- Linear State Space Models
- Introduction to Laplace Transforms
- Transfer Functions
- First-Order Behavior
- Integrating System
- Second-Order Behavior
- Lead-Lag Behavior
- Poles and Zeros
- Processes with Dead Time
- Padé Approximation for Dead Time
- Converting State Space Models to Transfer Functions
- Matlab and Simulink
- Student Exercises
3.11 Padé Approximation for Dead Time
As discussed in the previous section the transfer function for a pure time delay is e–qs, where q is the time delay. Some control system design techniques require a rational transfer function; the Padé approximation for dead time is often used in this case.
A first-order Padé approximation is
A second-order Padé approximation is
Example 3.8: Application of the Padé Approximations for Dead Time
Consider the first-order + dead time transfer function, where the time-delay dominates the time constant
The first-order Padé approximation yields the transfer function
and the second-order Padé approximation yields
A comparison of the step responses of g(s), g1(s), and g2(s) is shown in Figure 3-16. Notice that the first-order approximation has an inverse response, while the second-order approximation has a "double inverse response." The reader should find that there is a single positive zero for g1(s), and there are two positive, complex-conjugate zeros of the numerator transfer function of g2(s).
Figure 3-16. Comparison of first-order + dead time response with first and second-order Padé approximations for dead time.
Most ordinary differential equation numerical integrators require pure differential equations (with no time delays). If you have a system of differential equations that has time delays, the Padé approximation can be used to convert them to delay-free differential equations, which can then be numerically integrated.
One of the many advantages to using SIMULINK is that time delays are easily handled so that no approximation is required.