H-∞ optimal actuator location

There is often freedom in choosing the location of actuators on systems governed by partial differential equations. The actuator locations should be selected in order to optimize the performance criterion of interest. The main focus of this thesis is to consider H-∞-performance with state-feedback. That is, both the controller and the actuator locations are chosen to minimize the effect of disturbances on the output of a full-information plant. Optimal H-∞-disturbance attenuation as a function of actuator location is used as the cost function. It is shown that the corresponding actuator location problem is well-posed. In practice, approximations are used to determine the optimal actuator location. Conditions for the convergence of optimal performance and the corresponding actuator location to the exact performance and location are provided. Examples are provided to illustrate that convergence may fail when these conditions are not satisfied. Systems of large model order arise in a number of situations; including approximation of partial differential equation models and power systems. The system descriptions are sparse when given in descriptor form but not when converted to standard first-order form. Numerical calculation of H-∞-attenuation involves iteratively solving large H-∞-algebraic Riccati equations (H-∞-AREs) given in the descriptor form. An iterative algorithm that preserves the sparsity of the system description to calculate the solutions of large H-∞-AREs is proposed. It is shown that the performance of our proposed algorithm is similar to a Schur method in many cases. However, on several examples, our algorithm is both faster and more accurate than other methods. The calculation of H-∞-optimal actuator locations is an additional layer of optimization over the calculation of optimal attenuation. An optimization algorithm to calculate H-∞-optimal actuator locations using a derivative-free method is proposed. The results are illustrated using several examples motivated by partial differential equation models that arise in control of vibration and diffusion

Optimal actuator design for the Euler-Bernoulli vibration model based on LQR performance and shape calculus

A method for optimal actuator design in vibration control is presented. The optimal actuator, parametrized as a characteristic function, is found by means of the topological derivative of the LQR cost. An abstract framework is proposed based on the theory for infinite-dimensional optimization of both the actuator shape and the associated control problem. A numerical realization of the optimality condition is developed for the actuator shape using a level-set method for topological derivatives. A numerical example illustrating the design of actuator for Euler-Bernoulli beam model is provided

Infinite-dimensional Kalman Filtering and Sensor Placement Problem

One of the central problems in engineering is to estimate the state of a stochastic dynamic system from limited noisy measurements. A Kalman filter is commonly used for state estimation, which produces an unbiased optimal estimate that minimizes the variance of the estimation error. Many physical processes, such as diffusion and beam vibrations, can be described by partial differential equations. These governing equations may be reformulated mathematically as infinite-dimensional dynamic systems. In this work, the derivation of the Kalman filter for infinite-dimensional linear dynamic systems is reviewed, and the sensor placement problem for Kalman filtering is considered. The optimality criterion for sensor selection and location is to minimize the steady-state error variance, which is shown to be the nuclear norm of an operator that solves an algebraic Riccati equation. Three partial differential equation models are examined: one-dimensional diffusion, simply supported Euler-Bernoulli beam with Kelvin-Voigt damping, and two-dimensional diffusion on an L-shaped region. Optimal sensor locations are calculated for the three models. The sensor noise effects on the state estimation are investigated with the assumption that all the selected sensors are placed optimally. Results show that using multiple low quality sensors can lead to as good an estimate as using a single high quality sensor, provided that enough sensors are used. In particular, for the one-dimensional diffusion equation, approximately proportional relations between the square root of sensor noise variance and the estimation error are observed in simulations

Optimal sensor placement: A robust approach

We address the problem of optimally placing sensor networks for convection-diffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. The paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the well-posedness of the optimization problem and finalizes with a range of numerical tests

H2-Optimal Sensor Location

Optimal sensor placement is an important problem with many applications; placing thermostats in rooms, installing pressure sensors in chemical columns or attaching vibration detection devices to structures are just a few of the examples. Frequently, this placement problem is encountered while noise is present. The H_2-optimal control is a strategy designed for systems that have exogenous disturbing inputs. Therefore, one approach for the optimal sensor location problem is to combine it with the H_2-optimal control. In this work the H_2-optimal control is explained and combined with the sensor placement problem to create the H_2-optimal sensor location problem. The problem is examined for the one-dimensional beam equation and the two-dimensional diffusion equation in an L-shaped region. The optimal sensor location is calculated numerically for both models and multiple scenarios are considered where the location of the disturbance and the actuator are varied. The effect of different model parameters such as the weight of the state and the disturbance are investigated. The results show that the optimal sensor location tends to be close to the disturbance location

