Understanding Process Dynamics in Control Systems
Explore the importance of terms like gain, time-constant, integrator, and time-delay in identifying and tuning control systems. Learn how to represent time-constant systems mathematically and derive transfer functions. See examples of applying these concepts to a simulated heated tank system. Gain insights into reading system characteristics from step responses. Dive into the mathematical modeling of dynamic systems for practical applications.
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Course: Process Control, NMBU Dec 2017 - April 2018 Process dynamics By Finn Aakre Haugen, PhD, TechTeach (finnhaugen@hotmail.com) F. Haugen. Process Control. NMBU. 2017. 1
Terms Gain Time-constant Integrator (or accumulator) Time-delay Why are these terms important? To give names to dynamic properties of physical systems To make you identify and understand dynamic properties Can be used in controller tuning - using model-based methods (e.g. Skogestad s method - to be described later in this course) F. Haugen. Process Control. NMBU. 2017. 2
Definition of time-constant systems Time-constant systems can be represented with the following differential equation, where u is the system input and y is the output: T is the time-constant. K is the gain. From this differential equation we can calculate the following transfer function from input u to output y: This transfer function is the standard transfer function of a time- constant system. F. Haugen. Process Control. NMBU. 2017. 3
K and T in the step response By applying a step at the input of the system, you can read off K and T from the step response at the output. Step response: Input step ys U K = Output / Input = ys/U = delta y / delta u (at steady-state!) T is the 63% response time t F. Haugen. Process Control. NMBU. 2017. 4
Simulator: Time-constant F. Haugen. Process Control. NMBU. 2017. 5
Example: Liquid tank with heating Simulator: Heated tank (Mathematical model on next slide.) F. Haugen. Process Control. NMBU. 2017. 6
Mathematical model of heated tank Energy balance: From this differential equation we can derive the following transfer functions, assuming neglected heat transfer (Uh=0) (Delta indicates deviation from operating point ): Time-constant and gains: If the heat transfer is neglected (Uh=0), the time-constant is simply mass divided by mass flow: Let's see if m/F is equal to the "experimental" time-constant as read off on the simulator: Heated tank F. Haugen. Process Control. NMBU. 2017. 7
Definition of integrator systems Integrator systems can be represented with the following integral equation, where u is the system input and y is the output: Ki is the integrator gain. An integrator can be termed accumulator as it accumulates the inputs: y(tk) = Ki * [u(t0) + u(t1) + + u(t0)]*dT The above integral equation corresponds to this diff. equation: The transfer function from input to output is F. Haugen. Process Control. NMBU. 2017. 8
Step response of an integrator The step response is a ramp: Input (step) Output (ramp) F. Haugen. Process Control. NMBU. 2017. 9
Simulators: Integrator Liquid tank (Mathematical model on next slide.) F. Haugen. Process Control. NMBU. 2017. 10
Mathematical model of liquid tank Mass balance (assume valve is closed): A * dh/dt = qin qout = qin Kp*up (pump) dh/dt = (1/A) * (qin qout) = (1/A) * (qin Kp*up) which is on the standard form of an integrator (except in our example we have two input signals, qin and qout) F. Haugen. Process Control. NMBU. 2017. 11
Time-delay (or transport-delay, dead-time) Example: Conveyor belt (Outflow is equal to time-delayed inflow.) Transfer function: F. Haugen. Process Control. NMBU. 2017. 12
Simulator: Time-delay F. Haugen. Process Control. NMBU. 2017. 13
Very hard questions: 1. Is there a time-delay between readings at sensor FT1 and sensor FT2? FT1 FT2 Isolation Pipeline Liquid TT1 TT2 2. Is there a time-delay between readings at sensor TT1 and sensor TT2? F. Haugen. Process Control. NMBU. 2017. 14
Combined dynamics Example: Wood-chip tank Level control of wood-chip tank How will you characterize the dynamics of this system? F. Haugen. Process Control. NMBU. 2017. 15
