Sequential convex relaxation for robust static output feedback structured control

We analyse the very general class of uncertain systems that have Linear Fractional Representations (LFRs), and uncertainty blocks in a convex set with a finite number of vertices. For these systems we design static output feedback controllers. In the general case, computing a robust static output feedback controller with optimal performance gives rise to a bilinear matrix inequality (BMI). In this article we show how this BMI problem can be efficiently rewritten to fit in the framework of sequential convex relaxation, a method that searches simultaneously for a feasible controller and one with good performance. As such, our approach does not rely on being supplied with a feasible initial solution to the BMI. This sets it apart from methods that depend on a good initial, feasible starting point to progress from there using an alternating optimization scheme. In addition to using the proposed method, the controller matrices can be of a predetermined fixed structure. Alternatively, an L1 constraint can be easily added to the optimization problem as a convex variant of a cardinality constraint, in order to induce sparsity on the controller matrices.Team Raf Van de Pla

Subspace identification of distributed clusters of homogeneous systems

This note studies the identification of a network comprised of interconnected clusters of LTI systems. Each cluster consists of homogeneous dynamical systems, and its interconnections with the rest of the network are unmeasurable. A subspace identification method is proposed for identifying a single cluster using only local input and output data. With the topology of the concerned cluster being available, all the LTI systems within the cluster are decoupled by taking a transformation on the state, input and output data. To deal with the unmeasurable interconnections between the concerned cluster and the rest of the network, the Markov parameters of the decoupled LTI systems are identified first by solving a nuclear-norm regularized convex optimization, following the state-space realization of a single LTI system within the cluster by solving another nuclearnorm regularized optimization problem. The effectiveness of the proposed identification method is demonstrated by a simulation example.Accepted Author ManuscriptTeam Raf Van de Pla

N2SID: Nuclear norm subspace identification of innovation models

The identification of multivariable state space models in innovation form is solved in a subspace identification framework using convex nuclear norm optimization. The convex optimization approach allows to include constraints on the unknown matrices in the data-equation characterizing subspace identification methods, such as the lower triangular block-Toeplitz of weighting matrices constructed from the Markov parameters of the unknown observer. The classical use of instrumental variables to remove the influence of the innovation term on the data equation in subspace identification is avoided. The avoidance of the instrumental variable projection step has the potential to improve the accuracy of the estimated model predictions, especially for short data length sequencesAccepted Author ManuscriptTeam Raf Van de Pla

Kronecker-ARX models in identifying (2D) spatial-temporal systems

In this paper we address the identification of (2D) spatial-temporal dynamical systems governed by the Vector Auto-Regressive (VAR) form. The coefficient-matrices of the VAR model are parametrized as sums of Kronecker products. When the number of terms in the sum is small compared to the size of the matrix, such a Kronecker representation leads to high data compression. Estimating in least-squares sense the coefficient-matrices gives rise to a bilinear estimation problem, which is tackled using a three-stage algorithm. A numerical example demonstrates the advantages of the new modeling paradigm. It leads to comparable performances with the unstructured least-squares estimation of VAR models. However, the number of parameters in the new modeling paradigm grows linearly w.r.t. the number of nodes in the 2D sensor network instead of quadratically in the full unstructured matrix case.Team Raf Van de Pla

