A Data-driven Approach to Robust Control of Multivariable Systems by Convex Optimization

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

Sum-of-Squares approach to feedback control of laminar wake flows

A novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation of long-time averages of flow quantities is presented. It applies to reduced-order finite-dimensional models of fluid flows, expressed as a set of first-order nonlinear ordinary differential equations with the right-hand side being a polynomial function in the state variables and in the controls. The key idea, first discussed in Chernyshenko et al. 2014, Philos. T. Roy. Soc. 372(2020), is that the difficulties of treating and optimising long-time averages of a cost are relaxed by using the upper/lower bounds of such averages as the objective function. In this setting, control design reduces to finding a feedback controller that optimises the bound, subject to a polynomial inequality constraint involving the cost function, the nonlinear system, the controller itself and a tunable polynomial function. A numerically tractable approach to the solution of such optimisation problems, based on Sum-of-Squares techniques and semidefinite programming, is proposed. To showcase the methodology, the mitigation of the fluctuation kinetic energy in the unsteady wake behind a circular cylinder in the laminar regime at Re=100, via controlled angular motions of the surface, is numerically investigated. A compact reduced-order model that resolves the long-term behaviour of the fluid flow and the effects of actuation, is derived using Proper Orthogonal Decomposition and Galerkin projection. In a full-information setting, feedback controllers are then designed to reduce the long-time average of the kinetic energy associated with the limit cycle. These controllers are then implemented in direct numerical simulations of the actuated flow. Control performance, energy efficiency, and physical control mechanisms identified are analysed. Key elements, implications and future work are discussed

Robust explicit MPC design under finite precision arithmetic

We propose a design methodology for explicit Model Predictive Control (MPC) that guarantees hard constraint satisfaction in the presence of finite precision arithmetic errors. The implementation of complex digital control techniques, like MPC, is becoming increasingly adopted in embedded systems, where reduced precision computation techniques are embraced to achieve fast execution and low power consumption. However, in a low precision implementation, constraint satisfaction is not guaranteed if infinite precision is assumed during the algorithm design. To enforce constraint satisfaction under numerical errors, we use forward error analysis to compute an error bound on the output of the embedded controller. We treat this error as a state disturbance and use this to inform the design of a constraint-tightening robust controller. Benchmarks with a classical control problem, namely an inverted pendulum, show how it is possible to guarantee, by design, constraint satisfaction for embedded systems featuring low precision, fixed-point computations

Sparse Iterative Learning Control with Application to a Wafer Stage: Achieving Performance, Resource Efficiency, and Task Flexibility

Trial-varying disturbances are a key concern in Iterative Learning Control (ILC) and may lead to inefficient and expensive implementations and severe performance deterioration. The aim of this paper is to develop a general framework for optimization-based ILC that allows for enforcing additional structure, including sparsity. The proposed method enforces sparsity in a generalized setting through convex relaxations using $\ell_1$ norms. The proposed ILC framework is applied to the optimization of sampling sequences for resource efficient implementation, trial-varying disturbance attenuation, and basis function selection. The framework has a large potential in control applications such as mechatronics, as is confirmed through an application on a wafer stage.Comment: 12 pages, 14 figure

Blockwise Subspace Identification for Active Noise Control

In this paper, a subspace identification solution is provided for active noise control (ANC) problems. The solution is related to so-called block updating methods, where instead of updating the (feedforward) controller on a sample by sample base, it is updated each time based on a block of N samples. The use of the subspace identification based ANC methods enables non-iterative derivation and updating of MIMO compact state space models for the controller. The robustness property of subspace identification methods forms the basis of an accurate model updating mechanism, using small size data batches. The design of a feedforward controller via the proposed approach is illustrated for an acoustic duct benchmark problem, supplied by TNO Institute of Applied Physics (TNO-TPD), the Netherlands. We also show how to cope with intrinsic feedback. A comparison study with various ANC schemes, such as block filtered-U, demonstrates the increased robustness of a subspace derived controlle

Integrated system identification/control design with frequency weightings.