Optimal Actuator Design for Nonlinear Systems

For systems modeled by partial differential equations (PDE's), the location and shape of the actuators can be regarded as a design variable and included as part of the controller synthesis procedure. Optimal actuator location is a special case of optimal design. Appropriate actuator location and design can improve performance and significantly reduce the cost of the control in a distributed parameter system. For linear partial differential equations, the existence of an optimal actuator design for a number of cost functions has been established. However, many dynamics are affected by nonlinearities and linearization of the PDE can neglect some important aspects of the model. The existing literature uses the finite-dimensional approximation of nonlinear PDE's to address the optimal actuator design problem. There are new theoretical results on the optimal actuator design of nonlinear PDE's in Banach spaces. This thesis describes new results on optimal actuator design for abstract nonlinear systems on reflexive Banach spaces. Two classes of PDE's have been studied. In the first class, semi-linear systems, a weak continuity assumption on nonlinearities is imposed to establish optimality results. Two examples are provided for this class including nonlinear railway track model and nonlinear wave equation in two space dimensions. The second class is nonlinear parabolic PDE's. For this class, the weak continuity assumption is relaxed at the cost of imposing assumptions on the linear part of the system. The examples provided for this class are Kuramoto-Sivashinsky equation and nonlinear diffusion in two space dimensions. Furthermore, a thorough study of optimal actuator location for nonlinear railway track model was conducted. The study begins with investigating the well-posedness and stability of solutions to this model. It is shown that under certain conditions on inputs, solutions to the railway track model are stable. Further on, using optimization algorithms and numerical schemes, an optimal input and actuator location are computed. Several simulations are run for various physical parameters. The simulations show that the optimal actuator location is not at the center of the track, contrary to a common belief. They also show that an optimally-located actuator significantly improves the performance of the control system

Unified control/structure design and modeling research

To demonstrate the applicability of the control theory for distributed systems to large flexible space structures, research was focused on a model of a space antenna which consists of a rigid hub, flexible ribs, and a mesh reflecting surface. The space antenna model used is discussed along with the finite element approximation of the distributed model. The basic control problem is to design an optimal or near-optimal compensator to suppress the linear vibrations and rigid-body displacements of the structure. The application of an infinite dimensional Linear Quadratic Gaussian (LQG) control theory to flexible structure is discussed. Two basic approaches for robustness enhancement were investigated: loop transfer recovery and sensitivity optimization. A third approach synthesized from elements of these two basic approaches is currently under development. The control driven finite element approximation of flexible structures is discussed. Three sets of finite element basic vectors for computing functional control gains are compared. The possibility of constructing a finite element scheme to approximate the infinite dimensional Hamiltonian system directly, instead of indirectly is discussed

Sensor Choice for Minimum Error Variance Estimation

© 2016 IEEE, Morris, K. A., & Özer, A. ö. (2014). Modeling and Stabilizability of Voltage-Actuated Piezoelectric Beams with Magnetic Effects. SIAM Journal on Control and Optimization, 52(4), 2371–2398. https://doi.org/10.1137/130918319A Kalman filter is optimal in that the variance of the error is minimized by the estimator. It is shown here, in an infinite-dimensional context, that the solution to an operator Riccati equation minimizes the steady-state error variance. This extends a result previously known for lumped parameter systems to distributed parameter systems. It is shown then that minimizing the trace of the Riccati operator is a reasonable criterion for choosing sensor locations. It is then shown that multiple inaccurate sensors, that is, those with large noise variance, can provide as good an estimate as a single highly accurate (but probably more expensive) sensor. Optimal sensor location is then combined with estimator design. A framework for calculation of the best sensor locations using approximations is established and sensor location as well as choice is investigated with three examples. Simulations indicate that the sensor locations do affect the quality of the estimation and that multiple low quality sensors can lead to better estimation than a single high quality sensor.NSERC Discovery Grant US AFOSR grant || FA 9550-16-1-006