UNIVERSIT DEGLI STUDI DI SALERNO Bachelor Degree in

UNIVERSITÀ DEGLI STUDI DI SALERNO Bachelor Degree in Chemical Engineering Course: Process Instrumentation and Control (Strumentazione e Controllo dei Processi Chimici) TRANSFER FUNCTION Applications of Laplace Transform Rev. 3. 7 – May 8, 2019

INTRODUCTION Laplace Transform: Purpose and applications for the Dynamic Behavior and the Automatic Control • solution of dynamic mathematical models expressed in terms of linear ordinary differenzial equations (ODE). • concept of transfer function • dynamic responses: Ø unit impulse Ø step Ø harmonic response • classification and prediction of the dynamic behavior of linear systems see: Ch. 7, 8, 9 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 2

LAPLACE TRANSFORM Definition: where: f(t) is a known function of the time, defined for t 0, s=a+jb is a complex variable called Laplace abscissa, F(s) is the Laplace transform of f(t). F(s) exists if the integral takes a finite value. In particular, the integral is bounded for specified values of the real part of s. For every f(t) a convergence abscissa, c, has to be defined for F(s). The parallel line to the imaginary axes passing though c defines the domain of definition of F(s) as the right half-plane starting from c. For example: if f(t)=u(t) (unit step input change), the integral assumes finite values for every positive values of s; f f(t)=e 2 t, integral is finite for Re(s) >2. 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 3

DOMAIN OF DEFINITION L f(t) Domain of definition of f(t) X : t + Codomain of f(t) Y : f(t) 31/10/2020 L-1 Domain of definition of F(s) s C : Re(s) > c (convergence abscissa) Codomain of F(s) S C Processes Instrumentation and Control – Prof. M. Miccio 4

LAPLACE TRANSFORM The Laplace transform is a linear operation: g(t)= a 1 f 1(t)+a 2 f 2(t) (Principle of Superposition) L[g(t)]= L [a 1 f 1(t)+a 2 f 2(t)]=a 1 L[f 1(t)]+a 2 L[f 2(t)]= • NOTE: h(t)=[f(t)]2 is a nonlinear operation. Thus: 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 5

LAPLACE TRANSFORM TABLE Time Domain Laplace Domain f(t) F(s) e (–at) 1/ (s+a) • • af(t) a F(s) (t) (impulse) 1 sinh (ωt) ω/(s 2 -ω2) au(t) (step) a/s cosh(ωt) s/(s 2 -ω2) at (ramp) a/s 2 df(t)/dt s. F(s)-f(0) tn n!/s(n+1) sin(ωt) ω/(ω2+s 2) cos(ωt) s/(ω2+s 2) d 2 f(t)/dt 2 f(t-θ) 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio F(s)/s e-θs F(s) 6

UNIT STEP or HEAVISIDE STEP FUNCTION u(t) = 0 per t<0 u(t) = 1 per t>0 L[u(t)]=1/s also denoted by H or 1 or �� t Oliver Heaviside (1850– 1925) f(t) A STEP L[A u(t)] = AL[u(t)] = A/s f(t)= Au(t) A>0 t EXAMPLE on Temperature: ACTUAL 31/10/2020 ΔTi IDEALIZED Processes Instrumentation and Control – Prof. M. Miccio 7

IMPULSE DIRAC’ S DELTA δ(t)=0 per t 0 δ=∞ per t=0 Paul Dirac 1902 -1984 • L[δ(t)]=1 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 8

OSCILLATING or HARMONIC FUNCTION SINE g(t)=(sinωt)u(t) g(t)=0 per t<0 g(t)=sinωt per t>0 31/10/2020 L[sinωt]=ω/(s 2+ω2) Strumentazione e Controllo dei Processi Chimici - Prof M. 9

Table of Laplace Transforms Functions u(t) Unit spet u(t)=0 for t<0 u(t)=1 for t>o tu(t) Diagrams Transforms 1/s Functions Diagrams Transforms e(-at)u(t) 1/(s+a) Exponential decay 1/s 2 tne(-at)u(t) n!/(s+a)n+1 n!/s(n+1) (sinkt)u(t) k/(s 2+k 2) Ramp f(t)=0 for t<0 f(t)=t for t>0 tnu(t) Power law Sine g(t)=0 for t<0 g(t)=sinkt for t>0 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 10

