Part II Subsystem decoupling
3.3 LTI subsystem decoupling algorithm
3.3.3 Academic example
corresponding to theith mode. The magnitude of these elements are measures of controllability.
In case of the blend calculation problem this means that the blended input matrix Bku should contain non-zero values at the locations corresponding to the targeted mode, while it’s other elements should be small, possibly zero. This is clearly achievable if the row vectors of Bd are far from the subspace spanned by the rows of Bc. Similar reasoning corresponds to the output decoupling, based on observability and the columns ofC.
Remark 6. The case of unstable subsystems needs some further discussion. The minimum sensitivity characterized by theH−index can be calculated for unstable subsystems as well. On the other hand theH∞ norm is defined only for stable systems (i.e., poles having negative real part). At the same time, unstable systems that have no poles on the imaginary axis have an L∞ norm (also known as the peak gain). This peak gain can be computed using (2.27) after mirroring the unstable poles over the imaginary axis [140]. This modification is used in the decoupling algorithm for the unstable subsystems.
Remark 7. The proposed decoupling method is an appealing tool for subsystem identification.
In this case based on a preliminary model specific input and output transformations are synthe-sized to excite only a certain part of the dynamics and to reduce the identification problem to a SISO one. This way the approach scales down the complexity of the identification problem, by reducing the dimension of the unknown parameter vector. The effect of the neglected dynam-ics can be characterized by an additional disturbance term. Preliminary results of decoupled parameter identification are discussed in [TB91]. For less accurate preliminary models robust decoupling transformations can be synthesized, following the methods in Chapter 4.
It is important to review the achievable theoretical bounds on the decoupling performance.
Consider the maximization of the minimal sensitivity for the transfer corresponding to the controlled subsystem. The decoupling is carried out by one-norm blend vectors ku and kTy. By applying matrix norm identities on an arbitrary transfer function matrix Gny×nu, one gets:
√1nu||G||i∞≤σ(¯ G), (3.14) where||·||i∞is the maximal row sum. This shows that with a normalized blending, the maximum achievable singular value is bounded from above. Consequently by applying one-norm input and output blends, no higher H− index can be achieved than the maximum singular value of the given subsystem.
Concerning the suppression of the subsystem to be decoupled, the highest suppression rate can be achieved if the blended inputs or outputs are as close as possible to the corresponding null spaces. A blending approach proposed by [98] carries out decoupling based on a null space transformation. This approach is certainly valid, however, the existence of null space and its sensitivity for model parameters and changes might hinder its application for real problems.
The system consists of two stable modes (subsystems), where the first complex one should be controlled, and the real one should be decoupled. The controllability and observability properties of the modes can be quantified by the eigenvalues of their controllability and observability Gramians. The controllability Gramian denoted by W is calculated as the positive definite solution of the following Lyapunov equation:
AW +W AT +BBT = 0. (3.16)
The smallest eigenvalue of the controllability Gramian is a worst-case measure of the systems controllability, and it is inversely related to the amount of energy required to move the system’s state vector to the direction which is the most difficult to control [29]. If the system is not con-trollable, then the corresponding minimum eigenvalue is zero, which would require an infinitely large control effort to move the system in this direction of the state-space. In this example the Gramians corresponding to the subsystems {·}cand {·}d, have the eigenvalues
λc(Wc) =
0.4096 0.5904
, andλd(Wd) = 0.3714. (3.17) The observability GramianV is similarly defined as the positive definite solution of
ATV +V A+CTC= 0. (3.18)
Its eigenvalues are inversely proportional to the observation energy of the system. For the given subsystems the observability Gramians have eigenvalues as
λc(V) =
0.9209 1.2916
, λd(V) = 0.5179. (3.19)
The blend vectors ku and ky are calculated based on Sections 3.3.1 and 3.3.2. The fre-quency interval where the decoupling should be achieved was selected to be between 0 and ωn rad/s, where the latter stands for the natural frequency of the mode to be controlled, i.e., in (2.18)
¯ω = 0 and ¯ω = ωn. The blend vectors are kuT =
−0.9018 −0.1167 −0.4160T
and kTy =
−0.6921 0.7218T
, respectively. Figure 3.6 shows the maximum singular values of the subsystems, which are corresponding to the highest achievable sensitivity by suitable blends ac-cording to (3.14). Note that the subsystem corresponding to the undesired dynamics has higher steady state gain than the one to be controlled. As the lower subfigure shows after applying the input and output blends, this theoretically maximal sensitivity was retained, while the transfer through the other (undesired) mode was significantly reduced.
