H₂ Optimal Model Order Reduction Methods For Continuous Time-invariant Systems

With the development of science and technology,the complexity and the order of dy-namical systems are increasing rapidly,which lead to great difficulties for the simulation,optimization,design and control process of these dynamical systems.In order to achieve the simulation process in reasonable time,it is essential to present efficient and feasible methods to improve the computing speed.Model order reduction provides an effective method to this scientific problem.In this paper,based on the Gramians of systems,the H₂optimal model order reduction methods for continuous time-invariant systems are explored by using the projection methods,the geometric properties of Riemannian mani-folds and Riemannian optimization techniques.The specific research contents include the following aspects:???Based on the cross-Gramian,the two-sided projection H₂optimal model order reduction method of single-input single-output?SISO?linear time-invariant systems is ex-plored.The cross-Gramian of linear time-invariant systems can provide the controllability and observability information at the same time.Using the two-sided reduction technique and the cross-Gramian,we obtain the cost function of the H₂optimal model order re-duction problem for linear time-invariant systems,and derive the cross-Gramian-based H₂optimal first-order necessary conditions.The first-order necessary conditions can be viewed as the generalization of Wilson's conditions.Theoretical analysis shows that the resulting order-reduced system satisfies the cross-Gramian-based H₂optimal first-order necessary conditions.Thus,we obtain the local optimal solution in the sense of the H₂norm.???Applying the Riemannian geometric properties?such as the tangent space,the Riemannian gradient,the Riemannian metric,the retraction and the vector transport?and the structure of the Stiefel manifold,we explore the Riemannian optimization model order reduction methods for linear time-invariant systems.First,the general multi-input multi-output?MIMO?linear time-invariant systems are discussed,including the symmetric and non-symmetric cases.The general MIMO linear time-invariant system is decomposed into a set of SISO subsystems,and thus the original H₂optimal model order reduction problem for the general MIMO linear time-invariant system is turned into the problems for the SISO subsystems.Since the transformational matrices are column-orthogonal,we obtain the optimal solutions by using the Riemannian gradient of the cost functions on the Stiefel manifold.Combining with the compactness of the Stiefel manifold,we prove the convergence of the proposed algorithm.In order to improve the convergence speed,we further explore the Stiefel-based conjugate gradient model order reduction method for linear time-invariant systems.For this,we introduce another important geometric concept—vector transport.On this basis,the conjugate gradient on the Stiefel manifold is derived.All the order-reduced systems generated by these two methods can preserve the asymptotical stability of the original systems.???Based on the one-sided projection technique,the H₂optimal model order re-duction problem for bilinear time-invariant systems is discussed.When we utilize the one-sided reduction process to construct the H₂optimal order-reduced systems,this opti-mization problem has orthogonal constraint.Thus,this problem is treated as the uncon-strained Riemannian optimization problem and we then solve this problem.On the Stiefel manifold,the direction of the Riemannian gradient of the cost function is the steepest ascent direction.Thus,we perform the linear-search along the negative gradient direction and then the H₂optimal order-reduced system in this case is obtained.Besides,making full use of the geometric notions and the Riemannian gradient of the cost function,we prove the global convergence of this algorithm.???We explore an approach to the H₂optimal model order reduction on finite interval for bilinear time-invariant systems.For the bilinear time-invariant system,we define a new norm—H_2,?norm,which can be viewed as the variant of the H₂norm for bilinear time-invariant systems.Theoretical analysis shows the H_2,?norm of bilinear time-invariant systems can be efficiently computed by the frequency-limited Gramians.Further,we derive the H_2,?error related to the original system and its order-reduced system.Besides,the first-order necessary conditions to be H_2,?optimality are obtained.Based on the above work,we propose the H_2,?model order reduction method to construct the order-reduced system.Thereby,the model order reduction method on the finite interval for linear time-invariant systems is generalized.???Finally,we summarize the main research work of this paper,and give our future work.