Optimal H2 Moment Matching-Based Model Reduction for Linear Systems through (Non)convex Optimization
Abstract
:1. Introduction
- (i)
- We first formulate the model reduction problem in the family of models matching moments, parameterized in a set of free parameters. Then, a suitable nonconvex optimization framework is derived, where the objective function is the approximation error norm, written in terms of the controllability and observability Gramians of a minimal realization of the error system. We also write the necessary first-order Gramian-based optimality conditions, i.e., the KKT system;
- (ii)
- For the formulated model reduction problem, we propose two numerical optimization algorithms. The first method is using a gradient update for solving the KKT system, leading to a simple iteration involving matrix multiplications. However, with this update, the stability of the approximation is achieved asymptotically. The second method is based on a partial minimization approach. We show that, for the evaluation of the gradient of the objective function, we need to solve two Lyapunov equations yielding the Gramians. We also prove that the gradient of the objective function is Lipschitz continuous. Therefore, a gradient-based algorithm is developed, ensuring the convergence to a local optimal solution due to the smoothness of the objective function. Although the gradient evaluation is expensive, each iteration provides a stable reduced model;
- (iii)
- Finally, we propose a convex SDP relaxation of the original nonconvex optimization problem by assuming that the error system admits a block-diagonal observability Gramian. We also derive sufficient conditions to guarantee block diagonalization.
2. Preliminaries
2.1. Linear Systems
2.2. Sylvester Equation-Based Moment Matching
2.3. The Computation of the Moments
2.4. -Norm Based on the Gramians of Linear Systems
3. Model Reduction by Moment Matching and Optimization
- (i)
- the -norm of the error system is minimal;
- (ii)
- the model is stable, i.e., ;
- (iii)
- .
3.1. Optimization Formulation of the proposed model reduction problem
3.2. KKT Approach
3.3. Partial Minimization Approach
3.4. SDP Approach
4. Numerical Optimization Algorithms for Problem 1
4.1. Gradient Type Method for the KKT System
Initialization of the Algorithm
4.2. Gradient Method for the Partial Minimization Problem
4.3. Convex SDP Relaxation
5. Illustrative Examples
5.1. CD Player
5.2. Heat Equation
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Block Diagonal Gramians for General LTI Systems
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Gradient Method for Problem 1 | IRKA [17] | Balanced Truncation | ||||||
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Norm | Gradient Norm | max Re Pole | Norm | Gradient Norm | max Re Pole | Norm | max Re Pole | |
2 | ||||||||
6 |
Interp. Points by: | Gradient Method for Problem 1 | IRKA [17] | Balanced Truncation | ||||||
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Norm | Gradient Norm | max Re Pole | Norm | Gradient Norm | max Re Pole | Norm | max Re Pole | ||
6 | IRKA | ||||||||
BT |
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Necoara, I.; Ionescu, T.-C. Optimal H2 Moment Matching-Based Model Reduction for Linear Systems through (Non)convex Optimization. Mathematics 2022, 10, 1765. https://doi.org/10.3390/math10101765
Necoara I, Ionescu T-C. Optimal H2 Moment Matching-Based Model Reduction for Linear Systems through (Non)convex Optimization. Mathematics. 2022; 10(10):1765. https://doi.org/10.3390/math10101765
Chicago/Turabian StyleNecoara, Ion, and Tudor-Corneliu Ionescu. 2022. "Optimal H2 Moment Matching-Based Model Reduction for Linear Systems through (Non)convex Optimization" Mathematics 10, no. 10: 1765. https://doi.org/10.3390/math10101765