Table of Laplace Transforms (cont’ed) Functions Diagrams Transforms cos kt u(t) s / (s 2+k 2) e–at sinkt u(t) k /(s+a)2+k 2 sinh kt u(t) k / (s 2 -k 2) e–at coskt u(t) (s+a)/ ((s+a)2+k 2) cosh kt u(t) s /( s 2 -k 2) DIRAC DELTA FUNCTION (UNIT IMPULSE) 1 δ(t)=0 for t 0 δ=∞ for t=0 31/10/2020 Process Instrumentation and Control – Prof. M. Miccio • 11

DELAY OR LAG • • see: Ch. 7 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 12

UNIT PULSE f(t)=0 for t<0 f(t)=1/A for 0<t<A f(t)=0 for t>A • • f(t) 1/A • A t 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 13

LEAD • 31/10/2020 • Processes Instrumentation and Control – Prof. M. Miccio 14

FUNDAMENTAL THEOREMS Let us consider: • Initial-Value Theorem The value of the function at the initial condition is equal to the value of the limit of the transform function multiplied by the variable s as s apporaches to infinity. Final-Value Theorem The final value of the function is equal to the limit of the transform function multiplied by the variable s as s approaches to zero. Derivation Theorem The Laplace transform of the derivative is equal to the trasform fuction multiplied by the variable s minus the value of the function at the state 0 (initial condition). Integration Theorem The Laplace transform of the integral is equal to the transform function divided by the variable s. 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 15

INVERSE LAPLACE TRANSFORM • L-1 f(t) NOTE: Also the inverse Laplace transform is based on an integral and, hence, is a linear operation! 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 16

INVERSE LAPLACE TRASFORMS OF SELECTED EXPRESSIONS LAPLACE TRASFORM TIME FUNCTION f(t) 31/10/2020 Processes Instrumentation and Control – Prof. M. 17

INVERSE LAPLACE TRASFORMS OF SELECTED EXPRESSIONS (cont. ed) TRASFORM LAPLACE TIME FUNCTION f(t) 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 18

PARTIAL FRACTION (or HEAVISIDE) EXPANSION Hyp. : a proper rational function • Q(s)/P(s) can be expanded as a series of fractions: • where r 1(s), r 2(s), …, rn(s) are polynomials with a lower degree, i. e. linear polynomials, quadratic polynomials, etc. • The values of constants C 1, C 2, …, Cn are calculated by equation (1). • The inverse Laplace transform is obtained from each partial fraction. • The unknown function f(t) is evaluated by: • ☺ The inversion procedure of each fraction is performed by means of tables of Laplace transform and of inverse Laplace transform. 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 19

PARTIAL FRACTION (or HEAVISIDE) EXPANSION EXAMPLE N. 1 § 8. 2 Stephanopoulos, “Chemical process control: an Introduction to theory and practice” EXAMPLE N. 2 Rational proper function �� web page See also file frattisemplici. pdf 31/10/2020 20

Application to the resolution of linear ODEs with constant coefficients 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 21

CHARACTERISTIC EQUATION P(s) = 0 Hyp. : a rational function For a rational function 2 cases are possible: • P(s) has distinct roots • • P(s) has multiple roots NOTE: • P(s) takes the name of characteristic polynomial • The roots of the characteristic equation are called POLES. • The solutions of the equation Q(s)=0 are called ZEROS. 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 22

DEVIATION VARIABLE The deviation variable is defined as a new variable (indifferently for Input, State, Output) calculated as the difference between the current value and the steady state value. e. g. , for the output: • see: § 6. 3 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 31/10/2020 Process Instrumentation and Control - Prof M. Miccio 23

THE ADVANTAGES OF USING THE DEVIATION VARIABLES PROBLEM! When we pass to the Laplace domain, the derivation theorem introduces constant values that are not equal to zero value. SOLUTION! We introduce the deviation variables as a change of variables putting the initial condition as null. When we pass to the Laplace domain in the successive step, the derivation theorem generates constant values equal to zero value. 31/10/2020 Process Instrumentation and Control - Prof M. Miccio 24

TRANSFER FUNCTION Given a linear dynamic system SISO (Single-Input, Single-Output): Dynamic System INPUT f(t) INPUT ODE model DEFINITION (for linear dynamic systems): G(s) OUTPUT y(t) OUTPUT Block Diagram in the Laplace domain TF = (Laplace Transform of the Output) / (Laplace Transform of the Input) OUTPUT INPUT TRANSFER FUNCTION 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 25

TRANSFER FUNCTION L[δ(t)]=1 If f(t)=δ(t) the TF coincides with the impulse response the Laplace Domain OUTPUT INPUT G(s) TRANSFER FUNCTION Block Diagram in the Laplace domain 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 26

