Comfort Control Techniques for the Users of a Room - Comfort Control in Buildings - page 152

advanced control techniques. Furthermore, although important advances have been
performed in control theory, PID controllers are still used to solve a large amount
of control problems. In most cases derivative action is not used, so the controller
is actually a PI controller, although under certain conditions derivative action can
lead to substantial improvement (Guzmán 2006 ). A detailed explanation about PID
controllers can be found in Åström and Hägglund ( 2005 ). There are several forms
of PID controllers. The most typical one is given by Eq. 5.1 :

⊛ ⊞

t

1
T i

d e ( t )

⊝ e ⊠

u ( t ) = K ( t ) + e ( t ) d t + T d (5.1)

d t

0

and its associated transfer function in the Laplace variable s is shown in Eq. 5.2 :

K 1 T d s

1
T i s +

C PID ( s ) = + (5.2)

In the previous equations, K is the proportional gain, T i the integral time, and T d is
the derivative time. Their meaning can be found in classical handbooks (Åström and
Hägglund 2005 ).
In addition, most physical processes are constrained by several reasons, such as
physical limits, security levels, or performance criteria. In these cases, a controller
with integral action, such as PI and PID controllers, can originate some undesirable
behaviours which are called the integrator windup effect. In order to avoid this effect,
all practical PID controllers are provided with some facility for avoiding windup of
the integrator. There are several ways (Åström and Wittenmark 1997 ) to do this, one
of them (see Fig. 5.2 ) consists of providing an extra feedback pat h which contains the
actuator error, that is, the difference between the actuator output, u , and the controller
output, u , and feeding this error back to the integrator through a gain which is equal
to 1 T t . Therefore, the actuator error is zero when it is not saturated, and when

the actuator is saturated the extra feedback path resets the integrator, and thus, the
controller output, u , is at the saturation limit. More specifically, the integrator is reset
to an appropriate value with a time constant equivalent to T t and which is known as
tracking-time constant (Åström and Wittenmark 1997 ). The principal advantage of
this anti-windup scheme lies in the fact that it can be used with any actuator whose
output is measured.

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5.2.2 Model Predictive Control

MPC constitutes a wide control field which is developed using common ideas.
Moreover, it integrates diverse disciplines as optimal and stochastic control, dead-
time processes control, multivariable control or constrained control (Camacho and
Bordons 2004 ). MPC was originated in the late seventies (Cutler and Ramaker 1979 ;