Optimal Control With Disturbance Estimation

The paper deals with a very common situation in many control systems and this is the fact that, for zero control action, the controlled variable is nonzero. This is often caused by the existence of another process input which is uncontrolled. Classic controllers do not take into account the second input, so deviation variables are considered or some feedforward controller is used to compensate the variable. The authors propose a solution, that the process is considered as a system with two inputs and single output (TISO). Here, the uncontrolled input is estimated with the state observer and the controller is designed as the multivariable controller. A Linear-quadratic (LQ) state-feedback control and model predictive control (MPC) of simple thermal process simulations are provided to demonstrate the proposed control strategy. INTRODUCTION Control theory is frequently using models in the form of transfer functions, which from the definition, consider zero initial conditions (Åström and Murray 2010; Nise 2010; Ogata 1995; Skogestad and Postlethwaite 2005). This means practically that for zero control action the controlled variable will be zero as well. Unfortunately, this is not true for many practical applications. Even a P controller will not work very well and the situation is even worse for advanced controllers based on state space process models. These models are similar to P or PD controller formulations – without integral control action. One solution is subtracting working point variables and introducing deviation variables – “zero initial condition” will be met. Integral control action, to ensure offset-free reference tracking, is another interesting problem to solve (Maeder and Morari 2010; Dušek et al. 2015). But why not use the natural process model with disturbance variables and their dynamical effects, directly for the controller design? Then the control task can be solved as a multivariable control problem when only some of the process inputs are used as control variables, while the others are considered as disturbances. Disturbance modelling and state estimation for offset free reference tracking control problems, was published in (Muske and Badgwell 2002; Pannocchia and Rawlings 2003; Tatjewski 2014). Authors propose to estimate the disturbance variable by the augmented state observer. Extended formulation of a standard LQ state-feedback controller and predictive controller, so that the disturbance information is an integral part of the controller, is presented in the paper. A simple thermal process with electrical heating, ambient temperature effect and temperature sensor is modeled analytically by the first principle approach. The model has two inputs and one output. One of the inputs is heating power, while the other is ambient temperature. The output is the temperature sensor measured temperature. A discrete time linear time invariant process model is used for LQ controller design with infinity horizon and asymptotic set point tracking and predictive controller with finite horizon and special formulation of the cost function. Deviations of future states from desired states, calculated from the future set point knowledge, are considered instead of the future control errors which are commonly used in the literature (Camacho and Bordons 2007; Kouvaritakis and Cannon 2015; Maciejowski 2002; Rawlings and Mayne 2009; Rossiter 2003). PROCESS MODEL WITH OFFSETS Let us consider the controlled process with variable um as the control variable (control action) and ym as controlled variable. Disturbance (offset) variables u0 and y0 are considered as process input and additive disturbance on the process output – see block diagram in Fig. 1. Figure 1: Process model Discrete time state space process model can be written as, x(k + 1) = Ax(k) + bum(k) + b0u0 ym(k) = cx(k) + y0 (1) If we know the steady state input and both the offsets, the steady state output can be calculated as um(k) u0 y(k) ym(k) y0