RATIONAL TRANSFER FUNCTION Hyp. : a rational TF • This is true for feasible systems in engineering: • • EXAMPLE: 31/10/2020 Q(s), P(s) are polynomials with real coefficients degree Q(s) = q degree P(s) = p q<p (TF is strictly a proper rational funtion) P(s) is the characteristic polynomial P(s)= 0 is the characteristic equation POLES are the roots of the characteristic equation • q=0, p=n, q<p Processes Instrumentation and Control – Prof. M. Miccio 27

DYNAMIC RESPONSE OF A BOUNDED INPUT The dynamic response depends on the nature of both the TF and the input (type of forcing function). Forcing functions with a bounded asymptotic behavior are assumed. The Heaviside expansion is applied. For the main feasible systems in engineering it will be: • FOR COMMON INPUTS • IT ADDS OTHER PARTIAL FRACTIONS AND, THUS, FURTHER “ADDITIONAL POLES” WHICH DO NOT PRODUCE UNBOUNDED y(t) WHEN t→∞ FOR THE MORE COMMON INPUT THE POLES OF THE SYSTEM DETERMINE THE QUALITATIVE BEHAVIOUR OF THE RESPONSE y(t) THE INVERSE PROCEDURE GIVES US THE RESPONSE OF THE SYSTEM y(t) 31/10/2020 For example: if f(t)=u(t) Processes Instrumentation and Control – Prof. M. Miccio 28

EXAMPLES OF DYNAMIC RESPONSE DISTURBANCE at t=t 0 with a “BOUNDED“ INPUT. For example: f(t) = Au(t-t 0) Responses of unstable systems see: Ch. 1 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 29

BIBO STABILITY § 9. 4 Stephanopoulos, “Chemical process control : an Introduction to theory and practice” Definition of BIBO stability (Bounded Input Bounded Output) A dynamic system is defined as stable if its response (output) is mathematically bounded for every input that is bounded, whatever its initial condition. NOTES: A “bounded” input is a variable whose values are included in an interval with a supremum and a infimum (i. e. , sine function, step function). The ramp function is not bounded. The BIBO stability definition is valid for linear systems only. f(t) Scalar dynamic system: f e y are functions of the time 31/10/2020 LINEAR DYNAMIC SYSTEM y(t) Vector dynamic system: f e y are vectors of functions of the time Processes Instrumentation and Control – Prof. M. Miccio 30

MARGINAL STABILITY Definition: An input-output system is defined marginally stable if only certain bounded inputs will result in a bounded output. http: //en. wikipedia. org/wiki/Marginal_stability f(t) LINEAR DYNAMIC SYSTEM y(t) • 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. 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 31

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 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 • 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. 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 32

SELF-REGULATING SYSTEM Definition: A self-regulating system 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. f(t) LINEAR DYNAMIC SYSTEM y(t) see: Ch. 1 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 31/10/2020 Strumentazione e Controllo dei Processi Chimici - Prof M. 33

RATIONAL TRANSFER FUNCTIONS Position of poles in complex plane P 6 § 9. 4 Stephanopoulos, “Chemical process control : an Introduction to theory and practice” 31/10/2020 P 6* Processes Instrumentation and Control – Prof. M. Miccio 34

TRANSFER FUNCTION and STABILITY real single pole Im Re -P 1 e(-p 1 t)u(t) b. EXPONENTIAL DECAY (P 1) Im P 2 Re e(p 2 t)u(t) a. EXPONENTIAL GROWTH (P 2) 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 35

TRANSFER FUNCTION and STABILITY multiple real poles Im y(t) tne(-at)u(t) -P 3 t Re Example of a double pole (n=2) 31/10/2020 Processes Instrumentation and Control – Prof. M. Miccio 36

TRANSFER FUNCTION and STABILITY poles in the origin y(t) Im u(t) P 5 t Re Im y(t) t∙u(t) P 5 31/10/2020 C 5 Re Processes Instrumentation and Control – Prof. M. Miccio t 37

TRANSFER FUNCTION and STABILITY complex and conjugate poles Im y(t) -P 4* Im y(t) P 4 * 31/10/2020 t Re Re Processes Instrumentation and Control – Prof. M. Miccio t 38

TRANSFER FUNCTION and STABILITY imaginary and conjugate poles Im P 6 * 31/10/2020 Re Processes Instrumentation and Control – Prof. M. Miccio SUSTAINED OSCILLATIONS (P 6) 39