Part II Subsystem decoupling
3.4 The decoupling of multiple modes
3.4.3 Examples for the decoupling of multiple modes
This section demonstrates the effectiveness of the decoupling technique proposed in Section 3.4.2. The first example is the continuation of the academic example from Section 3.3.3, while the second one builds connections to a real world control problem involving high precision wafer scanners through a highly simplified system model.
Academic example
Take the system in (3.15), and by the original method in Section 3.3 synthesize independent (kui andkyi) input and output blend vector pairs for each mode. The resulting transformation vectors are
Ku =
ku1 ku2
=
−0.9018 −0.2428
−0.1167 0.5046
−0.4160 0.8285
, Ky =
ky1 ky2
=
−0.6921 −0.6878 0.7218 −0.7259
. (3.26) Note that ku1 and ky1 are the same as the ones in Section 3.3.3. Each pair of these transfor-mations will assure that the transfer through the targeted subsystem is maximized, while it is minimized through the other one. However, they do not try to minimize the couplings between the various input and output channels whenKu and Ky are applied to the system.
To overcome this issue, a new set of transformations can be synthesized based on (3.25).
These yield Kud =
kud1 kud2
=
−0.2180 −0.1874
−0.8967 0.7140
−0.3852 0.6746
, Kyd =
kyd1 kyd2
=
−0.7431 −0.7644 0.6692 −0.6448
, (3.27)
−100
−50 0
1.output
1. input
KyTG(s)Ku KyTdG(s)Kud
2. input
0.1 1 10 100
−60
−40
−20 0
frequency [rad/s]
2.output
0.1 1 10 100
frequency [rad/s]
Figure 3.11: Step response for theKyTGKu system
where the d subscript denotes diagonalization. Denote the system in (3.15) by G. Then Figure 3.11 shows the bode magnitude plots for the two pair of input and output transformations.
Note that by applying the Kud and Kyd blend matrices, the transfer is reduced through all channels. This is an expected behavior. For the diagonal entries Ku and Ky guarantee an optimal amplification (maximal transfer). The new set of blend vectors are rotated and point to a new direction in the space what leads to a sensitivity reduction through these channels. On the other hand these rotations are necessary to look for input and output directions where the couplings are reduced on the off-diagonal channels. The key is the ratio of sensitivity reduction between the diagonal and off-diagonal terms. This level of transfer reduction and so the rotations of the original blend vectors can be tuned by weighting factors corresponding to the coupling terms in the objective function and the synthesis LMI conditions.
Motion systems
This section has been motivated by the interesting discussions with Roland T´oth. I thank him for drawing my attention to a new area of applications of the developed results. Modern wafer scanners are state-of-the-art motion systems which are complex machines used in the semiconductor industry to produce integrated circuits by lithography. A key component of such a device is the wafer stage motion system, which is responsible for positioning a silicon wafer with a photosensitive layer at a given location where a light beam creates a pattern on the wafer. For the next generation of such devices the wafer stages are required to be controlled with high accuracy (<10−9m), and with high accelerations (>50 m/s2) [97]. This section first provides a high level description of the motion system used in wafer scanners, and then based on a simple mechanical example it draws connections between their control and the decoupling algorithm discussed in this chapter. The presentation of the device itself and the corresponding mathematical models are heavily based on [97, 60] and [105].
The motion system of the wafer scanner is a commutated magnetically levitated planar actuator, which consists of a moving magnet array (translator) and a stationary coil set. The magnets in the array are glued to an aluminum plate. The planar actuator is an active magnetic bearing what has to be controlled in six degrees of freedom. The coil set is used to generate the necessary forces and torques for the movement of the plate [105]. In [60] a special commutation algorithm is presented, which allows to control each force and torque component independently.
As a result of the lightweight design, one of the main limitations of the approach is the limited stiffness of the mechanical plate. When a spatially non-uniform distribution of forces acts on the magnet array, mechanical deformations are induced to the moving body. Unless
m1
m2 m3
k1 k2
k3
d2 d1
F1 p1
F2 p2
F3 p3
Figure 3.12: A mass-spring-damper system consisting of three masses
properly suppressed, the resulting deformations can severely reduce the attained positioning and so scanning accuracy [97]. It might be desirable to
1. control the rigid body motion without exciting the flexible dynamics,
2. or by additional controllers control the shape of the plate through its flexible modes.
For this reason, in the forthcoming the focus is on the dynamics of the moving plate. Mea-surement equations are neglected. The state-space model corresponding to the mechanical de-scription of the plate is derived from FEM simulations, and with modal coordinates it has the form
˙ pm,r
˙ pm,f
¨ pm,r
¨ pm,f
=
0 I
0 0 0 −Ω2
0 0
0 −2ZΩ
pm,r pm,f
˙ pm,r
˙ pm,f
+
0 0 BTm,r Bm,fT
Fn, (3.28)
whereFncollects the magnetic forces acting on the plate and theBmmatrices are given in terms of the corresponding mode shapes of the plate. The state vector can be decomposed into rigid and flexible states denoted by{·}rand{·}f, respectively. Furthermore,Z = diag(ξ1, ..., ξnf)∈Rnf×nf contains the damping coefficients, while the Ω∈Rnf×nf matrix collects the natural frequencies for the flexible modes as it is discussed next. Further details about the modeling of this specific type of device can be found in [97]. Due to the modal description, the motion of the translator points is described by the summation of sub-motions of the modes.
