Scaling and parameterizing a controller
Controller scaling and parameterization are described. Techniques that can be improved by employing the scaling and parameterization include, but are not limited to, controller design, tuning and optimization. The scaling and parameterization methods described here apply to transfer function based controllers, including PID controllers. The parameterization methods also applies to state feedback and state observer based controllers, as well as linear active disturbance rejection controllers.
1. A method for processing controller parameters, comprising:
selecting a controller to parameterize;
identifying two or more gains for the controller;
mathematically relating the two or more gains to a single controller tuning parameter to yield a parameterized controller, wherein the mathematically relating includes generating, for the two or more gains, respective two or more functions, wherein an only variable of the respective two or more functions is the single controller tuning parameter, wherein the single controller tuning parameter is a control system bandwidth ω c of the controller, and wherein the control system bandwidth ω c defines a common location of N poles of an Nth order closed-loop system comprising the parameterized controller and a plant, where N is an integer; and
relating the parameterized controller with the single controller tuning parameter to the plant to yield the closed-loop system.
2. The method of claim 1 , wherein the selecting the controller comprises selecting a controller that is a proportional/integral/derivative (PID) controller having parameters that include one or more of proportional gain k p , integral gain k i , or derivative gain k d , wherein the one or more of the proportional gain k p , the integral gain k i or the derivative gain k d are defined as functions of only the control system bandwidth ω c .
3. The method of claim 1 , further comprising relating a model of the controller G c to a model of the plant G p in terms of the single controller tuning parameter ω c, , a Laplace Transform variable s, and a coefficient ξ according to:
G p (s)
1
s
+
1
1
s
1
s
2
+
2
ξ
s
+
1
1
s
(
s
+
1
)
1
s
2
G c (s, ω c )
ω
c
(
s
+
1
)
s
ω c
ω
c
2
s
2
+
2
ξ
s
+
1
s
(
s
+
2
ω
c
)
ω
c
2
(
s
+
1
)
s
+
2
ω
c
ω
c
2
s
s
+
2
ω
c
.
4. The method of claim 1 , further comprising obtaining a model of the controller G c via a loop shaping design having a form of equation (36):
G
c
(
s
)
=
(
s
+
ω
1
s
)
m
1
s
ω
c
+
1
1
(
s
ω
2
+
1
)
n
G
p
-
1
(
s
)
(
36
)
wherein n is such that
1
s
ω
c
+
1
1
(
s
ω
c
+
1
)
n
G
p
-
1
(
s
)
is proper and G p is a model of the plant, ω c is the control system bandwidth of the controller, ω 1 is a lower constraint, ω 2 is an upper constraint, and n and m are both integers.
5. The method of claim 1 , wherein the selecting the controller comprises selecting a controller that is a state feedback (SF) controller.
6. The method of claim 5 , wherein the closed-loop system includes a state observer (SO) having an observer gain L.
7. The method of claim 1 , wherein the closed-loop system includes a linear extended state observer (LESO).
8. The method of claim 1 , wherein the controller is a linear active disturbance rejection controller (LADRC).
9. The method of claim 1 , wherein selecting the controller comprises selecting a transfer function based (TFB) controller.
10. The method of claim 6 , further comprising parameterizing the observer gain L to be a function of a single tuning parameter ω o , wherein ω o is an observer bandwidth.
11. The method of claim 7 , further comprising parameterizing the LESO to be a function of a single tuning parameter ω o , wherein ω o is an observer bandwidth, and wherein M observer poles are placed at −ω o for an Mth order plant, where M is an integer.
12. A method for processing controller parameters, comprising:
selecting a controller for control of an Nth order plant, wherein the controller is defined by two or more controller coefficients;
placing N poles of the controller at a common location defined by a controller bandwidth ω c , of the controller, where N is an integer;
parameterizing the controller to generate a parameterized controller, wherein the parameterizing includes defining the two or more controller coefficients using different algebraic functions expressed in terms of only the controller bandwidth ω c ;
relating the parameterized controller to the Nth order plant to yield a closed-loop system comprising the parameterized controller and the Nth order plant; and
configuring the parameterized controller for control of the Nth order plant by selecting a value for the controller bandwidth.
13. The method of claim 1 , wherein the N poles of an Nth order closed-loop system are located at −ω c .
14. The method of claim 13 , further comprising tuning the controller for control of the plant by optimizing the control system bandwidth ω c .
15. The method of claim 5 , wherein the state feedback controller has a form of equation (4):
u=r+K{circumflex over (x)} (4)
where u is a control input, r is a reference, K is a state feedback matrix, and {circumflex over (x)} is an estimated state vector.
16. The method of claim 6 , wherein the state observer has a form of equation (6):
{circumflex over (x)}=A{circumflex over (x)}+Bu+L ( y−ŷ ) (6)
where {circumflex over (x)} is a derivative of the estimated state vector, A is a state matrix, B is an input matrix, L is an observer gain, y is an output, and ŷ is an observed output.
17. The method of claim 7 , wherein the LESO has a form of equation (43) for a 2 nd order plant and a form of equation (60) for an n th order plant:
ż=Az+Bu+L ( y−ŷ) (43)
{
z
.
1
=
z
2
-
β
1
(
z
1
-
y
(
t
)
)
z
.
2
=
z
3
-
β
2
(
z
1
-
y
(
t
)
)
⋯
z
.
n
=
z
n
+
1
-
β
n
(
z
1
-
y
(
t
)
)
+
b
0
u
z
.
n
+
1
=
-
β
n
+
1
(
z
1
-
y
(
t
)
)
(
60
)
where z is an observer state, ż is a derivative of the observer state, A is a state matrix, B is an input matrix, L is an observer gain, y is an output, ŷ is an observed output, and β i , i=1, 2, ...n+1, are coefficients of the observer gain.
18. The method of claim 8 , wherein the LADRC has a form of equations (43), (44), (45), (47), (49) and (54) for a 2 nd order plant and a form of equations (60), (62), (64), (65), and (66) for an n th order plant:
ż=Az+Bu+L ( y−ŷ )
ŷ=Cz (43)
L=[β 1 β 2 β 3 ] T (44)
u
=
-
z
3
+
u
0
b
0
(
45
)
u 0 =k p ( r−z 1 )− k d z 2 (47)
k d =2ξω c and k p =ω c 2 (49)
β 1 =3ω o , β 2 =3ω o 2 , β 3 =ω o 3 (54)
{
z
.
1
=
z
2
-
β
1
(
z
1
-
y
(
t
)
)
z
.
2
=
z
3
-
β
2
(
z
1
-
y
(
t
)
)
⋯
z
.
n
=
z
n
+
1
-
β
n
(
z
1
-
y
(
t
)
)
+
b
0
u
z
.
n
+
1
=
-
β
n
+
1
(
z
1
-
y
(
t
)
)
(
60
)
u
=
z
n
+
1
+
u
0
b
0
(
62
)
u 0 =k p ( r−z 1 )− k d 1 z 2 − . . . −k d n−1 z n (64)
s n +k d n−1 s n−1 + . . . +k d 1 s+k p =( s+ω c ) n (65)
s n +β 1 s n−1 + . . . +β n−1 s+β n =( s+ω o ) n (66)
where z is an observer state, {circumflex over (z)} is a derivative of the observer state, A is a state matrix, B is an input matrix, L is an observer gain, y is an output, ŷ is an observed output, β n are coefficients of the observer gain, u is a control input, {circumflex over (x)} is an estimated state vector, k are gains, ω c is the control system bandwidth, and ω o is an observer bandwidth.
19. The method of claim 1 , wherein the selecting the controller comprises:
selecting a known controller for a second plant that is analogous to the first plant;
identifying one or more scalable parameters of the known controller, the one or more scalable parameters comprising at least one of a frequency or a gain; and
scaling the one or more scalable parameters of the known controller to yield the controller.
20. The method of claim 19 , wherein the scaling comprises scaling the one or more scalable parameters based on at least one of an identified frequency scale or an identified gain scale of the second plant.
21. The method of claim 12 , wherein the placing the N poles comprises placing the N poles at ω c .