Parametric Robust Structured Control Design

Apkarian, Pierre; Dao, Minh N.; Noll, Dominikus

doi:10.1109/tac.2015.2396644

Cited by 121 publications

(103 citation statements)

References 42 publications

(81 reference statements)

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“…The solution δ * ∈ ∆ of (1.3) represents a worst possible choice of the parameters δ ∈ ∆, which may be an important element in analyzing performance and robustness of the system, see e.g. [1].Our second criterion is similar in nature, as it allows to verify whether the uncertain system (1.1) is robustly stable over a given parameter range ∆. This can be tested by maximizing the spectral abscissa of the system A-matrix over the parameter range…”

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confidence: 99%

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Computing the structured distance to instability

Apkarian¹,

Noll²,

Ravanbod³

2015

2015 Proceedings of the Conference on Control and Its Applications

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We analyze robust stability and performance of dynamical systems with real uncertain parameters. We compute criteria like the distance to instability, the worst-case spectral abscissa, or the worst-case H∞-norm, which quantify the degree of robustness of such a system when parameters vary in a given set ∆. As computing these indices is NP-hard, we present a heuristic which finds good lower bounds fast and reliably. Posterior certification is then obtained by an intelligent global strategy. A test bench of 87 systems with up to 70 states 39 uncertain parameters with up to 11 repetitions demonstrates the potential of our approach.1 Problem Specification.Robustness specifications limit the loss of performance and stability in a system where differences between the mathematical model and reality crop up. Robustness against real parametric uncertainties is among the most challenging calls in this regard. Already deciding whether a given system with uncertain real parameters δ is robustly stable over a given parameter range δ ∈ ∆ is NP-hard, and this is aggravated when it comes to deciding whether a given level of H 2 -or H ∞ -performance is guaranteed over that range. In this work we address this type of uncertainty by computing three key criteria, which quantify the degree of parametric robustness of a system. These are (a) the worst-case H ∞ -norm, and (b) the worst-case spectral abscissa over a given parameter range, and (c) the distance to instability, or stability margin, of a system with uncertain parameters.Consider a Linear Fractional Transform [23] with real parametric uncertainties as in Figure 1, whereand x ∈ R n is the state, w ∈ R m1 a vector of exogenous inputs, and z ∈ R p1 a vector of regulated outputs. The uncertainty channel is defined as p = ∆q, where the time-invariant uncertain matrix ∆ has the blockdiagonal form2) * ONERA, Control System Department, Toulouse, France † Université de Toulouse, Institut de Mathématiques with δ 1 , . . . , δ m representing real uncertain parameters, and r i giving the number of repetitions of δ i . Here we assume without loss that δ = 0 ∈ ∆ represents the nominal parameter value, and we consider δ ∈ ∆ in one-to-one correspondence with the matrix ∆ in (1.2). For practical applications it is generally sufficient to consider the caseTo analyze the performance of (1.1) in the presence of the uncertain δ ∈ R m we compute the worst-casewhere · ∞ is the H ∞ -norm, and where T wz (s, δ) is the transfer function z(s) = F u (P (s), ∆)w(s), obtained by closing the loop between (1.1) and p = ∆q with (1.2) in Figure 1. The solution δ * ∈ ∆ of (1.3) represents a worst possible choice of the parameters δ ∈ ∆, which may be an important element in analyzing performance and robustness of the system, see e.g. [1].Our second criterion is similar in nature, as it allows to verify whether the uncertain system (1.1) is robustly stable over a given parameter range ∆. This can be tested by maximizing the spectral abscissa of the system A-matrix over the parameter range−1 C q , and w...

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confidence: 99%

mentioning

confidence: 99%

Computing the structured distance to instability

Apkarian¹,

Noll²,

Ravanbod³

2015

2015 Proceedings of the Conference on Control and Its Applications

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“…In order to design a control law that accommodates the requirements given in the previous section, the synthesis problem is recast into the

H_{\infty} false/ μ

framework. For the necessary mathematical background related to

H_{\infty} false/ μ

concepts, including the recent nonsmooth approaches, the reader is invited to refer to the nonexhaustive list of references . In this context, the synthesis problem is turned into an optimization problem of the worst‐case

L_{2}

induced norm between different closed‐loop signals scaled by so‐called weighting functions.…”

Section: Performance Objectives and Controller Designmentioning

confidence: 99%

Robust microvibration mitigation and pointing performance analysis for high stability spacecraft

Preda

Cieslak

Henry

et al. 2018

Intl J Robust & Nonlinear

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Summary This paper deals with the development of a mixed active‐passive microvibration mitigation solution capable of attenuating the transmitted vibrations generated by reaction wheels to a satellite structure. A dedicated simulation environment, provided by the European Space Agency and Airbus Defence and Space industries, serves as a support for testing the proposed solution at satellite level. This paper covers modeling, control system design, and worst‐case analysis for a typical satellite observation mission that requires high pointing stability. Combined with a novel disturbance model for the reaction wheel perturbations, the pointing performance and stability requirements are reformulated as bounds on the worst‐case L2 system gains. Subsequently, the active microvibration controller is tuned to manage the conflicting design goals and optimize different trade‐offs between robustness and performance. Finally, robust stability margins and worst‐case performance bounds with respect to various system uncertainties, time‐varying reaction wheel spin rates, actuator saturation, and time delays are obtained using the structured singular value, integral quadratic constraints, and time‐domain nonlinear simulations.

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“…As in , bundling is required in response to the nonsmoothness of the criteria, but for this to take effect, we first have to deal with the nonsmooth constraints

\overset{σ}{false} ({normalΔ}_{j}) \leq 1

. We propose to reduce them to simpler box‐constraints by a change of variables.…”

Section: Approachmentioning

confidence: 99%

“…For the proof, we refer to . For the spectral abscissa, the situation is more complicated, as α may in general even fail to be locally Lipschitz .…”

Section: Convergencementioning

confidence: 99%

Worst‐case stability and performance with mixed parametric and dynamic uncertainties

Apkarian

Noll

2016

Intl J Robust & Nonlinear

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Summary This work deals with computing the worst‐case stability and the worst‐case H∞ performance of linear time‐invariant systems subject to mixed real‐parametric and complex‐dynamic uncertainties in a compact parameter set. Our novel algorithmic approach is tailored to the properties of the nonsmooth worst‐case functions associated with stability and performance, and this leads to a fast and reliable optimization method, which finds good lower bounds of μ. We justify our approach theoretically by proving a local convergence certificate. Because computing μ is known to be NP‐hard, our technique should be used in tandem with a classical μ upper bound to assess global optimality. Extensive testing indicates that the technique is practically attractive. Copyright © 2016 John Wiley & Sons, Ltd.

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Parametric Robust Structured Control Design

Cited by 121 publications

References 42 publications

Computing the structured distance to instability

Computing the structured distance to instability

Robust microvibration mitigation and pointing performance analysis for high stability spacecraft

Worst‐case stability and performance with mixed parametric and dynamic uncertainties

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