Part III Performance Limitations in Feedback Control with
8.2 Non-minimum Phase Dynamics in Ship Response
150 8 Linear Performance Limitations
or|T(jω)|must be greater or equal to 0.5), and both input and output distur- bances may affect the output if they have enough energy at these frequencies.
Furthermore, the ability to reject both disturbances depends on the frequency separation between di(jω) anddo(jω) and the plantG(jω). All these are ex- amples of trade-offsassociated with the design. ◦ ◦ ◦ The above motivation example evidences the delicate interplay between the different parts of the control system, and the basic limitations and trade-offs the designer usually faces. Note that no particular characteristics of the plant (e.g. delays, unstable poles and non-minimum phase zeros) were mentioned.
These characteristics and limited control action (constraints imposed by the actuators) can indeed aggravate the limitations and design trade-offs. For the case of stabiliser design for displacement ships, the non-minimum phase characteristic of the plant is one of the main effects imposing limitations on the performance of the control system.
8.2 Non-minimum Phase Dynamics in Ship Response 151 response,i.e. if the final value ofy(t) due to the step input is positive, then the initial response will be negative and vise versa. It also follows from the integral (8.5) that the closer the zero is to the imaginary axis, the larger will be the initial inverse response. This is because there will be more positive area to compensate due to the exponential decaying at a slower rate with twhen qbecomes smaller.
The NMP characteristic of a physical dynamic system often arises from the interaction of opposite fast and slow dynamic effects. This can be seen in the following. Let us follow our simple example with one zeros=q >0, and factorize the closed loop system as follows:
T(s) =N(s)(1−q−1s)
D(s) = N(s) D(s)
T1(s)
−N(s)q−1s D(s)
T2(s)
. (8.6)
In this expression, we see that the response of the system can be represented by the interaction of two systems; namelyT1 andT2, whose responses oppose each other. Also, we can see thatT2 has an extrasin the numerator, which will result in a larger bandwidth forT2having a faster response than T1.
In a displacement ship, the roll response to the rudder command generally presents NMP characteristics. Figure 8.3 shows a simulation example of roll response to a step-like change in rudder angle to make a port turn (left turn) in calm water. In this figure we can see the initial inverse response, which is due to the rudder-induced force Yrudder. The roll response to this rudder command is fast because of the smaller moment of inertia and damping in roll with respect to that in yaw. Since the rudder also produces a yaw moment (the main function of the rudder), the ship will eventually start to turn. However, the dynamics associated with the yaw motion are usually slower than those of roll due to the larger moment of inertia and damping. Once there exists a rate of turn, a reaction force Yhyd is induced due to hydrodynamic effects.
This force is larger than that produced by the rudder and is the main force producing the turn [133]. Also, as depicted in Figure 8.3, this force produces a roll moment opposite and larger to that induced by the rudder. Figure 8.4 depicts the latter effect of the heel angle attained due to the rate of turn of a frigate performing a port turn similar to that shown in Figure 8.3.
The rest of the oscillations in the roll response depicted in Figure 8.4 appear due to a pair of complex conjugate poles on the rudder-to-roll response.
These poles are attributed to the inertia, restoring forces and hydrodynamic damping associated with the roll motion.
From the linear manoeuvring model (low-frequency model) given in Sec- tion 4.3.4, we can find the rudder-to-roll and rudder-to-yaw transfer functions, which are of the form
152 8 Linear Performance Limitations
0 10 20 30 40 50
-4 -2 0 2 4 6 8 10
CG CG CG CG
Yrudder Yrudder
Yrudder Yrudder
Yhyd Yhyd Ycent
Aft view Aft view
Begin of turn Steady turn
WL
φ <0 φ >0
φ[deg]
t[s]
Fig. 8.3.Roll dynamics during a turn to port. From [173].
Gφα(s) = Kroll(q1−s)(q2+s) (p1+s)(p2+s)(s2+ 2ξφωφs+ωφ2)
Gψα(s) = Kyaw(q3+s)(s2+ 2ξqωqs+ωq2) s(p1+s)(p2+s)(s2+ 2ξφωφs+ωφ2)
(8.7)
8.2 Non-minimum Phase Dynamics in Ship Response 153
φ >0
Fig. 8.4. Steady heel angle during a port turn of a Naval vessel. ( cCrown Copy- right/MOD. Reproduced with the permission of the Controller of Her Majesty’s Stationery Office, United Kingdom.)
Figure 8.5 shows the frequency responses, and Figure 8.6 shows the poles and zeros of these transfer functions for the benchmark example at two speeds.
From Figure 8.6, we see that the location of the NMP zeroq1in (8.7) is affected by the speed. We can also see from this figure that the damping of the complex poles in the roll response increases with the speed.
If we isolate the roll from all other motion components, what is left is a second-order system (only the resonant poles in Figure 8.6). This second-order approximation is of the form
Gφα(s)≈ φ(s)
τα4(s)= Kτω2φ
s2+ 2ξφωφs+ω2φ, (8.8) where τα4(s) is the roll moment induced by the rudder and the non- dimensional damping coefficientξφ is given by
ξφ= Kp
2#
ρg∇GM t(I44−Kp˙), (8.9) and theroll natural frequency is
ωφ=
ρg∇GM t I44−Kp˙
, (8.10)
and the low frequency gain is
154 8 Linear Performance Limitations
Fig. 8.5.Frequency response for the rudder-to-roll and rudder-to-yaw responses.
Kτ= 1
ρg∇GM t. (8.11)
The roll damping coefficientξφis typically between 0.1 and 0.2, and thus the response oscillates.
As we shall see in the rest of this chapter, the presence of NMP dynamics, in general, limits the performance of the control system. The behaviour of a particular ship depends on the location of the centre of gravityv(LCG,VCG—
see Figure B.2), location of the rudder relative to the centre of gravity, and also the hydrodynamic characteristics of the hull,i.e.roll-sway-yaw couplings.
We will return to this when we specifically analyze the applicability of rudder stabilisers in a later section. We will next proceed with the introduction of the tools for such analysis.