US 12,235,638 B2 · 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 Kp, Ki and Kd is performed in a closed control loop with negative feedback separately in time, i.e. independently of each other in iteration steps k for Kp, m for Ki and n for Kd (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.

An adaptive tuning method for parameters of a digital PID controller, which is characterized by a cyclic sequence of operations to adjust in equal time intervals only one PID parameter Kp, Ki, or Kd at any time point t in a closed control loop by analog feedback of the actual value xt by means of a pass counter and three additional indices k for Kp, m for Ki, and n for Kd used as iteration steps, and:

1) A tuning equation to calculate an adjustment step value dKp k for a proportional coefficient Kp in iteration step k at time t:
dK
p
k
=
(
d
2
⁢
y
t
-
K
i
m
-
1
·
de
t
·
dt
-
K
d
n
-
1
·
d
3
⁢
e
t
dt
)
·
de
t
(
de
t
)
2
-
(
dy
t
-
K
i
m
-
1
·
e
t
·
dt
-
K
d
n
-
1
·
d
2
⁢
e
t
dt
)
·
d
2
⁢
e
t
(
de
t
)
2
,
where:
dt is the sampling time of a digital PID controller;
dyt is a control variable change at time t, which is determined as dyt=yt−yt−1, ∀t≥1 on the condition that dy0=y0:
d2yt is the 2nd order differential of the control variable yt at time t, which is calculated as d2yt=dyt−dy−1, ∀t≥1 on the condition that d2y0=dy0;
et is the control error between the setpoint w and the actual value xt at time t, which is calculated as et=w−xt, ∀t≥0;
det is a 1st order differential of the control error et at time t, which is calculated as det=et−et−1, ∀t≥1 on the condition that de0=e0;
d2et is a 2nd order differential of the control error et at time t, which is calculated as d2et=et−2·et−1+et−2, ∀t≥2 on the condition that d2e0=e0 and d2e1=e1−2·e0;
d3et is a 3rd order differential of the control error et at time t, which is calculated as d3et=et−3·et−1·et−2−et−3;
∀t≥3 on the condition that d3e0=e0, d3e1=e1−3·e0, and d3e2=e2−3·e1+3·e0;
Ki m−1 is the actual integral action coefficient Ki at time t, which was modified in iteration step m−1;
∀m>1 on the condition that Ki 0 was assigned a certain initial value for Ki;
Kd n−1 is the actual derivative action coefficient Kd at time t, which was modified in iteration step n−1, ∀n>1 on the condition that Kd 0 was assigned a certain initial value for Kd,
2) a tuning equation to calculate the adjustment speed αp k for the proportional coefficient Kp in iteration step k at time t:
α
p
k
=
K
p
k
-
1
·
de
t
+
K
d
n
-
1
·
d
2
⁢
e
t
dt
-
dy
t
dK
p
k
·
de
t
·
dt
,
α
p
k
∈
[
0.001
,
e
t
4
2.71828
∈
[
0.0001
,
1
]
]
,
where:
Kp k−1 is the actual proportional coefficient Kp at time t, which was modified in iteration step k−1;
∀k>1 on the condition that Kp 0 was assigned a certain initial value for Kp,
3) a tuning equation of the proportional coefficient Kp in a negative direction for iteration step k:
4) a tuning equation to calculate the adjustment step value dKi m for an integral action coefficient Ki in iteration step m at time t:
dK
i
m
=
(
d
2
⁢
y
t
-
K
p
k
·
d
2
⁢
e
t
-
K
d
n
-
1
·
d
3
⁢
e
t
dt
)
·
e
t
e
t
2
·
dt
-
(
dy
t
-
K
p
k
·
de
t
-
K
d
n
-
1
·
d
2
⁢
e
t
dt
)
·
de
t
e
t
2
·
dt
,
5) a tuning equation to calculate the adjustment speed αi m for the integral action coefficient Ki in iteration step m at time t:
α
i
m
=
dy
t
-
K
i
m
-
1
·
e
t
·
dt
-
K
d
n
-
1
·
d
2
⁢
e
t
dt
dK
i
m
·
e
t
·
dt
,
α
i
m
∈
[
0.001
,
4
·
❘
"\[LeftBracketingBar]"
e
t
3
❘
"\[RightBracketingBar]"
2.71828
∈
[
0.0001
,
1
]
]
6) a tuning equation of the integral action coefficient Ki in a positive direction for iteration step m:
7) a tuning equation to calculate the adjustment step value dKd n for a derivative action coefficient Kd in iteration step n at time t:
dK
d
n
=
(
(
d
2
⁢
y
t
-
K
p
k
·
d
2
⁢
e
t
-
K
i
m
·
de
t
·
dt
)
·
d
2
⁢
e
t
(
d
2
⁢
e
t
)
2
-
(
dy
t
-
K
p
k
·
de
t
-
K
i
m
·
e
t
·
dt
)
·
d
3
⁢
e
t
(
d
2
⁢
e
t
)
2
)
·
dt
,
8) a tuning equation to calculate the adjustment speed αd n for the derivative action coefficient Kd in iteration step n at time t:
α
d
n
=
K
p
k
·
de
t
+
K
d
n
-
1
·
d
2
⁢
e
t
dt
-
dy
t
dK
d
n
·
d
2
⁢
e
t
·
dt
2
,
α
d
n
∈
[
0.001
,
❘
"\[LeftBracketingBar]"
e
t
5
❘
"\[RightBracketingBar]"
13.5914
∈
[
0.0001
,
1
]
]
9) a tuning equation of the derivative action coefficient Kd in a negative direction for iteration step n:
Kdn=Kdn−1−αdn·dKdn,
αdn·dKdn∈[−0.5,+0.5]
and using the above tuning equations derived by reverse engineering of the digital PID controller to automatically adjust the PID controller parameters;
and control a controlled system based on the adjusted parameters of the PID controller.

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Adaptive tuning method for a digital PID controller

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Adaptive tuning method for a digital PID controller

Life of this patent

Abstract

Claims2