UNIVERSIT À DEGLI STUDI DI SALERNO

1. UNIVERSITÀ DEGLI STUDI DI SALERNO Bachelor Degree in Chemical Engineering Course: Process Instrumentation and Control (Strumentazione e ControllodeiProcessiChimici) TRANSFER FUNCTION Applications of Laplace Transform Rev. 3.7 – May 8, 2019

2. INTRODUCTION see: Ch.7, 8, 9 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” Laplace Transform: Purpose and applicationsfor the Dynamic Behavior and the Automatic Control • solution of dynamicmathematical models expressed in terms of linear ordinarydifferenzialequations (ODE). • concept of transfer function • dynamicresponses: • unitimpulse • step • harmonicresponse • classification and prediction of the dynamicbehavior of linear systems Processes Instrumentation and Control – Prof. M. Miccio

3. LAPLACE TRANSFORM Definition: • where: • f(t)is a knownfunction of the time, defined for t 0, • s=a+jbis a complexvariablecalledLaplace abscissa, • F(s) is the Laplace transformof f(t). • F(s) existsif the integral takes a finite value. • In particular, the integralisbounded for specifiedvalues of the real part of s. • For everyf(t) a convergenceabscissa, c, has to be defined for F(s). The parallel line to the imaginaryaxes passing thoughcdefines 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 integralassumes finite values for every positive values of s; • ff(t)=e2t, integralis finite for Re(s) >2. Processes Instrumentation and Control – Prof. M. Miccio

4. L f(t) DOMAIN OF DEFINITION L-1 Domain of definition of f(t) X : t  + Codomain of f(t) Y : f(t)   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

5. LAPLACE TRANSFORM The Laplace transform is a linear operation: g(t)= a1f1(t)+a2f2(t) (Principle of Superposition) L[g(t)]= L [a1f1(t)+a2f2(t)]=a1L[f1(t)]+a2L[f2(t)]= NOTE: h(t)=[f(t)]2 is a nonlinear operation. Thus: Processes Instrumentation and Control – Prof. M. Miccio

6. LAPLACE TRANSFORM TABLE Processes Instrumentation and Control – Prof. M. Miccio

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

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

9. OSCILLATING or HARMONICFUNCTION SINE g(t)=(sinωt)u(t) g(t)=0 per t<0 g(t)=sinωt per t>0 L[sinωt]=ω/(s2+ω2) Strumentazione e Controllo dei Processi Chimici - Prof M. Miccio

10. Functions Functions Diagrams Diagrams Transforms Transforms Table of Laplace Transforms Processes Instrumentation and Control – Prof. M. Miccio

11. Table of Laplace Transforms (cont’ed) Functions Transforms Diagrams Functions Diagrams Transforms Process Instrumentation and Control – Prof. M. Miccio

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

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

14. LEAD Processes Instrumentation and Control – Prof. M. Miccio

15. 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 isequal to the trasformfuctionmultiplied by the variablesminus the value of the functionat the state 0 (initialcondition). Integration Theorem The Laplace transform of the integral is equal to the transform function divided by the variable s. Processes Instrumentation and Control – Prof. M. Miccio

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

17. INVERSE LAPLACE TRASFORMS OF SELECTED EXPRESSIONS Processes Instrumentation and Control – Prof. M. Miccio

18. INVERSE LAPLACE TRASFORMS OF SELECTED EXPRESSIONS (cont.ed) Processes Instrumentation and Control – Prof. M. Miccio

19. PARTIAL FRACTION (or HEAVISIDE) EXPANSION Hyp.: a properrationalfunction • Q(s)/P(s) can be expanded as a series of fractions: where r1(s), r2(s),…, rn(s) are polynomials with a lower degree, i.e. linear polynomials, quadraticpolynomials, etc. • The values of constants C1, C2, …, Cn are calculated by equation (1). • The inverse Laplace transformisobtained from eachpartialfraction. • The unknownfunctionf(t) isevaluated by: ☺ The inversion procedure of eachfractionisperformed by means of tables of Laplace transformand of inverse Laplace transform. Processes Instrumentation and Control – Prof. M. Miccio

20. PARTIAL FRACTION (or HEAVISIDE) EXPANSION • EXAMPLE N.1 • § 8.2 Stephanopoulos, “Chemical process control: an Introduction to theory and practice” EXAMPLE N.2 Rationalproperfunction 💻 web page Seealso file frattisemplici.pdf

21. Application to the resolutionof linear ODEs with constant coefficients Processes Instrumentation and Control – Prof. M. Miccio

22. P(s) hasdistinct roots CHARACTERISTIC EQUATIONP(s) = 0 Hyp.: a rationalfunction For a rationalfunction2 cases are possible: • P(s) has multiple roots • NOTE: • P(s) takes the name of characteristicpolynomial • The roots of the characteristicequationare calledPOLES. • The solutions of the equationQ(s)=0 are calledZEROS. Processes Instrumentation and Control – Prof. M. Miccio

