CA2370772A1 - Method of tuning a controller and a controller using the method - Google Patents
Method of tuning a controller and a controller using the method Download PDFInfo
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- CA2370772A1 CA2370772A1 CA 2370772 CA2370772A CA2370772A1 CA 2370772 A1 CA2370772 A1 CA 2370772A1 CA 2370772 CA2370772 CA 2370772 CA 2370772 A CA2370772 A CA 2370772A CA 2370772 A1 CA2370772 A1 CA 2370772A1
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- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B11/00—Automatic controllers
- G05B11/01—Automatic controllers electric
- G05B11/36—Automatic controllers electric with provision for obtaining particular characteristics, e.g. proportional, integral, differential
- G05B11/42—Automatic controllers electric with provision for obtaining particular characteristics, e.g. proportional, integral, differential for obtaining a characteristic which is both proportional and time-dependent, e.g. P.I., P.I.D.
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- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
- G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
- G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
- G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
Abstract
In tuning a controller for a process in a feedback control system, a method is provided for bringing the system into asymmetric self-excited oscillations for measuring the frequency of the oscillations, average over the period value of the process output signal and average over the period control signal and tuning the controller in dependence of the measurements obtained. An element having a non-linear characteristic is introduced into the system in series with the process and set point signal is applied to excite asymmetric self-excited oscillations in the system. An algorithm and formulas are given for identification of the process model having the form of first order plus dead time transfer function. PI controller settings are given as a function of the dead time /
time constant ratio. An apparatus for performing the method is disclosed.
time constant ratio. An apparatus for performing the method is disclosed.
Description
Descriution FIELD OF THE INVENTION
Despite the success of relay feedback system in autotune identification, it is well known that a relay based identification can lead to significant errors in the ultimate gain and ultimate frequency. The errors come from the linear approximation (describing function method) to a nonlinear element. The square type of output from the relay is approximated with the principal harmonic from the Fourier series (Derek P. Atherton, "Nonlinear Control Engineering", Van Nostrand Reinhod: Nev York, 1982) and the ultimate gain is estimated accordingly. Several attempts were proposed to overcome this inaccuracy but didn't overcome the main source of inaccuracy - linear approximation of the relay element due to the use of describing function method model. The present invention completely eliminates this source of inaccuracy - on account of application of a precise model of the oscillatory process - via the use of the locus of a perturbed relay system (LPRS) method (Igor Boiko, "Input-output analysis of limit cycling relay feedback control systems," Proc. of 1999 American Control Conference, San Diego, USA, Omnipress, 1999, pp.542-546; Igor Boiko;
"Application of the locus of a perturbed relay system to sliding mode relay control design," Proc. of 2000 IEEE International Conference on Control Applications, Anchorage, AK, USA, 2000, pp.
542-547; Igor Boiko, "Frequency Domain Approach to Analysis of Chattering and Disturbance Rejection in Sliding Mode Control," Proc. of World Multiconference on Systemics, Cybernetics and Informatics, Orlando, Florida, USA, Vol. XV, Part II, pp. 299-303) instead of describing function method. The present invention defines a method and an apparatus for bringing the system (comprising the process, nonlinear element and external source of constant set point signal) into asymmetric oscillations mode (further referred to as asymmetric oscillatory experiment) for determining (measuring) quantities essential for the tuning of the controller.
The invention includes all variations and combinations (P, PI, PD, PID, etc.) of the control types of PID controller but not limited to those types of controllers.
BACKGROUND OF THE INVENTION
Autotuning of PID controllers based on relay feedback tests received a lot of attention recently (W.
L. Luyben, "Derivation of Transfer Functions for Highly Nonlinear Distillation Columns", Ind.
Erg. Chem. Res. 26, 1987, pp.2490-2495; Tore Hagglund, Karl J. Astrom, "Industrial Adaptive Controllers Based on Frequency Response Techniques", Automatica 27, 1991, pp.599-609). It identifies the important dynamic information, ultimate gain and ultimate frequency, in a straightforward manner. The success of this type of autotuners lies on the fact that it is simple and reliable. The appealing feature of the relay feedback autotuning has lead to a number of commercial autotuners (Tore Hagglund, Karl J. Astrom, "Industrial Adaptive Controllers Based on Frequency Response Techniques", Automatica 27, 1991, pp.599-609) and industrial applications (H. S.
Papastathopoulou, W. L. Luyben, "Tuning Controllers on Distillation Columns with the Distillate-Bottoms Structure", Ind. Eng. Chem. Res. 29, 1990, pp.1859-1868).
Luyben (W. L. Luyben, "Derivation of Transfer Functions for Highly Nonlinear Distillation Columns", Ind. Eng. Chem. Res. 26, 1987, pp.2490-2495) pioneers the use of relay feedback tests for system identification. The ultimate gain and ultimate frequency from the relay feedback test are used to fit a typical transfer function (e.g., first-, second- or third order plus time delay system).
This identification procedure is called the ATV method. It was applied successfully to highly nonlinear process, e.g., high purity distillation column. Despite the apparent success of autotune identification, it can lead to signification errors in the ultimate gain and ultimate frequency approximation (e.g., 5-20% error in R. C. Chiang, S. H. Shen, C. C. Yu, "Derivation of Transfer Function from Relay Feedback Systems", Ind. Eng Chem. Res. 31, 1992, pp.855-860) for typical transfer functions in process control system.
The present invention completely eliminates the source of inaccuracy that comes from the linear approximation to the nonlinear element - via the use of the LPRS method instead of describing function method. The LPRS describes a relay system just like the transfer function describes a linear system. The present invention defines a method and an apparatus for bringing the system into asymmetric oscillations mode for measuring quantities essential for tuning a controller. More accurate description of the oscillations in the relay system allows for more precise identification of the parameters of the process model and a better quality of tuning a controller.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1. The system that comprises the nonlinear element, the process, and the source of the set point signal.
Despite the success of relay feedback system in autotune identification, it is well known that a relay based identification can lead to significant errors in the ultimate gain and ultimate frequency. The errors come from the linear approximation (describing function method) to a nonlinear element. The square type of output from the relay is approximated with the principal harmonic from the Fourier series (Derek P. Atherton, "Nonlinear Control Engineering", Van Nostrand Reinhod: Nev York, 1982) and the ultimate gain is estimated accordingly. Several attempts were proposed to overcome this inaccuracy but didn't overcome the main source of inaccuracy - linear approximation of the relay element due to the use of describing function method model. The present invention completely eliminates this source of inaccuracy - on account of application of a precise model of the oscillatory process - via the use of the locus of a perturbed relay system (LPRS) method (Igor Boiko, "Input-output analysis of limit cycling relay feedback control systems," Proc. of 1999 American Control Conference, San Diego, USA, Omnipress, 1999, pp.542-546; Igor Boiko;
"Application of the locus of a perturbed relay system to sliding mode relay control design," Proc. of 2000 IEEE International Conference on Control Applications, Anchorage, AK, USA, 2000, pp.
542-547; Igor Boiko, "Frequency Domain Approach to Analysis of Chattering and Disturbance Rejection in Sliding Mode Control," Proc. of World Multiconference on Systemics, Cybernetics and Informatics, Orlando, Florida, USA, Vol. XV, Part II, pp. 299-303) instead of describing function method. The present invention defines a method and an apparatus for bringing the system (comprising the process, nonlinear element and external source of constant set point signal) into asymmetric oscillations mode (further referred to as asymmetric oscillatory experiment) for determining (measuring) quantities essential for the tuning of the controller.
The invention includes all variations and combinations (P, PI, PD, PID, etc.) of the control types of PID controller but not limited to those types of controllers.
BACKGROUND OF THE INVENTION
Autotuning of PID controllers based on relay feedback tests received a lot of attention recently (W.
L. Luyben, "Derivation of Transfer Functions for Highly Nonlinear Distillation Columns", Ind.
Erg. Chem. Res. 26, 1987, pp.2490-2495; Tore Hagglund, Karl J. Astrom, "Industrial Adaptive Controllers Based on Frequency Response Techniques", Automatica 27, 1991, pp.599-609). It identifies the important dynamic information, ultimate gain and ultimate frequency, in a straightforward manner. The success of this type of autotuners lies on the fact that it is simple and reliable. The appealing feature of the relay feedback autotuning has lead to a number of commercial autotuners (Tore Hagglund, Karl J. Astrom, "Industrial Adaptive Controllers Based on Frequency Response Techniques", Automatica 27, 1991, pp.599-609) and industrial applications (H. S.
Papastathopoulou, W. L. Luyben, "Tuning Controllers on Distillation Columns with the Distillate-Bottoms Structure", Ind. Eng. Chem. Res. 29, 1990, pp.1859-1868).
Luyben (W. L. Luyben, "Derivation of Transfer Functions for Highly Nonlinear Distillation Columns", Ind. Eng. Chem. Res. 26, 1987, pp.2490-2495) pioneers the use of relay feedback tests for system identification. The ultimate gain and ultimate frequency from the relay feedback test are used to fit a typical transfer function (e.g., first-, second- or third order plus time delay system).
This identification procedure is called the ATV method. It was applied successfully to highly nonlinear process, e.g., high purity distillation column. Despite the apparent success of autotune identification, it can lead to signification errors in the ultimate gain and ultimate frequency approximation (e.g., 5-20% error in R. C. Chiang, S. H. Shen, C. C. Yu, "Derivation of Transfer Function from Relay Feedback Systems", Ind. Eng Chem. Res. 31, 1992, pp.855-860) for typical transfer functions in process control system.
The present invention completely eliminates the source of inaccuracy that comes from the linear approximation to the nonlinear element - via the use of the LPRS method instead of describing function method. The LPRS describes a relay system just like the transfer function describes a linear system. The present invention defines a method and an apparatus for bringing the system into asymmetric oscillations mode for measuring quantities essential for tuning a controller. More accurate description of the oscillations in the relay system allows for more precise identification of the parameters of the process model and a better quality of tuning a controller.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1. The system that comprises the nonlinear element, the process, and the source of the set point signal.
FIGS. 2A and 2B. Input-output relationship for symmetric hysteresis relay and asymmetric hysteresis relay.
FIG: 3. Block diagram of a relay feedback system.
FiG. 4. LPRS and determination of the frequency of oscillations.
FIG. 5. Block diagram of the controller and the process.
FIG. 6. Block diagram of the SIMULINK° model of the autotuning system.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
Refernng to the drawings, a description will be given of an embodiment of a controller autotuning method according to the present invention.
The PID-control stands for proportional, integrating and derivative control.
It is very common for controlling industrial processes. PID-controllers are manufactured by various manufacturers in large quantities. Usually the controllers are based on microprocessors and proportional, integrating and derivative functions are normally implemented within a software.
Nevertheless, the principal structure of a conventional PID-controller is retained and without loss of generality it is possible to consider a PID-controller as a parallel connection of three channels:
proportional with gain Kp, integrating with gain K; and derivative with gain Kd. As a result, transfer function of the PID-controller is:
Wpid~SJ- Kp + KT ~S -~ Kd S
Choice of gains Kp, KI and Kd values is a subject of tuning if the controller is implemented as a PID-controller. There are established methods of tuning a PID-controller in dependence on the parameters of the process, for example Ziegler-Nichols's method of manual tuning, Hagglund-Astrom's relay feedback autotuning method. There are also a number of other methods of manual and automatic tuning. All those methods can be divided into parametric and non-parametric.
Parametric methods are based on a certain dynamic model of the process with unknown parameters.
The process undergoes a test or a number of tests aimed at the process model parameters identification. Once the process model parameters are identified, the controller is tuned in accordance with known from the automatic control theory rules - to provide stability and required performance to the closed-loop system (comprising the process, the controller, the comparison device, and the feedback). Non-parametric methods are based on the tests on the process, which are aimed at the measurement of some general characteristics of the process, for example ultimate gain and ultimate frequency at both Ziegler-Nichols's method of manual tuning and Hagglund-Astrom's relay feedback autotuning method.
Generally, parametric methods can provide a better tuning quality (due to possibility of the use of more precise model of the process) but require more complex tests on the process. Therefore, there is a need for comparatively simple yet precise method of tuning (manual and automatic), which can be embedded into software of local controllers or distributed control system (DCS) or be implemented as a software for a personal computer used by an engineer who is supposed to tune the controllers.
Usually, methods of tuning that utilize Hagglund-Astrom's relay feedback test for estimating the parameters of oscillations are based on describing function method (Derek P.
Atherton, "Nonlinear Control Engineering", Van Nostrand Reinhod: New York, 1982). The use of this method is limited to harmonic oscillations in the system, which is normally not the case.
The present invention is based on the model of oscillations provided by the LPRS method (Igor Boiko, "Input-output analysis of limit cycling relay feedback control systems," Proc. of 1999 American Control Conference, San Diego, USA, Omnipress, 1999, pp.542-546; Igor Boiko, "Application of the locus of a perturbed relay system to sliding mode relay control design," Proc. of 2000 IEEE International Conference on Control Applications, Anchorage, AK, USA, 2000, pp.
542-547; Igor Boiko, "Frequency Domain Approach to Analysis of Chattering and Disturbance Rejection in Sliding Mode Control," Proc. of World Multiconfe.rence on Systemics, Cybernetics and Informatics, Orlando, Florida, USA, Vol. ~V, Part II, pp. 299-303), which doesn't use the above limiting hypothesis. The present invention provides a relatively simple parametric method of controller tuning. The method uses a modified Hagglund-Astrom's relay feedback test as means to identify parameters of a process model: A process can be modeled by a transfer function with a dead time (time delay) or without it or have a matrix state space description.
According to the present invention, a method is provided where the process has a transfer function Wp(s) or a matrix state space description and the system (Fig. 1 ) - via introduction a nonlinear element 2 in series with the process 1 and applying a set point signal 3 to the closed-loop system - is brought in asymmetric self excited oscillations mode for measuring the frequency of the oscillations, average over the period value of the process output signal and average over the period control signal whereupon the controller is tuned in dependence on the measurements obtained. An element having a non-linear (relay) characteristic (Fig. 2A or 2B) is introduced into the system in series with the process and set point signal is applied to excite asymmetric self excited oscillations in the system. If the nonlinear element has an asymmetric relay characteristic the system should be transformed into an equivalent relay system with a symmetric relay characteristic - with the use of known from the automatic control theory techniques.
It is proved (Igor Boiko, "Input-output analysis of limit cycling relay feedback control systems,"
Proc. of 1999 American Control Conference, San Diego, USA, Omnipress, 1999, pp.542-546; Igor Boiko, "Frequency Domain Approach to Analysis of Chattering and Disturbance Rejection in Sliding Mode Control," Proc. of World Multiconference on Systemics, Cybernetics and Informatics, Orlando, Florida, USA, Vol. XV, Part II, pp. 299-303) that asymmetric self excited oscillations in the relay feedback system (Fig. 3) comprising the process transfer function 1, the relay 2, the set point signal 3, the feedback 4, and the comparison device 5 can be described by the LPRS. The LPRS is a characteristic of a relay feedback system that has the following definition:
J(u~) _ -0.5 lim 6" + j -'~, lim y(t)~,_o (1) ~o-~~ u" 4c t~-~
where f~ is the set point, 6o and uo are constant terms of error signal 6(t) and control u(t) respectively, c is the amplitude of the relay, ~ is the frequency of the oscillations, which can be varied by means of varying the hysteresis b of the relay.
The LPRS is related with a transfer function of the linear part of a relay feedback system, and for a given transfer function W(s) of the linear part of a relay feedback system the LPRS J(~) can be calculated via the use of one of the following formulas and techniques (Igor Boiko, "Input-output analysis of limit cycling relay feedback control systems," Proc. of 1999 American Control Conference, San Diego, USA, Omnipress, 1999, pp.542-546; Igor Boiko, "Application of the locus of a perturbed relay system to sliding mode relay control design," Proc. of 2000 IEEE International Conference on Control Applications, Anchorage, AK, USA, 2000, pp. 542-547;
Igor Boiko, "Frequency Domain Approach to Analysis of Chattering and Disturbance Rejection in Sliding Mode Control," Proc. of World Multiconference on Systemics, Cybernetics and Informatics, Orlando, Florida, USA, Vol. XV, Part II, pp. 299-303):
a) The LPRS can be calculated as a series of transfer function values at multiple frequencies -according to the formula:
J(a~) _ ~k"'(-1)~'+'ReW(kw)+ j~ 1 ImW~(2k-1)tvJ (2) h=, ~-! 2k -1 where W(s) is the transfer function of the linear part of the system (of the process in this case), m=D
for type zero (non-integrating process) and m=1 for non-zero type servo system (integrating process). For realization of this technique, summation should be done from k=1 to k=MI and M~
where MI and MZ are sufficiently large numbers for the finite series being a good approximation of the infinite one.
b) The LPRS can be calculated with the use of the following technique.
Firstly, transfer function is represented as a sum of m transfer functions of 1 S' and 2"d order elements (expanded into partial fractions):
W(s)=W~(s) + W2(s) + W3(s) + ... + Wm(s) Secondly, for each transfer function, respective LPRSs (the partial LPRSs) are calculated with the use of formulas of Table 1.
Tahle 1. Formulas of LPRS J(CV) Transfer function LPRS J(w) W(s) Kls 0 j~Kl(8~) Kl(Ts+1) O.SK(1-acosech tx);j0.25~rKth(txl2), tx-~(T~) Kl(( T~s+I)( T2s+1)JO. SKjl -Til(T~-TZ) a~ cosech al- T2/( Tz-T~) gel cosech tx~)J
j0.25~cK1(TI-T~ jT, th(cel2) - Ti th(a2/2)J, a,~(T w), txa~l(Ta~) K/(s~+2~s+1) 0.5 Kj(1-(B+yC)l(sinZ/3+sh2cx)J
j0.25~K(shca ysin~3) l (cha'+-cos~
a~c~l~, ia~t(1-~)liz/~ ~~/~~
B~cos,(3sha+psin~icht~ C=asin,Ochc~-J3cosjishtx K sl(s2+2~s+1) 0.5 K j~ (B+yC) - ~lw cos/3shtxJ l(sinZj3+sh~a)J
j0.25 K ~'(I-~)-~~~ sin,l3/ (chcx+cos,~
a-~~1~, /3-~c(I-~)'i2/~, . y-~~~
B=~ecos~l3sha+,(isin,l3ch~ C=c~sin/3cha-/3cos~3sha, Ksl(s+1) 0.5 K ja(sha+ tech a)lsh?a -~j0.25~ca1(I
+ cha)J, a-~rl~
Kslj(Tls+1)(TZS+l)J D.5 Kl(T2-T,) j as cosech ct2-al cosech ~x~J
j0.25 K Jrl(T2-T~) j th(cell2) - th(azl2)J, al=~l(T ~), a2 ~cl(T2~) Thirdly, the LPRS is calculated as a sum of all partial LPRSs:
J(w)°J~(~) + JZ(~) + J3(~) + ... + Jm(~) c) For matrix state space description, the LPRS for type 0 servo systems is calculated with the use of formula (3):
J(to) _ -O. SC(A-' + 2~ (1- a 1°' A)-'e°'AJB
w (3) +j 4C(1+e~A)-'(1-e~A)A-'B
where A, B and C are matrices of the following state space description of a relay system:
x=Ax+Bu y=Cx +IifQ=fo-Y>b,h>0 u=
-lif 6--fo-y<-b,d~<0 where A is an nxn matrix, B is an nxl matrix, C is an 1 xn matrix, f0 is the set point, c: is the error signal, 2b is the hysteresis of the relay function, x is the state vector, y is the process output, a is the control, n is the order of the system;
or the LPRS for type 1 (integrating process) servo systems is calculated with the use of formula (4):
J(~) = 0.25CA-'((I-DZ)-'~Dz -(I+4~ A)D+D3 -I]+D-IjB+
a' + j'~ CA-'~ ~ +A-'C(1-DZ)-' 8 ~
~(3DZ -3D-Dj +I)-D+IJ)B, ~A
where D = a ~ , A, B and C are matrices of the following state space description of a relay system:
X=Ax+Bu .Y=Cx-fo !~+lif ~=-y>b,~>D
u--lifer=-y<-b,d~<0 where A is an (n-1)x(n-1) matrix, B is an (n-I)xl matrix, C is an 1 x(n-I) matrix, n is the order of the system.
Any of the three techniques presented above can be used for the LPRS
calculation. If the LPRS is calculated (Fig. 4) the frequency of oscillations .S2 and the equivalent gain of the relay k" can be easily determined. In a relay feedback system, the following equalities are true (directly follow from the LPRS definition above; detailed consideration is given in the paper:
I. Boiko, "Input-output analysis of limit cycling relay feedback control systems," Proc. of 1999 American Control Conference, San Diego, USA, Omnipress, 1999, pp.542-546):
Im J(Sa) _ - ~b- , (5) __ 1 k" 2Re J(SZ) (6) The frequency of oscillations S2 corresponds to the point of intersection of the LPRS 1 and the line 2 parallel to the real axis that lies below it at the distance of ~$/(4c).
Therefore, by measuring frequency of oscillations ,f~, average over the period process output yo and average over the period control signal uo we can calculate average over the period value of the error signal:6o fo yo, ~e equivalent gain of the relay k"= uola-o (although this is not an exact value the experiments prove that it is a very good approximation even if uo and ~o are not small; the smaller uo and ~o the more precise value of k,~ we obtain), the static gain of the process K= y~luo, and identify one point of the LPRS - at frequency S2.
J(.~) =-os (fn yo)luo - j ~bl(4c) If the model,of the process contains only 2 unknown parameters (beside static gain 1~, those two parameters can be found from complex equation (7), which corresponds to finding a point of the LPRS at frequency X52.
If the process model contains more than three unknown parameters, two or more asymmetric oscillatory experiments - each with different hysteresis b of the relay element- should be carried out. As a result, each experiment provides one point of the LPR.S, and N
experiments provide N
points of the LPRS on the complex plane enabling up to 2N unknown parameters (beside static gain K) to be determined.
One particular process model is worth an individual consideration. The first reason is that it is a good approximation of many processes. The second reason is - this process model allows for a simple semi-analytical solution. The process transfer function Wp(s) is sought to be of 1 S' order with a dead time:
Wp(s)=K exp( zs) l (Ts+1) (8) The LPRS for this transfer function is given by:
_r' Y
J(co) = K (1- a eYCOSech a) + j'-' K ( 2e .-a . _ I) (9) 2 4 I+e-a where K is a static gain, T is a time constant, zis the dead time, c~~tl~T, y=tlT
With the measured values ,Sl, y~ and u~ and known f~, b and c, parameters K, T
and zof the approximating transfer function are calculated as per the following algorithm:
(a) at first the static gain K is calculated as:
K-Yo a (10) (b) then the following equation is solved for ~x Ya=1.._-~~. (11) .fo a ' (c) after that time constant T is calculated as:
T aSl ' (12) (d) and finally dead time 2-is calculated as:
i = T Ink ~ (ea + I)j (13) The most time consuming part of the above algorithm is solving equation (11).
This is a nonlinear algebraic equation and all known methods can be applied for its solution.
t1 With the parameters K, T and zof the approximating transfer function identified, PI controller can be easily designed. The following tuning algorithm/values are proposed.
Proportional gain Kp and integrator gain Ki in PI control are calculated as follows. For desired overshoot being a constraint, proportional gain Kp and integrator gain K; are sought as a solution of the parameter optimization (minimization) problem with settling time being an objective function. This allows to obtain a minimal settling time at the step response as well as appropriate stability margins at any law of set point change.
A simplified solution of this problem is proposed by this invention too. It is proposed that quasi-optimal settings.are used instead of optimal settings. Those are obtained as a solution of the above formulated optimization problem and respective approximation. Gains Kp and K, as functions of desired overshoot are approximated. At first normalized values of KP and KI
denoted as K°n and K°;
should be calculated as follows:
For overshoot 20% integrator gain is calculated as K°; =1.60z/l;
(14) for overshoot 10% K°;=1.80a/I'; (15) for overshoot 5% K°1=1.952Yf. (16) Normalized proportional gain K°p is to be taken from Table 2 with in-between values determined via interpolation.
Table 2. Quasi-optimal settings of PI controller (proportional gain K°p) Overshootz/T z~f z~!' z/f zll Z/I' zlT 2/I' z/f zlT z/f [%] =0.1 =0.2 =0.3 =0.4 =0.5 =0.6 =0.7 =0.8 =0.9 =1.0 =1.5 20 K"p 3.7022.5642.0071.6831.4731.3291.2251.1461.0860.915 =7.177 Kp 3.0582.1201.6731.4191.2581.1481.0681.0080.9630.833 =5.957 5 K' 2.6241.8231.4831.2941.1701.0821.0140.9640.9240.808 p =5.203 Finally, Kp and K; are calculated as Kp = K°,, /K and K; K°;/K
where K is the static gain of the process determined by (10). Formulas (14)-(16) and Table 2 give quasi-optimal normalized values of PI controller settings for a desired overshoot.
In some cases an external unknown constant or slowly changing disturbance (static load) is applied to the process. In that case the static gain of the process is calculated on the basis of two asymmetric oscillatory experiments - each with different average over the period control signal - as a quotient of the increment of average over the period process output signal and increment of average over the period control signal.
Sometimes the process has a nonlinear character. In this case multiple asymmetric oscillatory experiments are to be performed with decreasing values of the output amplitude of the relay - with the purpose to obtain a better local approximation of the process. Parameters of the process transfer function corresponding to a local linear approximation of the process are found as a solution of equations (7), (10) where the process model is expressed as a formula of the LPRS and contains the parameters to be identified.
More complex models of the process can also be used. 1.f the process model has more than 3 unknown parameters, multiple asymmetric oscillatory experiments are performed with different values of hysteresis bk (k=1,2...) of the relay - with the purpose to identify several points of the LPRS:
ReJ(.fl~= _ ~ .~ou yox (I~) o~
Im J(.s2kj=-~kl(4c) (18) where S2~ , yap, uok are .fl, yo, uo corresponding to k th asymmetric oscillatory experiment.
Each experiment allows for identification of one point of the LPRS and consequently of two parameters (beside the static gain). As a result, 2N+1 unknown parameters can be identified from N
asymmetric oscillatory experiments via solution of 2N+1 nonlinear algebraic equations (10), (17), (18) with the unknown parameters expressed through a formula of the LPRS.
Therefore, the number of asymmetric oscillatory experiments can be planned accordingly, depending on the number of unknown parameters.
Alternatively, parameters of process transfer function are to be found as least squares criterion (or with the use of another criterion) approximation of the LPRS - if the number of unknown parameters of the process is less than 2N+1 (where N is the number of asymmetric oscillatory experiments). In other words, the LPRS represented via certain process model parameters is fitted to the LPRS points obtained through the asymmetric oscillatory experiments. .
Eventually, the designed self tuning PID (or another type) controller is supposed to be realized as a processor based (micro-computer or controller) device and all above formulas, the nonlinear element, the tuning rules are realized as computer programs with the use of applicable programming languages. The preferred embodiment of the controller is depicted in Fig. 5.
The controller 1 has two AlD converters 2 and 3 on its input for the process output and set point signals respectively (alternatively it may have only one A/D converter for the process output signal, and the set point may be realized within the controller so$ware), a processor (CPU) 4, a read-only memory (ROM) 5 for program storage, a random access memory (RAM) 6 for buffering the data, an addressldatalcontrol bus 7 for data transfer to/from the processor, and an D/A
converter 8 that converts digital control signal generated by the controller into analog format. The analog control signal is applied to the process 9 (to a control valve, etc.). All elements of the controller interact with each other in a known manner. Some elements of the controller listed above (for example A/D
and D/A converters) may be missing as well as the controller m.ay also contain elements other than listed above - depending on specific requirements and features of the control system.
EXAMPLE
The following example illustrates an application of the method as well as is realized with the software, which actually implements the described algorithm and formulas. , Let the process be described by the following transfer function, which is considered unknown to the autotuner and is different from the process model used by the autotuner:
W(s)=O.Sexp(0.6s)l(0.8s2+2.4s+1) The objective is to design a PI controller for this process with the use of first order plus dead time transfer function as an approximation of the process dynamics.
Simulations of the asymmetric oscillatory experiment and of the tuned system are done with the use of software SIMULINK~ (of MathWorks). The block diagram is depicted in Fig. 6.
Blocks Transport Delay l and Transfer Fcn 2 realize process model. The control is switched from the relay control (blocks Sign 3 and Gain 4) for the asymmetric oscillatory experiment to PI control (blocks Gainl 5, Integrator 6, Gain2 7, Suml 8) by the Switch 9 depending on the value of block Constant 10 ( 1 for the relay control and -1 for the PI control). Error signal and control signal are saved as data files named Error (block To Workspace 11) and Control (block To Workspace) 12) respectively. They are processed for calculation of the average output value, average control value and the frequency of oscillations. Process output and control signal can be monitored on Scope 13 and Scope) 14 respectively. Set point is realized as an input step function (block Step 15). Error signal is realized as difference between the set point signal and process output by block Sum 16.
Let us use first order plus dead time transfer function for the identification:
Wp(s) =Kexp( zs)l(Ts + 1) Let us choose set point value (final value of the step function) f~=0. l, amplitude of the relay c=1 and hysteresis b=0 and run the asymmetric oscillatory experiment. The following values of the oscillatory process are measured:
Frequency of oscillations S2=1.903, Average value of the process output y~=0.0734, Average value of the control signal u0=0.1455.
The following three equations should be solved for K, T, and Z.
1 fo -yo Re J(S~K, T, 2)= _ 2 uo , Im J(Sl,K,T, z)=-~bl(4c), K Yo l uo where the formula of J(~) is given by (9).
According to the algorithm described above, the following values of the process parameters are obtained from the above three equations:
K=0.5050, T=2.5285, 2=0.9573.
Calculate the settings of the PI controller for the desired overshoot 10% and the above values of the identified parameters. As per formula (15) and Table 2 (with the use of linear interpolation), K;=1.349 and Kp=3.503. Simulation of the system with the designed PI
controller produces a step response with overshoot of about 12.5% and settling time about 2.05s (at level ~12.5%). Error between the desired overshoot (10%) and the actual overshoot (12.5%) is mainly due to the use of an approximate model of the process but is also due to the use of the quasi-optimal values of the PI
controller settings instead of the optimal values.
FIG: 3. Block diagram of a relay feedback system.
FiG. 4. LPRS and determination of the frequency of oscillations.
FIG. 5. Block diagram of the controller and the process.
FIG. 6. Block diagram of the SIMULINK° model of the autotuning system.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
Refernng to the drawings, a description will be given of an embodiment of a controller autotuning method according to the present invention.
The PID-control stands for proportional, integrating and derivative control.
It is very common for controlling industrial processes. PID-controllers are manufactured by various manufacturers in large quantities. Usually the controllers are based on microprocessors and proportional, integrating and derivative functions are normally implemented within a software.
Nevertheless, the principal structure of a conventional PID-controller is retained and without loss of generality it is possible to consider a PID-controller as a parallel connection of three channels:
proportional with gain Kp, integrating with gain K; and derivative with gain Kd. As a result, transfer function of the PID-controller is:
Wpid~SJ- Kp + KT ~S -~ Kd S
Choice of gains Kp, KI and Kd values is a subject of tuning if the controller is implemented as a PID-controller. There are established methods of tuning a PID-controller in dependence on the parameters of the process, for example Ziegler-Nichols's method of manual tuning, Hagglund-Astrom's relay feedback autotuning method. There are also a number of other methods of manual and automatic tuning. All those methods can be divided into parametric and non-parametric.
Parametric methods are based on a certain dynamic model of the process with unknown parameters.
The process undergoes a test or a number of tests aimed at the process model parameters identification. Once the process model parameters are identified, the controller is tuned in accordance with known from the automatic control theory rules - to provide stability and required performance to the closed-loop system (comprising the process, the controller, the comparison device, and the feedback). Non-parametric methods are based on the tests on the process, which are aimed at the measurement of some general characteristics of the process, for example ultimate gain and ultimate frequency at both Ziegler-Nichols's method of manual tuning and Hagglund-Astrom's relay feedback autotuning method.
Generally, parametric methods can provide a better tuning quality (due to possibility of the use of more precise model of the process) but require more complex tests on the process. Therefore, there is a need for comparatively simple yet precise method of tuning (manual and automatic), which can be embedded into software of local controllers or distributed control system (DCS) or be implemented as a software for a personal computer used by an engineer who is supposed to tune the controllers.
Usually, methods of tuning that utilize Hagglund-Astrom's relay feedback test for estimating the parameters of oscillations are based on describing function method (Derek P.
Atherton, "Nonlinear Control Engineering", Van Nostrand Reinhod: New York, 1982). The use of this method is limited to harmonic oscillations in the system, which is normally not the case.
The present invention is based on the model of oscillations provided by the LPRS method (Igor Boiko, "Input-output analysis of limit cycling relay feedback control systems," Proc. of 1999 American Control Conference, San Diego, USA, Omnipress, 1999, pp.542-546; Igor Boiko, "Application of the locus of a perturbed relay system to sliding mode relay control design," Proc. of 2000 IEEE International Conference on Control Applications, Anchorage, AK, USA, 2000, pp.
542-547; Igor Boiko, "Frequency Domain Approach to Analysis of Chattering and Disturbance Rejection in Sliding Mode Control," Proc. of World Multiconfe.rence on Systemics, Cybernetics and Informatics, Orlando, Florida, USA, Vol. ~V, Part II, pp. 299-303), which doesn't use the above limiting hypothesis. The present invention provides a relatively simple parametric method of controller tuning. The method uses a modified Hagglund-Astrom's relay feedback test as means to identify parameters of a process model: A process can be modeled by a transfer function with a dead time (time delay) or without it or have a matrix state space description.
According to the present invention, a method is provided where the process has a transfer function Wp(s) or a matrix state space description and the system (Fig. 1 ) - via introduction a nonlinear element 2 in series with the process 1 and applying a set point signal 3 to the closed-loop system - is brought in asymmetric self excited oscillations mode for measuring the frequency of the oscillations, average over the period value of the process output signal and average over the period control signal whereupon the controller is tuned in dependence on the measurements obtained. An element having a non-linear (relay) characteristic (Fig. 2A or 2B) is introduced into the system in series with the process and set point signal is applied to excite asymmetric self excited oscillations in the system. If the nonlinear element has an asymmetric relay characteristic the system should be transformed into an equivalent relay system with a symmetric relay characteristic - with the use of known from the automatic control theory techniques.
It is proved (Igor Boiko, "Input-output analysis of limit cycling relay feedback control systems,"
Proc. of 1999 American Control Conference, San Diego, USA, Omnipress, 1999, pp.542-546; Igor Boiko, "Frequency Domain Approach to Analysis of Chattering and Disturbance Rejection in Sliding Mode Control," Proc. of World Multiconference on Systemics, Cybernetics and Informatics, Orlando, Florida, USA, Vol. XV, Part II, pp. 299-303) that asymmetric self excited oscillations in the relay feedback system (Fig. 3) comprising the process transfer function 1, the relay 2, the set point signal 3, the feedback 4, and the comparison device 5 can be described by the LPRS. The LPRS is a characteristic of a relay feedback system that has the following definition:
J(u~) _ -0.5 lim 6" + j -'~, lim y(t)~,_o (1) ~o-~~ u" 4c t~-~
where f~ is the set point, 6o and uo are constant terms of error signal 6(t) and control u(t) respectively, c is the amplitude of the relay, ~ is the frequency of the oscillations, which can be varied by means of varying the hysteresis b of the relay.
The LPRS is related with a transfer function of the linear part of a relay feedback system, and for a given transfer function W(s) of the linear part of a relay feedback system the LPRS J(~) can be calculated via the use of one of the following formulas and techniques (Igor Boiko, "Input-output analysis of limit cycling relay feedback control systems," Proc. of 1999 American Control Conference, San Diego, USA, Omnipress, 1999, pp.542-546; Igor Boiko, "Application of the locus of a perturbed relay system to sliding mode relay control design," Proc. of 2000 IEEE International Conference on Control Applications, Anchorage, AK, USA, 2000, pp. 542-547;
Igor Boiko, "Frequency Domain Approach to Analysis of Chattering and Disturbance Rejection in Sliding Mode Control," Proc. of World Multiconference on Systemics, Cybernetics and Informatics, Orlando, Florida, USA, Vol. XV, Part II, pp. 299-303):
a) The LPRS can be calculated as a series of transfer function values at multiple frequencies -according to the formula:
J(a~) _ ~k"'(-1)~'+'ReW(kw)+ j~ 1 ImW~(2k-1)tvJ (2) h=, ~-! 2k -1 where W(s) is the transfer function of the linear part of the system (of the process in this case), m=D
for type zero (non-integrating process) and m=1 for non-zero type servo system (integrating process). For realization of this technique, summation should be done from k=1 to k=MI and M~
where MI and MZ are sufficiently large numbers for the finite series being a good approximation of the infinite one.
b) The LPRS can be calculated with the use of the following technique.
Firstly, transfer function is represented as a sum of m transfer functions of 1 S' and 2"d order elements (expanded into partial fractions):
W(s)=W~(s) + W2(s) + W3(s) + ... + Wm(s) Secondly, for each transfer function, respective LPRSs (the partial LPRSs) are calculated with the use of formulas of Table 1.
Tahle 1. Formulas of LPRS J(CV) Transfer function LPRS J(w) W(s) Kls 0 j~Kl(8~) Kl(Ts+1) O.SK(1-acosech tx);j0.25~rKth(txl2), tx-~(T~) Kl(( T~s+I)( T2s+1)JO. SKjl -Til(T~-TZ) a~ cosech al- T2/( Tz-T~) gel cosech tx~)J
j0.25~cK1(TI-T~ jT, th(cel2) - Ti th(a2/2)J, a,~(T w), txa~l(Ta~) K/(s~+2~s+1) 0.5 Kj(1-(B+yC)l(sinZ/3+sh2cx)J
j0.25~K(shca ysin~3) l (cha'+-cos~
a~c~l~, ia~t(1-~)liz/~ ~~/~~
B~cos,(3sha+psin~icht~ C=asin,Ochc~-J3cosjishtx K sl(s2+2~s+1) 0.5 K j~ (B+yC) - ~lw cos/3shtxJ l(sinZj3+sh~a)J
j0.25 K ~'(I-~)-~~~ sin,l3/ (chcx+cos,~
a-~~1~, /3-~c(I-~)'i2/~, . y-~~~
B=~ecos~l3sha+,(isin,l3ch~ C=c~sin/3cha-/3cos~3sha, Ksl(s+1) 0.5 K ja(sha+ tech a)lsh?a -~j0.25~ca1(I
+ cha)J, a-~rl~
Kslj(Tls+1)(TZS+l)J D.5 Kl(T2-T,) j as cosech ct2-al cosech ~x~J
j0.25 K Jrl(T2-T~) j th(cell2) - th(azl2)J, al=~l(T ~), a2 ~cl(T2~) Thirdly, the LPRS is calculated as a sum of all partial LPRSs:
J(w)°J~(~) + JZ(~) + J3(~) + ... + Jm(~) c) For matrix state space description, the LPRS for type 0 servo systems is calculated with the use of formula (3):
J(to) _ -O. SC(A-' + 2~ (1- a 1°' A)-'e°'AJB
w (3) +j 4C(1+e~A)-'(1-e~A)A-'B
where A, B and C are matrices of the following state space description of a relay system:
x=Ax+Bu y=Cx +IifQ=fo-Y>b,h>0 u=
-lif 6--fo-y<-b,d~<0 where A is an nxn matrix, B is an nxl matrix, C is an 1 xn matrix, f0 is the set point, c: is the error signal, 2b is the hysteresis of the relay function, x is the state vector, y is the process output, a is the control, n is the order of the system;
or the LPRS for type 1 (integrating process) servo systems is calculated with the use of formula (4):
J(~) = 0.25CA-'((I-DZ)-'~Dz -(I+4~ A)D+D3 -I]+D-IjB+
a' + j'~ CA-'~ ~ +A-'C(1-DZ)-' 8 ~
~(3DZ -3D-Dj +I)-D+IJ)B, ~A
where D = a ~ , A, B and C are matrices of the following state space description of a relay system:
X=Ax+Bu .Y=Cx-fo !~+lif ~=-y>b,~>D
u--lifer=-y<-b,d~<0 where A is an (n-1)x(n-1) matrix, B is an (n-I)xl matrix, C is an 1 x(n-I) matrix, n is the order of the system.
Any of the three techniques presented above can be used for the LPRS
calculation. If the LPRS is calculated (Fig. 4) the frequency of oscillations .S2 and the equivalent gain of the relay k" can be easily determined. In a relay feedback system, the following equalities are true (directly follow from the LPRS definition above; detailed consideration is given in the paper:
I. Boiko, "Input-output analysis of limit cycling relay feedback control systems," Proc. of 1999 American Control Conference, San Diego, USA, Omnipress, 1999, pp.542-546):
Im J(Sa) _ - ~b- , (5) __ 1 k" 2Re J(SZ) (6) The frequency of oscillations S2 corresponds to the point of intersection of the LPRS 1 and the line 2 parallel to the real axis that lies below it at the distance of ~$/(4c).
Therefore, by measuring frequency of oscillations ,f~, average over the period process output yo and average over the period control signal uo we can calculate average over the period value of the error signal:6o fo yo, ~e equivalent gain of the relay k"= uola-o (although this is not an exact value the experiments prove that it is a very good approximation even if uo and ~o are not small; the smaller uo and ~o the more precise value of k,~ we obtain), the static gain of the process K= y~luo, and identify one point of the LPRS - at frequency S2.
J(.~) =-os (fn yo)luo - j ~bl(4c) If the model,of the process contains only 2 unknown parameters (beside static gain 1~, those two parameters can be found from complex equation (7), which corresponds to finding a point of the LPRS at frequency X52.
If the process model contains more than three unknown parameters, two or more asymmetric oscillatory experiments - each with different hysteresis b of the relay element- should be carried out. As a result, each experiment provides one point of the LPR.S, and N
experiments provide N
points of the LPRS on the complex plane enabling up to 2N unknown parameters (beside static gain K) to be determined.
One particular process model is worth an individual consideration. The first reason is that it is a good approximation of many processes. The second reason is - this process model allows for a simple semi-analytical solution. The process transfer function Wp(s) is sought to be of 1 S' order with a dead time:
Wp(s)=K exp( zs) l (Ts+1) (8) The LPRS for this transfer function is given by:
_r' Y
J(co) = K (1- a eYCOSech a) + j'-' K ( 2e .-a . _ I) (9) 2 4 I+e-a where K is a static gain, T is a time constant, zis the dead time, c~~tl~T, y=tlT
With the measured values ,Sl, y~ and u~ and known f~, b and c, parameters K, T
and zof the approximating transfer function are calculated as per the following algorithm:
(a) at first the static gain K is calculated as:
K-Yo a (10) (b) then the following equation is solved for ~x Ya=1.._-~~. (11) .fo a ' (c) after that time constant T is calculated as:
T aSl ' (12) (d) and finally dead time 2-is calculated as:
i = T Ink ~ (ea + I)j (13) The most time consuming part of the above algorithm is solving equation (11).
This is a nonlinear algebraic equation and all known methods can be applied for its solution.
t1 With the parameters K, T and zof the approximating transfer function identified, PI controller can be easily designed. The following tuning algorithm/values are proposed.
Proportional gain Kp and integrator gain Ki in PI control are calculated as follows. For desired overshoot being a constraint, proportional gain Kp and integrator gain K; are sought as a solution of the parameter optimization (minimization) problem with settling time being an objective function. This allows to obtain a minimal settling time at the step response as well as appropriate stability margins at any law of set point change.
A simplified solution of this problem is proposed by this invention too. It is proposed that quasi-optimal settings.are used instead of optimal settings. Those are obtained as a solution of the above formulated optimization problem and respective approximation. Gains Kp and K, as functions of desired overshoot are approximated. At first normalized values of KP and KI
denoted as K°n and K°;
should be calculated as follows:
For overshoot 20% integrator gain is calculated as K°; =1.60z/l;
(14) for overshoot 10% K°;=1.80a/I'; (15) for overshoot 5% K°1=1.952Yf. (16) Normalized proportional gain K°p is to be taken from Table 2 with in-between values determined via interpolation.
Table 2. Quasi-optimal settings of PI controller (proportional gain K°p) Overshootz/T z~f z~!' z/f zll Z/I' zlT 2/I' z/f zlT z/f [%] =0.1 =0.2 =0.3 =0.4 =0.5 =0.6 =0.7 =0.8 =0.9 =1.0 =1.5 20 K"p 3.7022.5642.0071.6831.4731.3291.2251.1461.0860.915 =7.177 Kp 3.0582.1201.6731.4191.2581.1481.0681.0080.9630.833 =5.957 5 K' 2.6241.8231.4831.2941.1701.0821.0140.9640.9240.808 p =5.203 Finally, Kp and K; are calculated as Kp = K°,, /K and K; K°;/K
where K is the static gain of the process determined by (10). Formulas (14)-(16) and Table 2 give quasi-optimal normalized values of PI controller settings for a desired overshoot.
In some cases an external unknown constant or slowly changing disturbance (static load) is applied to the process. In that case the static gain of the process is calculated on the basis of two asymmetric oscillatory experiments - each with different average over the period control signal - as a quotient of the increment of average over the period process output signal and increment of average over the period control signal.
Sometimes the process has a nonlinear character. In this case multiple asymmetric oscillatory experiments are to be performed with decreasing values of the output amplitude of the relay - with the purpose to obtain a better local approximation of the process. Parameters of the process transfer function corresponding to a local linear approximation of the process are found as a solution of equations (7), (10) where the process model is expressed as a formula of the LPRS and contains the parameters to be identified.
More complex models of the process can also be used. 1.f the process model has more than 3 unknown parameters, multiple asymmetric oscillatory experiments are performed with different values of hysteresis bk (k=1,2...) of the relay - with the purpose to identify several points of the LPRS:
ReJ(.fl~= _ ~ .~ou yox (I~) o~
Im J(.s2kj=-~kl(4c) (18) where S2~ , yap, uok are .fl, yo, uo corresponding to k th asymmetric oscillatory experiment.
Each experiment allows for identification of one point of the LPRS and consequently of two parameters (beside the static gain). As a result, 2N+1 unknown parameters can be identified from N
asymmetric oscillatory experiments via solution of 2N+1 nonlinear algebraic equations (10), (17), (18) with the unknown parameters expressed through a formula of the LPRS.
Therefore, the number of asymmetric oscillatory experiments can be planned accordingly, depending on the number of unknown parameters.
Alternatively, parameters of process transfer function are to be found as least squares criterion (or with the use of another criterion) approximation of the LPRS - if the number of unknown parameters of the process is less than 2N+1 (where N is the number of asymmetric oscillatory experiments). In other words, the LPRS represented via certain process model parameters is fitted to the LPRS points obtained through the asymmetric oscillatory experiments. .
Eventually, the designed self tuning PID (or another type) controller is supposed to be realized as a processor based (micro-computer or controller) device and all above formulas, the nonlinear element, the tuning rules are realized as computer programs with the use of applicable programming languages. The preferred embodiment of the controller is depicted in Fig. 5.
The controller 1 has two AlD converters 2 and 3 on its input for the process output and set point signals respectively (alternatively it may have only one A/D converter for the process output signal, and the set point may be realized within the controller so$ware), a processor (CPU) 4, a read-only memory (ROM) 5 for program storage, a random access memory (RAM) 6 for buffering the data, an addressldatalcontrol bus 7 for data transfer to/from the processor, and an D/A
converter 8 that converts digital control signal generated by the controller into analog format. The analog control signal is applied to the process 9 (to a control valve, etc.). All elements of the controller interact with each other in a known manner. Some elements of the controller listed above (for example A/D
and D/A converters) may be missing as well as the controller m.ay also contain elements other than listed above - depending on specific requirements and features of the control system.
EXAMPLE
The following example illustrates an application of the method as well as is realized with the software, which actually implements the described algorithm and formulas. , Let the process be described by the following transfer function, which is considered unknown to the autotuner and is different from the process model used by the autotuner:
W(s)=O.Sexp(0.6s)l(0.8s2+2.4s+1) The objective is to design a PI controller for this process with the use of first order plus dead time transfer function as an approximation of the process dynamics.
Simulations of the asymmetric oscillatory experiment and of the tuned system are done with the use of software SIMULINK~ (of MathWorks). The block diagram is depicted in Fig. 6.
Blocks Transport Delay l and Transfer Fcn 2 realize process model. The control is switched from the relay control (blocks Sign 3 and Gain 4) for the asymmetric oscillatory experiment to PI control (blocks Gainl 5, Integrator 6, Gain2 7, Suml 8) by the Switch 9 depending on the value of block Constant 10 ( 1 for the relay control and -1 for the PI control). Error signal and control signal are saved as data files named Error (block To Workspace 11) and Control (block To Workspace) 12) respectively. They are processed for calculation of the average output value, average control value and the frequency of oscillations. Process output and control signal can be monitored on Scope 13 and Scope) 14 respectively. Set point is realized as an input step function (block Step 15). Error signal is realized as difference between the set point signal and process output by block Sum 16.
