Passivity and its connection to the H∞ stability
An introduction to the passivity
Passivity [1] is a type of dissipativity [2] with a specific definition of the supply rate that accumulates in the storage. The framework of passivity follows a bilateral “input” and the “output” compared to others that treated them as flows of signals uni-directionally. This feature makes it a close companion with port modeling (bonding graph) and inherits its flexibility of combining multiple systems such as cascading or paralleling. In this article, we will first discuss the difference between passivity and Lyapunov stability with control (CLF). Then an example from [1] will be given to demonstrate the definition of passivity. The property of a system composed of multiple passive sub-systems was also discussed. At last, the connection between the passivity and the H∞ stability will be revealed.
You may want to refer to some previous articles that cover the basics: Seeking Nonlinear H∞ controller for Differential-Drive robot and Dissipativity of the H∞ system
Why should we care
The Lyapunov analysis itself was already a powerful tool that not only validates the stability of a system but also serves as a controller design guide. Refer to as a Control-Lyapunov function (CLF), the latter interpretation goes with an actuated system and utilizes Lyapunov analysis reversely with the problem stated as follows. Notice that the Lyapunov-based analysis is limited in a case of a passive system or a controlled system with an accessible input.
What is the feasible control input that satisfied Lyapunov stability?
On the other hand, the term “input” for a passivity analysis was treated as a disturbance rather than some controllable value. This leads to a different analogy and provides an extended definition of stability under disturbances. One significant difference between the two concepts is that the rate of Lyapunov function must not be increasing while the “storage” can both be increasing or decreasing.
Dissipative system [2]
Briefly rewind the definition of a dissipative system, with a nonlinear system modeled by the dynamic part “f” and the output function “h” shown as below.
Such a system is dissipative, if there exists a positive storage function “V” that satisfied the following relation with the supply rate “S”.
Like-wise in Lyapunov analysis, given a dissipative system, finding a valid storage function and supply rate often requires a decent amount of iteration, however, starting with the total energy as the storage function is always a good choice for a physical system.
Passive system [1]
With the supply rate “S” defined as the following equation with positive coefficients “ε, δ, ρ” and a positive function “ψ” that vanishes at origin, a dissipative system is called passive. Sharing the same dimension, the input (disturbance) and the output were denoted as “w” and “z” respectively, common pairs for physical systems would be: [force, speed] in mechanical, [pressure, flow] in fluid or acoustic, and [voltage, current] in the electrical domain.
From another perspective, the passivity implies a storage rate smaller than the input “wz” minus the part caused by input “w”, output “z”, and the storage itself “x”.
Example
A passive system will be introduced by the following example of a second-order electric circuit and hopefully helps understand the physical interpretation of each term. For simplicity, the inductance of “L” and the capacitance of “C” will be set to identity.
With the storage defined as physical energy, it is straightforward to show the dissipativity with the supply rate defined physically. One step further, since the supply rate also satisfied the definition given earlier, the system was also proven to be passive:
Two of the special cases will be stated here, starting with a simple one:
The case with “R₂=0, R₁=R₃=∞” is called a “lossless” system such that the rate of storage equals the supply rate, “ε=δ=ρ=0”.
Followed by a slightly different condition that will be used later:
The case with “R₁=R₃=∞” is called an “output strictly passive” system such that the rate of storage equals the supply rate minus the loss caused by output, “δ>0”.
Interconnecting passive systems
As mentioned in the intro, due to the bilateral construction of the supply rate for a passive system, the extension for the passivity of interconnected multiple systems was greatly simplified. In this section, an important fact of passivity will be stated and proved by utilizing such characteristics as follows.
A system composed of two passive sub-systems either coupled as cascaded or parallel results in a passive system.
Cascaded here refers to a distributed input to each sub-system and the output was held the same. For example, in a cascaded electrical system, the total voltage taken as input is the sum of each stage while the current as the output was identical for each stage. You might feel uncanny to brutally force the output the same, but it turns out fine with further arguments which will be provided in the future. Only the input cascading case will be discussed, a parallel case follows a similar argument. Start by defining the relations of input and output as shown:
Proof: The candidate of the storge for a cascaded system “Vc” can be selected as a naive summation from the sub-systems and satisfy the passivity shown in the derivation below with the cascading relation incorporated. The proof ends here and a similar procedure can be conducted for a parallel case (let’s skip that).
The blue-colored part follows the derivation shown below:
The discussion of passivity will stop here and let’s move our focus to the linkage between passivity and H∞ stability.
Passivity and the H∞ stability
Following the definition of a passive system, two interesting relations [1] will be revealed in this section. Assume a passive system with storage function “V”:
Without input (disturbance), “V” is also the Lyapunov function and the passive system was also Lyapunov stable around the equilibrium.
Proof: The proof comes directly from the definition of passivity:
An “output strictly passive” system stated previously was also H∞ stable
Proof: Similar to passivity, an H∞ stability implies some form of stability under external disturbances. Starting from the definition of an “output strictly passive” system, the following derivation shows a limited L₂-gain < 1/δ, thus proving the statement.
Furthermore, without disturbances “w=0" and a non-zero output function except the origin “z*z>0”, this derivation also shows asymptotic stability around the equilibrium as an enhancement for the first statement since “dV/dt < 0”.
References
[1] H. K. Khalil, Nonlinear Control. Pearson Education Limited, 2014.
[2] H. Márquez, Nonlinear Control Systems: Analysis and Design. Wiley, 2003.
[3] A. van der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control. Springer International Publishing, 2018.