CONTINUOUS-TIME VS. DISCRETE-TIME IDENTIFICATION MODELS USED FOR ADAPTIVE CONTROL OF NONLINEAR PROCESS

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1 CONINUOUS-IME VS. DISCREE-IME IDENIFICAION MODELS USED FOR ADAPIVE CONROL OF NONLINEAR PROCESS Jiri Vojtesek and Petr Dostal Faculty of Applied Informatics omas Bata Uniersity in Zlin Nam. GM 5555, Zlin, Czech Republic {ojtesek,dostalp}@fai.utb.cz KEYWORDS Simulation, Mathematical Model, Adaptie control, Continuous-time model, Delta-model, Continuous Stirred-tank Reactor. ABSRAC An adaptie 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 adaptiity of the controller is a recursie 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. he goal of this contribution is to compare identification models that work in continuous and discrete time. he control synthesis uses polynomial approach that satisfies basic control requirements such as a stability, a disturbance attenuation and a reference signal tracking. he control response could be tuned by the choice of the root position in the Pole-placement method. Moreoer, this control method could be easily programmable that is big adantage while we use this method in simulation software such as Matlab etc. INRODUCION he adaptie control (Åström and Wittenmark, 989) is not new control approach but it is still used because it produces good control results. Adantage of this method can be found in ery good theoretical background and ariety of modifications (Bobal et al., 005). he 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 (F) (Bobal et al., 005). Parameters of this ELM are then identified recursiely 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. he recursie identification of the continuous-time (C) model (Wahlberg, 990) is a bit more complicated than identification of the discrete-time (D) model where the computation uses measured or simulated alues of input and output ariables in discrete time interals. his approach could be inaccurate for bigger alues of the sampling period. One solution can be found in the use of so called delta-models (Middleton and Goodwin, 004) that are special types of D models where parameters of input and output ariables are related to the sampling period. It was proed that parameters of the delta-model approach to parameters of the C model for sufficiently small sampling period (Stericker and Sinha, 993). his combination of the continuoustime control synthesis with the discrete-time identification is called Hybrid adaptie control and some applications can be found for example in (Vojtesek and Dostal, 005) and (Vojtesek and Dostal, 0). he second way is to use the C control synthesis and also the C recursie identification. he C online estimation is not as simple as a D estimation because deriaties of the input and output ariable are immeasurable. his negatie feature could be soled for example with the use of differential filters (Dostal et al., 00). he 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. Moreoer, the two degrees-of-freedom (DOF) configuration has good results in the reference signal tracking (Kucera, 993). he continuous stirred-tank reactor (CSR) is typical nonlinear equipment used in the chemical and biochemical industry for production of arious chemicals (Ingham et al., 000). he mathematical model of this nonlinear system is described by the set of nonlinear ordinary differential equations (ODEs) which can be soled mathematically for example by the Runge-Kutta s method. his mathematical model than seres 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, ersion ADAPIVE CONROL he adaptie approach (Åström and Wittenmark, 989) takes its philosophy in the nature, where plants, animals or een human beings adapt their behaior to the actual conditions and enironment they lie in. here could be arious adaptie control techniques but the one Proceedings 30th European Conference on Modelling and Simulation ECMS horsten Claus, Frank Herrmann, Michael Manitz, Olier Rose (Editors) ISBN: / ISBN: (CD)

2 which is used in this work adapt parameters of the controller to actual state of the controlled system. his done ia recursie identification of the system s ELM and parameters of the controller are then recomputed according to identified parameters of the ELM. he design of the controller starts with the choice of the ELM. We can use for example transfer functions (F) that are generally described in the C form: b( s) G( s) () a s 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. he 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-ime Identification Model As G(s) is also relation of the Laplace transform of the output ariable, Y(s), to the input ariable, U(s), the ELM in the () could be also rewritten to the form ( σ) ( σ) a y t b u t () where u(t) denotes the input ariable, y(t) is the output ariable and σ is the