Continuous-Time Vs. Discrete-Time Identification Models Used For Adaptive Control Of Nonlinear Process

An adaptive control is a technique where the controller adopts a structure or parameters somehow to the control conditions and the state of the controlled system. One way how we can fulfil the adaptivity of the controller is a recursive identification of the controlled system which satisfies that parameters of the controller changes according to parameters of the controlled system during the whole control process. The goal of this contribution is to compare identification models that work in continuous and discrete time. The control synthesis uses polynomial approach that satisfies basic control requirements such as a stability, a disturbance attenuation and a reference signal tracking. The control response could be tuned by the choice of the root position in the Pole-placement method. Moreover, this control method could be easily programmable that is big advantage while we use this method in simulation software such as Matlab etc. INTRODUCTION The adaptive control (Åström and Wittenmark, 1989) is not new control approach but it is still used because it produces good control results. Advantage of this method can be found in very good theoretical background and variety of modifications (Bobal et al., 2005). The approach used here is based on the choice of the External Linear Model (ELM) which describes controlled, originally nonlinear, process in the linear way for example by the discrete or the continuous transfer function (TF) (Bobal et al., 2005). Parameters of this ELM are then identified recursively during the control and parameters of the controller are recomputed according to them. Results of control synthesis are the structure and relations for computing controller’s parameters that reflect identified parameters of ELM. The recursive identification of the continuous-time (CT) model (Wahlberg, 1990) is a bit more complicated than identification of the discrete-time (DT) model where the computation uses measured or simulated values of input and output variables in discrete time intervals. This approach could be inaccurate for bigger values of the sampling period. One solution can be found in the use of so called delta-models (Middleton and Goodwin, 2004) that are special types of DT models where parameters of input and output variables are related to the sampling period. It was proved that parameters of the delta-model approach to parameters of the CT model for sufficiently small sampling period (Stericker and Sinha, 1993). This combination of the continuoustime control synthesis with the discrete-time identification is called “Hybrid adaptive control” and some applications can be found for example in (Vojtesek and Dostal, 2005) and (Vojtesek and Dostal, 2011). The second way is to use the CT control synthesis and also the CT recursive identification. The CT online estimation is not as simple as a DT estimation because derivatives of the input and output variable are immeasurable. This negative feature could be solved for example with the use of differential filters (Dostal et al., 2001). The control synthesis uses polynomial approach which satisfies basic control system requirements such as a stability of the control loop, a reference signal tracking and a disturbance attenuation. Moreover, the two degrees-of-freedom (2DOF) configuration has good results in the reference signal tracking (Kucera, 1993). The continuous stirred-tank reactor (CSTR) is typical nonlinear equipment used in the chemical and biochemical industry for production of various chemicals (Ingham et al., 2000). The mathematical model of this nonlinear system is described by the set of nonlinear ordinary differential equations (ODEs) which can be solved mathematically for example by the Runge-Kutta’s method. This mathematical model than serves as a testing model for simulation analyses proposed in the theoretical part. All results in this paper are simulations made in the mathematical software Matlab, version 7.0.1. ADAPTIVE CONTROL The adaptive approach (Åström and Wittenmark, 1989) takes its philosophy in the nature, where plants, animals or even human beings “adapt” their behavior to the actual conditions and environment they live in. There could be various adaptive control techniques but the one Proceedings 30th European Conference on Modelling and Simulation ©ECMS Thorsten Claus, Frank Herrmann, Michael Manitz, Oliver Rose (Editors) ISBN: 978-0-9932440-2-5 / ISBN: 978-0-9932440-3-2 (CD) which is used in this work adapt parameters of the controller to actual state of the controlled system. This done via recursive identification of the system’s ELM and parameters of the controller are then recomputed according to identified parameters of the ELM. The design of the controller starts with the choice of the ELM. We can use for example transfer functions (TF) that are generally described in the CT form: ( ) ( ) ( ) b s G s a s = (1) where polynomials a(s) and b(s) will be later used in the computation of controller’s parameters. It is good to do the static and dynamic analysis of the controlled system before the design of the controller. The static analysis helps with the choice of the optimal working point where we can obtain for example the best concentration of the product or minimal costs. On the other hand, the dynamic analysis of the system can be used for example for the choice of the ELM’s order. Continuous-Time Identification Model As G(s) is also relation of the Laplace transform of the output variable, Y(s), to the input variable, U(s), the ELM in the (1) could be also rewritten to the form ( ) ( ) ( ) ( ) a y t b u t σ σ ⋅ = ⋅ (2) where u(t) denotes the input variable, y(t) is the output variable and σ is the differentiation operator. The identification of CT model in (2) is problem because the derivatives of the input and the output variables are immeasurable. If we replace these derivations by the filtered ones denoted by uf and yf and computed from ( ) ( ) ( ) ( ) ( ) ( ) f f c u t u t c y t y t σ σ ⋅ = ⋅ = (3) for a new stable polynomial c(σ) that fulfils condition ( ) ( ) deg deg c a σ σ ≥ , the Laplace transform of (3) is then ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1