Journal of Process Control 24 (2014) 1402–1411 Contents lists available at ScienceDirect Journal of Process Control j our na l ho me pa g e: www.elsevier.com/locate/jprocont A new calculation method of feedback controller gain for bilinear paper-making process with disturbance M. Hamdy ∗ , I. Hamdan Department of Industrial Electronics and Control Engineering, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt a r t i c l e i n f o Article history: Received 30 December 2013 Received in revised form 16 June 2014 Accepted 16 June 2014 Available online 28 July 2014 Keywords: Multivariable bilinear systems Paper-making process Robust H∞ control State feedback control Lyapunov stability analysis LMI a b s t r a c t This paper presents a new method to calculate the feedback control gain for a class of multivariable bilinear system, and also applied this method on the control of two sections of paper-making process with disturbance. The robust H∞ control problem is to design a state feedback controller such that the robust stability and a prescribed H∞ performance of the resulting closed-loop system are ensured. The controller turns out to be robust with respect to the disturbance in the plant. Utilizing the Schur complement and some variable transformations, the stability conditions of the multivariable bilinear systems are formulated in terms of Lyapunov function via the form of linear matrix inequality (LMI). The gain of controller will be calculated via LMI. Finally, the application examples of a headbox section and a dryer section of paper-making process are used to illustrate the applicability of the proposed method. © 2014 Elsevier Ltd. All rights reserved. 1. Introduction During last recent years, bilinear systems have been widely applied to a wide variety of fields, for example, engineering, biology, and economics. A bilinear system exists between linear and nonlinear systems, and its dynamic is simpler than that of nonlinear systems. Also, a bilinear model can obviously represent the dynamics of a nonlinear system more accurately than a linear one. In many practical systems bilinear systems arise as natural models where the bilinearity of states and control variables appears naturally, for example paper-making process, distillation columns, DC motors, induction motor drives, mechanical brake systems, nuclear reactors, dynamics of heat exchanger with controlled flow, some processes in elasticity, modeling and control of a small furnace, blood pressure, cardiac regulator, behavior of sense organ, water balance, temperature regulation in human body, and a growth of a national economy [1,2]. Problems of science and technology, as well as advances in nonlinear analysis and differential geometry, led to the development of nonlinear control system theory. In an important class of nonlinear control systems, the control u(t) is used as a multiplicative coefficient, ˙ x(t ) = f (x(t )) + g(x(t ))u(t ), where f(x(t)) and g(x(t)) are differentiable vector functions. They include a class of control systems in which f(x(t)) = Ax(t) and g(x(t)) = B + Nx(t), linear functions, so ˙ x(t ) = Ax(t ) + (B + Nx(t ))u(t ), which is called a bilinear system [3]. Bilinear systems involve products of state and control, which means that the term Ax(t) is a linear in state, Bu(t) is a linear in control but not jointly linear in state and control in the term Nx(t)u(t). In practice, due to changes in environmental conditions, aging, etc., disturbances occur during the modeling of a bilinear system. Therefore, disturbances ought to be integrated into models of bilinear systems. A disturbance signal is an unwanted input signal that affects the system’s output. Actually, there will almost always be disturbances in a system. In order to minimize the effect of the disturbance signal we will need to reduce the effect of the disturbance input on the regulated output to within a prescribed level. ∗ Corresponding author. Tel.: +20 1221374213. E-mail address: mhamdy72@hotmail.com (M. Hamdy). http://dx.doi.org/10.1016/j.jprocont.2014.06.009 0959-1524/© 2014 Elsevier Ltd. All rights reserved.