Stability of Feedback Systems

1
Stability of Feedback Systems

• SYSTEMS AND CONTROL I ECE 09.321
• 10/29/07 Lecture 13
• ROWAN UNIVERSITY
• College of Engineering
• Prof. John Colton
• DEPARTMENT OF ELECTRICAL COMPUTER ENGINEERING
• Fall 2007 - Semester One

2
Some Administrative Items

• Quiz 3 will be on Wednesday 10/31 950 AM to
  1040 AM.
• Material through Routh Hurwitz criterion
• Through section 6.2 of textbook
• Through problem set 8
• Problem Set 8 is due 10/31. Clarifications
  emailed
• Monday lab emailed

3
Stability of Linear Feedback Control Systems

• Stability Defined
• Routh Hurwitz Stability Criterion
• Root Locus
• Frequency Response Methods
• Gain and phase margin
• Nyquist Criterion

4
Routh Hurwitz Stability Criterion
General procedure Characteristic equation ?(s)
an s n an-1 s n -1 a1 s a0 0 Form
Routh array s n an an-2 an-4 an-6 .
s n 1 an-1 an-3 an-5 an-7 s n 2
bn-1 bn-3 bn-5 s n 3 cn-1 cn-3
cn-5 s 1 ln-1 s 0 mn-1
5
Routh Hurwitz Stability Criterion
General procedure Characteristic equation ?(s)
an s n an-1 s n -1 a1 s a0 0 Form
Routh array s n an an-2 an-4 an-6 .
s n 1 an-1 an-3 an-5 an-7 s n 2
bn-1 bn-3 bn-5 s n 3 cn-1 cn-3
cn-5 s 1 ln-1 s 0 mn-1
an an-2 bn-1 (1/
an-1 ) an-1 an-3 (an-1 an-2 an
an-3 )/ an-1
an an-4 bn-3 (1/
an-1 ) an-1 an-5 (an-1 an-4 an
an-5 )/ an-1
an an-4 bn-5 (1/
an-1 ) an-1 an-5 (an-1 an-6 an
an-7 )/ an-1
an-1 an-3 cn-1 (1/
bn-1 ) bn-1 bn-3 (an-3 bn-1 an-1
bn-3 )/ an-1
6
Routh Hurwitz Stability Criterion
General procedure Characteristic equation ?(s)
an s n an-1 s n -1 a1 s a0 0 Form
Routh array s n an an-2 an-4 an-6 .
s n 1 an-1 an-3 an-5 an-7 s n 2
bn-1 bn-3 bn-5 s n 3 cn-1 cn-3
cn-5 s 1 ln-1 s 0 mn-1
an an-2 bn-1 (1/
an-1 ) an-1 an-3 (an-1 an-2 an
an-3 )/ an-1
an an-4 bn-3 (1/
an-1 ) an-1 an-5 (an-1 an-4 an
an-5 )/ an-1
an an-4 bn-5 (1/
an-1 ) an-1 an-5 (an-1 an-6 an
an-7 )/ an-1
an-1 an-3 cn-1 (1/
bn-1 ) bn-1 bn-3 (an-3 bn-1 an-1
bn-3 )/ an-1
7
Routh Hurwitz Stability Criterion
General procedure Characteristic equation ?(s)
an s n an-1 s n -1 a1 s a0 0 Form
Routh array s n an an-2 an-4 an-6 .
s n 1 an-1 an-3 an-5 an-7 s n 2
bn-1 bn-3 bn-5 s n 3 cn-1 cn-3
cn-5 s 1 ln-1 s 0 mn-1
an an-2 bn-1 (1/
an-1 ) an-1 an-3 (an-1 an-2 an
an-3 )/ an-1
an an-4 bn-3 (1/
an-1 ) an-1 an-5 (an-1 an-4 an
an-5 )/ an-1
an an-6 bn-5 (1/
an-1 ) an-1 an-7 (an-1 an-6 an
an-7 )/ an-1
an-1 an-3 cn-1 (1/
bn-1 ) bn-1 bn-3 (an-3 bn-1 an-1
bn-3 )/ an-1
8
Routh Hurwitz Stability Criterion
General procedure Characteristic equation ?(s)
an s n an-1 s n -1 a1 s a0 0 Form
Routh array s n an an-2 an-4 an-6 .
