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Introduction to Model Order Reduction

Introduction to Model Order Reduction. I.4 - System Properties Stability, Passivity. Luca Daniel. Thanks to Joel Phillips, Jacob White. Outline. Review of Laplace Domain Transfer Function Stability of State-Space Models Passivity of State-Space Models Positive-Realness Bounded-Realness.

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Introduction to Model Order Reduction

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  1. Introduction to Model Order Reduction I.4 - System Properties Stability, Passivity Luca Daniel Thanks to Joel Phillips, Jacob White

  2. Outline • Review of Laplace Domain Transfer Function • Stability of State-Space Models • Passivity of State-Space Models • Positive-Realness • Bounded-Realness

  3. An Aside on Transfer Functions – Laplace Transform Rewrite the ODE in transformed variables  Transfer Function

  4. An Aside on Transfer Functions – Meaning of H(s) For Stable Systems, H(jw) is the frequency response Sinusoid Sinusoid with shifted phase and amplitude

  5. An Aside on Transfer Functions – EigenAnalysis Transfer Function Apply Eigendecomposition

  6. Outline • Review of Laplace Domain Transfer Function • Stability of State-Space Models • Passivity of State-Space Models • Positive-Realness • Bounded-Realness

  7. Stability of State-Space Models • Consider a state-space model in isolation (it is not part of a larger system) Model Input Output • For well-behaved (e.g., bounded) inputs, when will the outputs be well-behaved (e.g. bounded) as well?

  8. Stability of State-Space Models • From systems theory, the model will be bounded-input/bounded-output (BIBO) stable if the transfer function has no poles in the open left half-plane. • Recall • The poles of occur where is singular • Equivalently, for non-singular , has a partial-fraction expansion where the (poles) are the eigenvalues of . For stability, these eigenvalues must not have positive real part; otherwise the impulse response will contain a growing exponential

  9. Descriptor Systems • What about ? • For , the poles come from the eigenvalue problem • For non-singular E, we can transform to this form. For singular E, the poles occur when is singular. The poles are determined by the generalized eigenvalue problem

  10. Outline • Review of Laplace Domain Transfer Function • Stability of State-Space Models • Passivity of State-Space Models • Positive-Realness • Bounded-Realness

  11. Passivity • Passive systems do not generate energy. We cannot extract out more energy than is stored. A passive system does not provideenergy that is not in its storage elements.

  12. Need to preserve passivity of passive interconnect PCB, package, IC interconnects Analog or digital IP blocks - + - + D Q Z(f) Picture by J. Phillips C Picture by M. Chou Note: passive! Hence, need to guarantee passivity of the model otherwise can generate energy and the simulation will explode!! Would like to capture the results of the accurate interconnect field solver analysis into a small model for the impedance at some ports.

  13. Interconnected Systems • In reality, reduced models are only useful when connected together with models of other components in a composite simulation • Consider a state-space model connected to external circuitry (possibly with feedback!) ROM • Can we assure that the simulation of the composite system will be well-behaved? At least preclude non-physical behavior of the reduced model?

  14. - - - - + + + + - - - - + + + + D D D D Q Q Q Q C C C C Interconnecting Passive Systems • The interconnection of stable models is not necessarily stable. • BUT the interconnection of passive models is a passive model (and hence also stable).

  15. Outline • Review of Laplace Domain Transfer Function • Stability of State-Space Models • Passivity of State-Space Models • Positive-Realness • Bounded-Realness

  16. + + - - Positive Realness & Passivity • For systems with immittance (impedance or admittance) matrix representation, positive-realness of the transfer function is equivalent to passivity ROM

  17. Passivity condition on transfer function • For systems with immittance matrix representation, passivity is equivalent to positive-realness of the transfer function (no unstable poles) (impulse response is real) (no negative resistors)

  18. Passivity condition on transfer function • For systems with immittance matrix representation, passivity is equivalent to positive-realness of the transfer function (no unstable poles) (impulse response is real) (no negative resistors) It means its real part is a positive for any frequency. Note: it is a global property!!!! FOR ANY FREQUENCY

  19. Positive real transfer function in the complex plane for different frequencies Transfer Function Active region Passive region

  20. Sufficient conditions for passivity • Sufficient conditions for passivity: i.e. E is negative semidefinite • Note that these are NOT necessary conditions

  21. Sufficient conditions for passivity • Sufficient conditions for passivity: i.e. A is negative semidefinite • Note that these are NOT necessary conditions

  22. Heat In Example Finite Difference System from on Poisson Equation (heat problem) We already know the Finite Difference matrices is positive semidefinite. Hence A or E=A-1 are negative semidefinite.

  23. Sufficient conditions for passivity • Sufficient conditions for passivity: i.e. E is positive semidefinite i.e. A is negative semidefinite • Note that these are NOT necessary conditions (common misconception)

  24. + + - - Example. hState-Space Model from MNA of R, L, C circuits When using MNA For immittance systems in MNA form A is Negative Semidefinite E is Positive Semidefinite

  25. Necessary and Sufficient Condition for PassivityThe Positive Real (KYP) Lemma A stable system (A,B,C,D) is positive real if and only if there exists X=XH0 such that the linear matrix inequality is satisfied If D=0 the system is positive real if

  26. Positive-Real Lemma (other form) • Lur’e equations : • The system is positive-real if and only if is positive semidefinite • A dual set of equations can be written for a with • A similar set of equations exists for bounded-real models

  27. Outline • Review of Laplace Domain Transfer Function • Stability of State-Space Models • Passivity of State-Space Models • Positive-Realness • Bounded-Realness

  28. Bounded Realness & Passivity • For systems with scattering matrix representation, bounded-realness of the transfer function is equivalent to passivity ROM

  29. Passivity condition on transfer function • For systems with scattering matrix representation, passivity is equivalent to bounded-realness of the transfer function (no unstable poles) (impulse response is real) (no negative resistors)

  30. Passivity condition on transfer function • For systems with scattering matrix representation, passivity is equivalent to bounded-realness of the transfer function (no unstable poles) (impulse response is real) (no negative resistors) It means ||H(s)||2 < 1 is bounded for any frequency. Note: it is a global property!!!! FOR ANY FREQUENCY

  31. Bounded real transfer function in the complex plane for different frequencies +j Transfer Function 1 Passive region -1 -j Active region

  32. Summary. System Properties • Review of Laplace Domain Transfer Function • Stability of State-Space Models • Passivity of State-Space Models • Positive-Realness • Bounded-Realness

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