Abstract
Three algorithms for efficient solution of optimal control problems for high-dimensional systems are presented. Each bypasses the intermediate (and, unnecessary) step of open-loop model reduction. Each also bypasses the solution of the full Riccati equation corresponding to the LQR problem, which is numerically intractable for large n. Motivation for this effort comes from the field of model-based flow control, where open-loop model reduction often fails to capture the dynamics of interest (governed by the Navier–Stokes equation). Our minimum control energy method is a simplified expression for the well-known minimum-energy stabilizing control feedback that depends only on the left eigenvectors corresponding to the unstable eigenvalues of the system matrix A. Our Adjoint of the Direct-Adjoint method is based on the repeated iterative computation of the adjoint of a forward problem, itself defined to be the direct-adjoint vector pair associated with the LQR problem. Our oppositely-shifted subspace iteration (OSSI, the main new result of the present paper) method is based on our new subspace iteration method for computing the Schur vectors corresponding, notably, to the \(m\ll n\) central eigenvalues (near the imaginary axis) of the Hamiltonian matrix related to the Riccati equation of interest. Prototype OSSI implementations are tested on a low-order control problem to illustrate its behavior.
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Notes
Numerically, the Schur decomposition is the method of choice for large-scale problems. In the analysis presented in Sect. 2, however, it is more convenient to consider the eigen decomposition.
A Hamiltonian matrix of this form satisfies a symmetric root property: for every eigenvalue of Z in the LHP, \(\lambda \), there is a corresponding eigenvalue of Z in the RHP, \(-\lambda ^*\), where \((\cdot )^*\) denotes the complex conjugate.
The columns of Y are referred to as the left or adjoint eigenvectors of A.
We take \(\varLambda _{c}=-\varLambda _u^H\) following the first paragraph of Sect. 2, noting that all eigenvalues in \(\varLambda _u\) are unstable.
If \(\varLambda \) is diagonal, the product \(\varLambda V\) corresponds to scaling the i’th row of V by \(\lambda _i\) for all i, whereas the product \(V\varLambda \) corresponds to scaling the i’th column of V by \(\lambda _i\) for all i.
If all unstable eigenvalues of A are distinct, then \(d_i^{(k)}\ne 0\) for \(i\ne k\); P necessarily becomes diagonal in this case in the \(\epsilon \rightarrow 0\) limit, and its columns may be normalized such that \(P\rightarrow I\). If some of the unstable eigenvalues of A are repeated, then there are other solutions as well. However, \(P\rightarrow I\) is a valid solution in either case in the \(\epsilon \rightarrow 0\) limit.
In future work, it would be valuable to consider the myriad of subtle issues that arise when coupling the implicit OSSI algorithm developed in Sect. 4.2 with approximate inverses such as those arising in the multigrid setting.
Recall that the EE method is stable when all eigenvalues \(\lambda \), scaled by h, are contained in a unit disk centered at \(-1\) in the complex plane of \(h\lambda \).
That is, in a manner immediately replacing V with \(\underline{Q}\).
Various approaches are available to “stretch” the eigenvalue spectrum of A in order to mitigate this timestep restriction. For example, one may replace the matrix A to which the iteration is applied with an appropriately-designed low-order polynomial (of order p) in A; such a polynomial has the same eigenvectors as A. This approach helps to increase the gap between \(\lambda _m\) and \(\lambda _{m+1}\) [see (33a)] while decreasing the interval between \(\lambda _{m+1}\) and \(\lambda _{n}\) [see (33b)], both of which facilitate faster convergence. However, this approach also increases the number of function evaluations \(A{\mathbf{v}}\) that must be calculated per iteration by a factor of p, and is thus generally not worthwhile unless the range of the eigenvalue spectrum of A is, a priori, known fairly accurately.
Note, however, that those vectors still being worked on in V still need to be orthogonalized at every iteration against the converged Schur vectors that have been removed from the iteration [10].
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Bewley, T., Luchini, P. & Pralits, J. Methods for solution of large optimal control problems that bypass open-loop model reduction. Meccanica 51, 2997–3014 (2016). https://doi.org/10.1007/s11012-016-0547-3
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DOI: https://doi.org/10.1007/s11012-016-0547-3