Abstract-We propose a decentralized subspace algorithm for 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 the 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 a subspace identification algorithm that identifies a 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.

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 trade-off 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 graphtheoretic observability approaches cannot provide conclusions about one's 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.

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We consider the problem of computing an approximate banded solution of the continuous-time Lyapunov equation AX+XA 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.

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Abstract: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. , 1999). 25. G. Brusa-Zappellini, A. Riccardi, V. Biliotti, C. Del Vecchio, P. Salinari, P. Stefanini, P. Mantegazza, R. Biasi, M. Andrighettoni, C. Franchini, and D. Gallieni, "Adaptive secondary mirror for the 6.5-m conversion of the multiple mirror telescope: first laboratory testing results," Proc. SPIE 3726, 38-49 (1999).

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. , a TADM has been used to correct static wavefront aberrations. In the above cited paper, a static (steady-state) model of the TADM has been identified and, on the basis of this model, a control action for the TADM has been derived as the solution of a constrained least-squares problem. However, this control strategy requires that the time between two consecutive control iterations is approximately equal to the settling time (or the rise time) of the TADM. Consequently, this wavefront correction strategy is relatively slow and its performance might be additionally degraded in the case of time-varying wavefront aberrations.To achieve fast correction of both static and timevarying wavefront aberrations, the time between control iterations has to be significantly smaller than the TADM's settling time. In such cases, a dynamical model of a TADM has to be developed to accurately correct wavefront aberrations [9,10]. Once this dynamical model has been obtained, model-based control strategies [11,12] can be employed to maximize the performance of the wavefront correction. Apart from the control perspective, a dynamical model of a TADM is important because it can be used to simulate the dynamical behavior of the AO system before the real system has been built.A dynamical model of a TADM must meet two requirements. First, it must accurately capture the TADM's dynamics. Second, to be used for control, it must be relatively simple [13,14] and preferably low dimensional (i.e., it must have a relatively small number of states). However, the dynamics of TADMs are governed by the thermoelastic system of partial differential equations (PDEs) [15]. Furthermore, in the case of the TADMs that have been proposed in [3,7], the thermoelastic system of PDEs must be coupled with the biharmonic plate equation [16]. The dynamical model based on these PDEs is infinite-dimensional and as such is too complex to be used for control. To apply the model-based control strategies of [11,12], a more compact, finite-dimensional model must be developed. One way to develop such a model would be to discretize the system of PDEs and corresponding boundary conditions using the finite element method (FEM) [17]. However, the FEM can be applied only if all physical parameters of the TADM are known. Furthermore, the FEM model is usually high dimensional and thus is still relatively complex to be used for control.In this Letter, we follow another way of model building that is based on system identification techniques [18]. Accordingly, from experimental data, we identify a loworder, state-space model of a recently developed TADM...

We develop a simple and effective control method for accurate control of deformable mirrors (DMs). For a desired DM surface profile and using batches of observed surface profile data, the proposed method adaptively determines both a DM model (influence matrix) and control actions that produce the desired surface profile with good accuracy. In the first iteration, the developed method estimates a DM influence matrix by solving a multivariable least-squares problem. This matrix is then used to compute the control actions by solving a constrained least-squares problem. Then, the computed actions are randomly perturbed and applied to the DM to generate a new batch of surface profile data. The new data batch is used to estimate a new influence matrix that is then used to re-compute control actions. This procedure is repeated until convergence is achieved. The method is experimentally tested on a Boston Micromachines DM with 140 micro-electronic-mechanical-system actuators. Our experimental results show that the developed control approach can achieve accurate correction despite significant DM nonlinearities. Using only a few control iterations, the developed method is able to produce a surface profile root-mean-square error that varies from 5 − 30 [nm] for most of the tested Zernike wave-front modes without using direct feedback control. These results can additionally be improved by using larger data batches and more iterations or by combining the developed approach with feedback control. Finally, as we experimentally demonstrate, the developed method can be used to estimate a DM model that can effectively be used for a single-step open-loop DM control.