4.5 High-level control system design
4.5.3 Control system design
4.5.3.1 Longitudinal velocity controller
Recall from Eq. (4-53), the diagonal plant transfer function along the ݔ axis (longitudinal force input / longitudinal velocity output) is:
ݔଵ= (ܿଵݏ ଶ) ݏ(ݏଶ+ ܽଵܽଶ) ݑଵ or ܸ௫= ݉ ൫ݏଶݏ+ ߱ ௭,ଶ ൯ܨ௫ Therefore the longitudinal plant transfer function is:
ܩ௫= ݉ ൫ݏଶݏ+ ߱
and the off-diagonal terms in Eq.(4-53)can be considered as disturbance to the system (which was shown to be are small in the frequency range above the plant bandwidth).
߱௭, is the nominal yaw rate which normally ranges between 0 to 0.3, so the term ߱௭,ଶ is usually much smaller than the ݏ term in Eq. (4-58) and could be ignored, then, the plant transfer function is further simplified as:
ܩ௫= ݉ ݏ1 (4-59)
This simplified plant model is independent from operating point (i.e. the vehicle is neutral steer (Milliken & Milliken, 1995)). To verify this simplification, the frequency response of the plant model based on the Eq.(4-58)with two different nominal yaw rates (߱௭,= 0.3) as well as the simplified plant model based on Eq.(4-59)is shown in Figure 4-9.
Figure 4-9: Longitudinal Plant Transfer function
By defining the plant model ܩ௫ as Eq. (4-59), the longitudinal control problem is to design a feedback controller ܭ௫, as shown in Figure 4-10, to provide internal
10-2 10-1 100 101 102 -150 -100 -50 0 50 100 Longitudinal Plant Frequency (rad/s) G a in (d B )
Simplified Plant Model Plant model @z,n=0.3
stability as well as control performances in the presence of model uncertainty and disturbance. To design the high-level longitudinal (and also lateral and yaw rate) motion controllers, we employ the Youla parameterisation loop shaping method which is one of the novelties of this thesis.
Figure 4-10: Closed loop longitudinal motion control
Considering the fact that the plant has a first order dynamics, we take the Youla parameter as the inverse of the plant multiply to one second order filter with adjustable poles and zeros such as:
ܻ௫ =ܩ1 ௫ቈ (߬ଶݏ+ 1) (߬ଵݏ+ 1)ଶ= ݉ ݏ(߬ଶݏ+ 1) (߬ଵݏ+ 1)ଶ , ߬ଵ, ߬ଶ> 0 (4-60) The proposed Youla parameter is stable and has two tuneable parameters ߬ଵ and ߬ଶ which can be employed to shape of the loop gain |ܮ| such that to be large at low frequencies below control bandwidth, and small at high frequencies above bandwidth.
The closed loop transfer function (complementary sensitivity) is: ܶ௫= ܩ௫ܻ௫ =(߬(߬ଶݏ+ 1)
ଵݏ+ 1)ଶ (4-61)
And the sensitivity transfer function is: ܵ௫ = 1 − ܶ௫ = 1 −(߬(߬ଶݏ+ 1) ଵݏ+ 1)ଶ = (߬ଵݏ+ 1)ଶ− (߬ଶݏ+ 1) (߬ଵݏ+ 1)ଶ = ߬ଵଶݏଶ+ 2߬ଵݏ− ߬ଶݏ (߬ଵݏ+ 1)ଶ
By selectingܸ߬ݔ=߬2 = 2 ∗ ߬1, ܶ௫ = 2߬ೣݏ+ 1 ൫߬ೣݏ+ 1൯ ଶ (4-62) ܵ௫ = ߬ೣ ଶݏଶ ൫߬ೣݏ+ 1൯ ଶ (4-63)
The sensitivity function ܵ௫ (which is the transfer function from reference input ܸ௫, to tracking error ݁) has two zeros at the origin, therefore the asymptotic tracking of step and ramp input is guaranteed (Doyle, Francis, & Tannenbaum, 1992).
The plant has a pole atݏ= 0, and from Eqs.(4-61)and(4-62)
ܵ௫(0) = 0ܽ݊݀ܶ௫(0) = 1
Therefore, from the interpolation condition, the internal stability of the system is verified.
And finally, the controllerܭ௫ can be derived from Eq.(4-41)as: ܭ௫ = ݉ ݏ(߬߬ ଶݏ+ 1) ೣଶݏଶ =݉ (2߬߬ଵݏ+ 1) ೣଶݏ =2݉߬ ೣ +1ݏ߬݉ ೣଶ
which is a PI controller with the proportional and integrator gain of:
ܭ = 2݉ /߬ೣ and ܭூ= ݉ /߬ೣଶ (4-64)
The time constant ߬௫ could be employed as a tuning knob to perform the required control performances. The ܵ and ܶ shape for two arbitrary values of ߬௫= 1 and ߬௫= 0.1 is shown in Figure 4-11. For ߬௫= 1, ܵ and ܶ cross each other at frequency below the frequency of 6.28 rad/sec (plant bandwidth) and for the value of߬௫= 0.1 they crossing at the frequency above that.