differentiation operator. he identification of C model in () is problem because the deriaties of the input and the output ariables are immeasurable. If we replace these deriations by the filtered ones denoted by u f and y f and computed from c( σ ) uf ( t) u( t) (3) c σ y t y t f for a new stable polynomial c(σ) that fulfils condition deg c( σ ) deg a( σ ), the Laplace transform of (3) is then c( s) U f ( s) U( s) + o ( s) (4) c s Y s Y s + o s f where polynomials o (s) and o (s) includes initial conditions of filtered ariables. If we substitute (4) into the Laplace transform of the Equation (), the relation for the Laplace transform of the filtered output ariable, Y f (s) is b s Yf ( s) U f ( s) +Ψ ( s) (5) a s and Ψ(s) is a rational function which contains initial conditions of both filtered and unfiltered ariables. he dynamics of the differential filters c(s) in (4) must be faster than the dynamics of the controlled system (Dostal et al., 00). It is good to choose the parameters of this polynomial sufficiently small. he alues of filtered alues are taken in the discrete time moment t k k for k 0,,, N. is sampling period and the regression ector has n+m parts where deg a n and deg b m, i.e. ϕ () ( n ) C tk yf tk, yf tk,, yf tk, () (,,,, ) m uf tk uf tk uf ( tk), he ector of parameters ( t ) [ a, a,, a, b, b,, b ] C k 0 n 0 m (6) θ (7) is computed from the differential equation ( n ) yf ( tk) C ( tk) C ( tk) +Ψ( tk) where Ψ ( t k ) includes immeasurable errors. θ ϕ (8) Discrete-ime Identification Model he second approach used for example in (Vojtesek and Dostal, 0) uses so called delta-models for identification. he delta-models are special types of D models where input and output ariables are related to the sampling period. A new complex ariable γ is defined generally as (Mukhopadhyay et al., 99) z γ (9) β z+ β where denotes a sampling period and β is an optional parameter and it holds 0 β. It is clear, that there could be an infinite number of delta-models but so called Forward delta-model for β 0 was used here. he complex ariable γ is then z γ (0) Some works compares parameters C s. delta-model and it was proed for example in (Stericker and Sinha, 993), that parameters of the delta-model approaches to the C ones for sufficiently small sampling period. he C model () can be rewritten to ( ) ( ) a y t b u t () where a () and b () are discrete polynomials and their coefficients are different from those in C model but we suppose, that they are close to them. he regression ector is in this case ϕ( k ) [ y( k ),, y( k n), () u ( k+ m n),, u ( k n) ]

3 he ector of parameters is generally [ a,..., a, b,..., b ] θ (3) n 0 m 0 and its parameters are computed again from the differential equation ( ) + y k θ k ϕ k e k (4) for e(k) as a general random immeasurable component. Both identification methods with the C model and the delta-model was discussed in this work. Recursie Identification Vectors of C and delta parameters must be identified recursiely to satisfy the adaptiity condition. his could be done for example by the Recursie Least- Squares (RLS) method (Fikar and Mikles 999) which is simple, easily programmable method that could be modified with exponential, directional etc. forgetting factors. hese forgetting factors helps with the accuracy in the more complex systems. he RLS method used for estimation of ectors of parameters ˆC Τ θ Τ or θ ˆ in (7) and (4) could be described generally by the set of equations: P γ ε y ϕ ˆ θ ( k ) + ϕ P( k ) ϕ L γ P ( k ) ϕ P λ ( k ) ˆ θ ˆ P( k ) ϕ ϕ P( k ) ( k ) λ ( k ) + ϕ P ( k ) ϕ λ ( k ) θ ( k ) + L ε (5) where φ is regression ector, ε denotes a prediction error, P is a coariance matrix and λ and λ are forgetting factors. For example constant exponential forgetting (Fikar and Mikles 999) uses λ and K λ γ ε (6) where K is a ery small alue (e.g. K 0.00). his RLS modification was used in this work for the online estimation. DESIGN OF HE CONROLLER he controller is designed with the use of the polynomial synthesis (Kucera, 993). here are seeral adantages of this approach. At first, they can work with the controller in the polynomial description, for example in the form of the transfer function (). he result of the synthesis is not only the structure, but also the relations for computing of controller s parameters. Moreoer, this method satisfies basic control requirements. he control scheme with two degrees-of-freedom (DOF) (Grimble, 994) is shown in Figure. Figure : DOF control configuration he signal w is reference signal (i.e. wanted alue), denotes disturbance, u is an input and y an output ariable. he block G(s) in Figure represents the controlled system described by the F (), blocks Q(s) and R(s) are feedback and feedforward parts of the controller again in the form of F, generally: Q s q s r s ; R( s) p s p s (7) Degrees of polynomials