Note that (3.28) describes the equations of motion of a simple mass-sprig-damper system by modal coordinates. This fact is used in the upcoming to develop connections between the wafer stage motion system and the decoupling transformations, based on the simple example given in Figure 3.12. The highly simplified system model consists of three masses which are interconnected through springs and dampers. The example is characterized by a system of second-order differential equations
m1 0 0
0 m2 0
0 0 m3
¨ p1
¨ p2
¨ p3
+
d1 −d1 0
−d1 d1+d2 −d2
0 −d2 d2
˙ p1
˙ p2
˙ p3
+
k1+k3 −k1 −k3
−k1 k1+k2 −k2
−k3 −k2 k2+k3
p1 p2
p3
=
F1 F2
F3
(3.29) where pi and Fi denote the position of the mi mass, and the force exerted on the mass, respec-tively. With a more compact notation it can be written as
Mnp¨n+Dnp˙n+Knpn=BnFn, (3.30) where Bn characterizes the effect of the external forces to the nodes, and Bn = I for this specific example. The n index denotes the nodal description of the system dynamics, where the pn coordinate vector characterizes the displacement of the given structural locations. The connection between the nodal and modal system description is discussed in [43, 97], and it is
briefly repeated here. After multiplying (3.30) from the left by Mn−1, it is straightforward to rewrite it in the state-space form
p˙n
¨ pn
=
0 I
−Mn−1Kn −Mn−1Dn pn
˙ pn
+
0 Mn−1Bn
Fn. (3.31)
The modal coordinate representation can be obtained by the transformation of the nodal models. This transformation can be derived using the modal matrix, which is determined next.
In case of free vibrations (3.30) can be written as
Mnp¨n+Knpn= 0, (3.32)
where the corresponding solution has the form pn = ϕejωt and ¨pn = −ω2ϕejωt. Introducing these into (3.32) gives
(Kn−ω2Mn)ϕejωt= 0.
For this set of homogeneous equations a nontrivial solution exists if
det(Kn−ω2Mn) = 0. (3.33)
The determinant is satisfied for a set of nvalues ofω frequencies, denoted asω1, ω2, ..., ωn. The frequency ωi is called the ith natural frequency, and the corresponding ϕi vector the ith mode shape. The matrix of natural frequencies and the modal matrix are in the form
Ω = diag(ω1, ω2, ..., ωn), and Φ =
ϕ1 ϕ2 ...ϕn , respectively. By introducing a new variable as
pn= Φpm, (3.30) can be rewritten to
Mmp¨m+Dmp˙m+Kmpm= ΦTBnFn, (3.34) where Mm= ΦTMnΦ, Dm= ΦTDnΦ andKm= ΦTKnΦ. The multiplication of (3.34) byMm−1 from the left gives
¨
pm+ 2ZΩ ˙pm+ Ω2pm=BmFn, (3.35) where Ω2=Mm−1Km, 2ZΩ =Mm−1Dm, withZ being the diagonal matrix of modal damping and Bm=Mm−1ΦTBn. Rewriting this equation in state-space form yields
p˙m
¨ pm
=
0 I
−Ω2 −2ZΩ pm
˙ pm
+
0 Bm
Fn. (3.36)
Introducing the pm=
pTm,r pTm,fT
notation for separating the rigid body (translational) mode from the flexible (oscillatory) modes gives (3.28). For further details about the mechanical modeling the reader is invited to consult with [43, 97].
For evaluating the proposed diagonalizing algorithm, the numerical values of the parameters in (3.29) were: m1 = 1, m2 = 1.5, m3 = 2 kg, k1 = 1, k2 = 2, k3 = 2 N/m, and d1 = 0.01, d2 = 0.02, d3 = 0.03 Ns/m. By transforming the model into the (3.28) modal form, and by reordering its states the subsystem form in (2.5) is achieved, where the system has three modes (one translational and two flexible). Next, by Algorithm 3 an input decoupling Kud transformation is synthesized, which turns the transformed input matrix in (3.28) diagonally dominant
BmKud =
0.7849 0.0001 −0.0002 0.0053 0.8016 −0.0015 0.0110 0.0008 −0.8657
. (3.37)
From the previous discussion it is obvious that Bn = Mm−1ΦT−1
Bm = I, and it means a perfect decoupling. The effectiveness of the proposed approach is quantified by
||I− |BmKud| ||2= 0.216, (3.38) as the largest singular value of the difference between the perfectly decoupled and the BmKud input matrix. The result means that the largest error in the worst-case input direction is on the order of 20% compared to the theoretical optimum. Suppose that the available measurements are the (pm) modal displacements¶. ApplyKud to the system. The corresponding step response is given in Figure 3.13, where the first output corresponds to the rigid body motion, while the second two to the flexible dynamics. It clearly shows that diagonal dominance is achieved. The inverse based decoupling approach is a frequently used method for control synthesis is case of motion systems. However, the appealing properties of the proposed diagonalization algorithm may turn it a useful technique for such control design as well, especially under known or unknown parameter variations in the plant model, as it is discussed in the next chapters.
−40
−30
−20
−100
1.output
1. input 2. input 2. input
−0.6
−0.4
−0.2 0
2.output
0 2 4 6 8 10
0 0.2 0.4
time [s]
3.output
0 2 4 6 8 10
time [s]
0 2 4 6 8 10
time [s]
Figure 3.13: Step response for the blended mass-spring-damper system: G(s)Kud
¶In real systems these can be estimated. The point of this specific example is the independent control of the system modes.