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

Optimal Actuator Location for Semi-Linear Systems

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 exist. First order necessary optimality conditions are derived. The results are applied to optimal actuator location and controller design in a nonlinear railway track model.NSER

Estimation for Linear and Semi-linear Infinite-dimensional Systems

Estimating the state of a system that is not fully known or that is exposed to noise has been an intensely studied problem in recent mathematical history. Such systems are often modelled by either ordinary differential equations, which evolve in finite-dimensional state-spaces, or partial differential equations, the state-space of which is infinite-dimensional. The Kalman filter is a minimal mean squared error estimator for linear finite-dimensional and linear infinite-dimensional systems disturbed by Wiener processes, which are stochastic processes representing the noise. For nonlinear finite-dimensional systems the extended Kalman filter is a widely used extension thereof which relies on linearization of the system. In all cases the Kalman filter consists of a differential or integral equation coupled with a Riccati equation, which is an equation that determines the optimal estimator gain. This thesis proposes an estimator for semi-linear infinite-dimensional systems. It is shown that under some conditions such a system can also be coupled with a Riccati equation. To motivate this result, the Kalman filter for finite-dimensional and infinite-dimensional systems is reviewed, as well as the corresponding theory for both stochastic processes and infinite-dimensional systems. Important results concerning the infinite-dimensional Riccati equation are outlined and existence of solutions for a class of semi-linear infinite-dimensional systems is established. Finally the well-posedness of the coupling between a semi-linear infinite-dimensional system with a Riccati equation is proven using a fixed point argument

Optimal Locations of Sensors and Actuators for Control of a Pedestrian Bridge

Lightweight structures have been increasing in popularity in structural designs. However, they are more prone to disturbances. Therefore, a controller device can be placed on the structure to control the excessive vibrations. Past works have revealed that there exist corresponding optimal locations on the structure for placing the crucial parts of the controller device, the sensor and the actuator, that result in optimal performance of the controller. The physically practical collocated sensor/actuator design controller device can be moved around and deployed to different structures. The ideally more optimal non-collocated sensor/actuator design allows the sensor and the actuator to be placed separately in their corresponding optimal locations, but it may be physically impractical to implement. Hence, this motivates the study of the optimal locations, and to compare the performances of the non-collocated and the collocated sensor/actuator designs for a lightweight aluminum pedestrian bridge subject to pedestrian walking disturbances. The structure is modelled using the Euler-Bernoulli beam theory, and modal and Hermite basis finite element approximations are applied. The linear-quadratic performance objective control (LQ control) is reviewed and applied. Since approximations are applied, a mapping for the state energy weight in the LQ control performance objective functional from the original functional space to a generic approximation functional space is presented in this thesis. In the preliminary problem in this thesis, influences of the state weights and the disturbances' spatial distributions on the non-collocated and the collocated sensor/actuator designed linear-quadratic Gaussian (LQG) controllers' optimal locations and comparisons of the performances at their optimal locations are studied on a simplified system model with a Gaussian temporally distributed disturbance. Numerical implementation of disturbances is presented, and numerical complications are discussed and provided with solutions. The comparisons of the non-collocated and collocated sensor/actuator designs for a more realistic bridge model are made using three different state weights. The realistic bridge model is approximated using the Hermite basis finite element approximation. The H2-controller is reviewed and applied. The actuator device dynamics and its noise, a reliable pedestrian loading, and a low pass filter are included in this model to consider more realistic disturbances. The results suggest that the physically more practical collocated sensor/actuator design can achieve similar performances as the ideally more optimal non-collocated sensor/actuator design at their corresponding optimal locations