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

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

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

An Aside on Transfer Functions – EigenAnalysis Transfer Function Apply Eigendecomposition

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?

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

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

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.

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.

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?

- - - - + + + + - - - - + + + + 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).

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

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)

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

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

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

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

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.

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)

+ + - - 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

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

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

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

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)

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

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

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

Introduction to Model Order Reduction