Multi-Variable Valve Control
4.3 Damping Control
4.3.1 Problem Statement
4.3 Damping Control 59
Fig. 4.7 The results using Independent pressure and velocity control
Fig. 4.8 Linearization process of normal mode
in the flow pertaining to pressure and orifice area. The nonlinear equations of inlet and outlet flows are expressed as:
Qa = Cq Av
/
2(ps − pa)
ρ Qb = Cq Av
/ 2 pb
ρ (4.18)
The system is linearized under the following assumptions:
(1) The flow-pressure gain of the meter-in valve is set as 0 since the inlet flow of the meter-in valve is controlled by electronic PC and the flow across the valve is independent of pressure drops.
(2) Valve dynamics are ignored since its response is much faster than the rest of the system.
Consequently, the inlet flow is transferred to a constant flow Qa which eliminates the nonlinearity of inlet. There are some investigations to prove the validity of this simplified model, such as Heybroek from Linkoping University [36], have proved the validity of this simplified model. The outlet flow Qb is linearized by multiplying the pressure pb and the flow-pressure gain Kcb, of which the flow-pressure gain is pertaining to pressure and orifice area as follows:
Kcb = ∂qb
∂pb = Cq Av
√2ρ· pb
(4.19)
Under these reasonable assumptions, the derivations of the equations are provided by Eqs. (4.20)–(4.24).
Ab =κ Aa (4.20)
Qa − AaVc = Va βe
spa (4.21)
mts2 Xc = Aa pa −κ Aa pb − Bps Xc − F1 (4.22)
κ Aas Xc − Qb = Vb
βe
spb (4.23)
Qb = Kcb pb (4.24)
Defining the three parameters listed below:
γ = 1 + κ2 Va
Vb
, ωo = Aa
/ βe
mtVa
, ωb = βe Kcb
Vb
(4.25)
Then the transfer function is given by:
( s3 ω2 0ωb
+ s2 ωo 2 + γs
ωb
+ 1 + BPVa
A2 aβe
s + BpVa ωb A2 aβe
s2 )
Vc
= (
1 + s ωb
)( 1 Aa
Qa − Va A2 aβe
s F1
)
(4.26)
4.3 Damping Control 61
The denominator polynomials of the system transfer function may be factored.
The interesting case is when the system has a resonance and a break frequency [35, 37], see Eq. (4.27).
Dp = s3 ω02 ωb
+ ( 1
ω0 2 + BpVa ωb A2 aβe
)
· s2 ω0 2 +
(BPVa
A2 aβe
+ γ ωb
) s + 1
= (
1 + s ωn
)(s2 ω2 h + 2ξhs
ωh
+ 1 )
(4.27)
Eqs. (4.26) and (4.27), can also be applied to the conventional system, but the difference resides in the value of flow-pressure gain Kcb. Due to the larger opening area of the meter-out valve in the IMCS, this parameter is substantially higher in the IMCS than that of the traditional system. It is concluded that the system has a pair of conjugate poles, a real pole and two zeros. In general, the locations of the poles and zeros determine the dynamic performance of the system.
Taking the boom motion system as an example, the pole locations pertaining to the opening area of meter-out valve are depicted in Fig. 4.9 by solving Eq. (4.26) with the parameters from Table 4.1. The studied system has provided global parameters. State variables are constantly changing during the control process. They are evaluated by measured data or calculated by the input and output of controller. Here, for theoretical analysis, the state variables are provided directly. The locations of poles are acquired in in MATLAB platform based on the two types of parameters simultaneously. To represent different orifice areas, the pressures are supposed to rise from 0.2 MPa, with a gradient of 0.02 MPa for each evaluation. The three poles are placed on the left half-plane of the pole-zero map, as shown in Fig. 4.9. Then, their definitions are as follows:
p1,2 = −a ± bj,a > 0,b > 0 (4.28)
p3 = −c,c > 0 (4.29)
It is inferred that the influence of negative real pole p3 on the system dynamic can be ignored because:
(1) As the opening of the meter-out valve increases, this real pole moves away from the imaginary axis. When the ratio of the real part of this pole to –a (e.g., c/a) exceeds l, which is typically between 3 and 5 [38], it could be defined that the pole is far away from the imaginary axis and conjugate poles;
(2) As the opening area of the valve is not large enough, this real pole is not far away from the conjugate poles. Nevertheless, it should be noted according to Eq. (4.27) that there exists a real zero (z1 = −ωb), and the zero z1 is near to the pole p3 according to Eq. (4.28). This indicates that they achieve pole-zero cancellation;
Fig. 4.9 Pole locations with respect to valve area
Consequently, the real pole p3 does not affect the dynamics of the system, and thus conjugate poles p1,2 become the dominated poles, whose locations actually affect the system response including overshoot and settling time the system damping ξh is calculated by the angle between the conjugate poles and the negative real axis as Eq. (4.30). The angle, marked as θ, is easy to obtain by the movement of the poles in the pole-zero map.
θ= arccosζh = arctan /
1 − ζ h 2 ζh
⇒ζh = a
√a2 + b2 (4.30)
According to Eq. (4.30), a larger angle θ leads to a high damping. With the param- eters listed in Table 4.1, the relationship between damping and flow-pressure gain Kcb is illustrated in Fig. 4.10. As the opening area of the valve increases, the system damping of the system firstly increases and then continually decreases. The damping of IMCS is located in the lowest region, while that of the conventional system is nearby located in the highest region. Therefore, it can explains that the reason that the oscillations of cylinder velocity in the former system are more violent than the latter one, as the results shown in Fig. 4.7c and d.
The relationship between the pole location and the equivalent load mass mt in IMCS is exhibited in Fig. 4.11. As the equivalent load mass mt increases, the real pole p3 is also far away from the imaginary axis. Moreover, the ratio of the real pole p3 to −a (formulated as c/a) exceeds 10. Consequently, conjugate poles p1,2 are always the dominated poles, and the damping as function of the equivalent load mass mt is depicted in Fig. 4.12. It is concluded that the damping of heavy load systems
4.3 Damping Control 63
Fig. 4.10 Damping as a function of the flow-pressure gain
is lower than that of light load systems, which would explain the comparison results between boom and arm in Fig. 4.7a and b.
Fig. 4.11 Pole locations with respect to equivalent load mass
Fig. 4.12 Damping as a function of the equivalent load mass