modelPseudoInversion1

Information

This example demonstrates the mixing of two fluid streams, A and B, assuming constant specific heat capacities c_p. The mixing equation is:

m_flow_A * T_A + m_flow_B * T_B = (m_flow_A + m_flow_B) * T_mix;
  

It can be easily solved for one unknown. This test demonstrates the "TFS way" of determining the mass flow rate m_flow_B such that the mixing temperature T_mix = 25 °C.

Instead of using a nonlinear equation solver (e.g., Inversion1), the TFS approach uses the time integration solver to solve the "nonlinear equation".

With a reasonable start value integrator.y_start and integrator gain integrator.k, the time integration solver will converge. Specifically, convergence occurs for y_start >= 0, k > 0 or y_start < 0, k < 0, and divergence otherwise.

A larger integrator gain k results in faster convergence.

Mathematically, we introduce an auxiliary equation:

der(m_flow_B) = f(m_flow_B)

Because the time integrator is usually implicit, it also solves a nonlinear equation system:

m_flow_B_k+1 = m_flow_B_k + F(f(m_flow_B_k+1))

For example, using implicit Euler:

m_flow_B_k+1 = m_flow_B_k + dt * f(m_flow_B_k+1)

This effectively iterates in time. An explicit solver would limit the integrator gain k and is therefore not recommended when aiming for high gains. In essence, this defines a custom algorithm for solving nonlinear equation systems.

The advantages of this approach are:

• Start values can be more explicitly defined than with, for example, Inversion1.
• The approach can handle limitations and resembles a model-based feed-forward control, instead of an explicit inversion G⁻¹(s).

The disadvantages are:

• It introduces "pseudo" transients that might be undesirable.
• The time integration solver may have a lower convergence order. Step-size adaptive solvers may be strong, but using a controller essentially defines a custom nonlinear solver, which is usually less robust than established algorithms. Large gains or small time constants may result in stiff systems that are difficult to solve.
• The model-based feed-forward should ideally be stable, fast, without steady-state error, and should not introduce oscillations.
• It cannot produce an exact inverse model like G⁻¹(s).

For design purposes, inversion is beneficial when only the quasi-stationary solution is sought. For dynamic simulations, especially with limitations, the model-based feed-forward approach can be very effective.

Components

Revisions

• 2026, by Raphael Gebhart (raphael.gebhart@dlr.de):
  Initial version.