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Parametric robust structured control design

P. Apkarian, M. N. Dao, D. Noll

arXiv:1405.4202v1math.OC

TL;DR

Parametric robust synthesis is difficult because it involves nonconvex, nonsmooth design and NP-hard analysis problems. The paper develops a nonsmooth method using dynamically generated inner approximations and structured controllers. Its test bench and missile case study indicate that the method can address challenging designs with parametric uncertainty.

  • Problem

    Parametric robust synthesis is challenging because each analysis evaluation is NP-hard and the underlying design program is nonconvex and nonsmooth.

    Source: §I · p. 1

  • Method

    The method combines nonsmooth optimization tailored to semi-infinite minima with dynamically generated inner approximations of the uncertain-parameter set.

    Source: §I · p. 2, §VII · p. 22

  • Results

    The test bench reports better worst-case performance on a larger uncertainty box with simpler controllers than DKSYN, while remaining competitive in execution time.

    Source: A. Algorithm testing · p. 17

  • Takeaways & Limitations

    The test bench and case study indicate that the approach is a practical tool for challenging structured-control designs with parametric uncertainty.

    Source: §VII · p. 22

Abstract

from arXiv · show

We present a new approach to parametric robust controller design, where we compute controllers of arbitrary order and structure which minimize the worst-case $H_\infty$ norm over a pre-specified set of uncertain parameters. At the core of our method is a nonsmooth minimization method tailored to functions which are semi-infinite minima of smooth functions. A rich test bench and a more detailed example illustrate the potential of the technique, which can deal with complex problems involving multiple possibly repeated uncertain parameters.

I. Introduction

Parametric robust synthesis is especially difficult because each analysis evaluation is already NP-hard, while the underlying design program is nonconvex and nonsmooth. The paper introduces a nonsmooth synthesis approach using dynamically generated inner approximations, alongside worst-case analysis and convergence results.

I. Introduction

  • Parametric robust synthesis is harder than robustness analysis because each synthesis step involves an already NP-hard analysis problem, and its design program is nonconvex and nonsmooth.Existing synthesis tools may be conservative and produce controllers that are too complicated for practice.Source: §I · p. 1
  • Dynamic inner approximation iteratively builds a smaller scenario set for structured-controller design and can be paired with outer analysis for a posteriori robustness and performance certificates.The approach aims to avoid or reduce the lack of immediate certificates associated with inner approximations.Source: §I · p. 2
  • The paper presents a nonsmooth method for semi-infinite minimum objectives and reports rigorous worst-case analysis and convergence results for a new resolution algorithm.Global-minimum convergence is not algorithmically feasible because the problems are NP-hard; the stated convergence guarantee is to local minima from arbitrary starting points.Source: §I · p. 2

Notation

  • The notation defines the Clarke subdifferential for locally Lipschitz functions and uses Fl and Fu for lower and upper linear fractional transformations.For Hermitian matrices, ≻0 denotes positive definiteness and ⪰0 denotes positive semidefiniteness.Source: Notation · p. 3

A. Setup

  • The setup models real uncertain parameters, including repeated parameters, and seeks a structured output-feedback controller that is robustly stable and optimizes performance over a compact convex uncertainty set.The controller structure is parameterized, with examples including PIDs, reduced-order controllers, and observer-based controllers.Source: A. Setup · pp. 3–4

B. Dynamic inner approximation

Static gridding can impose a high computational burden and become intractable as the number of uncertain parameters grows. Dynamic inner approximation instead updates a smaller set of active scenarios, while the overall approach uses locally optimal solutions and seeks global certificates afterward.

B. Dynamic inner approximation

  • Straightforward static gridding becomes intractable for sizable uncertainty dimension and can burden multi-model H∞ optimization with many scenarios.The resulting computational burden makes the approach prone to failure.Source: B. Dynamic inner approximation · p. 5
  • Dynamic inner approximation generates a substantially smaller scenario set and updates its active scenarios by searching for problematic parameter configurations.The authors describe the procedure as rapidly converging and less prone to failure than static gridding.Source: B. Dynamic inner approximation · p. 5
  • The method relaxes robust stability and performance through locally optimal solutions, with a global certificate sought only in the final analysis step.The synthesis and search programs are handled using nonsmooth optimization; the final certificate is not supplied by the local optimization alone.Source: B. Dynamic inner approximation · p. 5

III. Nonsmooth min-max versus min-min programs

The algorithm distinguishes familiar min-max optimization over controller parameters from the min-min programs that arise when optimizing over uncertain parameters. Its steps alternate controller synthesis with searches for destabilizing or performance-degrading scenarios, then check robustness and distance to failure.

