Vasileios Tzoumas

Minimal Actuator Placement With Bounds on Control Effort

We address the problem of minimal actuator placement in a linear system subject to an average control energy bound. First, following the recent work of Olshevsky, we prove that this is NP-hard. Then, we provide an efficient algorithm which, for a given range of problem parameters, approximates up to a multiplicative factor of O(log n), with n being the network size, any optimal actuator set that meets the same energy criteria; this is the best approximation factor one can achieve in polynomial time in the worst case. Moreover, the algorithm uses a perturbed version of the involved control energy metric, which we prove to be supermodular. Next, we focus on the related problem of cardinality-constrained actuator placement for minimum control effort, where the optimal actuator set is selected so that an average input energy metric is minimized. While this is also an NP-hard problem, we use our proposed algorithm to efficiently approximate its solutions as well. Finally, we run our algorithms over large random networks to illustrate their efficiency.

Minimal actuator placement with optimal control constraints

We introduce the problem of minimal actuator placement in a linear control system so that a bound on the minimum control effort for a given state transfer is satisfied while controllability is ensured. We first show that this is an NP-hard problem following the recent work of Olshevsky [1]. Next, we prove that this problem has a supermodular structure. Afterwards, we provide an efficient algorithm that approximates up to a multiplicative factor of O(log n), where n is the size of the multi-agent network, any optimal actuator set that meets the specified energy criterion. Moreover, we show that this is the best approximation factor one can achieve in polynomial-time for the worst case. Finally, we test this algorithm over large Erdös-Rényi random networks to further demonstrate its efficiency.

Sensor placement for optimal Kalman filtering: Fundamental limits, submodularity, and algorithms

In this paper, we focus on sensor placement in linear dynamic estimation, where the objective is to place a small number of sensors in a system of interdependent states so to design an estimator with a desired estimation performance. In particular, we consider a linear time-variant system that is corrupted with process and measurement noise, and study how the selection of its sensors affects the estimation error of the corresponding Kalman filter over a finite observation interval. Our contributions are threefold: First, we prove that the minimum mean square error of the Kalman filter decreases only linearly as the number of sensors increases. That is, adding extra sensors so to reduce this estimation error is ineffective, a fundamental design limit. Similarly, we prove that the number of sensors grows linearly with the systems size for fixed minimum mean square error and number of output measurements over an observation interval; this is another fundamental limit, especially for systems where the systems size is large. Second, we prove that the log det of the error covariance of the Kalman filter, which captures the volume of the corresponding confidence ellipsoid, with respect to the systems initial condition and process noise is a supermodular and non-increasing set function in the choice of the sensor set. Therefore, it exhibits the diminishing returns property. Third, we provide an efficient approximation algorithm that selects a small number sensors so to optimize the Kalman filter with respect to this estimation error -the worst-case performance guarantees of this algorithm are provided as well.

Minimal reachability problems

In this paper, we address a collection of state space reachability problems, for linear time-invariant systems, using a minimal number of actuators. In particular, we design a zero-one diagonal input matrix B, with a minimal number of non-zero entries, so that a specified state vector is reachable from a given initial state. Moreover, we design a B so that a system can be steered either into a given subset, or sufficiently close to a desired state. This work extends the results of [1] and [2], where a zero-one diagonal or column matrix B is constructed so that the involved system is controllable. Specifically, we prove that the first two of our aforementioned problems are NP-hard; these results hold for a zero-one column matrix B as well. Then, we provide efficient algorithms for their general solution, along with their worst case approximation guarantees. Finally, we illustrate their performance over large random networks.

Near-optimal sensor scheduling for batch state estimation: Complexity, algorithms, and limits

In this paper, we focus on batch state estimation for linear systems. This problem is important in applications such as environmental field estimation, robotic navigation, and target tracking. Its difficulty lies on that limited operational resources among the sensors, e.g., shared communication bandwidth or battery power, constrain the number of sensors that can be active at each measurement step. As a result, sensor scheduling algorithms must be employed. Notwithstanding, current sensor scheduling algorithms for batch state estimation scale poorly with the system size and the time horizon. In addition, current sensor scheduling algorithms for Kalman filtering, although they scale better, provide no performance guarantees or approximation bounds for the minimization of the batch state estimation error. In this paper, one of our main contributions is to provide an algorithm that enjoys both the estimation accuracy of the batch state scheduling algorithms and the low time complexity of the Kalman filtering scheduling algorithms. In particular: 1) our algorithm is near-optimal: it achieves a solution up to a multiplicative factor 1/2 from the optimal solution, and this factor is close to the best approximation factor 1/e one can achieve in polynomial time for this problem; 2) our algorithm has (polynomial) time complexity that is not only lower than that of the current algorithms for batch state estimation; it is also lower than, or similar to, that of the current algorithms for Kalman filtering. We achieve these results by proving two properties for our batch state estimation error metric, which quantifies the square error of the minimum variance linear estimator of the batch state vector: a) it is supermodular in the choice of the sensors; b) it has a sparsity pattern (it involves matrices that are block tri-diagonal) that facilitates its evaluation at each sensor set.

Scheduling nonlinear sensors for stochastic process estimation

In this paper, we focus on activating only a few sensors, among many available, to estimate the batch state of a stochastic process of interest. This problem is important in applications such as target tracking and simultaneous localization and mapping (SLAM), and in general, in problems where we need to have a good estimate of the trajectory taken so far, e.g., for linearisation purposes. It is challenging since it involves stochastic systems whose evolution is largely unknown, sensors with nonlinear measurements, and limited operational resources that constrain the number of active sensors at each measurement step. We provide an algorithm applicable to general stochastic processes and nonlinear measurements whose time complexity is linear in the planning horizon and whose performance is up to a multiplicative factor 1/2 away from the optimal performance. This is notable because the algorithm offers a significant computational advantage over the polynomial-time algorithm that achieves the best approximation factor 1/e. In addition, for important classes of Gaussian processes and nonlinear measurements corrupted with Gaussian noise, our algorithm enjoys the same time complexity as the state-of-the-art algorithms for linear systems and measurements. We achieve our results by proving two properties for the entropy of the batch state vector conditioned on the measurements: a) it is supermodular in the choice of the sensors; b) it has a sparsity pattern (involves block tri-diagonal matrices) that facilitates its evaluation at each sensor set.