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

Open and closed loop transfer functions.

BIBO stability

by M. Miccio

rev. 3.5 of March 12, 2015

DEFINITIONS

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Proper vs. improper rational transfer function


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.

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.

BIBO Stability

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Definition of BIBO Stability

An unconstrained linear dynamic system is said to be stable if the output response is bounded for all bounded inputs.

Marginal Stability


•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

http://en.wikipedia.org/wiki/Marginal_stability

SUMMARY


System Open-Loop Behavior

Transfer Function Propervs.improper

1st order self-regulating

Proper

purely capacitive 1st order

non self-regulating

Proper

2nd order self-regulating

Proper

dead time self-regulating

not applicable

PID controller = = =

Improper

Summary of Linear Dynamic Systems


1s

K)s(G

p

p

s

K)s(G

'p

stde)s(G

BIBO Stability


• 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

BIBO Stability of Linear Dynamic OL Systems


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

BIBO Stability of Linear Dynamic OL Systems


Note that the poles are on the imaginary axis

Marginally stableMarginally stable

step input

CLOSED LOOP



Closed Loop Block Diagramin the time domain

Finalcontrol element

PROCESS

SENSOR

PID ControllerySP(t) ε(t) o(t) m(t)

d(t)

ym(t)

y(t)+

-

Kc


Closed Loop Block Diagram:

shut-down of the controller


PROCESS

SENSOR

PID ControllerySP(t) ε(t) m(t)

d(t)

ym(t)

y(t)+

-

practically, the same as setting Kc = 0 o(t) = 0


Closed Loop Block Diagram:

manual mode of the controller Open Loop operation


PROCESS

SENSOR

PID ControllerySP(t) ε(t) o(t) m(t)

d(t)

ym(t)

y(t)+

-


Open Loop Transfer FunctionLaplace domain


PROCESS

SENSOR

PID ControllerySP(s) ε(s) o(s) m(s)

d(s)

ym(s)

y(s)+

-

Definition: GOL(s)=GcGfGpGm

see:Ch.14 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall,1984

Forward path

Feedback path


Closed Loop Transfer Function


PROCESS

SENSOR


d(s)

ym(s)

y(s)+

-

Servo Problem: y(s)=GCL,SP(s)ySP(s)(Hyp.: d(s) = 0)

Regulator Problem: y(s)=GCL,load(s)d(s)(Hyp.: ySP(s) = 0)

where the Open Loop Transfer Function:

see:Ch.14 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984

mpfcOL

OL

pload,CL

OL

pfcSP,CL

GGGG)s(G

)s(G1

G)s(G

)s(G1

GGG)s(G


Closed Loop Transfer Function


PROCESS

SENSOR


d(s)

ym(s)

y(s)+

-

from thePrinciple of Superimposition:

where theOpen Loop Transfer Function:

see:Ch.14 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984

mpfcOL

OL

pSP

OL

pfc

GGGG)s(G

)s(d)s(G1

G)s(y

)s(G1

GGG)s(y


Sensitivity Function

Closed Loop Transfer Function:

Sensitivity Function:

Example 1:

Closed Loop Transfer Function:

Sensitivity Function:

Hyp.: Gm=1

1s2

1s

2

1

2s

1s

1s1

1

1)s(S

1s1

1

1s1

)s(G

1s

1)s(G

)s(G1

1

G

G

G

G

GG

G

G

)s(S

)s(G1

)s(G)s(G

SP,CL

OL

OLOL

SP,CL

SP,CL

OL

OL

OL

SP,CL

SP,CL

OL

OLSP,CL

d

d

Dynamical Systems in SeriesLaplace domain


)s(G)s(GGG

)s(f)s(G)s(G)s(y

definitionby )s(y)s(G)s(y

definitionby )s(f)s(G)s(y

21

2

1ii12

212

122

11

G1 G2y2y1

f G1 Gnyn

Dynamical Systems in ParallelLaplace domain


Block Diagram:graphical conventions,

block manipulation


Block_Diagrams(D.Cooper).pdf

Closed Loop BIBO Stability

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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 GOL(s)

The characteristic equation of the closed-loop linear dynamic system is:

1 + GOL(s) = 0

poles of the characteristic equation Closed Loop BIBO STABILITY

BIBO Stability of Linear Dynamical CL Systems


Closed-loop poles (x)and time response:

stable system

unstable system

step input

Non-miminum phase systemsLet’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

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

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