UNIVERSIT DEGLI STUDI DI SALERNO Dipartimento di Ingegneria

UNIVERSITÁ DEGLI STUDI DI SALERNO Dipartimento di Ingegneria Industriale Open and closed loop transfer functions. BIBO stability Prof. Ing. Michele MICCIO • Dip. Ingegneria Industriale (DIIn) • Prodal Scarl (Fisciano) rev. 4. 5 of March 5, 2019 21/02/2021 Prof M. Miccio 1

Definitions 21/02/2021 Prof M. Miccio 2

Transfer Function For a SISO (Single-Input, Single-Output) linear dynamical system in the Laplace domain: TF = (Laplace Trasform of output) / (Laplace Trasform of input) output input mathematical form: Transfer Function block diagram: 21/02/2021 G(s) Prof M. Miccio 3

Proper vs. improper rational transfer function A rational function is: Ø proper if the degree of the numerator does not exceed the degree of the denominator Ø strictly proper if the degree of the numerator is less than the degree of the denominator Ø improper otherwise Examples In the first example, the numerator is a second-degree polynomial and the denominator is a third-degree polynomial, so the rational is strictly 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 proper. 21/02/2021 Prof M. Miccio 4 4

Self-Regulating Processes 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. INPUT 21/02/2021 Dynamical System Prof M. Miccio OUTPUT 5

BIBO Stability Definition of BIBO Stability An unconstrained linear dynamic system is said to be stable if the output response is bounded for all bounded inputs. INPUT 21/02/2021 Dynamical System Prof M. Miccio OUTPUT 6 6

Marginal Stability 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 • This latter case occurs when there are poles with single multiplicity on the stability boundary, i. e. the imaginary axis. Ø A marginally stable system may exhibit an output response that neither decays nor grows, but remains strictly constant or displays a sustained oscillation. 21/02/2021 Prof M. Miccio 7 7

Review 21/02/2021 Prof M. Miccio 8

Summary of Linear Dynamic Systems 21/02/2021 System Open. Loop Behavior Transfer Function 1 st order selfregulating Strictly proper purely capacitive 1 st order non selfregulating Strictly proper 2 nd order selfregulating Strictly proper dead time selfregulating Not applicable PID controller not applicable Improper Prof M. Miccio Proper vs. improper 9

Open-loop BIBO stability For systems with proper rational transfer 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 or if it has one or more poles with multiplicity larger than 1 on the imaginary axis 3. a system is marginally stable if it has one or more poles with unit multiplicity on the imaginary axis and all remaining poles with a negative real part • 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 a few processes, such as exothermic chemical reactors, that can be open-loop unstable. 21/02/2021 Prof M. Miccio 10

BIBO Stability of Linear Dynamic OL Systems The location of the poles of a proper rational 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. Im Stability boundary Re Stable Region From Romagnoli & Palazoglu “Introduction to Process Control” 21/02/2021 Unstable Region (2005), Prof M. Miccio 11

BIBO Stability: Example of Linear 2 nd order OL Systems Unit step input R(s) = 1/s u(t) b=9 a=9 t a=2 Marginally stable Note the poles on the imaginary axis! a=0 a=6 21/02/2021 Prof M. Miccio 12

Block Diagram Algebra 21/02/2021 Prof M. Miccio 13

Block Diagram and Block Algebra Laplace domain graphical conventions, block manipulation, etc. Block_Diagrams(D. Cooper). pdf 21/02/2021 Prof M. Miccio 14

Dynamical Systems in Series Laplace domain f 21/02/2021 Prof M. Miccio 15

Dynamical Systems in Parallel Laplace domain divider 21/02/2021 summer Prof M. Miccio 16

Block Diagram Models (DIDO) from prof. Pribeiro 21/02/2021 Prof M. Miccio 17

Block Diagram Models 21/02/2021 Prof M. Miccio 18

Block Diagram Models 21/02/2021 Original Diagram Equivalent Diagram Prof M. Miccio 19

Block Diagram Models 21/02/2021 Original Diagram Equivalent Diagram Prof M. Miccio 20

Block Diagram Models 21/02/2021 Original Diagram Equivalent Diagram Prof M. Miccio 21

Block Diagram Models 21/02/2021 Prof M. Miccio 22

Block Diagram Models Example 2. 7 21/02/2021 Prof M. Miccio 23

Block Diagram Models 21/02/2021 Prof M. Miccio Example 2. 7 24

Closed Loop 21/02/2021 Prof M. Miccio 25

Closed Loop Block Diagram in the time domain d(t) Kc y. SP(t) + ε(t) PID Controller - ym(t) 21/02/2021 o(t) Final m(t) control element y(t) PROCESS SENSOR Prof M. Miccio 26

From Closed Loop to Open Loop: shut-down of the controller d(t) Kc y. SP(t) + ε(t) PID Controller - ym(t) Final m(t) control element PROCESS SENSOR practically, the same as setting Kc = 0 21/02/2021 y(t) Prof M. Miccio o(t) = 0 27

From Closed Loop to Open Loop: manual mode of the controller d(t) Kc y. SP(t) + ε(t) PID Controller - ym(t) o(t) Final m(t) control element y(t) PROCESS SENSOR Open Loop operation 21/02/2021 Prof M. Miccio 28

Open Loop Transfer Function Laplace domain Definition: GOL(s)=Gc. Gf. Gp. Gm d(s) Forward path y. SP(s) + ε(s) PID Controller - ym(s) Final m(s) o(s) control element y(s) PROCESS SENSOR Feedback path see: Ch. 14 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984 21/02/2021 Prof M. Miccio 29

Closed Loop Transfer Function d(s) y. SP(s) + ε(s) PID Controller − ym(s) Servo Problem: o(s) Final m(s) control element y(s) PROCESS SENSOR y(s)=GCL, SP(s) y. SP(s) (Hyp. : d(s) = 0) Regulator Problem: y(s)=GCL, load(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 21/02/2021 Prof M. Miccio 30

Closed Loop Transfer Function d(s) y. SP(s) + ε(s) PID Controller − ym(s) o(s) Final m(s) control element y(s) PROCESS SENSOR from the Principle of Superimposition: where the Open Loop Transfer Function is see: Ch. 14 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984 21/02/2021 Prof M. Miccio 31

Closed Loop BIBO Stability Hy. : • no dead time in GOL(s) So, the characteristic equation of the closed-loop linear dynamic system is: 1 + GOL(s) = 0 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. poles of the characteristic equation Closed Loop BIBO STABILITY 21/02/2021 Prof M. Miccio 32 32

BIBO Stability: Example of Linear 3 rd order CL System step input Closed-loop poles (x) Time response stable system unstable system 21/02/2021 Prof M. Miccio 33

Minimum vs. Non-miminum phase systems Stable systems without dead time, which do not have zeros in the right half plane, are called minimum phase systems. A minimum phase system is one which: Ø has a positive gain Ø is free of any dead time Ø all poles have negative o null real part Ø all zeroes (if any) have negative o null real part 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 have a minimum phase value. Remarks: 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. 21/02/2021 Prof M. Miccio adapted from Chau, pag. 157 and from Bolzern, Scattolini e Schiavoni, par. 6. 6. 4 34

Sensitivity Function Hyp. : Gm=1 Closed Loop Transfer Function: DEFINITION of Sensitivity Function: Example 1: Closed Loop Transfer Function: Sensitivity Function: 21/02/2021 Prof M. Miccio 35