p(s), q(s) and r(s) must hold properness condition: deg q s deg p s ; deg r s deg p s (8) he condition for the reference signal tracking is satisfied if the polynomial p(s) in the denominator of the controller s transfer functions (7) is diided into p( s) f ( s) p ( s) (9) where f(s) is a least common diisor of the reference and the disturbance transfer functions. If we hae these F in the form of the step function, f(s) s and (7) could be rewritten into Q s q s r s ; R( s) s p s s p s (0) Parameters of controller s polynomials are computed from the set of polynomial equations a( s) s p ( s) + b( s) q( s) d( s) () t s s+ b s r s d s that are in the literature called Diophantine equations (Kucera, 993) and they can be soled by the Method of uncertain coefficients. Polynomial t(s) in equation () is an auxiliary stable polynomial and coefficients of this polynomial are not used for computing of coefficients of the polynomial r(s). Polynomials a(s) and b(s) in () are known from the recursie identification and the polynomial d(s) on the right side of Diophantine equations () is stable optional polynomial which could affect the quality of the control. Degrees of controller s polynomials p ( s), q(s) and r(s) and the degree of the stable polynomial d(s) are deg p ( s) deg a( s) deg r( s) 0 () deg q s deg a s deg d s deg a s

4 he simplest way how to choose the stable optional polynomial d(s) define the Pole-placement method he polynomial d(s) is then diided into deg d( s) i (3) i ( + ) d s s s where roots s i are generally in the complex form s i α i + ω i j and the stability is satisfied for α i < 0. If we want to obtain an aperiodic output response, ω i must be ω i 0 and (3) is then d( s) ( s+ α ) deg d (4) One disadantage of this method is that it is ery general and it proides for example for deg d(s) 4 four simple roots, two double roots, one single and one triple root but no recommendation for the choice of these roots. Our preious experiment (Vojtesek and Dostal, 0) hae shown, that it is good co connect the choice of this polynomial somehow with the controlled system. he Spectral factorization could be used for this task and it means that the polynomial d(s) is diided into two parts deg d deg n d s n s s+ α (5) where one part is classic pole-placement method and n(s) comes from the Spectral factorization of the polynomial a(s) in the denominator of the controlled system s transfer function (): * * n s n s a s a s (6) he use of Spectral factorization satisfies that the polynomial n(s) is always stable een if the polynomial a(s) is unstable. his could happen for example by inaccurate estimation at the beginning of the control when an estimator does not hae enough information about the system. SIMULAION EXPERIMEN he adaptie approach was tested by simulations on the mathematical model of the Continuous Stirred-ank Reactor (CSR) with so called Van der Vusse reaction inside (Chen et al., 995). his reaction can be described by the following scheme: k k A B C (7) k3 A D and the mathematical model of this system comes from material and heat balances inside the reactor. he result is the set of four nonlinear ordinary differential equations (ODE): dca qr ( c 0 c ) k c k 3 c dt Vr dcb qr c B + k c A k c B dt V r A A A A (8) dr qr hr ArU ( r0 r) ( c r) dt V r ρrc + pr Vrρrc pr (8) dc ( Qc + AU r ( r c) ) dt m c c pc State ariables are in this case concentrations c A, c B and temperatures of the reactant r and the cooling c. here could be theoretically four input ariables a olumetric flow rate of the reactant, q r, a heat remoal of the cooling, Q c, an input concentration c A0 and an input temperature of the reactant, r0. he last two are only theoretical and could not be used as an input ariable from the practical point of iew. he scheme of this chemical reactor is in Figure. Figure : Continuous Stirred-tank Reactor (CSR) with Van der Vusse reaction inside Other ariables are supposed to be constant during the control because of the simplification. he olume of the reactor is denoted as V r, A r is the heat exchange surface, ρ r is used for the density of the reactant, U is the heat transfer coefficient, c pc and c pr are specific heat capacities of the cooling and the reactant a m c is the weight of the cooling mass. Values of these fixed parameters are in able (Chen et al., 995). able : Parameters of the reactor k min - k min - mol - E /R K h -400 kj.kmol - h kj.kmol - V r 0.0 m 3 c pr 3.0 kj.kg -.K - U 67. kj.min - m - K - c A0 5. kmol.m -3 r K k min - E /R K E 3 /R 8560 K h 000 kj.kmol - ρ r 934. kg.m -3 c pc.0 kj.kg -.K - A r 0.5 m c B0 0 kmol.m -3 m c 5 kg he steady-state analysis (Vojtesek and Dostal, 005) has shown that the optimal working point is in this case defined by the olumetric flow rate of the reactant q r s m 3.min - and heat remoal of the coolant Q c s kj.min -.

