Review Transactions of the Institute of Measurement and Control 1–7 Ó The Author(s) 2016 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0142331216681664 tim.sagepub.com Error minimizing linear regulation for discrete-time systems Rahat Ali, Mohammad Bilal Malik, Fahad Mumtaz Malik and Muwahida Liaqat Abstract This paper presents error minimizing linear regulator (EMLR) for discrete-time, time invariant linear systems. The control objective is tracking of refer- ence signal and rejection of disturbances. EMLR design is based on minimization of a quadratic cost function using canonical form of plant model. The proposed control scheme is based on analytical solution applicable for arbitrary initial conditions. In addition, the control design does not require for- mulation of Linear Matrix Inequalities and Riccati equations, which are otherwise typical requirements of regulators. Stability analysis of the closed loop system using EMLR is presented along with simulation results for regulation of discretized model of underactuated fourth-order ball and beam system. Keywords Discrete-time linear systems, output regulation, error minimization, optimal control, deadbeat control Introduction Output regulation theory, which deals with tracking of refer- ence signals and rejection of disturbance signals, both gener- ated by an exosystem, is one of the well-established areas of control systems research. The asymptotic stability of the closed loop system, along with tracking of reference signal and disturbance rejection under mild conditions of stablizabil- ity and detectability, has been discussed in Davison (1973), Francis and Wonham (1975) and Francis (1977). The classical output regulation theory emphasized on asymptotic regulation. Subsequent research was aimed at optimal regulation with minimization of regulation error sig- nal energy. In this context, Davison and Scherzinger (1987) introduced model based controller with constraint on initial conditions. H N criterion has been used for obtaining optimal output regulation in Mita et al. (2000) and filtering problem for discrete-time system is discussed in Cao et al. (2016). H 2 framework has also been used for finding optimal and subop- timal output regulators. H 2 suboptimal control has been designed for multivariable continuous-time systems in Lin et al. (1998) and Saberi et al. (2000). Linear quadratic (LQ), a special case of H 2 framework, has also been used for obtaining optimal, and suboptimal reg- ulators (Jia and Huo, 2015; Ullah and Muwahida, 2015; Terra et al., 2014). An optimal regulator based on quadratic cost function has been presented in Lindquist and Yakubovich (1999). In Masoud et al. (2011), continuous-time linear quadratic regulation (LQR) has been achieved. The control scheme is based on the minimization of error based performance index using internal model (IM) principle with reference and disturbance model information (Francis and Wonham, 1976; Fahad et al., 2011). In Saberi et al. (2003), optimal regulator based on Linear Matrix Inequalities (LMIs) was proposed; however, the optimal solution could be obtained for initial conditions belonging to a particular set. Balandina and Koganb (2009) addressed the issue of initial conditions for LMI-based design framework. Discrete-time optimal regulators for sampled-data output regulation have also been presented by researchers in the past (Fahad et al., 2013; Muwahida et al., 2014). Lin et al. (1988) proposed a discrete-time solution for multivariable system. In this case, state feedback gains are parameterized and three different estimator structures are used for finding a sub opti- mal solution. The methodology of Masoud et al. (2011) has been explored for discrete-time systems by Rahat et al. (2013), wherein the calculation of controller gains is done in continuous-time domain and control input is sampled at high sampling rate. Furthermore, a sampled-data control scheme was suggested in Rahat et al. (2014) for linear systems in which the control input is calculated by minimizing a discrete-time LQ performance index based on IM principle. The discrete-time optimal regulator techniques require solu- tion of Algebraic Riccati Equation (ARE) for calculation of controller gain. In Saberi et al. (2003), LMI based discrete- time optimal output regulator is presented. The proposed Department of Electrical Engineering, College of Electrical and Mechanical Engineering, National University of Sciences and Technology, Pakistan Corresponding author: Fahad Mumtaz Malik, Department of Electrical Engineering, College of Electrical and Mechanical Engineering, National University of Sciences and Technology, College of EME Peshawar Road, Rawalpindi, Islamabad, 46000, Pakistan. Email: malikfahadmumtaz@yahoo.com