Part II Subsystem decoupling
5.5 Academic example
This section compares the proposed parameter-varying decoupling methods, based on a single academic example, which is a slight modification of the LTI one, presented in (3.15). More
complex aerospace examples are provided in Chapter 6. The G(ϱ) parameter-varying system is given as
A(ϱ) =
Ac 0 0 Ad
=
−0.4 1.6 0
−1.6 −0.4 0
0 0 −1.4
,
B(ϱ) =
0.7 −0.1 0.3
−0.4 −0.2 0.1
−0.6 −0.2 0.8
+
−sin(ϱ) cos(ϱ) 0
−sin(ϱ) cos(ϱ) 0
−sin(ϱ) cos(ϱ) 0
,
C(ϱ) =
0 0.8 −0.8
−0.8 −0.7 −0.9
+
sin(ϱ) sin(ϱ) sin(ϱ) cos(ϱ) cos(ϱ) cos(ϱ)
, D= 0,
(5.36)
with π/2 ≤ ϱ ≤ 3π/2. A subsystem with complex eigenvalues should be controlled, while a real pole corresponds to the undesired dynamics. By evaluating (5.36) over a ϱ ∈ [π/2, 3π/2]
parameter grid with 0.1 increments, a state-space array of 32 LTI systems is generated. This set of models is the basis for the blend synthesis by the polytopic and gridded approaches as well.
These are discussed next.
Polytopic LPV design
For the polytopic design, a crucial step is the bounding Θ polytope generation. The Tensor Product tool generates the vertex models for a gridded LPV model, given as a set of state-space systems. The toolbox is freely available at [127], with theoretical background discussed in [11, 123]. The (5.36) LPV model can be described in the from of
G(ϱ) = X3 i=1
ξi(ϱ)Gi, (5.37)
by a polytope of three vertex models, with corresponding ξ(ϱ) weights shown in Figure 5.3.
π/2 π 3π/2
0 0.2 0.4 0.6 0.8 1
%
ξi(%) ξ1
ξ2 ξ3
Figure 5.3: LPV academic example: Polytopic weights
The next step is to design the Ku(ϱ) and Ky(ϱ) input and output blend matrix functions, according to Proposition 8. Once they are found, (5.12) directly provides the ku(ϱ) :R→ R3, and ky(ϱ) : R → R2 blend vector functions for the gridded LPV models, as the parameter-varying left singular vectors corresponding to the only non-zero element of S(ϱ). The resulted vector functions are shown in the left subfigure of Figure 5.4. Note that, slight variations are observable, due to the parameter variation.
The decoupling performance is assessed in the frequency and time domains as well. For the frequency domain evaluation, the LPV model is evaluated at the grid points, and the singular value curves are used to characterize the quality of the decoupling. Note that, the evaluation
−1
−0.5 0 ku(%)
Polytopic design
ku1(%) ku2(%) ku3(%)
Grid-based design
π/2 π 3π/2
−1
−0.50.501
% ky(%)
ky1(%) ky1(%)
π/2 π 3π/2
%
Figure 5.4: LPV academic example: parameter dependent ku(ϱ) :R→R3, and ky(ϱ) :R→R2 input and output blends
of the LPV models at various grid points leads to groups of singular value plots. For better visualization, only the lower and upper bounds of these groups are plotted, the remaining curves lie in the shaded areas. The dynamics of Gc(ϱ) and Gd(ϱ) are coupled through their inputs and outputs, which in terms of singular values means that the subsystem to be decoupled has higher gains over a particular frequency range, thanGc(ϱ). This is shown in the left subfigure of Figure 5.5.
The decoupling performance at the vertex points is evaluated by computing theku,i andky,i blend vectors from the resultingKu,i andKy,i blend matrices separately. This is necessary, be-cause the bounding convex polytope may involve models, that are not in the model set described by (5.36), i.e. the bounding is conservative. The decoupling of the vertex point models is shown in the middle subfigure of Figure 5.5. The right subfigure shows the case, when the ku(ϱ) and ky(ϱ) decoupling vector functions are computed according to Proposition 8, and applied to the Gc(ϱ) andGd(ϱ) subsystems. Despite that the two areas are overlapping, a one by one evaluation at every grid point reveals that, decoupling is achieved, and the transfer through the controlled subsystem is always higher. In order to provide an insight to the minimum level of suppression, the case where the singular value curves are closest to each other are plotted by black dashed line. This corresponds to a∼5 dB suppression rate, and occurs atϱ≈1.2π.
In order to provide a comprehensive analysis of the decoupling performance, time domain simulations were carried out as well. For comparison, a constant blend vector pair has been designed for the plant at ϱ=π, and then applied to the subsystems. The corresponding results are shown in the left and middle columns of Figure 5.6. During the simulation the system was excited through the blended input by a constant step signal at 1 s. The ϱ(t) parameter variation is governed by a multisine signal (where the frequency of the first two components are the natural frequencies of the modes, while the third one models a slow parameter variation) as the upper subfigure shows. As a direct consequence, the polytopic input and output vector functions were continuously changing. The value of the ky(t) output transformation is plotted in the second subfigure on the left. The time domain simulation results show a close agreement with the frequency domain properties discussed before. The
¯yc(t) and
¯yd(t) blended outputs of the controlled and decoupled subsystems are shown in the lower two subfigures of Figure 5.6.
