Study of robust stability of indoor temperature control system

The stages of design and study of the indoor temperature control system have been considered. An assessment of the possibilities of the indoor temperature control of a discrete model of the control system has been carried out. The transfer function of the control object with interval coefficients of the numerator and denominator polynomials taking into account possible ranges of change of the greatest time constant responsible for the enclosing structures has been obtained. The applied graphical method of studying the obtained system has provided an output of the family of root trajectories on a complex plane and visually demonstrated absolute and robust absolute stability of the automatic temperature control system. The simulation carried out in the MATLAB system has confirmed the correctness of the results obtained.


Introduction
The existing heating systems provide maintaining indoor air temperature depending on the ambient temperature, external wind speed, solar radiation, heater power and other parameters. The physical principles of the heating system functioning may be different, but, in the general case, the system is used that contains three main elements: a heat source, the system of heat transfer, the equipment of heat dosing and a controller. Gas, oil, coal, thermal sources, as well as solar energy are used to warm up the heat transfer medium. In boiler houses, autonomous systems are used, the controllers of which can be analog or digital. Analog controllers have a great number of drawbacks, the main of which is low stability. Such systems are rather simple, where the simplest sensitive elements (bimetallic plate) are used [1]. Digital controllers possess higher accuracy of maintaining temperature and are able to respond to weather changes during several minutes, and this allows to save energy [2,3]. The digital control system makes it possible to control indoor temperature more precisely in comparison with an analog (thermostat) system where the heating inertia can cause the room temperature to fluctuate, approximately by 1,5°C, that results in thermal discomfort [4]. The contemporary systems of indoor temperature automatic control are the systems which function despite the existence of many types of uncertainties [5][6][7][8], for which it is necessary to apply special methods of study that would take into account a model stability to uncertainties (robustness). The efficiency and quality of functioning of technological processes control systems are largely determined by the correct choice of the controller parameters. The most common types of controllers are PID controllers which contain proportional, integral, and differential components of signals in their structure which are set by the way of selecting constant coefficients, corresponding to these components. Functional design of PID controllers is such that signal generating blocks are connected in parallel. This allows estimation of the influence of each component on the dynamics of the system both separately and in combination, which simplifies the process of tuning under production conditions. In [9] it is suggested to use fuzzy logic controllers. The application of neural networks [10,11] is also considered, in which it is proposed to use model predictive control (MPC). In [12,13], implementation of such networks based on deeper learning is proposed. In [14], the application of a PID controller for ACS using solar radiation is considered. In [15], it is proposed to use the smart home and building (SHaB) control system which takes into account the incident solar radiation which provides reduction in energy consumption of the air conditioning systems. The MATLAB computer simulation system [16][17][18] has become widespread for the study of temperature regimes. The temperature controller for closed premises, based on the theory of robust control, which takes into account the influence of tolerances of the model parameters on the analysis of the system stability, is considered in [19][20][21]. Papers [22][23][24][25][26][27][28][29][30] demonstrate the use of PID controllers to provide the robust stability of systems with different kinds of uncertainties.

The problem statements
At present, there are a great number of publications related to the mathematical modelling of the temperature regimes of buildings which are residential, livestock, and industrial premises. In papers [8][9][10], various mathematical models of the temperature control systems inside residential buildings are considered. Insolation of buildings by solar radiation is considered in papers [14,15].
A typical transfer function of the heated room in these publications is written as follows . (1) where K is a function of the following thermophysical values: Qthermal energy coming from the heater; Whereas, time constant of the heated room also depends on the following magnitudes: -weight of the room; -heat transfer coefficient of the enclosing structures (walls); area of enclosing structures (walls); -specific heat capacity. The transfer function of the object includes the time constant of the temperature sensor . Analysis of the transfer function and the time constant of the heated room shows that the system under study contains the following uncertainties [6,7]: a) exogenous (depends on the ambient temperature); b) endogenous (depends on the supplied thermal energy and the weight of the heated room).
The task is to synthesize the controller and study the heating system as a discrete one taking into account the influence of temperature and other uncertainties on the robust stability. The synthesis of the controller and the simulation of the heating system should be performed in the MATLAB software product.

