Unstable systems have always posed challenges to the researchers as well as engineers who have worked on it. Some of the typical systems that include unstable features can be found in modelling and control of aeroelastic aircrafts, magnetically levitated apparatus engaged as a heart assist device, and modelling of multi-input multi-output system of NASA Highly Manoeuvrable Aircraft Technology (HIMAT). Plenty of engineering applications are often represented by complex, high-order models that are troublesome to analyze, control and design. Their reduced models are more appropriate to provide deep insight and can also contribute for designing lower order controllers. The main motive behind various model reduction techniques is to obtain an appropriate lower order model such that it retains the input-output behavior and inherent characteristics of the original system with reduced error. To deal with unstable systems, the common practice is to partition the original system into stable and unstable parts. Model order reduction is performed on the stable part. At the end, the reduced stable part is combined with the unstable part to obtain the overall reduced model. Although, this technique requires less computation, it is not helpful for the model reduction of the unstable part. Further, if a system has only unstable poles and zeroes then it is completely impossible to perform the model reduction on that system using this approach. Another approach is to shift all the poles or the eigenvalues of an unstable system into the stable region, and perform model order reduction of the linearly transformed system. This pole shifting technique has been applied with several methods to reduce the unstable systems. However, these techniques cannot guarantee that the number of right half plane poles is equal to that of the original unstable system. Hence, a straightforward, new and yet powerful pole shifting method using the concept of weighted harmonic mean is developed to reduce the unstable system having non-minimum phase feature retaining the input-output characteristics of the system. So, the higher-order unstable system is first converted to a stable higher-order model using a linear transformation. A new hybrid metaheuristic search tool combining the advantages of chaotic particle swarm optimization (CPSO) and grey wolf optimizer (GWO) is then employed to perform the necessary order reduction of the stable, transformed higher-order model preserving their minimum phase, stability, dc gain and both time and frequency domain features. The stable reduced-order system is further transformed back to yield an unstable model. The proposed method is adequately supported with the help of a relevant test system. Sufficient number of metaheuristic techniques are also used for comparison.
