that stability and $H_{\infty}$ performance of the PIE system implies that of the ODE-PDE system. We then construct a dual PIE system and show that asymptotic stability and $H_{\infty}$ performance of the dual system is equivalent to that of the primal PIE system. Next, we pose a convex dual formulation of the stability and $H_{\infty}$-performance problems using the Linear PI Inequality (LPI) framework. LPIs are a generalization of LMIs to Partial Integral (PI) operators and can be solved using PIETOOLS, a MATLAB toolbox. Next, we use our duality results to formulate the stabilization and $H_{\infty}$-optimal state-feedback control problems as LPIs. Finally, we illustrate the accuracy and scalability of the algorithms by constructing controllers for several numerical examples

Optimization based control design techniques for distributed parameter systems

The study presents optimization based control design techniques for the systems that are governed by partial differential equations. A control technique is developed for systems that are actuated at the boundary. The principles of dynamic inversion and constrained optimization theory are used to formulate a feedback controller. This control technique is demonstrated for heat equations and thermal convection loops. This technique is extended to address a practical issue of parameter uncertainty in a class of systems. An estimator is defined for unknown parameters in the system. The Lyapunov stability theory is used to derive an update law of these parameters. The estimator is used to design an adaptive controller for the system. A second control technique is presented for a class of second order systems that are actuated in-domain. The technique of proper orthogonal decomposition is used first to develop an approximate model. This model is then used to design optimal feedback controller. Approximate dynamic programming based neural network architecture is used to synthesize a sub-optimal controller. This control technique is demonstrated to stabilize the heave dynamics of a flexible aircraft wings. The third technique is focused on the optimal control of stationary thermally convected fluid flows from the numerical point of view. To overcome the computational requirement, optimization is carried out using reduced order model. The technique of proper orthogonal decomposition is used to develop reduced order model. An example of chemical vapor deposition reactor is considered to examine this control technique --Abstract, page iii