Unified Approach to Analog and Digital Two-Degree-of-Freedom PI Controller Tuning for Integrating Plants with Time Delay

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1 Acta Montanistica Slovaca Ročník 6 0), číslo, Unified Approach to Analog and Digital wo-degree-of-freedom PI Controller uning for Integrating Plants with ime Delay Miluse Viteckova, Antonin Vitecek, Marek Babiuch 3 he article describes the unified approach to the tuning of the analog and digital PI controller with two-degree-of-freedom by the multiple dominant pole method for integrating plants with a time delay on the basis of D-transform he approach is fully analytical and it enables so tuning that the servo and regulatory responses are non-oscillatory without overshoots he uses are shown in the example Keywords: DOF controller, PI controller, integrating plant, time delay Introduction he tuning of the PI controller for integrating plants with a time delay belongs to demanding problems of the control theory and practice It is caused by the type of the control system q = Predisposition to instability and presence of big overshoots are main problems [,, 3, 4] he integrating plus time delay IPD) plant with the L-transfer function G P s) = k s e ds ) is considered, where: k - the plant gain, d - the time delay, s - the complex variable in the L-transform Further the D-transform is used It is defined by the relations Xγ) = D{xk )} = k=0 xk ) = D {Xγ)} = Xγ) γ + ) k dγ π j C xk ) γ + ) k ) where: Xγ) - the D-transform of the original xk ), D and D - the direct and inverse D-transform operators, γ - the complex variable in the D-transform, k - the relative discrete time k = 0,,,), - the sampling period he closed integration path C lays within convergence domain of the transform Xγ) and contains its all singular points he conditions for the original are the same like for Z-transform More detailed information can be found in publications [5, 6] he simple transfer relations among complex variables, transforms and transfer functions hold: complex variables s = lim γ, z = γ + 3) 0 transforms of variables transfer functions Xs) = lim 0 Xγ), Gs) = lim 0 Gγ), Xz) = Xγ) γ= z Gz) = Gγ) γ= z prof Ing Miluse Viteckova, CSc, VSB - echnical University of Ostrava, Czech Republic, Department of Control Systems and Instrumentation, miluseviteckova@vsbcz prof Ing Antonin Vitecek, CSc, Drhc, VSB - echnical University of Ostrava, Czech Republic, Department of Control Systems and Instrumentation, antoninvitecek@vsbcz 3 Ing Marek Babiuch, PhD, VSB - echnical University of Ostrava, Czech Republic, Department of Control Systems and Instrumentation, marekbabiuch@vsbcz Review and revised version ) 4) 5) 89

2 Miluse Viteckova et al: Unified Approach to Analog and Digital wo-degree-of-freedom PI Controller uning for Integrating Plants with ime Delay where: Xs) - the L-transform of original xt), Xz) - the Z-transform of original xk ), Gs) - the L-transfer function, Gz) - the Z-transfer function, Gγ) - the D-transfer function, t - continuous time, z - the complex variable in the Z-transform In the case of the D/A converter in a control system with a digital controller which corresponds to a zero-order hold is used the plant D-transfer function is given by the formula [5, 6] G P γ) = γ { { } } γ + D L GP s) 6) s t=k he D-transfer function of the plant ) in accordance with 6) is given by relation where the relative discrete time delay G P γ) = k γ γ + ) d 7) d = d meanwhile the integer number is supposed he control system with the conventional PI controller with one-degree-of-freedom) is shown in Fig, where: W - the desired variable transform, Y - the controlled variable transform, V - the disturbance variable transform, E - the error transform, G C - the controller transfer function, G P - the plant transfer function Fig : Control system with conventional controller he error D-transfer function for the conventional PI controller G C γ) = K P + γ + ) I γ 8) and the IPD plant 7) is given by the formula where: K P - the controller gain, I - the integral time For the unit step the linear control area is given by the relation [5, 6] G w eγ) = Eγ) W γ) = I γ I γ + k K P [ I + )γ + ] γ + ) d 9) k=0 W γ) = γ + γ ek ) = lim γ 0 Eγ) = lim γ 0 [G w eγ)w γ)] = 0 0) he relation 0) has very important interpretation It expresses the fact, that the areas under and over the half line determined by the desired value are equal, ie one overshoot appears in the servo response at least herefore it is impossible to tune the conventional PI controller so that the servo response is without the overshoot If the servo response without the overshoot is desired, then the use of the PI controller with two-degree-offreedom DOF) is elegant solution [3, 4] 90

