PI-control design of continuum models of production systems governed by scalar hyperbolic partial differential equation Xiaodong Xu ∗ , Dong Ni ∗∗ , Yuan Yuan ∗ , Stevan Dubljevic ∗ ∗ Department of Chemical & Materials Engineering, University of Alberta, Canada, T6G 2V4, Stevan.Dubljevic@ualberta.ca ∗∗ College of Control Science and Engineering, Zhejiang University, 38 Zheda Road, Hangzhou, 310027, China, dni@zju.edu.cn Abstract: A production system which produces plenty of items in many steps can be modelled as a continuous flow problem governed by a nonlinear and nonlocal hyperbolic partial differential equation. One way to adjust the output of such process is by manipulating the start rate. This paper considers the control and regulation by proportional-integral (PI) controllers for the continuum production systems. In the considered system, the input and output are located at the boundary of production system. In particular, the closed-loop stability of the linearized continuum production model with the designed PI-controller is proved using spectral analysis and Lyapunov theory. Numerical results demonstrate successful tracking for step inputs in the demand rate. Keywords: Optimal state estimation, Inequality constraints, Linear time-invariant system. 1. INTRODUCTION Recent years, several continuum models were introduced to simulate the average behavior of production systems at an aggregate level, see Armbruster et al. (2006), La Marca et al. (2010). The continuum model is governed by scalar hyperbolic partial differential equations and the model description is appropriate, for this work a semiconductor factory producing a large number of items in many steps. The mathematical variable used to describe the production flow is a density variable ρ(x,t) denoting the production density at stage x at a time t. In this work, without loss of generality, we scale x ∈ [0, 1], where x = 0 denotes the beginning of the production flow line and x = 1 denotes the end. It should be noted that the velocity of the production flow along entire system is a constant. This explains that in a real world factory, all parts move through the factory with the same speed. In a serial production system, velocity through the factory is dependent on all items and machines downstream. The problem of control systems described by hyperbolic partial differential equations has received a considerable amount of attentions Aksikas et al. (2008), Bastin and Coron (2016), Xu and Dubljevic (2016). For a complete and detailed review, readers should refer Luo et al. (2012). Numerous, available results have been provided to guar- antee asymptotical and exponential stability of the closed- loop hyperbolic systems Coron et al. (2007), Dos Santos et al. (2008), Coron and Bastin (2015) and in addition to realize the regulation Xu et al. (2017), Paunonen (2017). The backstepping methods have been explored for the regulation of infinite-dimensional systems in Deutscher (2015), Deutscher (2017), Xu and Dubljevic (2017b). In this work, our objective is to design a PI-controller to ensure the stability of the equilibrium of the nonlinear close-loop production system and the output regulation to a desired set-point. The idea of using output feedback control for infinite-dimensional systems is motivated by Pohjolainen (1982), Deutscher (2011), Xu and Dublje- vic (2017a). In Blandin et al. (2017), an internal state feedback controller has been designed to stabilize the en- tropy solutions around a constant equilibrium in the L 1 and L ∞ . A nonlocal stabilization boundary controller has been developed in Perrollaz (2013) to obtain asymptotic stability of the constant equilibrium in the L 2 . In this paper, a 1-D scalar hyperbolic PDE is considered and we are interested in a boundary PI-controller design to asymptotically stabilize the system around its equilibrium profile. Inspired by Coron et al. (2007) and Trinh et al. (2017), we construct a new Lyapunov functional to prove exponential stabilization of the closed-loop system with the designed PI-controller. The paper is organized as follows: Section 2 introduces the statement of the considered model and problem. Then, Lypunov direct method is provided and applied to the linearized model. Section 3 gives the discussion of the stability from the point of frequency domain. Section 4 im- plements the PI-controller through numerical simulations. Finally, our conclusions are given in Section 5. 2. STATEMENT OF THE PROBLEM AND RESULT In this work, the time evaluation of the product density governed by the following 1-D hyperbolic PDE is consid- ered: Preprints, 10th IFAC International Symposium on Advanced Control of Chemical Processes Shenyang, Liaoning, China, July 25-27, 2018 Copyright © 2018 IFAC 578