Part II Subsystem decoupling
4.6 Numerical Examples
efficient characterization of the same problem. The dynamic multipliers are allowing a more precise characterization of the underlying problem on the expense of more optimization variables and so heavier computational demands. These findings are supported by the numerical results in Section 4.6.2. Regarding that example, the necessary computation times (on an Intel(R) Core(TM) i5-9300H (2.40GHz) CPU and 8 GB RAM computer) for calculating a full rank Ku
blend matrix for the three different methods are such that ts < tp < td, where ts = 0.27, tp = 0.55,td= 2.82 seconds.
in the form of
Πd(jω) =
X(jω)I 0
0 −X(jω)I)
. (4.60)
A hard factorization of (4.60) was used for the analysis problem, which is given as Md=
1 0 0 −1
, Ψd(s) =
s+10.2
s+5.102 0
0 s+5.102s+10.2
. (4.61)
For |δ| ≤1 parametric uncertainties [87] suggests a multiplier in the form Πp(jω) =
X(jω) Y(jω) Y(jω)∗ −X(jω)
, (4.62)
where X(jω) =X(jω)∗ ≥ 0 andY(jω) = −Y(jω)∗ are bounded and measurable matrix func-tions. A hard factorization of Πp is provided by its J-spectral factorization, described in the Appendix B of [94]. By selecting Πp(jω) as
Πp(s) =
51.02
s+51.02 s
s+0.04017 s
s−0.04017 −51.02 s+51.02
, (4.63)
it’s J-spectral factorization leads to
Mp = 1 0
0 −1
, Ψp(s) =
"0.70711(s+10.57) s+5.102
0.70711(s+2.759) (s+5.102)
−70711(s+2.759) (s+5.102)
0.70711(s+10.57) (s+5.102)
#
. (4.64)
In the forthcoming sensitivity analysis (4.61) and (4.64) were used for describing the model uncertainties. The nominalG(s) system is shown in Figure 4.5, along with a shaded area where the Gp(s) perturbed plant can take its values. The upper bound of this area was found by the wcgain, worst case gain computing MATLAB function. Then by MATLAB’s usample function samples were taken such that the uncertainties are uniformly distributed in their variation intervals. Then a frequency-wise lower bound was calculated as the minimum values of the corresponding sets at each frequency.
Lemmas 5 and 9 provided the minimum and maximum sensitivities of theG(s) nominal sys-tem over finite frequency ranges. Dotted lines show their calculated values when the investigated frequency range was increased from [0 10−3] to [0 102] rad/s in 100 steps.
Theorem 4 allows the calculation of||G||[Ω]∆− and ||G||[Ω]∆∞ over a finite frequency range. The upper bound of the frequency range was again increased from 10−3 to 102 rad/s in 100 steps.
Theorem 3 and Theorem 4 gives the same result for the 0 ∞
frequency range. However, if G(s) would be strictly proper (with zero gain at high frequency), then only Theorem 4 could be applied over a finite frequency range to calculate the minimum sensitivity.
4.6.2 Academic example: subsystem decoupling
This section provides a comparison for the algorithms with different uncertainty handling method-ologies discussed so far. For this purpose a simple academic example is used, where 1-1 poles of the subsystems are uncertain. It suits for a simple and easily tractable comparison, while more detailed real life examples (with larger system dimensions, and various uncertainty types) are discussed later in Chapter 6.
0.001 0.01 0.1 1 10 100
−40
−20 0 20 40
frequency [rad/s]
magnitude[dB]
G(s)
Lemma 5 - 9
Theorem 3 - Lemma 14 Theorem 4
Figure 4.5: Robust sensitivity analysis example
The system is given asGp=Gc+Gd, where theGi =
Ai Bi Ci Di
subsystems are defined as
Gc=
−8.09 0 0 0 2.03 1.78 0 −4.04 0 −1.27 −0.28 1.77 0 0 −1.74δc 1.18 0.6 −1.87
−1.05 −1.37 0.07 0 0 0
−0.42 −0.29 0.45 0 0 0
1.4 1.27 −0.32 0 0 0
,
Gd=
−6.72 −8.23 0 1.35 −1.25 0.64
8.23 −6.72 0 0 1.99 0
0 0 −6.65δd 0.47 0.6 1.56
−0.94 −0.29 −0.49 0 0 0
0.22 0 −0.27 0 0 0
1.06 1.56 −0.13 0 0 0
.
These subsystems have nu = 3 inputs, ny = 3 outputs and nx = 3 states. The uncertainties enter the system as the third states of each subsystem are allowed to vary in certain parameter ranges. For the controlled subsystem δc∈
1 2.3 with nominal value 2.2. For the subsystem to be decouple δd ∈
0.01 1.01 with nominal value 0.4. In the example we want to achieve decoupling over the
0, ω¯ frequency range, where ¯ω was selected as the maximum natural frequency (10.625 rad/s) of the targeted nominal subsystemGc.
Polytopic approach
The polytope corresponding to the uncertain system Gp is modeled by four vertex points. This polytope is represented in Figure 4.6, and the uncertain system can be described as the convex combinations of these vertex models. The approach assumes that δc and δd vary independently in the prescribed intervals, which is the case exactly.
Θ Gp(¯δc,
¯δd) Gp(
¯δc,
¯δd) Gp(¯δc,δ¯d) Gp(
¯δc,δ¯d)
Figure 4.6: Robust academic example: the resulting polytope for two uncertain parameters (δc
and δd)
Static IQC multiplier approach
In case of the static IQC approach, various choices for the Π =
Q S ST R
multiplier are possible.
