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A Data-driven Approach to Robust Control of Multivariable Systems by Convex Optimization
Alireza Karimi, Christoph Kammer
TL;DR
The paper addresses robust control design directly from measured frequency data, avoiding a parametric plant model while accounting for measurement noise. It uses a fully parameterized, convexified frequency-domain framework for multivariable control; simulations report norms generally about the same as or lower than fixed-structure alternatives, though solutions are local and initialization-dependent.
Problem
Robust data-driven control design uses measured data without a parametric plant model, but measurement noise remains a source of uncertainty.
Source: §1 · p. 1Method
The approach uses frequency-response data and fully parameterized matrix-polynomial controllers, convexifying performance constraints around an initial controller.
Source: §1 · p. 2, §8 · p. 11Results
The data-driven method generally achieves about the same or a lower norm than second-order state-space controllers optimized with HIFOO or hinfstruct.
Source: §6 · p. 7Takeaways & Limitations
The framework extends frequency-domain data-driven design beyond fixed LP or PID structures to H2, H∞, and loop-shaping specifications.
Source: §1 · p. 2The results are local and depend on the algorithm's initialization.
Source: §1 · p. 2
Abstract
from arXiv · showhide
The frequency-domain data of a multivariable system in different operating points is used to design a robust controller with respect to the measurement noise and multimodel uncertainty. The controller is fully parametrized in terms of matrix polynomial functions and can be formulated as a centralized, decentralized or distributed controller. All standard performance specifications like $H_2$, $H_\infty$ and loop shaping are considered in a unified framework for continuous- and discrete-time systems. The control problem is formulated as a convex-concave optimization problem and then convexified by linearization of the concave part around an initial controller. The performance criterion converges monotonically to a local optimal solution in an iterative algorithm. The effectiveness of the method is compared with fixed-structure controllers using non-smooth optimization and with full-order optimal controllers via simulation examples. Finally, the experimental data of a gyroscope is used to design a data-driven controller that is successfully applied on the real system.
1 Introduction
Measured data and faster optimization make data-driven control an alternative to model-based design, without requiring a parametric plant model. This paper extends frequency-domain convex design to fully parameterized multivariable controllers and multiple performance specifications, while its solutions remain local and initialization-dependent.
1 Introduction
- Data-driven design is increasingly practical as optimization becomes faster and more measured data become available; it computes controller parameters from data without a parametric plant model.Measurement noise is the stated uncertainty source, and its influence can be reduced with more measurement data.Source: §1 · p. 1
- Related frequency-domain methods include manually tuned graphical designs, non-convex fixed-structure optimization, and linearized MIMO controller approaches, including methods with closed-loop stability conditions.The paper notes that its convexification uses an approximation also used in prior work; like other fixed-structure methods, its results are local and depend on initialization.Source: §1 · pp. 1–2
- Earlier convex frequency-domain methods covered SISO LP controllers and MIMO decoupling LP controllers, while necessary and sufficient data-driven H∞ existence conditions were proposed for SISO systems.These prior designs are narrower in controller structure or system scope than the proposed approach.Source: §1 · p. 1
- The proposed approach fully parameterizes frequency-domain multivariable controllers beyond LP or PID structures and unifies H2, H∞, mixed H2/H∞, and loop-shaping specifications under multimodel uncertainty.It also gives a closed-loop stability proof based on the Nyquist criterion.Source: §1 · p. 2
2 Preliminaries
The preliminaries represent the plant by a multivariable frequency response identified from sampled input/output data. Controllers are expressed as ratios of polynomial matrices, a structure that supports several control architectures and includes MIMO PID as a special case.