s n 1 an-1 an-3 an-5 an-5 s n 2
bn-1 bn-3 bn-5 s n 3 cn-1 cn-3
cn-5 s 1 ln-1 s 0 mn-1
an an-2 bn-1 (1/
an-1 ) an-1 an-3 (an-1 an-2 an
an-3 )/ an-1
an an-4 bn-3 (1/
an-1 ) an-1 an-5 (an-1 an-4 an
an-5 )/ an-1
an an-4 bn-5 (1/
an-1 ) an-1 an-5 (an-1 an-6 an
an-7 )/ an-1
This pattern is continued until the rest of the b
array coefficients are zero
9
Routh Hurwitz Stability Criterion
General procedure Characteristic equation ?(s)
an s n an-1 s n -1 a1 s a0 0 Form
Routh array s n an an-2 an-4 an-6 .
s n 1 an-1 an-3 an-5 an-7 s n 2
bn-1 bn-3 bn-5 s n 3 cn-1 cn-3
cn-5 s 1 ln-1 s 0 mn-1
an-1 an-3 cn-1 (1/
bn-1 ) bn-1 bn-3 (an-3 bn-1 an-1
bn-3 )/ bn-1
an-1 an-5 cn-3 (1/
bn-1 ) bn-1 bn-5 (an-5 bn-1 an-1
bn-5 )/ bn-1
an-1 an-7 cn-5 (1/
bn-1 ) bn-1 bn-7 (an-1 bn-6 an
bn-7 )/ bn-1
This pattern is continued until the rest of the c
array elements are zero
10
Routh Hurwitz Stability Criterion
General procedure Characteristic equation ?(s)
an s n an-1 s n -1 a1 s a0 0 Form
Routh array s n an an-2 an-4 an-6 .
s n 1 an-1 an-3 an-5 an-7 s n 2
bn-1 bn-3 bn-5 s n 3 cn-1 cn-3
cn-5 s 1 ln-1 s 0 mn-1
an-1 an-3 cn-1 (1/
bn-1 ) bn-1 bn-3 (an-3 bn-1 an-1
bn-3 )/ bn-1
an-1 an-5 cn-3 (1/
bn-1 ) bn-1 bn-5 (an-5 bn-1 an-1
bn-5 )/ bn-1
an-1 an-7 cn-5 (1/
bn-1 ) bn-1 bn-7 (an-1 bn-6 an
bn-7 )/ bn-1
This pattern is continued until the rest of the c
array elements are zero
11
Routh Hurwitz Stability Criterion
General procedure Characteristic equation ?(s)
an s n an-1 s n -1 a1 s a0 0 Form
Routh array s n an an-2 an-4 an-6 .
s n 1 an-1 an-3 an-5 an-7 s n 2
bn-1 bn-3 bn-5 s n 3 cn-1 cn-3
cn-5 s 1 ln-1 s 0 mn-1
an-1 an-3 cn-1 (1/
bn-1 ) bn-1 bn-3 (an-3 bn-1 an-1
bn-3 )/ bn-1
an-1 an-5 cn-3 (1/
bn-1 ) bn-1 bn-5 (an-5 bn-1 an-1
bn-5 )/ bn-1
an-1 an-7 cn-5 (1/
bn-1 ) bn-1 bn-7 (an-7 bn-1 an-1
bn-7 )/ bn-1
This pattern is continued until the rest of the c
array elements are zero
12
Routh Hurwitz Stability Criterion
General procedure Characteristic equation ?(s)
an s n an-1 s n -1 a1 s a0 0 Form
Routh array s n an an-2 an-4 an-6 .
s n 1 an-1 an-3 an-5 an-7 s n 2
bn-1 bn-3 bn-5 s n 3 cn-1 cn-3
cn-5 s 1 ln-1 s 0 mn-1
an-1 an-3 cn-1 (1/
bn-1 ) bn-1 bn-3 (an-3 bn-1 an-1
bn-3 )/ bn-1
an-1 an-5 cn-3 (1/
bn-1 ) bn-1 bn-5 (an-5 bn-1 an-1
bn-5 )/ bn-1
an-1 an-7 cn-5 (1/
bn-1 ) bn-1 bn-7 (an-7 bn-1 an-1
bn-7 )/ bn-1
This pattern is continued until the rest of the c
array elements are zero
13
Routh Hurwitz Stability Criterion
General procedure Characteristic equation ?(s)
an s n an-1 s n -1 a1 s a0 0 Form
Routh array s n an an-2 an-4 an-6 .
s n 1 an-1 an-3 an-5 an-7 s n 2
bn-1 bn-3 bn-5 s n 3 cn-1 cn-3
cn-5 s 1 ln-1 s 0 mn-1
Continue this process until the final s 1 and s 0
array rows are complete
The Routh Hurwitz criterion says the number of
sign reversals in the first array column is
equal to the number of roots that do not have
negative real parts
14
Board Example Case 1 no 0s in first column
?(s) s 5 s 4 10s 2 72s 2 152s 240
0 Routh array s 5 1 10 152 s 4
1 72 240 s 3 62 88 s 2
70.6 240 s 1 122.6 s 0
240 The first column has two sign changes
therefore, there are two roots that do not
have negative real parts
15
Board Example Case 1 no 0s in first column
?(s) s 5 s 4 10s 2 72s 2 152s 240
0 Routh array s 5 1 10 152 s 4
1 72 240 s 3 62 88 s 2
70.6 240 s 1 122.6 s 0
240 The first column has two sign changes
therefore, there are two roots that do not
have negative real parts
16
Example Case 1 no 0s in first column
?(s) s 6 3s 5 2s 4 9s 3 5s 2 12s 20
0 Simplification The coefficients of any row
can be multiplied or divided by any
positive number without changing the sign of any
elements in the first column Routh array s 6
1 2 5 20 s 5 3 9
12 s 4 s 3 s 2 s 1 s 0

17
Example Case 1 no 0s in first column
?(s) s 6 3s 5 2s 4 9s 3 5s 2 12s 20
0 Simplification The coefficients of any row
can be multiplied or divided by any
positive number without changing the sign of any
elements in the first column Routh array s 6
1 2 5 20 s 5 1 3
4 /3 s 4 -1 1 20 s 3 4 24 s 2 s 1
s 0
18
Example Case 1 no 0s in first column
?(s) s 6 3s 5 2s 4 9s 3 5s 2 12s 20
0 Simplification The coefficients of any row
can be multiplied or divided by any
positive number without changing the sign of any
elements in the first column Routh array s 6
1 2 5 20 s 5 1 3
4 /3 s 4 -1 1 20 s 3 1 6 /4 s 2
7 20 s 1 22/7 s 0 20
Two sign reversals in first column mean there are
two roots that do not have negative real parts
19
Example Case 2 0 in first column, non-zero in
associated row
If there is a zero in the first column, but other
elements in that row are nonzero, it may be
replaced with a small positive number e that is
allowed to approach zero after completing the
array. ?(s) s 5 2s 4 2s 3 4s 2 11 s
10 0 Routh array s 5 1 2
11 s 4 2 4 10 s 3 e 6
0 s 2 c1
10 0 c1 4 e 12 / e ? 12/ e s 1
d1 0 0 d1 6 c1 10 e / c1 ? 6 s 0
10 0 0 Two sign changes in first
column say that there are two roots that do not
have negative real parts.
20
Example Case 2 0 in first column, non-zero in
associated row
If there is a zero in the first column, but other
elements in that row are nonzero, it may be
replaced with a small positive number e that is
allowed to approach zero after completing the
array. ?(s) s 5 2s 4 2s 3 4s 2 11 s
10 0 Routh array s 5 1 2
11 s 4 2 4 10 s 3 0 6
0 s 2
c1 10 0 c1 4 e 12 / e ? 12/ e s
1 d1 0 0 d1 6 c1 10 e / c1 ?
6 s 0 10 0 0 Two sign changes in
first column say that there are two roots that do
not have negative real parts.
21
Example Case 2 0 in first column, non-zero in
associated row
If there is a zero in the first column, but other
elements in that row are nonzero, it may be
replaced with a small positive number e that is
allowed to approach zero after completing the
array. ?(s) s 5 2s 4 2s 3 4s 2 11 s
10 0 Routh array s 5 1 2
11 s 4 2 4 10 s 3 e 6
0 s 2
c1 10 0 c1 4 e 12 / e ? 12/ e s
1 d1 0 0 d1 6 c1 10 e / c1 ?
6 s 0 10 0 0 Two sign changes in
first column say that there are two roots that do
not have negative real parts.
22
Example Case 2 0 in first column, non-zero in
associated row
If there is a zero in the first column, but other
elements in that row are nonzero, it may be
replaced with a small positive number e that is
allowed to approach zero after completing the
array. ?(s) s 5 2s 4 2s 3 4s 2 11 s
10 0 Routh array s 5 1 2
11 s 4 2 4 10 s 3 e 6
0 s 2
c1 10 0 c1 4 e 12 / e ? 12/ e s
1 d1 0 0 d1 6 c1 10 e / c1 ?
6 s 0 10 0 0 Two sign changes in
first column say that there are two roots that do
not have negative real parts.
23
Example Case 3 all zero row
If all elements in an array row are zero, the
polynomial contains roots that are symmetrically
located about the origin in the s plane. e.g. (s
j?0 ) (s j?0 ) e.g. (s s0 ) (s s0 ) The
procedure is to form an auxiliary equation U(s)
0 using the row immediately preceding the zero
row in the Routh array, differentiate it and
replace the all zero row with the coefficients
from U(s) 0, then continue forming the Routh
array. The roots of the auxiliary equation U(s)
0 are also roots of ?(s) 0 Example ?(s)
s 4 2s 3 11s 2 18 s 18 0 s 4 1
11 18 s 3 1 9
/2 s 2 1 9 /2 s 1
2 U(s) s 2 9 0 roots s j3
U(s) 2s 0 0 s 0 9 The roots s j3
are also roots of the original equation ?(s) 0
24
Example Case 3 all zero row
If all elements in an array row are zero, the
polynomial contains roots that are symmetrically
located about the origin in the s plane. e.g. (s
j?0 ) (s j?0 ) e.g. (s s0 ) (s s0 ) The
procedure is to form an auxiliary equation U(s)
0 using the row immediately preceding the zero
row in the Routh array, differentiate it and
replace the all zero row with the coefficients
from U(s) 0, then continue forming the Routh
array. The roots of the auxiliary equation U(s)
0 are also roots of ?(s) 0 Example ?(s)
s 4 2s 3 11s 2 18 s 18 0 s 4 1
11 18 s 3 1 9
/2 s 2 1 9 /2 s 1
2 U(s) s 2 9 0 roots s j3
U(s) 2s 0 0 s 0 9 The roots s j3
are also roots of the original equation ?(s) 0
25
Example Case 3 all zero row
If all elements in an array row are zero, the
polynomial contains roots that are symmetrically
located about the origin in the s plane. e.g. (s
j?0 ) (s j?0 ) e.g. (s s0 ) (s s0 ) The
procedure is to form an auxiliary equation U(s)
0 using the row immediately preceding the zero
row in the Routh array, differentiate it and
replace the all zero row with the coefficients
from U(s) 0, then continue forming the Routh
array. The roots of the auxiliary equation U(s)
0 are also roots of ?(s) 0 Example ?(s)
s 4 2s 3 11s 2 18 s 18 0 s 4 1
11 18 s 3 2 18
s 2 1 9 /2 s 1 2
U(s) s 2 9 0 roots s j3
U(s) 2s 0 0 s 0 9 The roots s j3
are also roots of the original equation ?(s) 0
26
Example Case 3 all zero row
If all elements in an array row are zero, the
polynomial contains roots that are symmetrically
located about the origin in the s plane. e.g. (s
j?0 ) (s j?0 ) e.g. (s s0 ) (s s0 ) The
procedure is to form an auxiliary equation U(s)
0 using the row immediately preceding the zero
row in the Routh array, differentiate it and
replace the all zero row with the coefficients
from U(s) 0, then continue forming the Routh
array. The roots of the auxiliary equation
U(s) 0 are also roots of ?(s) 0 Example
?(s) s 4 2s 3 11s 2 18 s 18 0 s 4
1 11 18 s 3 1 9
/2 s 2 1 9 /2 s
1 2 U(s) s 2 9 0 roots s
j3 U(s) 2s 0 0 s 0 9 The roots s
j3 are also roots of the original equation
?(s) 0
27
Example Case 3 all zero row
If all elements in an array row are zero, the
polynomial contains roots that are symmetrically
located about the origin in the s plane. e.g. (s
j?0 ) (s j?0 ) e.g. (s s0 ) (s s0 ) The
procedure is to form an auxiliary equation U(s)
0 using the row immediately preceding the zero
row in the Routh array, differentiate it and
replace the all zero row with the coefficients
from U(s) 0, then continue forming the Routh
array. The roots of the auxiliary equation
U(s) 0 are also roots of ?(s) 0 Example
?(s) s 4 2s 3 11s 2 18 s 18 0 s 4
1 11 18 s 3 1 9
/2 s 2 2 18 /2 s
1 2 U(s) s 2 9 0 roots s
j3 U(s) 2s 0 0 s 0 9 The roots s
j3 are also roots of the original equation
?(s) 0
28
Example Case 3 all zero row
If all elements in an array row are zero, the
polynomial contains roots that are symmetrically
located about the origin in the s plane. e.g. (s
j?0 ) (s j?0 ) e.g. (s s0 ) (s s0 ) The
procedure is to form an auxiliary equation U(s)
0 using the row immediately preceding the zero
row in the Routh array, differentiate it and
replace the all zero row with the coefficients
from U(s) 0, then continue forming the Routh
array. The roots of the auxiliary equation
U(s) 0 are also roots of ?(s) 0 Example
?(s) s 4 2s 3 11s 2 18 s 18 0 s 4
1 11 18 s 3 1 9
/2 s 2 1 9 /2 s
1 0 U(s) s 2 9 0 roots s
j3 U(s) 2s 0 0 s 0 9 The roots s
j3 are also roots of the original equation
?(s) 0
29
Example Case 3 all zero row
If all elements in an array row are zero, the
polynomial contains roots that are symmetrically
located about the origin in the s plane. e.g. (s
j?0 ) (s j?0 ) e.g. (s s0 ) (s s0 ) The
procedure is to form an auxiliary equation U(s)
0 using the row immediately preceding the zero
row in the Routh array, differentiate it and
replace the all zero row with the coefficients
from U(s) 0, then continue forming the Routh
array. The roots of the auxiliary equation
U(s) 0 are also roots of ?(s) 0 Example
?(s) s 4 2s 3 11s 2 18 s 18 0 s 4
1 11 18 s 3 1 9
/2 s 2 1 9 /2 s
1 2 U(s) s 2 9 0 roots s
j3 U(s) 2s 0 0 s 0 The roots s
j3 are also roots of the original equation ?(s)
0
30
Example Case 3 all zero row
If all elements in an array row are zero, the
polynomial contains roots that are symmetrically
located about the origin in the s plane. e.g. (s
