- Background
- 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
- Summary
- References
- Student Exercises
This chapter is from the book
This chapter is from the book
This chapter is from the book
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.