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III. GAIN SCHEDULING CONTROL DESIGN BASED ON FLOW ADAPTATION
In the previous section, we developed a linear parameter varying model for radiator instead of the high-order nonlinear model (1). To control this system, among various possible control structures, gain scheduling approach is selected which is a very useful technique for reducing the effects f parameter variations [15]. Therefore, the name of flow adaptation indicates to this fact that controller parameters re dependent on the estimated radiator flow.
The main idea for design of adaptive controller is to transform the system model (25) to a system independent of he operating point. Then, the controller would be designed based on the transformed linear time invariant (LTI) system. The block diagram of this controller is shown in Fig. 8.
Fig.8. Block diagram of the closed loop system with linear transformation
Function g is chosen such that to cancel out the moving ole of the radiator and places a pole instead in the desired position. This position corresponds to the farthest position f the radiator pole which happens in high flows or high demand condition. Therefore, the simplest candidate for the linear transfer function g is a phase-lead structure, (26).
In which Khd and τhd correspond to the gain and time constant of radiator in the high demand situation when the low rate is maximum. Consequently, the transformed system would behave always similar to the high demand situation. y choosing the high demand as the desired situation, we give the closed loop system the prospect to have the dominant poles as far as possible from the origin, and as a result as fast as possible.
The controller for the transformed LTI system is a fixed PI controller then. The parameters of this controller are calculated based on Ziegler-Nichols step response method [6]. To this end, the transformed second order system is approximated y a first-order system with a time delay, (27). The choice f PI controller is to track a step reference with zero steady state error.
The time delay and time constant of the above model can e found by a simple step response time analysis of the transformed second-order model:
In which q (t) = u (t) is the unit step input. The apparent time constant and time delay are calculated based on the time when 0.63 and 0.05 of final Ta is achieved, respectively. The positive solution of the following equation gives the time delay when χ = 0.95 and the time constant when χ= 0.37.
Having τ and L calculated, the parameters of the regulator obtained by Ziegler-Nichols step response method would be the integration time Ti = 3L and proportional gain Kc = with a = and k = Khd × Ka which is the static gain.
A. Simulation Results
The proposed controller parameterized based on radiator parameters, is applied to the simulation models of room and radiator. Parameters of the PI controller are found based on the parameter values in table II as Kc = 0.01 and Ti = 400. Ambient temperature is considered as the only source of disturbance for the system. In a partly cloudy weather condition, the effect of intermittent sunshine is modeled by a fluctuating outdoor temperature. A random binary signal is added to a sinusoid with the period of two hours to model the ambient temperature.
Simulation results with the designed controller and the corresponding ambient temperature are depicted in Fig. 9 and Fig. 10. The results are compared to the case with fixed PI controllers designed for both high and low heat demand conditions.
Fig.9. (Top) ambient temperature, (bottom) room temperature for three controllers. The results of simulation with flow adaptive controller together with two fixed PI controllers are shown. The PI controller designed for the high demand situation encounters instability in the low heat demand condition.