Of course the decoupling has its own price, which can be revealed by calculating the eigenval-ues of the controllability and observability Gramians corresponding to the blended subsystems.
These are found to be
λc(fWc) =
0.3179 0.5214
, andλd(Wfd) = 0.0192, λc(Vec) =
0.6887 1.1298
, andλd(Ved) = 0.0033, (3.20) where {e·}denotes that they are the Gramians corresponding to the blended subsystems. The worst-case change in the controllability Gramians can be quantified by
1−λc,min(Wfc)
λc,min(Wc) ≈0.22, and 1−λd,min(fWd)
λd,min(Wd) ≈0.95, (3.21)
that is the smallest eigenvalue of the Gramian corresponding to the controlled subsystem has been reduced by approximately 22%, while for the decoupled one it has been shrunk by 95%.
This immediately shows that an applied control action will be less effective on the undesired dynamical part of the system if the synthesized input and output transformations are in use.
However, for this one has to sacrifice a certain amount of controllability of the mode that should
CHAPTER 3. DECOUPLING OF LTI SYSTEMS T. Ba´ar
−40
−20 0
magnitude[dB]
¯
σ(Gc(s))
¯
σ(Gd(s))
0.01 0.1 1 10 100
−40
−20 0
frequency [rad/s]
magnitude[dB]
kyTGc(s)ku kyTGd(s)ku
Figure 3.6: Above: The maximum singular values of the subsystems before blending. Below:
The singular values of the blended SISO subsystems.
−0.7 −0.65 −0.6 −0.55 −0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0.2
0.4
infeasible region
β
γ
Pareto front max: β s.t.: β < min: γ s.t.: γ > achieved optimum
Figure 3.7: Pareto front for the output blend calculation of the academic example be controlled. Similarly the undesired dynamics are made unobservable, and so their effects are suppressed in the blended measurements. This can be achieved on the expense of reducing the observability of the controlled mode also. This is closely related to the game-theoretic approach discussed during the input blend design.
This game-theoretic interpretation of the solution has also been calculated for the academic example and summarized in Figure 3.7 for better understanding the trade-off. The yellow curve represents the Pareto-front computed by using theϵ-constrained method as in (3.8) with different values of ϵ. These are the Pareto-efficient solutions. The blue and the red curves refer to the corresponding single objective optimization problems. In these only one of the two objectives has been considered and solved for different values of ϵ, without any information about the
”other” subsystem. The resulting blends are then applied for the complementary subsystems to see the achieved effects. These are the solutions for the non-cooperative games, where the two players are aiming to minimize their respective costs only [104]. Notice that theβ maximization can achieve a high minimum sensitivity (≈ 0.95), however, the corresponding γ value is also high (≈10). On the other hand, minimizing only γ can lead to small values, but the resulting blend matrix also decreases the β significantly. Lastly, the solution to the optimization in (3.7) is denoted by the red star. To evaluate this result, the contour plots of the scalarized objective function−β2+γ2 are also given in Figure 3.7. Along each line, the cost is constant, while they
0 20 40 60 80 100 120 140 160 180 200 220 0
20 40 60 80
supression rate [dB]
numberofmodels
Figure 3.8: Batch test: the distribution of the achieved suppression rates
increase from the bottom left to the upper right corner. Accordingly, one can depict that the red star has actually the minimal cost with the highestβ value among the Pareto-efficient solutions.