A. Classification of the programs in Algorithm 1

  • The algorithm alternates structured multi-model controller synthesis with searches for destabilizing and performance-degrading uncertainty scenarios.New scenarios are added to the active set, and step 6 checks distance to instability and performance degradation before termination.Source: A. Classification of the programs in… · p. 6
  • Programs (7) and (8) become min-min problems when written as minimizations because of the minus sign.The H∞ program in step 2 is instead min-max, with a semi-infinite maximum over frequency.Source: A. Classification of the programs in… · p. 6
  • Minimizing upper-C1 min-min objectives is expected to move iterates away from nonsmoothness, while lower-C1 min-max objectives call for nonconvex bundle methods.The paper proposes handling the former much like smooth nonlinear programs and the latter with tailored methods.Source: A. Classification of the programs in… · p. 7
  • Distance-to-instability and distance-to-performance-boundary searches also have min-min structure, while well-posedness can be checked during optimization or in the final certificate.The authors report that a posterior certificate can cover well-posedness, so checking it at every step may not be necessary.Source: A. Classification of the programs in… · pp. 5, 7, 8

B. Highlighting the difference between min-max and min-min

The section contrasts semi-infinite min-max programs with min-min programs, whose disjunctive structure remains even when there are finitely many branches. For local optimization, the authors propose following one active branch at nonsmooth iterates; global optimization, by contrast, can incur combinatorial explosion.

B. Highlighting the difference between min-max and min-min

  • Semi-infinite min-max programs require tailored methods, a difficulty illustrated by the H∞ program in step 2.When the index set is finite, a min-max program can be rewritten as a classical nonlinear program using an additional variable.Source: B. Highlighting the difference between… · p. 8
  • Min-min programs retain disjunctive structure even with finitely many branches, making global optimization prone to combinatorial explosion.The section introduces active indices and branches to describe which functions attain the inner minimum or maximum.Source: B. Highlighting the difference between… · p. 8
  • At a nonsmooth iterate with several active branches, a local min-min search can select one branch and continue as if the objective were smooth.The authors say their later analysis supports this intuition and that experiments show good results with a good heuristic.Source: B. Highlighting the difference between… · p. 8

IV. Computing subgradients

The section develops subgradient computation for the H∞ norm and spectral abscissa. For the H∞ norm, the authors characterize Clarke subgradients using active frequencies and singular vectors; they also establish regularity properties of the objective.

A. Case of the H∞-norm

  • Closing the uncertainty loop yields the transfer function used to compute h− subgradients, and the paper notes that simple, single-frequency activity gives differentiability.For that case, the gradient uses the unique left and right singular vectors associated with the maximum singular value.Source: A. Case of the H∞-norm · pp. 9–10
  • The H∞ objective h− is Clarke subdifferentiable throughout its internally stable domain, with its subdifferential formed from active frequencies and associated singular vectors.The matrices Yω are Hermitian weights, while Qω and Pω contain left and right singular vectors for the maximum singular value.Source: A. Case of the H∞-norm · pp. 9–10
  • The H∞ norm is lower-C1 on the internally stable domain, so its negative h− is upper-C1 there.The proof represents the norm using a compact index set involving frequency and unit left and right singular vectors.Source: A. Case of the H∞-norm · p. 10

B. Case of the spectral abscissa

The spectral-abscissa objective is less regular than the H∞ objective: it is not locally Lipschitz everywhere. The section gives subdifferential results under conditions on active eigenvalues and notes a local regularity result when those eigenvalues are simple.

