Simulation Of 1DOF And 2DOFAdaptive Control Of The Water Tank Model

The water tank is a typical nonlinear system representing several types of systems like barrels, reservoirs, tanks etc. for liquid. The simulation, similarly as in other cases, can help with the understanding of the system’s static and dynamic behavior and it is starting point for the controller design. The adaptive control in this work is based on the choice of the external linear model of the nonlinear system, parameters of which are identified recursively and the controller’s parameters are also recomputed recursively according to the identified system. Basic control requirements are satisfied with the use of polynomial approach together with the Pole-placement method and spectral factorization. The resulting controller has also one tuning parameter which affects mainly the speed of the control and overshoots. Two control schemes with one degree-of-freedom and two degrees-of-freedom are tested and compared in the work. INTRODUCTION The adaptive control belongs to relatively new control strategies (Dusek and Honc, 2009), (Honc and Dusek, 2013b) which can be used for controlling of wide range of technological processes like heat exchangers, chemical reactors, flow control, water level in the tank etc. The big advantage of this method can be found in strong theoretical background (Åström 1989) and a number of improvements. This control strategy comes from the feature known from the nature where animals and plants “adopt” their behavior to the actual living environment. Adaptation in the control field is done mainly by the change of the structure, parameters of the controller etc. One way how we can also influence the result of the control is the choice of the control structure (Bobal et al. 2005). The most common control scheme is one degree-of-freedom (1DOF) control configuration, which has controller only in the feedback part (Grimble 1994). Better results can be sometimes obtained with the use of two degrees-of-freedom (2DOF) structure (Grimble 1994), where one part of the controller lies in the feedback and the second part is in the feedforward part. This control configuration could improve control results especially at the beginning of the control. The controlled system and also the controller act in the 1DOF and 2DOF control schemes in the form of the continuous-time transfer function. Relations for computation of the controller’s parameters uses polynomial approach (Kucera 1993) together with Poleplacement method and Spectral factorization. These methods satisfy basic control requirements such as stability, reference signal tracking and disturbance attenuation. Moreover, the polynomial synthesis produces not only the structure of the controller but also relations for controller’s parameters computing. The system under the consideration is the real model of the water tank which is one part of the Process Control Teaching system PCT40 from Armfield (Armfield 2005). This four-liter model offers various types for control including continuous control of the water tank. All experiments in this work are based on the modelling and simulation techniques which are often used nowadays because of time and mainly cost savings (Ingham et al. 2000). The mathematical model of the controlled water tank is described by the one nonlinear ordinary differential equation which is then solved numerically. The paper is divided into four main parts. The adaptive approach is introduced in the first chapter. Then, the model of the controlled water tank is introduced together with the mathematical model and simulation results of the steady-state and dynamic analyses. The third part presents simulation results of control for both 1DOF and 2DOF control configurations and compares these two controllers. The last part is conclusion which summarizes main results and form recommendations for controlling of the real process. All simulations were done in the mathematical software Matlab, version 7.0.1 which is suitable for this task and also widely used (Honc and Dusek, 2013a). ADAPTIVE CONTROL The adaptivity in the control strategy here is satisfied by the recursive estimation of the External Linear Model (ELM) as a linear representation of originally nonlinear system parameters of which are estimated recursively Proceedings 29th European Conference on Modelling and Simulation ©ECMS Valeri M. Mladenov, Petia Georgieva, Grisha Spasov, Galidiya Petrova (Editors) ISBN: 978-0-9932440-0-1 / ISBN: 978-0-9932440-1-8 (CD) during the control and parameters of the controller are recomputed in each step according to identified parameters too (Bobal et al. 2005). The ELM is usually in the form of continuous-time (CT) or discrete-time (DT) transfer functions, where CT models are more accurate. On the other hand, DT models are better for online