System Dynamics and Control

HW3 – Nichols plots and frequency domain specifications FORMULAE FOR SECOND ORDER SPECIFICATIONS:

If the higher order closed-loop system has two dominant poles (and no dominant zeros) it can be approximated by a second order

system. Here are the formulae for transient specifications for second order systems or approximate second order systems:

1. Natural frequency: 𝜔𝑛 (distance of pole to origin)

2. Damping ratio: 𝜁 (related to angle of the pole)

3. The closed-loop dominant poles: 𝑝1, 𝑝2 = −𝜎 ± 𝑗𝜔𝑑 = −𝜔𝑛𝜁 ± 𝑗𝜔𝑛√1 − 𝜁 2

4. Angle of the poles with negative real axis: 𝛽 = arccos 𝜁 = cos−1 𝜁

5. The damped frequency of oscillations is the imaginary part of the roots: 𝜔𝑑

6. The exponential time constant is the reciprocal of the real part of the roots: 𝜏 = 1

𝜎 =

1

𝜔𝑛𝜁

7. Distance of the pole from origin is √𝜎2 + 𝜔𝑑 2 = 𝜔𝑛 = natural frequency.

8. Settling time: 𝑇𝑠 = 4

𝜎 =

4

𝜔𝑛𝜁

9. Peak time: 𝑇𝑝 = 𝜋

𝜔𝑑

10. Maximum overshoot: 𝑀𝑝 = 𝑒

− 𝜋𝜁

√1− 𝜁2 = 𝑒−𝜋 cot 𝛽 = 𝑒

−𝜋 𝜎

𝜔𝑑

11. To get percent overshoot multiply 𝑀𝑝 with 100.

Similar characteristics in the frequency domain are:

1. Resonant frequency of the closed loop system: 𝜔𝑟 = 𝜔𝑛√1 − 2𝜁 2

2. Resonant peak amplitude the closed loop system: 𝑀𝑟 = 1

2𝜁√1−𝜁2

Note that there is no resonant frequency or peak when the damping ratio is greater than 1

√2 = 0.707

3. Cutoff frequency the closed loop system: 𝜔𝑐 = 𝜔𝑛√1 − 2𝜁 2 + √1 + (1 − 2𝜁2)2

Cutoff frequency tells about the bandwidth and is related to speed of the closed loop system.

Higher cut-off frequency → higher bandwidth → lower settling time, rise-time, peak time.

4. Damping ratio of the closed loop system 𝜁 is related to the phase margin of the open loop system:

𝜙 = tan−1 2

√√4+ 1

𝜁4 −2

(𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠) ≈ 100𝜁 (𝑖𝑛 𝑑𝑒𝑔𝑟𝑒𝑒𝑠)

The approximate formula for phase margin, 100𝜁, works only for phase margins up to about 60 degrees.

Errors specifications:

1. For a typical system with a plant G and controller K.

The error is 𝐸(𝑠) = 1

1+𝐺𝐾 𝑅

The steady state error is 𝑒(∞) = lim 𝑠→0

𝐸(𝑠)𝑠 = lim 𝑠→0

𝑠𝑅

1+𝐺𝐾

2. Steady state error for a step input R(s) = 1/s is:

𝑒(∞) = lim 𝑠→0

1

1 + 𝐺𝐾 =

1

1 + lim 𝑠→0

𝐺𝐾 =

1

1 + 𝑑𝑐 𝑔𝑎𝑖𝑛 𝑜𝑓 𝐺𝐾 =

1

1 + 𝐾𝑝

3. Steady state error for a ramp input R(s) = 1/s2 is:

𝑒(∞) = lim 𝑠→0

1

𝑠(1 + 𝐺𝐾) =

1

lim 𝑠→0

𝑠𝐺𝐾 =

1

𝐾𝑣

If GK has no integrator 𝐾𝑣 is 0 (ramp error is infinite). If GK has 1 integrator 𝐾𝑣 is finite, it is the x-intercept (frequency

intercept) of the asymptote line from the left half of the bode(GK). If GK has more integrators 𝐾𝑣 is infinite.

