(386g) Dissipativity Learning and Verification from Online Trajectories of Nonlinear Dynamics

Dissipativity is an input-output behavioral characterization of nonlinear dynamics [1]. Specifically, it is defined as the property that there exists a state-dependent storage function whose accumulation/depletion is bound by a rate dependent on inputs and outputs, called supply rate. Once the dissipative property of the nonlinear dynamics is determined, the resulting controller can be easily synthesized, based on the fact that interconnected dissipative systems (plant and controller) is still dissipative. For process systems, such dissipativity (or passivity with a particular form of supply rate expressed as an inner product of inputs and outputs) is usually deducted from the thermodynamical laws based on analyzing their transport-reaction models, through which the storage and supply rate functions can be related to energy and entropy concepts [2, 3].

Recently, dissipativity learning control (DLC) has been proposed in the author’s previous works [4, 5] as an input-output data-driven model-free control framework. Instead of using first-principles analysis, in DLC, the dissipative properties are learned from input and output trajectories. Specifically, the supply rate function is parameterized by in the form of an inner product of dissipativity parameters as the system property and trajectory-dependent dual dissipativity parameters. By collecting samples, the range of dual dissipativity parameters (called dual dissipativity set) can be inferred, whose dual cone gives the dissipativity set, namely the set of dissipativity parameters. Based on this information, an optimal controller can be synthesized to optimize a performance metric, e.g., an L₂-gain from disturbances to inputs/outputs.

However, the previously proposed dissipativity learning approach in DLC is not economical in the data size, since every trajectory is only utilized as a single point in the sample, and thus the entire learning requires a large number of trajectories. In contrast, traditional system identification approaches need only a single trajectory that is persistently excited [6]. In view of this gap, this work proposes an improved approach to estimate dissipativity and verify its validity with a single trajectory. Specifically,

1. Rather than sampling trajectories starting from the origin (where the storage is zero), a single excited trajectory can be utilized. At every time instant on this single trajectory, the rate of change in the storage is supposed to be upper bounded. This upper bound is proportional to the magnitudes of the dissipativity parameters and the dual dissipativity parameters at the time, where the coefficient ε is a hyperparameter that is tunable but desirably verified by data.
2. Based on the above upper-bound technique, one can randomly sample time instants on the trajectory. Each time instant (instead of each trajectory) gives a sample point of the dual dissipativity parameters. The dissipativity set becomes an ε-strict dual cone of the dual dissipativity set, which can be inferred, e.g., through principal component analysis (PCA) from the samples of dual dissipativity parameters.
3. To estimate ε, empirical L₂-gain from the manipulated inputs to the outputs is first approximated, based on the ratios calculated on both the overall trajectory and every local time interval. Then, the storage change is set as if the supply rate is a quadratic form defined according to the empirical L₂-gain. In turn, the ε value can be verified by a batch of resampled data.

The proposed approach is motivated by a relevant work [6], which focused on the estimation of scalar passivity indices rather than the parameters in a general expression of dissipativity. The application to a benchmark two-phase reactor case study demonstrates that the DLC framework using the proposed approach generates a well-performing output feedback controller.

References

[1] Brogliato, B., et al. (2007). Dissipative systems analysis and control: Theory and applications. Springer.

[2] Ydstie, B. E. (2002). Computers & Chemical Engineering, 26, 1037-1048.

[3] Bao, J., Lee, P. L., & Ydstie, B. E. (2007). Process control: The passive systems approach. Springer.

[4] Tang, W., & Daoutidis, P. (2019). Computers & Chemical Engineering, 130, 106576.

[5] Tang, W., & Daoutidis, P. (2021). Systems & Control Letters, 147, 104831.

[6] Isermann, R., & Münchhof, M. (2011). Identification of dynamic systems: An introduction with applications. Springer.

[7] Welikala, S., Lin, H., & Antsaklis, P. J. (2022). 61st Conference on Decision and Control (pp. 267-272). IEEE.