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Combining Prior Knowledge and Data for Robust Controller Design

Julian Berberich, Carsten W. Scherer, Frank Allgöwer

arXiv:2009.05253v4eess.SYmath.OC

TL;DR

The paper addresses how to design robust controllers for unknown LTI systems when both noisy data and prior knowledge are available. It combines prior and data-learned multipliers within an LMI-based robust-control framework, yielding guaranteed stability and performance across consistent uncertainties. Numerical examples show that combining both information sources can reduce conservatism and improve performance relative to data-only approaches.

  • Problem

    Existing data-driven control methods are often black-box approaches that do not systematically incorporate prior plant knowledge, despite finite noisy data and difficult identification guarantees.

  • Method

    The framework represents the partially known system as an LFT and combines prior-knowledge multipliers with multipliers learned from noisy data using a general disturbance description.

  • Results

    The framework produces robust stability and performance guarantees for all uncertainties consistent with the prior knowledge and data, with extensions to output-feedback and nonlinear uncertainties.

  • Takeaways & Limitations

    Combining prior knowledge and data can shrink the admissible uncertainty set, reduce conservatism, and improve robust-controller performance compared with data-only design.

Abstract

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We present a framework for systematically combining data of an unknown linear time-invariant system with prior knowledge on the system matrices or on the uncertainty for robust controller design. Our approach leads to linear matrix inequality (LMI) based feasibility criteria which guarantee stability and performance robustly for all closed-loop systems consistent with the prior knowledge and the available data. The design procedures rely on a combination of multipliers inferred via prior knowledge and learnt from measured data, where for the latter a novel and unifying disturbance description is employed. While large parts of the paper focus on linear systems and input-state measurements, we also provide extensions to robust output-feedback design based on noisy input-output data and against nonlinear uncertainties. We illustrate through numerical examples that our approach provides a flexible framework for simultaneously leveraging prior knowledge and data, thereby reducing conservatism and improving performance significantly if compared to black-box approaches to data-driven control.

I. INTRODUCTION

The paper addresses the open problem of combining prior knowledge with measured data for robust controller design. It develops a flexible LTI framework that uses uncertainty descriptions and data to provide robust stability and performance guarantees.

  • I. INTRODUCTION: Data-driven controller design can avoid sequential identification and model-based control, but finite noisy data and limited theoretical guarantees remain important challenges.Existing methods are often black-box approaches that cannot systematically incorporate plant knowledge.
  • I. INTRODUCTION: Prior information can include uncertainty structure, parameter bounds, system-matrix knowledge, or loop-shaping information, while data-driven methods alone generally cannot use these forms flexibly.The problem setting assumes an uncertain LTI system with known interconnection matrices and an unknown true uncertainty.
  • I. INTRODUCTION: The proposed framework combines measured LTI-system data with prior knowledge through an LFT representation, prior multipliers, and data-learned multipliers.The framework models known and unknown components separately and uses a general disturbance description for noisy trajectories.
  • I. INTRODUCTION: Robust controllers are designed with stability and performance guarantees for all uncertainties consistent with the prior knowledge and available data.The framework supports state-feedback, output-feedback, and extensions involving nonlinear uncertainties.
  • I. INTRODUCTION: The paper illustrates the approach using a flexible satellite example in which torque input and disturbance act on a system with uncertain physical parameters.The satellite dynamics are represented as an LFT, and sampled input-state measurements may be affected by bounded disturbance.
  • I. INTRODUCTION: The framework is intended to bridge purely model-based and purely data-driven control, including settings with only data or only prior model knowledge.Additional prior knowledge may shrink the combined uncertainty set and reduce conservatism, while no-prior and no-data cases are included as special cases.

III. MULTIPLIERS FROM PRIOR KNOWLEDGE AND DATA

The framework constructs uncertainty multipliers by combining prior knowledge with information learned from noisy data, yielding multipliers for all uncertainties consistent with both sources.

  • Prior and data-derived multipliers are constructed separately and then combined for uncertainties consistent with the prior knowledge and measurements.The combined construction first translates prior multipliers and then learns additional multipliers from disturbed data.

A. Multipliers from prior knowledge

Prior structural and bound information is translated into multipliers for a transformed full-block uncertainty without discarding its structure.

  • Prior uncertainty information is represented by a family of transformed multipliers for ˜∆.The resulting multiplier class accounts for prior bounds while retaining the structural form induced by Bw.
  • Non-strict matrix inequalities allow the transformed multipliers to preserve structural information about the uncertainty channel.The paper identifies this translation of structural uncertainty knowledge as a key step of the framework.

B. Learning multipliers from data

Noisy input-state data provide additional multiplier constraints, which are combined with prior multipliers to characterize the transformed uncertainties consistent with both sources.

