ADAPTIVE TUNING METHOD FOR A DIGITAL PID CONTROLLER

Abstract

The aim of the invention is rapid automatic tuning the parameters of a digital proportional-integral-derivative (PID) controller by analog feedback of an actual value for automation of technological processes with programmable logic controllers (PLCs).

The proposed invention is based on the use of nine tuning equations derived by reverse engineering of a PID controller.

Adjusting the PID controller parameters K.sub.p, K.sub.i and K.sub.d is performed in a closed control loop with negative feedback separately in time, i.e. independently of each other in iteration steps k for K.sub.p, m for K.sub.i and n for K.sub.d (see FIG. 1).

The adaptive tuning method is compact, independent of other methods and algorithms, mathematically balanced (i.e. minimal computational resource requirements), and easy to implement.

Setting up a PID controller by this method does not require a preliminary evaluation of a controlled system and the creation of its mathematical model. This implies its universal applicability.

Claims

1. The subject matter of the invention is an adaptive tuning method for parameters of a digital PID controller, which is characterized by using the nine tuning equations derived by reverse engineering of a PID controller.

2. An adaptive tuning method for parameters of a digital PID controller according to claim 1, which is characterized by adaptive tuning of the PID controller parameters K.sub.p, K.sub.i and K.sub.d in the following: the adaptive tuning is performed in a closed control loop separately in time by means of a pass counter and three additional indices k for K.sub.p, m for K.sub.i, and n for K.sub.d that are used as iteration steps in order to modify only one PID parameter at any time point t; the separate tuning of PID parameters over time is caused by the derived tuning equations.

3. An adaptive tuning method for parameters of a digital PID controller according to claim 2, which is characterized in that the adaptive tuning of the PID controller parameters K.sub.p, K.sub.i and K.sub.d is performed cyclically in equal time intervals and comprises the following sequence of steps: S01: Start S02: Reset a time index t to 0 and set a time counter to 0 seconds S03: Reset an iteration step k for K.sub.p to 0 S04: Reset an iteration step m for K.sub.i to 0 S05: Reset an iteration step n for K.sub.d to 0 S06: Set the proportional coefficient K.sub.p to an initial value when k=0 S07: Set the integral action coefficient K to an initial value when m=0 S08: Set the derivative action coefficient K.sub.d to an initial value when n=0 S09: Increase an iteration step k for K.sub.p by 1 S10: Increase an iteration step m for K.sub.i by 1 S11: Increase an iteration step n for K.sub.d by 1 S12: Reset a pass counter to 0 S13: Set a control variable y.sub.t to 0 at time t S14: Set a control variable change dy.sub.t to 0 at time t S15: Set a control variable change dy.sub.t+1 to 0 for the time t+1 S16: Increase a time index t by 1 and a time counter by a sampling time dt S17: Calculate an actual control error e.sub.t between a setpoint w and an actual value x.sub.t at time t as e.sub.t=w−x.sub.t S18: If a time index t>3, go to step S19, otherwise go back to step S13 S19: If an absolute value of the actual control error e.sub.t falls below a specified threshold, go to step S41, otherwise go to step S20 S20: Evaluate a pass counter between 0 and 2: if its value is 0, go to step S21; if its value is 1, go to step S25; or if its value is 2, go to step S29 S21: Calculate an adjustment step value dK.sub.p k for the proportional coefficient K.sub.p in iteration step k at time t ${dK}_{p_{k}} = \frac{(d^{2} y_{t} - K_{i_{m - 1}} .Math. {de}_{t} .Math. dt - K_{d_{n - 1}} .Math. \frac{d^{3} e_{t}}{dt}) .Math. {de}_{t}}{{({de}_{t})}_{2}} -- \frac{({dy}_{t} - K_{i_{m - 1}} .Math. e_{t} .Math. dt - K_{d_{n - 1}} .Math. \frac{d^{2} e_{t}}{dt}) .Math. d^{2} e_{t}}{{({de}_{t})}_{2}},$ in which: d.sup.2y.sub.t is a 2.sup.nd order differential of the control variable y.sub.t at time t, which is calculated as d.sup.2y.sub.t=dy.sub.t-dy.sub.t−1; de.sub.t is a 1st order differential of the control error e.sub.t at time t, which is calculated as de.sub.t=e.sub.t- e.sub.t−1; d.sup.2e.sub.t is a 2.sup.nd order differential of the control error e.sub.t at time t, which is calculated as d.sup.2e.sub.t=e.sub.t-2⋅e.sub.t−1+e.sub.t−2; d.sup.3e.sub.t is a 3.sup.rd order differential of the control error e.sub.t at time t, which is calculated as d.sup.3e.sub.t=e.sub.t-3 ⋅e.sub.t−1+3⋅e.sub.t−2-e.sub.t−3; K.sub.i m−7 is the actual integral action coefficient K.sub.i at time t, which was modified in iteration step m−1; K.sub.d n−1 is the actual derivative action coefficient K.sub.d at time t, which was modified in iteration step n-1 S22: Calculate an adjustment speed α.sub.p k for the proportional coefficient K.sub.p in iteration step k at time t: $α_{p_{k}} = \frac{K_{p_{k - 1}} .Math. {de}_{t} + K_{d_{n - 1}} .Math. \frac{d^{2} e_{t}}{dt} - {dy}_{t}}{{dK}_{p_{k}} .Math. {de}_{t}} .Math. dt,$ $α_{p_{k}} \in [0.001, \frac{e_{t}^{4}}{2.71828} \in [0.0001, 1]],$ in which: K.sub.p k−1 is the actual proportional coefficient K.sub.p at time t, which was modified in iteration step k−1 S23: Adjust the proportional coefficient K.sub.p in iteration step k: K.sub.p.sub.k=K.sub.p.sub.k−1−α.sub.p.sub.k⋅dK.sub.p.sub.k, α.sub.p.sub.k⋅dK.sub.p.sub.k∈[−0.5,+0.5] S24: Go to step S32 S25: Calculate an adjustment step value dK.sub.i m for the integral action coefficient K.sub.i in iteration step m at time t: ${dK}_{i_{m}} = \frac{(d^{2} y_{t} - K_{p_{k}} .Math. d^{2} e_{t} - K_{d_{n - 1}} .Math. \frac{d^{3} e_{t}}{dt}) .Math. e_{t}}{e_{t}^{2} .Math. dt} -- \frac{({dy}_{t} - K_{p_{k}} .Math. {de}_{t} - K_{d_{n - 1}} .Math. \frac{d^{2} e_{t}}{dt}) .Math. {de}_{t}}{e_{t}^{2} .Math. dt}$ S26: Calculate an adjustment speed α.sub.i m for the integral action coefficient K.sub.i in iteration step m at time t: $α_{i_{m}} = \frac{{dy}_{t} - K_{i_{m - 1}} .Math. e_{t} .Math. dt - K_{d_{n - 1}} .Math. \frac{d^{2} e_{t}}{dt}}{{dK}_{i_{m}} .Math. e_{t}} .Math. dt,$ $α_{i_{m}} \in [0.001, \frac{4 .Math. .Math. e_{t}^{3} .Math.}{2.71828} \in [0.0001, 1]]$ S27: Adjust the integral action coefficient K in iteration step m: K.sub.i.sub.m=K.sub.i.sub.m−1+α.sub.i.sub.m ⋅dK.sub.i.sub.m, α.sub.i.sub.m⋅dK.sub.i.sub.m∈[−0.5+0.5] S28: Go to step S32 S29: Calculate an adjustment step value dK.sub.d n for the derivative action coefficient K.sub.d in iteration step n at time t: ${dK}_{d_{n}} = (\frac{(d^{2} y_{t} - K_{p_{k}} .Math. d^{2} e_{t} - K_{i_{m}} .Math. {de}_{t} .Math. dt) .Math. d^{2} e_{t}}{{(d^{2} e_{t})}_{2}} -- \frac{({dy}_{t} - K_{p_{k}} .Math. {de}_{t} - K_{i_{m}} .Math. e_{t} .Math. dt) .Math. d^{3} e_{t}}{{(d^{2} e_{t})}_{2}}) .Math. dt$ S30: Calculate an adjustment speed α.sub.d n for the derivative action coefficient K.sub.d in iteration step n at time t: $α_{d_{n}} = \frac{K_{p_{k}} .Math. {de}_{t} + K_{d_{n - 1}} .Math. \frac{d^{2} e_{t}}{dt} - {dy}_{t}}{{dK}_{d_{n}} .Math. d^{2} e_{t}} .Math. {dt}^{2},$ $α_{d_{n}} \in [0.001, \frac{.Math. e_{t}^{5} .Math.}{13.5914} \in [0.0001, 1]]$ S31: Adjust the derivative action coefficient K.sub.d in iteration step n: K.sub.d.sub.n=K.sub.d.sub.n−1-α.sub.d.sub.n⋅dK.sub.d.sub.n, α.sub.d.sub.n⋅dK.sub.d.sub.n∈[−0.5,+0.5] S32: increase a pass counter by 1 S33: If a pass counter <3, go to step S38, otherwise go to step S34 S34: Increase an iteration step k for K.sub.p by 1 S35: Increase an iteration step m for K.sub.i by 1 S36: Increase an iteration step n for K.sub.d by 1 S37: Reset a pass counter to 0 S38: Calculate a control variable change dy.sub.t+1 according to the PID velocity algorithm for the time t+1 using the actual values of the PID controller parameters K.sub.p, K.sub.i, and K.sub.d at time t: ${dy}_{t + 1} = (K_{p} + K_{i} .Math. dt + \frac{K_{d}}{dt}) .Math. e_{t} - (K_{p} + 2 .Math. \frac{K_{d}}{dt}) .Math. e_{t - 1} + \frac{K_{d}}{dt} .Math. e_{t - 2}$ S39: Modify the control variable y.sub.t at time t: y.sub.t=y.sub.t−1+dy.sub.t+1 S40: Go back to step S16 S41: End.

