Lecture 31
Gain Scheduling and Dynamic Inversion
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Topicsz A Brief Overview of Gain Scheduling
Philosophy Steps Issues
z Dynamic Inversion (DI) Design Philosophy Steps Advantages
Gain Scheduling An Accepted Practice in Industry
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Philosophy of Gain Scheduling
z Design local linear controllers, based on linearization of the nonlinear system at several expected operating points (based on the operating values of the scheduling variable(s)).
z Obtain a nonlinear controller for the nonlinear system by interpolating (scheduling) the linear gains, based on the actual value of the scheduling variable(s).
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Practice in Industryz The scheduling variable may or may not be one/more
state variable(s). Usual practice is to choose a slowly varying meaningful output variable as the scheduling variable. In flight control applications, usually dynamic pressure and Mach number are chosen as the scheduling variables
z The robustness, performance, stability etc. are usually justified from extensive simulation studies.
z The design results in a semi-global controller, depending on the selection of the operating points and validity of linear gains.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Steps of Gain Schedulingz Linearize the dynamics about an
operating point
z Design the controller gain based on it
z Repeat this procedure about several operating points (within the domain of interest) and store the gains
z Interpolate the gains online and obtain a nonlinear controller
X AX BU= +
U K X= K
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Issuesz Is the linearized system good enough?z How many operating points to consider?z Does the interpolated gain lead to stability?
There are examples of loss of stability because of scheduling!
z How easy it is to select a set of schedulingvariables?
z What if the parameters of the model are updated? z Higher dimensional scheduling vector leads to
complexity in interpolations.z Loss of Minimum-phase property: A serious issue!
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Reference
J. S. Shamma and M. Athans: Gain Scheduling:
Potential Hazards and Possible Remedies, IEEE
Control System Magazine, June 1992, pp.101-107.
Dynamic Inversion: A Promising Substitute for Gain Scheduling
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Philosophy ofDynamic Inversion
x
yxy
?
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Philosophy of Dynamic Inversionz Carryout a co-ordinate transformation such
that the problem appears to be linear in the transformed co-ordinates.
z Design the control for the linear-looking system using the linear control design techniques.
z Obtain the controller for the original system using an inverse transformation.
z Intuitively, the control design is carried out by enforcing a stable linear error dynamics.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Problem
z System Dynamics:
z Goal (Tracking):
z Special Class:(control affine & square)
( )( )
,
, ,n m p
X f X U
Y h X
X U Y
==
R R R
( )( )
*
*
,
Assumption: is smooth
Y Y t t
Y t
( ) ( )( ), non-singular Y
X f X g X U
p m g X t
= + =
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Dynamic Inversion Design
z Derive the output dynamics:
z Define Error of Tracking:
( ) ( ){ }( ) ( )Y Y
hY XXh f X g X UX
f X g X U
= = +
= +
( ) ( ) ( )*E t Y t Y t
( ) ( )( )
X f X g X U
Y h X
= + =Known:
1 1
1
1
n
p p
n
h hx x
hX
h hx x
"
# % #
"
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Dynamic inversion
z Select a fixed gain such that:
z Carry out the algebra:
z Solve for the controller:
0K >
( ) ( ) ( ){ }1 * *Y YU g X Y K Y Y f X=
( ) ( )( ) ( ) ( )
* *
* *
0
Y Y
Y Y K Y Y
f X g X U Y K Y Y
+ =+ =
00 0,KtE K E E e E t+ = =
( )Usually 1/ , 0i iK diag = >
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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A Demonstrative Toy Problemz System Dynamics:z Objective:z Solution:
Desired output: Desired error dynamics:
Control solution:
( ) ( )2 21x x x x u= + + +( ) 0 as y x t=
* *0 , 0y t y t= = ( ) ( ) ( )( ) ( )
( ) ( ){ }* *
2 2
0, 1/ 0
0 0 0
1 0
y y k y y k
x k x
x x x u k x
+ = = > + =
+ + + + =
( )2211u x x k xx = + +
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Is a first-order error dynamics always enforced?
Answer: No!The order of the error dynamics is dictated by the relative degree of the problem, which is defined as the number of times the output needs to be differentiated so that the control variable appears explicitly.If a second-order error dynamics needs to be enforced, then the corresponding equation is
0V PE K E K E+ + = ( ) ( ) ( )2Usually 2 , , , 0i i iV i n P n i nK diag K diag = = >
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Example
z System Dynamics:
z Objective:
z Output:
( ) ( )2 2 21 1 2 1 1 22 1
2 3 2 5 1
3
x x x x x x u
x x
= + + + +=
[ ]1 2 0,Tx x as t ( )
( ) ( )2
2 1
2 2 21 1 2 1 1 2
selecting a proper is crucial!3
3 3 2 3 2 5 1
y x yy x x
y x x x x x x u
== =
= = + + + +
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Examplecontd.z Solution:
Desired output: Desired error dynamics:
Control solution:
* * *0 0y t y y t= = = ( )*e y y2
22 2 2
2 0
2 0n n
n n
e e ex x x
+ + =
+ + =
0 1, 0n < < >
( ) ( )2 21 2 1 1 22 21 21 3 2 3 2 6
15 1 n nu x x x x x
x x = + + + +
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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When Does the DynamicInversion Fail?
:
1. Differentiate repeatedly until the input appears. 2. Design to cancel the nonlinearity.
Q : Is it always possible
Fundamental Principle of Dynamic Inversion
y uu
to design this way?
Ans: Not necessarily ! It is possible only if the relative degree is "well-defined".
u
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Undefined Relative degree
0
Undefined relative degree : It may so happen that upon sucessive
differentiation of , appears . However , the coefficient of mayvanish at , whereas it is non-zero at points arbitrarly close to
y u uX
( )
( )2
0
0
1
2 1 221
1 1 1 2
1 2 1 2 1 1
.In such cases, the relative degree is undefined at .
02Ex : 1,
, : Some nonlinear functionThen 2 2
2 2 2 ,
XX
xxu
x x x
y xy x x x xy x x x x x x x
= +
== == + = +
[ ] 222u x+
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Undefined Relative degree( )
( ) ( )N
( )2 2
21 1 2 2 1
1
1
1
1 2
1
2 , 2 + 2
As 0 , ( ) 0.
Hence, at 0 the relative degree is .
: If one choses then ,and the coeffic
,
y x x x x x ug Xf X yy
x g xyx
y xy x xy x x x u
= +
= ==
== == = +
Note
not defined
ient of 1 0 globally.In this case the relative degree is .globally
u = well defined
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Advantages of DI designz Simple design: No need of tedious gain scheduling
(hence, sometimes dynamic inversion is known as a universal gain scheduling design).
z Easy online implementation: It leads to a closed form solution for the controller.
z Asymptotic (rather exponential) stability is guaranteed for the Error dynamics (subjected to the control availability, this is true globally).
z No problem if parameters are updated (the updated values can simply be used in the formula).
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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