5 he input ariable for the dynamic study was the change of the heat remoal of the coolant, ΔQ c, and the output ariable was the change of reactant s temperature, r, s Qc t Qc s u() t 00 [%], y s () t () t [ K] (9) Q c he dynamic behaior was obsered for arious step changes of the input ariable from the range Δu(t) <-00%; +00%> and results are in Figure 3. y(t) [K] % -75% -50% -5% 5% 50% 75% 00% Figure 3: Results of the dynamic analysis for the arious changes of the input ariable Δu(t) It was already mentioned, that the dynamic analysis could help us with the choice of the ELM. It can be seen, that resulted step responses could be approximated by the second order F with relatie order one. he F () is then b s bs+ b G s 0 a s s + as + a0 (30) As the ELM (30) is of the second order, the F of the controller for both identification methods are according to (0) and () Q s qs + qs q ; R( s) r s ps+ p s ps+ p 0 0 (3) and the stable polynomial d(s) on the right side of () is of the fourth degree, i.e. d( s) n( s) ( s+ α ) (3) where n(s) comes from the Spectral factorization of (6) Finally, we hae one tuning parameter the position of the root α. All simulations hae same parameters the sampling period was 0.3 min, the initial coariance matrix P(0) has on the diagonal 0 6 and starting ectors of parameters for the identification was chosen ˆ θ ( 0) ˆ ( 0) [ 0., 0., 0., 0.] C θ. he simulation took 750 min and there were done 5 changes of the reference signal w(t). Our preious experiments hae shown that we can obtain better control results if the first change of the reference signal is exponential function instead of the step function. he input signal u(t) was limited to the alues u(t) <-75%; +75%> due to physical limitations. Control with C Identification Model he first simulation experiment was done for C identification model. he degree of the polynomial c(s) was chosen as deg c(s) deg a(s) and c s s + cs+ c0 s +.4s (33) Filtered input and output ariables are then ( ) ( ) f f 0 f ( ) ( ) f f 0 f y () t + c y () t + c y () t y() t u () t + cu () t + c u () t u() t (34) he ector of parameters and the regression ector are for ELM (30) ϕ () θ ( t ) [ a, a, b, b ] () t y t, y t, u t, u t C k f k f k f k f k C where parameters ( t ) C k k 0 0 (35) θ are estimated recursiely by RLS method with constant exponential forgetting described in the theoretical part. here were done three simulation studies for different α and results are shown in Figure 4 and Figure 5. w(t), y(t) [K] w, y (α 0.05), y (α 0.08), y (α 0.4) Figure 4: Courses of the reference signal, w(t), and the output ariable, y(t), for arious alues of the parameter α, results for the C identification model Figure 4 clearly shows that increasing alue of the parameter α affects mainly the speed of the output response an increasing alue of α produces quicker output response. It is worth to notice, that the change of the reference signal from the positie to the negatie alue causes problems for smaller alues of α. he output response then do not reach the reference signal. On the other hand, the control with the biggest alue of α 0.4 produces ery good results also with this negatie step changes of the reference signal. Figure 5 shows that the controller computes also ery smooth course of the action alue, u(t), what is also important from the practical point of iew. he action signal is represented by some action of the actuators and

6 quick changes of the input ariable could affect the lifetime of them. u(t)[%] u (α 0.05), u (α 0.08), u (α 0.4) Figure 5: he course of the input ariable, u(t), for arious alues of the parameter α, results for the C identification model Control with Delta-model Identification he second approach uses delta-models for identification which means that ector of parameters and data ector are [ ] θ [ a, a, b, b ] ϕ ( k ) y ( k ), y ( k ), u ( k ), u ( k ) 0 0 where -alues of the input and the output ariables are Parameters of ˆ ( k ) yk yk ( ) + yk ( ) y yk ( ) yk ( ) y ( k ) y ( k ) y( k ) uk ( ) uk ( ) u ( k ) u ( k ) u( k ) (37) θ are again estimated recursiely by the RLS method with constant exponential forgetting. w(t), y(t) [K] w, y (α 0.0), y (α 0.05), y (α 0.4) Figure 6: Courses of the reference signal, w(t), and the output ariable, y(t), for arious alues of the parameter α, results for the delta identification model (36) here were done simulation experiments for the same alues of the parameter α 0.05, 0.08 and 0.4 and the same changes of the reference signal as in preious case due to comparability. Results are shown in Figure 6 and Figure 7. u(t)[%] u (α 0.05), u (α 0.08), u (α 0.4) Figure 7: he course of the input ariable, u(t), for arious alues of the parameter α, results for the delta identification model It can be seen that the use of delta-models for the identification can also produce good control results. he effect of α is the same as in preious case, i.e. quicker output response can be obtained for bigger alues of α. he output response in this case hae a ery small oershoots for the biggest alue of α 0.4 but does not hae problem with negatie changes of the