3 Acta Montanistica Slovaca Ročník 6 0), číslo, wo-degree-of-freedom Controller uning he DOF PI controller in the γ-complex variable domain can be described by the relation { Uγ) = K P bwγ) Y γ) + γ + } [Wγ) Y γ)] I γ which can be modify in the form Fig ) ) Uγ) = G F γ)g C γ)wγ) G C γ)y γ) ) G F γ) = b I + )γ + I + )γ + where: G C γ) - the conventional PI controller D-transfer function 8), G F γ) - the input filter D-transfer function, b - the set-point weight for proportional term 0 b ) For b = the relation ) describes the conventional PI controller 8), G F = and W = W 3) Fig : Control system with DOF controller From Fig it is obvious that the conventional PI controller 8) must be tuned from the point of view of the desired regulatory response and the input filter 3) weight b must be chosen from the point of view of the desired servo response [3, 4] In the article for both cases the multiple dominant pole method is used his method supposes that the nondominant poles and zeros have an insignificant influence on the control process For more detail, see [7, 8] For the conventional PI controller the characteristic polynomial of the control system in Fig has the form [see the denominator of the D-transfer function 9)] Nγ) = I γ γ + ) d + k K P [ I + )γ + ] 4) he triple dominant pole γ3 the three equations It is obtained and the adjustable controller parameters I and KP can be obtained by solution of d i Nγ) dγ i = 0 for i = 0,, 5) ) γ3 d ) = d + ) d + 6) K P ) = k [d + ) γ 3 + ]γ 3 γ 3 + ) d 7) I ) = d + ) γ3 ) + d + 3) γ3 + [d + ) γ3 + 8) ]γ 3 For the triple dominant pole 6) computation the positive + sign must be considered For the computed adjustable controller parameters 7) and 8) the control system D-transfer function can be approximated by the dominant pole 6) G w yγ) = Y γ) W γ) I + )γ + ) 3 γ + ) γ3 γ + d 9) 9

4 Miluse Viteckova et al: Unified Approach to Analog and Digital wo-degree-of-freedom PI Controller uning for Integrating Plants with ime Delay he big overshoot in the servo response is caused by the numerator of the D-transfer function 9) and therefore it is suitable to compensate for it he control system D-transfer function for the DOF PI controller in accordance with Fig and the relations ), 3) and 9) has the form G wy γ) = Y s) Ws) = G Fγ)G w yγ) b I + )γ + I + )γ + d I ) + )γ + 3 γ + ) γ3 γ + 0) From the relation 0) follows that the input filter denominator compensates for the numerator of the control system D-transfer function It causes the overshoot removing and simultaneously the servo response slowdown By the suitable choice of the set-point weight b it is possible by the input filter numerator to compensate for one binomial of the denominator of the control system D-transfer function, ie bi + )γ + = γ + b ) = γ 3 I γ 3 he dynamics of the whole control system will be simpler and the servo response accelerates in this case he relations 6) - 8) and ) directly hold for the conventional digital PI controller with the Z-transfer function G C z) = K P + ) z ) I z and the input filter with the Z-transfer function ) ) G F z) = b I + )z b I I + )z I 3) he corresponding relations for the conventional analog PI controller with the L-transfer function G C s) = K P + ) I s 4) and the input filter with the L-transfer function G F z) = b Is + I s + 5) can be obtained for the limit transition 0 It is obtained s 3 = lim K P = lim 0 K P ) = I γ3 ) = 6) 0 d ) k d e = 046 7) k d = lim I ) = 3 + ) d = 588d 0 8) b = lim b ) = = ) he relations 6) - 8) and ) for the digital DOF PI controller have unpleasant forms for practical use hey can be fundamentally simplified but it is somewhat inaccurate For the sufficient small sampling period the values for the digital controller can be approximated by the values for the corresponding analog controller if the time delay is increased to the half of the sampling period [] On the basis of the relations 7) - 9) it can be obtained KP ) k d + )e 046 = k d + ) 30) I 3 + ) d + ) = 588 d + ) 3) 9

