## Supported systems

A YopSystem is a symbolic representation of a model. It handles semi-expicit differential algebraic equations (DAEs) of differential index 1 on the form:

dx = f(t, x, z, u, w, p)  % Differential equation
0 == g(t, x, z, u, w, p)  % Algebraic equation


where t represents the independent variable (typically time), x the state variable, z the algebraic variable, u the control input, w external inputs (exogenous variable), and p model parameters.

## Declaration

To create a YopSystem it is necessary to specify the dimension of the variable vectors. This is done using the following syntax:

mySystem = YopSystem(...
'states', numberOfStates, ...
'algebraics', numberOfAlgebraics, ...
'controls', numberOfControlInputs, ...
'externals', numberOfExternalInputs, ...
'parameters', numberOfParameters ...
)


If any of dimensions is 0, that key-value pair can be omitted.

If the model is implemented in a function, and that function has the argument order [dx, alg, signals] = f(t, x, z, u, w, p), this can included in the command which will automatically set the system dynamics:

mySystem = YopSystem(...
'states', numberOfStates, ...
'algebraics', numberOfAlgebraics, ...
'controls', numberOfControlInputs, ...
'externals', numberOfExternalInputs, ...
'parameters', numberOfParameters, ...
'model', @myModelFunction ...
)


When running the command the output looks like (provided a model function handle was given):

>> mySystem

mySystem =

YopSystem with properties:



#### YopSystem variables

The shorthand properties t, x, z, u, w, p, ode, ae, y contains the symbolic variables and expressions representing the model, we may refer to these as the system variables. The same information is contained in the other entries, these are used internally, but can of course be used by the user, if that is preferred. The shorthand properties can be changed by modifying the file yopCustomPropertyNames.m.

### The 'model' entry

The entry 'model' in the declaration assumes a function handle on the following form:

function [dx, alg, signals] = myModelFunction(t, x, z, u, w, p)


dx is the evaluation of dx = f(t, x, z, u, w, p), and alg the evaluation of g(t, x, z, u, w, p) , both of these must be column vectors. signals represent any signal specified by the user. Opposite to dx and alg, this can be of arbitrary type.

### When your system do not require all of the arguments

If your system do not need one or several of the inputs, then that entry can be omitted, except for the independent variable. For instance if your system only takes states and controls as input, the declaration looks as follows:

mySystem = YopSystem(...
'states', numberOfStates, ...
'controls', numberOfControlInputs, ...
'model', @myModelFunction ...
);


This means that correspoding outputs should also be removed (i.e. alg, as the system lacks an algebraic variable):

function [dx, ~] = myModelFunction(t, x, u)


where ~ can be replaced by a signals if desired.

### When the model file do not match the argument order

If your model file do not match the specified argument order, there are two alternative ways of specifying the system dynamics. The simplest way is to use an anonymous function to reorder the arguments:

mySystem = YopSystem(...
'states', numberOfStates, ...
'algebraics', numberOfAlgebraics, ...
'controls', numberOfControlInputs, ...
'externals', numberOfExternalInputs, ...
'parameters', numberOfParameters, ...
);


The other way of doing it is to omit the model entry. Then you use the symbolic variables to obtain expressions, either by using the variables to compute the output expressions, or using them to make a call to a model file:

mySystem = YopSystem(...
'states', numberOfStates, ...
'algebraics', numberOfAlgebraics, ...
'controls', numberOfControlInputs, ...
'externals', numberOfExternalInputs, ...
'parameters', numberOfParameters ...
);

% Order arbitrary, but default used for convenience
[dx, alg, signals] = myModelFunction(mySystem.t, mySystem.x, mySystem.z, ...
mySystem.u, mySystem.w, mySystem.p);

% Set model expressions
mySystem.set('ode', dx);
mySystem.set('alg', alg);

% Not necessary, but can be convenient
mySystem.set('y', signals);