1. I.4 - System Properties Stability, Passivity
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
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 provide energy that is not in its storage elements.

12. Need to preserve passivity of passive interconnect
Analog or digital IP blocks
PCB, package, IC interconnects - + - + D Q
Z(f) C
Picture by
J. Phillips
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. 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). Q D C - + Q D C - + Q D C - + Q D C - +

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
Active
region
Passive region

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

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

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

23. 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 Passivity The 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 -1
Passive region -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