FOR LTI SYSTEMS
vik:=|Wd,i(jωk)|,i= 1, ..., nω. Minimize γk subject to vik+γk ≥ |wu0,N+1,i(jωk, θN+1k)|
|zu,N+1,i(jωk, θN+1k)| , i= 1, ..., τ, vτ+i k+γk≥ |d0,N+1,i(jωk, θN+1k)|, i= 1, ..., nd,
If for all k= 1, ..., nω γk ≤0 holds, then, the algorithm is finished with controller KN uncertainty model W(ΘN) and performance level ¯γs(KN). Otherwise, let W(ΘN+1) be constructed by the over-bounding method presented in Step B2 of Algorithm 1 for points vik+ max(0, γk). Let EN+1=ENS
EN+1,N =N+ 1, and continue from Step B.
4.5. SOLUTIONS WITH CONVEX OPTIMIZATION
Define real diagonal matrices
Vk= diag{|W∆(jωk)|,|Wd(jωk)|} ∈Rnwu+nd×nwu+nd, (4.14) diagonal of complex vectors
Alk= diag
au,lk,1
|zu,l,1(jωk)|, ..., au,lk,τ
|zu,l,τ(jωk)|, ad,lk,1, ..., ad,lk,nd
∈Cnwu+nd×τ+nd, and diagonal of complex matrices
Blk= diag
Bu,1(jωk)
|zu,l,1(jωk)|, ..., Bu,τ(jωk)
|zu,l,τ(jωk)|, Bd,1(jωk), ..., Bd,nd(jωk)
∈Cnwu+nd×nθ(τ+nd), and let Θlk = diag{θlk, ..., θlk} ∈ Cnθ(τ+nd)×τ+nd be the τ +nd times repeated block-diagonal matrix. The consistency conditions formulated in Theorem 4.1 are provided for the case of additive uncertainty structure of Figure 4.4 in the following theorem.
Theorem 4.3 Given stable and stable invertible weighting functions W∆ and Wd as defined in Section 4.1.1 and given set EN of measurement data. Then, for every exper-iment l= 1, ..., N there exists a perturbation ∆l∈BS∆u and a disturbance dl ∈BLn2d
that satisfy consistency condition (4.4) for all k and l, if and only if there exist θlk k= 1, ..., nω and l= 1, ..., N such that
Vk2 (Alk+BlkΘlk)∗ Alk+BlkΘlk Inwu+nd
≥0 (4.15)
In Step B2 of Algorithm 1, constraints (4.12) and (4.13) should be replaced by (4.15).
Note that Vk defined by (4.14) is nothing but
Vk = diag{v1kIwu,1, ..., vτ kIwu,τ, vτ+1, ..., vτ+nd}, (4.16) where vik >0 are the candidate magnitudes of the uncertainty weighting functions at frequency ωk. Recall that VI,k = diag{Vk, Ir} has already been defined in Step B2 of Section 4.4. Criterion (4.11) can always be rewritten to the complex LMI of the form
Inw
γ2kInd+nr
VI,k2 VI,k2 MD(jωk)∗ MD(jωk)VI,k2 I
>0 (4.17)
by applying Schur complement. Minimization of γk2 subject to constraints (4.15) and (4.17) is a generalized eigenvalue problem in variables θlk,vik2 and γk2.
Step D of Algorithm 1 (update of the uncertainty model) gives also a convex problem.
The consistency of modelW(ΘN) withEN+1 is checked as follows. Solve the following optimization problem point-wise for each ωk. Let vik := |W∆,i(jωk)|, i = 1, ..., τ and vik :=|Wd,i(jωk)|, i= 1, ..., nd. Minimize γk subject to
Vk2+γk2I (AN+1k+BN+1kΘN+1k)∗ AN+1k+BN+1kΘN+1k Inwu+nd
≥0
FOR LTI SYSTEMS
Finally, we can conceive theoretical conditions for the model based upper-bound
¯
γs(K)being equal to the true performance level ¯γ(K). For avoiding exactness problems due to sampling, and for simplifying notation, the conditions are stated in terms of continuous-frequency representation. Suppose that experimental data consist of input-output pairs(y(jω), u(jω)), ω∈R, of the worst-case experiment
Ewc={(y(jω), u(jω))|(T,d,¯r) = arg¯ sup
T ∈ST,d∈S¯ d¯,¯r∈BL2
kz¯pk2 } i.e. the measured performance of the worst-case experiment equals toγ¯(K).