Description

BRIEF DESCRIPTION OF DRAWINGS

[0020] All features that accentuate novelty of the invention are described in detail in the claims attached. However, the essence of the invention is shown in the following detailed description with references to drawings of the best mode for carrying out the invention:

[0021] FIG. 1. Time scale for tuning of a PID controller

[0022] FIG. 2A. Flowchart of the adaptive tuning method for a PID controller (start)

[0023] FIG. 2B. Flowchart of the adaptive tuning method for a PID controller (end)

[0024] FIG. 3. Closed control loop

[0025] FIG. 4. Reaction of controlled system no. 1 to a step function

[0026] FIG. 5. Reaction of controlled system no. 2 to a step function

[0027] FIG. 6. Reaction of controlled system no. 3 to a step function

[0028] FIG. 7. Reaction of controlled system no. 4 to a step function

[0029] FIG. 8. Reaction of controlled system no. 5 to a step function

BEST MODE FOR CARRYING OUT THE INVENTION

[0030] The following detailed description with accompanying drawings refers to the best mode for carrying out the invention, which should not be considered as a stripped-down form of the invention object. All amendments and supplements contained in the claims are disclosed in the relevant claims.

[0031] The best mode for carrying out the invention is described below.

[0032] The proposed invention relates to the adaptive tuning of a PID controller described by a standard equation as follows [3]:

[00001] $\begin{matrix} y_{t} = K_{p} .Math. e_{t} + K_{i} .Math. \int e_{t} dt + K_{d} .Math. \frac{{de}_{t}}{dt}, & (1) \end{matrix}$

in which:

[0033] y.sub.t is a control variable at time t;

[0034] e.sub.t is a control error between a setpoint w and an actual value x.sub.t at time t, which is calculated as e.sub.t=w-x.sub.t;

[0035] K.sub.p is a proportional coefficient;

[0036] K.sub.i is an integral action coefficient;

[0037] K.sub.d is a derivative action coefficient.

[0038] The PID controller (1) is a basis for reverse engineering to derive the tuning equations for parameters K.sub.p, K.sub.i, and K.sub.d. This derivation method is characterized by the following sequence of steps:

[0039] Step 01: Eliminate an integrator represented explicitly in a PID controller. This is achieved by differentiating both sides of equation (1) according to the time t[3]:

[00002] $\begin{matrix} {dy}_{t} = K_{p} .Math. {de}_{t} + K_{i} .Math. e_{t} .Math. dt + K_{d} .Math. \frac{d^{2} e_{t}}{dt} & (2) \end{matrix}$

[0040] Step 02: Fixing two of the three PID controller parameters at any time point t. To maximally simplify the derivation method of tuning equations, two of the three PID controller parameters K.sub.p, K.sub.i, and K.sub.d are used as fixed values in succession. That is, K.sub.i and K.sub.d are used as fixed values in tuning equations for K.sub.p at time t. K.sub.p and K.sub.d are used as fixed values in tuning equations for K.sub.i at time t+1. And K.sub.p and K.sub.i are used as fixed values in tuning equations for K.sub.d at time t+2. Thus, the adaptive tuning of all three PID controller parameters K.sub.p, K.sub.i, and K.sub.d is performed separately in time, i.e. only one PID parameter is modified at any time point t. To separate the parameter modifications in the tuning method over time and determine the iteration steps, three additional indices k for K.sub.p, m for K.sub.i, and n for K.sub.d are used (see FIG. 1). With these indices the PID controller (2) takes its final form:

[00003] $\begin{matrix} {dy}_{t} = K_{p_{k}} .Math. {de}_{t} + K_{i_{m}} .Math. e_{t} .Math. dt + K_{d_{n}} .Math. \frac{d^{2} e_{t}}{dt} & (3) \end{matrix}$

