# MPC

Model Predictive Control (MPC) computes a feedback control by repeatedly solving (open loop) optimal control problems

# Model Predictive Control

**Coordination:**Prof. Dr. Lars Grüne

**Contact:**Prof. Dr. Lars Grüne

**Represented in MODUS since**September 2016

**Members with experience in this area:**

**The method "Model Predictive Control":**

## What is this?

Model Predictive Control (MPC) is a control method in which a feedback control (i.e., a state dependent control input) is computed by repeatedly solving open loop optimal control problems based on current measurements of the system state. The initial piece of the resulting time dependent optimal control function is implemented at the system until the next optimization time instant.

## What is it good for?

The advantage of MPC compared to other feedback design methods is that optimization objectives and constraints on control inputs and states can be explicitly incorporated in the computation. Its disadvantage is the computational effort of repeatedly solving optimal control problems. For this reason, MPC was traditionally mainly used in chemical process control, in which the optimization criterion consists in penalizing the distance of the state (temperature, concentrations, pressure, ...) to a desired set point, a so called stabilization problem. In this application the maintenance of constraints (maximal temperature, maximal pressure, ...) is particularly important. At the same time the dynamics of chemical processes is usually so slow that the optimization can be performed online even with relatively slow optimization algorithms. Due to the enormous progress in optimization algorithms and available computer hardware MPC can nowadays be applied to much faster processes, for instance for controlling electric motors or combustion engines. From a theoretical point of view the method is interesting because in recent years it was shown that for many classes of optimization criteria MPC yields an approximately optimal solution for *infinite horizon* optimal control problems. These are in general very difficult to solve with other methods, while the *finite horizon* optimal control problems which need to be solved in each step of the MPC algorithms are relatively easy to solve. MPC thus provides an approximate solution method for infinite horizon optimal control problems, which is also interesting for economic applications.

## Where have we applied it?

The members of MODUS have in particular made important contributions to the qualitative and quantitative analysis of the solution trajectories delivered by the MPC methods. Moreover, MPC was and is applied to the following problems:

- distributed control of smart grids with battery storage and a high penetration of renewable energy
- energy efficient heating, ventilation and air conditioning (HVAC) of buildings
- efficient start up of gas power plants
- solution of various economic optimal control problems
- control of probability density functions via the Fokker-Planck equation
- solution of various optimal control problems governed by partial differential equations