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Modern Control

GUC Faculty of Engineering and Material Science Department of Mechatronics

Tutorial #9

Stability

Modern Control MCTR 702 Dr. Ayman Ali El-Badawy



Stability

Internal

• Eigen values stability

• Lyapunov stability

External

• Bounded input –bounded output (BIBO)

stability

Stability types



Eigen values stability



Problem 1:

Asses the stability of the following system Using Eigen values stability analysis

414

10A

Eigenvalues:

-2 + 3.16i

-2 - 3.16i System is asymptotically stable



414

10A

2 + 3.16i

2 - 3.16i

Eigenvalues: System is unstable

40

10A

0

-4

Eigenvalues: System is marginally stable due to zero

pole (and no positive poles).

Stable in the sense of Lyapunov (i.s.l)

014

10A

Eigenvalues: 0 + 3.74i

0 - 3.74i

System is marginally stable due to

zero real part poles (and no positive

real poles).

Stable in the sense of Lyapunov

(i.s.l)



Lyapunov Stability Theorem

And to asses the stability of a system using Lyapunov theorem, we use Lyapunov equation

System A matrix



Problem 2:

Asses the stability of the following system Using Lyapunov stability analysis

414

10A

10

01

414

10

41

140

2212

1211

2212

1211

pp

pp

pp

pp

10

01Q

2212

1211

pp

ppP P is symmetric



10

01

414

414

44

1414

221222

121112

22122111

2212

ppp

ppp

pppp

p P-

10

01

82144

14428

2212221211

22121112

PP P PP

P PP P-

Solving for P11 , P12, P22, we get

0.1340.0357

0.03572.02P

And its eigenvalues are 2.0185, and 0.1333, which show that the P matrix is Positive Definite, and hence the system is Asymptotically stable.



0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

x1 (m)

x2 (

m/s

ec)

Phase plane



414

10A

By performing the same procedures as the previous problem

10

01

82144

14428

2212221211

22121112

PP P PP

P PP P-

Solving for P11 , P12, P22, we get

0.134-0.0357

0.03572.02-P

And its eigenvalues are -2.02, and -0.133, which show that the P matrix is Negative Definite, and hence the system is unstable.



-400 -200 0 200 400 600 800 1000 1200 1400 1600-1000

-500

0

500

1000

1500

2000

2500

3000

x1 (m)

x2 (

m/s

ec)

Phase plane



40

10A


10

01

824

40

22121211

1211

PP PP

PP

Here, the no. of unknowns is less than the no. of equations, hence there is no possibility for a unique P matrix, and Lyapunov test fails.




10

01

214

1428

122211

221112

P PP

PP P

Here, the no. of unknowns is less than the no. of equations, hence there is no possibility for a unique P matrix, and Lyapunov test fails.

014

10A



External test

Bounded-input, bounded-output (BIBO) stability

In this stability test, we apply a bounded input to the system (impulse or step input), And then we check the output. If the output is bounded, then the system is BIBO stable.

Important note:



Problem 3:

solution

The system is bounded-input, bounded-output stable because the impulse response

2( ) (1 ) th t t e

The three-dimensional controller canonical form realization is specified by

CCF CCF CCF

0 1 0 0

0 0 1 0 2 1 1

8 4 2 1

A B C

2

3 2 2 2

2 ( 2)( 1) ( 1)( )

2 4 8 ( 2)( 2) ( 2)

s s s s sH s

s s s s s s



The observability matrix

CCF

CCF CCF

2

CCF CCF

2 1 1

8 2 3

24 4 8

C

Q C A

C A

modern control - ems.guc.edu.egems.guc.edu.eg/download.ashx?id=446&file=tut9_446.pdf · dr....

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