2 Preliminaries
- The plant is an LTI-MIMO system represented by G(e^jω) ∈ C^n×m, identified from m sets of sampled input/output data using Fourier analysis.At least m experiments are needed; each data column represents one experiment, and the response is assumed bounded except at a finite set of unit-circle poles.Source: §2 · p. 2
- The controller is K = XY^-1, with X and Y polynomial matrices in s or z, allowing continuous- or discrete-time controller design.Fixed known terms in the controller can encode desired performance, for example through the internal model principle.Source: §2 · p. 2
- For discrete-time design, polynomial coefficients contain the controller parameters; a diagonal Y structure can produce lower-order controllers and simplify inversion and implementation.Y(e^jω) must be invertible over the design frequencies outside the specified boundary-root set.Source: §2 · p. 2
- The controller structure covers centralized, decentralized, and distributed designs, with MIMO PID as a special case.Source: §2 · p. 2
3 Control Performance
The section reformulates standard performance specifications as frequency-domain constraints and convexifies their concave terms by linearization around a chosen controller. It applies this approach to H∞, H2, and loop-shaping objectives, while noting that loop shaping alone does not necessarily guarantee closed-loop stability.
3 Control Performance
- Convex-concave performance constraints are convexified by Taylor-linearizing the quadratic term P*P around a known matrix Pc, yielding a sufficient convex approximation.The approximation follows from the nonnegative quadratic form (P − Pc)*(P − Pc).Source: §3 · p. 3
3.1 H∞performance
- The H∞ mixed-sensitivity specification is converted to a spectral-norm constraint and then to a convex optimization problem with LMIs using the linearized approximation.The resulting convex constraint is stated to be sufficient for the spectral constraint for any chosen initial controller.Source: §3.1 · p. 3
3.2 H2 performance
- For H2 performance, the formulation introduces a positive matrix function Γ(ω), substitutes K = XY^-1, and linearizes the quadratic term to obtain an LMI.For a finite frequency set, Γ(ω) can be replaced by a matrix variable at each frequency; it may also be approximated by a finite-order polynomial.Source: §3.2 · p. 3
3.3 Loop shaping
- Loop shaping minimizes the 2- or infinity-norm distance between the loop transfer function GK and a desired loop transfer function using a convexified formulation.The resulting loop-shaping controller does not necessarily guarantee closed-loop stability.Source: §3.3 · p. 4
4 Robust Controller Design
The stability analysis gives conditions under which the convexified design preserves closed-loop stability. The framework also accommodates multiple operating-point models and measurement-noise uncertainty through frequency-domain constraints.
4.1 Stability analysis
- A bounded weighted sensitivity norm alone does not ensure stability; the theorem guarantees it with a feasible LMI and conditions on Y and the initial controller.The initial and final controllers must share their stability-boundary poles, and det(Y) must be nonzero over the frequency set.Source: §4.1 · p. 4
4.2 Multimodel uncertainty
- For multimodel uncertainty, impose a separate set of constraints for each operating-point model, including the stability constraint.The models may have different orders and may include pure input/output delays.Source: §4.2 · p. 5
4.3 Frequency-domain uncertainty
- Measurement-noise uncertainty is represented as a nominal frequency response plus W1ΔW2, with W1 and W2 encoding noise magnitude and direction.The robust stability condition is ∥W2KSW1∥∞ < 1 when W1 is invertible over the frequency set.Source: §4.3 · pp. 5–6
5 Implementation Issues
Implementation replaces infinitely many frequency constraints with a finite grid and relies on a stabilizing initial controller. Repeated convex approximations then produce a stable, monotonically improving sequence that converges to a local optimum.
5.1 Frequency gridding
- Frequency constraints are evaluated on a finite grid, usually logarithmically spaced, with denser sampling possible near low-damped resonances.The problem complexity scales linearly with the number of frequency samples, and negative-frequency constraints follow from Hermitian symmetry.Source: §5.1 · p. 6
5.2 Initial controller
- The algorithm requires a stabilizing initial controller; for unstable plants, the paper assumes one was available for data acquisition.A poor initial controller can make the optimization infeasible or cause numerical problems; relaxation or re-initialization can help.Source: §5.2 · p. 6
5.3 Iterative algorithm
- Each convex approximation uses the previous controller as its new initialization, preserving closed-loop stability while the nonnegative objective decreases toward a local optimum.The iterations can stop when the change in the performance criterion is sufficiently small.Source: §5.3 · p. 7
6 Simulation Results
In simulations on 10 Compleib plants, the data-driven method generally achieves about the same or a lower mixed-sensitivity norm than the compared fixed-structure methods. Solver time scales mainly with frequency-grid size, not controller order.
