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 locations of observations for improving the estimation at the initial time, based on Kalman smoother is discussed. The obtained results are applied to a linear advection-diffusion model.
In this paper, we study the approximate controllability for the stochastic heat equation over measurable sets, and the optimal actuator location of the minimum norm controls. We formulate a relaxed optimization problem for both actuator location and its corresponding minimum norm control into a two-person zero sum game problem and develop a sufficient and necessary condition for the optimal solution via Nash equilibrium. At last, we prove that the relaxed optimal solution is an optimal actuator location for the classical problem.
In this paper, we study optimal actuator location of the minimum norm controls for a multi-dimensional heat equation with control defined in the space $L^p(0,T;L^2(Ω))$. The actuator domain $ω$ is quite general in the sense that it is required only to have a prescribed Lebesgue measure. A relaxation problem is formulated and is transformed into a two-person zero-sum game problem. By the game theory, we develop a necessary and sufficient condition and the existence of relaxed optimal actuator location for $p\in[2,+\infty]$, which is characterized by the Nash equilibrium of the associated game p…
Optimal Actuator Design across Ranks and Control Horizons
Sorbonne Université, Université Paris Cité, CNRS, Inria, Laboratoire Jacques-Louis Lions (LJLL)（索邦大学，巴黎 Cité 大学，法国国家科学研究中心，法国国家信息与自动化研究所，雅克-路易·利翁斯实验室）；Chair for Dynamics, Control, Machine Learning and Numerics (AvH Professorship), Department of Mathematics, FAU Erlangen-Nürnberg（埃尔朗根-纽伦堡大学数学系动力学、控制、机器学习和数值计算讲席（阿·冯·洪堡教授职位））；Chair of Computational Mathematics, Fundación Deusto, Bilbao, Spain（西班牙毕尔巴鄂德乌斯托基金会计算数学讲席）；Departamento de Matemáticas, Universidad Autónoma de Madrid, Spain（西班牙马德里自治大学数学系）
Actuator location and design are important choices in controller design for distributed parameter systems. Semi-linear partial differential equations model a wide spectrum of physical systems with distributed parameters. It is shown that under certain conditions on the nonlinearity and the cost function, an optimal control input together with an optimal actuator choice exists. First-order necessary optimality conditions are derived. The results are applied to optimal actuator and controller design in a nonlinear railway track model as well as semi-linear wave models.
It is shown that an internal control based on a moving indicator function is able to stabilize the state of parabolic equations evolving in rectangular domains. For proving the stabilizability result, we start with a control obtained from an oblique projection feedback based on a finite number of static actuators, then we used the continuity of the state when the control varies in relaxation metric to construct a switching control where at each given instant of time only one of the static actuators is active, finally we construct the moving control by traveling between the static actuators. Nu…
Stabilization of a class of time-varying parabolic equations with uncertain input data using Receding Horizon Control (RHC) is investigated. The diffusion coefficient and the initial function are prescribed as random fields. We consider both cases, uniform and log-normal distributions of the diffusion coefficient. The controls are chosen to be finite-dimensional and enter into the system as a linear combination of finitely many indicator functions (actuators) supported in open subsets of the spatial domain. Under suitable regularity assumptions, we study the expected (averaged) stabilizability…
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…
The present work is concerned with the stabilization of a general class of time-varying linear parabolic equations by means of a finite-dimensional receding horizon control (RHC). The stability and suboptimality of the unconstrained receding horizon framework is studied. The analysis allows the choice of the squared $\ell_1$-norm as control cost. This leads to a nonsmooth infinite-horizon problem which provides stabilizing optimal controls with a low number of active actuators over time. Numerical experiments are given which validate the theoretical results and illustrate the qualitative diffe…
In this paper, we consider the optimal controller design problem against data injection attacks on actuators for an uncertain control system. We consider attacks that aim at maximizing the attack impact while remaining stealthy in the finite horizon. To this end, we use the Conditional Value-at-Risk to characterize the risk associated with the impact of attacks. The worst-case attack impact is characterized using the recently proposed output-to-output $\ell_2$-gain (OOG). We formulate the design problem and observe that it is non-convex and hard to solve. Using the framework of scenario-based …
The stabilizability to trajectories of the Schlögl model is investigated in the norm of the natural state space for strong solutions, which is strictly contained in the standard pivot space of square integrable functions. As actuators a finite number of indicator functions are used and the control input is subject to a bound constraint. A stabilizing saturated explicit feedback control is proposed, where the set of actuators and the input bound are independent of the targeted trajectory. Further, the existence of open-loop optimal stabilizing constrained controls and related first-order optima…
We propose a dual dynamic integer programming (DDIP) framework for solving multi-scale mixed-integer model predictive control (MPC) problems. Such problems arise in applications that involve long horizons and/or fine temporal discretizations as well as mixed-integer states and controls (e.g., scheduling logic and discrete actuators). The approach uses a nested cutting-plane scheme that performs forward and backward sweeps along the time horizon to adaptively approximate cost-to-go functions. The DDIP scheme proposed can handle general MPC formulations with mixed-integer controls and states and…
Due to increasing railway use, the capacity at railway yards and maintenance locations is becoming limiting to accommodate existing rolling stock. To reduce capacity issues at maintenance locations during nighttime, railway undertakings consider performing more daytime maintenance, but the choice at which locations personnel needs to be stationed for daytime maintenance is not straightforward. Among other things, it depends on the planned rolling stock circulation and the maintenance activities that need to be performed. This paper presents the Maintenance Location Choice Problem (MLCP) and pr…
