Model Reduction Theory And Numerical Algorithms For Negative Imaginary Systems

Systems tend to be increasingly complex due to advancing technology.Mathemat-ical modeling of these systems are often high-order transfer functions,which pose seri-ous difficulties in system analysis and synthesis.Therefore,an important issue concerns model reduction of high-order systems to simplify further analysis and synthesis.This thesis mainly studies the model reduction problems and design the numerical algorithms for negative imaginary systems.A negative imaginary system is a kind of dissipative system,which relative order can be two.Many practical systems can be modelled as negative imaginary systems,for example,the lightly damped flexible structure with a collocated position sensor and force actuator.For negative imaginary systems,it is desirable to preserve the negative imaginary structure during the reduction.Meanwhile,the model reduction error is optimized.A direct application of the classical model reduction methods can not preserve the negative imaginary structure for the reduced-order systems.Therefore,this research follows the question:How to preserve the negative imaginary structure for the reduced-order systems with a samll approximation error?The main research results of this dissertation are divided into two parts,the model reduction methods for the asymptotically stable negative imaginary systems and the model reduction methods for the negative imaginary systems with poles at the origin.The main research contents of this dissertation are summarized as follows:· H? model reduction problem is studied for asymptotically stable negative imag-inary systems.Sufficient conditions in terms of matrix inequalities are derived for the construction of an H? reduced-order negative imaginary system.Iterative algorithms are provided to find the desired reduced-order system and to minimize the H? approximation error bound.Finally,an RLC network example is given to demonstrate the effectiveness of the proposed model reduction method.This study is detailed in Chapter 2.· H2 and mixed H2/H? model reduction problems are studied for asymptotical-ly stable negative imaginary systems.Sufficient conditions are derived for the construction of H2 and mixed H2/H? reduced-order negative imaginary system-s.Iterative algorithms are provided to find the reduced-order negative imaginary systems and to minimize the H2 approximation error bound.Compared with the single approximation error indicator,the mixed H2/H? model reduction method has the significance of achieving a smaller approximation error.This study is detailed in Chapter 3.· In order to further improve the approximate accuracy,a Stiefel manifold based H2 model reduction problem is studied for the asymptotically stable negative imagi-nary systems.By using the Galerkin projection,the H2 model reduction problem is formulated as a minimization problem over the Stiefel manifold.First-order necessary condition is derived for the construction of a local optimal reduced-order system.A gradient descent algorithm is provided to search for the desired reduced-order system.Moreover,the convergence property of the gradient de-scent algorithm is analyzed.Finally,the advantages of the proposed method are demonstrated.This study is detailed in Chapter 4.· Moment matching model reduction problem for negative imaginary systems with poles at the origin is studied.Firstly,the original system is split into an asymp-totically stable subsystem,a lossless negative imaginary subsystem and an aver-age subsystem.Then,moment matching model reduction is implemented on the asymptotically stable subsystem and the lossless negative imaginary subsystem.The resulting reduced-order system preserves the negative imaginary structure and the poles at the origin.Compared with the existing model reduction method-s,the proposed moment matching model reduction method has wider application,This study is detailed in Chapter 5.· H? model reduction problem for interval frequency negative imaginary systems is studied.Necessary and sufficient conditions are derived for the construction of an H? reduced-order interval frequency negative imaginary system.An im-proved iterative algorithm is provided to find the desired reduced-order system and to minimize the H? approximation error bound over the given frequency in-terval.The proposed method is further clarified via the application to the electrical circuits,such as high order Sallen-key low pass filter,piezoeletric tube scanner and RLC circuit.This study is detailed in Chapter 6.