23. 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” Process Instrumentation and Control - Prof M. Miccio

24. THE ADVANTAGES OF USINGTHE 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. Process Instrumentation and Control - Prof M. Miccio

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

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

27. RATIONAL TRANSFER FUNCTION Hyp.: a rational TF • Thisistrue for feasiblesystems in engineering: • Q(s), P(s) are polynomials with realcoefficients • degreeQ(s) = q • degreeP(s) = p • q<p (TF isstrictly a properrationalfuntion) • P(s) is the characteristicpolynomial • P(s)= 0 is the characteristicequation • POLES are the roots of the characteristicequation EXAMPLE: q=0, p=n, q<p Processes Instrumentation and Control – Prof. M. Miccio

28. DYNAMIC RESPONSE OF A BOUNDED INPUT The dynamic response depends on the nature of both the TF and the input (type of forcingfunction). Forcing functions with a boundedasymptoticbehavior are assumed. The Heavisideexpansionisapplied. For the mainfeasible systems in engineering itwill 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) For example: if f(t)=u(t) THE INVERSE PROCEDURE GIVES US THE RESPONSE OF THE SYSTEM y(t) Processes Instrumentation and Control – Prof. M. Miccio

29. DISTURBANCE at t=t0 with a “BOUNDED“ INPUT. For example: f(t) = Au(t-t0) EXAMPLES OF DYNAMIC RESPONSE Responses of unstable systems see: Ch.1 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” Processes Instrumentation and Control – Prof. M. Miccio

30. f(t) y(t) LINEAR DYNAMICSYSTEM BIBO STABILITY § 9.4 Stephanopoulos, “Chemical process control : an Introduction to theory and practice” Definition of BIBO stability (Bounded Input Bounded Output) A dynamicsystemisdefinedasstableifitsresponse (output) ismathematicallybounded for everyinputthatisbounded, whateveritsinitialcondition. NOTES: A “bounded” input is a variablewhosevalues are included in an interval with a supremum and a infimum (i.e., sine function, stepfunction). The rampfunctionisnotbounded. The BIBO stabilitydefinitionisvalid for linear systemsonly. Scalar dynamic system: f e y are functions of the time Vector dynamic system: f e y are vectors of functions of the time Processes Instrumentation and Control – Prof. M. Miccio

31. f(t) y(t) LINEAR DYNAMICSYSTEM 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 • Thislatter case occurswhenthere are poles with single multiplicityon the stabilityboundary, i.e. the imaginaryaxis. • A marginallystablesystemmayexhibit an output responsethatneitherdecaysnorgrows, butremainsstrictlyconstant or displays a sustainedoscillation. Processes Instrumentation and Control – Prof. M. Miccio

32. OPEN-LOOP BIBO STABILITY • For systems with properrational transfer functions • BIBO Stability Theorem: • a system is (asymptotically) stable if all of its poles have negative real parts • a system is unstableif at least one pole has a positive real part • a system is marginally stableif 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 stableor 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. Processes Instrumentation and Control – Prof. M. Miccio

33. f(t) y(t) LINEAR DYNAMICSYSTEM 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. see: Ch.1 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” Strumentazione e Controllo dei Processi Chimici - Prof M. Miccio

34. P6 P6* RATIONAL TRANSFER FUNCTIONSPosition of poles in complex plane § 9.4 Stephanopoulos, “Chemical process control : an Introduction to theory and practice” Processes Instrumentation and Control – Prof. M. Miccio

35. Im Im Re Re -P1 P2 TRANSFER FUNCTION and STABILITYreal single pole • e(-p1t)u(t) b. EXPONENTIAL DECAY (P1) • e(p2t)u(t) a. EXPONENTIAL GROWTH (P2) Processes Instrumentation and Control – Prof. M. Miccio

36. Im Re -P3 TRANSFER FUNCTION and STABILITYmultiple realpoles Example of a double pole (n=2) y(t) • tne(-at)u(t) t Processes Instrumentation and Control – Prof. M. Miccio

37. Im Im y(t) C5 Re Re t TRANSFER FUNCTION and STABILITYpoles in the origin u(t) P5 P5 y(t) t∙u(t) t Processes Instrumentation and Control – Prof. M. Miccio

38. Im Im Re Re TRANSFER FUNCTION and STABILITYcomplex and conjugatepoles ++ y(t) -P4 P4 t -P4* P4* y(t) t Processes Instrumentation and Control – Prof. M. Miccio

39. Im Re TRANSFER FUNCTION and STABILITYimaginary and conjugatepoles + = + P6 P6* SUSTAINED OSCILLATIONS (P6) Processes Instrumentation and Control – Prof. M. Miccio