Sebastian Trimpe

Event-based state estimation with variance-based triggering

An event-based state estimation scenario is considered where multiple distributed sensors sporadically transmit observations of a linear process to a time-varying Kalman filter via a common bus. The triggering decision is based on the estimation variance: each sensor runs a copy of the Kalman filter and transmits its measurement only if the associated measurement prediction variance exceeds a tolerable threshold. The resulting variance iteration is a new type of Riccati equation, with switching between modes that correspond to the available measurements and depend on the variance at the previous step. Convergence of the switching Riccati equation to periodic solutions is observed in simulations, and proven for the case of an unstable scalar system (under certain assumptions). The proposed method can be implemented in two different ways: as an event-based scheme where transmit decisions are made online, or as a time-based periodic transmit schedule if a periodic solution to the switching Riccati equation is found.

An Experimental Demonstration of a Distributed and Event-Based State Estimation Algorithm ?

Abstract A distributed state estimation algorithm that makes use of model-based predictions to reduce communication requirements in a networked control architecture is tested on an unstable system. A cube balancing on one of its edges serves as the test platform, and six rotating bodies on the cubes inner faces constitute the agents in the control network. Each agent carries a computational unit, which runs estimation and control algorithms, and is associated with local sensors and an actuator. Measurement data is shared among the agents over a broadcast network. Each agent maintains two estimates of the system state: the first reflecting the common knowledge in the network, and the second additionally including all local sensor information. An agents sensor measurement is only broadcast if it deviates from the common estimate of that measurement by more than a specified threshold. Experimental results show that the number of communicated measurements required for stabilizing the system can be significantly reduced with this event-based communication protocol.

Event-Based State Estimation with Switching Static-Gain Observers ⋆

An event-based state estimation problem is considered where the state of a dynamic system is observed from multiple distributed sensors that sporadically transmit their measurements to a remote estimator over a common bus. The common bus allows each sensor to run a copy of the remote estimator and to make the triggering decision based on this estimate: a measurement is transmitted only if its prediction by the estimator deviates by more than a tunable threshold. The event-based estimator is a switching observer that mimics a Luenberger observer with full communication of all measurements. It is proven that the difference between the event-based estimator and its full communication counterpart is bounded. The reduction of average sensor communication rates achieved by using the event-based state estimator for feedback control is demonstrated in experiments on a balancing cube.

Accelerometer-based tilt estimation of a rigid body with only rotational degrees of freedom

An estimation algorithm is developed for determining pitch and roll angles (tilt) of a rigid body fixed at a pivot point using multiple accelerometers. The estimate is independent of the rigid body dynamics; the method is applicable both in static conditions and for any dynamic motion of the body. No dynamic model is required for the estimator; only the mounting positions of the sensors need to be known. The proposed estimator is the optimal linear estimate in a least-squares sense if knowledge of the system dynamics is not used. The estimate may be used as a basis for further filtering and fusion techniques, such as sensor fusion with rate gyro data. The estimation algorithm is applied to the problem of state estimation for the Balancing Cube, a rigid structure that can actively balance on its corners. Experimental results are provided.

The Balancing Cube: A Dynamic Sculpture As Test Bed for Distributed Estimation and Control

The balancing cube is a dynamic sculpture that can balance autonomously on any of its edges or corners (see Figures 14). When standing on a corner, the cube represents a three-dimensional (3-D) inverted pendulum with multiple actuation, sensing, and control units that are interconnected over a communication network. The main structural components are the cube body (a rigid aluminum structure with a cubic shape) and six identical rotating arms located on each of the cubes inner faces. The rotating arms are self-contained units carrying sensors, actuation, a computer, and a battery. Due to their modular design, these units are referred to as modules. As they rotate, they shift the overall center of mass of the system, exert forces on the cube structure, and can, as a result, influence the cubes motion. The modules constitute the agents in the distributed and networked control system; their joint objective is the stabilization of the cube. A video of the cube can be found on the project Web site [1].