identification. Two CT control system configurations were tested for this model – the one degree-of-freedom (1DOF) and the two degrees-of-freedom (2DOF) control configurations. Control System Synthesis The first control structure with one degree-of-freedom has controller only in the feedback part – see Figure 1. Figure 1: 1DOF control configuration The block G(s) represents the transfer function of the controlled system, in our case ELM chosen for example from the dynamic analysis. The second block Q(s) is feedback controller again in the form of the transfer function. Signal w is reference signal, e.g. wanted value of the output variable y and e denotes control error, e = w – y. The computed output signal u from the controller is action value and v is random error. The polynomial approach together with the Poleplacement method and spectral factorization are employed in the controller design. The transfer function of the controlled system is then with the use of polynomial approach ( ) ( ) ( ) b s G s a s = (1) where ( ) ( ) deg deg b s a s ≤ due to properness condition and the transfer function of the controller is ( ) ( ) ( ) q s Q s p s = (2) and ( ) ( ) deg deg q s p s ≤ again due to properness. The asymptotic tracking of the reference signal is satisfied if the polynomial p(s) include the least common divisor of the Laplace transform of the reference signal w and the random error v. If we consider these signals from the ring of step functions, the least common divisor is s and polynomial p(s) in the denominator of (2) is ( ) ( ) p s s p s = ⋅ . Polynomials a(s) and b(s) are known in each step from the recursive identification and the task of the controller is to compute parameters of polynomials p(s) and q(s). These parameters are computed from the so called Diophantic equation (Kucera 1993) ( ) ( ) ( ) ( ) ( ) a s s p s b s q s d s ⋅ ⋅ + ⋅ = (3) by the Method of uncertain coefficients which compares coefficients of individual s-powers. Degrees of the polynomials ( ) p s and q(s) are ( ) ( ) ( ) ( ) deg deg 1 deg deg p s a s q s a s = − = . (4) The polynomial d(s) on the right side of equation (3) is stable optional polynomial and the degree of this polynomial is ( ) ( ) ( ) deg deg deg 1 d s a s p s = + + (5) The simplest way how to choose this polynomial is to use the Pole-placement method which defines this polynomial with several poles, number of which depends on the degree of this polynomial, i.e. ( ) ( ) d d s s α = + (6) with α as a tuning constant with condition α > 0. Disadvantage of this method is that there is no general rule how to choose α. Our experiments (Vojtesek and Dostal 2012) have shown that it is good to connect the choice of the polynomial d(s) with the parameters of the controlled system, for example with the use of spectral factorization of the polynomial a(s) in the numerator of G(s) in (1) known from the recursive identification ( ) ( ) ( ) ( ) * * n s n s a s a s ⋅ = ⋅ (7) The polynomial d(s) in (6) is then divided into two parts – the first, n(s), is computed from the spectral factorization (7) and the second from the Poleplacement method in (6), i.e. ( ) ( ) ( ) deg d n d s n s s α − = ⋅ + (8) The second control configuration with two degrees-offreedom (2DOF) has the controller divided into feedback part Q(s) and feedforward part R(s) – see Figure 2. Figure 2: 2DOF control configuration The main benefit of this control configuration can be found in the better reference signal tracking which is satisfied by the feedforward part of the controller R(s). Transfer functions of the controller are generally ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ; q s q s r s r s Q s R s p s s p s p s s p s = = = = ⋅ ⋅ (9) for ( ) ( ) deg deg q s p s ≤ and ( ) ( ) deg deg r s p s ≤ and polynomials ( ) p s , q(s) and r(s) are computed from Diophantine equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) a s s p s b s q s d s t s s b s r s d s ⋅ ⋅ + ⋅ = ⋅ + ⋅ = (10) with the same stable polynomial d(s) on the right side which is again constructed by the Pole-placement method and the spectral factorization described above. Polynomial t(s) in the second Diophantine equation is an auxiliary stable polynomial. Coefficients of this polynomial are not used for computing of coefficients of the polynomial r(s). Recursive Identification Important part of the adaptive approach here is the online recursive identification of the ELM. The simple Recursive Least-Squares (RLS) method (Fikar and Mikles 1999) is used in this work for this task. The big advantage of this method is that it can be easily expanded by the additional “forgetting” techniques and also programmable in standard program languages. The RLS method used for estimation of the vector of parameters δ̂ Τ θ could be described by the set of equations: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1