The problem solution
To solve the formulated problem, it is necessary to perform the synthesis of a discrete heating system. Therefore, it is required to transform its analog transfer function into a discrete form, and then, with the help of the pidtune program, synthesize a digital PID controller that will provide a sufficiently large phase margin. The next step is to conduct a study of this heating system taking into account the main uncertainties in the system, i.e. investigate the system with interval coefficients of the transfer function As an example, consider the transfer function of the temperature control system [1] .
Since it is supposed to implement a digital temperature controller, then it is necessary to convert the transfer function of the control object (1) into a digital form. For this purpose, we will use, as well as for all subsequent calculations, MATLAB-Control System Toolbox application program package. We assume the discrete period equal to 1 s and calculate the discrete transfer function taking into account the zero-order extrapolator. It has the following form (3) The widespread use of PID controllers in industrial installations is determined along with the possibility of the simplified parametric tuning, suitability for solving many practical problems at low cost of hardware facilities, as well as application of clearly evident, deeply investigated principles for correcting the behavior of dynamic systems based on the classical control theory. However, despite the results achieved in the field of designing PID controllers and the gathered experience of their practical application, empirical methods often continue to be used in choosing the tuning parameters, without any proper theoretical justification and the application of up-to-date computer technologies.
At the same time, the problem of parametric tuning of PID controllers continues to attract much attention. The reason consists in difficulties arising in the control of non-linear systems, objects with transport delay, as well as in the case of incomplete dynamics of the processes under non-stationary operational conditions. In many cases, the tuning of the controller parameters is performed by the way of visual assessment of the transient processes without any complex quantitative assessments using computer technologies.
The efficiency and quality of estimating parameters of the controllers tuning can be improved due to up-to-date means of simulation and application of special tools of various computing environments focused on solving the problems in this object domain. In the Matlab integrated environment, a PID tuner is such a tool which is implemented with the help of the pidtool function. The application of the transfer functions makes it possible to use the PID tuner to work with non-minimum-phase systems and the systems with transport delay.
When performing the synthesis of a discrete PIDF controller it is necessary to take into consideration obtaining the robustness of the system under the conditions of interval changes in its parameters. As it is known, the maximum value of the logarithmic amplitudefrequency characteristic (LAFC) of the closed-loop control system can serve as an indirect sign of robustness, hence, the maximum margins in module and phase of an open-loop system [17]. Taking into account the need for astatism of the control system, we will assume the transfer function of the PID controller as a control device. Since the controller should be digital, we will choose its standard PIDF type and calculate its parameters using pidtool program. This program makes it possible to set the desired phase shift and, by the way of changing the transfer coefficient of the proportional channel, achieve the desired nature of the transient processes and speed of their running. Figure 1 presents the logarithmic amplitude and phase frequency characteristics obtained during calculations. As a result of calculations, the following discrete transfer function of the controller has been obtained: . ( In this case, the controller parameters have the following values: KP =0,7419 is the transfer coefficient of the proportional channel, KI = 0,00281 is the transfer coefficient of the integral channel, KD =17,29 is the transfer coefficient of the differential channel. The controller parameters provide a gain margin of 41.9 dB, a phase margin of 78 degrees.
To study transient processes, we use the Simulink program and the block diagram of the analog-to-digital model of the system presented in Figure 2. The transition function is presented in Figure 3.
For further assessment of the robustness achieved during the design process, we subject the obtained transfer function to a bilinear transformation by the Tastin method with correction. As a result, we obtain the following transfer function T, C t, c  (6) 4 Estimation of the robust stability of the control system First, we check the absolute stability of this system by using a graphical approach [20] based on constructing a root locus for the transfer function (6) with nominal values of the numerator and denominator coefficients.
The constructed root locus for the transfer function with nominal coefficients is presented in Figure 4. It can be seen from the presented figure that the system under study is absolutely stable. To study the robust stability of this system it is necessary to determine the spread of the values of the coefficients of the polynomials of the numerator and denominator of the transfer function. These values will be obtained from the numerical values of the coefficients of the polynomials of the numerators and denominators of the transfer functions obtained by varying the value time constant , which contains all the characteristics describing the enclosed structure of the heated room. The choice of the coefficients of these transfer functions is feasible based on the magnitude of the change in the value time constant within the range of ±25%. In this case, the transfer function of the system with interval parameters will be written in the following form (7) The constructed root locus for the transfer function with interval coefficients (7) is presented in Figure 5. It can be seen from the presented figure that the dominant branches of the root locus have acquired a blurred outline, and the trajectories of the root locus branches do not fall on the real positive axis, which indicates the robust absolute stability of the system under study.

Conclusion
The proposed approach to the implementation of the room temperature servo control system has shown that it is possible to use PID controller which under practical conditions is easier to implement with the help of standard procedures in MATLAB, and, in addition, provides the robust absolute stability of the temperature control system under study.