Theorem 4.4 Given a controller K and closed-loop experimental data set El = Ewc
generated by K. Suppose, the structured uncertainty has at most two blocks: ∆uS∆u
withτ ≤2, and the uncertainty structure is additive, i.e.,U11= 0 andU12= 0. Suppose that Assumption 4.2 holds. Then
¯
γ(K) = max
ω γ(ω)
where γ(ω) is the optimum of the following minimization problem: Minimizeγ(ω) sub-ject to(4.17)and (4.15), where all variables and transfer matrices are taken atωinstead of ωk.
Proof. Fromτ ≤2it follows that extended perturbation block∆ain the robust perfor-mance problem has at most three complex blocks, which implies that µ upper-bound is exact: infDσ(D¯ LM DR−1) = µ∆a(M). Assumption 4.2 ensures that zp = ¯zp which implies that the calculated boundmaxωγ(ω)is at least ¯γ(K), i.e. it is an upper-bound.
However, this upper-bound is exact, since the additive structure allows a convex char-acterization of the optimization problem, for this reason there existΘl andvi such that kzpk2 is the worst-case performance of the model.
A simple illustrative example for the D-K-W iteration with standard (non-skew) µ-synthesis is presented in Appendix A.
Remarks on Algorithm 1
On model validation methods
The measurement data define a set of consistency constraints on the uncertainty model.
Time-domain model validation approaches, e.g. [108] and [124], have the drawback that they cannot handle long data series. This is because in case of dynamic uncertainty, the number of variables is at least as high as the length of the experiment. Frequency-domain methods [123, 25, 28] and [79] have the advantage that the problem reduces to a series of small tractable feasibility problems. The number of variables increases linearly with the number of experiments.
A further problem arises from the structure of the uncertainty model. Structured uncertainty model in LFT interconnection leads to a model validation problem, which is a NP-hard problem in general. There are, however, many approaches for computing
4.5. SOLUTIONS WITH CONVEX OPTIMIZATION approximate solutions in the model validation context. Methods based on S-procedure [124, 98], scaling-approach [28] and skewµapproach [30, 73, 95] are some examples. In the latter approach, the model validation problem is reformulated as a skew µanalysis problem and there is no direct connection to the skew µsynthesis of the dissertation.
Concerning control synthesis, the available robust design methods have their limits in case of too sophisticated uncertainty descriptions. The upper bounds on the SSV, in case of mixed parametric and dynamic uncertainties, are not necessarily tight [45, 89].
Based on the above discussion, the perturbation model in the dissertation is re-stricted to contain only LTI components. The special case of η(θ) =η is particularly advantageous in computational point of view, because the algorithm consists of only convex optimization steps subject to LMI constraints.
Implementation
The controller synthesis step and the search for the scalings can be implemented by the minor modification of the existing routines in the Robust Control Toolbox for Matlab [12] designed for D-K iteration. The computational burden of steps D and K is the same as in the µ-synthesis. Step W requires approximately the same complexity of calculations as step D.
We note that it is numerically more advantageous to introduce a variable for the last block ofDLandDR, and the search for scalings is followed by a normalization step.
Generalization to other class of systems
Frequency-domain consistency constraints are applicable in case of LTI perturbation models. Nominal model G23, however, does not play a role in the constraints, it can be replaced, therefore, by LPV or nonlinear models. The only requirement is that a robust controller can be designed for the structured uncertainty model. A formulation of the algorithm for LPV nominal models is presented in Chapter 5.
For the sake of simplicity the plant is restricted to be stable in the present work.
This assumption is necessary for the used frequency-domain model validation method.
The proposed scheme can be extended, however, to unstable systems along the idea presented in [54]. The extension can be the subject of future research.
Convergence of iterations
Standard D-K iteration is not guaranteed to converge to the global optimum, but it has been shown to work on practical problems with LTI perturbations [99]. The same can be said about the presented D-K-W iteration [RGB09, RGB10, RG10]. Regarding the whole iterative algorithm, the uncertainty set increases with every new falsifying experimental data. In the presented applications (Sections 4.6, 5.6 and Chapter 6) the algorithm stopped after 2-3 iterations.
FOR LTI SYSTEMS