Figure 4-11: S and T variation with parameterࢇ
A driver’s steering wheel input bandwidth is measured from the number of degrees of steering wheel angle input per second. For example if a driver is capable of applying 180 degrees/second of steering, then his/her bandwidth, in hertz, is computed as follows,
݂ = 180݀݁݃ݎ݁݁ݏ/ sec ∗ (180݀݁݃ݎ݁݁ݏ) ∗ (ߨݎܽ݀ 2 ∗ ߨ) = 0.5ܪݖ1
The average driver has a bandwidth of less than 1 Hz, however, the bandwidth of the advanced drivers could be more than 1 Hz bandwidth, whilst, the professional drivers are capable of applying steering inputs four times faster than the average drivers. The high level controllers in this work should respond quicker than the fastest drivers’ inputs, hence, the bandwidths of these controllers are selected to be 3 Hz. The speed of the response of these high level controllers will be further evaluated, and if necessary, adjusted and validated in the final chapter of this thesis.
By selecting ܸ߬ݔ= 0.1, the control bandwidth (crossover frequency) is set to around 20 rad/sec (≈ 3.2ܪݖ) and the longitudinal controller ܭ௫ becomes:
ܸ߬ݔ= 1
ܸ߬ݔ= .1
ܭ௫= 20݉ + 100݉ 1ݏ (4-65)
where݉ is the vehicle mass as indicated in appendix A.
Figure 4-12: Open loop, Closed loop and sensitivity transfer functions for࣎ࢂ࢞=0.1 To investigate the behaviour of the control system, a close-up plot of the frequency response of the open loop ܮ, closed loop ܶ and sensitivity ܵ transfer functions are shown in Figure 4-12. The following conclusions are justified16:
1. The sensitivity transfer function |S| first crosses -3 dB from below at frequency around 15 rad/sec, the open loop transfer function |L| first crosses 1 (0 dB) from above at frequency around 20 rad/sec and the closed loop transfer function |T| first crosses -3 dB from above at frequency around 25 rad/sec, so:
ω ≈ 15ݎܽ݀/ s < ωୡ≈ 20ݎܽ݀/ s < ω ≈ 25ݎܽ݀/ݏ
The crossover frequency ωୡis between ω and ω, therefore minimum 90° phase margin is guaranteed. Moreover, during this interval the
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sensitivity transfer function ܵ remains negative, so the control performance is not degraded while the frequency is increasing. The crossover frequency 20 rad/s (≈ 3.2ܪݖ) is selected as the control system bandwidth.
2. The open loop gain at low frequencies (below control bandwidth) is high whereas the gain at high frequency (after control bandwidth) is low, therefore, the control system has good robustness, command tracking and disturbance attenuation performance at low frequencies and good noise rejection performance at high frequencies.
3. The value of|ܵ| and |ܶ| at crossover frequency (the point that ܵ = ܶ) are less than zero dB, therefore the stability of the closed loop system is guaranteed.
4. The maximum value of ܵ and ܶ (ܯௌ and ܯ்) are less than 2, so the minimum of 60° phase margin and 6dB gain margin is also guaranteed and the control performance is met.
5. Finally, the value of|ܵ|at plant (dynamics) bandwidth is less than -10 dB. This ensures a good stability margin and robustness even in the presence of plant uncertainties and disturbance.
The frequency response of the system including the plant ܩ௫, Youla parameter ܻ௫, closed loop ܶ௫, sensitivity ܵ௫, controller ܭ௫ and open loop ܮ௫ transfer functions are shown in Figure 4-13, which confirms all the previous conclusions about the control system.
Figure 4-13: Longitudinal Control Transfer Functions
To investigate the closed loop system performance in time domain, the step response of the system with the existence of step disturbance (applied at sec 3) is shown in Figure 4-14. The magnitude of disturbance is set to %50 of reference value (which is quite high). The transient response of the system is sufficiently fast and well damped: the overshoot is less than %20 and the settling time is 0.4 sec and the disturbance is properly attenuated.
Figure 4-14: Closed loop step response with disturbance
4.5.3.2 Lateral velocity (sideslip) controller:
In a similar fashion to the longitudinal motion, the simplified lateral motion transfer function can be derived as:
ܩ௬ = ݉ ݏ1 (4-66)
and the off-diagonal terms in Eq.(4-54)can be considered as disturbance to the system (which was shown to be small in the frequency range above the plant bandwidth).
The control design procedure is similar to the longitudinal controller design mentioned in the previous chapter, and leads to a PI controller with the proportional and integral coefficients as:
ܭ = 2݉ /߬ and ܭூ= ݉ /߬ଶ (4-67)
where ܽ is a parameter which can be used for tuning the lateral motion controller. By selecting ܸ߬ݕ= 0.1, the lateral motion control system transfer functions is:
ܭ௬ = 20݉ + 100݉ 1ݏ (4-68)
which is similar to longitudinal motion control. Therefore, their response in the frequency domain and time domain as well as the stability and performance of the system is the same as longitudinal motion control.
4.5.3.3 Yaw rate controller
The plant transfer function as:
ܩ௭=ܫ1
௭ݏ (4-69)
and there is no disturbance (due to off-diagonal term) in exist the system: ݀௭= 0
The control design procedure is the same as the longitudinal and lateral controllers, which lead to a PI controller with the proportional and integral coefficients as:
ܭ = 2ܫ௭/߬ఠ and ܭூ= ܫ௭/߬ఠଶ (4-70)
where ߬߱ݖis a parameter which can be used for tuning the yaw rate controller. By selecting߬߱ݖ= 0.1, the lateral motion control system transfer functions is:
ܭ௬ = 20ܫ௭+ 100ܫ௭ 1ݏ (4-71)
which is similar to longitudinal motion control. Therefore, their response in the frequency domain and time domain as well as the stability and performance of the system is similar to the longitudinal and lateral motion controls.