B. Case of the spectral abscissa

  • The spectral abscissa is not locally Lipschitz everywhere, and derogatory active eigenvalues may cause this failure.The section identifies an eigenvalue as active when its real part equals the spectral abscissa.Source: B. Case of the spectral abscissa · pp. 10–11
  • When all active eigenvalues are semi-simple, a− is Clarke subdifferentiable nearby, with its subdifferential built from their eigenvectors and positive-semidefinite weights.The weights across active eigenvalues have traces summing to one.Source: B. Case of the spectral abscissa · p. 11
  • For structured uncertainty, subgradient components can be computed as sums of inner products over active eigenvalues.The corresponding selection of rows and columns is determined by the structure of Δ.Source: B. Case of the spectral abscissa · p. 11
  • When every active eigenvalue is simple, a− is upper-C1 in a neighborhood.In this case, each weight Yi reduces to a scalar, enabling a fast implementation using LU decomposition.Source: B. Case of the spectral abscissa · p. 11

V. Algorithm for min-min programs

The section presents a descent algorithm for min-min programs that stays close to smooth optimization while addressing their upper nonsmoothness.

V. Algorithm for min-min programs

  • The descent algorithm targets the min-min programs (7) and (8), starting from a feasible initial guess and stopping when the current iterate is a KKT point.The algorithm then enters an inner loop to find the next serious iterate.Source: §V · pp. 11–12
  • The method is designed to retain the structure of standard smooth optimization while ensuring convergence for min-min programs with upper nonsmoothness.It uses a standard model based on the Clarke directional derivative as a substitute for a first-order model.Source: §V · p. 11
  • Its step-finding subroutine uses lower approximations of the objective to construct a working model.The supplied passages identify this model and subroutine but do not give enough detail to reconstruct every step.Source: §V · p. 12

A. Practical aspects of Algorithm 2

The practical subroutine often reduces to a standard backtracking line search, but uses cutting and aggregate planes when a strict model requires additional subgradients.

A. Practical aspects of Algorithm 2

  • When the objective is certified upper-C1, the subroutine reduces to a projected-gradient line search, with the acceptance test becoming the usual Armijo test.In this case, cutting-plane and aggregate-plane steps are redundant, as is the test quotient, which equals 1.Source: A. Practical aspects of Algorithm 2 · pp. 12–13
  • For a strict standard model that is not certified upper-C1, step 4 may be needed when the initial subgradient does not match the model, prompting a new subgradient and bundle-style updates.The aggregate subgradient limits the working set, and the tangent program is then solved with a QP solver.Source: A. Practical aspects of Algorithm 2 · pp. 12–13
  • For the negative spectral abscissa, the cautious variant may require step 4 and leads to a low-dimensional semidefinite program.When the Clarke subdifferential is a singleton, the strict-model subroutine is expected to reduce to a standard line search.Source: A. Practical aspects of Algorithm 2 · pp. 12–13
  • Practical stopping tests declare optimality after a sufficiently small accepted step, five consecutive unsuccessful backtracks, or exceeding the line-search limit.The experiments use tol1 = 10^-4, tol2 = 10^-4, and kmax = 50.Source: A. Practical aspects of Algorithm 2 · p. 14
  • The tangent-program parameter acts like a step size: decreasing it makes inner-loop steps smaller, while increasing the memorized value permits larger steps in the next inner loop.With a single subgradient and an unconstrained tangent program, the step length equals the parameter times the subgradient norm.Source: A. Practical aspects of Algorithm 2 · p. 14

B. Convergence analysis for the negative H∞-norm

For the negative H∞-norm, the algorithm converges to a KKT point; upper-C1 structure also supports a standard line search and, under additional analysis, convergence to a single point.

B. Convergence analysis for the negative H∞-norm

  • The sequence generated by Algorithm 2 with standard line search for program (8) converges to a KKT point.Feasibility of the iterates and strictness of the standard model support the KKT conclusion.Source: B. Convergence analysis for the… · p. 14
  • Because the negative H∞-norm is upper-C1 on its domain, the step-finding subroutine reduces to a line search, and boundedness of the parameter set removes the need for a separate bounded-iterate hypothesis.Upper-C1 functions have a strict standard model, although strictness is a weaker property.Source: B. Convergence analysis for the… · pp. 14–15
  • Analytic dependence on the parameter makes the negative H∞-norm subanalytic, and the Łojasiewicz inequality then assures convergence to a single KKT point.A strict standard model need not imply upper-C1 behavior, as illustrated by the lightning function.Source: B. Convergence analysis for the… · p. 15

C. Convergence analysis for the negative spectral abscissa

For the negative spectral abscissa, convergence is established conditionally, with assumptions that are difficult to verify even though they appear to hold in practice.