  • Measured trajectories affected by a bounded disturbance yield a learned uncertainty constraint containing the true uncertainty.The learned information is expressed through quadratic constraints suitable for robust-control synthesis.
  • The data-derived multipliers depend on measured data, known prior model parts, and the disturbance description.They provide a constraint without explicitly identifying the underlying system.
  • The learned transformed set does not generally encode that the transformed uncertainty has the structural form Bw∆.That structural information is supplied by the prior multipliers.
  • Combining prior and learned multipliers produces a set containing all available information on the transformed true uncertainty.Under the stated assumptions, the combined set equals the transformed image of the combined original uncertainty set.
  • Prior multipliers remain beneficial when structural information is available, even when their uncertainty bounds are conservative.The learned multipliers alone do not capture the structural form of the transformed uncertainty.
  • The framework generalizes a purely data-driven approach by incorporating uncertainty structure or bounds, flexible disturbance multipliers, input-output data, and nonlinear uncertainties.With no prior knowledge and simple disturbance multipliers, the results reduce to the referenced data-driven method.

IV. ROBUST CONTROLLER DESIGN USING PRIOR KNOWLEDGE AND DATA

The combined uncertainty description supports LMI-based robust controller synthesis with stability and H2-performance guarantees, including extensions to output-feedback design from noisy input-output data.

  • A. Robust H2-performance: The framework designs controllers with guaranteed robust H2-performance for every uncertainty in the combined set.The synthesis uses the transformed combined uncertainty representation and targets the channel d 7→e.
  • A. Robust H2-performance: Theorem 1 guarantees stability and a closed-loop H2-norm below γ when its conditions hold.The guarantee applies to all ∆ ∈ ∆com under the stated assumptions.
  • A. Robust H2-performance: The robust synthesis conditions can be transformed into LMIs using the Schur complement and the variable substitution L = KX.This converts the matrix inequality involving the state-feedback gain into a linear feasibility condition.
  • A. Robust H2-performance: The same framework can robustly stabilize the system without an H2 constraint by omitting the performance condition and setting Bd = 0.The results also extend to other disturbance-to-performance channels and robust quadratic performance settings.
  • A. Robust H2-performance: The controller design combines finite noisy data directly with prior knowledge, avoiding an additional system-identification estimation step.The paper contrasts this with the difficulty of obtaining comparable finite-data error bounds through system identification.
  • B. Robust output-feedback design: Noisy input-output measurements can be converted into a structured robust-control problem for dynamic output-feedback design.The construction uses an extended state so that state-feedback controllers correspond to dynamic output-feedback controllers.
  • B. Robust output-feedback design: Output-feedback design is not always applicable without prior knowledge: it requires p = 1 or, when p > 1, the condition pl = n.The paper identifies overcoming this limitation as future research.

V. PRIOR KNOWLEDGE DESCRIPTIONS

The framework represents prior uncertainty knowledge through multiplier classes that encode bounds and structure. Larger multiplier classes can tighten the represented uncertainty set, but require more computation, while specific conditions enable structural conclusions.

  • Uncertainty descriptions: Prior uncertainty descriptions can use full-block or repeated-scalar multiplier constructions, including norm-bound special cases.The framework also permits more flexible descriptions such as convex-hull multipliers.
  • Multiplier classes: The repeated-scalar multiplier class is larger than the full-block class, so it can represent a smaller uncertainty set at higher computational cost.In general, enlarging the multiplier class shrinks the corresponding uncertainty set and is typically beneficial for robust controller design.
  • Structural implications: Any full-block uncertainty satisfying the repeated-scalar multiplier description must have the form ∆j = δjI.Thus, repeated diagonal structure can be incorporated through an appropriate multiplier choice even when the uncertainty blocks are treated as unstructured.
  • Structural implications: Negative definiteness of the left-upper multiplier block is crucial for handling prior knowledge about uncertainty structure.The condition is satisfied, for example, by norm-bound-based full-block and repeated-scalar multiplier classes.
  • Dual descriptions: The framework uses dual uncertainty bounds involving ∆j^⊤ because such descriptions arise when multipliers are learned from data.Under mild inertia assumptions, dualization can transform bounds on ∆j^⊤ into bounds on ∆j and vice versa.

B. Prior knowledge on the disturbance (Assumption 2)

The disturbance description supports multiple multiplier classes for pointwise, quadratic, convex-hull, periodic, and other structured prior information. These classes can be combined to tighten the disturbance set, with a trade-off between conservatism and computational cost.

