Aleksandar Haber

Moving Horizon Estimation for Large-Scale Interconnected Systems

We present computationally efficient centralized and distributed moving horizon estimation (MHE) methods for large-scale interconnected systems, that are described by sparse banded or sparse multibanded system matrices. Both of these MHE methods are developed by approximating a solution of the MHE problem using the Chebyshev approximation method. By exploiting the sparsity of this approximate solution we derive a centralized MHE method, which computational complexity and storage requirements scale linearly with the number of local subsystems of an interconnected system. Furthermore, on the basis of the approximate solution of the MHE problem, we develop a novel, distributed MHE method. This distributed MHE method estimates the state of a local subsystem using only local input-output data. In contrast to the existing distributed algorithms for the state estimation of large-scale systems, the proposed distributed MHE method is not relying on the consensus algorithms and has a simple analytic form. We have studied the stability of the proposed MHE methods and we have performed numerical simulations that confirm our theoretical results.

Subspace Identification of Large-Scale Interconnected Systems

We propose a decentralized subspace algorithm for the identification of large-scale, interconnected systems that are described by sparse (multi) banded state-space matrices. First, we prove that the state of a local subsystem can be approximated by a linear combination of inputs and outputs of local subsystems that are in its neighborhood. Furthermore, we prove that for interconnected systems with well-conditioned, finite-time observability Gramians (or observability matrices), the size of this neighborhood is relatively small. On the basis of these results, we develop the subspace identification algorithm that identifies the state-space model of a local subsystem from the local input-output data. Consequently, the developed algorithm is computationally feasible for interconnected systems with a large number of local subsystems. Numerical results confirm the effectiveness of the new identification algorithm.

Brief paper: Linear computational complexity robust ILC for lifted systems

In this paper we propose a new methodology to synthesize and implement robust monotonically convergent ILC for lifted systems, with the computational complexity that is linear in the trial length. Starting from the model uncertainty of the local sample to sample LTI or LTV models, and using the randomized algorithm, we compute the bound on the model uncertainty of the ILC system representation in the trial domain (lifted ILC). Based on this computed uncertainty bound, we design weighting matrices of the Norm Optimal ILC, such that the robust monotonic convergence condition is satisfied. Since we compute the uncertainty bound, rather than assuming its value in the trial domain, we reduce the conservatism of the robust design. The linear computational complexity of the algorithms for computation of the uncertainty bound and implementation of the Norm Optimal ILC law, is achieved through exploiting the sequentially semi-separable structure of the lifted system matrices. Therefore the framework proposed in this paper is especially suitable for the LTI and LTV uncertain systems with a large number of samples in the trial. We have performed numerical experiments to demonstrate the robustness and linear computational complexity of the proposed method.

Identification of a dynamical model of a thermally actuated deformable mirror

Using the subspace identification technique, we identify a finite dimensional, dynamical model of a recently developed prototype of a thermally actuated deformable mirror (TADM). The main advantage of the identified model over the models described by partial differential equations is its low complexity and low dimensionality. Consequently, the identified model can be easily used for high-performance feedback or feed-forward control. The experimental results show good agreement between the dynamical response predicted by the model and the measured response of the TADM.

Predictive control of thermally induced wavefront aberrations

In this paper we experimentally demonstrate the proof of concept for predictive control of thermally induced wavefront aberrations in optical systems. On the basis of the model of thermally induced wavefront aberrations and using only past wavefront measurements, the proposed adaptive optics controller is able to predict and to compensate the future aberrations. Furthermore, the proposed controller is able to correct wavefront aberrations even when some parameters of the prediction model are unknown. The proposed control strategy can be used in high power optical systems, such as optical lithography machines, where the predictive correction of thermally induced wavefront aberrations is a crucial issue.