[0041] Step 03: Derive an equation to calculate an adjustment step value dK.sub.p k for the proportional coefficient K.sub.p. For this purpose, the parameter K.sub.p k is expressed from (3), and the derived equation is differentiated according to the time t:

[00004] $\begin{matrix} {dK}_{p_{k}} = \frac{(d^{2} y_{t} - K_{i_{m - 1}} .Math. {de}_{t} .Math. dt - K_{d_{n - 1}} .Math. \frac{d^{3} e_{t}}{dt}) .Math. {de}_{t}}{{({de}_{t})}_{2}} -- \frac{({dy}_{t} - K_{i_{m - 1}} .Math. e_{t} .Math. dt - K_{d_{n - 1}} .Math. \frac{d^{2} e_{t}}{dt}) .Math. d^{2} e_{t}}{{({de}_{t})}_{2}}, & (4) \end{matrix}$

in which:

[0042] dy.sub.t is a control variable change at time t, which is determined as dy.sub.t=y.sub.t-y.sub.t−1;

[0043] d.sup.2y.sub.t is a 2.sup.nd order differential of the control variable y.sub.t at time t, which is calculated as d.sup.2y.sub.t=dy.sub.t-dy.sub.t−1;

[0044] de.sub.t is a 1.sup.st order differential of the control error e.sub.t at time t, which is calculated as de.sub.t=e.sub.t-e.sub.t−1;

[0045] d.sup.2e.sub.t is a 2.sup.nd order differential of the control error e.sub.t at time t, which is calculated as d.sup.2e.sub.t=e.sub.t-2⋅e.sub.t−1+e.sub.t−2;

[0046] d.sup.3e.sub.t is a 3.sup.rd order differential of the control error e.sub.t at time t, which is calculated as d.sup.3e.sub.t=e.sub.t-3⋅e.sub.t−1⋅e.sub.t−2-e.sub.t−3;

[0047] K.sub.i m−1 i is the actual integral action coefficient K.sub.i at time t, which was modified in iteration step m−1;

[0048] K.sub.d n−1 is the actual derivative action coefficient K.sub.d at time t, which was modified in iteration step n−1.

[0049] Step 04: Determine a rule to adjust the proportional coefficient K.sub.p in iteration step k as follows:

K.sub.p.sub.k=K.sub.p.sub.k−1−α.sub.p.sub.k⋅dK.sub.p.sub.k,60 .sub.p.sub.k⋅dK.sub.p.sub.k∈[−0.5, +0.5],tm (5)

in which:

[0050] K.sub.p k−1 is the actual proportional coefficient K.sub.p at time t, which was modified in iteration step k−1;

[0051] α.sub.p k is an adjustment speed for the proportional coefficient K.sub.p in iteration step k.

[0052] Equation (5) limits the maximum modification of the parameter K.sub.p up to ±0.5 to prevent uncontrollability of the tuning method.

[0053] Step 05: Derive an equation to calculate an adjustment speed α.sub.p k for the proportional coefficient K.sub.p. For this purpose, equation (5) is substituted into (3), and the control error e.sub.t is expressed from the derived equation. Considering that a limit of e.sub.tas t.fwdarw.+∞ equals zero, α.sub.p k is expressed from the derived equation:

[00005] $\begin{matrix} α_{p_{k}} = \frac{K_{p_{k - 1}} .Math. {de}_{t} + K_{d_{n - 1}} .Math. \frac{d^{2} e_{t}}{dt} - {dy}_{t}}{{dK}_{p_{k}} .Math. {de}_{t}} .Math. dt, & (6) \end{matrix}$ $α_{p_{k}} \in [0.001, \frac{e_{t}^{4}}{2.718282} \in [0.0001, 1]]$

[0054] This equation is characterized by an additional limitation of the parameter α.sub.p k in the range from 0.0001 to 1 depending on the control error e.sub.t. This ensures a smooth attenuation of the adaptive tuning method in the final phase. In addition, this prevents an abrupt modification of the proportional coefficient K.sub.p when the control error e.sub.t approaches zero.

[0055] Step 06: Derive an equation to calculate an adjustment step value dK.sub.i m for the integral action coefficient K.sub.i. For this purpose, the parameter K.sub.i m is expressed from (3), and the derived equation is differentiated according to the time t:

[00006] $\begin{matrix} {dK}_{i_{m}} = \frac{(d^{2} y_{t} - K_{p_{k}} .Math. d^{2} e_{t} .Math. dt - K_{d_{n - 1}} .Math. \frac{d^{3} e_{t}}{dt}) .Math. e_{t}}{e_{t}^{2} .Math. dt} -- \frac{({dy}_{t} - K_{p_{k}} .Math. {de}_{t} - K_{d_{n - 1}} .Math. \frac{d^{2} e_{t}}{dt}) .Math. {de}_{t}}{e_{t}^{2} .Math. dt} & (7) \end{matrix}$

[0056] Step 07: Determine a rule to adjust the integral action coefficient K.sub.i in iteration step m as follows:

K.sub.i.sub.m=K.sub.i.sub.m−1+α.sub.i.sub.m⋅dK.sub.i.sub.m, α.sub.i.sub.m⋅dK.sub.i.sub.m∈[−0.5, +0.5], (8)

in which:

[0057] α.sub.i m is an adjustment speed for the integral action coefficient K.sub.i in iteration step m.

[0058] Equation (8) limits the maximum modification of the parameter K.sub.i up to ±0.5 to prevent uncontrollability of the tuning method.

[0059] Step 08: Derive an equation to calculate an adjustment speed α.sub.i m for the integral action coefficient K.sub.i. For this purpose, equation (8) is substituted into (3), and the 1.sup.st order differential of the control error de.sub.t is expressed from the derived equation. Considering that a limit of de.sub.t as t.fwdarw.+∞ equals zero, α.sub.i m is expressed from the derived equation:

[00007] $\begin{matrix} α_{i_{m}} = \frac{{dy}_{t} - K_{i_{m - 1}} .Math. e_{t} .Math. dt - K_{d_{n - 1}} .Math. \frac{d^{2} e_{t}}{dt}}{{dK}_{i_{m}} .Math. e_{t}} .Math. dt, & (9) \end{matrix}$ $α_{i_{m}} \in [0.001, \frac{4 .Math. .Math. e_{t}^{3} .Math.}{2.71828} \in [0.0001, 1]]$

[0060] This equation is characterized by an additional limitation of the parameter α.sub.i m in the range from 0.0001 to 1 depending on the control error e.sub.t. This ensures a smooth attenuation of the adaptive tuning method in the final phase. In addition, this prevents an abrupt modification of the integral action coefficient K.sub.i when the control error e.sub.t approaches zero.

[0061] Step 09: Derive an equation to calculate an adjustment step value dK.sub.d n for the derivative action coefficient K.sub.d. For this purpose, the parameter K.sub.d n is expressed from (3), and the derived equation is differentiated according to the time t:

[00008] $\begin{matrix} {dK}_{d_{n}} = (\frac{(d^{2} y_{t} - K_{p_{k}} .Math. d^{2} e_{t} - K_{i_{m}} .Math. {de}_{t} .Math. dt) .Math. d^{2} e_{t}}{{(d^{2} e_{t})}_{2}} -- \frac{({dy}_{t} - K_{p_{k}} .Math. {de}_{t} - K_{i_{m}} .Math. e_{t} .Math. dt) .Math. d^{3} e_{t}}{{(d^{2} e_{t})}_{2}}) .Math. dt & (10) \end{matrix}$

[0062] Step 10: Determine a rule to adjust the derivative action coefficient K.sub.d in iteration step n as follows:

K.sub.d.sub.n=K.sub.d.sub.n−1−α.sub.d.sub.n⋅dK.sub.d.sub.n, α.sub.d.sub.n⋅dK.sub.d.sub.n∈[−0.5 +0.5], 11)

in which:

[0063] α.sub.d n is an adjustment speed for the derivative action coefficient K.sub.d in iteration step n.

[0064] Equation (11) limits the maximum modification of the parameter K.sub.d up to ±0.5 to prevent uncontrollability of the tuning method.

[0065] Step 11: Derive an equation to calculate an adjustment speed α.sub.d n for the derivative action coefficient K.sub.d. For this purpose, equation (11) is substituted into (3), and the control error e.sub.t is expressed from the derived equation. Considering that a limit of e.sub.t as t.fwdarw.+∞ equals zero, α.sub.d n is expressed from the derived equation:

[00009] $\begin{matrix} α_{d_{n}} = \frac{K_{p_{k}} .Math. {de}_{t} + K_{d_{n - 1}} .Math. \frac{d^{2} e_{t}}{dt} - {dy}_{t}}{{dK}_{d_{n}} .Math. d^{2} e_{t}} .Math. {dt}^{2}, & (12) \end{matrix}$ $α_{d_{n}} \in [0.001, \frac{.Math. e_{t}^{5} .Math.}{13.5914} \in [0.0001, 1]]$

[0066] This equation is characterized by an additional limitation of the parameter α.sub.d n in the range from 0.0001 to 1 depending on the control error e.sub.t. This ensures a smooth attenuation of the adaptive tuning method in the final phase. In addition, this prevents an abrupt modification of the derivative action coefficient K.sub.d when the control error e.sub.t approaches zero.

[0067] Step 12: Select a digital PID controller. The PID velocity algorithm is the most suitable variant for this adaptive tuning method (see [4], p. 1085):

[00010] $\begin{matrix} \begin{matrix} y_{t} = y_{t - 1} + {dy}_{t + 1} = \\ = y_{t - 1} + (K_{p} + K_{i} .Math. dt + \frac{K_{d}}{dt}) .Math. e_{t} - \\ (K_{p} + 2 .Math. \frac{K_{d}}{dt}) .Math. e_{t - 1} + \frac{K_{d}}{dt} .Math. e_{t - 2}, \end{matrix} y_{t} \in [0, 100 %], y_{t \leq 0} = {dy}_{t \leq 0} = 0, & (13) \end{matrix}$

in which:

[0068] dy.sub.t+1 is a control variable change for the time t+1;

[0069] dt is a sampling time of a digital PID controller.

[0070] In this equation the actual values of the PID controller parameters K.sub.p, K.sub.i, and K.sub.d are always used at time t.

[0071] A choice of the PID velocity algorithm is caused by the following criterion:

[0072] Direct integration of control errors e.sub.t into the control variable y.sub.t. In practice, this allows the control variable y.sub.t to be forcibly modified as needed without explicitly correcting an integrator for seamless functionality (as distinct from the PID position algorithm)

[0073] Finally, a flowchart shown in drawings FIG. 2A and FIG. 2B integrates and arranges the equations for automatic tuning of a digital PID controller as a sequence of steps to illustrate the entirety and completeness of the proposed invention description.

[0074] Demonstration of the Invention

[0075] To illustrate the description, the invention is demonstrated on some mathematical models of controlled systems.

[0076] FIG. 3 shows a closed control loop with negative feedback that consists of a PID controller and a controlled system.

[0077] To demonstrate the invention, the adaptive digital PID controller developed for a PLC in the programming language SCL (Structured Control Language [5], see Appendix A) was used with some transfer functions as controlled systems (see Table I).

TABLE-US-00001 TABLE I Transfer functions G(s) of controlled systems in the Laplace s-domain No. Transfer functions G(s) Reference 1 [00011] $G (s) = \frac{5 0 0 0}{(s + 1) (s + 5) (s + 1 0 0)}$ [6] 2 [00012] $G (s) = \frac{1 6 0 0 0 0}{7.2 2 s^{3} + 8.2 65 s^{2} + 3 81600 s + 1600 0 0}$ [7] 3 [00013] $G (s) = \frac{1}{s^{2} + s + 1}$ [8] 4 [00014] $G (s) = \frac{0.1 s + 10}{0.0 004 s^{4} + 0.0 45 s^{3} + 0.5 55 s^{2} + 1.4 1 s + 1}$ [9] 5 [00015] $G (s) = \frac{0.0 5 1 8 7 9 3 6 - 3.5 9 4 \times 1 0^{- 6} s}{0.0 0 02979 s^{2} + 0.0 1 0 11916 s + 0.00 9 2}$ [10]

[0078] Since the transfer functions G(s) in the Laplace s-domain cannot be used explicitly in a PLC, they are to be converted beforehand into equivalent equations of the time domain. For this purpose, the transfer functions G(s) are first converted by the MATLAB function c2d from the Laplace s-domain into similar discrete transfer functions in the Z-domain with a sampling time dt=0.1 s (see Table II).

TABLE-US-00002 TABLE II Equivalent transfer functions in the z-domain with a sampling time dt = 0.1 s No. Discrete transfer functions in the z-domain G(z) 1 [00016] $G (z) = \frac{0.0 5 4 8 8 + 0.2462 z^{- 1} + 0.0 7307 z^{- 2} + 0.0 0 02884 z^{- 3}}{1 - 1.5 11 z^{- 1} + 0.5 489 z^{- 2} - 2.4 9 2 \times 1 0^{- 5} z^{- 3}}$ 2 [00017] $G (z) = \frac{0.0 2 0 55 + 0.0421 z^{- 1} + 0.0 4063 z^{- 2} + 0.0 1885 z^{- 3}}{1 + 0 08548 z^{- 1} - 0.0 7151 z^{- 2} - 0.8 918 z^{- 3}}$ 3 [00018] $G (z) = \frac{0.0 0 1 6 2 5 + 0.0 06338 z^{- 1} + 0.0 01546 z^{- 2}}{1 - 1.8 95 z^{- 1} + 0.9 048 z^{- 2}}$ 4 [00019] $G (z) = \frac{\begin{matrix} 0008 0 9 + 0 06928 z^{- 1} + 0 05295 z^{- 2} + \\ 00036 z^{- 3} - 9.1 8 2 \times 1 0^{- 8} z^{- 4} \end{matrix}}{\begin{matrix} 1 - 2059 z^{- 1} + 1327 z^{- 2} - \\ 02546 z^{- 3} + 1 3 0 1 \times 1 0^{- 5} z^{- 4} \end{matrix}}$ 5 [00020] $G (z) = \frac{0.1 4 6 4 + 0.3 096 z^{- 1} + 0.0 2883 z^{- 2}}{1 - 0.9 475 z^{- 1} + 0.0 3348 z^{- 2}}$

[0079] The transfer functions in the -domain are then converted into recurrent equations of the time domain as polynomials as follows (for details, see [4], pp. 443-444):

x.sub.t=b.sub.1⋅x.sub.t−1+b.sub.2⋅x.sub.t−2+b.sub.3⋅x.sub.t−3+b.sub.4⋅x.sub.t−4+α.sub.0⋅y.sub.t+α.sub.1⋅y.sub.t−1++α.sub.2⋅y.sub.t−2+α.sub.3⋅y.sub.t−3+α.sub.4⋅y.sub.t−4, (14)

in which:

[0080] y.sub.t is a control variable of a PID controller at time t;

[0081] x.sub.t is a controlled system response on the control variable as a simulated sensor value at time t.

[0082] All polynomial parameters (14) for the simulated controlled systems are given in Table III.

TABLE-US-00003 TABLE III Polynomial parameters (14) for transfer functions of the controlled systems No. b.sub.1 b.sub.2 b.sub.3 b.sub.4 a.sub.0 a.sub.1 a.sub.2 a.sub.3 a.sub.4 1 1.511 −0.5489 2.492 × 0 5.488 × 0.2462 7.307 × 2.884 × 0 10.sup.−5 10.sup.−2 10.sup.−2 10.sup.−4 2 −8.548 × 7.151 × 0.8918 0 2.055 × 4.21 × 4.063 × 1.885 × 0 10.sup.−2 10.sup.−2 10.sup.−2 10.sup.−2 10.sup.−2 10.sup.−2 3 1.895 −0.9048 0 0 1.625 × 6.338 × 1.546 × 0 0 10.sup.−3 10.sup.−3 10.sup.−3 4 2.059 −1.327 0.2546 −1.301 × 8.09 × 6.928 × 5.295 × 3.6 × −9.182 × 10.sup.−5 10.sup.−3 10.sup.−2 10.sup.−2 10.sup.−3 10.sup.−8 5 0.9475 −3.348 × 0 0 0.1464 0.3096 2.883 × 0 0 10.sup.−2 10.sup.−2

[0083] The simulation was performed on a computer-aided PLC simulator as a closed control loop (see FIG. 3). The results shown in Table IV were obtained for all mathematical models of the controlled systems from Table I. Here T.sub.i is a reset time, which is determined as T.sub.i=K.sub.p/K.sub.i, and T.sub.d is a derivative time, which is determined as T.sub.d=K.sub.d/K.sub.p. All experiments were performed with initial parameters K.sub.p=1, K.sub.i=1, K.sub.d=1, and a sampling time dt=0.1 s. A step function 0.fwdarw.1 was used as an activation trigger.

TABLE-US-00004 TABLE IV Parameters of a digital PID controller found using the adaptive tuning method Parameters of a PID controller for controlled systems No. K.sub.p T.sub.i [s] T.sub.d [S] FIG. 1 0.167542959333063 1.274823369 0.381714789 4 2 0.998936489520022 0.791433896 0.945415624 5 3 1.16564861359731 0.999456194 0.999827146 6 4 0.330603372851095 1.025824996 0.260850227 7 5 0.128946021372936 1.275031550 0.390702812 8

References

[0084] [1] Vladimir Bobal et. al., “AUTO-TUNING OF DIGITAL PID CONTROLLERS USING RECURSIVE IDENTIFICATION”, Adaptive systems in Control and Signal Processing, Jun. 16, 1995 (1995-06-16), pp. 359-364, XP055754038, Great Britain, ISBN: 978-0-08-042375-3.

[0085] [2] Sukede Abhijeet Kishorsingh et al., “Auto tuning of PID controller”, 2015 International Conference on Industrial Instrumentation and Control (ICIC), IEEE, May 28-30, 2015, pp. 1459-1462, XP033170865.

[0086] [3] “Three Types of PID Equations”, http://bestune.50megs.com/typeABC.htm

[0087] [4] Lutz H., Wendt W., “Taschenbuch der Regelungstechnik mit MATLAB and Simulink”, 10., ergänzte Auflage, Verlag Europa-Lehrmittel, Haan-Gruiten, 2014.

[0088] [5] International standard IEC 61131-3:2013. Programmable controllers—Part 3: Programming languages.

[0089] [6] Lin Feng, Brandt Robert D., Saikalis George, “Self-tuning of PID Controllers by Adaptive Interaction”, Proceedings of the 2000 American Control Conference, pp. 3676-3681.

[0090] [7] Y. Chen et al., “Design of PID Controller of Feed Servo-System Based on Intelligent Fuzzy Control”, Key Engineering Materials, Vol. 693, pp. 1728-1733, 2016.

[0091] [8] X. Wang et al., “Simulation Research of CNC Machine Servo System Based on Adaptive Fuzzy Control”, Advanced Materials Research, Vol. 819, pp. 181-185, 2013.

[0092] [9] T. Boone et al., “PID Controller Tuning Based on the Guardian Map Technique”, International Journal of Systems Applications, Engineering& Development, Vol. 9, pp. 192-196, 2015.

[0093] [10] Dipraj, Dr. A. K. Pandey, “Speed Control of D.C. Servo Motor By Fuzzy Controller”, International Journal of Scientific& Technology Research, Vol. 1, Issue 8, pp. 139-142, 2012.

INDUSTRIAL APPLICABILITY

[0094] This invention is preferably used in automation systems of industrial facilities with programmable logic controllers, where the individual tuning of PID controller parameters is required to regulate the technological processes in production.

TABLE-US-00005 APPENDIX A A source code of the adaptive digital PID controller 001 FUNCTION_BLOCK ″A-PID_CONTROL″ 002 TITLE = A-PID controller 003 AUTHOR : Valentin_Dimakov 004 FAMILY : PID_CONTROL 005 NAME : ′A-PID_CONTROL′ 006 VERSION : 13.44 007 // FUNCTION 008 // Digital PID controller with automatic tuning of parameters 009 // 010 // Called blocks: none 011 012 VAR_INPUT 013 MAN_ON : Bool := FALSE; // Switch-over between manual & automatic mode (0=A/1=M) 014 AUTO_ON : Bool := FALSE; // Activate automatic mode for the A-PID controller 015 INV_CONTROL : Bool := FALSE; // Control direction (0 = SP > PV, 1 = PV > SP) 016 CYCLE : Time := T#100MS; // Sampling time dt for the controller [10 ms. .10 s] 017 SP : LReal := 0.0; // Setpoint w <temperature, pressure, etc.> 018 PV : LReal := 0.0; // Actual value x.sub.t <temperature, pressure, etc.> 019 LMN_LLM : LReal := 0.0; // Lower limit for the control variable y.sub.t [0..99 %] 020 LMN_HLM : LReal := 100.0; // Upper limit for control variable y.sub.t [LMN_LLM..100 %] 021 END_VAR 022 023 VAR_OUTPUT 024 CTRL_ERR : LReal := 0.0; // Actual control error e.sub.t 025 LMN : LReal := 0.0; // Control variable y.sub.t [0..100 %] 026 ERR_CODE : USInt := 0; // Error code of the A-PID controller < > 0, 0 = no error 027 END_VAR 028 029 VAR_IN_OUT 030 SELF_TUN_ON : Bool := FALSE; // Activate auto-tuning for the A-PID controller 031 GAIN : LReal := 1.0; // Proportional coefficient K.sub.p [0.01..30] 032 TI : LTime := LT#1S; // Reset time T.sub.i [CYCLE. .100 m] 033 TD : LTime := LT#1S; // Derivative time T.sub.d [0..60 s] 034 TUN_ERR_TOLER : LReal := 0.01; // Threshold value to stop auto-tuning [0..100] 035 TUN_COMPL_TM : Time := T#3S; // Delay to stop auto-tuning [1 s..1 m] 036 LMN_MAN : LReal := 0.0; // Control variable for the manual mode [0..100 %] 037 END_VAR 038 039 VAR 040 Kp : LReal := 1.0; // Proportional coefficient K.sub.p 041 Ki : LReal := 1.0; // Integral action coefficient K.sub.i 042 Kd : LReal := 1.0; // Derivative action coefficient K.sub.d 043 PASS_NO : USInt := 0; // Pass counter for auto-tuning [0..2] 044 045 e: STRUCT // Control errors at different times 046 t : LReal; // Control error e.sub.t at time t 047 t1 : LReal; // Control error e.sub.t−1 at time t−1 048 t2 : LReal; // Control error e.sub.t−2 at time t−2 049 t3 : LReal; // Control error e.sub.t−3 at time t−3 050 t4 : LReal; // Control error e.sub.t−4 at time t−4 051 sqr : LReal; // Control error squared e.sup.2.sub.t at time t 052 END_STRUCT; 053 054 y: STRUCT // Control variables 055 out: LReal; // Internal control variable y.sub.t [0..100 %] 056 END_STRUCT; 057 058 d: STRUCT // Calculated 1.sup.st order differentials 059 e : LReal; // 1.sup.st order differential de.sub.t of the control error e.sub.t 060 Kp : LReal; // Adjustment step value dK.sub.p for the proportional coefficient K.sub.p 061 Ki : LReal; // Adjustment step value dK.sub.i for the integral action coefficient K.sub.i 062 Kd : LReal; // Adjustment step value dK.sub.d for the derivative action coefficient K.sub.d 063 y : LReal; // Control variable change dy.sub.t+1 for the time t+1 064 y_t1 : LReal; // Control variable change dy.sub.t in previous cycle 065 END_STRUCT; 066 067 d2: STRUCT // Calculated 2.sup.nd order differentials 068 e: LReal; // 2.sup.nd order differential d.sup.2e.sub.t of the control error e.sub.t 069 y: LReal; // 2.sup.nd order differential d.sup.2y.sub.t of the control variable y.sub.t 070 END_STRUCT; 071 072 d3: STRUCT // Calculated 3.sup.rd order differentials 073 e: LReal; // 3.sup.rd order differential d.sup.3e.sub.t of the control error e.sub.t 074 END_STRUCT; 075 076 a: STRUCT // Adjustment speeds for parameters of the A-PID controller 077 Kp: LReal := 1.0; // Adjustment speed a.sub.p for the proportional coefficient K.sub.p 078 Ki: LReal := 1.0; // Adjustment speed a.sub.i for the integral action coefficient K.sub.i 079 Kd: LReal := 1.0; // Adjustment speed a.sub.d for the derivative action coefficient K.sub.d 080 END_STRUCT; 081 082 T_TUN_MON: TON_TIME; // Timer to stop auto-tuning for the A-PID controller 083 END_VAR 084 085 VAR_TEMP 086 LT_CYCLE : LTime; // Sampling time dt for the A-PID controller 087 Ts : LReal; // Sampling time dt for the A-PID controller [sec] 088 dKp : LReal; // Adjustment value for the proportional coefficient K.sub.p 089 dKi : LReal; // Adjustment value for the integral action coefficient K.sub.i 090 dKd : LReal; // Adjustment value for the derivative action coefficient K.sub.d 091 a_mx_Kp : LReal; // Upper limit of the adjustment speed a.sub.p for parameter K.sub.p 092 a_mx_Ki : LReal; // Upper limit of the adjustment speed a.sub.i for parameter K.sub.i 093 a_mx_Kd : LReal; // Upper limit of the adjustment speed a.sub.d for parameter K.sub.d 094 fact_1 : LReal; // 1.sup.st factor in an equation 095 fact_2 : LReal; // 2.sup.nd factor in an equation 096 divisor : LReal; // Divisor in an equation 097 098 r: STRUCT // Time parameters converted to seconds 099 TI: LReal; // Reset time T.sub.i [sec] 100 TD: LReal; // Derivative time T.sub.d [sec] 101 END_STRUCT; 102 END_VAR 103 104 VAR CONSTANT 105 GAIN_MN : LReal := 0.01; // Lower limit for the proportional coefficient K.sub.p 106 GAIN_MX : LReal := 30.0; // Upper limit for the proportional coefficient K.sub.p 107 TI_MK : LTime := LT#100M; // Upper limit for the reset time T.sub.i 108 TD_MK : LTime := LT#1M; // Upper limit for the derivative time T.sub.d 109 LMN_MN : LReal := 0.0; // Lower limit for the control variable y.sub.t [%] 110 LMN_MK : LReal := 100.0; // Upper limit for the control variable y.sub.t [%] 111 TUN_ACCURACY : LReal := 1.0E−07; // Computational accuracy for auto-tuning 112 TUN_ERR_TOLER_MN : LReal := 0.0; // Minimum control error e.sub.t to stop auto-tuning 113 TUN_ERR_TOLER_MK : LReal := 100.0; // Maximum control error e.sub.t to stop auto-tuning 114 TUN_COMPL_TM_MN : Time := T#1S; // Minimum delay to stop auto-tuning 115 TUN_COMPL_TM_MK : Time := T#1M; // Maximum delay to stop auto-tuning 116 CF_MN : LReal := 0.0001; // Lower limit for an adjustment speed 117 CF_MK : LReal := 1.0; // Upper limit for an adjustment speed 118 END_VAR 119 120 BEGIN 121 // Reset an error code of the A-PID controller 122 #ERR_CODE := 0; 123 124 IF #CYCLE < T#10MS OR #CYCLE > T#10S THEN 125 // E01 = Sampling time CYCLE is out of the range [10 ms. .10 s] 126 #ERR_CODE := 1; 127 #y.out := 0.0; 128 ELSIF #LMN_LLM > #LMN_HLM THEN 129 // E02 = Lower limit for the control variable LMN_LLM > upper limit LMN_HLM 130 #ERR_CODE := 2; 131 #y.out := 0.0; 132 ELSIF #LMN_LLM < #LMN_MN THEN 133 // E03 = Lower limit for the control variable LMN_LLM < 0 % 134 #ERR_CODE := 3; 135 #y.out := 0.0; 136 ELSIF #LMN_HLM > #LMN_MK THEN 137 // E04 = Upper limit for the control variable LMN_HLM > 100 % 138 #ERR_CODE := 4; 139 #y.out := 0.0; 140 ELSE 141 // Convert the sampling time dt to seconds 142 #Ts := DINT_TO_LREAL(TIME_TO_DINT(#CYCLE)) / 1000.0; 143 144 // Convert the sampling time dt to IEC high resolution time 145 #LT_CYCLE := TIME_TO_LTIME(#CYCLE); 146 147 // Check the permissible values of the A-PID controller parameters 148 #GAIN := LIMIT(IN := #GAIN, MN := #GAIN_MN, MX := #GAIN_MK); 149 #TI := LIMIT(IN := #TI, MN := #LT_CYCLE, MX := #TI_MK); 150 #TD := LIMIT(IN := #TD, MN := LT#0NS, MX := #TD_ME); 151 #TUN_ERR_TOLER := LIMIT(IN := #TUN_ERR_TOLER, MN := #TUN_ERR_TOLER_MN, 152 MX := #TUN_ERR_TOLER_MX); 153 #TUN_COMPL_TM := LIMIT(IN := #TUN_COMPL_TM, MN := #TUN_COMPL_TM_MN, 154 MX := #TUN_COMPL_TM_MX); 155 #LMN_MAN := LIMIT(IN := #LMN_MAN, MN := #LMN_MN, MX := #LMN_MX) ; 156 157 // Save the previous control errors 158 #e.t4 := #e.t3; 159 #e.t3 := #e.t2; 160 #e.t2 := #e.t1; 161 #e.t1 := #e.t; 162 163 // Calculate a control error e.sub.t according to the specified control direction 164 IF #INV_CONTROL THEN 165 #e.t := #PV − #SP; 166 ELSE 167 #e.t := #SP − #PV; 168 END_IF; 169 170 // Output an actual control error e.sub.t 171 IF #INV_CONTROL THEN 172 #CTRL_ERR := −#e.t; 173 ELSE 174 #CTRL_ERR := #e.t; 175 END_IF; 176 177 // Activate the A-PID controller in automatic mode 178 IF #AUTO_ON AND NOT #MAN_ON THEN 179 // Stop condition for auto-tuning of the A-PID controller 180 #T_TUN_MON(IN:= #SELF_TUN_ON AND ABS(#e.t) <= #TUN_ERR_TOLER, PT: = #TUN_COMPL_TM); 181 IF #T_TUN_MON.Q THEN 182 #SELF_TUN_ON := FALSE; 183 #PASS_NO := 0; 184 END_IF; 185 186 // Convert a reset time T.sub.i to seconds 187 #r.TI := LINT_TO_LREAL(LTIME_TO_LINT(#TI)) / 1.0E+9; 188 189 // Calculate an integral action coefficient K.sub.i 190 #Ki := #GAIN / #r.TI; 191 192 // Convert a derivative time T.sub.d to seconds 193 #r.TD := LINT_TO_LREAL(LTIME_TO_LINT(#TD)) / 1.0E+9; 194 195 // Calculate a derivative action coefficient K.sub.d 196 #Kd := #GAIN * #r.TD; 197 198 // Save a proportional coefficient K.sub.p 199 #Kp := #GAIN; 200 201 (************************************************************************ 202 * AUTO-TUNING OF THE A-PID CONTROLLER PARAMETERS * 203 ************************************************************************) 204 IF #SELF_TUN_ON AND ABS (#e.t4) > 0.0 AND ABS (#d.y_t1) > 0.0 THEN 205 // Calculate a 2.sup.nd order differential d.sup.2y.sub.t for the control variable y.sub.t 206 #d2.y := #d.y - #d.y_t1; 207 208 // Calculate a 1.sup.st order differential de.sub.t for a control error e.sub.t 209 #d.e := #e.t − #e.t1; 210 211 // Calculate a 2.sup.nd order differential d.sup.2e.sub.t for a control error e.sub.t 212 #d2.e := #e.t − 2.0 * #e.t1 + #e.t2; 213 214 // Calculate a 3.sup.rd order differential d.sup.2e.sub.t for a control error e.sub.t 215 #d3.e := #e.t − 3.0 * #e.t1 + 3.0 * #e.t2 − #e.t3; 216 217 // Calculate upper limits for the adjustment speeds of controller parameters 218 #e.sgr := #e.t * #e.t; 219 #a_mx_Kp := LIMIT(IN := #e.sgr * #e.sgr / 2.71828, MN := #CF_MN, MX := #CF_ME); 220 #a_mx_Ki := LIMIT(IN := 4.0 * ABS(#e.sgr * #e.t) / 2.71828, 221 MN := #CF_MN, MX := #CF_ME); 222 #a_mx_Kd := LIMIT(IN := ABS(#e.t * #e.sgr * #e.sgr) / 13.5914, 223 MN := #CF_MN, MX := #CF_ME); 224 (************************************************************************ 225 * AUTO-TUNING OF THE PROPORTIONAL PART * 226 ************************************************************************) 227 // Perform auto-tuning for the proportional coefficient K.sub.p 228 IF #PASS_NO = 0 THEN 229 // Calculate an adjustment step value dK.sub.p for the proportional coefficient K.sub.p 230 IF ABS(#d.e) > #TUN_ACCURACY THEN 231 #fact_1 := #d2.y − #Ki * #d.e * #Ts − #Kd * #d3.e / #Ts; 232 #fact_2 := #d.y − #Ki * #e.t * #Ts − #Kd * #d2.e / #Ts; 233 #d.Kp := (#fact_1 * #d.e − #fact_2 * #d2.e) / (#d.e * #d.e); 234 ELSE 235 #d.Kp := 0.0; 236 END_IF; 237 238 // Calculate an adjustment speed a.sub.p for the proportional coefficient