  • Disturbance descriptions: The disturbance framework covers quadratic full-block bounds, pointwise Euclidean norm bounds, convex-hull descriptions, and approximately periodic or LTI-generated disturbances.Toeplitz constraints can encode approximate membership in the kernel associated with periodic or LTI-generated disturbance sequences.
  • Multiplier classes: For pointwise Euclidean bounds, diagonal multipliers assign one nonnegative scalar multiplier to each data point.This differs from the single scalar multiplier used by the quadratic full-block description.
  • Multiplier combinations: Convex-hull multipliers describe disturbances lying in the convex hull of known disturbance matrices and can be combined with other multiplier classes.The framework also permits combinations with periodicity information or bounds applying to subsets of the available data.
  • Comparison of classes: For infinity-norm disturbance bounds, the multiplier classes satisfy Pquad ⊆ Pdiag ⊆ Pcon, making convex-hull multipliers the tightest of the three.The stated ordering follows the corresponding parameter choices that make all three classes valid under the same pointwise bound.
  • Comparison of classes: Pdiag and Pcon prevent deterioration when additional data points are added under pointwise disturbance bounds, unlike the translated quadratic class Pquad.The trade-off is computational: Pquad uses one scalar decision variable, whereas Pdiag and Pcon use N and 2ndN variables, respectively.
  • Computational scope: Pcon is restricted to systems with low spatial disturbance dimension and few data points, while Pdiag remains applicable to medium-sized problems.This boundary follows from the substantially larger number of decision variables required by convex-hull multipliers.
  • Data usage: Multiple trajectories can be stacked and used whenever the concatenated data satisfy the system data equation, including cases where long trajectories are difficult to generate.This is particularly relevant for unstable systems.

VI. TOWARDS TIGHT DESIGN CONDITIONS

The framework can reduce conservatism by adding multiplier refinements, and it is tight in several polyhedral or unconstrained-prior settings. Its main limitation is reliance on common quadratic Lyapunov functions and unresolved relaxation gaps for general uncertainty sets.

  • Limitations: Reliance on a common quadratic Lyapunov function is an additional source of conservatism, motivating parameter-dependent Lyapunov functions as future work.The paper attributes other conservatism sources to inherited classical multiplier relaxations rather than additional discrepancies introduced by the framework.
  • Reducing conservatism: Additional multipliers for individual, full, or transformed uncertainties can progressively reduce conservatism and improve guaranteed performance.The refinements may use S-procedure, convex-hull, Lagrange, or SOS-based multiplier constructions.
  • Tightness: The framework is necessary and sufficient for robust stability and performance with a common quadratic Lyapunov function in several settings.One such setting has no prior uncertainty knowledge and disturbance bounds captured by the quadratic description, under a suitable constraint qualification.
  • Tightness: For polyhedral uncertainty descriptions, convex-hull multipliers can yield tight controller-design conditions.This applies when the relevant prior or disturbance sets are polytopes and the uncertainty is full and unstructured, including specified prior-only or data-only cases.
  • General uncertainty sets: For non-polytopic LMI or semi-algebraic sets, SOS relaxations can construct suitable multiplier classes, but their relaxation gaps remain unquantified.Relaxation families may be asymptotically exact at the cost of large computational complexity.

VII. EXTENSION TO NONLINEAR UNCERTAINTIES

The framework extends to systems with static nonlinear uncertainties by combining data-learned multipliers for parametric uncertainty with prior multipliers for the nonlinear component. An LMI feasibility condition then guarantees stability and robust quadratic performance under the stated assumptions.

  • Nonlinear uncertainty model: Static nonlinear uncertainties are incorporated through a convex cone of multipliers admitting an LMI representation.Norm, sector, and slope bounds provide examples of nonlinear uncertainty descriptions covered by this assumption.
  • Combining data and prior knowledge: Data affected by disturbances can be used to learn multipliers for the parametric uncertainty while treating the nonlinear uncertainty as a known LFT component.The resulting uncertainty set remains compatible with the framework's data-consistency construction.
  • Controller synthesis: Theorem 2 combines learned multipliers for parametric uncertainty with nonlinear-uncertainty multipliers to design state-feedback controllers.The theorem's feasibility condition uses Y, K, and both multiplier classes.
  • Guarantees: Feasibility of the synthesis condition guarantees stability and robust quadratic performance for all admissible parametric and nonlinear uncertainties.The guarantee applies to the performance index specified in the theorem and can be reformulated as an LMI using L = KY.
  • Guarantees: The performance guarantee holds over an infinite time horizon for arbitrary disturbance inputs, without requiring the disturbances that generate the data to satisfy the same bound.The design condition is formulated for a dual system because the learned uncertainty description involves the transpose of the transformed uncertainty.

VIII. NUMERICAL EXAMPLES

The numerical examples evaluate how prior knowledge, measured data, and disturbance descriptions affect robust controller performance. Combining prior knowledge and data provides the best possible performance bounds across the tested noise range.

