Fixed-Point Blockset

Integrator Realizations

This section presents realizations for trapezoidal, backward, and forward integration. For each realization, the transfer function and difference equation, block parameters, and model design are discussed.

Trapezoidal Integration

The FixPt Integrator: Trapezoidal realization is a masked subsystem that performs discrete-time integration using the Trapezoidal method. For this method, integration is approximated by the z-domain transfer function

where Ts is the sampling period. The realization is shown below.

As shown in the figure, the transfer function yields the difference equation

where k is the current time step, k - 1 is the previous time step, y(k) is the current output, y(k - 1) is the output from the previous time step, u(k) is the current input, and u(k - 1) is the input from the previous time step.

Parameters and Dialog Box

The dialog box and parameter descriptions for the trapezoidal integrator realization are given below.

Sample time

Base data type

Accumulator data type

Model Design Review

A brief review of the model design is given below. The design criteria reflect the rules presented in Design Rules.

• The gains involve multiplications which are a size-growing operation. In most cases, it is desirable for gains and inputs to use the word size given by the Base data type or smaller. The output can be left at the Accumulator data type for extra precision in subsequent operations. Alternatively, if the output were stored in RAM, or used by a size-growing operation, it could be reduced to the Base data type.
• The FixPt Sum block converts inputs to the output data type before performing the actual addition. Given this order of operation, using the Accumulator data type often gives better precision.
• The FixPt Conversion block forces the output to the Base data type before storage in RAM (i.e., before input to the unit delay). Casting the output in the feedforward part of the realization prevents subsequent operations from being burdened with a large data type.

Design Rules

Backward Integration