CY
C. Yu
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1
In this paper, a unified identification framework called constrained subspace method for structured state-space models (COSMOS) is presented, where the structure is defined by a user specified linear or polynomial parametrization. The new approach operates directly from the input and output data, which differs from the traditional two-step method that first obtains a state-space realization followed by the systemparameter estimation. The new identification framework relies on a subspace inspired linear regression problem which may not yield a consistent estimate in the presence of process noise. To alleviate this problem, the linear regression formulation is imposed by structured and low rank constraints in terms of a finite set of system Markov parameters and the user specified model parameters. The non-convex nature of the constrained optimization problem is dealt with by transforming the problem into a difference-of-convex optimization problem, which is then handled by the sequential convex programming strategy. Numerical simulation examples show that the proposed identification method is more robust than the classical prediction-error method (PEM) initialized by random initial values in converging to local minima, but at the cost of heavier computational burden.
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In this paper, a unified identification framework called constrained subspace method for structured state-space models (COSMOS) is presented, where the structure is defined by a user specified linear or polynomial parametrization. The new approach operates directly from the input and output data, which differs from the traditional two-step method that first obtains a state-space realization followed by the systemparameter estimation. The new identification framework relies on a subspace inspired linear regression problem which may not yield a consistent estimate in the presence of process noise. To alleviate this problem, the linear regression formulation is imposed by structured and low rank constraints in terms of a finite set of system Markov parameters and the user specified model parameters. The non-convex nature of the constrained optimization problem is dealt with by transforming the problem into a difference-of-convex optimization problem, which is then handled by the sequential convex programming strategy. Numerical simulation examples show that the proposed identification method is more robust than the classical prediction-error method (PEM) initialized by random initial values in converging to local minima, but at the cost of heavier computational burden.
This paper considers the identification of a network consisting of discrete-time LTI systems that are interconnected by their unmeasurable states. For a large-scale network, the computational burden prevents a centralized solution. To cope with this problem, a subspace-based local identification approach using local observations is presented, which consists of subspace intersection operations in both the temporal and spatial domains. Sufficient conditions are provided for the consistent identification of the presented identification approach. Finally, the implementation of this approach on 1D network is specially investigated and numerical simulations are provided to show its effectiveness.
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This paper considers the identification of a network consisting of discrete-time LTI systems that are interconnected by their unmeasurable states. For a large-scale network, the computational burden prevents a centralized solution. To cope with this problem, a subspace-based local identification approach using local observations is presented, which consists of subspace intersection operations in both the temporal and spatial domains. Sufficient conditions are provided for the consistent identification of the presented identification approach. Finally, the implementation of this approach on 1D network is specially investigated and numerical simulations are provided to show its effectiveness.
Affinely parametrized state-space models
Ways to maximize the Likelihood Function
Using Maximum Likelihood (or Prediction Error) methods to identify linear state space model is a prime technique. The likelihood function is a nonconvex function and care must be exercised in the numerical maximization. Here the focus will be on affine parameterizations which allow some special techniques and algorithms. Three approaches to formulate and perform the maximization are described in this contribution: (1) The standard and well known Gauss-Newton iterative search, (2) a scheme based on the EM (expectation-maximization) technique, which becomes especially simple in the affine parameterization case, and (3) a new approach based on lifting the problem to a higher dimension in the parameter space and introducing rank constraints.
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Using Maximum Likelihood (or Prediction Error) methods to identify linear state space model is a prime technique. The likelihood function is a nonconvex function and care must be exercised in the numerical maximization. Here the focus will be on affine parameterizations which allow some special techniques and algorithms. Three approaches to formulate and perform the maximization are described in this contribution: (1) The standard and well known Gauss-Newton iterative search, (2) a scheme based on the EM (expectation-maximization) technique, which becomes especially simple in the affine parameterization case, and (3) a new approach based on lifting the problem to a higher dimension in the parameter space and introducing rank constraints.
The objective of adaptive optics (AO) system control is to design an output feedback controller to reduce the adverse effect of the phase aberration caused by the atmospheric turbulence. As the size of the telescope or AO system becomes larger and larger, how to improve the efficiency of the controller execution becomes an urgent but challenging problem. To this end, this paper presents a structured and sparse controller design method for the large-scale AO systems. A Kronecker structured turbulent phase model, inspired by the frozen-flow movement of the atmospheric turbulence, is developed first, following the design of a sparse controller gain under the H₂-norm optimal control framework. Based on the Kronecker structured system matrices and the sparse controller gain, the obtained dynamical controller has a linear execution complexity in the dimension of the turbulent phase, which is even lower than the standard matrix-vector multiplication method. Since the proposed method is a preliminary result, which cannot be directly used in a telescope today, its performance is demonstrated by numerical simulations only.
