The Hx control theory has received a lot of attention in the last decade within the research community due to the robustness characteristics supplied by its controllers. These features make it of a priori interest to be used in controlling such solar refrigeration systems. The basic idea is to minimize the ratio between the energy of the error vector and the energy of the exogenous signals [353]. The sub-optimum solution of the problem based on the S/T or S/KS/T mixed sensitivity problem for building up the generalized plant, allows the controller to be obtained by just designing a nominal model and some suitable weighting matrices.
In this application [119], the Hx control is used to regulate the absorption machine inlet temperature which establishes the evaporator and absorber pressures. It is also worth mentioning that the controller contains a feedforward action to treat system disturbances. The results of the application are tested both in simulation and in real plant experiments.
The feedback controller design problem for this system can be formulated as an Hx optimization problem with suitable features of disturbance rejection and robustness. The optimal Hx problem has not been solved yet but there is a solution for the suboptimal problem where the value of the energy ratio is decreased as much as possible by means of an iteration process. This synthesis process is used in this work and implemented in various well-known software packages [26].
A configuration for building up the generalized plant is the S/T mixed sensitivity problem [353], which is described in Fig. 7.22, where P(s) is the generalized plant and K(s) is the controller. The terms WS(s) and WT(s) constitute weighting functions which allow to specify the range of frequencies of most importance for the corresponding closed-loop transfer function.
Once the nominal model GN(z) has been chosen, the magnitude of the multiplicative output uncertainty can be estimated as follows:
(7.13)
where Gi(z) stands for the different non-nominal systems at each operation point where the controller is required to work effectively.
As shown in [281], the weighting function WT(s) must be designed with the following conditions: stable, minimum phase and with module greater than the maximum singular value of the uncertainty previously calculated for each non-nominal model and frequency, that is,
(7.14)
In the case of WS(s), the following form is proposed:
(7.15)
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Fig. 7.22 S/T mixed sensitivity problem |
where each of the parameters is designed in the following way:
• aW is the function gain at high frequency. A suitable value is approximately about 0.5.
• is the function gain at low frequency. A suitable low value for these parameter is 10-4.
• as is the crossover frequency of the function; as an initial value, a decade below the crossover frequency of the function WT(s) previously designed. It is proposed to change as according to the expression as = 10(kw-1)aT in order to shape the desired speed of the output response [335]. The parameter kw is employed to vary the value of as once the value of aT(s) has been obtained. The initial value of as is obtained for kw equal to zero, while a value of this parameter equal to one shows that as is equal to aT. Therefore, the final selection of this frequency is determined by an adimensional parameter where the value must be higher as the desired response speed increases [281].
