Event-Based Control and Estimation with Stochastic Disturbances

This thesis deals with event-based control and estimation strategies, motivated by certain bottlenecks in the control loop. Two kinds of implementation constraints are considered: closing one or several control loops over a data network, and sensors that report measurements only as intervals (e.g. with quantization). The proposed strategies depend critically on _events_, when a data packet is sent or when a change in the measurement signal is received. The value of events is that they communicate new information about stochastic process disturbances. A data network in the control loop imposes constraints on the event timing, modelled as a minimum time between packets. A thresholdbased control strategy is suggested and shown to be optimal for firstorder systems with impulse control. Different ways to find the optimal threshold are investigated for single and multiple control loops sharing one network. The major gain compared to linear time invariant (LTI) control is with a single loop a greatly reduced communication rate, which with multiple loops can be traded for a similarly reduced regulation error. With the bottleneck that sensors report only intervals, both the theoretical and practical control problems become more complex. We focus on the estimation problem, where the optimal solution is known but untractable. Two simplifications are explored to find a realistic state estimator: reformulation to a mixed stochastic/worst case scenario and joint maximum a posteriori estimation. The latter approach is simplified and evaluated experimentally on a moving cart with quantized position measurements controlled by a low-end microcontroller. The examples considered demonstrate that event-based control considerably outperforms LTI control, when the bottleneck addressed is a genuine performance constraint on the latter

Logarithmic Concave Observers

The problem of (online) state estimation for dynamical systems arises frequently in control. The well known Kalman Filter comprises a coherent theory for the case of linear systems with Gaussian noise, but as soon as either condition is relaxed the picture becomes much less clear. This thesis investigates the case when process disturbances and measurements are relaxed from Gaussian to log-concave. The range of systems that can be analyzed is broadened while still retaining enough structure that many desirable properties are preserved. Strongly log-concave functions are introduced as a means to quantify the Gaussian-like properties of log-concave functions. The main contribution of the thesis is two fundamental theorems, one giving a bound on covariance and the other describing how (strong) log-concavity is preserved and propagated. Applying the theorems to log-concave observers, they are found to have much in common with the Kalman Filter. It is shown that a Kalman Filter can be constructed that gives a conservative bound on the error covariance of the log-concave Bayesian Observer. Event based control is one case where measurements are far from (uncorrelated) Gaussian, but often log-concave. An example of control and state estimation for such a system is pursued throughout the thesis. Using proper consideration a Kalman Filter is found that gives a reasonable approximation of the optimal log-concave observer

Sporadic Event-Based Control using Path Constraints and Moments

Control is traditionally applied using periodic sensing and actuation. In some applications, it is beneficial to use instead event based control, to communicate or make a change only when necessary. There are no known general closed form solutions to such event based control problems. We consider stationary event-based control problems with mixed continuous/discrete time dynamics and stochastic disturbances. The system is modelled by a set of path constraints, which are converted into constraints on trajectories’ moments up to some order N; upper and lower bounds on the control objective for any system that meets the constraints are derived using sum-of-squares techniques and convex semidefinite programming. Joint optimization of upper bound and controller parameters is non-convex in general; approaches to such controller optimization are investigated, including local optimization using bilinear matrix inequalities. Examples show that the bounds are significantly tighter than earlier results obtained using quadratic value functions

Stochastic Event-Based Control and Estimation

Digital controllers are traditionally implemented using periodic sampling, computation, and actuation events. As more control systems are implemented to share limited network and CPU bandwidth with other tasks, it is becoming increasingly attractive to use some form of event-based control instead, where precious events are used only when needed. Forms of event-based control have been used in practice for a very long time, but mostly in an ad-hoc way. Though optimal solutions to most event-based control problems are unknown, it should still be viable to compare performance between suggested approaches in a reasonable manner. This thesis investigates an event-based variation on the stochastic linear-quadratic (LQ) control problem, with a fixed cost per control event. The sporadic constraint of an enforced minimum inter-event time is introduced, yielding a mixed continuous-/discrete-time formulation. The quantitative trade-off between event rate and control performance is compared between periodic and sporadic control. Example problems for first-order plants are investigated, for a single control loop and for multiple loops closed over a shared medium. Path constraints are introduced to model and analyze higher-order event-based control systems. This component-based approach to stochastic hybrid systems allows to express continuous- and discrete-time dynamics, state and switching constraints, control laws, and stochastic disturbances in the same model. Sum-of-squares techniques are then used to find bounds on control objectives using convex semidefinite programming. The thesis also considers state estimation for discrete time linear stochastic systems from measurements with convex set uncertainty. The Bayesian observer is considered given log-concave process disturbances and measurement likelihoods. Strong log-concavity is introduced, and it is shown that the observer preserves log-concavity, and propagates strong log-concavity like inverse covariance in a Kalman filter. A recursive state estimator is developed for systems with both stochastic and set-bounded process and measurement noise terms. A time-varying linear filter gain is optimized using convex semidefinite programming and ellipsoidal over-approximation, given a relative weight on the two kinds of error

