4.4 Robustness Margins
4.4.2 Robustness in Mixed Sensitivity Design
This chapter closes with a brief discussion of the inherent robustness properties of the mixed sensitivity design which was proposed in Section 4.2.3. As robustness is solely determined by feedback, it suffices to consider the single-degree-of-freedom interconnection of Figure 4.8a which is repeated here as Figure 4.14 for convenience.
P Pd
WuDu−1
C WeD−e1
De Dd
w1 z1 z2 w2
−
Figure 4.14:Generalized closed-loop for single-degree-of-freedom mixed sensitivity design.
As the generalized closed-loop interconnection is guaranteed to satisfy a norm bound kGCLk< γ, the relation to robustness analysis is immediate: Uncertainties can be included to close the fictitious loops with
w1
w2
=∆11 ∆12
∆21 ∆22
| {z }
∆
z1
z2
. (4.29)
According to the small-gain theorem [e. g., Zhou et al. 1995, Theorem 9.1, p. 212, origi-nally formulated by Desoer & Vidyasagar 1975, Sec. III.2, pp. 40], the interconnection is guaranteed to be stable as long as γk∆k < 1. Figure 4.15 shows the corresponding generalized closed-loop interconnection, resulting in an “allowable” uncertain plantP∆. In particular, the original plant model can be perturbed with an additive uncertainty (PdDd∆22+De∆12)WuD−1u . Such uncertainty is representative of unmodeled high fre-quency dynamics, as well as uncertain non-minimum phase zeros, including a change in the number of non-minimum phase zeros [Zhou et al. 1995, Sec 9.3.4]. Similarly, the allowable inverse multiplicative uncertainty (PdDd∆21+De∆11)WeD−1e is representative of low-frequency parameter errors as well as uncertain unstable poles including a change in the number of unstable poles [ibid.]. While this demonstrates “general robustness”, not much insight is gained due to the complexity of the involved expressions.
A more insightful interpretation can be provided if instead load disturbances are consid-ered (Pd=P andDd=Du). In this case, the “allowable uncertainty” shown in Figure 4.16 is obtained. The uncertainty ∆ is allowed to be fully populated, so any structured un-certainty is included. Fixing ∆11= diag(δ1, . . . , δny) and ∆22= diag(δny+1, . . . , δny+nu) to diagonal uncertainties and ∆21= ∆12= 0 as is done in the margin tests which were previously discussed is hence permissible. That is, the loop is also guaranteed to be stable
P
C −
(PdDd∆22+De∆12)WuD−1u
(PdDd∆21+De∆11)WeD−1e P∆
Figure 4.15:Stable closed-loop with mixed-sensitivity controller and unstructured uncertainty
∆ =∆11∆12
∆21∆22
withk∆k
<1/γ.
P
C −
Du∆22WuDu−1
De∆12WuD−1u
Du∆21WeDe−1
De∆11WeD−1e P∆
Figure 4.16: Stable closed-loop with mixed-sensitivity controller, Pd = P, Dd = Du, and unstructured uncertainty ∆ =∆11∆12
∆21∆22
withk∆k
<1/γ. for
γ
δ1
...
δny+nu
<1⇔γkδik<1 i= 1, . . . , ny+nu (4.30) In this case, owing to their diagonal structure,Du, Wu, De, andWeall commute with the uncertainties such that the scalingsDuand Dedrop out, i. e.,
Du∆22WuD−1u = diag(δny+1, . . . , δny+nu)Wu, De∆11WeD−1e = diag(δ1, . . . , δny)We.
Assume the standard weights Wu andWeare used as proposed in Section 4.2.3, i. e.,Wu
has, in each channel, at least unit gain andWe has, in each channel, at least a gain of 0.5.
Then, the closed-loop (shown in Figure 4.17) can tolerate multiplicative perturbations with a size of1/γ in each input channel, and inverse multiplicative perturbations of size1/2γ in
each output channel. It should be noted thatγ >1 in this case, asγ≥ kGCLk>kTik ≥1.
