# US 8,255,066 B2 · Method and apparatus for tuning a PID controller

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- Number: US 8,255,066 B2
- Document ID: US8255066B2
- Status: Granted
- Kind: B2
- Application number: 12/780,822
- Filed: 2010-05-14
- Published: 2010-11-18
- Granted: 2012-08-28
- Earliest claimed: 2009-05-18
- Assignees: IMB Controls Inc.
- Inventors: Igor Boiko; Mikhail Boiko; Natalia Boiko; Alexander Boiko
- Web page: https://patentgrep.com/patent/US8255066B2

## CPC

- G05B 23/0256: {injecting test signals and analyzing monitored process response, e.g. injecting the test signal while interrupting the normal operation of the monitored system; superimposing the test signal onto a control signal during normal operation of the monitored system}
- G05B 19/042: using digital processors (G05B19/05 takes precedence)
- G05B 2219/41025: Detect oscillation, unstability of servo and change gain to stabilize again
- G05B 2219/41077: Self tuning, test run, detect, compute optimal backlash, deformation compensation
- G05B: CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS (systems for controlling or regulating non-electric variables G05D; systems for regulating electric or magnetic variables G05F; control devices or systems insofar as characterised by mechanical features only G05G)

## Abstract

In tuning a PID controller for a process in a feedback control system, a method is provided for bringing the system into symmetric self-excited oscillations for measuring the frequency and the amplitude of the oscillations, and tuning the controller in dependence on the measurements obtained. A control algorithm referred to as the modified relay feedback test is introduced into a system in series with a process to generate self-excited oscillations. Tuning includes the steps of selecting a desired gain margin or phase margin, generating oscillations with the algorithm parameter corresponding to a selected gain or phase margin, measurements of the amplitude and the frequency of these oscillations, and computing PID controller tuning parameters. Data and formulas are given for the computation of the specific parameter of the modified relay feedback test and for tuning the parameters of the PID controller. An apparatus for performing the method is disclosed.

## Claims

### Claim 1

A method of tuning a proportional-integral-derivative controller, further referred to as a PID controller, having a transfer function
W
c
⁡
(
s
)
=
K
c
(
1
+
1
T
ic
⁢
s
+
T
dc
⁢
s
)
,
where s is a Laplace variable and Kc, Tic, and Tdc are tuning parameters, for a process in a feedback control system comprising the steps of:
(a) selection of a desired gain margin γm of the system for tuning for a gain margin specification or desired phase margin φm, of the system for tuning for a phase margin specification;
(b) determining the values of the parameters c1, c2, and c3 that define optimal tuning rules of the PID controller from the desired gain margin γm of the system for tuning for a gain margin specification or the desired phase margin φm of the system for tuning for a phase margin specification selected at the previous step;
(c) bringing the control system into symmetric self-excited oscillations of a frequency at which a phase response of the process is equal to the desired value equal to ψ−180°, where −90°<ψ<90°;
(d) measurement of a frequency of the oscillations Ω0, and an amplitude of the oscillations α0;
(e) calculation of the tuning parameters of the PID controller in dependence on the measurements obtained at the previous step.

### Claim 2 (depends on 1)

The method as recited in claim 1, wherein said parameters c1, c2, and c3 that define the tuning rules are selected from the following table for tuning for the gain margin specification γm=2 for a proportional, proportional-integral, and PID controllers, respectively,
Controller
c1
c2
c3
Proportional
0.50
0
0
Proportional-integral
0.49
0.80
0
PID
0.46
0.50
0.12

### Claim 3 (depends on 1)

The method as recited in claim 1, wherein said parameters c1, c2, and c3 that define the tuning rules are selected from the following table for tuning for the phase margin specification Φm=45° for a proportional, proportional-integral, and PID controller, respectively,
Controller
c1
c2
c3
Proportional
1.0
0
0
Proportional-integral
0.98
0.80
0
PID
0.92
0.50
0.12

### Claim 4 (depends on 1)

The method as recited in claim 1, wherein said parameters c2, and c3 that define the tuning rules are selected from the table for a proportional-integral and PID controller, respectively,
Controller
c2
c3
Proportional-
0.80
0
integral
PID
0.50
0.12
plus or minus 20% and parameter c1 is calculated for tuning for a gain margin specification as follows:
c
1
=
1
γ
m
⁢
1
+
(
2
⁢
π
⁢
⁢
c
3
-
1
2
⁢
π
⁢
⁢
c
2
)
2
.

### Claim 5 (depends on 1)

The method as recited in claim 1, wherein said parameters c2, and c3 that define the tuning rules are selected from the table for a proportional-integral and PID controller, respectively,
Controller
c2
c3
Proportional-
0.80
0
integral
PID
0.50
0.12
plus or minus 20%, and parameter c1 is calculated for tuning for a phase margin specification as follows:
c
1
=
1
1
+
(
2
⁢
π
⁢
⁢
c
3
-
1
2
⁢
π
⁢
⁢
c
2
)
2
.

### Claim 6 (depends on 1)

The method as recited in claim 1, wherein said bringing the control system into symmetric self-excited oscillations is done via the introduction into the system the following control algorithm further referred to as modified relay feedback test with parameter β, in series with the process, so that the error signal σ is the difference between the set point and the process control variable, the error signal is an input to the modified relay feedback test, and the output u of the modified relay feedback test is an input to the process:
u
⁡
(
t
)
=
{
h
if
⁢
⁢
σ
⁡
(
t
)
≥
Δ
1
⁢
⁢
or
⁢
⁢
(
σ
⁡
(
t
)
>
-
Δ
2
⁢
⁢
and
⁢
⁢
u
⁡
(
t
-
)
=
h
)
-
h
if
⁢
⁢
σ
⁡
(
t
)
≤
Δ
2
⁢
⁢
or
⁢
⁢
(
σ
⁡
(
t
)
<
Δ
1
⁢
⁢
and
⁢
⁢
u
⁡
(
t
-
)
=
-
h
)
where t is current time, Δ1=βσmax, Δ2=−βσmin, σmax and σmin are last singular points of the error signal σ(t) corresponding to the last maximum and minimum values of σ(t), respectively, after crossing the zero level, h is the control amplitude, u(t−) is the output of the modified relay feedback test at time immediately preceding current time, and parameter βof the modified relay feedback test is computed as β=sinψ, ψ is a parameter.

### Claim 7 (depends on 6)

The method as recited in claim 6, wherein said parameter ψ is calculated as follows:
ψ
=
-
arc
⁢
⁢
tg
(
2
⁢
π
⁢
⁢
c
3
-
1
2
⁢
π
⁢
⁢
c
2
)
for tuning for the gain margin specification.

### Claim 8 (depends on 6)

The method as recited in claim 6, wherein said parameter ψ is calculated as follows:
ψ
=
ϕ
m
+
arc
⁢
⁢
tg
(
1
2
⁢
π
⁢
⁢
c
2
-
2
⁢
π
⁢
⁢
c
3
)
for tuning for the phase margin specification.

### Claim 9 (depends on 6)

The method as recited in claim 6, wherein said tuning parameters are calculated as follows:
K
c
=
c
1
⁢
4
⁢
h
π
⁢
⁢
a
0
,
⁢
T
ic
=
c
2
⁢
2
⁢
π
Ω
0
,
⁢
T
dc
=
c
3
⁢
2
⁢
π
Ω
0
.

### Claim 10 (depends on 6)

The method as recited in claim 6, wherein said output of the modified relay feedback test and the process control variable are increments from the corresponding values in a steady operating point to which in a preferred embodiment the control system is brought by means of not optimally tuned but stable PID controller.