reference signal compared with the C model. he course of the input ariable, u(t), is ery similar to the preious case. We can see only some problems at the ery beginning of the control which is typical for the adaptie controll that starts from the general ector of parameters ˆ θ ( 0). It takes some time to approach to right alues, but once they are reached, results are good. he quality of the control for both control techniques was ealuated by the control quality criteria S u that displays how big are changes of the input ariable u(t) and the output criteria S y that sums the square of the control error e w y, i.e. u N ( () ( ) ) [ ] S u i u i i N ( () ()) i Sy w i y i K f for N (38) where f is final time which is in this case f 450 min. Values of these quality criteria was computed for each simulation study and results are shown in able. able : Values of quality criteria S u and S y C model Delta model S u [-] S y [K ] S u [-] S y [K ] α α α

7 he choice of the optimal alue of the tuning parameter α in both strategies depends what is important for us from the control point of iew. able shows that if the output ariable is more important, the bigger alue of α is better. his can be also clearly seen from graphs in Figure 4 and Figure 6. Oppositely, if we want the less changes of the input ariable, the control with lower alue of α is good choice. As all results in this paper comes from the simulation it is worth to mention the computation requirements. he simulation of the control with the delta identification model takes in Matlab about 0 seconds. On the other hand, the C identification model is more computationally demanding and the simulation for the same parameters took.5 minutes. As a result, computation with C model is nine times more demanding than control with delta identification model. In fact it is not big problem because the sampling period was 0.3 min, which is 0 seconds that is enough time for the identification and the computation of new parameters of the controller een for C model. CONCLUSIONS he goal of this paper was to show two on-line recursie identification methods used in the adaptie control. he first one is continuous-time identification that uses differential filters. his method is more computationally demanding but offers more accurate results. he next identification method is based on delta-models that are special types of D models where input and output ariables are related to the sampling period which could shift parameters of the delta-model close to parameters of the C model. As a result, this method is quicker and the output responses are ery close to those from the C model. Used adaptie approach uses polynomial approach with the Poleplacement method and the Spectral factorization that satisfies basic control requirements. Moreoer, this adaptie controller could be tuned by the choice of the position of the root in the Pole-placement method and the main effect is in the speed of the control. All approaches were tested on the mathematical model of the CSR as a typical member of the nonlinear systems with lumped parameters. he future work will head to the erification of simulated results on the real model of this or similar system. REFERENCES Åström, K. J., B. Wittenmark 989. Adaptie Control. Addison Wesley. Reading. MA. Bobal, V.; J. Böhm; J. Fessl; J. Machacek Digital Selftuning Controllers: Algorithms. Implementation and Applications. Adanced extbooks in Control and Signal Processing. Springer-Verlag London Limited. 005, ISBN Dostal, P.; V. Bobal; M. Blaha, M. 00. One Approach to Adaptie Control of Nonlinear Processes. In: Proc. IFAC Workshop on Adaptation and Learning in Control and Signal Processing ALCOSP 00, Cernobbio-Como, Italy, 00. p Fikar, M.; J. Mikles 999. System Identification. 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Control-heory and Adanced echnology. ol Vojtesek, J.; P. Dostal, 005. From steady-state and dynamic analysis to adaptie control of the CSR reactor. In: Proc. of 9th European Conference on Modelling and Simulation ECMS 005. Riga, Latia, p Vojtesek, J.; P. Dostal. 0. wo ypes of External Linear Models Used for Adaptie Control of Continuous Stirred ank Reactor. In: Proceedings 5th European Conference on Modelling and Simulation ECMS 0. Nicosia: p ISBN Wahlberg, B he Effects of Rapid Sampling in System Identification, Automatica, ol. 6, AUHOR BIOGRAPHIES JIRI VOJESEK was born in Zlin. Czech Republic and studied at the omas Bata Uniersity in Zlin, where he got his master degree in chemical and process engineering in 00. He has finished his Ph.D. focused on Modern control methods for chemical reactors in 007 and become Associatie professor at the omas Bata Uniersity in Zlin in 05. His contact is ojtesek@fai.utb.cz. PER DOSAL studied at the echnical Uniersity of Pardubice. He obtained his PhD. degree in echnical Cybernetics in 979 and he became professor in Process Control in 000. His research interest are modelling and simulation of continuous-time chemical processes. polynomial methods. optimal. adaptie and robust control. You can contact him on address dostalp@fai.utb.cz.