(284b) Control Invariant Set Computation for Nonlinear Systems: A Distributed Approach

n many engineered systems, the ability of the system states to stay within a prescribed safe set – usually in the form of state constraints – is hindered by limits on the available control energy. Thus, to avoid unsafe and catastrophic events during operation, it is highly desirable to determine the set of states for which a control input exists that keeps the state of the system within the prescribed safe set forever. This set of states is known as the control invariant set. Control invariant sets play a critical role in control system design and analysis of constrained dynamical systems. For example, control invariant sets have been found to be important in addressing stability, constraint satisfaction and feasibility issues in model predictive control strategies. It is well known, however, that the task of determining control invariant sets is not a trivial one.

Some results which address computational issues and algorithmic procedures of control invariant sets exist for linear [1] and nonlinear systems [2]. However, the applicability of these algorithms, even for linear systems, is in general subject to the ‘curse of dimensionality’. Chemical processes are usually nonlinear and described by a large-scale network of interconnected process units which make the applicability of the existing algorithms to determine the control invariant sets of these systems difficult, if not impossible. There is a strong and long-lasting need to develop control invariant set computing algorithms that is applicable to large-scale process networks.

In this work, we present a distributed control invariant set computing algorithm. The proposed algorithm exploits the structure of the interconnections within a process network and decomposes the entire process network into smaller subsystems. After decomposing the process network, a distributed approach with iterations is then developed to compute the control invariant set of the entire system. The proposed algorithm adopts graph-based algorithms in the computing of the control invariant sets [3]. Specifically, in this presentation, we will share our results for process networks with subsystems connected in series. The process network is first decomposed into several subsystems with overlapping states. Subsequently, the dynamics of the subsystems is represented using weighted digraphs and then analyzed using well known graph theoretical algorithms in a sequential manner. The interaction between two subsystems are taken into account and the control invariant set is refined iteratively. Results of a few examples will be shown to demonstrate that the proposed distributed control invariant set computing can achieve the result of a centralized algorithm.

Figures 1 below shows the results of the proposed algorithm applied to a simple 3-dimensional example. It shows that the invariant set calculated by the proposed distributed computing converges to the actual control invariant set within 4 iterations.

We believe that the proposed distributed computing algorithm paves a promising way to overcome the ‘curse of dimensionality’ in control invariant set computing. It is also expected that the proposed approach can be extended to more general process networks.

References

[1] Maidens, J.N., Kaynama, S., Mitchell, I.M., Oishi, M.M., Dumont, G.A., 2013. Lagrangian methods for approximating the viability kernel in high-dimensional systems. Automatica 49 (7), 2017–2029

[2] Homer, T., Mahmood, M., Mhaskar, P., 2020. A trajectory-based method for constructing null controllable regions. Int. J. Robust Nonlinear Control 30 (2), 776–786.

[3] Decardi-Nelson, B., & Liu, J. (2021). Computing robust control invariant sets of constrained nonlinear systems: A graph algorithm approach. Computers & Chemical Engineering, 145, 107177.