Optimal Control and Time Scales

My main research field is optimal control. My thesis was derived from a project dealing with the optimal control of the hot water in a building, using a large underground water storage. In fact, this storage had, thermally, an inter-seasonal capacity. My soon-to-be colleague used a particular method to drive this storage: he only considered the energy balance after a long time. My work was to find a rationale for this scheme.
It turned out that the hot water usage in the attached building, and the underground storage, lived in two very distinct time scale. From the point of view of the long term storage, it was efficient to consider the variations of its thermal capacity over a long period. This amounts to integrating (or averaging) the thermal flux over a long horizon, with respect to the variation of the water usage in the building. In fact, the system was managed using averaging in optimal control.

However, this method used an explicit two time scales formulation for the time varying features of the system. This means that there are two time variables in the model. This is obviously not physical. So I was desperately looking for a real life way of computing local averages. This is how my focus then shifted to multiresolution approximations and wavelets. It turns out that, because of boundary effects, the only relevant basis for ordinary differential equations is the Haar basis. Numerical averaging is obtained by using averages of the dynamics over adjacent rectangular windows. A preprint for this work is available. Current work is about the generalization of this method to two time scaled systems.

On the applicative side, I have focused on energy systems. The systems that I have studied share a common feature: the cost bears only on the control, which is the energy injected into the system. As a consequence, the dynamics is involved only if there is some kind of state constraint.

A first example of energy systems features the control of hybrid automotive vehicles. The vehicles use alternatively a classic thermal engine and an electric one. In this example, the trajectory, and consequently the requested power, is known in advance. We only pay for the fuel, and the control consists in choosing between fuel and electricity over the optimization horizon. Only the final state is constrained. This aim of the study is to determine the impact of various neglected dynamics. It is shown that this brings little improvement to the performance. A theoretical paper was derived from this work, which generalizes the robustness of optimal control subject to functional errors bounded in magnitude (regular perturbations) to the case where the control is constrained. Applications involved taking into account the temperature of the thermal engine and the temperature of the exhaust catalyst.

Another example is the thermal control of a building which must satisfy constraints relative to the comfort of the occupants of the building. The system is controlled by various electric apparatus. Time varying inputs influence the behavior of the system, essentially due to the weather conditions and the varying cost of electricity. We pay for the electricity, and the state constrain bears on the inner temperature. This was the opportunity for Paul Malisani, together with the lab team, to show that interior point methods are efficient in finding optimal controls with constraints on the state and the control. The theoretical paper is here. On the applicative side, this kind of control was proven to be effective in regulating the load of the electrical system for modern buildings.

This led to a fruitful collaboration with Center for Energy Efficiency. A full solution involved Model Predictive Control, an observer and a low level controller was first studied. It was also shown that efficient decomposition coordination for the optimal control of large, multi-zone buildings can be found even in presence of state constraints (work in progress).

Another feature of this kind of energy systems is the presence of multiple time scales. Indeed, the air temperature and the temperature of the heavy structures of the building (concrete, for instance) live in different time scales. Additionally, weather chronicles feature multiple time scale cycles. It was first shown that suitable methods must be used to identify such systems (available is a theoretical paper and an applicative one). Even when starting from complex physical models and reducing them by balanced realizations, the multiple time scale feature remains. This is further enhanced when deep probes into the ground are used to store thermal energy across the seasons. Together with the multiple time scales of the system's sollicitions, this raises the question of the optimal control of multiple time scale systems subject to multiple time scale excitation (work in progress).