Smith Predictor Based Autotuners for Time-dela Sstems ROMAN PROKOP, JIŘÍ KORBEL, RADEK MATUŠŮ Facult of Applied Informatics Tomas Bata Universit in Zlín Nám. TGM 5555, 76 Zlín CZECH REPUBLIC prokop@fai.utb.cz http://www.fai.utb.cz Abstract: This paper presents a set of autotuners for single input-output sstems with time dela. Autotuners represent a combination of rela feedback identification and some control design method. In this contribution, models with up to three parameters are estimated b means of a single asmmetrical rela experiment. Then a stable low order transfer function with a time dela term is identified. Controller parameters are analticall derived from a general solution of Diophantine equations in the ring of proper and stable rational functions R PS. This approach covers a generalization of PID controllers and it enables to define a scalar positive parameter for further tuning of the control performance. The Smith predictor scheme is applied for sstems with a time dela term. The simulations are performed in the Matlab environment and a toolbox for automatic design and simulation was developed. Ke-Words: Algebraic control design, Rela experiment, Autotuning, Pole-placement problem, Smith predictor Introduction The development of various autotuning principles was started b a simple smmetrical rela feedback experiment proposed b Åström and Hägglund in [] in 984. The ultimate gain and ultimate frequenc are then used for adjusting of parameters b original Ziegler-Nichols rules. From that time, man studies have been reported to extend and improve autotuners principles; see e.g. [] - [4], [9], []. Over time, the direct estimation of transfer function parameters instead of critical values began to appear. The extension in rela utilization was performed in e.g. [8] - [] b an asmmetr and hsteresis of a rela. Nowadas, almost all commercial industrial PID controllers provide the feature of autotuning. In this paper, a novel combination for autotunig method of PI and PID like controllers is proposed and developed. The basic autotuning principle combines an asmmetrical rela identification experiment and a control design performed in the ring of proper and stable rational functions R PS. The factorization approach proposed in [] was generalized to a wide spectrum of control problems in [], [5] - [7]. The pole placement problem in R PS ring is formulated through a Diophantine equation and the pole is analticall tuned according to the required response of the closed loop. Naturall, there exist man principles of control design sntheses which can be used for autotuning principles, e.g. [], [8], [9], [8]. This contribution deals with two simplest SISO linear dnamic sstems with a dela term. The first model of the first order (stable) plus dead time (FOPDT) is supposed in the form: K Gs () = e Ts + Θs () Similarl, the second order model plus dead time (SOPDT) is assumed in the form: K Gs () = e ( Ts + ) Θs () The contribution is organized as follows. Section outlines a background of algebraic control design, see [5] - [9] for details. In section the principle of the Smith predictor is introduced. Section 4 presents some facts about rela identification for autotuning principles. The developed Matlab program environment for design and simulations is described in Section 5. Finall, section 6 presents simulation results in two examples of SISO sstems. Algebraic Control Design The control design is based on the fractional approach; see [], [], [5]. An transfer function G(s) of a (continuous-time) linear sstem is expressed as a ratio of two elements of R PS. The set R PS means the ring of (Hurwitz) stable and proper rational functions. Traditional transfer functions as a ratio of two polnomials can be easil transformed ISBN: 978--684-7- 9
