# Negative Imaginary Theory

Flexible structure systems arise in many important applications such as ground and aerospace vehicles, atomic force microscopes, rotating flexible spacecraft, rotary cranes, robotics and flexible link manipulators, hard disk drives and other nano-positioning systems. In control systems design for these flexible systems, it is important to consider the effect of highly resonant modes. Such resonant modes are known to adversely affect the stability and performance of flexible structure control systems and are often very sensitive to changes in environmental variables. These can lead to vibrational effects that limit the ability of control systems in achieving desired levels of performance. These problems are simplified to some extent by using force actuators combined with colocated measurements of velocity, position, or acceleration. Using force actuators combined with colocated measurements of velocity can be studied using positive real systems theory, which has received great attention since 1962. Using force actuators combined with colocated measurements of position and acceleration can be studied using negative imaginary (NI) systems theory.

The motivations and objectives of this research lead to the following contributions:

##### 1. Providing a systematic method for checking if a given system is NI or not.

We derive a method to check for the NI and strictly NI properties in both the single-input-single-output as well as the multi-input-multi-output cases. The proposed methods are based on spectral conditions on a corresponding Hamiltonian matrix obtained for a given transfer function matrix. Under some technical conditions, the transfer function matrix satisfies the NI property if and only if the corresponding Hamiltonian matrix has no pure imaginary eigenvalues with odd multiplicity. Also, the transfer function matrix satisfies the SNI property if and only if the corresponding Hamiltonian matrix has no pure imaginary eigenvalues except at the origin. These results may be useful in both the analysis of NI systems and in the synthesis of NI controllers using optimization techniques. Also, spectral conditions on the Hamiltonian matrix tend to have fewer numerical problems when compared to the LMI conditions.

##### 2. Developing a method for enforcing NI dynamics on mathematical system models to satisfy an NI Property.

We provide two methods for enforcing NI dynamics on mathematical models, given that it is known that the underlying dynamics ought to belong to the NI system class. The first method is based on a study of the spectral properties of Hamiltonian matrices. A test for checking the negativity of the imaginary part of the corresponding transfer function matrix is first developed. If an associated Hamiltonian matrix has pure imaginary axis eigenvalues, the mathematical model loses the NI property in some frequency bands. In such cases, a first-order perturbation method is proposed for the precise characterization of the frequency bands where the NI property is violated and this characterization is then used in an iterative perturbation scheme aimed at displacing the imaginary eigenvalues of the Hamiltonian matrix away from the imaginary axis, thus restoring the NI dynamics. In the second method, the direct spectral properties of the imaginary part of a transfer function are used to identify the frequency bands where the NI properties are violated. A discrete frequency scheme is then proposed to restore the NI system properties in the mathematical model.

##### 3. Generalizing the existing NI definition to include flexible structures with free body motion.

A generalized NI system framework is presented. A new NI system definition is given, which allows for flexible structure systems with colocated force actuators and position sensors and with free body motion. This definition extends the existing definitions of NI systems.

##### 4. Deriving stability conditions for NI systems with poles at the origin.

A necessary and sufficient condition is provided for the stability of a positive feedback control system where the plant is NI according to the new definition and the controller is strictly negative imaginary (SNI). This general stability result captures all previous NI stability results which have been developed. The stability conditions in this thesis, are given purely in terms of properties of the plant and controller transfer function matrices, although the proof relies on state-space techniques. Furthermore, the stability conditions given are independent of the plant and controller system order.

##### 5. Providing an Algebraic Riccati Equation analysis of NI systems and static NI controller synthesis.

We provide a systematic method to design controllers for NI systems with guaranteed robust stability and performance. Unlike the state feedback controller synthesis problem Riccati equation approach is developed and used to present a systematic method for designing a controller to force the closed-loop system to satisfy the NI property.