Dissipativity of the H∞ system

A physical interpretation of H∞ stability

In complement to the series of posts about the design of an H∞ controller, the dissipativity of the H∞ controlled system will be revealed in this article. The concept of dissipative will fill in the gaps between Lyapunov stability analysis and the H∞ performance and provides an insightful interpretation. Some background knowledge of dynamic systems will be sufficient for this topic.

Dissipative system [1]

Starting with a nonlinear system modeled by the dynamic part “f” and the output function “h” formulated as below.

Nonlinear dynamic system with the input “u” and the output “z”, the output function “h” was not a measurement but serves as a variable in the supply rate “S” which will be introduced later.

Such a system is dissipative, if there exists a positive storage function “V” that satisfied the following relation with the supply rate “S”.

Definition of a “Dissipative system” in integral and differential expression

As you may notice, a dissipative system also satisfied the Lyapunov stability under the case with a zero supply rate “S=0”. Unlike a Lyapunov analysis that simply demands a decreasing Lyapunov function, the dissipativity requires a changing rate of the storage function limited by the supply rate (even if it is a positive one!).

Mass-Spring-Damper system

Let’s try out this definition with our old friend, a Mass-Spring-Damper system.

Mass-Spring-Damper system modeled with position “x1” and speed “x2”

By choosing the total energy (both kinematic and potential) in Joul as the storage function “V” and the input power in Watt as the supply rate “S”. One can immediately verify the following relation between the supply “S” and the rate of storage function “V”, thus, a dissipative system.

A natural selection of storage “V” and supply rate “S” from physical quantity

Dissipativity in the H∞ system

An H∞ design is aimed toward a limited gain of the input to output energy (L₂-gain) of a system. Refer to this previous article, by satisfying the Hamilton Jacobi Inequality [2].

The relation between the in/output and the associated Lyapunov function for an H∞ system

The conclusion can be stated:

An H∞ system is dissipative with the supply rate “S” defined as the difference between the scaled input power “ε²” and the output power “z²”.

References

[1] H. Márquez, Nonlinear Control Systems: Analysis and Design. Wiley, 2003.

[2] A. van der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control. Springer International Publishing, 2018.