4. To reduce the error due to reference command (and disturbance dy) of frequency 𝜔 by a factor of F: |𝐺(𝑗𝜔)| > 𝐹 + 1 .

5. To reduce the error due to a sinusoidal noise command of frequency 𝜔 by a factor of F: |𝐺(𝑗𝜔)| < 1

1+𝐹 .

1. Most of the specifications for the system of the unity

feedback architecture shown on the right are boiled down to

Bode/Nyquist/Nichols plots of the GK(s).

a. If there was a sensor block 𝐻(𝑠) in the feedback

path, whose Bode/Nyquist/Nichols plots will you plot to check stability of the closed loop system?

b. If there was a prefilter block 𝐹(𝑠) after the reference command, whose Bode/Nyquist/Nichols plots will you plot to

check stability of the closed loop system?

2. Here is a Bode plot of some open loop system GK, plot the Nichols plot by hand. Show the gain and phase margins in there.

Is the closed loop system stable?

3. Mark the following No-go regions in the Bode plot of GK in Q2:

a. If you want error due to reference R in the frequency range 0 to 1 rad/s to be <1% of the R amplitude.

b. If you want error due to noise N in the frequency range 10 to 100 rad/s to be <1% of the N amplitude.

c. If you want error due to disturbance dY in the frequency range 0 to 1 rad/s to be <1% of the dY amplitude.

d. If you want error due to disturbance dU in the frequency range 0 to 1 rad/s to be <1% of the dU amplitude.

Here is the Bode of the plant G(s):

R(s) E(s)

-

+ 𝐺(𝑠) Y(s

) 𝐾

-100

-50

0

50

M a g n itu

d e (

d B

)

10 -1

10 0

10 1

10 2

-270

-180

-90

0

P h a s e (

d e g )

Bode Diagram

Gm = 2.92 dB (at 2.24 rad/sec) , Pm = 24.1 deg (at 1.63 rad/sec)

Frequency (rad/sec)

4. A Controller 𝐾 is used to control the angle 𝑦, of a motor by providing the required voltage to the motor.

The bode plot of the openloop system 𝐺𝐾 is shown below:

a. What happens to the phase margin of the system as the time delay increases? Explain why.

b. If there is time delay between the controller and the plant, what is that maximum time delay this controller can

handle before it becomes unstable. (Do not do Pade Approximations). Show all your calculations.

c. If there is time delay introduced by the sensor that measures the angle, then what is the maximum time delay this

system can handle before it becomes unstable. (Do not do Pade Approximations)

-100

-50

0

50

M a g n itu

d e (

d B

)

10 -2

10 -1

10 0

10 1

10 2

-180

-135

-90

P h a s e (

d e g )

Bode Diagram

Gm = Inf dB (at Inf rad/sec) , Pm = 48.5 deg (at 0.882 rad/sec)

Frequency (rad/sec)

5. Practice: closed-loop specifications from open loop plots.

Figure on the side shows the Nichols chart of a transfer

function GK(s). Magnified view of a portion of the

graph is also shown below, with some points labelled

on it. Answer the questions below.

a. What is the gain margin as read from the Nichols chart?

b. What is the phase margin as read from the Nichols chart?

c. What is the resonant peak amplitude and the resonant frequency of the closed loop system as read from the

Nichol’s chart?

d. What is 𝜔𝑐 , the cutoff frequency of the closed loop system as read from the Nichol’s chart?

e. Calculate the damping ratio, 𝜁 of the closed-loop poles from the phase margin.

f. Using the 𝜁 and 𝜔𝑐 calculate the % overshoot and settling time of the step response of the closed loop system.

g. What is GK(s) in the limit as s goes to zero? Hence what is the steady state error for a step input?

h. Extra credit: What is sGK in the limit as s goes to zero? Hence what is the steady state error for a ramp input? You

are provided the info that at the top-right of the chart, the gain is 43.2dB, the phase is -90 deg. and the frequency

is 0.01 rad/s.

R(s) E(s)

-

+ 𝐺(𝑠) Y(s

) 𝐾