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The mathematical model of the heating coil is obtained from mass and energy
balances. The control input to the model is the water inlet temperature i
#w.Further
inputs which are assumed to be constant are the air inlet temperature i
#a,the
air mass flow P ma and the water mass flow P mw. The model outputs are the outlet
temperatures of air
o
#a and water
o
#w. The tubes of the heating coil are pided
into segments, see Fig. 2.In(1–5), temperatures are denoted by #, massflows by P m.
The subscripts w, a, i , o and t denote “water”, “air”, “inlet”, “outlet” and “tube”
respectively. For one segment, the following set of first order differential equationsD C
Details concerning the parameters k1 to k4 and can be found in [5]. The model
of the heating coil is found by interconnecting the segments. In the present case ten
segments were used to model the heating coil.
The relation between mixing ratio of the fluids and the valve gear position was
obtained via measurements.
2.2.2 Room
Due to a large number of influencing factors, rooms are extremely difficult to
model. Therefore the room is represented by a first order element with dead time
as suggested in [3]:
a denote room air inlet and outlet temperature respectively.
The gain K, the time constant T r
and the dead time T r
d were identified from3 Describing function method
The basic idea of the describing function method is simple, see e.g. [2]. Though it
is a powerful technique to predict oscillations in nonlinear feedback loops. Figure 3
shows the structure of the nonlinear temperature control loop, where R.j!/ denotes
the frequency response of the controller to be designed. N.A;!/ is the describing
function which is composed of valve gear, valve, hydraulic circuit, heating coil and
room. As usual, A and ! denote the amplitude and the frequency of a potential limit
cycle
h A sin !t; (7)
where h is the deviation of the valve gear position h from its (constant) operating
point. As indicated in Fig. 3, it is the key issue to find intersection points of the
describing surface and the controller frequency response. In case of at least one
intersection point, limit cycles are likely to occur, otherwise no limit cycle is
expected. Figure 4 shows the describing surface for the pilot plant operated in the
vicinity of an operating point of #a D 17:5ıC and a valve gear position h of 50%.
The algorithm to compute N.A;!/ is outlined in [4].
3.1 Controller design
The starting point is a predefined standard PI-controller R1.s/ with fixed controller
parameters. The practical application of this controller in the pilot plant yields
unacceptable behaviour. Instead of a desired steady state temperature of 17:5ıC,
the room air temperature #a shows an undesired oscillation with an amplitude
of approximately 1:5ıC, see Fig. 5. This poor performance is confirmed by the
describing function method, i.e. the frequency response R1.j!/ intersects the
describing surface, see Fig. 4.
Based on the linear algebraic method [1] which is applied to a linearized plant
model P.s/, an alternative controller R2.s/ is computed. Specifications for the
closed loop system are expressed in terms of maximum peak and rise timeNote that the design procedure is an iterative process. If the frequency response
R2.j!/ intersects 1=N.A;!/, the design has to be repeated with more conser-
vative specifications. The frequency response of the final controller R2.s/ has no
intersection point with the describing surface (see Fig. 4), consequently it is unlikely
that limit cycleswill occur. Experimental results shown in Fig. 6 confirmthe absence
of limit cycles and reveal very satisfactory overall performance of the feedback loop.4 Conclusion