into the fractional form simpl b dividing, both the polnomial denominator and numerator b the same stable polnomial of the appropriate order. Then all transfer functions can be expressed b the ratio: bs () n bs () ( s+ m) Bs () Gs () = = = as () as () As () n ( s+ m) () n= max(deg( a),deg( b)), m> (4) Then, all feedback stabilizing controllers for the feedback sstem depicted in Fig. are given b a general solution of the Diophantine equation: AP + BQ = (5) which can be expressed with Z free in R PS : Q Q AZ = P P + BZ In contrast of polnomial design, all controllers are proper and can be utilized. Fig. : One-degree of freedom (DOF) control loop The Diophantine equation for designing the feedforward controller depicted in Fig. is: (6) FS w + BR= (7) with parametric solution: R R FZ w = P P + BZ (8) F w = s ; m s+ m > (9) The similar conclusion is valid also for the load disturbance d = G d / F d. The load disturbance attenuation is then achieved b divisibilit of P b F d. More precisel, for tracking and attenuation in the closed loop according to Fig. the multiple of AP must be divisible b the least common multiple of denominators of all input signals. The divisibilit in R PS is defined through unstable zeros and it can be achieved b a suitable choice of a rational function Z in (6), see [], [5] for details. The derivation of controller parameters can be found in [7], [9] also with aperiodic tuning, similar adjusting is solved in [4]. Time dela sstems are studied in [8], []. Smith Predictors The Smith predictor was designed in the late 95s for sstems with time dela, see [6], deep insight into time dela sstems can be found in [5]. The basic classical interpretation of the Smith predictor is depicted in Fig.. The time dela term e -Ɵs has a negative influence to feedback stabilit which follows from the frequenc analsis. The feedback signal for the main controller C(s) in Fig. is a predicted value of the output. It means that the signal (t) inputs into the control error instead of the delaed (t-ɵ), it explains the name predictor. The Smith predictor launched the high development of Internal model controllers (IMC), where the plant model is present in the feedback loop, see [7]. When the transfer function G(s) is stable then the feedback sstem in Fig. is equivalent to the IMC version depicted in Fig. 4. Fig. : Two-degree of freedom (DOF) control loop Asmptotic tracking is then ensured b the divisibilit of the denominator P in (6) b the denominator of the w = G w / F w. The most frequent case is a stepwise with the denominator in the form: Fig. : Smith predictor classical version The main advantage of the Smith predictor is that the controller C(s) can be designed according to dela-free part G(s) of the plant. However, there are two main weak points in this sophisticated scheme. The first one is that the signal v(t) is zero onl in the case when the transfer function G(s) is the same in the outer and inner loops in Fig.. The second weakness is that the transfer function must be stable. In the case of autotuning, the approximated transfer ISBN: 978--684-7-
function of the plant can alwas be incorporated into the feedback. Fig. 4: Smith predictor IMC version 4 Rela Feedback Estimation The estimation of the process or ultimate parameters is a crucial point in all autotuning principles. The rela feedback test can utilize various tpes of rela for the parameter estimation procedure. The classical rela feedback test [] was proposed for stable processes b smmetrical rela without hsteresis and the scheme is depicted in Fig. 5. Following sustained oscillation are then used for determining the critical (ultimate) values. The control parameters (PI or PID) are then generated in standard manner. Fig. 5: Block diagram of an autotuning principle Asmmetrical relas with or without hsteresis bring furtherr progress [], [8], [9], []. After the rela feedback test, the estimation of process parameters can be performed. A tpical data response of such experiment is depicted in Fig. 6. The rela asmmetr is required for the process gain estimation ( ) while a smmetrical rela would cause the zero division in the appropriate formula. In this paper, an asmmetrical rela with hsteresis is used. This rela test enables to estimate transfer function parameters as well as a time dela term. For the purpose of the aperiodic tuning the time dela is not exploited. The process gain can be computed b the relation (see []): T K = td () dt T u() t dt ( ) where T is a chosen suitable time for at least ten oscillations in the rela experiment. For the first order model (), the time constant and time dela term are given b []: T 6 K u T = π π a () T Θ = πt ε π arctg arctg π T a ε where a and T are depicted in Fig. 6 and ε is the hsteresis. In the case of second order model (), the gain is given b (), the time constant and time dela term can be estimated according to [] b the relation: T T = π T Θ πt ε = π arctg arctg π T a ε 5 Fig. 6: Asmmetrical rela oscillation 4 K u π a Simulation