MPC, system identification, load-introduced disturbance, LASSO, convex optimization
A dynamic model of a building's temperature is necessary for model-based control of building HVAC (Heating Ventilation and Air Conditioning) systems. Due to the complexity of thermal dynamics, system identification from data is considered advantageous, from which a particular challenge may arise: temperature is affected by large, unknown disturbances, especially the cooling load induced by the occupants. Many system identification methods ignore these disturbances, which can produce highly erroneous results, or use a specialized test building to measure the occupant induced load. In this paper we propose a method that simultaneously identifies a dynamic model of a building's temperature in the presence of large, unmeasured disturbances, and a transformed version of the unmeasured disturbance from easily measurable input-output data. The proposed method, which we call SPDIR (Simultaneous Plant and Disturbance Identication through Regularization) is based on solving a l1-regularized least-squares problem, where l1 penalty encourages the identified transformed disturbance to be sparse. The motivation for this is that the disturbance, which consists mostly of internal load due to occupants, is often piecewise-constant. We show that this makes the transformed disturbances an approximately sparse signal, motivating the use of l1 regularization. The selection of regularization parameter can be crucial to the identification accuracy of the l1 regularized-least-square problems. There are a number of heuristics for the classic Lasso (least absolute shrinkage and selection operator). However, the conditions required for their application do not hold in our case. We therefore propose a distinct heuristic to choose λ. The method proposed here can enforce properties of the system that are known from the physics of the thermal processes, including stability and signs of DC gains for certain input-output pairs. Such properties are interpreted into convex constraints and augmented to the proposed problem. Thus the our method consists of solving a convex optimization problem, which we prove to be feasible and regular. Our method uses data collected during regular operation of a building and does not need data collected when the building is empty. Even when data from unoccupied periods is available, assuming the disturbance to be zero during that time is not desirable since doing so will prevent the disturbance from absorbing model mismatch. We test our method via simulations, and results indicate that the method can accurately estimate the thermal dynamic model and transformed disturbance with both open-loop and closed-loop data, even when the disturbance does not satisfy the piecewise constant property. The main advantages of the method is posing the estimation problem as a convex optimization problem with constraints from physical insights about the system and the disturbances, without requiring specially collected data. Previous methods lacked both convexity and/or physically meaningful constraints. The main limitation is that the identified disturbance is a linear transformation of the true disturbance with unknown coefficients.