K.sub.p 239 #divisor := #d.Kp * #d.e; 240 IF ABS(#divisor) > #TUN_ACCURACY THEN 241 #a.Kp := (#Kp * #d.e + #Kd * #d2.e / #Ts − #d.y) * #Ts / #divisor; 242 #a.Kp := LIMIT(IN := #a.Kp, MN := #CF_MN, MX := #a_mx_Kp); 243 ELSE 244 #a.Kp := #CF_MN; 245 END_IF; 246 247 // Adjust the proportional coefficient K.sub.p 248 #dKp := LIMIT(IN := #a.Kp * #d.Kp, MN := −0.5, MX := 0.5); 249 #GAIN := LIMIT(IN := #GAIN − #dKp , MN := #GAIN_MN, MX := #GAIN_MX); 250 #Kp := #GAIN; 251 END_IF; 252 253 (************************************************************************ 254 * AUTO-TUNING OF THE INTEGRAL PART * 255 ************************************************************************) 256 // Perform auto-tuning for the integral action coefficient K.sub.i 257 IF #PASS_NO = 1 THEN 258 // Calculate an adjustment step value dK.sub.i for integral action coefficient K.sub.i 259 IF ABS(#e.t) > #TUN_ACCURACY THEN 260 #fact_1 := #d2.y − #Kp * #d2.e − #Kd * #d3.e / #Ts; 261 #fact_2 := #d.y − #Kp * #d.e − #Kd * #d2.e / #Ts; 262 #d.Ki := (#fact_1 * #e.t − #fact_2 * #d.e) / (#e.t * #e.t * #Ts); 263 ELSE 264 #d.Ki := 0.0; 265 END_IF; 266 267 // Calculate an adjustment speed a.sub.i for the integral action coefficient K.sub.i 268 #divisor := #d.Ki * #e.t; 269 IF ABS(#divisor) > #TUN_ACCURACY THEN 270 #a.Ki := (#d.y − #Ki * #e.t * #Ts − #Kd * #d2.e / #Ts) * #Ts / #divisor; 271 #a.Ki := LIMIT(IN := #a.Ki, MN := #CF_MN, MX := #a_mx_Ki); 272 ELSE 273 #a.Ki := #CF_MN; 274 END_IF; 275 276 // Adjust the integral action coefficient K.sub.i 277 #dKi := LIMIT(IN := #a.Ki * #d.Ki, MN := −0.5, MX := 0.5); 278 #Ki := LIMIT(IN := #Ki + #dKi, 279 MN := #GAIN * 1.0E+9 / LINT_TO_LREAL(LTIME_TO_LINT(#TI_MX)), 280 MX := #GAIN / #Ts); 281 282 // Convert K.sub.i to a reset time T.sub.i [sec] 283 #r.TI := #GAIN / #Ki; 284 285 // Convert a reset time T.sub.i [sec] to IEC high resolution time 286 #TI := LINT_TO_LTIME(LREAL_TO_LINT(#r.TI * 1.0E+9)); 287 END_IF; 288 289 (************************************************************************ 290 * AUTO-TUNING OF THE DERIVATIVE PART * 291 ************************************************************************) 292 // Perform auto-tuning for the derivative action coefficient K.sub.d 293 IF #PASS_NO = 2 THEN 294 // Calculate an adjustment step value dK.sub.d for the derivative action coeff. K.sub.d 295 IF ABS(#d2.e) > #TUN_ACCURACY THEN 296 #fact_1 := #d2.y − #Kp * #d2.e − #Ki * #d.e * #Ts; 297 #fact_2 := #d.y − #Kp * #d.e − #Ki * #e.t * #Ts; 298 #d.Kd := (#fact_1 * #d2.e − #fact_2 * #d3.e) * #Ts / (#d2.e * #d2.e); 299 ELSE 300 #d.Kd := 0.0; 301 END_IF; 302 303 // Calculate an adjustment speed a.sub.d for the derivative action coefficient K.sub.d 304 #divisor := #d.Kd * #d2.e; 305 IF ABS(#divisor) >#TUN_ACCURACY THEN 306 #a.Kd := (#Kp * #d.e + #Kd * #d2.e / #Ts − #d.y) * #Ts * #Ts / #divisor; 307 #a.Kd := LIMIT(IN := #a.Kd, MN := #CF_MN, MX := #a_mx_Kd); 308 ELSE 309 #a.Kd := #CF_MN; 310 END_IF; 311 312 // Adjust the derivative action coefficient K.sub.d 313 #dKd := LIMIT(IN := #a.Kd * #d.Kd, MN := −0.5, MX := 0.5); 314 #Kd := LIMIT(IN := #Kd − #dKd, MN := 0.0, 315 MX := #GAIN * LINT_TO_LREAL(LTIME_TO_LINT(#TD_MX))/ 1.0E+9); 316 317 // Convert K.sub.d to a derivative time T.sub.d [sec] 318 #r.TD := #Kd / #GAIN; 319 320 // Convert a derivative time T.sub.d [sec] to IEC high resolution time 321 #TD := LINT_TO_LTIME(LREAL_TO_LINT(#r.TD * 1.0E+9)); 322 END_IF; 323 324 // Increase a pass counter by one for auto-tuning 325 #PASS_NO := #PASS_NO + 1; 326 327 // Reset a pass counter if it is greater than 2 328 IF #PASS_NO > 2 THEN 329 #PASS_NO := 0; 330 END_IF; 331 ELSE 332 #PASS_NO := 0; 333 END_IF; 334 335 (**************************************************** 336 * DRIVE CONTROL * 337 ****************************************************) 338 // Save a control variable change dy.sub.t 339 #d.y_t1 := #d.y; 340 341 // Calculate a control variable change dy.sub.t+i for the time t+1 342 #d.y := (#Kp + #Ki * #Ts + #Kd / #Ts) * #e.t − (#Kp + 2.0 * #Kd / #Ts) * #e.t1 + 343 #Kd / #Ts * #e.t2; 344 345 // Modify the control variable y.sub.t at time t 346 #y.out := LIMIT(IN := #y.out + #d.y, MN := #LMN_LLM, MX := #LMN_HLM); 347 ELSE 348 IF #MAN_ON THEN 349 // Use the control variable for manual mode 350 #y.out := LIMIT(IN := #LMN_MAN, MN := #LMN_MN, MX := #LMN_MX); 351 ELSE 352 // Reset a control variable y.sub.t at standstill 353 #y.out := #LMN_MN; 354 END_IF; 355 // Reset the internal controller variables 356 #d.y := #d.y_t1 := 0.0; 357 #e.t := #e.t1 := #e.t2 := #e.t3 := 0.0; 358 #PASS_NO := 0; 359 END_IF; 360 END_IF; 361 362 // Move the actual control variable y.sub.t to the control variable for manual mode 363 #LMN_MAN := #y.out; 364 365 // Output a control variable y.sub.t 366 #LMN := #y.out; 367 368 END_FUNCTION_BLOCK

ADAPTIVE TUNING METHOD FOR A DIGITAL PID CONTROLLER

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G05B2219/13103

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G05B6/02

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G05B13/024

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G05B13/0245

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International classification

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G05B6/02

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Abstract

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