An invariant subspace method for large-scale algebraic Riccati equation. (English) Zbl 1227.65053
The linear time-invariant dynamical system
\[ \begin{cases} \dot x(t) = Ax(t) + Bu(t),\quad x(0)=x_0,\\ y(t) ~=~ Cx(t), \end{cases} \tag{S} \]
where \(A\), \(B\), \(C\) are matrices, is considered. In practice the square matrix \(A\) is \(n \times n\), and \(n\) is very large (of the order \(10^5\) or \(10^6\)). The authors are interested in the feedback control of the system (S), the corresponding cost functional being quadratic in an infinite horizon. Therefore, a new family of low-rank approximations of the solution of the algebraic Riccati equation is introduced. It is based on invariant subspaces of the Hamiltonian matrix. The stabilizing property of the feedback is obtained. In particular, the exact stabilizing solution of the Bernoulli equation is obtained. Numerical examples are presented.
\[ \begin{cases} \dot x(t) = Ax(t) + Bu(t),\quad x(0)=x_0,\\ y(t) ~=~ Cx(t), \end{cases} \tag{S} \]
where \(A\), \(B\), \(C\) are matrices, is considered. In practice the square matrix \(A\) is \(n \times n\), and \(n\) is very large (of the order \(10^5\) or \(10^6\)). The authors are interested in the feedback control of the system (S), the corresponding cost functional being quadratic in an infinite horizon. Therefore, a new family of low-rank approximations of the solution of the algebraic Riccati equation is introduced. It is based on invariant subspaces of the Hamiltonian matrix. The stabilizing property of the feedback is obtained. In particular, the exact stabilizing solution of the Bernoulli equation is obtained. Numerical examples are presented.
Reviewer: Viorel Arnăutu (Iaşi)
MSC:
| 65K10 | Numerical optimization and variational techniques |
| 93C05 | Linear systems in control theory |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 93B52 | Feedback control |
Keywords:
feedback control; algebraic Riccati equation; invariant subspace; stabilization; low-rank approximation; linear time-invariant dynamical system; Hamiltonian matrix; numerical examplesReferences:
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