Lecture 13: More About Bode Plots
Non-Minimum Phase Systems
Systems with Delay
Polar Plots
Dorf
Cleveland State University
Mechanical Engineering
Hanz Richter, PhD
MCE441 – p.1/13
Among all systems having the same magnitude plot, those with the least phase shift range are called minimum phase . The others are called nonminimum phase . The TF of a MP system can be determined from the magnitude alone.
Fact: Systems having rhp zeros are nonminimum phase. But they’re not the only ones.
Fact: Every rational transfer function (ratio of polynomials) has a high-frequency magnitude asymptote with slope − 20( n − m ) dB/dec, where n is the number of poles and m is the number of zeros.
Fact: Every rational, minimum phase transfer function has high-frequency phase asymptote at − 90( n − m ) .
Use these facts to detect nonminimum phase systems from their
Bode plot (phase-magnitude consistency). Can be done from an experimental Bode plot (no model).
MCE441 – p.2/13

Compare the Bode plots of
1+ T s
1+ T
1 s and
1
−
T s
1+ T
1 s when 0 < T < T
1
.
Bode Diagram
−45
−90
0
−1
−2
−3
−4
−5
−6
0
−135
−180
10
−1
**** Minimum Phase
___ Non Minimum Phase
10
0
Frequency (rad/sec)
10
1
As we see, the nonminimum phase system introduces extra phase lag for the same attenuation.
MCE441 – p.3/13
Compare the step responses of s 2
1+ s
+ s +1 and s 2
1
− s
+ s +1
.
0.6
0.4
0.2
0
−0.2
−0.4
0
1.4
1.2
1
0.8
2 4
**** MP : (s+1)/(s^2+s+1)
__ NMP : (−s+1)/(s^2+s+1)
6 Time (sec) 8 10 12
When hitting the system with a step, the NMP system actually responds in a direction opposite to the intended one! (out-of-phase at the high frequencies involved in the jump).
MCE441 – p.4/13

722 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 4, JULY 2000
Rongjun Zhang, Member, IEEE, Yaobin Chen, Member, IEEE, Zengqi Sun, Senior Member, IEEE,
Fuchun Sun, Member, IEEE, and Hanzhen Xu
Abstract—This paper addresses the path control problem for a ship steering in restricted waters using sliding mode techniques.
The ship’s dynamic equations and numerical computation of the bank disturbance forces are briefly described. The ship’s path control system is presented which leads to a nonminimum phase system. Therefore, the traditional input–output linearization strategy in geometric control theory is modified to convert the controller design of a fourth-order nonlinear system to that of a second-order linear system. A continuous sliding mode controller is designed to ensure the system’s robustness and better performance. Simulation and experimental studies validate the controller design. The simulation results demonstrate the effectiveness of the proposed controller and its practicality in the steering control and navigation of the ship.
Index Terms—Control systems, differential geometry, dynamics, marine vehicle control, navigation, simulation, stability, variable structure systems.
Other control techniques have also been introduced to improve the safety of a ship steering in restricted waters, such as linear quadratic Gaussian (LQG) [9], stochastic control [13], and adaptive control [12]. It is worth mentioning that Kallstrom and Astrom developed a controller that makes the ship move as desired and rejects the disturbances adaptively [8]. The adaptive controller, although mostly designed based on the energy cost, in fact has no advantage on fuel consumption to most of the ships [15], and presents less robustness against speed and environment changes [20]. In the literature, disturbance forces acting on a ship moving in restricted waters are usually assumed as stochastic noises. However, the difference between the stochastic noises, including the white or colored noises, and the disturbance forces are significant from the point of view of hydrodynamics [24]. Thus the assumption may lead to the unreliability in the control simulation results. This paper will present
MCE441 – p.5/13
Transport Lag , Delay , or Pure Delay is commonly found in actual systems. The measurement seen by the controller is corresponds to a past event.
MCE441 – p.6/13

Let x ( t ) be the original signal and y ( t ) = x ( t − T ) be the delayed signal.
By using the definition of Laplace transform we can show that
Y ( s ) = e − sT
, so
Y ( s )
X ( s )
= e − sT
A system with delay introduces a TF which is not a ratio of polynomials.
Sometimes we can use a series expansion: the Padé approximation.
e −
T s ≈
1 −
1 +
T s
2
T s
2
+
+
( T s ) 2
2
( T s ) 2
8
−
+
( T s ) 3
3
( T s ) 3
48
+ ...
+ ...
First order approx: e
−
T s ≈ 1 − T s
Second order approx: e
−
T s ≈
1
T s +1
MCE441 – p.7/13
| G ( jw ) | = | e −
T wj | = | cos wT − j sin wT | = 1
∠ G ( jw ) = ∠ e −
T wj = − wT
A time delay introduces a lot of phase lag!!
There is no phase-magnitude consistency, therefore a system with delay is also NMP. Systems with delay need to be compensated with predictive action.
MCE441 – p.8/13

Sketch a Bode plot for e − s
G ( s ) =
1 + s
MCE441 – p.9/13
MCE441 – p.10/13

Find the mystery TF from the following Bode plot:
Bode Diagram for Mystery Transfer Function
150
−50
−180
−225
−270
−315
−360
10
−1
100
50
0
10
0
10
1
10
2
Frequency (rad/sec)
Check with Matlab: bode(num,den)
MCE441 – p.11/13
A polar plot is a magnitude-angle chart in polar coordinates, for w = 0 to ∞ . A Nyquist plot also includes the w = 0 to −∞ portion, which is symmetrical to the positive frequency part.
Nyquist Diagram
1.5
0
−0.5
1
0.5
−1
−1.5
−1 −0.5
negative freq positive freq
0
Real Axis
0.5
1
MCE441 – p.12/13

Draw an approximate polar plot from the mystery TF considered earlier.
MCE441 – p.13/13