Event-Based Control over Networks: Some Research Questions and Preliminary Results

The paper discusses some research questions related to event-based control over networks and presents preliminary results regarding event-based minimum-variance control of first-order systems with specified minimum inter-event times

Scheduling of Event-Triggered Controllers on a Shared Network

We consider a system where a number of independent, time-triggered or event-triggered control loops are closed over a shared communication network. Each plant is described by a first-order linear stochastic system. In the event-triggered case, a sensor at each plant frequently samples the output but attempts to communicate only when the magnitude of the output is above a threshold. Once access to the network has been gained, the network is busy for T seconds (corresponding to the communication delay from sensor to actuator), after which the control action is applied to the plant. Using numerical methods, we compute the minimum-variance control performance under various common MAC-protocols, including TDMA, FDMA, and CSMA (with random, dynamic-priority, or static-priority access). The results show that event-triggered control under CSMA gives the best performance throughout

A Framework for Nonlinear Model Predictive Control in JModelica.org

Nonlinear Model Predictive Control (NMPC) is a control strategy based on repeatedly solving an optimal control problem. In this paper we present a new MPC framework for the JModelica.org platform, developed specifically for use in NMPC schemes. The new framework utilizes the fact that the optimal control problem to be solved does not change between solutions, thus decreasing the computation time needed to solve it. The new framework is compared to the old optimization framework in JModelica.org in regards to computation time and solution obtained through a benchmark on a combined cycle power plant. The results show that the new framework obtains the same solution as the old framework, but in less than half the time

Recursive State Estimation for Linear Systems with Mixed Stochastic and Set-Bounded Disturbances

Recursive state estimation is considered for discrete time linear systems with mixed process and measurement disturbances that have stochastic and (convex) set-bounded terms. The state estimate is formed as a linear combination of initial guess and measurements, giving an estimation error of the same mixed type (and causing minimal interference between the two kinds of error). An ellipsoidal over-approximation to the set-bounded estimation error term allows to formulate a linear matrix inequality (LMI) for optimization of the filtergain, considering both parts of the estimation error in the objective. With purely stochastic disturbances, the standard Kalman Filter is recovered. The state estimator is shown to work well for an event based estimation example, where measurements are very coarsely quantized

NMPC Application using JModelica.org: Features and Performance

Abstract In the past JModelica.org was successfully applied for generating optimal trajectories. Using it for Nonlinear Model Predictive Control (NMPC) is the natural next step and sets high requirements on calculation time. To improve real time capabilities warmstarting of the optimization and elimination of algebraic variables based on Block Lower Triangular (BLT) form were implemented. In performance comparisons, using the example of steam temperature control, a speed-up of the optimization time by a factor of five and of two respectively was measured. The increased efficiency allows application of NMPC to faster systems than before

A simple model for the interference between event-based control loops using a shared medium

Traditionally, control loops are closed using periodic sensing and actuation. When communication resources are scarce, however, much may be gained from transmitting only when something important has happened in the loop. This paper presents a simple model of the interference between event-based control loops caused by sharing a common medium, based on the coupled dynamics of a Markov chain representing the state of the medium, and the processes modeled as integrators disturbed by white noise. The model is simple enough to derive the stationary state distribution at low computational cost using mostly standard linear time-invariant system theory (applied in the spatial dimension), while capturing important aspects of the problem. Control laws are optimized to minimize state variance using the limited communication resources. The results are compared to the less restrictive but unrealistic case of aperiodic control, and to simulations of the system without simplifications