New Results in Negative Imaginary Systems Theory

Abstract

Negative imaginary (NI) systems property naturally arises in flexible structures with colocated force actuators and position sensors. NI systems theory provides a robust control approach to these systems and has attracted attention among control theorists since it was introduced. NI systems theory has found its application in many areas, such as nanopositioning in atomic force microscopy. This thesis provides some new results in NI systems theory.

We provide a definition of nonlinear NI systems to allow for systems with free body motion. Roughly speaking, a system is said to be nonlinear NI if there exists a positive semidefinite storage function, whose time derivative is less than or equal to the inner product of the system input and the time derivative of the system output. Under some assumptions, we show that the interconnection of a nonlinear NI system and a nonlinear output strictly negative imaginary (OSNI) system is asymptotically stable. We also provide a separate stability result for nonlinear NI plants with positive definite storage functions, which corresponds to the case that the systems do not have free body motion.

The problem of state feedback equivalence to NI systems is investigated. The necessary and sufficient conditions under which a linear system can be made NI using state feedback control is that there exists an output transformation such that the resulting system is of relative degree less than or equal to two and is weakly minimum phase. Similar results are also provided for the state feedback equivalence to OSNI systems and strongly strictly negative imaginary systems. For an input-affine nonlinear system, we provide sufficient conditions under which it can be made NI using state feedback. We also provide necessary and sufficient conditions for a nonlinear system in a particular normal form to be state feedback equivalent to an NI system. These results lead to alternative approaches to address the stabilization problems for systems with NI uncertainty.

We provide control frameworks for the output feedback consensus of three types of networked nonlinear NI systems. They are networked identical nonlinear NI systems, networked heterogeneous nonlinear NI systems, and networked heterogeneous nonlinear NI systems with free body motion. In these three cases, we use OSNI systems as controllers, whose outputs are distributed to the plants according to the network topology.

We extend the NI systems theory to switched systems. A system is said to be switched NI if each subsystem is NI and the storage functions of the subsystems in the switching sequence are nonincreasing. Under some assumptions, the interconnection of a switched NI system and a switched OSNI system is asymptotically stable.

For the digital control of NI systems, we provide a new discrete-time NI systems definition for general nonlinear systems. This definition is automatically satisfied for a ZOH sampled continuous-time NI system. We provide linear matrix inequality conditions and frequency-domain conditions for a linear discrete-time system to be NI. Under certain assumptions, asymptotic stability is achieved for the interconnection of a discrete-time NI system and a discrete-time step advanced OSNI system, and also the interconnection of a discrete-time step advanced NI (SANI) system and a discrete-time OSNI system.

Hybrid integrator-gain systems (HIGS), are investigated as controllers for NI systems due to their own NI property. We show that a single HIGS, a multi-HIGS, and the cascade of two HIGS are all nonlinear NI systems. They can be applied to achieve feedback stability for linear NI plants. This motivates the application of a multi-HIGS on a microelectromechanical nanopositioner, which improves the system performance. Also, a discrete-time HIGS is shown to be an SANI system, which can be applied to stabilize a discrete-time NI system.

This thesis includes several examples to illustrate the results. Show more