CONTROLLER SYNTHESIS FOR STRICTLY NEGATIVE IMAGINARY SYSTEMS VIA RICCATI EQUATIONS AND LINEAR MATRIX INEQUALITIES

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Abstract

Systems with a negative imaginary (NI) frequency response for positive frequencies are called NI systems. This property appears in several engineering applications such as nanopositioning systems; gantry crane systems; flexible structures with possibly free body dynamics; hard disk drive servo systems; active filters; descriptor systems and multi-agent networked systems. Particularly, the spillover unmodeled dynamics of flexible structures with collocated position sensors and force actuator have the NI property and controllers based on passivity are inapplicable. Commonly, the unmodeled dynamics are lightly damped, i.e. they are oscillatory. Evidently, there is an interest in designing controllers to reduce the oscillations/vibrations and to guarantee stability against the unmodeled dynamics. Traditional control techniques such as Linear Quadratic Gaussian, $\mathcal H_\infty$ and $\mathcal H_2$ may have poor stability margins or may be conservative because of the oscillatory behavior of flexible structures. Interestingly, the positive feedback interconnection of two NI systems is stable if one of them is a subclass of NI, known as weakly strict negative imaginary (WSNI) and also some gain conditions are satisfied. Hence, when the unmodeled dynamics have the NI property, the designed controller should render the nominal closed-loop WSNI and satisfy the gain conditions in order to guarantee stability of the uncertain system. This problem is called the ``WSNI synthesis problem". The synthesis of such controller is the main objective of this thesis. Firstly, a set of necessary and sufficient conditions are developed for the characterization of WSNI systems. This result supports both the analysis and synthesis of WSNI systems. Then, a set of sufficient conditions, based on the solution of algebraic Riccati equations (AREs), is provided for the solution of the WSNI synthesis problem. This solution is a dynamic output feedback controller and a formula for such controller is derived. However, the computation of the provided solution is numerically challenging because one of the AREs has a singular Hamiltonian. In order to avoid these numerical challenges, it is necessary to restrict our attention to a subclass of WSNI systems known as strongly strict negative imaginary (SSNI). The problem of synthesizing a controller to render the closed-loop SSNI is called the ``SSNI synthesis problem". By investigating further the relations between strict negative imaginary systems and strict positive real systems, it is shown that SSNI systems are equivalent, under a transformation, to classes of SPR systems. Then, necessary and sufficient conditions are developed for the solution of the SSNI synthesis problem. This solution is given in terms of simple linear matrix inequalities (LMIs). These LMIs include the dc value of the closed-loop to guaranteed robust stability of the closed-loop in the negative imaginary framework. Furthermore, a new solution for the SPR synthesis problem is provided, which will be useful when established solutions in the literature for this problem are not feasible. Finally, two numerical examples are given to illustrate the findings and usefulness of the results.

Keyword(s)

Control systems; Controller synthesis; Dynamic output feedback; Dynamical systems; Flexible structures; LMI; Linear algebra; NI; Nanopositioning systems; Negative imaginary systems; PR; Riccati equations; Robust control; SSNI; Static state feedback; Uncertain systems; Vibration control; WSNI; WSPR; negative-imaginary systems

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