# 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, $$T$$, 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, then`T`

is a SISO`genss`

model.If

`location`

is a vector signal, or specifies multiple analysis points, then`T`

is a MIMO`genss`

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:

$$T=\frac{L}{1-L}=S-1.$$

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`

.

$$\begin{array}{l}y=-GK(y+dy)\\ \to y=-GKy-GKdy\\ \to (I+GK)y=-GKdy\\ \to y=\underset{T}{\underbrace{-{(I+GK)}^{-1}GK}}dy.\end{array}$$

Here, *I* is an identity matrix of the same size as *G**K*. 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:

$$T=\left[\begin{array}{cc}{T}_{du\to u}& {T}_{dy\to u}\\ {T}_{du\to y}& {T}_{dy\to y}\end{array}\right].$$

## 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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