Using Equations (4.26) and (4.27), the closed-loop system has, for each channel, guaranteed margins
GMoutputlower = 2γ
2γ+ 1, GMoutputupper = 2γ
2γ−1, PMoutput=±2 arcsin 1 4γ
(4.31)
GMinputlower = 1− 1
γ, GMinputupper= 1 +1
γ, PMinput=±2 arcsin 1 2γ
. (4.32) For example, a performance index γ = 2 guarantees an upper output gain margin of 2.5 dB, an upper input gain margin of 3.5 dB, and phase margins of 14◦at the output and 29◦ at the input. These bounds are conservative, such that dedicated robustness tests as discussed in the previous section can provide better results and should always be carried out after a controller was designed. The existence of guaranteed margins nevertheless demonstrates the inherent robustness of the proposed mixed sensitivity design and is in stark contrast, e. g., to LQG control which has no robustness guarantees [Doyle 1978].
P
C −
1 γ
δny+1
...
δny+nu
1
2γ
δ1
...
δny
P∆
Figure 4.17:Stable closed loop with mixed-sensitivity controller,Pd=P,Dd=Du, and scalar uncertaintiesδiwithkδik<1 i= 1, . . . , ny+nu.
It should further be noted that the guaranteed input margins are related to the assumption of disturbances occurring at the plant input [cf. Doyle & Stein 1979]. That is, they can only be guaranteed when the disturbance model equals the plant model. For a general disturbance model, it is nevertheless possible to derive guaranteed input margins by shifting the uncertainty to the plant input [e. g. Skogestad & Postlethwaite 2005, p. 295].
Considering again ∆21= ∆12= 0, structured diagonal uncertainties ∆11,∆22, and (for simplicity)Dd=Du, the additive uncertainty
PdDd∆22WuD−1u =Pd diag(δny+1, . . . , δny+nu)Wu
can be expressed as a multiplicative input uncertainty P†Pd diag(δny+1, . . . , δny+nu)Wu.
Thus, at every frequency, an additional factor 1/σmax(P†Pd) appears which reduces the guaranteed admissible uncertainty. The guarantees of the previously discussed case are, however, asymptotically recovered asPd is selected to resembleP. Thus, the disturbance model should include dominant high-gain dynamics ofP such as resonances in order to achieve robustness.
5 Application Examples
This chapter provides detailed application examples for the techniques which where devel-oped in this thesis. First, the applicability and effectiveness of the proposed model order reduction methods of Chapter 3 are demonstrated on two different high-fidelity industry-grade models of unmanned aeroservoelastic aircraft. Further, a large-scale aerodynamics model with over 20000 states is considered, both as a proof-of-concept and to demonstrate the versatility of the proposed oblique projection technique. Finally, two comprehensive design studies for aeroservoelastic control applications are provided. The first study is concerned with active flutter suppression on a small unmanned aircraft and envelope expansion into the naturally unstable flight regime. The second study introduces a flight control design concept for a large aircraft that integrates structural damping augmentation.
These design studies substantiate the systematic design guidelines of Chapter 4. Parts of this chapter were published as research papers by Theis, Takarics, Pfifer, Balas & Werner [2015c], Theis, Seiler & Werner [2018], Theis, Pfifer & Seiler [2016a], and Theis, Pfifer, Balas & Werner [2015a].
5.1 Model Order Reduction for Aeroservoelastic Aircraft
Design of flight control systems for aeroservoelastic unmanned aircraft requires accurate, yet low-order models. The modal interpolation method is demonstrated on a high fidelity longitudinal dynamics model of the Body Freedom Flutter Vehicle. The Body Freedom Flutter Vehicle is a high aspect ratio flying wing with a span of 3 m that was deliberately designed by Lockheed Martin and the U. S. Air Force Research Laboratory to flutter at very low airspeed [Burnett et al. 2010, Holm-Hansen et al. 2010]. Lockheed Martin and the Air Force Research Laboratory also developed a larger flying wing aircraft with a span of 8,5 m, designated the X56A multi utility technology testbed (MUTT). The X56A is currently used as a research platform by NASA [Ryan et al. 2014]. The oblique projection method is demonstrated on a model of this aircraft. For both of these MIMO systems, the LMI-based state-of-the-art methods fail to produce a reduced-order model.