Let us use first order plus dead time transfer function for the identification:
Wp(s) =Kexp( zs)l(Ts + 1) Let us choose set point value (final value of the step function) f~=0. l, amplitude of the relay c=1 and hysteresis b=0 and run the asymmetric oscillatory experiment. The following values of the oscillatory process are measured:
Frequency of oscillations S2=1.903, Average value of the process output y~=0.0734, Average value of the control signal u0=0.1455.
The following three equations should be solved for K, T, and Z.
1 fo -yo Re J(S~K, T, 2)= _ 2 uo , Im J(Sl,K,T, z)=-~bl(4c), K Yo l uo where the formula of J(~) is given by (9).
According to the algorithm described above, the following values of the process parameters are obtained from the above three equations:
K=0.5050, T=2.5285, 2=0.9573.
Calculate the settings of the PI controller for the desired overshoot 10% and the above values of the identified parameters. As per formula (15) and Table 2 (with the use of linear interpolation), K;=1.349 and Kp=3.503. Simulation of the system with the designed PI
controller produces a step response with overshoot of about 12.5% and settling time about 2.05s (at level ~12.5%). Error between the desired overshoot (10%) and the actual overshoot (12.5%) is mainly due to the use of an approximate model of the process but is also due to the use of the quasi-optimal values of the PI
controller settings instead of the optimal values.
Claims (21)
1. A method of tuning a controller for a process in a feedback control system comprising the steps of:
(a) bringing the system into asymmetric self-excited oscillations (further referred to as asymmetric oscillatory experiment) via the introduction into the system an element having a nonlinear characteristic in series with the process, and applying a constant set point signal f0 to the system, so that the error signal is the difference between the set point and the process output signal, the error signal is an input of the nonlinear element, and the output of the nonlinear element is a control signal for the process;
(b) measurement of the frequency of the oscillations .OMEGA., average (over the period of the oscillations) value of the process output signal y0, and average (over the period of the oscillations) control signal u0;
(c) choosing the process model, parameters of which have unknown values;
(d) identification of the parameters of the process model;
(e) tuning the controller in dependence on the identified parameters of the process model.
(a) bringing the system into asymmetric self-excited oscillations (further referred to as asymmetric oscillatory experiment) via the introduction into the system an element having a nonlinear characteristic in series with the process, and applying a constant set point signal f0 to the system, so that the error signal is the difference between the set point and the process output signal, the error signal is an input of the nonlinear element, and the output of the nonlinear element is a control signal for the process;
(b) measurement of the frequency of the oscillations .OMEGA., average (over the period of the oscillations) value of the process output signal y0, and average (over the period of the oscillations) control signal u0;
(c) choosing the process model, parameters of which have unknown values;
(d) identification of the parameters of the process model;
(e) tuning the controller in dependence on the identified parameters of the process model.
2. The method as recited in Claim 1, wherein the nonlinear element is the hysteresis relay characteristic with positive, zero or negative hysteresis;
3. The method as recited in Claim 1, wherein the original process model with unknown parameters is transformed into the form of a formula of the locus of a perturbed relay system (LPRS) being a complex function, which has same unknown parameters;
4. The method as recited in Claim 1, wherein with the measurement obtained, the process model parameters identification step further includes the steps of:
(a) calculation of the static gain of the process as:
K=y0/u0 (b) and calculation of one point of the LPRS J(.OMEGA.) of the process at the frequency of the oscillations .OMEGA. (which is the measured value of the LPRS at the frequency .OMEGA.) as:
Im J(.OMEGA.)=-.pi.b/(4c) where f0 if the set point, y0 is the average value of the process output signal, u0 is the average value of the control signal, b is a half of the hysteresis of the relay, c is the amplitude of the relay.
(a) calculation of the static gain of the process as:
K=y0/u0 (b) and calculation of one point of the LPRS J(.OMEGA.) of the process at the frequency of the oscillations .OMEGA. (which is the measured value of the LPRS at the frequency .OMEGA.) as:
Im J(.OMEGA.)=-.pi.b/(4c) where f0 if the set point, y0 is the average value of the process output signal, u0 is the average value of the control signal, b is a half of the hysteresis of the relay, c is the amplitude of the relay.
5. The method as recited in Claim 1, wherein two unknown process model parameters (beside the static gain) are identified as the values that provide equality of the LPRS
calculated on the basis of the chosen process model (at the frequency of the oscillations) to the point of the LPRS calculated through the measurements obtained via the asymmetric oscillatory experiment.
calculated on the basis of the chosen process model (at the frequency of the oscillations) to the point of the LPRS calculated through the measurements obtained via the asymmetric oscillatory experiment.
6. The method as recited in Claim 1, wherein multiple asymmetric oscillatory experiments are performed with different values of the hysteresis of the relay - with the purpose to measure several points of the LPRS.
7. The method as recited in Claim 1, wherein in case of nonlinear character of the process, multiple asymmetric oscillatory experiments are performed with decreasing values of the output amplitude of the relay - with the purpose to obtain a better local model of the process.
8. The method as recited in Claim 1, wherein a combination of multiple asymmetric oscillatory experiments with different values of the hysteresis of the relay and decreasing values of the output amplitude of the relay are performed - with the purpose to measure several points of the LPRS and obtain a better local model of the process in case of its nonlinear character.
9. The method as recited in Claim 6 or 8, wherein with the measurement obtained, the process model parameters identification step further includes the steps of:
(a) calculation of the static gain of the process as:
(b) and calculation of N points of the LPRS J(.OMEGA.k) of the process at the frequencies of the oscillations .OMEGA.k, k=1,2, ...,N (which are the measured values of the LPRS
at frequencies .OMEGA.k, k=1;2, ...,N) as:
Im J(.OMEGA.k)=-.pi.bk/(4c) where .OMEGA.k , y0k., u0k bk (k=1,2...,N) are values of .OMEGA. y0,, u0, and b for k-th asymmetric oscillatory experiment, f0 if the set point, y0 is the average value of the process output signal, u0 is the average value of the control signal, b is a half of the hysteresis of the relay, c is the amplitude of the relay.
(a) calculation of the static gain of the process as:
(b) and calculation of N points of the LPRS J(.OMEGA.k) of the process at the frequencies of the oscillations .OMEGA.k, k=1,2, ...,N (which are the measured values of the LPRS
at frequencies .OMEGA.k, k=1;2, ...,N) as:
Im J(.OMEGA.k)=-.pi.bk/(4c) where .OMEGA.k , y0k., u0k bk (k=1,2...,N) are values of .OMEGA. y0,, u0, and b for k-th asymmetric oscillatory experiment, f0 if the set point, y0 is the average value of the process output signal, u0 is the average value of the control signal, b is a half of the hysteresis of the relay, c is the amplitude of the relay.
10. The method as recited in Claim 6 or 8, wherein 2N unknown parameters (beside the static gain) of the process model are found as the solution of 2N nonlinear algebraic equations, which are obtained as conditions of equality of the N measured (via the asymmetric oscillatory experiments) points of the LPRS the process to the N points of the LPRS calculated on the basis of the chosen process model (at the frequencies of the oscillations).
11. The method as recited in Claim 6 or 8, wherein the unknown parameters (beside the static gain) of the process model are found through the least squares criterion (or with the use of another criterion) fitting of the LPRS expressed via certain process model parameters to the LPRS points obtained through the asymmetric oscillatory experiments.
12. The method as recited in Claim 3, wherein the LPRS for a process model in the form of transfer function Wp(s) of the process is calculated with the use of the following formula:
J(.omega.)=.SIGMA.k m(-1)k+1 Re W p(k.omega.)+ j.SIGMA.Im W p[(2k-1).omega.]/(2k-1) k=1 k=1 where m=0 for type 0 servo systems (non-integrating process) and m=1 for type 1 servo systems (integrating process), .omega. is a frequency, M1, M2 are numbers sufficiently large for the above formula being an adequate approximation of respective infinite series.
