ROM Framework

Inputs and Outputs (u/y)

Inputs and outputs represent the communication that the ROM has with the outside world (i.e., other blocks to which it might be connected) through explicit ports.

INPUT ports are used to send the input signals to the ROM.
OUTPUT ports are used by the ROM to expose the calculated output signals to the outside world.

The ports are user-defined. The same physical problem can be modeled differently depending on the user perspective, such as how the ROM is configured to communicate with the outside world, and how the ROM is intended to be deployed.

State Variables (x)

State variables are defined as the set of variables used to describe the mathematical state of a dynamical system.

In the absence of any external input (e.g., force or torque), the state of a system reveals enough about the system to predict future behavior. A linear, time-invariant system with no external outputs is described by this set of equations:

\{\dot{x}\} = A \{x\}, \{x\} = \{\begin{matrix} x_{1} \\ ⋮ \\ x_{n} \end{matrix}\}

where:

x is the system state vector.
$x_{i} (i = 1, \dots, n)$

Typical choices of state variables for different physical systems include:


System	State Variables
Mechanical	Position of a mechanical part. Velocity of a mechanical part.
Thermodynamical	Internal Energy, Enthalpy, Entropy, Temperature, Pressure, and Volume.
Electrical	Voltage across nodes. Currents through components.

Since romAI can be used for all the above-mentioned physics, you can choose the state variables based on what is most suitable for your problem. There is not one unique set of state variables that will lead to good results. Some sets will lead to better results than others.

Moreover, states variables might not be needed at all if the system exhibits a zero-order behavior. In this case, romAI can still be used to generate a static ROM.

Characteristic Parameters (ϴ)

These are a set of parameters that define the equations that link inputs and states to outputs.

Observe the case of a linear, time-invariant system with external inputs:

\{\dot{x}\} = A \{x\} + B \{u\}, A = [\begin{matrix} a_{1, 1} & \dots & a_{1, n} \\ ⋮ & ⋱ & ⋮ \\ a_{n, 1} & \dots & a_{n, n} \end{matrix}], B = [\begin{matrix} b_{1, 1} & \dots & b_{1, m} \\ ⋮ & ⋱ & ⋮ \\ b_{n, 1} & \dots & m_{n, m} \end{matrix}]

The characteristic parameters are the coefficient a_(i,j) and b_(i,j) of the matrices A and B. These parameters are automatically computed by the algorithm leveraged in romAI.

More generally, the romAI algorithm automatically determines the mathematical equations that are in play and that connect inputs, state variables and outputs.

These equations generally can be written:

\begin{array}{l} \dot{x} = f (x, u, θ, t) \\ y = g (x, u, θ, t) \end{array}

where f() and g() are nonlinear functions.