Sparse solution of the Lyapunov equation for large-scale interconnected systems

We consider the problem of computing an approximate banded solution of the continuous-time Lyapunov equation A ? X ? + X ? A ? T = P ? , where the coefficient matrices A ? and P ? are large, symmetric banded matrices. The (sparsity) pattern of A ? describes the interconnection structure of a large-scale interconnected system. Recently, it has been shown that the entries of the solution X ? are spatially localized or decaying away from a banded pattern. We show that the decay of the entries of X ? is faster if the condition number of A ? is smaller. By exploiting the decay of entries of X ? , we develop two computationally efficient methods for approximating X ? by a banded matrix. For a well-conditioned and sparse banded A ? , the computational and memory complexities of the methods scale linearly with the state dimension. We perform extensive numerical experiments that confirm this, and that demonstrate the effectiveness of the developed methods. The methods proposed in this paper can be generalized to (sparsity) patterns of A ? and P ? that are more general than banded matrices. The results of this paper open the possibility for developing computationally efficient methods for approximating the solution of the large-scale Riccati equation by a sparse matrix.

Iterative learning control of a membrane deformable mirror for optimal wavefront correction

We present an iterative learning control (ILC) algorithm for controlling the shape of a membrane deformable mirror (DM). We furthermore give a physical interpretation of the design parameters of the ILC algorithm. On the basis of this insight, we derive a simple tuning procedure for the ILC algorithm that, in practice, guarantees stable and fast convergence of the membrane to the desired shape. In order to demonstrate the performance of the algorithm, we have built an experimental setup that consists of a commercial membrane DM, a wavefront sensor, and a real-time controller. The experimental results show that, by using the ILC algorithm, we are able to achieve a relatively small error between the real and desired shape of the DM while at the same time we are able to control the saturation of the actuators. Moreover, we show that the ILC algorithm outperforms other control algorithms available in the literature.

An innovative and efficient method to control the shape of push-pull membrane deformable mirror

We carry out performance characterisation of a commercial push and pull deformable mirror with 48 actuators (Adaptica Srl). We present a detailed description of the system as well as a statistical approach on the identification of the mirror influence function. A new efficient control algorithm to induce the desired wavefront shape is also developed and comparison with other control algorithms present in literature has been made to prove the efficiency of the new approach.

Fast and Robust Iterative Learning Control for Lifted Systems

Abstract In this paper we propose a new methodology to synthesize and implement a robust ILC controller for lifted systems. The control law is obtained as the solution of the bounded data uncertainty (BDU) least squares problem, using the bounded model uncertainty in the trial domain. The uncertainty bounds in the trial domain are directly obtained from the bounds on the model uncertainties of the local sample to sample LTI or LTV models, using the randomized algorithm. This way we avoid introduction of unnecessary conservatism in the lifted robust ILC design. Efficient solutions, with the linear complexity in the size of the trial length, are presented for both the BDU least squares problem and for the computation of the uncertainty bounds in the trial domain. The linear complexity is achieved through exploiting the sequentially semi-separable structure of the lifted system matrices. Therefore the proposed framework is especially suitable for the LTI and LTV uncertain systems with large number of samples in the trial.

State Observation and Sensor Selection for Nonlinear Networks

A large variety of dynamical systems, such as chemical and biomolecular systems, can be seen as networks of nonlinear entities. Prediction, control, and identification of such nonlinear networks require knowledge of the state of the system. However, network states are usually unknown, and only a fraction of the state variables are directly measurable. The observability problem concerns reconstructing the network state from this limited information. Here, we propose a general optimization-based approach for observing the states of nonlinear networks and for optimally selecting the observed variables. Our results reveal several fundamental limitations in network observability, such as the tradeoff between the fraction of observed variables and the observation length on one side, and the estimation error on the other side. We also show that, owing to the crucial role played by the dynamics, purely graph-theoretic observability approaches cannot provide conclusions about ones practical ability to estimate the states. We demonstrate the effectiveness of our methods by finding the key components in biological and combustion reaction networks from which we determine the full system state. Our results can lead to the design of novel sensing principles that can greatly advance prediction and control of the dynamics of such networks.