CDS 101/110: Lecture 7.2Loop Analysis of Feedback Systems

November 9 2016

Goals:• Review Nyquist Diagrams (including examples).• Gain margin and phase margin• Some analysis of Nyquist stability criterion systems• Note: 2-page Bode Plot addition to on-line PDF lecture slide

Reading: • Åström and Murray, Feedback Systems, Chapter 10, Sections 10.1-10.3,

2

Closed Loop Stability Analysis

Closed loop transfer function:

𝐺𝐺𝑦𝑦𝑦𝑦 = 𝑃𝑃𝑃𝑃1+𝑃𝑃𝑃𝑃

= 𝑛𝑛𝑝𝑝𝑛𝑛𝑐𝑐𝑑𝑑𝑝𝑝𝑑𝑑𝑐𝑐+𝑛𝑛𝑝𝑝𝑛𝑛𝑐𝑐

• Poles of 𝐺𝐺𝑦𝑦𝑦𝑦 determine stability.

• Zeros of (1 + 𝑃𝑃𝑃𝑃 ) are poles of 𝐺𝐺𝑦𝑦𝑦𝑦• Is there an easy way to check for RHP zeros of (1 + 𝑃𝑃𝑃𝑃 )?

“simple” negative unity feedback Q: If we know properties of 𝑃𝑃(𝑠𝑠) and C(𝑠𝑠), (the open loop transfer function) can we infer anything about closed loop behavior and performance?

𝐿𝐿 𝑠𝑠 = 𝑃𝑃 𝑠𝑠 𝑃𝑃 𝑠𝑠 =𝑛𝑛𝑃𝑃 𝑠𝑠𝑑𝑑𝑃𝑃(𝑠𝑠)

𝑛𝑛𝑃𝑃(𝑠𝑠)𝑑𝑑𝑃𝑃(𝑠𝑠)

C(s) P(s)++

d

ye u

-1

r r

The Nyquist plot allows us to check if (1 + 𝑃𝑃𝑃𝑃 ) has zeros in RHP, and therefore if 𝐺𝐺𝑦𝑦𝑦𝑦 is unstable

• useful if poles of 𝑑𝑑𝑃𝑃 𝑠𝑠 𝑑𝑑𝑃𝑃 𝑠𝑠 +𝑛𝑛𝑃𝑃 𝑠𝑠 𝑛𝑛𝑃𝑃 𝑠𝑠 not easily found.

• collateral benefit: new ideas on “robustness” of closed loop system.

• useful for time delay analysis of linear systems

3

Basic Nyquist Plot (review)(no poles of 𝐿𝐿(𝑠𝑠) on imaginary axis)

Nyquist Contour (Γ):• Start from 0, and move along positive

Imaginary axis (increasing frequency)• Follow semi-Circle, or arc at infinity, in

clockwise direction (connecting the endpoints of the imaginary axis)

• From −𝑖𝑖𝑖 to zero on imaginary axis• Note, portion of plot corresponding to 𝜔𝜔 < 0 is mirror image of 𝜔𝜔 > 0

•Nyquist “D” contour

•Take limit as R → ∞

Nyquist Plot• Formed by tracing 𝑠𝑠 around the Nyquist

contour, Γ, and mapping through 𝐿𝐿(𝑠𝑠) to complex plane representing magnitude and phase of 𝐿𝐿 𝑠𝑠 .

• I.e., the image of 𝐿𝐿(𝑠𝑠) as 𝑠𝑠 traverses Γ is the Nyquist plot

• Goal: from complex analysis, we’re trying to find number of zeros (if any) in RHP, which leads to instability

R Γ𝑅𝑅𝑅𝑅(𝑠𝑠)

𝐼𝐼𝐼𝐼(𝑠𝑠)+𝑖𝑖𝑖

−𝑖𝑖𝑖

𝐿𝐿 𝑠𝑠

Real Axis

Imag

inar

y Ax

is

Nyquist Diagrams

-1.5 -1 -0.5 0 0.5 1 1.5 2-3

-2

-1

0

1

2

3From: U(1)

To: Y

(1)

𝜔𝜔=0𝜔𝜔=∞ 𝜔𝜔=-∞

𝐺𝐺𝑦𝑦𝑦𝑦 𝑠𝑠 =𝜔𝜔02

𝑠𝑠2 + 2𝜁𝜁𝜔𝜔0𝑠𝑠 + 𝜔𝜔02

4

Nyquist Criterion

Consequence:• If Z ≥ 1, then 1 + 𝐿𝐿 𝑠𝑠 has RHP

zeros, which means that 𝐺𝐺𝑦𝑦𝑦𝑦 𝑠𝑠 has RHP poles.