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The objective of adaptive optics (AO) system control is to design an output feedback controller to reduce the adverse effect of the phase aberration caused by the atmospheric turbulence. As the size of the telescope or AO system becomes larger and larger, how to improve the efficiency of the controller execution becomes an urgent but challenging problem. To this end, this paper presents a structured and sparse controller design method for the large-scale AO systems. A Kronecker structured turbulent phase model, inspired by the frozen-flow movement of the atmospheric turbulence, is developed first, following the design of a sparse controller gain under the H₂-norm optimal control framework. Based on the Kronecker structured system matrices and the sparse controller gain, the obtained dynamical controller has a linear execution complexity in the dimension of the turbulent phase, which is even lower than the standard matrix-vector multiplication method. Since the proposed method is a preliminary result, which cannot be directly used in a telescope today, its performance is demonstrated by numerical simulations only.
Identification of structured state-space (gray-box) model is popular for modeling physical and network systems. Due to the non-convex nature of the gray-box identification problem, good initial parameter estimates are crucial for successful applications. In this paper, the non-convex gray-box identification problem is reformulated as a structured low-rank matrix factorization problem by exploiting the rank and structured properties of a block Hankel matrix constructed by the system impulse response. To address the low-rank optimization problem, it is first transformed into a difference-of-convex (DC) formulation and then solved using the sequentially convex relaxation method. Compared with the classical gray-box identification methods like the prediction-error method (PEM), the new approach turns out to be more robust against converging to non-global minima, as supported by a simulation study. The developed identification can either be directly used for gray-box identification or provide an initial parameter estimate for the PEM.
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Identification of structured state-space (gray-box) model is popular for modeling physical and network systems. Due to the non-convex nature of the gray-box identification problem, good initial parameter estimates are crucial for successful applications. In this paper, the non-convex gray-box identification problem is reformulated as a structured low-rank matrix factorization problem by exploiting the rank and structured properties of a block Hankel matrix constructed by the system impulse response. To address the low-rank optimization problem, it is first transformed into a difference-of-convex (DC) formulation and then solved using the sequentially convex relaxation method. Compared with the classical gray-box identification methods like the prediction-error method (PEM), the new approach turns out to be more robust against converging to non-global minima, as supported by a simulation study. The developed identification can either be directly used for gray-box identification or provide an initial parameter estimate for the PEM.
Many recently developed data-driven fault estimation methods are restricted to minimum-phase systems so that their practical applications are limited. In this paper, the data-driven fault estimation for non-minimum phase (NMP) systems is studied, for which the main difficulty is that the unstable zeros of an NMP system will result in a growing fault-estimation error. To deal with this problem, the inverse of an NMP system is equivalently formulated as a mixed causal and anti-causal system, and the proposed fault estimator is the sum of a stable causal filter and a stable anti-causal filter. The proposed fault estimator is shown to be asymptotically unbiased and its performance is demonstrated by numerical simulations.
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Many recently developed data-driven fault estimation methods are restricted to minimum-phase systems so that their practical applications are limited. In this paper, the data-driven fault estimation for non-minimum phase (NMP) systems is studied, for which the main difficulty is that the unstable zeros of an NMP system will result in a growing fault-estimation error. To deal with this problem, the inverse of an NMP system is equivalently formulated as a mixed causal and anti-causal system, and the proposed fault estimator is the sum of a stable causal filter and a stable anti-causal filter. The proposed fault estimator is shown to be asymptotically unbiased and its performance is demonstrated by numerical simulations.
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.
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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.
Gray-box identification is prevalent in modeling physical and networked systems. However, due to the non-convex nature of the gray-box identification problem, good initial parameter estimates are crucial for a successful application. In this paper, a new identification method is proposed by exploiting the low-rank and structured Hankel matrix of impulse response. This identification problem is recasted into a difference-of-convex programming problem, which is then solved by the sequential convex programming approach with the associated initialization obtained by nuclear-norm optimization. The presented method aims to achieve the maximum impulse-response fitting while not requiring additional (non-convex) conditions to secure non-singularity of the similarity transformation relating the given state-space matrices to the gray-box parameterized ones. This overcomes a persistent shortcoming in a number of recent contributions on this topic, and the new method can be applied for the structured state-space realization even if the involved system parameters are unidentifiable. The method can be used both for directly estimating the gray-box parameters and for providing initial parameter estimates for further iterative search in a conventional gray-box identification setup.
