We study the problem of actuator and sensor placement in a linear advection partial differential equation (PDE). The problem is motivated by its application to actuator and sensor placement in building systems for the control and detection of a scalar quantity such as temperature and contaminants. We propose a gramian based approach to the problem of actuator and sensor placement. The special structure of the advection PDE is exploited to provide an explicit formula for the controllability and observability gramian in the form of a multiplication operator. The explicit formula for the gramian, as a function of actuator and sensor location, is used to provide test criteria for the suitability of a given sensor and actuator location. Furthermore, the solution obtained using gramian based criteria is interpreted in terms of the flow of the advective vector field. In particular, the almost everywhere stability property of the advective vector field is shown to play a crucial role in deciding the location of actuators and sensors. Simulation results are performed to support the main results of this paper.
Actuator placement is an active field of research which has received significant attention for its applications in complex dynamical networks. In this paper, we study the problem of finding a set of actuator placements minimizing the metric that measures the average energy consumed for state transfer by the controller, while satisfying a structural controllability requirement and a cardinality constraint on the number of actuators allowed. As no computationally efficient methods are known to solve such combinatorial set function optimization problems, two greedy algorithms, forward and reverse…
We consider linear systems subject to packet dropouts and obtain necessary and sufficient conditions for an arbitrary state transfer and state estimation over a finite time instance $T$. The data loss signal is modeled using the Bernoulli random variable. We leverage properties of the Hadamard product in our approach and use the derived necessary and sufficient conditions to compute the probability that an arbitrary state transfer is possible at a specified time instant. Similarly, the probability of finding an exact state estimate is found using the observability counterparts of the results. …
We consider the time-varying actuator placement in continuous time, where the goal is to maximize the trace of the controllability Grammian. A natural relaxation of the problem is to allow the binary $\{0,1\}$ variable indicating whether an actuator is used at a given time to take on values in the closed interval $[0,1]$. We show that all optimal solutions of both the original and the relaxed problems can be given via an explicit formula, and that, as long as the input matrix has no zero columns, the solutions sets of the original and relaxed problem coincide.
Actuator placement is a fundamental problem in control design for large-scale networks. In this paper, we study the problem of finding a set of actuator positions by minimizing a given metric, while satisfying a structural controllability requirement and a constraint on the number of actuators. We first extend the classical forward greedy algorithm for applications to graphs that are not necessarily strongly connected. We then improve this greedy algorithm by extending its horizon. This is done by evaluating the actuator position set expansions at the further steps of the classical greedy algo…
In this paper, we consider the problem of jointly performing online parameter estimation and optimal sensor placement for a partially observed infinite dimensional linear diffusion process. We present a novel solution to this problem in the form of a continuous-time, two-timescale stochastic gradient descent algorithm, which recursively seeks to maximise the log-likelihood with respect to the unknown model parameters, and to minimise the expected mean squared error of the hidden state estimate with respect to the sensor locations. We also provide extensive numerical results illustrating the pe…
This paper describes an optimization framework to control a distributed parameter system (DPS) using a team of mobile actuators. The framework simultaneously seeks optimal control of the DPS and optimal guidance of the mobile actuators such that a cost function associated with both the DPS and the mobile actuators is minimized subject to the dynamics of each. The cost incurred from controlling the DPS is linear-quadratic, which is transformed into an equivalent form as a quadratic term associated with an operator-valued Riccati equation. This equivalent form reduces the problem to seeking for …
The choice of the location of controllers and observations is of great importance for designing control systems and improving the estimations in various practical problems. For time-varying systems in Hilbert spaces, the existence and convergence of the optimal location based on linear-quadratic control on a finite-time horizon is studied. The optimal location of observations for improving the estimation of the state at the final time, based on Kalman filter, is considered as the dual problem to the LQoptimal problem of the control locations. Further, the existence and convergence of optimal l…
The numerical approximation of an optimal control problem with $L^1$-control of a Timoshenko beam is considered and analyzed by using the finite element method. From the practical point of view, inclusion of the $L^1$--norm in the cost functional is interesting in the case of beam vibration model, since the sparsity enforced by the $L^1$--norm is very useful for localizing actuators or control devices. The discretization of the control variables is performed by using piecewise constant functions. The states and the adjoint states are approximated by a \emph{locking free scheme} of linear finit…
In this paper, we consider the problem of optimally guiding a large-scale swarm of underwater vehicles that is tasked with the indirect control of an advection-diffusion environmental field. The microscopic vehicle dynamics are governed by a stochastic differential equation with drift. The drift terms model the self-propelled velocity of the vehicle and the velocity field of the currents. In the mean-field setting, the macroscopic vehicle dynamics are governed by a Kolmogorov forward equation in the form of a linear parabolic advection-diffusion equation. The environmental field is governed by…
There is a rising interest in Spatio-temporal systems described by Partial Differential Equations (PDEs) among the control community. Not only are these systems challenging to control, but the sizing and placement of their actuation is an NP-hard problem on its own. Recent methods either discretize the space before optimziation, or apply tools from linear systems theory under restrictive linearity assumptions. In this work we consider control and actuator placement as a coupled optimization problem, and derive an optimization algorithm on Hilbert spaces for nonlinear PDEs with an additive spat…
In this paper, we establish the almost sure convergence of two-timescale stochastic gradient descent algorithms in continuous time under general noise and stability conditions, extending well known results in discrete time. We analyse algorithms with additive noise and those with non-additive noise. In the non-additive case, our analysis is carried out under the assumption that the noise is a continuous-time Markov process, controlled by the algorithm states. The algorithms we consider can be applied to a broad class of bilevel optimisation problems. We study one such problem in detail, namely…
Sensor and actuator selection problems (SASP) are some of the core problems in dynamic systems design and control. These problems correspond to determining the optimal selection of sensors (measurements) or actuators (control nodes) such that certain estimation/control objectives can be achieved. While the literature on SASP is indeed inveterate, the vast majority of the work focuses on linear(ized) representation of the network dynamics, resulting in the placements of sensors or actuators (SA) that are valid for confined operating regions. As an alternative, herein we propose a new general fr…
We describe in this paper an optimal control strategy for shaping a large-scale swarm of particles using boundary global actuation. This problem arises as a key challenge in many swarm robotics applications, especially when the robots are passive particles that need to be guided by external control fields. The system is large-scale and underactuated, making the control strategy at the microscopic particle level infeasible. We consider the Kolmogorov forward equation associated to the stochastic process of the single particle to encode the macroscopic behaviour of the particles swarm. The contr…
Optimal experimental design (OED) is the general formalism of sensor placement and decisions about the data collection strategy for engineered or natural experiments. This approach is prevalent in many critical fields such as battery design, numerical weather prediction, geosciences, and environmental and urban studies. State-of-the-art computational methods for experimental design, however, do not accommodate correlation structure in observational errors produced by many expensive-to-operate devices such as X-ray machines or radar and satellite retrievals. Discarding evident data correlations…