  • A. Influence of prior knowledge and data: The experiments compare controller designs using prior knowledge alone, data alone, both sources, or exact model knowledge.Scenario 2 is equivalent to the approach in [23].
  • A. Influence of prior knowledge and data: For all selected noise levels, stabilizing controllers with guaranteed H2-norms exist under prior knowledge alone, combined information, and exact model knowledge.The purely data-driven design is stabilizing only for noise levels ¯d ≤ 0.02.
  • A. Influence of prior knowledge and data: As ¯d tends to zero, data-only and combined designs match the exact-model performance γ = 1.93, while increasing noise makes prior knowledge essential for stabilization.The combined design matches the data-only case at ¯d = 0 and the prior-knowledge-only case at ¯d = 0.2.
  • A. Influence of prior knowledge and data: For every ¯d ∈ [0, 0.2], combining prior knowledge and data gives the best possible performance bounds for the chosen multiplier classes.The combined design systematically trades off the information available from the two sources.
  • A. Influence of prior knowledge and data: The least-squares model-based controller has performance level 1.87 but achieves H2-performance 1.93 on the true system without theoretical guarantees.It also fails to robustly stabilize the prior uncertainty set, so closed-loop properties cannot be certified without further knowledge.

B. Comparison of different disturbance multipliers

The comparison shows that disturbance-multiplier choice materially affects guaranteed performance and how effectively additional data are exploited. More sophisticated descriptions outperform simple quadratic multipliers in the reported experiments.

  • B. Comparison of different disturbance multipliers: Convex-hull and diagonal multipliers generally outperform simple multipliers, even with N = 5 instead of N = 200 data points.All multiplier classes show performance approaching the nominal controller for small noise and the prior-knowledge-only robust controller for large noise.
  • B. Comparison of different disturbance multipliers: For fixed ¯d = 0.15, diagonal multipliers outperform quadratic multipliers for every considered data length.With diagonal multipliers, the robust closed-loop H2-norm is non-increasing as N grows.
  • B. Comparison of different disturbance multipliers: Quadratic multipliers achieve their best performance at only N = 40 data points, so increasing data length can deteriorate their guaranteed performance.This behavior results from the conservative overapproximation used in the simple disturbance description.
  • B. Comparison of different disturbance multipliers: The more sophisticated disturbance-bound description is instrumental for fully exploiting information in available data.The paper contrasts this with simple noise descriptions whose learnt uncertainty sets need not shrink when more data are included.

C. Data-driven H∞-loop-shaping

The framework uses prior frequency-response specifications and measured data to perform data-driven H∞ loop-shaping. In the example, the data-based robust design meets the specifications and closely matches nominal design based on complete model knowledge.

  • C. Data-driven H∞-loop-shaping: Dynamic filters encode a trade-off between small tracking errors and control inputs as prior loop-shaping knowledge.The example uses a low-pass filter w1 and a constant filter w2.
  • C. Data-driven H∞-loop-shaping: The design uses N = 100 measured samples with disturbance bound ¯d = 5 and no prior knowledge on the uncertainty.A static state-feedback controller is synthesized from these data using the framework's performance specification.
  • C. Data-driven H∞-loop-shaping: The robust design meets the loop-shaping specifications because the magnitude plots lie below the inverse filter dynamics w1 and w2.The comparison is made using Bode plots of the relevant open- and closed-loop transfer functions.
  • C. Data-driven H∞-loop-shaping: The data-based synthesis closely matches nominal design based on complete model knowledge.The example demonstrates that measured data can be used to perform loop-shaping with good closed-loop behavior.

APPENDIX

The appendix proves that prior and data-learnt uncertainty descriptions are equivalent after transformation into the full-block uncertainty representation. The argument establishes both set inclusions for prior, learnt, and combined uncertainties.

  • Proofs: The proof establishes ˜∆prior ⊇ Bw∆prior by showing every transformed prior uncertainty satisfies the transformed multiplier conditions.The argument uses arbitrary prior multipliers and the definition of the transformed uncertainty set.
  • Proofs: The reverse inclusion ˜∆prior ⊆ Bw∆prior follows by isolating each uncertainty block and using full column rank of Bj.The construction produces ∆j such that ˜∆j = Bj∆j.
  • Proofs: The transformed learnt uncertainty set is obtained by defining ˜∆ := Bw∆ and requiring M − ˜∆Z to belong to the transformed data-consistent set.This yields Bw∆ = ˜∆ ∈ ˜∆learnt.
  • Proofs: The combined transformed uncertainty set equals the intersection of the transformed prior and learnt sets.The proof uses the two individual inclusions to establish both directions for the combined set.
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