• 𝐺𝐺𝑦𝑦𝑦𝑦 𝑠𝑠 is unstable with simple unity feedback, and control 𝑃𝑃(𝑠𝑠)

Thm (Nyquist). Consider the Nyquist plot for loop transfer function L(s). Let

P # RHP poles of open loop L(s)N # clockwise encirclements of -1

(counterclockwise is negative)Z # RHP zeros of 1 + L(s)

ThenZ = N + P

Real Axis

Nyquist Diagrams

-6 -4 -2 0 2 4 6 8 10 12-6

-4

-2

0

2

4

6From: U(1)

To: Y

(1)

=0𝜔𝜔=+i∞

𝜔𝜔=-i∞

i𝜔𝜔 < 0

i𝜔𝜔 > 0

+

N=2

R Γ𝑅𝑅𝑅𝑅(𝑠𝑠)

𝐼𝐼𝐼𝐼(𝑠𝑠)+𝑖𝑖𝑖

−𝑖𝑖𝑖

-150

-100

-50

0

50

Mag

nitu

de (d

B)

10-1

100

101

102

-270

-180

-90

0

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)

5

Nyquist Plot Example #1

𝑃𝑃 𝑠𝑠 = 1𝑠𝑠+1 3 𝑃𝑃 𝑠𝑠 = 𝑘𝑘 𝑠𝑠+𝑎𝑎

𝑠𝑠+𝑏𝑏1 < 𝑎𝑎 < 𝑏𝑏

-80

-60

-40

-20

0

Mag

nitu

de (d

B)

10-1

100

101

-270

-180

-90

0

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)

𝑃𝑃 𝑠𝑠 =1

𝑠𝑠 + 1 3

10

11

12

13

14

Mag

nitu

de (d

B)

10-1

100

101

102

0

5

10

15

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)

𝑃𝑃 𝑠𝑠 = 𝑘𝑘𝑠𝑠 + 𝑎𝑎𝑠𝑠 + 𝑏𝑏

𝑘𝑘 = 5,𝑎𝑎 = 2, 𝑏𝑏 = 3

𝐿𝐿 𝑠𝑠 =𝑘𝑘(𝑠𝑠 + 𝑎𝑎)

𝑠𝑠 + 1 3(𝑠𝑠 + 𝑏𝑏)

𝑘𝑘 = 5,𝑎𝑎 = 2, 𝑏𝑏 = 3

6

Nyquist Plot Example #1

𝑃𝑃 𝑠𝑠 = 1𝑠𝑠+1 3

𝑃𝑃 𝑠𝑠 = 𝑘𝑘𝑠𝑠 + 𝑎𝑎𝑠𝑠 + 𝑏𝑏

1 < 𝑎𝑎 < 𝑏𝑏

𝑘𝑘 = 5,𝑎𝑎 = 2, 𝑏𝑏 = 3

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Nyquist Diagram

Real Axis

Imag

inar

y Ax

is

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Nyquist Diagram

Real Axis

Imag

inar

y Ax

is

𝑘𝑘 = 5𝑘𝑘 = 1

-2 -1 0 1 2 3 4 5 6 7

-5

-4

-3

-2

-1

0

1

2

3

4

5

Nyquist Diagram

Real Axis

Imag

inar

y Ax

is

𝑘𝑘 = 10

-4 -2 0 2 4 6 8 10 12 14

-10

-8

-6

-4

-2

0

2

4

6

8

10

Nyquist Diagram

Real Axis

Imag

inar

y Ax

is

Nyquist: •𝑃𝑃 = 0•𝑁𝑁 = +1•𝑍𝑍𝑅𝑅𝑅𝑅𝑃𝑃 = 1

7

Robust stability: gain and phase marginsNyquist plot tells us if closed loop is stable, but not how stable