6 Simulation Results
- For the mixed-sensitivity problem, the optimization is formed with N = 100 logarithmically spaced frequency points over [0.01, 500] rad/s, where 500 is much larger than all plants’ bandwidths.Performance is compared with hinfstruct and HIFOO on 10 plants drawn from the Compleib library.Source: §6 · p. 7
- The data-driven method generally achieves about the same or a lower mixed-sensitivity norm than HIFOO and hinfstruct on 10 Compleib plants.The paper attributes superior results to the matrix-polynomial controller having more parameters than a same-order state-space controller.Source: Table 1 · p. 7, §6 · p. 7
- One iteration’s solver time depends almost linearly on the number of frequency-grid points, while controller order has minimal impact.The number of iterations to convergence mostly depends on the initial controller, and a solution is generally reached in fewer than 25 iterations.Source: §6 · p. 7
7 Further Simulation Results
The examples use measured frequency responses for controller design, including cases where parametric plant models are available but not used. They compare controller performance in simulation and validate multimodel gyroscope control experimentally.
7.1 Fixed-structure controller design
- The proposed controller gives an open-loop response similar to hinfstruct and is closer to the desired response at low frequencies, with an objective value around 30 times smaller.The comparison uses the sum of the trace of Γk in the objective function.Source: §7.1 · p. 8, Figure 2 · p. 8
7.2 Mixed sensitivity problem
- For p = 5, the method achieves the global mixed-sensitivity optimum with 99 design parameters, substantially fewer than the 289-parameter full-order controller.Good results are also reported for lower-order controllers, without requiring a parametric model.Source: §7.2 · p. 9
7.3.1 Experimental setup
- The 3-DOF gyroscope objective is to track the blue and red gimbal angles while minimizing axis coupling; disk speed is treated as multimodel uncertainty.The frame is mechanically fixed, while the disk and gimbals are actuated and their angles measured by optical encoders.Source: §7.3.1 · pp. 9–10
7.3.3 Control design formulation
- A cascaded architecture uses feedback linearization in the inner loop, while a discrete-time controller is designed from the three identified frequency responses to shape the loop and reduce coupling.The design minimizes a weighted two-norm loop-shaping error and imposes H∞ constraints for each identified model.Source: §7.3.2 · p. 10, §7.3.3 · p. 10
7.3.2 Plant identification
- Identification at 300, 400, and 500 rpm yields three plant frequency responses, with coupling and resonance modes becoming stronger at higher disk speeds.A PRBS signal was applied to each reference channel for four periods at each disk speed.Source: §7.3.2 · p. 10, Figure 6 · p. 11
7.3.4 Experimental results
- 0.625 s for the blue and 0.486 s for the red gimbal angle are the measured rise times, close to the desired 0.55 s, with good decoupling across disk speeds.A slight overshoot is observed, especially for the red gimbal, likely due to system nonlinearities.Source: §7.3.4 · p. 11
8 Conclusions
Frequency-response measurements can be used to design controllers directly, without parametric identification. The proposed unified convex approximation covers several control objectives and converges iteratively to a local optimum from an initial stabilizing controller.
8 Conclusions
- Frequency-response data supports direct high-performance controller design without parametric identification, with measurement-noise uncertainty modeled through weighting frequency functions.The paper identifies the absence of unmodeled dynamics as an advantage of this approach.Source: §8 · p. 11
- A unified convex approximation covers H∞, H2, and loop-shaping control problems.The method uses a full controller parametrization and adds a closed-loop stability proof compared with other frequency-domain data-driven approaches.Source: §8 · p. 11
- The convex approximation relies on an initial stabilizing controller, and the iterative algorithm converges to a local optimum of the original non-convex problem.Source: §8 · p. 11