A Novel Principle for Relay-Based Autotuning

Transcription

1 A Novel Principle for Relay-Based Autotuning ROMAN PROOP*, JIŘÍ ORBEL*, ONDREJ LÍŠA** *Faculty of Applied Inforatics, oas Bata University in Zlín Ná..G.Masaryka 5555, 76 Zlín, CZECH REPUBLIC **Faculty of Mechanical Engineering, echnical University of ošice, Letná 9, 4 ošice SLOVA REPUBLIC Abstract: - his paper presents a new siple principle for aperiodic tuning of SISO controllers used in autotuning schees. Autotuners represent a cobination of relay feedback identification and soe control design ethod. In this contribution, odels with up to three paraeters are estiated by eans of a single asyetrical relay experient. hen a stable low order transfer function is identified. Subsequently, the controller is analytically derived fro general solutions of Diophantine equations in the ring of proper and stable rational functions R PS. his approach enables to define a scalar positive paraeter through a poleplaceent root of the characteristic closed loop equation. A first order identification yields a PI-like controllers while a second order identification generates PID ones. he analytical siple rule is derived for aperiodic control response and the scalar tuning paraeter > is then tuned according to identified tie constant of an approxiated transfer function. ey-words: Autotuning, Relay experient, Algebraic control design, Pole-placeent proble. Introduction Industrial processes are usually controlled by PID controllers, Yu in [] refers that ore than 97 % of control loops are of this type and ost of the are actually under PI control. he practical advantages of PID controllers can be seen in a siple structure, in an understandable principle and in control capabilities. Moreover, PID controllers have survived changes in technology fro pneuatic principles through analog and digital representation to DCS ones. It is widely known that PID controllers are quite resistant to changes in the controlled process without eaningful deterioration of the loop behavior. For 7 years, the Ziegler Nichols tuning rule has been glorified and vilified as well. Nevertheless, the Ziegler Nichols rule stay reains the ost frequent ethod of PID tuning. However, there are any liitations, drawbacks and infirities in the behavior of the Ziegler Nichols setting. A solution for qualified choice of controller paraeters can be seen in ore sophisticated, proper and autoatic tuning of PID controllers. he developent of various autotuning principles was started by a siple syetrical relay feedback experient proposed by Åströ and Hägglund in [] in the year 984. he ultiate gain and ultiate frequency are then used for adjusting of paraeters by original Ziegler-Nichols rules. During the period of ore than two decades, any studies have been reported to extend and iprove autotuners principles; see e.g. [], [4], [8], [9]. he extension in relay utilization was perfored in [], [5], [7], [4] by an asyetry and hysteresis of a relay. Over tie, the direct estiation of transfer function paraeters instead of critical values began to appear. Experients with asyetrical and dead-zone relay feedback are reported in []. Also, various control design principles and rules can be investigated in entioned references. Nowadays, alost all coercial industrial PID controllers provide the feature of autotuning. In this paper, a new cobination for autotunig ethod of PI and PID controllers with an aperiodic control rule is proposed and developed. he basic autotuning principle cobines an asyetrical relay identification experient and a control design perfored in the ring of proper and stable rational functions R PS. he factorization approach proposed in [] was generalized to a wide spectru of control probles in [], [], [5] - []. A different philosophy is reported in []. he pole ISBN:

2 placeent proble in R PS ring is forulated through a Diophantine equation and the pole is analytically tuned according to aperiodic response of the closed loop. he proposed ethod is copared by an equalization setting proposed in []. A general basic schee of the autotuning principle can be seen in Fig.. Relay Feedback Estiation he estiation of the process or ultiate paraeters is a crucial point in all autotuning principles. he relay feedback test can utilize various types of relay for the paraeter estiation procedure. he classical relay feedback test [], was proposed for stable processes by syetrical relay without hysteresis. Following sustained oscillation are then used for deterining the critical (ultiate) values. he control paraeters (PI or PID) are then generated in standard anner. PID w e u y Process _ Relay Fig.: Block diagra of an autotuning principle Asyetrical relays with or without hysteresis bring further progress [], [4]. After the relay feedback test, the estiation of process paraeters can be perfored. A typical data response of such relay experient is depicted in Fig.. he relay asyetry is required for the process gain estiation () while a syetrical relay would cause the zero division in the appropriate forula. In this paper, an asyetrical relay with hysteresis is used. his relay enables to estiate transfer function paraeters as well as a tie delay ter. For the purpose of the aperiodic tuning the tie delay is not exploited. u(t) y(t) Fig.: Asyetrical relay oscillation of stable process y a y u he odel for first order (stable) systes plus dead tie (FOPD) is supposed in the for: Θs e () s+ and the process gain can be coputed by the relation (see []): iy iy y( t) dt ; i,,,... u( t) dt () he tie constant and tie delay ters are given by []: y 6 u π π ay () y π ε Θ π arctg arctg π y ay ε where a y and y are depicted in Fig. and ε is the hysteresis. Siilarly, the second order odel plus dead tie (SOPD) is assued in the for: e ( s+ ) Θs (4) he gain is given by (), the tie constant and tie delay ter can be estiated according to [] by the relation: y 4 u π π ay (5) y π ε Θ π arctg arctg π y ay ε Algebraic Control Design he control design is based on the fractional approach; see e.g. [], [], [5] - [7]. Any transfer function G(s) of a (continuous-tie) linear syste is expressed as a ratio of two eleents of R PS. he set R PS eans the ring of (Hurwitz) stable and proper rational functions. raditional transfer functions as a ratio of two polynoials can be easily transfored into the fractional for siply by dividing, both the polynoial denoinator and nuerator by the sae stable polynoial of the appropriate order. hen all transfer functions can be expressed by the ratio: b( s) n b( s) ( s+ B( s) (6) a( s) a( s) A( s) n ( s+ ISBN:

3 n ax(deg( a),deg( b)), > (7) hen, all feedback stabilizing controllers for the feedback syste depicted in Fig. are given by a general solution of the Diophantine equation: AP+ BQ (8) which can be expressed with Z free in R PS : Q Q AZ (9) P P + BZ In contrast of polynoial design, all controllers (9) are proper and can be utilized. Fig.: Feedback (DOF) control loop Asyptotic tracking is then ensured by the divisibility of the denoinator P in (9) by the Gw denoinator of the reference w. he ost F w frequent case is a stepwise reference with the denoinator in the for: s F w ; s+ > () he siilar conclusion is valid also for the load Gd disturbance d. he load disturbance Fd attenuation is then achieved by divisibility of P by F d. More precisely, for tracking and attenuation in the closed loop according to Fig. the ultiple of AP ust be divisible by the least coon ultiple of denoinators of all input signals. he divisibility in R PS is defined through unstable zeros and it can achieved by a suitable choice of rational function Z in (9), see [], [5] for details. 4 PI and PID-Like Controllers Diophantine equation (8) for the first order systes () without the tie delay ter can be easily transfored into polynoial equation: ( s+ ) p + q s+ () with general solution: P + Z s+ s+ Q Z s+ () where Z is free in the ring R PS. Asyptotic tracking is achieved by the choice: Z () and the resulting PI controller is in the for: Q qs+ q C( s) (4) P s where paraeters q a q are given by: q q (5) For the SOPD the design equation (8) takes the for: ( s+ ) s p + ( qs + qs+ q) ( s+ (6) and after siilar anipulations the resulting PID controller gives the transfer function: Q qs + qs+ q C( s) (7) P s( s+ p ) with paraeters: p ; q (8) q ; q For both systes FOPD and SOPD the scalar paraeter > sees to be a suitable tuning knob influencing control behavior as well as robustness properties of the closed loop syste. Naturally, both derived controllers correspond to classical PI and PID ones. It is clear that (4) represents the PI controller: u( t) P e( t) + e( τ ) dτ (9) I and the conversion of paraeters is trivial. Relation (7) represents a PID in the standard fourparaeter for []: u( t) P e( t) + e( ) d D y f ( t) τ τ + I () τ y ( t) + y ( t) y ( t) f 5 Aperiodic uning f Over the past 6 years after the introduction of Ziegler-Nichols rule in 94, veheent research activity in controller tuning has been perfored. More than 4 tuning rules are referred in [], ore than rules for PI controllers. A siple and attractive choice for the tuning paraeter > can be easily obtained analytically. In the R PS expression, the closed-loop transfer function wy is for () and PI controller (4) given in a very siple for: ISBN:

4 wy BQ ( ) s+ BQ AP+ BQ ( s+ () he step response of () can be expressed by Laplace transfor: wy ks+ k h( t) L L s s( s+ () A B C L + +, s ( s+ s+ where A,B,C are calculated by coparing appropriate fractions in () and k -, k. he response h(t) in tie doain is then t t h( t) A+ Be + Cte (4) he overshoot or undershoot of this response is characterized by the first derivative condition t t t h ( t) Be + C( e te ) (5) Fro (5) tie of the extree of response h(t) is then easily calculated by the relation: C B B te (6) C C Since the aperiodic response eans that the extree does not exist for positive t e, it iplies t e < and after all substitutions of A,B,C, k, k relation (6) takes the siple for B < (7) C he denoinator of (7) ust be positive and less than and > which iplies the inequality: < < (8) Any positive paraeter fro (8) ensures aperiodic response. It is a question for further investigation and siulation what choice fro interval (8) is the best. For autotuning philosophy tie constant is always an estiation then the iddle value of (8) would be reasonable, it eans the choice 4 (9) Also another tuning principles for aperiodic tuning certainly exist. For the entioned algebraic synthesis, the equalization ethod developed by Gorez and lán in []. he idea goes out fro PI controller in the for (9). he tuning rule is very siple and it leads in relations:.4 () P I u where is a process gain and u is the ultiate period obtained fro the Ziegler-Nichols experient. However, the fulfillent of () by unique value of > is ipossible, see [6]. he exact fulfillent of both relations in () could be obtained in the case of two distinct roots in denoinator (), so (s+ )(s+ ) instead of (s+. 6 Exaples and Analysis he following exaples illustrate the situation where the estiated odel is always of the first order ones () without tie delay and the controller has a PI structure (4). Exaple : he first order syste governed by the transfer function G(s) was after the relay experient estiated by Gɶ ( s) in the for: Aplitude s+ G ɶ ( s).7s+ () Step Response Approx. syste ie (sec) Controlled syste Fig. 4 Step responses of noinal and estiated systes () he PI controller was then generated for three values of within the interval given by (8),.85;.78;.7, respectively. he responses are depicted in Fig.5. Reference and output value reference ie (s) Fig. 5 Control responses (Exaple ) his choice represents the lowest, iddle and upper liit values in derived interval (8). Responses in Fig. 5deonstrate that all ones have aperiodic behavior. Exaple : A second order (stable) syste G(s) without a tie delay was estiated by a first order odel in the above entioned relay experient. Both transfer functions have the for: ISBN:

5 ..5s G ɶ ( s) e (s+ ).89s+ () he above entioned relay experients enable to estiate the original syste by the first or second order transfer functions (), (4). Step responses without tie delay ters are depicted in Fig.6. he PI controller generated fro the approxiated ~ syste G ( s ) was designed by (7), (8)for three values of tuning paraeters > with respect the interval given by (8). he responses are shown in Fig.7. Also, the aperiodic responses are achieved. larger values of > iplicate larger overshoots and oscillations. As a consequence, for inaccurate relay identifications, lower values of > in interval (8) can be recoended. Aplitude.5.5 Step Response Aplitude Step Response Controlled syste Approx. syste w /o transport delay Approx. syste with transport delay ie (sec) Fig. 6 Step responses of systes ().5 Approx. syste with transport delay ie (sec) Controlled syste Approx. syste w /o transport delay Fig.: Step responses of systes (4) In all control siulations, the reference value is changed in / of the siulation horizon and the load disturbance is injected in the / of the siulation horizon. All siulations were perfored in Siulink environent. In the case of (), the best response is achieved for.4..5 Reference and output value reference ie (s) Fig. 7 Control responses (Exaple ) Exaple : A higher order syste (8 th order) with transfer function G(s) is supposed. Again, after ~ the relay experient, a first order estiation G ( s ) was identified, both governed by: 5.s G ɶ ( s) e 8 ( s+ ).5s+ () he step responses of systes () are shown in Fig. 8. Naturally, the step response of the estiated syste is quite different fro the noinal syste G(s). Again, PI controllers are generated fro (7), (8) and the tuning paraeter > can influence the control responses. Since the difference of controlled and estiated systes is considerable, it can be expected that not all values of and soe of > represent acceptable behavior. With respect of (8), three responses are shown in Fig.9. Generally, Reference and output value.5.5 reference ie (s) 7 Conclusions Fig.: Control responses his contribution gives a new cobination of relay feedback identification and a control design ethod. he estiation of a low order transfer function paraeters is perfored fro asyetric liit cycle data. he control synthesis is carried out through the solution of a linear Diophantine equation according to [], [5], [6]. his approach brings a scalar tuning paraeter which can be adjusted by various strategies. A first order estiated odel generates PI-like controllers while a second order odel generates a class of PID ones. he aperiodic tuning through the paraeter > is proposed by the analytic derivation. he ethodology is illustrated by several exaples of various orders and dynaics. ISBN:

6 Acknowlegeents his work was supported by the Ministry of Education, Youth and Sports of the Czech Republic under the Research Plan No. MSM 7885 and by the Europian Regional Developent Fund under the project CEBIA-ech No. CZ..5./.. /.89. References: [] Ch.Ch. Yu, Autotuning of PID Controllers. Springer, London, 999. [].J. Åströ and. Hägglund, Autoatic tuning of siple regulators with specification on phase and aplitude argins. Autoatica, Vol., 984, pp [].J. Åströ and. Hägglund, PID Controllers: heory, Design and uning. Research riangle Park, NC: Instruental Society of Aerica, 995. [4] R.F. Garcia and F.J.P. Castelo, A copleent to autotuning ethods on PID controllers, Iin: Preprints of IFAC Workshop PID, pp. - 4,. [5] R.R. Pecharroán and F.L. Pagola, Control design for PID controllers auto-tuning based on iproved identification, In: Preprints of IFAC Workshop PID, pp ,. [6] C.C.Hang,.J. Åströ and Q.C. Wang, Relay feedback auto-tuning of process controllers a tutorial review, Journal of Process Control, Vol., No6,. [7]. hyagarajan and Ch.Ch. Yu, Iproved autotuning using shape factor fro relay feedback, In: Preprints of IFAC World Congres,. [8] S. Majhi and D.P. Atherton, Autotuning and controller design for unstable tie delay processes, In: Preprints of UACC Conf an Control, 998, pp [9] F. Morilla, A. Gonzáles and N. Duro, Autotuning PID controllers in ters of relative daping, In: Preprints of IFAC Workshop PID,, pp [] M. Vítečková and A. Víteček Experientální identifikace etodou relé (in Czech), In: Autoatizácia a inforatizáci, 4. [] M. Vidyasagar, Control syste synthesis: a factorization approach. MI Press, Cabridge, M.A. (987). [] V. učera, Diophantine equations in control - A survey, Autoatica, Vol. 9, No. 6, 99, pp [] R. Gorez and P. lán, Nonodel-based explicit design relations for PID controllers, In: Preprints of IFAC Workshop PID,, pp [4] I. aya and D.P. Atherton, Paraeter estiation fro relay autotuning with asyetric liit cycle data, Journal of Process Control, Vol., No4,, pp [5] R. Prokop and J.P. Corriou, Design and analysis of siple robust controllers, Int. J. Control, Vol. 66, 997, pp [6] R. Prokop, J. orbel and Z. Prokopová, Relay feedback autotuning A polynoial approach, In: Preprints of 5th IFAC World Congress,. [7] R. Prokop, orbel, J., Prokopová, Z., Relay based autotuning with algebraic control design, In: Preprints of the rd European Conf. on odelling and Siulation, Madrid, 9, s [8] R Matušů and R. Prokop, Robust Stabilization of Interval Plants using ronecker Suation Method. In: Last rends on Systes, 4th WSEAS International Conference on Systes, Corfu Island, Greece,, pp [9] L. Pekař and R. Prokop, Non-delay depending stability of a tie-delay syste. In: Last rends on Systes, 4th WSEAS International Conference on Systes, Corfu Island, Greece,, pp [] L. Pekař and R. Prokop, Control of Delayed Integrating Processes Using wo Feedback Controllers: RMS Approach, In: Proceedings of the 7th WSEAS International Conference on Syste Science and Siulation in Engineering, Venice, 8, pp [] D. rokavec and A. Filasová, Pole assignent in robust state observer design. In: A&P Journal Plu,. Vol., č. (7), s [].Vyhlídal, Anisochronic first order odel and its application to internal odel control, In: Preprints of ASR Seinar. Prague,, pp. -. [] A. O Dwyer, Handbook of PI and PID controller tuning rules. London:Iperial College Press,. ISBN: