Cranking up the gain ☺
Steady—state error due to step input:
Type 0 system (no disturbance)
eR (∞) =
VVref
(s)
ref(s)
0.3162
0.3162
s+2
s+2
K
K
(s)
VV(s)
Type 1 system with disturbance
2
2 + 0.3162K
eR (∞) → 0
K→∞
as
Steady—state error due to step input:
eR (∞) = 0
Steady—state error due to step disturbance:
(s)
VVref
ref(s)
2.004 Fall ’07
K
0.3162
0.3162
s(s+2)
s(s+2)
X(s)
X(s)
Lecture 17 – Monday, Oct. 17
eD (∞) = −
eD (∞) → 0
as
1
K
K→∞
Cranking up the gain
Type 1 system (no disturbance)
Vref
ref (s)
0.3162
s(s+2)
K
X(s)
X(s)
Closed—loop transfer function
X(s)
0.3162K
= 2
Vref (s)
s + 2s + 0.3162K
Pole locations
p1 = −1 +
√
1 − 0.3162K
p2 = −1 −
√
1 − 0.3162K
System becomes underdamped⇒
⇒ step response overshoots if
1 − 0.3162K < 0 ⇔ K > 3.1626
2.004 Fall ’07
Lecture 17 – Monday, Oct. 17
Cranking up the gain: poles and step response
K increasing
K > 3.1626
K increasing
K < 3.1626
Root
Locus
2.004 Fall ’07
Lecture 17 – Monday, Oct. 17
Root locus for nonunity feedback systems
Forward
transfer
function
R(s)
Input
Ea(s)
Actuating
signal
(error)
+
-
KG(s)
Caveat: K>0
C(s)
Closed loop TF:
Output
R(s)
H(s)
Nise Figure 8.1
Feedback transfer
function
Figures by MIT OpenCourseWare.
KG(s)
1 + KG(s)H(s)
“Open loop” TF:
G(s)H(s)
Closed-loop pole locations
½
K = 1/ |G(s)H(s)| ;
1 + KG(s)H(s) = 0 ⇒
6 KG(s)H(s) = (2n + 1)180◦ .
2.004 Fall ’07
Lecture 17 – Monday, Oct. 17
C(s)
Root locus terminology
jω
Break-in point
j3
j2
RL imaginary axis
intercept
Breakaway point
X
-4
j1
Real-axis
segment
-3
X
-2
X
-1
Branches
Asymptote
real-axis intercept
Asymptote
X
0
-4
-3
X
-2
σ
X
-1
-j1
Asymptote
angle
2
1
σ
jω
-j1
Departure/Arrival angles
-j2
ζ = 0.45
j4
j4
s-plane
0
-2
-j2
X
j1
σ
X
-4 -3
X
-2
-1
-j1
-j3
-j6
-j4
-j5
Lecture 17 – Monday, Oct. 17
0 1
2
-j2
-j4
Figure by MIT OpenCourseWare.
s-plane
j2
j2
-5
2 + j4
j3
X
-10
jω
j5
j6
-j3
2.004 Fall ’07
j1
s-plane
s-plane
Asymptote
Asymptote
jω
Breakaway point
2 - j4
3
4
σ
Root-locus sketching rules
•
•
•
Rule 1: # branches = # poles
Rule 2: symmetrical about the real axis
Rule 3: real-axis segments are to the left of an odd number of realaxis finite poles/zeros
NG (s)
NH (s)
, H(s) =
.
DG (s)
DH (s)
X
X
6 (poles) −
6 (zeros).
G(s)H(s) =
Let
⇒
6
G(s) =
Recall angle condition for closed—loop pole:
6
KG(s)H(s) = (2n + 1)180◦ .
Complex—pole/zero contributions: cancel
because of symmetry
Real—pole/zero contributions: each is
0◦ from the left, 180◦ from the right;
total contributions from right must be
odd number of 190◦ ’s to satisfy angle condition.
Image removed due to copyright restrictions.
Please see: Fig. 8.8 in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, 2004.
2.004 Fall ’07
Lecture 17 – Monday, Oct. 17
Root-locus sketching rules
•
Rule 4: RL begins at poles, ends at zeros
Let G(s) =
⇒
NG (s)
,
DG (s)
Closed—loop TF(s) =
H(s) =
KNG (s)DH (s)
.
DG (s)DH (s) + KNG (s)NH (s)
If K → 0+ (small gain limit)
Closed—loop TF(s) ≈
If K → +∞ (large gain limit)
KNG (s)DH (s)
⇒
DG (s)DH (s) + ²
Closed—loop TF(s) ≈
closed—loop denominator is denominator of G(s)H(s)
⇒closed—loop poles are the poles of G(s)H(s).
Example
s-plane
-4
-3
X
-2
NH (s)
.
DH (s)
jω
KNG (s)DH (s)
⇒
² + KNG (s)NH (s)
closed—loop denominator is numerator of G(s)H(s)
⇒closed—loop poles are the zeros of G(s)H(s).
j1
σ
X
-1
-j1
Nise Figure 8.10
2.004 Fall ’07
Figure by MIT OpenCourseWare.
Lecture 17 – Monday, Oct. 17
Please see the following selections from
Mathworks, Inc. "Control System Toolbox Getting Started Guide."
http://www.mathworks.com/access/helpdesk/help/ pdf_doc/control/get_start.pdf
Ch. 4, pp. 3-18