These show that, the designed input and output transformations achieve larger amplification through the controlled subsystem, but the contribution of the undesired dynamical part to the overall response of the system is still significant. Nevertheless, compared to the constant transformations, some improvements are achieved. The peak amplitude of
¯yc(t) is higher, while the peak of
¯yd(t) is lower for the polytopic synthesis.
0.01 1 10
−100
−80
−60
−40
−20 0 20
frequency [rad/s]
magnitude[dB]
Maximum singular values of the subsystems
¯
σ(Gc(%))
¯
σ(Gd(%))
0.01 0.1 1 10
frequency [rad/s]
Decoupling of vertex models
ky,iT Gc,iku,i ky,iT Gd,iku,i
0.01 1 100
frequency [rad/s]
Polytopic LPV blend design
kTy(%)Gc(%)ku(%) kTy(%)Gd(%)ku(%)
Figure 5.5: LPV academic example: subsystem decoupling results by the polytopic approach
3 5
%(t)
Constant blend Polytopic approach Grid-based approach
−1 0 1 ky(t)
−3
−2
−10
¯yc
0 5 10 15
0 0.5
−0.5
time [s]
¯yd
0 5 10 15
time [s]
0 5 10 15
time [s]
Figure 5.6: LPV academic example: simulation results with various blends Gridded LPV design
For the gridded LPV synthesis, a quadratic parameter variation has been assumed. The input and output blend matrices are searched in the form
K(ϱ) =K0+ϱK1+ϱ2K2. (5.38) This selection assures that, all entries of the resulting K(ϱ) blend matrices are quadratic func-tions of ϱ.
Then theku(ϱ) andky(ϱ) input and output blend vector functions can be computed according to Propositions 9 and 10 and Algorithm 4. The resulting blend functions are smooth and continuous functions ofϱ, and they are given in Figure 5.4. Compared to the polytopic design,
−100
−50 0
10−1 1 101
frequency [rad/s]
magnitude[dB]
Constant blend design
ky,const.T Gc(%)ku,const.
ky,const.T Gd(%)ku,const.
10−2 1 102
frequency [rad/s]
Grid-based LPV blend design
kyT(%)Gc(%)ku(%) kyT(%)Gd(%)ku(%)
Figure 5.7: LPV academic example: subsystem decoupling with grid-based design note the stronger variation of the blend elements over the ϱ∈[π/2, 3π/2] interval.
Similarly for the polytopic case, decoupling performance is evaluated over the frequency and time domains as well. For frequency domain analysis, the singular values of the blended subsystems lie in the shaded intervals of Figure 5.7. Two subfigures are given. In order to show the necessity of parameter-varying decoupling, a constant blend vector pair has been designed for the plant at ϱ=π, and then applied to the subsystems. This is shown in the left subfigure. For the nominal models, decoupling is achieved (see the black dotted lines), however, they cannot ensure decoupling over the whole parameter range. This means that, there are cases when the amplification of the decoupled subsystem is higher, than the gain of the targeted one. This has to be checked for each grid point separately, but for a compact representation they are collectively plotted as two overlapping areas.
By using the grid-based decoupling method, the resultingku(ϱ) andky(ϱ) parameter depen-dent input and output transformations ensure the successful decoupling of the subsystems. This is shown in the right subfigure of Figure 5.7. Note that there is no overlapping, and the approx-imately 40 dB distance between the two areas can be recognized as a guaranteed suppression ratio of 100.
The performance of the grid-based decoupling algorithm has been tested in the same time domain simulation as the polytopic one, and the results are shown in the right column of Figure 5.6. The elements of the ky(t) output transformation is vary more significantly, as it has been expected based on Figure 5.4. However, notice that the magnitude of
¯yc(t) decreased compared to the polytopic case, but the shape of the response is similar. By looking at the fourth subfigure, the benefits of the grid-based approach becomes obvious. As the
¯yd(t) response shows, an almost perfect decoupling has been achieved, where the maximum amplitude of the decoupled dynamics is less than 2×10−4.
This section compared two decoupling synthesis techniques for LPV systems, relying on polytopic or grid-based modeling approaches. One needs to see that, the polytopic method needed an additional step during the design process, to calculate the vertex point models. But once they are found, the approach becomes significantly faster, because the synthesis inequalities need to be evaluated only over the vertex models. The grid-based technique has to be evaluated over all grid points, for the minimum and maximum parameter variation rates as well. This difference becomes more significant in the light of the alternating projection algorithm, where the evaluation of the synthesis LMIs has to be done for thenu−1 (or ny−1) projection sequences
(where each sequence contains multiple projections). In case of large nu, and ny and/or a dense grid, the computational burden of the grid-based method may significantly increase the computation time. However, on the other hand, the parameter dependent transformations have to be computed only once, and so based on the numerical results, the benefits of the grid-based technique may overpower its disadvantages. This is due to the fact that, the fit of the polytope may cover additional systems as well, which are not present in the actual dynamics. The grid-based approach allows the algorithm to tune the decoupling vector functions specifically to the plant dynamics.