C. Convergence analysis for the negative spectral abscissa

  • The analysis argues that semi-simple active eigenvalues support a strict standard model through directional eigenvalue expansions, subject to uniformity over directions.The negative spectral abscissa is expected to have a strict standard model as a rule, while upper-C1 behavior is expected at least when active eigenvalues are semi-simple.Source: C. Convergence analysis for the… · p. 15
  • With step 4 activated, Algorithm 2 converges to a unique KKT point if every accumulation point is simple or semi-simple and satisfies condition (22).The authors report that the required conditions are difficult to check but seem to hold most of the time.Source: C. Convergence analysis for the… · p. 16
  • Convergence certificates for spectral-abscissa minimization depend on additional hypotheses that are hard to verify directly, including assumptions associated with Theorem 2.The authors describe the observed behavior as consistent with upper-C1 behavior, but present this as an expectation rather than a general certificate.Source: C. Convergence analysis for the… · pp. 15–16

D. Multiple performance measures

The paper extends dynamic inner relaxation to multiple performance requirements and evaluates it against static gridding and µ-synthesis approaches. In the tests, it achieves stronger worst-case performance than DKSYN on a larger uncertainty box with simpler controllers, while the missile example shows robust performance across sampled models.

D. Multiple performance measures

  • The method extends to multiple H∞ and H2 performance requirements and pole-location spectral constraints through a max-over-requirements performance function.The authors state that the results carry over because the worst-case multi-objective performance retains the same min-min structure.Source: D. Multiple performance measures · p. 16

A. Algorithm testing

  • Static relaxation missed worst-case performance in tests 6, 9, 12, and 14, while its 5^m-point grids became intractable in higher dimensions.When gridding succeeds, it is equivalent to Algorithm 1; the dynamic method's active-scenario set rarely exceeded 10 in these tests.Source: A. Algorithm testing · p. 17
  • Algorithm 1 was certified in all tested cases using the mixed-µ upper bound, with ratios near one indicating certification of its predicted values.A ratio much greater than one can reflect either a suboptimal Algorithm 1 result or conservatism in the upper bound.Source: A. Algorithm testing · p. 17
  • Algorithm 1 achieved better worst-case performance on a larger uncertainty box with simpler controllers than DKSYN, and was competitive in execution time.The comparison covered complex and real µ synthesis; real µ synthesis gave no reasonable certificates except in test cases 8, 9, and 13.Source: A. Algorithm testing · p. 17

B. Tail fin controlled missile

  • In the missile example, controller orders above 12 did not improve performance much, and the augmented plant had order 23.The reported plant order includes flexible modes, detector and actuator dynamics, and weighting filters.Source: B. Tail fin controlled missile · p. 20
  • 5 iterations produced the missile example's final active set of 6 uncertainty scenarios, after a destabilizing uncertainty was found at the first iteration.Worst-case H∞ performance need not improve monotonically as problematic parameter configurations are discovered.Source: B. Tail fin controlled missile · p. 20
  • Step responses for 100 sampled models showed good tracking across the uncertainty range, alongside achieved rigid margins and flexible-mode attenuation.The authors also report robust roll-off and notching of flexible modes.Source: B. Tail fin controlled missile · p. 20, Figure 3 · p. 21, Figure 5 · p. 22

VII. Conclusion

The paper presents a structured-controller approach to parametric robust H∞ control using dynamic inner approximation and tailored nonsmooth optimization. Tests on challenging examples and a case study indicate it is a practical tool for parametric-uncertainty design.

VII. Conclusion

  • Dynamic inner approximation iteratively adapts parameter scenarios and is reported to work rapidly without introducing conservatism.The method uses nonsmooth optimization tailored to upper-C1 stability and performance functions.Source: §VII · p. 22
  • Global robustness and performance certificates are best obtained after design by applying analysis tools based on outer approximations.Source: §VII · p. 22
  • Tests on challenging examples and a case study indicate that the technique is a practical tool for parametric-uncertainty design problems.Source: §VII · p. 22
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