The need for a newpedagogical approach to

system modeling and analysisSoumitro Banerjee

Indian Institute of Technology, Kharagpur, India

The need for a newpedagogical approach tosystem modeling and analysis – p. 1/22

Important components ofCAS education

The ability to formulate mathematical models ofsystems;

The ability to analyze the behavior of systems basedon the mathematical description.

The need for a newpedagogical approach tosystem modeling and analysis – p. 2/22

The current practice

Modeling approaches aimed at deriving transferfunctions (block diagram, signal flow graph, etc.)

Analyzing stability and other properties based onstandard techniques in the s-domain.

Approaches restricted to linear systems only.

The need for a newpedagogical approach tosystem modeling and analysis – p. 3/22

The reality

Most physical systems are nonlinear.

Linearity is a very special case.

For most physical systems, linear description isactually a first order approximation in theneighborhood of an equilibrium point.

Most circuits and systems actually exhibit dynamicalphenomena that cannot be understood in terms oflinear description.

The need for a newpedagogical approach tosystem modeling and analysis – p. 4/22

The necessity

Students should learn the techniques of time-domainformulations (in addition to those in theLaplace-domain):

the techniques of obtaining the differentialequations of any given physical system;

understanding the dynamics in terms of thecharacter of the vector field.

The need for a newpedagogical approach tosystem modeling and analysis – p. 5/22

The time-domain methods

The systematic methods of obtaining the differentialequations:

The Lagrangian method

The Hamiltonian method

The graph theoretic technique

The bond graph technique.

Unfortunately, these approaches have not been adoptedin the mainstream EE or CAS education.

The need for a newpedagogical approach tosystem modeling and analysis – p. 6/22

The Lagrangian approach

. .2 CCC

LL

q1 q2 31

1 2E

ddt

(

∂L∂q̇i

)

− ∂L∂qi +∂ℜ∂q̇i

= 0.

L = T − V

T =1

2L1q̇1

2 +1

2L2q̇2

2

V =1

2C1q21 +

1

2C2q22 +

1

2C3(q1 − q2)

2 − E q1

ℜ =1

2R1q̇1

2 +1

2R2q̇2

2

The need for a newpedagogical approach tosystem modeling and analysis – p. 7/22

The Hamiltonian approach

q̇i =∂H

∂pi,

ṗi = −∂H

∂qi−

∂ℜ

∂q̇i.

q1 q2

q3

q1

q3

q1

q2

q3

q1

q2

E C

R

C

L

R1

2

1

2

+

+−

H = T + V =1

2L1p21 +

1

2C1q22 +

1

2C2q23 − E(q1 + q3).

ℜ =1

2R1(q̇1 − q̇2)

2 +1

2R2(q̇1 − q̇2 + q̇3)

2.

The need for a newpedagogical approach tosystem modeling and analysis – p. 8/22

The graph theoreticmethod

12

0

+

3

−

2

ea

1b

c

d

0

3

f

dvC1

dt=

1

C1iL −

1

R1C1vC1 −

1

R1C1vC2 +

1

R1C1E

dvC2

dt= −

1

R1C2vC1 −

1

C2

(

1

R1+

1

R2

)

vC2 +1

C2

(

1

R1+

1

R2

)

E

diL

dt= −

1

LvC1 +

1

LE.

The need for a newpedagogical approach tosystem modeling and analysis – p. 9/22

The bond graph method

R1 L2

L1 R2E C

R1 L1 L2

R2

(a) (b)

1 0 1 0

R: I: I: C:

R:

C

:E

2 4 6 8

SE1 3 5 7 9

ṗ4 = E − R1

(

p4

L1+

p6

L2

)

ṗ6 = R1

(

p4

L1+

p6

L2

)

−q8

C

q̇8 =p6

L2−

q8

R2C.

The need for a newpedagogical approach tosystem modeling and analysis – p. 10/22

Bond graph software

The need for a newpedagogical approach tosystem modeling and analysis – p. 11/22

The advantage

These methods are

equally applicable to linear and nonlinear systems

equally applicable to electrical, mechanical, andelectromechanical systems

The need for a newpedagogical approach tosystem modeling and analysis – p. 12/22

Understanding dynamics

x.

x

The vector field for ẍ − (1 − x2)ẋ + x = 0.

The need for a newpedagogical approach tosystem modeling and analysis – p. 13/22

Local linearization

δẋ

δẏ

=

∂f1∂x

∂f1∂y

∂f2∂x

∂f2∂y

δx

δy

where δx = x − x∗, δy = y − y∗.

↓

ẋ = Ax + Bu

The need for a newpedagogical approach tosystem modeling and analysis – p. 14/22

Eigenvalues andeigenvectors

eigenv

alue=

−1eigenvalue= −2

decays

as e

decays as e−2t

−t

The need for a newpedagogical approach tosystem modeling and analysis – p. 15/22

Higher dimensionalsystems

Eigenplane

The need for a newpedagogical approach tosystem modeling and analysis – p. 16/22

Nonlinear systems

y

(0,0) ( ,0)π π π(2 ,0)(−2 ,0) −π( ,0) x

The vector field of the simple pendulum.

ẋ = y

ẏ = −g

lsin x.

The need for a newpedagogical approach tosystem modeling and analysis – p. 17/22

Limit cycle

1.5

1

0.5

0

−0.5

−1

−1.5−1.5 −1 −0.5 0 0.5 1 1.5

x

y

−3 −2 −1 0 1 2 3

0

1

2

3

−3

−2

−1

x

y

The vector fields for the system ẍ − µ(1 − x2)ẋ + x = 0,(a) for µ < 0, and (b) for µ > 0.

The need for a newpedagogical approach tosystem modeling and analysis – p. 18/22

Quasiperiodicity

x1

Time

x

xx

1

23

A quasiperiodic orbitThe need for a newpedagogical approach tosystem modeling and analysis – p. 19/22

Chaos201510

50

−5

−10−15−20

0 5 10 15 20 25 30 35 40 45 50

Time

x 1

x y

z

−15 −10 −5 0 5 10 15 20−20

−100

1020

−20

10

20

30

40

50

A chaotic orbit.The need for a newpedagogical approach tosystem modeling and analysis – p. 20/22

The need of the hour

In view of the variety and complexity of the systems amodern engineer has to deal with, such exposure isvital.

New courses with such content should be introduced.

The need for a newpedagogical approach tosystem modeling and analysis – p. 21/22

Book is now available

The need for a newpedagogical approach tosystem modeling and analysis – p. 22/22

Important components of CAS educationThe current practiceThe realityThe necessityThe time-domain methodsThe Lagrangian approachThe Hamiltonian approachThe graph theoretic methodThe bond graph methodBond graph softwareThe advantageUnderstanding dynamicsLocal linearizationEigenvalues and eigenvectorsHigher dimensional systemsNonlinear systemsLimit cycleQuasiperiodicityChaosThe need of the hourBook is now available