Integrator

The integrator is another frequently used element. In continuous time it is defined by

$${\frac{1}{T_{N}}} \int_{0}^{t} \tau \tau$$ u_I(t) = (13)

$$\dot{u}_{I}^{} {\frac{1}{T_{N}}}$$ = (14)

To obtain a realization for discrete time systems the derivative is replaced by a difference quotient again.

$${\frac{u_{I}(k)-u_{I}(k-1)}{\Delta T}} {\frac{1}{T_{N}}} \Longrightarrow$$ = (15)

u_I(k) = u_I(k - 1) + K_I x_d(k) (16)

Equation 16 leads to a feed back of the latched adder's output, as shown in Figure 4. However, there is always a possibility for an overflow or underflow. To avoid this, an error treatment is necessary. If an error occurs then the sign changes in a wrong manner. Unfortunately, it is transmitted and therefore computed as the last bit. Thus, the whole word has to be latched to enable a substitution with a given replacement. This additional delay corresponds a dead time and worsens control quality.

Figure 4: Integrator

An error is impossible by the compromise of addends with a limited range. The usable range for the addends is -2^{n - 2}...2^{n - 2} - 1. An addition of two such operands can never lead to a over or under flow. Since the addend originated by x_d has a known range it has only to be scaled by an appropriate gain factor K_I. However the other addend is the fed back integrator part of the control input and uses the full value range. Therefore it has to be limited via a limiter. An integrator realization basing upon this approach avoids a dead time by the compromise of a halved value range. However, a big advantage of the bit serial approach is the possibility of scaling in steps of one bit, therefore the range can be simply extended.