Welcome to the 2022 ACC Workshop on Robust State Estimation with Applications to Networked Control Systems: An Unknown Input Observer Approach!
A state observer, or just observer, for short, is a deterministic dynamical system that can generate an estimate of the plant’s states using that plant’s inputs and outputs. Observers can be used to augment or replace sensors in a control system. The observer was first proposed and developed by Luenberger in the early sixties. The early observers require full knowledge of the inputs of the controlled plant. Observers that do not require full knowledge of the inputs have also been developed. The problem of designing observers for linear systems with both known and unknown inputs can be formulated as an unknown input observer (UIO) design problem. This problem was already studied by Basile and Marro in 1969. Since then, different UIO structures have been reported in the literature.
One important application of the UIO of current interest is in secure state estimation of networked control systems corrupted by malicious packet drops both in the communication between the plant sensors and the controller and that between the controller and the plant actuators. In a Networked Control System (NCS), the controlled system, actuators, and sensors are connected over a communication network with its limited bandwidth, delays, and losses. A major challenge in the NCS design is the system security. In particular, the problems of unknown input errors and packet drops during the signal transmission need to be addressed. If unknown input errors and sparse packet drops are sparse, their estimation can be formulated as a sparse vector recovery problem. The recovered sparse error vector can then be used to estimate the state of the plant. The estimated state, in turn, is used to synthesize combined observer-controller compensator. Detection and monitoring of malicious packet drops in the transmitted control signals can be accomplished using teh unknown input observer (UIO).
Another area of application of the UIO is in fault detection and isolation, which is also one of the topics of this workshop. We will present an unknown input estimator architecture that reconstructs sensor and actuator faults.
In this workshop, we also discuss digital implementation of observers designed in the continuous-time (CT) domain. In particular, we show that the exact discretization does not negatively impact the existence of the UIO and in some cases can facilitate the construction of the UIO when they do not exist for the CT systems. One of the conditions for the existence of an UIO for a continuous-time (CT) linear time-invariant (LTI) system model is a matrix rank condition. In many cases, while the original CT plant does not satisfy the matrix rank condition for the existence of an UIO, its discretized model satisfies the matrix rank condition for the existence of a discrete-time UIO. We characterize a class of systems showing this continuous-discrete UIO existence dichotomy.
We demonstrate that if a CT system satisfies the matrix rank condition, then the exact discretization of it will (with the possible exception of a countable set of sampling times) satisfy the matrix rank condition. We show that for a controllable single-input system, the exact discretization of it will (with the possible exception of a countable set of sampling times) satisfy the matrix rank condition, regardless of whether it holds for the original CT system. We also characterize a class of systems for which the matrix rank condition fails in the CT domain but holds in the discrete-time (DT) domain.
We test our proposed estimators on a CT nonlinear model on the double inverted pendulum on a cart (DIPC) corrupted by unknown input and output errors. The CT linearized DIPC model does not satisfy the matrix rank condition for the existence of an UIO but its discretized model does satisfy the discrete matrix rank condition. In both cases, we perform MATLAB-based animations.
In this workshop, the theory, design, and applications of estimators for the states and unknown inputs will be presented in a tutorial fashion. The workshop targets practicing engineers and graduate students. The emphasis will be on design in order to show how uncertain system control theory fits into practical applications. This workshop should be accessible for anyone with a basic knowledge of linear systems at the undergraduate level. No prior knowledge of observers is assumed.