AA247 MODAL ANALYSIS OF COMPLEX FIELDS Lecture 8

AA-247: MODAL ANALYSIS OF COMPLEX FIELDS Lecture 8 – Balanced POD and ERA

MOTIVATION As we discussed in the previous classes, POD methodology produces an orthogonal basis that best describes the variance/correlation of the data. However, even if a large fraction of energy is captured by the modes used for projection, the resulting low-order PODGalerkin models may still have completely different qualitative behavior: Transients can be poorly captured Stability of equilibria can be even different Energetic structures are dynamically significant, but their evolution can be influenced by low-energy modes Is there a better way find low-rank approximation of the dynamics of the system ?

ROADMAP LTI Systems Controllability and Observability Balanced Truncation Balanced POD Output Projection Connections with standard POD ERA

LINEAR TIME-INVARIANT SYSTEM

STATE-SPACE REPRESENTATION

TRANSFER FUNCTION AND IMPULSE RESPONSE

SOLVING LTI SYSTEMS

STABILITY The stability of the system is determined by the eigenvalues of A The union of the generalized eigenspaces associated to eigenvalues with negative real part gives us the stable subspace. The union of the eigenspaces associated to eigenvalues with zero real part gives us the neutral subspace. The orthogonal complement of the union of the aforementioned subspaces is the unstable subspace.

CONTROLLABILITY/REAC HABILITY Reachability Matrix

OBSERVABILITY Observability problem: find all states that can be determined if you have access to input and output history. Observability Matrix

GRAMIANS

ADJOINT MODEL Using the defined inner product: Using integration by parts: The first term is zero since the initial condition is zero and the system is stable. Since, this equality should hold for any x and z, we obtain

REACHABILITY GRAMIAN

OBSERVABILITY GRAMIAN

COMPUTING GRAMIANS

BALANCED TRUNCATION

BALANCED TRUNCATION, CONT’D

BALANCED POD

BALANCED POD, CONT’D

BALANCED POD, CONT’D

OUTPUT PROJECTION

CONNECTION WITH STANDARD POD

EIGENVALUE REALIZATION ALGORITHM (ERA)

EIGENVALUE REALIZATION ALGORITHM (ERA), CONT’D

EIGENVALUE REALIZATION ALGORITHM (ERA), CONT’D

EIGENVALUE REALIZATION ALGORITHM (ERA), CONT’D

MAIN REFERENCES Rowley, Clarence W. "Model reduction for fluids, using balanced proper orthogonal decomposition. " International Journal of Bifurcation and Chaos 15, no. 03 (2005): 9971013 Ahuja, Sunil, and Clarence W. Rowley. "Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators. " Journal of fluid mechanics 645 (2010): 447 -478. D. M. Luchtenberg, C. W. Rowley, Model Reduction Using Snapshot-Based Realizations, SIAM Journal on Control and Optimization (2012). Juang, J-N. , and Richard S. Pappa. "An eigensystem realization algorithm for modal parameter identification and model reduction. " Journal of guidance, control, and dynamics 8, no. 5 (1985): 620 -627. Ma, Zhanhua, Sunil Ahuja, and Clarence W. Rowley. "Reduced-order models for control of fluids using the eigensystem realization algorithm. " Theoretical and Computational Fluid Dynamics 25, no. 1 -4 (2011): 233 -247. Flinois, Thibault LB, and Aimee S. Morgans. "Feedback control of unstable flows: a direct modelling approach using the eigensystem realisation algorithm. " Journal of Fluid Mechanics 793 (2016): 41 -78. Flinois, Thibault LB, Aimee S. Morgans, and Peter J. Schmid. "Projection-free approximate balanced truncation of large unstable systems. " Physical Review E 92, no. 2 (2015): 023012.