Backstepping PDE Design: A Convex Optimization Approach

Abstract\u2014Backstepping design for boundary linear PDE is formulated as a convex optimization problem. Some classes of parabolic PDEs and a first-order hyperbolic PDE are studied, with particular attention to non-strict feedback structures. Based on the compactness of the Volterra and Fredholm-type operators involved, their Kernels are approximated via polynomial functions. The resulting Kernel-PDEs are optimized using Sumof- Squares (SOS) decomposition and solved via semidefinite programming, with sufficient precision to guarantee the stability of the system in the L2-norm. This formulation allows optimizing extra degrees of freedom where the Kernel-PDEs are included as constraints. Uniqueness and invertibility of the Fredholm-type transformation are proved for polynomial Kernels in the space of continuous functions. The effectiveness and limitations of the approach proposed are illustrated by numerical solutions of some Kernel-PDEs

Observer design for multivariable transport-reaction systems based on spatially distributed measurements

This paper is concerned with the design of observers for a class of one-dimensional multi-state transport-reaction systems considering distributed in-domain measurements over the spatial domain. A design based on the Lyapunov method is proposed for the stabilization of the estimation error dynamics. The approach uses positive definite matrices to parameterize a class of Lyapunov functionals that are positive in the Lebesgue space of integrable square functions. Thus, the stability conditions can be expressed as a set of LMI constraints which can be solved numerically using sum of squares (SOS) and standard semi-definite programming (SDP) tools. In order to evaluate the proposed methodology, a state observer is designed to estimate the variables of a nonisothermal tubular reactor model. Numerical simulations are presented to demonstrate the potentials of the proposed observer.Campus Arequip

Robust H

This paper addresses the problem of robust H∞ control design via the proportional-spatial derivative (P-sD) control approach for a class of nonlinear distributed parameter systems modeled by semilinear parabolic partial differential equations (PDEs). By using the Lyapunov direct method and the technique of integration by parts, a simple linear matrix inequality (LMI) based design method of the robust H∞ P-sD controller is developed such that the closed-loop PDE system is exponentially stable with a given decay rate and a prescribed H∞ performance of disturbance attenuation. Moreover, a suboptimal H∞ controller is proposed to minimize the attenuation level for a given decay rate. The proposed method is successfully employed to address the control problem of the FitzHugh-Nagumo (FHN) equation, and the achieved simulation results show its effectiveness

Delayed point control of a reaction–diffusion PDE under discrete-time point measurements

We consider stabilization problem for reaction–diffusion PDEs with point actuations subject to a known constant delay. The point measurements are sampled in time and transmitted through a communication network with a time-varying delay. To compensate the input delay, we construct an observer for the future value of the state. Using a time-varying observer gain, we ensure that the estimation error vanishes exponentially with a desired decay rate if the delays and sampling intervals are small enough while the number of sensors is large enough. The convergence conditions are obtained using a Lyapunov–Krasovskii functional, which leads to linear matrix inequalities (LMIs). We design output-feedback point controllers in the presence of input delays using the above observer. The boundary controller is constructed using the backstepping transformation, which leads to a target system containing the exponentially decaying estimation error. The in-domain point controller is designed and analysed using an improved Wirtinger-based inequality. We show that both controllers can guarantee the exponential stability of the closed-loop system with an arbitrary decay rate smaller than that of the observer’s estimation error

Delayed boundary control of a heat equation under discrete-time point measurements

We consider a reaction-diffusion PDE under continuously applied boundary control that contains a constant delay. The point measurements are sampled in time and transmitted through a network with a time-varying delay. We construct an observer that predicts the value of the state allowing to compensate for the constant boundary delay. Using a time-varying injection gain, we ensure that the estimation error vanishes exponentially with a desired decay rate if the delays and sampling intervals are small enough while the number of sensors is large enough. The stability conditions, obtained via a Lyapunov-Krasovskii functional, are formulated in terms of linear matrix inequalities. By applying the backstepping transformation to the future state estimation, we derive a boundary controller that guarantees the exponential stability of the closed-loop system with an arbitrary decay rate smaller than that of the observer. The results are demonstrated by an example