- 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
The main objective of this chapter was to develop an understanding of dynamic process behavior. Although Laplace transforms can be used to obtain analytical solutions to differential equations, we will not be using them for this purpose during the rest of this textbook. The concept of a transfer function is very useful for control system design and analysis. In future chapters we find that transfer functions allow the use of algebra rather than differential equations when analyzing control systems composed of a number of components (controller, valve, process, sensor, etc.).
The stability of a process is determined by the eigenvalues of the state space A matrix, or the poles of the Laplace transfer function: if all are negative, the system is stable. Complex eigenvalues (poles) correspond to underdamped (oscillatory) behavior, characterized by damping factors less than 1. Dynamic responses are also affected by the values of the zeros (roots of the numerator polynomial in s) of a transfer function. If zeros are positive (in the right-half-plane), step responses have a characteristic inverse response.
A first-order + dead time model is the most common process representation. Time-delays are represented by an irrational term (e–qs) in the Laplace domain. First and second-order Padé approximations are sometimes used for controller design; the approximations lead to right-half-plane zeros and inverse response.
The initial and final value theorems of Laplace transforms can be used to find the limiting behavior of transfer functions without having to obtain a full solution. The process gain is a measure of the long-term change in a process output for a given input change.
State space models are easily converted to Laplace transfer function models. The use of LTI objects (see Module 4) allows easy interconversion of models in MATLAB.