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## MIMO Control System

This example shows how to build a MIMO control system using `connect` to interconnect Numeric LTI models and tunable Control Design Blocks.

Consider the following two-input, two-output control system.

The plant G is a distillation column with two inputs and two outputs. The two inputs are the reflux L and boilup V. The two outputs are the concentrations of two chemicals, represented by the vector signal y = [y1,y2]. You can represent this plant model as:

`$G\left(s\right)=\frac{1}{75s+1}\left[\begin{array}{cc}87.8& -86.4\\ 108.2& -109.6\end{array}\right].$`

The vector setpoint signal r = [r1,r2] specifies the desired concentrations of the two chemicals. The vector error signal e represents the input to D, a static 2-by-2 decoupling matrix. CL and CV represent independent PI controllers that control the two inputs of G.

To create a two-input, two-output model representing this closed-loop control system:

1. Create a Numeric LTI model representing the 2-by-2 plant G.

```s = tf('s','TimeUnit','minutes'); G = [87.8 -86.4 ; 108.2 -109.6]/(75*s+1); G.InputName = {'L','V'}; G.OutputName = 'y'; ```

When you construct the closed-loop model, `connect` uses the input and output names to form connections between the block diagram components. Therefore, you must assign names to the inputs and outputs of the transfer function `G` in either of the following ways: .

• You can assign input and output names to individual signals by specifying signal names in a cell array, as in ```G.InputName = {'L','V'}```

• Alternatively, you can use vector signal naming, which the software automatically expands. For example, the command ```G.OutputName = 'y'``` assigns the names `'y(1)'` and `'y(2)'` to the outputs of `G`.

2. Create tunable Control Design Blocks representing the decoupling matrix D and the PI controllers CL and CV.

```D = tunableGain('Decoupler',eye(2)); D.u = 'e'; D.y = {'pL','pV'}; C_L = tunablePID('C_L','pi'); C_L.TimeUnit = 'minutes'; C_L.u = 'pL'; C_L.y = 'L'; C_V = tunablePID('C_V','pi'); C_V.TimeUnit = 'minutes'; C_V.u = 'pV'; C_V.y = 'V';```
 Note:   `u` and `y` are shorthand notations for the `InputName` and `OutputName` properties, respectively. Thus, for example, entering:```D.u = 'e'; D.y = {'pL','pV'};```is equivalent to entering:```D.InputName = 'e'; D.OutputName = {'pL','pV'};```
3. Create the summing junction.

The summing junction produces the error signals e by taking the difference between r and y.

`Sum = sumblk('e = r - y',2);`

`Sum` represents the transfer function for the summing junction described by the formula `'e = r - y'`. The second argument to `sumblk` specifies that the inputs and outputs of `Sum` are each vector signals of length 2. The software therefore automatically assigns the signal names `{'r(1)','r(2)','y(1)','y(2)'}` to `Sum.InputName` and `{'e(1)','e(2)'}` to `Sum.OutputName`.

4. Join all components to build the closed-loop system from r to y.

`CLry = connect(G,D,C_L,C_V,Sum,'r','y');`

The arguments to the `connect` function include all the components of the closed-loop system, in any order. `connect` automatically combines the components using the input and output names to join signals.

The last two arguments to `connect` specify the output and input signals of the closed-loop model, respectively. The resulting `genss` model `CLry` has two-inputs and two outputs.