This article presents one of the applications of fractional-order models in modelling the dynamics of air temperature changes in residential spaces. The characteristics of these models include the order of the derivative which is a real number and, in more general cases, a complex number. These models are a further generalization of the traditional dynamic model but, thanks to the fact that derivatives are not real numbers, they can be used to model dynamics of much more complex physical phenomena. This article undertakes to identify the parameters of the fractional-order model which models the dynamics of air temperature changes inside buildings. Measurement data which were registered during experiment were used in the identification process.

This paper describes a design scheme for terminal sliding mode controllers of certain types of non-linear dynamical systems. Two classes of such systems are considered: the dynamic behavior of the first class of systems is described by non-linear second-order matrix differential equations, and the other class is described by non-linear first-order matrix differential equations. These two classes of non-linear systems are not completely disjointed, and are, therefore, investigated together; however, they are certainly not equivalent. In both cases, the systems experience unknown disturbances which are considered bounded. Sliding surfaces are defined by equations combining the state of the system and the expected trajectory. The control laws are drawn to force the system trajectory from an initial condition to the defined sliding surface in finite time. After reaching the sliding surface, the system trajectory remains on it. The effectiveness of the approaches proposed is verified by a few computer simulation examples.

The paper presents a design scheme of the linear dynamic feedback controller for some non-linear systems. These systems are mathematically described by matrix non-linear differential equations of the first and second orders. A first-order form of the studied systems includes some types of differential-algebraic equations. The stability property of the non-linear systems with the linear controller is assured by an appropriate definition of the system output, and the linear dynamic compensator is an important part of the feedback control system. The order of the dynamic part is equal to the size of the system input and is independent of the size of the system state vector. The asymptotic stability in the Lyapunov sense is analysed and proved by the use of Lyapunov functionals and LaSalle’s invariance principle. Stabilisation in a wide range of controller parameters improves the system’s robustness.

The paper presents a method for reducing the order of the thermal dynamics model of a building or room, obtained by application of a lumped parameter method. The reduced model has the same dynamic characteristics as the primary model, while additionally it is an observable one. The described model order reduction method involves the identification and aggregation of non-distinguishable state variables. The paper presents a sample calculation confirming the validity of the deliberations. The proposed reduction method of state variables can be used with other dynamic systems provided that the assumptions concerning the structures of the system matrices are fulfilled. Practical application: The use of simplified but accurate mathematical models of the dynamics of heat in buildings enables the efficient design of high-performance control systems, e.g. the stabilisation of air temperature in the building. Simplified mathematical models also require less computing power and can be implemented in embedded control systems. This gives the ability to design, for example, distributed systems of predictive control, which requires mathematical models of controlled processes.