by Ka-lun Tung.Thesis (M.Phil.)--Chinese University of Hong Kong, 1995.Includes bibliographical references (leaves 168-[175]).Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Control with Uncertainties --- p.1Chapter 1.1.1 --- Adaptive Control --- p.2Chapter 1.1.2 --- H∞ Robust Control --- p.3Chapter 1.2 --- A Unified Framework: Adaptive Robust Control --- p.4Chapter 1.3 --- System Identification for Robust Control --- p.6Chapter 1.3.1 --- Choice of input signal --- p.7Chapter 1.4 --- Objectives and Contributions --- p.8Chapter 1.5 --- Thesis Outline --- p.9Chapter 2 --- Background on Robust Control --- p.11Chapter 2.1 --- Notation and Terminology --- p.12Chapter 2.1.1 --- Notation --- p.12Chapter 2.1.2 --- Linear System Terminology --- p.13Chapter 2.1.3 --- Norms --- p.15Chapter 2.1.4 --- More Terminology: A Standard Feedback Configuration --- p.17Chapter 2.2 --- Norms and Power for Signals and Systems --- p.18Chapter 2.3 --- Plant Uncertainty Model --- p.20Chapter 2.3.1 --- Multiplicative Unstructured Uncertainty --- p.21Chapter 2.3.2 --- Additive Unstructured Uncertainty --- p.22Chapter 2.3.3 --- Structured Uncertainty --- p.23Chapter 2.4 --- Motivation for H∞ Control Design --- p.23Chapter 2.4.1 --- Robust stabilization: Multiplicative Uncertainty and Weight- ing function W3 --- p.24Chapter 2.4.2 --- Robust stabilization: Additive Uncertainty and Weighting function W2 --- p.25Chapter 2.4.3 --- Tracking Problem --- p.26Chapter 2.4.4 --- Disturbance Rejection (or Sensitivity Minimization) --- p.27Chapter 2.5 --- The Robust Control Problem Statement --- p.28Chapter 2.5.1 --- The Mixed-Sensitivity Approach --- p.29Chapter 2.6 --- An Augmented Generalized Plant --- p.30Chapter 2.6.1 --- The Augmented Plant --- p.30Chapter 2.6.2 --- Adaptation of Augmented Plant to Sensitivity Minimiza- tion Problem --- p.32Chapter 2.6.3 --- Adaptation of Augmented Plant to Mixed-Sensitivity Prob- lem --- p.33Chapter 2.7 --- Using MATLAB Robust Control Toolbox --- p.34Chapter 3 --- Statistical Plant Set Estimation for Robust Control --- p.36Chapter 3.1 --- An Overview --- p.37Chapter 3.2 --- The Schroeder-phased Input Design --- p.39Chapter 3.3 --- The Statistical Additive Uncertainty Bounds --- p.40Chapter 3.4 --- Additive Uncertainty Characterization --- p.45Chapter 3.4.1 --- "Linear Programming Spectral Overbounding and Factor- ization Algorithm (LPSOF) [20,21]" --- p.45Chapter 4 --- Basic System Identification and Model Reduction Algorithms --- p.48Chapter 4.1 --- The Eigensystem Realization Algorithm --- p.49Chapter 4.1.1 --- Basic Algorithm --- p.49Chapter 4.1.2 --- Estimating Markov Parameters from Input/Output data: Observer/Kalman Filter Identification (OKID) --- p.51Chapter 4.2 --- The Frequency-Domain Identification via 2-norm Minimization --- p.54Chapter 4.3 --- Balanced Realization and Truncation --- p.55Chapter 4.4 --- Frequency Weighted Balanced Truncation --- p.56Chapter 5 --- Plant Model Reduction and Robust Control Design --- p.59Chapter 5.1 --- Problem Formulation --- p.59Chapter 5.2 --- Iterative Reweighting Scheme --- p.60Chapter 5.2.1 --- Rationale Behind the Scheme --- p.62Chapter 5.3 --- Integrated Model Reduction/ Robust Control Design with Iter- ated Reweighting --- p.63Chapter 5.4 --- A Design Example --- p.64Chapter 5.4.1 --- The Plant and Specification --- p.64Chapter 5.4.2 --- First Iteration --- p.65Chapter 5.4.3 --- Second Iteration --- p.67Chapter 5.5 --- Approximate Fractional Frequency Weighting --- p.69Chapter 5.5.1 --- Summary of Past Results --- p.69Chapter 5.5.2 --- Approximate Fractional Frequency Weighting