and Program Sstem () A Matlab program sstem was developed for engineering applications of auto-tuning principles. The estimated model is of a first or second order transfer function with time dela while the controlled sstem is of arbitrar order. The user can choose three cases for the time dela term. In the first case the time term is neglected, in the second one the term is approximated b the Padé expansion and the third case utilizes the Smith predictor control structure. The Main menu window of the program sstem can be seen in Fig. 7. In the first phase of the program routine, the controlled transfer function is defined and parameters for the rela experiment can be adjusted. Then, the rela experiment is performed and an ISBN: 978--684-7-
estimated transfer function in the form of () or () is identified. Then controller parameters are generated after pushing of the appropriate button. The second phase begins with the Design controller parameters button and the chosen control design is performed and the controller is derived and displaed. The control scheme depends on the choice for the DOF or DOF structure and on the choice of the treatment with the time dela term. During the third phase, after pushing the Start simulation button, the simulation routine is performed and required outputs are displaed. Various simulation parameters can be specified in the Simulink environment. In alll simulation a change of the step is performed in the second third of the simulation horizon and a step change in the load is injected in the last third. A tpical control loop of the case with the Smith predictor in Simulink is depicted in Fig. 8. Fig. 8: Control loop in Simulink (Smith predictor) 6 Examples and Simulations Example : The example represents a fifth order sstem with a time dela term with transfer function G() s = (s + ) 5 Fig. 7: Main Menu 5s e ( ) The first and second order estimation results in the following transfer functions: G.99.5s ( s) = e 5.88s + G.99 ( s) =.9s + 6.69s + Then controllers were designed for the identified models (4) with neglected time dela terms. The PI controller was derived for the value of m =. and the PID one was derived for m =.. Both controllers in the DOF structure have the transfer functions:.7s +. C () s = s (5).4s +.s +.. C () s =.5s + s The control responses for the first order approximation and design are depictedd in Fig. 9. In this case the difference of responses between neglected time dela term and with the use of the Smith predictor is remarkabl strong. While the standard feedback control response is quite poor and oscillating then the response with Smith predictor in the loop is smooth and aperiodic. However, the Padé approximation gives also acceptable behaviour. w,.5.5.5 5 5 e 8.49s 5 5 4 45 5 time dela neglected Pade approximation with Smith predictor Fig. 9: DOF - first order (4) Almost the same situation is showed in Fig. where the second order approximation and snthesis were utilized. However, comparison of Fig. 9 and Fig. shows that the first order snthesis is sufficient and the second order is redundant. ISBN: 978--684-7-
w, Fig. : DOF - second order Example : This example represents a case of higher order sstem without dela approximated b a law order sstem with a time dela term. A higher order sstem (8 th order) with transfer function G(s) is supposed: Gs () = ( s + ).5.5.5 time dela neglected Pade approximation with Smith predictor 5 5 5 5 4 45 5 8 (6) Again, after the rela experiment, a first order and second estimation gives the following transfer functions:.96 4.96s Gs () = e 4.s +.96 Gs () = e s + s+ 4.8 4.4 4s (7) The step responses of sstems (6) and (7) are compared in Fig.. Step Response estimated sstems is considerable, it can be expected that not all values of and some of m> represent acceptable behaviour. With respect of aperiodic tuning in [9], three responses are shown in Fig.. Generall, larger values of m> implicate larger overshoots and oscillations. As a consequence, for inaccurate rela identifications, lower values of m> can be recommended. The PI controller for m =.8 gives the transfer function:.7s +.5 Cs () = (8) s w,.5.5 m =.9.5 m =.78 m =.7 m =.9 with Smith pred. 5 5 5 Fig. : DOF first order The control responses for (6) and (8) with and without the Smith predictor are shown in Fig...5.5.5 w, Amplitude.5 Controlled sstem.5 First order identification Second order identification 5 5 5 5 Time (seconds) Fig. : Step responses of sstems Naturall, both step responses of the estimated sstems are quite different from the original sstem G(s). PI controllers are generated from (5) and the tuning parameter m> can influence the control behaviour. Since the difference of controlled and Fig. : DOF second order The second order identification and snthesis of example for m =.4 gives the PID controller: Cs () =.5 time dela neglected with Smith predictor 4 6 8 4 6 8 + +.s + s.8s.s.5 (9) The control responses of the combination (6), (9) are depicted in Fig.. ISBN: 978--684-7-
