4.4 LPV Identification of an FRF Matrix
4.4.1 Spatially-Varying Characteristics of FRFs
K11 K12 K13 K14 K15
K22 K23 K24 K25
K33 K34 K35
sym K44 K45 K55
. (4.24)
Given the fact that each operator Gij in (4.23) is determined individually, so is the controller operator Kij in (4.24). The involved work makes the plant structure unfavourable from a practical point of view.
Driven by the above two drawbacks of the conventional black-box identification technique, a novel identification approach is needed. The identified model should preserve the accu-racy achieved by the conventional technique to a certain extent. In addition, the resulting plant structure should lead to an efficient system identification and controller synthesis process, which involves a reasonable amount of effort, even when dealing with a complex or large-scale system.
4.4 LPV Identification of an FRF Matrix
To address the two issues related to the individual identification of the FRFs in (4.23), a spatial LPV model is proposed to capture the spatially-varying characteristics of the FRF matrix. To justify this approach, the system in Fig. 4.2 is taken as an example to demonstrate the varying properties of the FRFs, and how this variation leads to a spatial LPV representation.
4.4.1 Spatially-Varying Characteristics of FRFs
Due to the interconnection between subsystems through springs and dampers, the dynam-ics of the 5 subsystems in Fig. 4.2 are clearly not decoupled, but interacting with nearest
neighbours. This interaction can be observed in terms of mode shapes, where the rela-tive displacements of subsystems follow a certain pattern at resonant modes. The typical unforced mode shapes of the 5-node system in Fig. 4.2 at the first 3 natural frequencies are shown in Fig. 4.3. One important assumption can already be made: The dynamics of one subsystem have a functional dependence on its location within the interconnected system. If this assumption is true, the spatially-varying properties can be captured using an LPV model according to a certain spatial scheduling policy.
−1
−0.5 0 0.5 1
−1
−0.5 0 0.5 1
1 2 3 4 5
−1
−0.5 0 0.5 1
mode 1
mode 2
mode 3
amplitude
subsystems
Figure 4.3: First 3 mode shapes of the five-spring-mass-damper system
Consider the diagonal terms in (4.23) at first, the so-calledpoint FRFs, where the actuat-ing and sensactuat-ing subsystems coincide. If the spatial LPV characteristics can be established among the diagonal terms, it should be possible to use an LPV input/output model to generalize G11, G22, G33, G44, and G55 as functions of the spatial coordinates. Introduce an extended notion of operating points in contrast to the lumped LPV sense. In this case, determine the spatial coordinate si, where the response is measured, as the spatial operating point θis, i.e. θis =si (i= 1, . . . ,5). Let ˆp0(θis) be a smooth function of θis. The
diagonal terms can be represented in LPV form as
The spatial LPV modelG(ˆp0(θsi)) differs from (4.9) in terms of the scheduling parameters—
time-varying in a lumped system, time/space-varying in a spatially-distributed system.
The subscript 0 indicates the diagonal terms.
The true values of p0(θsi) can be first estimated by applying the conventional black-box identification technique to individually identify the LTI operators Gii (i = 1, . . . ,5) as described in steps 1) and 2) in Section 4.2.2, except that the fixed operating points are spatial variables here. Approximate p0(θsi) with a polynomial function ˆp0(θsi) of θsi. An LPV representation of the 5 diagonal terms in (4.25) can then be easily derived following step 3) in Section 4.2.2. It is evident that if a solution to (4.17) exists, the approximated polynomial coefficients ˆp0(θis) agree with the true values p0(θsi) (i= 1, . . . ,5).
To explore the advantages of using an LPV model to generalize the diagonal terms (4.25), now assume that identification experiments can not be implemented at all operating points due to certain constraints. For example, the LTI operators G11, G33, G44 and G55 can be experimentally identified, whereas it is not possible to perform actuating and/or sensing at subsystem 2, so that G22 needs to be determined without performing an experiment.
This problem can be solved by first constructing an LPV model G(ˆp0(θsi)) based on the information at operating points θs1, θ3s, θs4 and θs5, then interpolating θ2s in G(ˆp0(θis)) to obtain G22. Fig. 4.4 shows a comparison between two FRFs ofG22: one is its true value;
the other one is simulated from G(ˆp0(θ2s)). Although the interpolated operator is lightly damped at the 2nd mode, the resonant peaks at other modes are well preserved. The main dynamics are captured in spite of a degraded accuracy. The modelling error may be attributed to a functional dependence of ˆp0(θsi) on θsi of a higher complexity than the simple polynomial function assumed here.
The 1st off-diagonal terms are the transfer FRFs from the single excitation at subsystem j =i+ 1 to measured response at subsystem i(i= 1, . . . ,4). Applying similar procedures
where the same scheduling policy as for the diagonal terms is employed, i.e., the spatial operating points are the spatial coordinates where the responses are measured.
The identified spatial LPV model also allows the interpolation of operating points where the identification experiments are not performed. Suppose that the response at subsystem 1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 10−4
10−2 100 102
frequency (Hz)
magnitude
Figure 4.4: Comparison between two FRFs of G22: the blue solid curve shows its true value; the red dotted curve is simulated by interpolating the LPV modelG(ˆp0(θsi)) with θs2.
can not be measured. The unknown operator G12 is approximated by interpolating the LPV model, which is constructed from the identified local LTI systems G23,G34 andG45, with a polynomial dependence of ˆp1(θsi) onθsi (i= 1,2,3,4). The comparison between the true value and the simulation ofG12 as shown in Fig.4.5 suggests a satisfactory accuracy of the LPV modelling.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
10−4 10−3 10−2 10−1 100 101 102
frequency (Hz)
magnitude
Figure 4.5: Comparison between two FRFs of G12: the blue solid curve shows its true value; the red dotted curve is simulated by interpolating the LPV modelG(ˆp1(θsi)) with θs1.
Repeating the same procedures for the 2nd, 3rd and 4th off-diagonal operators,
an-other three sets of spatial LPV models are accordingly derived—G(ˆp2(θis)) for i= 1,2,3, G(ˆp3(θis)) for i= 1,2, and G(ˆp4(θsi)) for i= 1, respectively. Finally, the transfer function matrix G parametrized in a set of spatial LPV models is obtained as
G=
where the operating points are chosen to be the spatial coordinates of the locations where the responses are measured.
Certainly, (4.27) is not a unique way to explore the spatial-varying properties of the FRF matrix. Similar characteristics can also be detected when examining the rows of (4.23).
Operators on the first row of (4.23), G1j (j = 1, . . . ,5), denote the transfer function from a single input at subsystem j (j = 1, . . . ,5) to the response at subsystem 1, respectively.
Define a new scheduling policy as θsj = sj—the spatial coordinate of a subsystem where an excitation is applied. The first row of (4.23) is represented in a spatial LPV form as
G= the comparison between the true value of G12 and the approximated one obtained by insertingθs2into the LPV modelG(ˆp1(θsi)) calculated fromG11,G13,G14andG15, confirms the spatially-varying property among the operators on one row. Repeating the same procedures to the 2nd, 3rd and 4th rows in a sequel, the FRF matrix of an LPV form alternative to (4.27) is written as
G=
where the operating points are the spatial coordinates of subsystems at which the actua-tion is applied.