Modeling and state-space identification of deformable mirrors

To develop high-performance controllers for adaptive optics (AO) systems, it is essential to first derive sufficiently accurate state-space models of deformable mirrors (DMs). However, it is often challenging to develop realistic large-scale finite element (FE) state-space models that take into account the system damping, actuator dynamics, boundary conditions, and multi-physics phenomena affecting the system dynamics. Furthermore, it is challenging to establish a modeling framework capable of the automated and quick derivation of state-space models for different actuator configurations and system geometries. On the other hand, for accurate model-based control and system monitoring, it is often necessary to estimate state-space models from the experimental data. However, this is a challenging problem since the DM dynamics is inherently infinite-dimensional and it is characterized by a large number of eigenmodes and eigenfrequencies. In this paper, we provide modeling and estimation frameworks that address these challenges. We develop an FE state-space model of a faceplate DM that incorporates damping and actuator dynamics. We investigate the frequency and time domain responses for different model parameters. The state-space modeling process is completely automated using the LiveLink for MATLAB toolbox that is incorporated into the COMSOL Multiphysics software package. The developed state-space model is used to generate the estimation data. This data, together with a subspace identification algorithm, is used to estimate reduced-order DM models. We address the model-order selection and model validation problems. The results of this paper provide essential modeling and estimation tools to broad AO and mechatronics scientific communities. The developed Python, MATLAB, and COMSOL Multiphysics codes are available online.Team Raf Van de Pla

Local subspace identification of distributed homogeneous systems with general interconnection patterns

This paper studies the local identification of large-scale homogeneous systemswith general network topologies. The considered local system identification problem involves unmeasurable signals between neighboring subsystems. Compared with our previous work in Yu et al. (2014) which solves the local identification of 1D homogeneous systems, the main challenge of this work is how to deal with the general network topology. To overcome this problem, we first decompose the interested local system into separate subsystems using some state, input and output transform, namely the spatially lifted local system has block diagonal system matrices.We subsequently estimate the Markov parameters of the local system by solving a nuclear norm regularized optimization problem. To realize the state-space system model from the estimated Markov parameters, another nuclear norm regularized optimization problem is provided by taking into account of the inherent dependence of a redundant parameter vector. Finally, the overall identification procedure is summarized.Accepted Author ManuscriptTeam Michel Verhaege

Subspace identification of 1D spatially-varying systems using Sequentially Semi-Separable matrices

We consider the problem of identifying 1D spatially-varying systems that exhibit no temporal dynamics. The spatial dynamics are modeled via a mixed-causal, anti-causal state space model. The methodology is developed for identifying the input-output map of e.g a 1D flexible beam described by the Euler-Bernoulli beam equation and equipped with a large number of actuators and sensors. It is shown that the static input-output map between the lifted inputs and outputs possess a so-called Sequentially Semi-Separable (SSS) matrix structure. This structure is of key importance to derive algorithms with linear computational complexity for controller synthesis of large-scale systems. A nuclear norm subspace identification method of the N2SID class is developed for estimating these state space models from input-output data. To enable the method to deal with a large number of repeated experiments a dedicated Alternating Direction Method of Multipliers (ADMM) algorithm is derived. It is shown in this paper that a nuclear norm relaxation on the SSS structure can be imposed which improves the estimates of the system matrices.Accepted Author ManuscriptTeam Raf Van de Pla

Distributed system identification with ADMM

Accepted Author ManuscriptTeam Michel Verhaege

Convex optimization-based blind deconvolution for images taken with coherent illumination

A rank-constrained reformulation of the blind deconvolution problem on images taken with coherent illumination is proposed. Since in the reformulation the rank constraint is imposed on a matrix that is affine in the decision variables, we propose a novel convex heuristic for the blind deconvolution problem. The proposed heuristic allows for easy incorporation of prior information on the decision variables and the use of the phase diversity concept. The convex optimization problem can be iteratively re-parameterized to obtain better estimates. The proposed methods are demonstrated on numerically illustrative examples.Team Raf Van de Pla

Subspace identification of individual systems operating in a network (SI²ON)

Abstract:This note studies the identification of individual systems operating in a large-scale distributed network by considering the interconnection signals between neighboring systems to be unmeasurable. The unmeasurable interconnections act as unknown system inputs to the individual systems in a network, which poses a challenge for the identification problem. A subspace identification framework is proposed in this note for the consistent identification of individual systems using only local input and output information. The key step of this identification framework is the accurate estimation of the unknown system inputs of individual systems using local observations. Sufficient identifiability conditions are provided for the proposed identification framework and a simulation example is given to demonstrate its performance.Accepted Author ManuscriptTeam Raf Van de Pla