getCompSensitivity
Complementary sensitivity function from generalized model of control system
Description
returns the complementary
sensitivity measured at the specified location for a generalized model of
a control system.T
= getCompSensitivity(CL
,location
)
specifies additional loop openings for the complementary sensitivity function
calculation. Use an opening, for example, to calculate the complementary sensitivity
function of an inner loop, with the outer loop open.T
= getCompSensitivity(CL
,location
,opening
)
If opening
and location
list the same
point, the software opens the loop after adding the disturbance signal at the
point.
Examples
Complementary Sensitivity Function at a Location
Compute the complementary sensitivity at the plant output, X
, of the control system of the following illustration.
Create a model of the system by specifying and connecting a numeric LTI plant model G
, a tunable controller C
, and the AnalysisPoint
block X
. Use the AnalysisPoint
block to mark the location where you assess the complementary sensitivity, which in this example is the plant output.
G = tf([1],[1 5]); C = tunablePID('C','p'); C.Kp.Value = 3; X = AnalysisPoint('X'); CL = feedback(G*C,X);
CL
is a genss
model that represents the closed-loop response of the control system from r to y. Examine the Control Design Blocks of the model.
CL.Blocks
ans = struct with fields:
C: [1x1 tunablePID]
X: [1x1 AnalysisPoint]
The model's blocks include the AnalysisPoint
block, X
, that identifies the analysis-point location.
Calculate the complementary sensitivity, T, at X
.
T = getCompSensitivity(CL,'X')
Generalized continuous-time state-space model with 1 outputs, 1 inputs, 1 states, and the following blocks: C: Tunable PID controller, 1 occurrences. X: Analysis point, 1 channels, 1 occurrences. Type "ss(T)" to see the current value and "T.Blocks" to interact with the blocks.
getCompSensitivity
preserves the Control Design Blocks of CL
, and returns a genss
model. To get a numeric model, you can convert T
to transfer-function form, using the current value of the tunable block.
Tnum = tf(T)
Tnum = From input "X" to output "X": -3 ----- s + 8 Continuous-time transfer function.
Specify Additional Loop Opening for Complementary Sensitivity Function Calculation
In the multiloop system of the following illustration, calculate the inner-loop sensitivity at the output of G2
, with the outer loop open.
Create a model of the system by specifying and connecting the numeric plant models, tunable controllers, and AnalysisPoint
blocks. G1
and G2
are plant models, C1
and C2
are tunable controllers, and X1
and X2
are AnalysisPoint
blocks that mark potential loop-opening locations.
G1 = tf(10,[1 10]); G2 = tf([1 2],[1 0.2 10]); C1 = tunablePID('C','pi'); C2 = tunableGain('G',1); X1 = AnalysisPoint('X1'); X2 = AnalysisPoint('X2'); CL = feedback(G1*feedback(G2*C2,X2)*C1,X1);
Calculate the complementary sensitivity, , at X2
, with the outer loop open at X1
. Specifying X1
as the third input argument tells getCompSensitivity
to open the loop at that location.
T = getCompSensitivity(CL,'X2','X1'); tf(T)
ans = From input "X2" to output "X2": -s - 2 ---------------- s^2 + 1.2 s + 12 Continuous-time transfer function.
Input Arguments
CL
— Model of control system
generalized state-space model
Model of a control system, specified as a generalized state-space model
(genss
).
Locations at which you can perform sensitivity analysis or open loops are
marked by AnalysisPoint
blocks in
CL
. Use getPoints(CL)
to get the
list of such locations.
location
— Location
character vector | cell array of character vectors
Location at which you calculate the complementary sensitivity function, specified as a character vector or cell array of character vectors. To extract the complementary sensitivity function at multiple locations, use a cell array of character vectors.
Each specified location must match an analysis point in
CL
. Analysis points are marked using
AnalysisPoint
blocks. To get the list of available
analysis points in CL
, use
getPoints(CL)
.
Example: 'u'
or
{'u','y'}
opening
— Additional loop opening
character vector | cell array of character vectors
Additional loop opening used to calculate the complementary sensitivity function, specified as a character vector or cell array of character vectors. To open the loop at multiple locations, use a cell array of character vectors.
Each specified opening must match an analysis point in
CL
. Analysis points are marked using
AnalysisPoint
blocks. To get the list of available
analysis points in CL
, use
getPoints(CL)
.
Use an opening, for example, to calculate the complementary sensitivity function of an inner loop, with the outer loop open.
If opening
and location
list the
same point, the software opens the loop after adding the disturbance signal
at the
point.
Example: 'y_outer'
or
{'y_outer','y_outer2'}
Output Arguments
T
— Complementary sensitivity function
generalized state-space model
Complementary
sensitivity function of the control system,
T
, measured at location
,
returned as a generalized state-space model (genss
).
If
location
specifies a single analysis point, thenT
is a SISOgenss
model.If
location
is a vector signal, or specifies multiple analysis points, thenT
is a MIMOgenss
model.
More About
Complementary Sensitivity
The complementary sensitivity function, T, at a point is the closed-loop transfer function around the feedback loop measured at the specified location. It is related to the open-loop transfer function, L, and the sensitivity function, S, at the same point as follows:
Use getLoopTransfer
and getSensitivity
to compute
L and S.
Consider the following model:
The complementary sensitivity, T, at y
is
defined as the transfer function from dy
to
y
.
Observe that, in contrast to the sensitivity function, the disturbance,
dy
, is added after the measurement,
y
.
Here, I is an identity matrix of the same size as GK. The complementary sensitivity transfer function at
y
is equal to -1
times the closed-loop
transfer function from r
to y
.
Complementary sensitivity at multiple locations, for example, u
and y
, is defined as the MIMO transfer function from the
disturbances to measurements:
Version History
Introduced in R2014a
See Also
getPoints
| AnalysisPoint
| genss
| getLoopTransfer
| systune
| getIOTransfer
(Simulink Control Design) | getSensitivity
| getValue
| getCompSensitivity
(Simulink Control Design)
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