According to Appendix C.1, Π may be selected to describe structured uncertain parameters, or a polytopic uncertainty coming from the set ∆ ∈ co
¯δc, δ¯c,
¯δd, ¯δd . In this example the second approach is applied, characterizing a polytopic uncertainty set. This selection allows the comparison of the method to the polytopic solution. Furthermore, if the results agree, it provides a feedback for the validity of the underlying synthesis LMIs. The multipliers were selected asR∈Snv,Q=−R andS = 0.
Dynamic IQC multiplier approach
The application of dynamic IQCs necessitates a bit more lengthy discussion. The uncertain pole locations are described by the |δi| ≤ 1 parametric uncertainties. The corresponding dynamic multiplier suggested by [87] has the form of
Πδ(jω) =
X(jω) Y(jω) Y(jω)∗ −X(jω)
, (4.65)
where X(jω) =X(jω)∗ ≥ 0 andY(jω) = −Y(jω)∗ are bounded and measurable matrix func-tions. The SISO X(jω) and Y(jω) transfer functions were initially selected to over-bound the maximum singular value curves of the sampled uncertain subsystems, and then adjusted for better decoupling. The same Πδ was selected for both subsystems. A hard factorization of Πδ is provided by its J-spectral factorization, as described in the Appendix B of [94]. By selecting Πδ(jω) as
Πδ(s) = 80.9
s+80.9 s s+0.174 s−0.174s −80.9
s+80.9
, (4.66)
it’s J-spectral factorization is:
Jδ= 1 0
0 −1
, Ψˆδ(s) =
"0.70711(s+166.5)
s+80.9 0.70711(s−0.174)(s+39.43) (s+80.9)(s+0.174)
−70711(s+39.43) (s+80.9)
0.70711(s+166.5) (s+80.9)
#
. (4.67)
The dynamical models have been extended with the IQC descriptions and the design algorithms discussed in Section 4.4.3 were evaluated.
Results
Upon computing the blend vectors, the decoupling performance is evaluated in the frequency domain, based on the singular values of the blended subsystems, shown in Figure 4.7. Instead of specific singular value curves, shaded intervals are plotted denoting the ranges where the uncertain singular values may fall. Green and red colors correspond to the controlled and decoupled subsystems, respectively. Clearly the aim of decoupling is to separate the two sets of
Nominal Polytopic Static IQC Dynamic IQC -0.49847 -0.827 -0.8269 -0.83289 -0.037756 -0.32455 -0.32472 -0.38394 0.86608 0.45907 0.45912 0.39861
Table 4.1: Calculated input blends for the robust decoupling example Nominal Polytopic Static IQC Dynamic IQC
-0.61055 -0.24553 -0.24565 -0.24369 -0.31267 -0.7045 -0.70448 -0.74261 0.72764 0.66588 0.66586 0.62382
Table 4.2: Calculated output blends for the robust decoupling example
curves by maximizing the difference between these ranges, and having higher gains through the targeted subsystem than through the one to be decoupled.
First review the achievable decoupling performance based on nominal blend vectors (designed by techniques from Chapter 3). This is shown in the first (left) subfigure. Note that the two areas are overlapping, indicating that these input and output transformations do not yield satisfactory decoupling, i.e. in particular cases the gain of the decoupled subsystem might be higher than the controlled one’s. The situation is much different, if only the nominal plant is considered (no uncertainty is present, and δc and δd have their nominal values). In all subfigures, black dashed lines show the blended nominal models (where the upper one always corresponds to the controlled subsystem). For these nominal models, the suppression rate (the distance between the dashed curves) is highest for the nominal blend vectors. This is straightforward, i.e. if the solution has to be found for a set of plant models (the uncertain case), then the exact numerical properties of a particular model count to a less extent than in the nominal case (with a single model). The corresponding blend vectors are given in the first columns of Tables 4.1 and 4.2.
Moving on to the polytopic approach, one can clearly see that the shaded intervals are well separated, and so robust decoupling is achieved. However, for this we had to solve a larger problem, at each iteration step roughly involving four times more LMIs, as it has been discussed in subsection 4.2. As the corresponding section discusses, the huge advantage of the approach is the ease of implementation once the nominal algorithm is available.
As it has been discussed before, for the static IQC multiplier approach Π was selected in a form to describe a polytopic uncertainty set. The decoupling result is shown in the third subfigure of Figure 4.7. The second and third subfigures are almost identical, what verifies the theory behind the static multiplier approach. Comparing the blend vectors in Table 4.1 and Table 4.2 further confirms this close relationship.
The fourth subfigure shows the result corresponding to the dynamic IQC multiplier. The decoupling performance compared to the previous results is further enhanced. The shaded inter-val corresponding to the decoupled subsystem almost shrank into a single line. This is, because with the dynamic multipliers we were able to provide a more precise uncertainty description.
However, for this we had to select the precise form and exact numerical values of the weighting filters (basis functions). The corresponding blend vectors are given in Tables 4.1 and 4.2. In case of more complex uncertainty blocks, it is possible to apply numerous basis functions cor-responding to each uncertainty block, and than their relative weights (λc,i, λd,i) are outcome of the optimization process as well as the blend vectors. A great advantage of the static multi-plier approach is that one only needs to select the form of the multimulti-plier, and than its precise numerical values are computed by the optimization process along the transformation vectors.
−60
−40
−20 0 20
10−3 10−11 101102 frequency. [rad/s]
magnitude[dB]
Nominal
10−2 1 101102 frequency. [rad/s]
Polytopic
Control Decouple
10−2 1 101102 frequency. [rad/s]
Static IQC
10−2 1 101102103 frequency. [rad/s]
Dynamic IQC
Figure 4.7: Robust academic example: dashed line shows the blended nominal subsystems