5 Acta Montanistica Slovaca Ročník 6 0), číslo, For d d ) 3) the approximation of the controller gain 30) gives the error less than 3 % and approximation of the integral time 3) gives the error less than 5 % he set-point weight 9) can be approximated by the constant value 9) for d 4 with the error less than 5 % or for 3) by the approximate relation d 4) 33) d d b + ) d + ) = d ) which gives the error less than % he simplified relations are universal, for > 0 they hold for the digital DOF PI controller and for = 0 hold for the analog DOF PI controller It can see as well that the supposition that d must be an integer number is not substantial Example It is necessary to tune the analog and digital DOF PI controller for the IPD plant with the transfer function G P s) = 005 e 5s s so that the servo and regulatory step responses were non-oscillatory without overshoots the time delay is in seconds) Fig 3: Responses of control system with analog and digital conventional b = ) and DOF PI controller tuned by the multiple dominant pole method For the plant parameters k = 005 s and d = 5 s, the sampling period = 0 and = 5 s there can be obtained: Analog DOF PI controller he relation 7) - 9) or 30), 3) and 34) for = 0: K P = 845, I = 94, b = 093 he responses of the control system with the analog DOF PI controller for b = 093 and b = conventional) are shown in Fig 3 Digital DOF PI controller he accurate relations 6) - 8) and ) for = 5 = d = : γ 3 = 00845, K P = 437, I = 348, b =

6 Miluse Viteckova et al: Unified Approach to Analog and Digital wo-degree-of-freedom PI Controller uning for Integrating Plants with ime Delay Fig 4: Responses of control system with digital DOF PI controller tuned by the multiple dominant pole method he approximate relations 30), 3) and 34) for = 5: KP = 476, I = 3648, b = 066 he responses of the control system with the digital DOF PI controller for the accurate values b = 068 and b = conventional) are shown in Fig 3 and for the accurate and approximate values in Fig 4 he high value of the sampling period = 5 s d = ) was specially chosen in order to show that the approximate relations 30), 3) and 34) are quite satisfactory for practical use Conclussion he paper describes the unified approach to analog and digital DOF PI controller tuning by the multiple dominant pole method for the integrating plant with a time delay he approach is fully derived It ensures the servo and regulatory responses non-oscillatory without overshoot he described tuning method is relatively robust and very simple and therefore there are good assumptions for its practical use Acknowledgement his work was supported by research project GACR No 0/09/0894 References [] V Vasek, K Kolomaznik, and D Janacova, Optimization and automatic control of chromium recycling technology Proc 5th WSEAS Int Conf on Simulation, modeling and optimization Corfu, Greece, August 7-9, pp , 005 [] M Viteckova and A Vitecek, Basis of Automatic Control in Czech) VSB - U Ostrava, 008 [3] M Viteckova and A Vitecek, wo-degree of Freedom Controller uning for Integral Plus ime Delay Plants ICIC Express Letters Volume, Number 3, September 008, pp 5-9, 008 [4] M Viteckova and A Vitecek, DOF PI and PID controllers tuning Proc 9th IFAC Workshop on ime Delay Systems DS), 6 pp on CD), Prague, Czech Republic, June 7-9, 00 [5] A Feuer and G C Goodwin, Sampling in Digital Signal Processing and Control Birkhauser, Boston, 996 [6] R H Middleton and G C Goodwin, Digital Control and Estimation A Unified Approach Prentice-Hall, Englewood Cliffs, 990 [7] H Gorecki, S Fuksa, P Grabowski, and A Korytowski, Analysis and Synthesis of ime Delay Systems Polish Scientific Publishers - John Wiley & Sons, Warszawa-Chichester, 989 [8] M Viteckova and A Vitecek, New dominant poles controller tuning method for proportional non-oscillatory plants with time delay Proceedings of the Workshop on Intelligent Mining Systems KYUSHU University, pp 03-08, Fukuoka, 00 94