Modelling module

The main component of the HILO-MPC Modelling module is the Model class. For a more detailed description of the methods refer to the API.

Models are used to describe the dynamics of systems. The HILO-MPC Model class supports the following system properties:

• linear/nonlinear system dynamics

• continuous/discrete time dynamics

• time-invariant/time-variant systems

In addition to ordinary differential equations (ODEs), systems with continuous time dynamics can also be modelled using differential algebraic equations (DAEs). Systems with discrete time dynamics can be designed analogously. Optionally, measurement equations mapping the states of a model to some measurable quantity of the system can be supplied as well.

Note

Observers usually require measurement equations to be present. If no measurement equations were supplied during model setup the observers will assume that all states can be measured.

A general time-invariant continuous-time nonlinear system can be described by the following DAEs

\[\begin{split}\dot{x}(t)&=f(x(t),z(t),u(t_k),p),\\ 0&=g(x(t),z(t),u(t_k),p),\\ y&=h(x(t),z(t),u(t_k),p),\end{split}\]

where \(x\in\mathbb{R}^{n_x}\) is the state vector, \(z\in\mathbb{R}^{n_z}\) the vector of algebraic variables, \(u\in\mathbb{R}^{n_u}\) is the input vector and \(p\in\mathbb{R}^{n_p}\) is a vector of parameters. Since the system is time-invariant only the states \(x\), algebraic variables \(z\) and the input \(u\) depend on the time \(t\). Due to practical reasons the input \(u\) is assumed to be piecewise constant (hence the index \(k\)). The function \(f\colon\mathbb{R}^{n_x+n_z+n_u+n_p}\mapsto\mathbb{R}^{n_x}\) represents the ODEs of the model and the function \(g\colon\mathbb{R}^{n_x+n_z+n_u+n_p}\mapsto\mathbb{R}^{n_z}\) depicts the algebraic equations forming a semi-explicit DAE system overall.

Note

The HILO-MPC Model class only supports semi-explicit DAE systems of index 1. The index of 1 indicates that the DAE system can be transformed into a pure ODE system by differentiating the algebraic equations once.

The function \(h\colon\mathbb{R}^{n_x+n_z+n_u+n_p}\mapsto\mathbb{R}^{n_y}\) describes the measurement equations mapping the states and algebraic variables to some measurable quantities.

In the time-variant case the functions \(f\), \(g\) and \(h\) additionally depend explicitly on the time as well

\[\begin{split}\dot{x}(t)&=f(x(t),z(t),u(t_k),p,t),\\ 0&=g(x(t),z(t),u(t_k),p,t),\\ y&=h(x(t),z(t),u(t_k),p,t).\end{split}\]

A general time-invariant discrete-time nonlinear system can be described by the following set of equations

\[\begin{split}x_{k+1}&=\tilde{f}(x_k,z_k,u_k,p),\\ 0&=\tilde{g}(x_k,z_k,u_k,p),\\ y_k&=h(x_k,z_k,u_k,p),\end{split}\]

Model Setup

Running Simulations

Model Solution

Model Discretization

Model Linearization