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Gray-box identification is prevalent in modeling physical and networked systems. However, due to the non-convex nature of the gray-box identification problem, good initial parameter estimates are crucial for a successful application. In this paper, a new identification method is proposed by exploiting the low-rank and structured Hankel matrix of impulse response. This identification problem is recasted into a difference-of-convex programming problem, which is then solved by the sequential convex programming approach with the associated initialization obtained by nuclear-norm optimization. The presented method aims to achieve the maximum impulse-response fitting while not requiring additional (non-convex) conditions to secure non-singularity of the similarity transformation relating the given state-space matrices to the gray-box parameterized ones. This overcomes a persistent shortcoming in a number of recent contributions on this topic, and the new method can be applied for the structured state-space realization even if the involved system parameters are unidentifiable. The method can be used both for directly estimating the gray-box parameters and for providing initial parameter estimates for further iterative search in a conventional gray-box identification setup.
The continuous-time subspace identification using state-variable filtering has been investigated for a long time. Due to the simple orthogonal basis functions that were adopted by the existing methods, the identification performance is quite sensitive to the selection of the system-dynamic parameter associated with an orthogonal basis. To cope with this problem, a subspace identification method using generalized orthonormal (Takenaka-Malmquist) basis functions is developed, which has the potential to perform better than the existing state-variable filtering methods since the adopted Takenaka-Malmquist basis has more degree of freedom in selecting the system-dynamic parameters. As a price for the flexibility of the generalized orthonormal bases, the transformed state-space model is time-varying or parameter-varying which cannot be identified using traditional subspace identification methods. To this end, a new subspace identification algorithm is developed by exploiting the structural properties of the time-variant system matrices, which is then validated by numerical simulations.
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The continuous-time subspace identification using state-variable filtering has been investigated for a long time. Due to the simple orthogonal basis functions that were adopted by the existing methods, the identification performance is quite sensitive to the selection of the system-dynamic parameter associated with an orthogonal basis. To cope with this problem, a subspace identification method using generalized orthonormal (Takenaka-Malmquist) basis functions is developed, which has the potential to perform better than the existing state-variable filtering methods since the adopted Takenaka-Malmquist basis has more degree of freedom in selecting the system-dynamic parameters. As a price for the flexibility of the generalized orthonormal bases, the transformed state-space model is time-varying or parameter-varying which cannot be identified using traditional subspace identification methods. To this end, a new subspace identification algorithm is developed by exploiting the structural properties of the time-variant system matrices, which is then validated by numerical simulations.
This note considers the identification of large-scale 1D networks consisting of identical LTI dynamical systems. A subspace identification method is developed that only uses local input-output information and does not rely on knowledge about the local state interaction. The proposed identification method estimates the Markov parameters of a locally lifted system, following the state-space realization of a single subsystem. The Markov-parameter estimation is formulated as a rank minimization problem by exploiting the low-rank property and a two-layer Toeplitz structural property in the data equation, while the state-space realization of a single subsystem is formulated as a structured low-rank matrix factorization problem. The effectiveness of the proposed identification method is demonstrated by simulation examples.
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This note considers the identification of large-scale 1D networks consisting of identical LTI dynamical systems. A subspace identification method is developed that only uses local input-output information and does not rely on knowledge about the local state interaction. The proposed identification method estimates the Markov parameters of a locally lifted system, following the state-space realization of a single subsystem. The Markov-parameter estimation is formulated as a rank minimization problem by exploiting the low-rank property and a two-layer Toeplitz structural property in the data equation, while the state-space realization of a single subsystem is formulated as a structured low-rank matrix factorization problem. The effectiveness of the proposed identification method is demonstrated by simulation examples.
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.
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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.
The identification of a 1D heterogenous network with unmeasurable interconnections between neighboring systems is studied in this paper. For a large-scale networked system, it is usually computationally prohibitive to identify the global system in a centralized manner. To cope with this problem, the local identification of a network using local input-output data is considered in this paper, and a subspace identification method is developed for the identification of individual systems operating in the network. A simulation example is given to validate the proposed identification method.
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The identification of a 1D heterogenous network with unmeasurable interconnections between neighboring systems is studied in this paper. For a large-scale networked system, it is usually computationally prohibitive to identify the global system in a centralized manner. To cope with this problem, the local identification of a network using local input-output data is considered in this paper, and a subspace identification method is developed for the identification of individual systems operating in the network. A simulation example is given to validate the proposed identification method.
This note provides an instrumental-variable nuclear-norm subspace identification (IV-N2SID) method for the identification of state-space models with measurement noise. The key difference of the proposed method against the classical N2SID method is that the measurement-noise influence is suppressed by the IV technique rather than the regularization scheme for which the regularization parameter is usually difficult to tune.