Approach [40] --- p.70Chapter 5.5.3 --- Simulation Results --- p.71Chapter 5.6 --- Integrated System Identification/Control Design with Iterative Reweighting Scheme --- p.74Chapter 6 --- Controller Reduction and Robust Control Design --- p.82Chapter 6.1 --- Motivation for Controller Reduction --- p.83Chapter 6.2 --- Choice of Frequency Weightings for Controller Reduction --- p.84Chapter 6.2.1 --- Stability Margin Considerations --- p.84Chapter 6.2.2 --- Closed-Loop Transfer Function Considerations --- p.85Chapter 6.2.3 --- A New Way to Determine Frequency Weighting --- p.86Chapter 6.3 --- A Scheme for Iterative Frequency Weighted Controller Reduction (IFWCR) --- p.87Chapter 7 --- A Comparative Design Example --- p.90Chapter 7.1 --- Plant Model Reduction Approach --- p.90Chapter 7.2 --- Weighted Controller Reduction Approach --- p.94Chapter 7.2.1 --- A Full Order Controller --- p.94Chapter 7.2.2 --- Weighted Controller Reduction with Stability Considera- tions --- p.94Chapter 7.2.3 --- Iterative Weighted Controller Reduction --- p.96Chapter 7.3 --- Summary of Results --- p.101Chapter 7.4 --- Discussions of Results --- p.101Chapter 8 --- A Comparative Example on a Benchmark problem --- p.105Chapter 8.1 --- The Benchmark plant [54] --- p.106Chapter 8.1.1 --- Benchmark Format and Design Information --- p.106Chapter 8.1.2 --- Control Design Specifications --- p.107Chapter 8.2 --- Selection of Performance Weighting function --- p.108Chapter 8.2.1 --- Reciprocal Principle --- p.109Chapter 8.2.2 --- Selection of W1 --- p.110Chapter 8.2.3 --- Selection of W2 --- p.110Chapter 8.3 --- System Identification by ERA --- p.112Chapter 8.4 --- System Identification by Curve Fitting --- p.114Chapter 8.4.1 --- Spectral Estimate --- p.114Chapter 8.4.2 --- Curve Fitting Results --- p.114Chapter 8.5 --- Robust Control Design --- p.115Chapter 8.5.1 --- The selection of W1 weighting function --- p.115Chapter 8.5.2 --- Summary of Design Results --- p.116Chapter 8.6 --- Stress Level 1 --- p.117Chapter 8.6.1 --- System Identification Results --- p.117Chapter 8.6.2 --- Design Results --- p.119Chapter 8.6.3 --- Step Response --- p.121Chapter 8.7 --- Stress Level 2 --- p.124Chapter 8.7.1 --- System Identification Results --- p.124Chapter 8.7.2 --- Step Response --- p.125Chapter 8.8 --- Stress Level 3 --- p.128Chapter 8.8.1 --- System Identification Results --- p.128Chapter 8.8.2 --- Step Response --- p.129Chapter 8.9 --- Comparisons with Other Designs --- p.132Chapter 9 --- Conclusions and Recommendations for Further Research --- p.133Chapter 9.1 --- Conclusions --- p.133Chapter 9.2 --- Recommendations for Further Research --- p.135Chapter A --- Design Results of Stress Levels 2 and3 --- p.137Chapter A.1 --- Stress Level 2 --- p.137Chapter A.2 --- Stress Level 3 --- p.140Chapter B --- Step Responses with Reduced Order Controller --- p.142Chapter C --- Summary of Results of Other Groups on the Benchmark Prob- lem --- p.145Chapter C.1 --- Indirect and implicit adaptive predictive control [45] --- p.146Chapter C.2 --- H∞ Robust Control [51] --- p.150Chapter C.3 --- Robust Stability Degree Assignment [53] --- p.152Chapter C.4 --- Model Reference Adaptive Control [46] --- p.154Chapter C.5 --- Robust Pole Placement using ACSYDE (Automatic Control Sys- tem Design) [47] --- p.156Chapter C.6 --- Adaptive PI Control [48] --- p.157Chapter C.7 --- Adaptive Control with supervision [49] --- p.160Chapter C.8 --- Partial State Model Reference (PSRM) Control [50] --- p.162Chapter C.9 --- Contstrainted Receding Horizon Predictive Control (CRHPC) [52] --- p.165Bibliography --- p.16