Negative Imaginary $H_{2}$ Controller Synthesis Using Nonlinear Optimization

Negative imaginary (NI) systems theory has attracted considerable attention in the area of robust control of highly resonant flexible structures systems. These systems, often naturally, satisfy the NI property. In this paper, we present a control synthesis methodology for NI systems based on a nonlinear optimization techniques. In the presented method, a parametrized library of strictly negative imaginary (SNI) controllers is created and used in a standard numerical nonlinear optimization setup. The SNI controller library contains generic controllers that are widely used in the control highly resonant flexible structures such as positive position feedback (PPF) and integral resonant control (IRC). Sequential quadratic programming (SQP) techniques are used in the numerical optimization problem. The synthesized controller satisfies the SNI property as well as optimizing $H_{2}$ performance. As an application of these results, an example of controlling an Euler-Bernoulli beam with piezo-electric actuator and sensor is presented.

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