Arrays of LTI Models (Creating and Manipulating Models)

Creating and Manipulating Models

Special Cases for Operations on LTI Arrays

There are some special cases in coding operations on LTI arrays.

Consider

```
sysa = op(sys1,sys2)
```

where op is a symbol for the operation being applied. sys1 is an LTI array, and sysa (the result of the operation) is an LTI array with the same array dimensions as sys1. You can use shortcuts for coding sysa = op(sys1,sys2) in the following cases:

For operations that apply to LTI arrays, sys2 does not have to be an array. It can be a single LTI model (or a gain matrix) whose I/O dimensions satisfy the compatibility requirements for op (with those of each of the models in sys1). In this case, op applies sys2 to each model in sys1, and the kth model in sys satisfies

```
sysa(:,:,k) = op(sys1(:,:,k),sys2)
```

For arithmetic operations, such as +, *, /, and \, sys2 can be either a single SISO model, or an LTI array of SISO models, even when sys1 is an LTI array of MIMO models. This special case relies on MATLAB's scalar expansion capabilities for arithmetic operations.

When sys2 is a single SISO LTI model (or a scalar gain), op applies sys2 to sys1 on an entry-by-entry basis. The ijth entry in the kth model in sysa satisfies

    sysa(i,j,k) = op(sys1(i,j,k),sys2)

When sys2 is an LTI array of SISO models (or a multidimensional array of scalar gains), op applies sys2 to sys1 on an entry-by-entry basis for each model in sysa.

    sysa(i,j,k) = op(sys1(i,j,k),sys2(:,:,k))

Examples of Operations on LTI Arrays with Single LTI Models

Suppose you want to create an LTI array containing three models, where, for in the set , each model has the form

You can do this efficiently by first setting up an LTI array h containing the SISO models and then using concatenation to form the LTI array H of MIMO LTI models , . To do this, type

tau = [1.1 1.2 1.3];
for i=1:3                       % Form LTI array h of SISO models.
  h(:,:,i)=tf(1,[1 tau]); 
end
H = [h 0; -1 tf(1,[1 0])]; %Concatenation: array h & single models
size(H)
3x1 array of continuous-time transfer functions
Each transfer function has 2 output(s) and 2 input(s).

Similarly, you can use append to perform the diagonal appending of each model in the SISO LTI array h with a fixed single (SISO or MIMO) LTI model.

S = append(h,tf(1,[1 3])); % Append a single model to h.

specifies an LTI array S in which each model has the form

You can also combine an LTI array of MIMO models and a single MIMO LTI model using arithmetic operations. For example, if h is the LTI array of three SISO models defined above,

```
[h,h] + [tf(1,[1 0]);tf(1,[1 5])]
```

adds the single one-output, two-input LTI model [1/s 1/(s + 5)] to every model in the 3-by-1 LTI array of one-output, two-input models [h,h]. The result is a new 3-by-2 array of models.

Examples: Arithmetic Operations on LTI Arrays and SISO Models

Using the LTI array of one-output, two-input state-space models [h,h], defined in the previous example,

```
tf(1,[1 3]) + [h,h]
```

adds a single SISO transfer function model to each entry in each model of the LTI array of MIMO models [h,h].

Finally,

```
G = rand(1,1,3,1);
sysa = G + [h,h]
```

adds the array of scalars to each entry of each MIMO model in the LTI array [h,h] on a model-by-model basis. This last command is equivalent to the following for loop.

hh = [h,h];
for k = 1:3
  sysa(:,:,k) = G(1,1,k) + hh(:,:,k);
end

Dimension Requirements Other Operations on LTI Arrays