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Numerical Solutions for Stochastic Continuous-time Algebraic Riccati Equations

Tsung-Ming Huang, Yueh-Cheng Kuo, Ren-Cang Li, Wen-Wei Lin

Abstract

We are concerned with efficient numerical methods for stochastic continuous-time algebraic Riccati equations (SCARE). Such equations frequently arise from the state-dependent Riccati equation approach which is perhaps the only systematic way today to study nonlinear control problems. Often involved Riccati-type equations are of small scale, but have to be solved repeatedly in real time. Important applications include the 3D missile/target engagement, the F16 aircraft flight control, and the quadrotor optimal control, to name a few. A new inner-outer iterative method that combines the fixed-point strategy and the structure-preserving doubling algorithm (SDA) is proposed. It is proved that the method is monotonically convergent, and in particular, taking the zero matrix as initial, the method converges to the desired stabilizing solution. Previously, Newton's method has been called to solve SCARE, but it was mostly investigated from its theoretic aspect than numerical aspect in terms of robust and efficient numerical implementation. For that reason, we revisit Newton's method for SCARE, focusing on how to calculate each Newton iterative step efficiently so that Newton's method for SCARE can become practical. It is proposed to use our new inner-outer iterative method, which is provably convergent, to provide critical initial starting points for Newton's method to ensure its convergence. Finally several numerical experiments are conducted to validate the new method and robust implementation of Newton's method.

Numerical Solutions for Stochastic Continuous-time Algebraic Riccati Equations

Abstract

We are concerned with efficient numerical methods for stochastic continuous-time algebraic Riccati equations (SCARE). Such equations frequently arise from the state-dependent Riccati equation approach which is perhaps the only systematic way today to study nonlinear control problems. Often involved Riccati-type equations are of small scale, but have to be solved repeatedly in real time. Important applications include the 3D missile/target engagement, the F16 aircraft flight control, and the quadrotor optimal control, to name a few. A new inner-outer iterative method that combines the fixed-point strategy and the structure-preserving doubling algorithm (SDA) is proposed. It is proved that the method is monotonically convergent, and in particular, taking the zero matrix as initial, the method converges to the desired stabilizing solution. Previously, Newton's method has been called to solve SCARE, but it was mostly investigated from its theoretic aspect than numerical aspect in terms of robust and efficient numerical implementation. For that reason, we revisit Newton's method for SCARE, focusing on how to calculate each Newton iterative step efficiently so that Newton's method for SCARE can become practical. It is proposed to use our new inner-outer iterative method, which is provably convergent, to provide critical initial starting points for Newton's method to ensure its convergence. Finally several numerical experiments are conducted to validate the new method and robust implementation of Newton's method.
Paper Structure (16 sections, 16 theorems, 139 equations, 2 figures, 2 tables, 4 algorithms)

This paper contains 16 sections, 16 theorems, 139 equations, 2 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Properties of scare
  3. Fixed-point iteration via sda
  4. fpsda
  5. Convergence Analysis
  6. Newton's method
  7. Direct solution for \ref{['eq:NT4SCARE-step']}
  8. Fixed-point iteration for \ref{['eq:NT4SCARE-step']}
  9. Numerical experiments
  10. Test problems for validation
  11. Real-World Problems
  12. Conclusions
  13. Fixed-point iteration
  14. Structure-Preserving Doubling Algorithm for care
  15. A modified Newton's method
  16. ...and 1 more sections

Key Result

Theorem 1.1

If both Assumptions asm:stab and asm:dete hold, then scareeq:SCARE-0 has a unique positive semi-definite (psd) solution $X_*$, which is also stabilizing, such that is stable, i.e., the linear differential equation $\frac{\mathop{\mathrm{d\!}}\nolimits}{\mathop{\mathrm{d\!}}\nolimits t}S(t)=\mathscr{L}_{F_*}S(t)$ is exponentially stable, where

Figures (2)

  • Figure 1: Iterative histories of fpsda, $\hbox{\sc nt}_1$ (via Kronecker's reformulation of each Newton step equation), $\hbox{\sc nt}_2$ (via MATLAB's lyap to iteratively solve Newton step equations), and $\hbox{\sc nt}_3$ (via Smith's method to iteratively solve Newton step equations in an inner-outer fashion). The plotted histories of normalized residual by the variants of Newton's method include the portion by fpsda for calculating an initial.
  • Figure 2: Iterative histories of fpsda, $\hbox{\sc nt}_1$ (via Kronecker's reformulation of each Newton step equation), $\hbox{\sc nt}_2$ (via MATLAB's lyap to iteratively solve Newton step equations), and $\hbox{\sc nt}_3$ (via Smith's method to iteratively solve Newton step equations in an outer-inner fashion). The plotted histories of normalized residual by the variants of Newton's method include the portion by fpsda for calculating an initial.

Theorems & Definitions (35)

  • Theorem 1.1: drms:2013
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Remark 3.1
  • Lemma 3.1: laro:1995
  • ...and 25 more