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Process Control: Modeling, Design, and Simulation by B. Wayne Bequette

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M16.2. PID Controllers

Consider the continuous (analog) form of an ideal PID controller,

Equation M16.1

We use a constant sample time, Δt, and the index k to represent the value of the continuous time signal at discrete step k. For example, the error at discrete sample time k is equal to the continuous system value at time t = tk.

Equation M16.2

We approximate the integral term

Equation M16.3

and the derivative term is approximated using backward ...

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