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This note provides an instrumental-variable nuclear-norm subspace identification (IV-N2SID) method for the identification of state-space models with measurement noise. The key difference of the proposed method against the classical N2SID method is that the measurement-noise influence is suppressed by the IV technique rather than the regularization scheme for which the regularization parameter is usually difficult to tune.
In this paper, we study the deterministic blind identification of multiple channel state-space models having a common unknown input using measured output signals that are perturbed by additive white noise sequences. Different from traditional blind identification problems, the considered system is an autoregressive system rather than an FIR system; hence, the concerned identification problem is more challenging but possibly having a wider scope of application. Two blind identification methods are presented for multi-channel autoregressive systems. A cross-relation identification method is developed by exploiting the mutual references among different channels. It requires at least three channel systems with square and stably invertible transfer matrices. Moreover, a general subspace identification method is developed for which two channel systems are sufficient for the blind identification; however, it requires the additive noises to have identical variances and the transfer matrices having no transmission zeros. Finally, numerical simulations are carried out to demonstrate the performance of the proposed identification algorithms.
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In this paper, we study the deterministic blind identification of multiple channel state-space models having a common unknown input using measured output signals that are perturbed by additive white noise sequences. Different from traditional blind identification problems, the considered system is an autoregressive system rather than an FIR system; hence, the concerned identification problem is more challenging but possibly having a wider scope of application. Two blind identification methods are presented for multi-channel autoregressive systems. A cross-relation identification method is developed by exploiting the mutual references among different channels. It requires at least three channel systems with square and stably invertible transfer matrices. Moreover, a general subspace identification method is developed for which two channel systems are sufficient for the blind identification; however, it requires the additive noises to have identical variances and the transfer matrices having no transmission zeros. Finally, numerical simulations are carried out to demonstrate the performance of the proposed identification algorithms.
This paper studies the identification of ARMA systems with colored measurement noises using finite-level quantized observations. Compared with the case under colorless noises, this problem is more challenging. Our approach is to jointly design an adaptive quantizer and a recursive estimator to identify system parameters. Specifically, the quantizer uses the latest estimate to adjust its thresholds, and the estimator is updated by using quantized observations. To accommodate the temporal correlations of quantization errors and measurement noises, we construct a second-order statistics equivalent system, from which the original ARMA system is identified. The associated identifiability problem and convergence are analyzed as well. Finally, numerical simulations are performed to demonstrate the effectiveness of the proposed algorithm.
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This paper studies the identification of ARMA systems with colored measurement noises using finite-level quantized observations. Compared with the case under colorless noises, this problem is more challenging. Our approach is to jointly design an adaptive quantizer and a recursive estimator to identify system parameters. Specifically, the quantizer uses the latest estimate to adjust its thresholds, and the estimator is updated by using quantized observations. To accommodate the temporal correlations of quantization errors and measurement noises, we construct a second-order statistics equivalent system, from which the original ARMA system is identified. The associated identifiability problem and convergence are analyzed as well. Finally, numerical simulations are performed to demonstrate the effectiveness of the proposed algorithm.
This paper studies the local identification of large-scale homogeneous systems
with 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.
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This paper studies the local identification of large-scale homogeneous systems
with 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.
This paper studies the local subspace identification of 1D homogeneous networked systems. The main challenge lies at the unmeasurable interconnection signals between neighboring subsystems. Since there are many unknown inputs to the concerned local system, the corresponding identification problem is semi-blind. To cope with this problem, a nuclear norm optimization based subspace identification is presented, which is carried out for solving the Markov parameters of a locally lifted system, followed by determining the system matrices of a single subsystem. In the step of Markov parameter estimation, we form a nuclear norm regularized optimization problem which can well handle the adverse effects of the unknown system inputs as long as the number of unknown system inputs is relatively small. In the step of system realization, we again derive a nuclear norm regularized optimization formulation which can cope with the under-determinedness of the realization problem. In the end, the overall identification algorithm is summarized.
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This paper studies the local subspace identification of 1D homogeneous networked systems. The main challenge lies at the unmeasurable interconnection signals between neighboring subsystems. Since there are many unknown inputs to the concerned local system, the corresponding identification problem is semi-blind. To cope with this problem, a nuclear norm optimization based subspace identification is presented, which is carried out for solving the Markov parameters of a locally lifted system, followed by determining the system matrices of a single subsystem. In the step of Markov parameter estimation, we form a nuclear norm regularized optimization problem which can well handle the adverse effects of the unknown system inputs as long as the number of unknown system inputs is relatively small. In the step of system realization, we again derive a nuclear norm regularized optimization formulation which can cope with the under-determinedness of the realization problem. In the end, the overall identification algorithm is summarized.