2 lec Eng

Direct method of Stability analysis

u(t)

Input signal

System

y(t)

Output signal

W s

y(s)

b sm b sm 1

... b

s b

0

1

m 1

m

sn a sn 1

u(s)

a

... a

n 1

s a

n

0

1

The equation

a

sn a sn 1

... a

n 1

s a

n

0

is called

0

1

characteristic equation.

The roots of this equation 1 , 2 , ... , n are called poles of the system (or the transfer function)

Direct method of Stability analysis

Let	W s	s 1		1	1;
				The poles:
		s 1 2s 1				1
				2			.
						2

The impulse response for this system is

y t 2e 1 t 32 e 12 t

The exponential terms e i t control the response y t as t

The real parts of the poles must be negative

Direct method of Stability analysis

For asymptotic stability the roots of the characteristic equation must all be located in the left-half s-plane.

Direct method of Stability analysis

If there is at least one pole located in the right-half s-plane, the system is unstable:

If there is no one pole located in the right-half s-plane, but there are one or more poles directly on the imaginary axis, the system is neutral stable:

Direct method of Stability analysis

There is an exception.

In the case of multiple poles in the imaginary axis, the system is unstable:

Examples

1.	W s	1
		s2

s-plane

x2

4

3

2

1

0	Equilibrium
-1	position

-2

-3

-4
-15	-10	-5	0	5	10	15

x1

1.5

zero-input

1
			input
0.5			output
zero-state
00	1	2	3	4	5
		time, sec

	Examples
2.	1
	W s
		s s 1

s-plane

x2

3

2

1

0

			Equilibrium
-1			position
-2
-3
-6	-4	-2	0	2	4	6

x1

zero-input

zero-state

1
0.8		input
		output
0.6
0.4
0.2
00	2	4	6	8
		time, sec

	Examples
3.	1
	W s
		s 1 2s 1

s-plane

	5
	4
	3
	2
	1
x2	0
	-1
	-2		Equilibrium
			position
	-3
	-4
	-5
	-6	-4	-2	0	2	4	6
				x1

zero-input

zero-state

1.5
1		input
0.5		output
0
-0.50	5	10	15
		time, sec

Examples

	4.					1
		W s		1	s2		1	s 1
		2					2

s-plane

x2

4

3

2

1

0

-1		Equilibrium
-2		position
-3
-4
-2	-1	0	1	2	3	4

x1

zero-input

zero-state

1.5
1	input
	output
0.5
0
-0.50	5	10
	time, sec

Examples	W s				1
5.			1 2		1
				s			s 1
			2			2

s-plane

	20
	10
	0
x2	-10		Equilibrium
			position
	-20
	-30
	-40
	-10	-5	0	5	10	15	20
				x1

zero-input

zero-state

10

5		input
		output
0
-5
-10
-150	2	4	6	8
		time, sec