j?0 ) (s j?0 ) e.g. (s s0 ) (s s0 ) The
procedure is to form an auxiliary equation U(s)
0 using the row immediately preceding the zero
row in the Routh array, differentiate it and
replace the all zero row with the coefficients
from U(s) 0, then continue forming the Routh
array. The roots of the auxiliary equation U(s)
0 are also roots of ?(s) 0 Example ?(s)
s 4 2s 3 11s 2 18 s 18 0 s 4 1
11 18 s 3 1 9
/2 s 2 1 9 /2 s 1
2 U(s) s 2 9 0 roots s j3
U(s) 2s 0 0 s 0 9 The roots s j3
are also roots of the original equation ?(s) 0
31
Example Case 3 0 all zeros row
?(s) s 3 2s 2 4 s K 0 Routh array s
3 1 4 s 2 2 K s 1
(8 K)/2 0 s 0 K
0 For a stable system, 0 lt K lt 8 When K 8,
the Routh array has an all zero row, and the
system has two roots on the j? axis and is
therefore unstable. Where are the roots on the
j? axis? Aux eqn 2s 2 K 0 ? s 2 4 0 ? s
j2 ?(s) (s j2)(s j2)(s 2) 0
32
Example Case 3 0 all zeros row
?(s) s 3 2s 2 4 s K 0 Routh array s
3 1 4 s 2 2 K s 1
(8 K)/2 0 s 0 K
0 For a stable system, 0 lt K lt 8 When K 8,
the Routh array has an all zero row, and the
system has two roots on the j? axis and is
therefore unstable. Where are the roots on the
j? axis? Aux eqn 2s 2 K 0 ? s 2 4 0 ? s
j2 ?(s) (s j2)(s j2)(s 2) 0
33
Example Case 3 0 all zeros row
?(s) s 3 2s 2 4 s K 0 Routh array s
3 1 4 s 2 2 K s 1
(8 K)/2 0 s 0 K
0 For a stable system, 0 lt K lt 8 When K 8,
the Routh array has an all zero row, and the
system has two roots on the j? axis and is
therefore unstable. Where are the roots on the
j? axis? Aux eqn 2s 2 K 0 ? s 2 4 0 ? s
j2 ?(s) (s j2)(s j2)(s 2) 0
34
Example Case 3 0 all zeros row
?(s) s 3 2s 2 4 s K 0 Routh array s
3 1 4 s 2 2 K s 1
(8 K)/2 0 s 0 K
0 For a stable system, 0 lt K lt 8 When K 8,
the Routh array has an all zero row, and the
system has two roots on the j? axis and is
therefore unstable. Where are the roots on the
j? axis for K 8? Aux eqn 2s 2 K 0 ? s 2
4 0 ? s j2 ?(s) (s j2)(s j2)(s 2)
0
35
Example Case 3 0 all zeros row
?(s) s 3 2s 2 4 s K 0 Routh array s
3 1 4 s 2 2 K s 1
(8 K)/2 0 s 0 K
0 For a stable system, 0 lt K lt 8 When K 8,
the Routh array has an all zero row, and the
system has two roots on the j? axis and is
therefore unstable. Where are the roots on the
j? axis for K 8? Aux eqn 2s 2 K 0 ? s 2
4 0 ? s j2 ?(s) (s j2)(s j2)(s 2)
0
36
Example Multiple all zeros rows
?(s) s 5 s 4 2 s 3 2 s 2 s 1 0
(s 1)(s j)(s j)(s j)(s j) Routh
array s 5 1 2 1 s 4 1
2 1 s 3 0 0
U(s) s 4 2 s 2 1 0 (s 2 1) (s 2
1) s 1 s 0
37
Example Multiple all zeros rows
?(s) s 5 s 4 2 s 3 2 s 2 s 1 0
(s 1)(s j)(s j)(s j)(s j) Routh
array s 5 1 2 1 s 4 1
2 1 s 3 4 4
U(s) s 4 2 s 2 1 0 (s 2 1) (s 2
1) s 1 1 U(s 4s 3 4 s
0 s 0 4
The characteristic equation has two roots on the
j? axis, making it unstable. Continuing the
Routh array, there are no further sign reversals,
which means the remaining root has a negative
real part (s -1), as expected.
38
Example Multiple all zeros rows
?(s) s 5 s 4 2 s 3 2 s 2 s 1 0
(s 1)(s j)(s j)(s j)(s j) Routh
array s 5 1 2 1 s 4 1
2 1 s 3 4 4
U(s) s 4 2 s 2 1 0 (s 2 1) (s 2
1) s 1 1 U(s 4s 3 4 s
0 s 0 4