Efficient Joint Estimation of States and Sparse Control Inputs in Linear Dynamical Systems
Core Concepts
The paper presents two novel approaches, regularizerbased and Bayesian learningbased, to efficiently estimate the states and sparse control inputs of a linear dynamical system from lowdimensional measurements.
Abstract
The paper focuses on the problem of jointly estimating the states and sparse control inputs of a linear dynamical system (LDS) from lowdimensional (compressive) measurements.
The key highlights are:

Regularizerbased approach:
 Integrates sparsitypromoting priors into the maximum a posteriori (MAP) estimator of the system's states and inputs.
 Explores two priors: ℓ1regularized and reweighted ℓ2regularized optimization problems.
 Relies on Kalman filtering and smoothing for efficient implementation.

Bayesian learningbased approach:
 Uses hierarchical Gaussian priors with unknown variances (hyperparameters) to induce sparsity.
 Explores two techniques: sparse Bayesian learning (SBL) and variational Bayesian (VB) inference.
 Combines Bayesian learning with Kalman smoothing for joint state and input estimation.

Complexity analysis:
 Derives the time and memory complexities of the proposed algorithms.
 Shows that the Bayesian learningbased algorithms have lower complexity compared to the regularizerbased approach.

Extensions:
 Extends the approaches to handle jointly sparse control inputs.
 Provides a similar analysis for the joint sparsity case.
The proposed sparsityaware algorithms can accurately estimate the states and inputs even in the lowdimensional measurement regime, where conventional methods fail. The integration of sparse signal processing techniques with the Kalman smoothing framework is the key innovation of this work.
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Joint State and Sparse Input Estimation in Linear Dynamical Systems
Stats
The system dynamics are governed by the following equations:
xk+1 = Akxk + Bkuk + wk
yk = Ckxk + Dkuk + vk
where xk is the state, uk is the sparse input, yk is the measurement, and wk and vk are the process and measurement noises, respectively.
Quotes
"Sparsity constraints on the control inputs of a linear dynamical system naturally arise in several practical applications such as networked control, computer vision, seismic signal processing, and cyberphysical systems."
"By exploiting input sparsity using sparse recovery techniques and temporal correlation in the state sequence via Kalman smoothing, we achieve highly accurate state and input recovery, despite the system being underdetermined."
Deeper Inquiries
How can the proposed algorithms be extended to handle nonlinear dynamical systems or timevarying system matrices?
The proposed algorithms for joint state and sparse input estimation in linear dynamical systems (LDSs) can be extended to handle nonlinear dynamical systems by employing techniques such as the Extended Kalman Filter (EKF) or Unscented Kalman Filter (UKF). These methods allow for the approximation of nonlinear system dynamics by linearizing the system around the current estimate or using a set of sigma points to capture the mean and covariance of the state distribution.
For timevarying system matrices, the algorithms can be adapted by incorporating a timedependent formulation of the state transition and input matrices. This can be achieved by allowing the matrices (A_k), (B_k), (C_k), and (D_k) to vary with time, thus reflecting the dynamic nature of the system. The Bayesian framework can be maintained by updating the priors and likelihoods at each time step, ensuring that the estimation process remains robust to changes in the system dynamics.
Additionally, the sparsitypromoting priors can be modified to account for the nonlinearities and timevarying characteristics by introducing adaptive hyperparameters that adjust based on the observed data. This would enhance the flexibility of the algorithms, allowing them to better capture the underlying dynamics of the system while still promoting sparsity in the control inputs.
What are the theoretical performance guarantees of the presented approaches in terms of recovery error bounds or convergence rates?
The theoretical performance guarantees of the presented approaches, particularly the regularizerbased and Bayesian learningbased methods, can be analyzed through the lens of statistical learning theory and optimization. For instance, under certain conditions, it can be shown that the recovery error bounds for the estimated states and inputs are proportional to the noise levels in the measurements and the sparsity of the inputs.
Specifically, the use of sparsitypromoting priors, such as the ℓ1 norm in the regularized approach, can lead to recovery guarantees that are similar to those found in compressed sensing literature. These guarantees often state that if the number of measurements is sufficiently large relative to the sparsity level of the inputs, the algorithms can recover the true states and inputs with high probability.
Convergence rates can also be established for the iterative algorithms, such as those based on the Alternating Direction Method of Multipliers (ADMM). The convergence rates typically depend on the properties of the objective function, such as smoothness and strong convexity. For the Bayesian approaches, convergence can be analyzed in terms of the expected loglikelihood, with guarantees that the estimates will converge to the true parameters as the number of iterations increases, provided that the hyperparameters are updated appropriately.
Can the sparsitypromoting priors be further generalized to capture structured sparsity patterns in the control inputs, and how would that affect the estimation performance?
Yes, the sparsitypromoting priors can be generalized to capture structured sparsity patterns in the control inputs by employing group sparsity or block sparsity models. These models allow for the incorporation of prior knowledge about the relationships between different input components, enabling the algorithms to exploit the inherent structure in the data.
For instance, one could use a group ℓ1 norm, which encourages sparsity at the group level rather than individual components. This approach is particularly useful in scenarios where certain groups of inputs are expected to be active together, such as in networked control systems where inputs may correspond to specific nodes or agents.
Incorporating structured sparsity into the estimation framework can significantly enhance estimation performance by improving recovery accuracy and reducing the number of required measurements. The algorithms would be better equipped to distinguish between active and inactive groups of inputs, leading to more robust estimates of the states and inputs. Furthermore, structured sparsity can also lead to improved computational efficiency, as the algorithms can focus on estimating only the relevant groups rather than all individual components, thus reducing the dimensionality of the optimization problem.