7 Conclusion This contribution gives some rules for autotuning principles with a combination of rela feedback identification and a control design method. The estimation of a low order transfer function parameters is performed from asmmetric limit ccle data. The control snthesis is carried out through the solution of a linear Diophantine equation according to [], [5], [7], [9]. This approach brings a scalar tuning parameter which can be adjusted b various strategies. A first order estimated model generates PI-like controllers while a second order model generates a class of PID ones. In both cases also the Smith predictor influence was compared with neglecting of time dela terms and/or the Padé approximation. The methodolog supported b developed Matlab program sstem is illustrated b two examples. The results of all simulations prove that the Smith predictor structure brings a significant improvement of the aperiodic responses. The price for the improvement is a more complex structure of the feedback control sstem. Acknowledgements This work was supported b the European Regional Development Fund under the project CEBIA-Tech No. CZ..5./../.89. References: [] K.J. Åström and T. Hägglund, Automatic tuning of simple regulators with specification on phase and amplitude margins. Automatica, Vol., 984, pp.645-65. [] Ch.Ch. Yu, Autotuning of PID Controllers. Springer, London, 999. [] K.J. Åström and T. Hägglund, PID Controllers: Theor, Design and Tuning. Research Triangle Park, NC: Instrumental Societ of America, 995. [4] A. O Dwer, Handbook of PI and PID controller tuning rules. London: Imperial College Press,. [5] Q. Zhohg. Robust Control of Time-dela sstems. Springer, London, 6. [6] O.J.M. Smith, Feedback Control Sstems. McGraw-Hill Book Compan Inc., 958. [7] K.J. Åström and R.M. Murra, Feedback Sstems. Research Triangle Park, NC: Instrumental Societ of America, 995. [8] C.C. Hang, K.J. Åström and Q.C. Wang, Rela feedback auto-tuning of process controllers a tutorial review, Journal of Process Control, Vol., No. 6,. [9] S. Majhi and D.P. Atherton, Autotuning and controller design for unstable time dela processes, In: Preprints of UKACC Conf on Control, 998, pp. 769-774. [] M. Morari and E. Zafiriou, Robust Process Control. Prentice Hall, New Jerse, 989. [] M. Vítečková, and A. Víteček, Plant identification b rela methods. In: Engineering the future (edited b L. Dudas). Scio, Rijeka,, pp. 4-56. [] M. Vidasagar, Control Sstem Snthesis: A Factorization Approach. MIT Press, Cambridge, M.A., 987. [] V. Kučera, Diophantine equations in control - A surve, Automatica, Vol. 9, No. 6, 99, pp. 6-75. [4] R. Gorez and P. Klán, Nonmodel-based explicit design relations for PID controllers, In: Preprints of IFAC Workshop PID,, pp. 4-46. [5] R. Prokop and J.P. Corriou, Design and analsis of simple robust controllers, Int. J. Control, Vol. 66, 997, pp. 95-9. [6] R. Prokop, J. Korbel and Z. Prokopová, Rela feedback autotuning A polnomial approach, In: Preprints of 5th IFAC World Congress,. [7] R. Prokop, Korbel, J. and Prokopová, Z., Rela based autotuning with algebraic control design, In: Preprints of the rd European Conf. on modelling and Simulation, Madrid, 9, pp. 5-56. [8] L. Pekař and R. Prokop, Non-dela depending stabilit of a time-dela sstem. In: Last Trends on Sstems, 4th WSEAS International Conference on Sstems, Corfu Island, Greece,, pp. 7-75. [9] R. Prokop, J. Korbel and O. Líška, A novel principle for rela-based autotuning. International Journal of Mathematical Models and Methods in Applied Science,, Vol. 5, No. 7, s. 8-88. [] L. Pekař, R. Prokop and R. Matušů. Stabilit conditions for a retarded quasipolnomial and their applications. International Journal of Mathematics and Computers in Simulations, Vol. 4, No.,, pp. 9-98 ISBN: 978--684-7- 4