Gain margin• How much we can modify the loop gain

and still have the system be stable• Determined by the location where the loop

transfer function crosses 180˚ phase

Phase margin• How much “phase delay” can be addeded

while system remains stable• Determined by the phase at which the loop

transfer function has unity gain

Bode plot interpretation• Look for gain = 1, 180˚ phase crossings• MATLAB: margin(sys)

Nyquist Diagram

-1.5 -1 -0.5 0 0.5 1 1.5-3

-2

-1

0

1

2

3

Frequency (rad/sec)

Phas

e (d

eg);

Mag

nitu

de (d

B)

Bode Diagram

-100

-50

0

50

10-2 10-1 100 101-300

-200

-100

0

Gm=7.005 dB (at 0.34641 rad/sec), Pm=18.754 deg. (at 0.26853 rad/sec)

8

Example: Proportional + Integral* speed controller

Remarks• N = 0, P = 0 ⇒ Z = 0 (stable)• Need to zoom in to make sure there

are no net encirclements• Note that we don’t have to compute

closed loop response-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

-1.5

-1

-0.5

0

0.5

1

1.5

C(s) P(s)++

d

ye u

-1

r

+

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-1.5

-1

-0.5

0

0.5

1

1.5

Nyquist Diagram

Real Axis

Imag

inar

y Ax

is

-150

-100

-50

0

50

100

Mag

nitu

de (d

B)

10-4 10-3 10-2 10-1 100 101 102 103-270

-225

-180

-135

-90

-45

0

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec)

Example: cruise control

Effect of additional sensor dynamics• New speedometer has pole at s = 10 (very fast); problems develop in the field• What’s the problem? A: insufficient phase margin in original design (not robust)

Nyquist plots

-500 0 500 1000 1500 2000 2500

-1500

-1000

-500

0

500

1000

1500Nyquist Diagram

Real Axis

Imag

inar

y Ax

is

C(s) P(s)++

d

ye u

-1-G(s)

r

10

-150

-100

-50

0

50

100

Mag

nitu

de (

dB)

10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3-270

-225

-180

-135

-90

-45

0

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec)

Preview: control design

Approach: Increase phase margin• Increase phase margin by reducing gain ⇒ can accommodate new sensor dynamics• Tradeoff: lower gain at low frequencies ⇒ less bandwidth, larger steady state error

-4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Nyquist Diagram

Real Axis

Imag

inar

y Ax

is

+

-500 0 500 1000 1500 2000 2500-1500

-1000

-500

0

500

1000

1500

Nyquist Diagram

Real Axis

Imag

inar

y Ax

is

Nyquist plots

C(s) P(s)++

d

ye u

-1-G(s)

r

11

Comments and cautionsWhy is the Nyquist plot useful?• Old answer: easy way to compute stability (before computers and MATLAB)• Real answer: gives insight into stability and robustness; very useful for reasoning

about stability

Nyquist plots for systems with poles on the jω axis

Cautions with using MATLAB• MATLAB doesn’t generate portion of plot for poles on imaginary axis• These must be drawn in by hand (make sure to get the orientation right!)

•chose contour to avoid poles on axis

•need to carefully compute Nyquist plot at these points

•evaluate H(ε+0i) todetermine direction

𝜔𝜔=-i∞𝜔𝜔 =+i∞

𝜔𝜔=0+

𝜔𝜔=0-

12

Summary: Loop Analysis of Feedback Systems

• Nyquist criteria for loop stability• Gain, phase margin for robustness

r

R

-j∞

+j∞Nyquist Diagram

-1.5 -1 -0.5 0 0.5 1 1.5-3

-2

-1

0

1

2

3

Phas

e (d

eg);

Mag

nitu

de (d

B)

Bode Diagram

-100

-50

0

50

10-2 10-1 100 101-300

-200

-100

0

Gm=7.005 dB (at 0.34641 rad/sec), Pm=18.754 deg. (at 0.26853 rad/sec)

Thm (Nyquist). P # RHP poles of L(s)N # CW encirclementsZ # RHP zeros

Z = N + P

C(s) P(s)++

d

ye u

-1

r