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Chapter 5

Chapter 5. Transfer Functions and State Space Models. Overall Course Objectives. Develop the skills necessary to function as an industrial process control engineer. Skills Tuning loops Control loop design Control loop troubleshooting Command of the terminology Fundamental understanding

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Chapter 5

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  1. Chapter 5 Transfer Functions and State Space Models

  2. Overall Course Objectives • Develop the skills necessary to function as an industrial process control engineer. • Skills • Tuning loops • Control loop design • Control loop troubleshooting • Command of the terminology • Fundamental understanding • Process dynamics • Feedback control

  3. Transfer Functions • Provide valuable insight into process dynamics and the dynamics of feedback systems. • Provide a major portion of the terminology of the process control profession.

  4. Transfer Functions • Defined as G(s) = Y(s)/U(s) • Represents a normalized model of a process, i.e., can be used with any input. • Y(s) and U(s) are both written in deviation variable form. • The form of the transfer function indicates the dynamic behavior of the process.

  5. Derivation of a Transfer Function • Dynamic model of CST thermal mixer • Apply deviation variables • Equation in terms of deviation variables.

  6. Derivation of a Transfer Function • Apply Laplace transform to each term considering that only inlet and outlet temperatures change. • Determine the transfer function for the effect of inlet temperature changes on the outlet temperature. • Note that the response is first order.

  7. In-Class Exercise: Derive a Transfer Function

  8. Poles of the Transfer Function Indicate the Dynamic Response • For a, b, c, and d positive constants, transfer function indicates exponential decay, oscillatory response, and exponential growth, respectively.

  9. Poles on a Complex Plane

  10. Exponential Decay

  11. Damped Sinusoidal

  12. Exponentially Growing Sinusoidal Behavior (Unstable)

  13. What Kind of Dynamic Behavior?

  14. Unstable Behavior • If the output of a process grows without bound for a bounded input, the process is referred to a unstable. • If the real portion of any pole of a transfer function is positive, the process corresponding to the transfer function is unstable. • If any pole is located in the right half plane, the process is unstable.

  15. Routh Stability Criterion

  16. Routh Array for a 3rd Order System

  17. Routh Stability Analysis Example

  18. In-Class Exercise

  19. Zeros of a Transfer Function • The zeros of a transfer functions are the value of s that render N(s)=0. • If any of the zeros are positive, an inverse response is indicated. • If all the zeros are negative, overshoot can occur in certain situations

  20. Combining Transfer Functions • Consider the CST thermal mixer in which a heater is used to change the inlet temperature of stream 1 and a temperature sensor is used to measure the outlet temperature. • Assume that heater behaves as a first order process with a known time constant.

  21. Combining Transfer Functions • Transfer function for the actuator • Transfer function for the process • Transfer function for the sensor

  22. Combining Transfer Functions

  23. In-Class Exercise: Overall Transfer Function For a Self-Regulating Level

  24. Block Diagram Algebra • Series of transfer functions • Summation and subtraction • Divider

  25. Block Diagram Algebra

  26. Block Diagram Algebra

  27. In-Class Exercise: Block Diagram Algebra

  28. Solution of In-Class Exercise

  29. What if the Process Model is Nonlinear • Before transforming to the deviation variables, linearize the nonlinear equation. • Transform to the deviation variables. • Apply Laplace transform to each term in the equation. • Collect terms and form the desired transfer functions. • Or instead, use Equation 5.7.3.

  30. Transfer Function for a Nonlinear Process

  31. Advantages of Equation 5.7.3 • Equation 5.7.3 was derived based on linearing the nonlinear ODE, applying deviation variables, applying Laplace transforms and solving for Y(s)/U(s). • Therefore, Equation 5.7.3 is much easier to use than deriving the transfer function for both linear and nonlinear first-order ODEs.

  32. Application of Equation 5.7.3

  33. In-Class Exercise

  34. State Space Models • State space models are a system of linear ODEs that approximate a system of nonlinear ODEs at an operating point. • Similar to Equation 5.7.3, state space models can be conveniently generated using the definitions of the terms in the coefficient matrices and the nonlinear ODEs.

  35. State Space Models

  36. Poles of a State Space Model

  37. Overview • The transfer function of a process shows the characteristics of its dynamic behavior assuming a linear representation of the process.

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