Automatic LQR tuning based on Gaussian process global optimization

This paper proposes an automatic controller tuning framework based on linear optimal control combined with Bayesian optimization. With this framework, an initial set of controller gains is automatically improved according to a pre-defined performance objective evaluated from experimental data. The underlying Bayesian optimization algorithm is Entropy Search, which represents the latent objective as a Gaussian process and constructs an explicit belief over the location of the objective minimum. This is used to maximize the information gain from each experimental evaluation. Thus, this framework shall yield improved controllers with fewer evaluations compared to alternative approaches. A seven-degree-of-freedom robot arm balancing an inverted pole is used as the experimental demonstrator. Results of two- and four-dimensional tuning problems highlight the methods potential for automatic controller tuning on robotic platforms.

On the choice of the event trigger in event-based estimation

In event-based state estimation, the event trigger decides whether or not a measurement is used for updating the state estimate. In a remote estimation scenario, this allows for trading off estimation performance for communication, and thus saving resources. In this paper, popular event triggers for estimation, such as send-on-delta (SoD), measurement-based triggering (MBT), variance-based triggering (VBT), and relevant sampling (RS), are compared for the scenario of a scalar linear process with Gaussian noise. First, the analysis of the information pattern underlying the triggering decision reveals a fundamental advantage of triggers employing the real-time measurement in their decision (such as MBT, RS) over those that do not (VBT). Second, numerical simulation studies support this finding and, moreover, provide a quantitative evaluation of the triggers in terms of their average estimation versus communication performance.

Reduced communication state estimation for control of an unstable networked control system

A state estimation method is presented that allows the designer to trade off estimator performance for communication bandwidth in a networked control system. The method is based on a time-varying Kalman filter and a communication decision rule for each sensor: a sensor measurement is transmitted and used to update the Kalman filter if its associated prediction variance exceeds a certain tolerable bound. The resulting equation for the estimation error variance is deterministic, which enables its off-line analysis. If a periodic solution to the variance equation is found, it facilitates a straight-forward implementation of the communication decision: each sensor transmits its measurements with a fixed periodic sequence. This state estimation method is applied in the feedback control system of a cube balancing on one of its edges. Six rotating bodies on the cube stabilize the system and constitute the agents in the networked control system: each one is equipped with local actuation, sensing, and computation, and the agents share their sensor data over a broadcast network. Experimental results compare the performance of the reduced communication state estimation algorithm to a Kalman filter with full measurements.

A Self-Tuning LQR Approach Demonstrated on an Inverted Pendulum

Abstract An automatic controller tuning approach is presented that iteratively updates a linear quadratic regulator (LQR) design such that the resulting controller achieves improved closed-loop performance. In each iteration, an updated LQR gain is obtained by adjusting the weighting matrices of the associated quadratic cost. The performance of the resulting controller (measured in terms of another quadratic cost with fixed weights) is evaluated from experimental data obtained by testing the controller in closed-loop operation. The weight adjustment occurs through a stochastic optimization seeking to minimize the experimental cost. Simulation results of a stochastic linear system show that the self-tuning algorithm can recover optimal performance despite having imprecise model knowledge. Experiments on an inverted pendulum demonstrate that the method is effective in improving the systems balancing performance.

A limiting property of the matrix exponential with application to multi-loop control

A limiting property of the matrix exponential is proven: For a real square matrix, where the log norm of the upper-left n by n block approaches negative infinity in a limiting process, the matrix exponential goes to zero in the first n rows and n columns. This property is useful for simplification of dynamic systems that exhibit modes with sufficiently different time scales; for example, in multi-loop control systems with fast inner and slow outer feedback loops. For this case, we derive a time scale separation algorithm for a linear continuous-time model under the assumption of high-gain inner loop feedback, which yields a simplified discrete-time model at the slow time scale. The proposed technique is applied to the design of a two-loop control system for stabilizing an inverted pendulum. Experimental results are provided.