J(.omega.)=.SIGMA.k m(-1)k+1 Re W p(k.omega.)+ j.SIGMA.Im W p[(2k-1).omega.]/(2k-1) k=1 k=1 where m=0 for type 0 servo systems (non-integrating process) and m=1 for type 1 servo systems (integrating process), .omega. is a frequency, M1, M2 are numbers sufficiently large for the above formula being an adequate approximation of respective infinite series.
13. The method as recited in Claim 3, wherein the LPRS for a process model in the form of transfer function W p(s) of the process is calculated on the basis of expansion of the process transfer function W p(s) into partial fractions, calculation of the partial LPRSs with the use of formulas of the following Table and summation of the partial LPRSs.
Formulas of LPRS J(.omega.)
Formulas of LPRS J(.omega.)
14. The method as recited in Claim 4, wherein the LPRS for a process model in the form of state space description for type 0 (non-integrating process) servo systems is calculated with the use of the following formula:
where A, B and C are matrices of the following state space description of a relay system:
X = Ax+Bu y=Cx where A is an nXn matrix, B is an nx1 matrix, C is an 1Xn matrix, f0 is the set point, .sigma. is the error signal, 2b is the hysteresis of the relay function, x is the state vector, y is the process output, µ is the control, n is the order of the system;
or the LPRS for type 1 (integrating process) servo systems is calculated with the use of the following formula:
where D = , A, B and C are matrices of the following state space description of a relay system:
~ = Ax + Bu ~ = Cx-f0 where A is an (n-1)x(n-1) matrix, B is an (n-1)x1 matrix, C is an 1 x(n-1) matrix, n is the order of the system.
where A, B and C are matrices of the following state space description of a relay system:
X = Ax+Bu y=Cx where A is an nXn matrix, B is an nx1 matrix, C is an 1Xn matrix, f0 is the set point, .sigma. is the error signal, 2b is the hysteresis of the relay function, x is the state vector, y is the process output, µ is the control, n is the order of the system;
or the LPRS for type 1 (integrating process) servo systems is calculated with the use of the following formula:
where D = , A, B and C are matrices of the following state space description of a relay system:
~ = Ax + Bu ~ = Cx-f0 where A is an (n-1)x(n-1) matrix, B is an (n-1)x1 matrix, C is an 1 x(n-1) matrix, n is the order of the system.
15. The method as recited in Claim 1, wherein the chosen model of the process is given by the transfer function Wp(s) of 1st order with a dead time:
Wp(s)=K exp(-~s)/(Ts+1) where K is a static gain, T is a time constant, ~ is a dead time.
Wp(s)=K exp(-~s)/(Ts+1) where K is a static gain, T is a time constant, ~ is a dead time.
16. The method as recited in Claim 15, wherein the corresponding LPRS of the process is given by:
where .alpha.=.pi./.omega.T, .gamma.=~/T.
where .alpha.=.pi./.omega.T, .gamma.=~/T.
17. The method as recited in Claim 15, wherein with the measurements of the frequency of the oscillations .OMEGA., average output signal y0 and average control signal up obtained, calculation of the parameters K, T and ~ of the process transfer function comprises the following steps:
(a) calculating the static gain K as:
(b) solving the following equation for .alpha:
(c) calculating the time constant T as:
(d) calculating the dead time ~ as:
(a) calculating the static gain K as:
(b) solving the following equation for .alpha:
(c) calculating the time constant T as:
(d) calculating the dead time ~ as:
18. The method as recited in Claim 15, wherein the proportional gain K p and the integrator gain K i in PI control are calculated as follows. For desired overshoot being a constraint, proportional gain K p and integrator gain K i are sought as a solution of the parameter optimization (minimization) problem with settling time being an objective function.
19. The method as recited in claim 15, wherein for the desired overshoot being a given value, the calculation of the proportional gain K p and the integrator gain K i comprises the following steps:
(a) calculation of the normalized values of K p and K i (denoted as K0p and K0i) as follows:
For desired overshoot 20%, normalized integrator gain K0i is calculated as:
K0i =1.60~/T;
for desired overshoot 10%, K0i=1.80~/T;
for desired overshoot 5%, K0i=1.95~/T
with intermediate values determined with the use of interpolation.
Normalized proportional gain K0p is taken from the following Table with intermediate values determined with the use of interpolation.
Normalized proportional gain settings Overshoot ~/T ~/T ~/T ~/T ~/T ~/T ~/T ~/T ~/T ~/T ~/T
[%] =0.1 =0.2 =0.3 =0.4 =0.5 =0.6 =0.7 =0.8 =0.9 =1.0 =1.5
(a) calculation of the normalized values of K p and K i (denoted as K0p and K0i) as follows:
For desired overshoot 20%, normalized integrator gain K0i is calculated as:
K0i =1.60~/T;
for desired overshoot 10%, K0i=1.80~/T;
for desired overshoot 5%, K0i=1.95~/T
with intermediate values determined with the use of interpolation.
Normalized proportional gain K0p is taken from the following Table with intermediate values determined with the use of interpolation.
Normalized proportional gain settings Overshoot ~/T ~/T ~/T ~/T ~/T ~/T ~/T ~/T ~/T ~/T ~/T
[%] =0.1 =0.2 =0.3 =0.4 =0.5 =0.6 =0.7 =0.8 =0.9 =1.0 =1.5
20 K0p 3.702 2.564 2.007 1.683 1.473 1.329 1.225 1.146 1.086 0.915 =7.177 K0p 3.058 2.120 1.673 1.419 1.258 1.148 1.068 1.008 0.963 0.833 =5.957 5 K0p 2.624 1.823 1.483 1.294 1.170 1.082 1.014 0.964 0.924 0.808 =5.203 (b) Proportional gain K p and integrator gain K i are calculated as K p =
K0p/K and K i=K°i/K where K
is the static gain of the process.
20. The method as recited in Claim 4 or 9, wherein if an external unknown constant or slowly changing disturbance (static load) is applied to the process, each static gain of the process is calculated on the basis of a pair of asymmetric oscillatory experiments - each with different average control signal value - as a quotient of the increment of the average process output signal and increment of the average control signal.
K0p/K and K i=K°i/K where K
is the static gain of the process.
20. The method as recited in Claim 4 or 9, wherein if an external unknown constant or slowly changing disturbance (static load) is applied to the process, each static gain of the process is calculated on the basis of a pair of asymmetric oscillatory experiments - each with different average control signal value - as a quotient of the increment of the average process output signal and increment of the average control signal.
21. The controller using the method recited in Claim 1, wherein the controller is realized as a processor based device and all above formulas, the nonlinear element, and the tuning rules are realized as computer programs with the use of applicable programming languages.
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| CA 2370772 CA2370772A1 (en) | 2002-01-24 | 2002-01-24 | Method of tuning a controller and a controller using the method |
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Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| FR2944561A3 (en) * | 2009-04-16 | 2010-10-22 | Renault Sas | Method for adjusting regulator with state parameter i.e. particle filter/nitrogen oxide trap output gas temperature, in electronic control unit of internal combustion engine of motor vehicle, involves calculating parameters of corrector |
| EP3146395A4 (en) * | 2014-03-18 | 2018-10-24 | Honeywell Limited | Method and apparatus for specifying and visualizing robust tuning of model-based controllers |
| CN110188373A (en) * | 2019-04-11 | 2019-08-30 | 汉腾汽车有限公司 | A kind of positive relay switch control method of power battery master |
| CN114690620A (en) * | 2022-06-01 | 2022-07-01 | 湖南恩智测控技术有限公司 | Electronic load PID self-tuning method and system |
-
2002
- 2002-01-24 CA CA 2370772 patent/CA2370772A1/en not_active Abandoned
Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| FR2944561A3 (en) * | 2009-04-16 | 2010-10-22 | Renault Sas | Method for adjusting regulator with state parameter i.e. particle filter/nitrogen oxide trap output gas temperature, in electronic control unit of internal combustion engine of motor vehicle, involves calculating parameters of corrector |
| EP3146395A4 (en) * | 2014-03-18 | 2018-10-24 | Honeywell Limited | Method and apparatus for specifying and visualizing robust tuning of model-based controllers |
| CN110188373A (en) * | 2019-04-11 | 2019-08-30 | 汉腾汽车有限公司 | A kind of positive relay switch control method of power battery master |
| CN114690620A (en) * | 2022-06-01 | 2022-07-01 | 湖南恩智测控技术有限公司 | Electronic load PID self-tuning method and system |
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