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Multiobjective Robust Control with HIFOO 2.0
Suat Gumussoy, Didier Henrion, Marc Millstone, Michael L. Overton
TL;DR
Multiobjective control design must handle conflicting specifications across multiple plants, while existing methods can require high-order controllers or increased computation. The paper presents HIFOO 2.0, which searches locally over a fixed-order controller’s coefficients using nonsmooth optimization. Across benchmarks, it reports successful low-order stabilization and strong simultaneous H∞ performance, including first-order controllers matching reported optima in two academic problems.
Problem
Multiobjective control requires satisfying potentially conflicting specifications across plants, while established methods can produce high-order controllers or incur increased computational burden.
Method
HIFOO 2.0 formulates controller design as a nonsmooth, nonconvex optimization over controller coefficients and searches for local minimizers using a two-phase BFGS-based procedure.
Results
HIFOO successfully solves simultaneous stabilization with low-order controllers and achieves optimal H∞ performance with a single first-order controller in the second academic benchmark.
Takeaways & Limitations
HIFOO 2.0 provides a public-domain approach that lets users set controller order a priori while obtaining strong benchmark performance.
Takeaways & Limitations
HIFOO outputs may differ across randomly initialized runs, and its local-search algorithm provides no guarantee of behavior.
Abstract
from arXiv · showhide
Multiobjective control design is known to be a difficult problem both in theory and practice. Our approach is to search for locally optimal solutions of a nonsmooth optimization problem that is built to incorporate minimization objectives and constraints for multiple plants. We report on the success of this approach using our public-domain Matlab toolbox HIFOO 2.0, comparing our results with benchmarks in the literature.
1 Introduction
Multiobjective control must reconcile potentially conflicting specifications across channels and plants, while established approaches often impose high controller orders or substantial computational burdens. HIFOO 2.0 instead searches locally over controller coefficients with fixed order and no Lyapunov or lifting variables.
- Multiobjective control designs feedback laws meeting potentially conflicting specifications on multiple input/output channels.
- Lyapunov shaping reduces design to semidefinite programming when one quadratic Lyapunov function certifies all closed-loop specifications.The controller is recovered afterward through linear algebra.
- These standard formulations can produce controllers whose order equals the plant plus weighting-function orders, making implementation difficult.Alternative parametrization-based methods also typically yield very high-order controllers.
- Relaxing Lyapunov restrictions enables more flexible designs but increases computational burden; ROMULOC implements these techniques publicly in Matlab.
- HIFOO 2.0 fixes controller order and optimizes only controller coefficients in a small-dimensional, nonsmooth, nonconvex problem without large convex subproblems.It searches for local minima, so no guarantee is made about the algorithm’s behavior.
2 Problem Formulation
The formulation chooses one controller for multiple generalized plants and minimizes a common objective subject to plant-specific bounds. Supported objectives include spectral abscissa, H∞ norm, and reciprocal complex stability radius, enabling stabilization and performance trade-offs.
- A single controller is selected for N generalized plants whose signals include regulated outputs, exogenous inputs, measured inputs, and control inputs.
- The controller order n_K is fixed by the designer, while closed-loop metrics include H∞ norm, complex stability radius, and spectral abscissa.The complex stability radius measures the largest 2-norm perturbation tolerated while preserving stability.
- The optimization minimizes an objective over controller coefficients subject to g_j(K) ≤ β_j for each plant.
- Scenario 1: Scenario 1 minimizes the maximum closed-loop spectral abscissa across plants; a negative final objective means all closed-loop plants are stable.
- Scenario 2: Scenario 2 stabilizes all plants while minimizing the maximum closed-loop H∞ norm, but repeated H∞ evaluations are more time-consuming and require specified I/O channels.
- Alternative formulations: Alternative 3 maximizes the minimum complex stability radius, using it as a more robust stability measure when performance channels are unspecified.
3 Optimization method
HIFOO 2.0 searches for local minimizers of the constrained nonsmooth problem through a two-phase BFGS procedure. It first seeks stabilization, then optimizes a penalized objective, optionally followed by gradient sampling.
- HIFOO 2.0 searches for local minimizers using BFGS, with user-supplied and randomly generated initial controllers.
- Stabilization: The stabilization phase minimizes the maximum spectral abscissa for plants using H∞ or stability-radius objectives until a stabilizing controller is found.This supplies a finite-objective starting point for the optimization phase.
- Optimization: The optimization phase applies BFGS to an unconstrained penalized objective combining F with ρ times the constraint violations.
- Optimization: If violations remain, HIFOO increases ρ and repeats optimization; gradient sampling can then refine the result but generally takes more time.
4 User interface
The Matlab interface accepts plant models, controller order, initialization, objective types, bounds, and optional settings. A simple call solves the H∞ scenario by default, while options control structure, penalties, time, verbosity, and optimization speed.
- The simple Matlab call hifoo(P, order) returns a controller for the specified plants and controller order.Omitting order selects order 0, corresponding to static output feedback.
- By default, the call addresses Scenario 2 and minimizes the sum of closed-loop H∞ norms while stabilizing the plants.
- The general interface accepts an initialization, objective-function characters, upper bounds, and an options structure in flexible argument order.
- Options can impose controller sparsity, penalize controller size, augment H∞ objectives with reciprocal stability radius, limit CPU time, and select BFGS-only optimization.
- HIFOO uses HANSO and Matlab’s Control System Toolbox; gradient sampling additionally requires quadprog, but that component is optional.
5 Benchmarks
HIFOO 2.0 performs strongly across simultaneous stabilization and H∞ optimization benchmarks, often finding low-order controllers that match or outperform literature results. Its fixed-order design is especially effective on application and academic examples.
- Scenario 1: Simultaneous Stabilization: HIFOO successfully solves diverse simultaneous stabilization benchmarks with low-order single controllers and lets users specify controller order beforehand.The evaluation covers 31 problems: 11 application and 20 academic examples.
- Scenario 1: Simultaneous Stabilization: Application benchmarks show HIFOO outperforming existing methods on Lane-Keeping of Automated Vehicles and Helicopter Toy.For the Rationalized Helicopter Model, literature methods produce a 26th-order controller, whereas HIFOO stabilizes it with a static controller.
- Scenario 1: Simultaneous Stabilization: Academic stabilization results are better than or equivalent to existing methods for almost all problems, often using static or first-order controllers instead of high-order designs.The cited examples include Bredemann Ex.5.6, Chen et al., F.-Anaya et al. Ex.1 and Ex.2, and Gündes-Kabuli.
- Scenario 2: Simultaneous H∞ Optimization: HIFOO finds a single first-order controller matching optimal H∞ performance in the first simultaneous H∞ benchmark, while a previous single-controller result used fourth-order control with performance 1.806.It also finds a static controller with performance 1.530.
- Scenario 2: Simultaneous H∞ Optimization: HIFOO achieves optimal H∞ performance with a single first-order controller in the second benchmark, compared with a known sixth-order controller having performance 1.833.The three plants have respective full-order optimal H∞ norms 1.290, 1.245, and 1.038.
- Overall benchmark assessment: Across industrial application and academic test problems, the benchmark results demonstrate that HIFOO is effective for varied simultaneous-control examples.The conclusion identifies simultaneous stabilization and simultaneous H∞ optimization as supported use cases.