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# Documentation

## Numeric Model of SISO Feedback Loop

This example shows how to interconnect Numeric LTI Models representing multiple system components to build a single numeric model of a closed-loop system, using model arithmetic and interconnection commands.

Construct a model of the following single-loop control system.

The feedback loop includes a plant G(s), a controller C(s), and a representation of sensor dynamics, S(s). The system also includes a prefilter F(s).

1. Create model objects representing each of the components.

```G = zpk([],[-1,-1],1);
C = pid(2,1.3,0.3,0.5);
S = tf(5,[1 4]);
F = tf(1,[1 1]);```

The plant G is a zero-pole-gain (zpk) model with a double pole at s = –1. Model object C is a PID controller. The models F and S are transfer functions.

2. Connect the controller and plant models.

`H = G*C;`

To combine models using the multiplication operator *, enter the models in reverse order compared to the block diagram.

 Tip   Alternatively, construct H(s) using the series command: `H = series(C,G);`
3. Construct the unfiltered closed-loop response $T\left(s\right)=\frac{H}{1+HS}$.

`T = feedback(H,S);`
 Caution   Do not use model arithmetic to construct T algebraically:`T = H/(1+H*S)`This computation duplicates the poles of H, which inflates the model order and might lead to computational inaccuracy.
4. Construct the entire closed-loop system response from r to y.

`T_ry = T*F;`

T_ry is a Numeric LTI Model representing the aggregate closed-loop system. T_ry does not keep track of the coefficients of the components G, C, F, and S.

You can operate on T_ry with any Control System Toolbox™ control design or analysis commands.