1. Open and closed loop transfer functions. BIBO stability by M. Miccio rev. 3.5 of March 12, 2015

2. DEFINITIONS 22/10/2015 Prof M. Miccio 2

3. Proper vs. improper rational transfer function 22/10/2015 Prof M. Miccio 3 A rational function is proper if the degree of the numerator is less than the degree of the denominator, and improper otherwise. For example, look at these three rational expressions: In the first example, the numerator is a second-degree polynomial and the denominator is a third-degree polynomial, so the rational is proper. In the second example, the numerator is a fifth-degree polynomial and the denominator is a second-degree polynomial, so the expression is improper. In the third example, the numerator and denominator are both fourth-degree polynomials, so the rational function is improper.

4. Self-Regulating Processes 22/10/2015 Prof M. Miccio 4 Definition: A self-regulating dynamic process is such to seek a steady state operating point if all manipulated and disturbance variables, after a limited change, are held constant for a sufficient length of time.

5. BIBO Stability 22/10/2015 Prof M. Miccio 5 5 Definition of BIBO Stability An unconstrained linear dynamic system is said to be stable if the output response is bounded for all bounded inputs.

6. Marginal Stability 22/10/2015 Prof M. Miccio 6 6 This latter case occurs when there are poles with single multiplicity on the stability boundary, i.e. the imaginary axis. A marginally stable system may exhibit an output response that neither decays nor grows, but remains strictly constant or displays a sustained oscillation. Definition of Marginal Stability: An input-output dynamic system is defined marginally stable if only certain bounded inputs will result in a bounded output. http://en.wikipedia.org/wiki/Marginal_stability

7. SUMMARY 22/10/2015 Prof M. Miccio 7

8. SystemOpen- Loop Behavior Transfer FunctionProper vs. improper 1st orderself- regulating Proper purely capacitive 1st order non self- regulating Proper 2nd orderself- regulating Proper dead timeself- regulating not applicable PID controller= = = Improper Summary of Linear Dynamic Systems 22/10/2015 Prof M. Miccio 8

9. BIBO Stability 22/10/2015 Prof M. Miccio 9 9 Most industrial processes are stable without feedback control. Thus, they are said to be open-loop stable or self-regulating. An open-loop stable process will return to the original steady state after a transient disturbance (one that is not sustained) occurs. By contrast there are a few processes, such as exothermic chemical reactors, that can be open-loop unstable. For systems with proper rational functions BIBO Stability Theorem: 1.a system is (asymptotically) stable if all of its poles have negative real parts 2.a system is unstable if at least one pole has a positive real part 3.a system is marginally stable if it has one or more single poles on the imaginary axis and any remaining poles have negative real parts

10. BIBO Stability of Linear Dynamic OL Systems 22/10/2015 Prof M. Miccio 10 The location of the poles of a transfer function gives us the first criterion for checking the stability of a system. If the transfer function of a dynamic system has even one pole with a positive real part, the system is unstable. Re Im  Unstable Region Stable Region From Romagnoli & Palazoglu (2005), “Introduction to Process Control”  Stability boundary

11. BIBO Stability of Linear Dynamic OL Systems 22/10/2015 Prof M. Miccio 11 Note that the poles are on the imaginary axis Marginally stable step input

12. CLOSED LOOP 22/10/2015 Prof M. Miccio 12

13. 22/10/2015 Prof M. Miccio 13 Closed Loop Block Diagram in the time domain Final control element PROCESS SENSOR PID Controller y SP (t) ε(t) o(t) m(t) d(t) y m (t) y(t) + - KcKc

14. 22/10/2015 Prof M. Miccio 14 Closed Loop Block Diagram: shut-down of the controller Final control element PROCESS SENSOR PID Controller y SP (t)ε(t) m(t) d(t) y m (t) y(t) + - practically, the same as setting K c = 0  o(t) = 0

15. 22/10/2015 Prof M. Miccio 15 Closed Loop Block Diagram: manual mode of the controller  Open Loop operation Final control element PROCESS SENSOR PID Controller y SP (t) ε(t) o(t) m(t) d(t) y m (t) y(t) + -

16. 22/10/2015 Prof M. Miccio 16 Open Loop Transfer Function Laplace domain Final control element PROCESS SENSOR PID Controller y SP (s) ε(s) o(s) m(s) d(s) y m (s) y(s) + - Definition: G OL (s)=G c G f G p G m see: Ch.14 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall,1984 Forward path Feedback path

17. 22/10/2015 Prof M. Miccio 17 Closed Loop Transfer Function Final control element PROCESS SENSOR PID Controller y SP (s) ε(s) o(s) m(s) d(s) y m (s) y(s) + - Servo Problem: y(s)=G CL,SP (s)  y SP (s) (Hyp.: d(s) = 0) Regulator Problem: y(s)=G CL,load (s)  d(s) (Hyp.: y SP (s) = 0) where the Open Loop Transfer Function: see: Ch.14 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984

18. 22/10/2015 Prof M. Miccio 18 Closed Loop Transfer Function Final control element PROCESS SENSOR PID Controller y SP (s) ε(s) o(s) m(s) d(s) y m (s) y(s) + - from the Principle of Superimposition: where the Open Loop Transfer Function: see: Ch.14 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984

19. 22/10/2015 Prof M. Miccio 19 Sensitivity Function Closed Loop Transfer Function: Sensitivity Function : Example 1: Closed Loop Transfer Function: Sensitivity Function : Hyp.: G m =1

20. Dynamical Systems in Series Laplace domain 22/10/2015 Prof M. Miccio 20

21. Dynamical Systems in Parallel Laplace domain 22/10/2015 Prof M. Miccio 21

22. Block Diagram: graphical conventions, block manipulation 22/10/2015 Prof M. Miccio 22 Block_Diagrams(D.Cooper).pdf

23. Closed Loop BIBO Stability 22/10/2015 Prof M. Miccio 23 General BIBO Stability Criterion: The feedback control system is stable if and only if all roots of the characteristic equation (closed loop poles) are negative or have negative real parts. Hy.: no dead time in G OL (s) The characteristic equation of the closed-loop linear dynamic system is: 1 + G OL (s) = 0 poles of the characteristic equation  Closed Loop BIBO STABILITY

24. BIBO Stability of Linear Dynamical CL Systems 22/10/2015 Prof M. Miccio 24 Closed-loop poles ( x ) and time response: stable system  unstable system step input

25. Non-miminum phase systems Let’s look at the dead time. The important point is that the phase lag of the dead time increases without bound with respect to frequency. This is what is called a non-minimum phase system, as opposed to the first- and second-order transfer functions, which are minimum phase systems. A minimum phase system is one which: Has a positive gain Is free of any dead time all poles have negative o null real part all zeroes (if any) have negative o null real part A minimum phase system is less prone to closed loop instability. A non-minimum phase system exhibits more phase lag than another transfer function which has same AR plot.  adapted from Chau, pag. 157 and from Bolzern, Scattolini e Schiavoni, par. 6.6.4