# Stability analysis of hybrid discrete-continuous systems

I'm trying to derive the overall state-space system model for a hybrid system, in order to plot its eigenvalues.

The system is shown as follows:

Which is originally from this paper: Modeling and Sensitivity Study of Consensus Algorithm-Based Distributed Hierarchical Control for DC Microgrids.

The blue part is a discrete algorithm with sampling time of 0.1s. The red part is the control system in discrete-time with sampling time of 1e-4s. And the green part is the plant model which represents a DC microgrid with buck converters and loads, in continuous-time.

As described in this paper, following is the procedure to find the overall system dynamics:

1. Discretize the plant model using zero-order-hold (ZOH) method with the same sampling time of control system (1e-4s).
2. Combine the model in step 1 and the control system model.
3. Discretize the combined model in step 2 with the same sampling time of the algorithm (0.1 s).
4. Finally, combine the model in step 3 with the algorithm model to get the overall system model.

The procedure is depicted as shown below:

The procedure seems to make sense, but the question is:

Is the final system model accurate, given that the plant model and the control system model are now combined with a discretized system with sampling time of 0.1 s, and the fact that we lose some information in between sampling instants?

• If yes, can you explain how to derive the system dynamics with different sampling times and time domains?
• If not, what is an alternative approach to find the overall system dynamics?

Sorry if I'm making a mistake, I'm new to control theory.

• Since the coloured blocks already have an imposed sampling frequency, it means that the overall system has accounted for all the frequencies outside Nyquist to be irrelevant, therefore the result should match the system. Commented Jul 3, 2022 at 10:52

Is the final system model accurate Error tolerance specs were not defined but results were plotted.

The purpose of this paper was to find the minimum discrete data-sharing rate in a redundant network to avoid a single-point failure and network instability.

The data sampling rate is a major cost when the network is large with redundancy and depends on the grid error tolerance. It was derived using Laplacian Graph theory and the slowest Eigenvector or largest wavelength in the matrix. (don`t ask me how it works, but is shown how it converges ). There is a plot to show the sensitivity to network response with complex roots as the communication rate is changed by the data time interval, Tca.

Alternatives

Typically control loops use some form of linear feedback with gain constants for proportional, P, integral, I and derivative, D (PID) optimized for stability and speed by phase+gain margin. They are tested by impulse and step response. Outer loops with delay are well known to introduce instabilities and testing many-to-many nodes might be overbearing.

One rule of thumb is the step response (10% to 90%) rise time Tr= 0.35 / f-3dB (half-power bandwidth). Nyquist sampling theory tells us the number of samples per f cycle depends on the resolution error or S/N ratio.

Looking at the plots, I estimated the Tr=0.5s and the resonant frequency cycle for a RING= 2.5s had a settling time less than 10% error of 2.5s with 3-nodes and Tca=0.1s or 25 samples.

It seems they have good methods for analyzing a complex network to conserve data bandwidth, but needs more work to define all variables, tolerance errors, and disturbance responses, such as a single point of error in communication.