(243h) Lyapunov Stability and Process Flexibility Analysis: Analogies and New Approaches

Process flexibility, namely the ability of process systems to maintain feasible operation under uncertainties, is an important field of process systems engineering [1] with well-established solution methods (e.g., [2]). Flexibility analysis contains the following three aspects:

1. (Flexibility test problem) Given the process design and the range of uncertain parameters, can the process be operated feasibly (satisfy the operational constraints)?
2. (Flexibility index problem) Given the process design and the operational constraints, what is the range of uncertain parameters in which the operational constraints can be satisfied?
3. (Flexible process design) Given the operational constraints and the range of uncertainties, what is the optimal process design (with respect to an expected cost)?

In nonlinear control theory, Lyapunov stability analysis (e.g., [3]) also considers three questions:

1. (Rate of convergence estimation) Given the Lyapunov function and the domain of interest in the state space, what is the rate of convergence of the Lyapunov function?
2. (Domain of attraction estimation) Given the Lyapunov function and a required rate of convergence, what is the largest domain inside which this rate of convergence is guaranteed?
3. (Design of Lyapunov function) Given the domain in the state space and a required rate of convergence, what is the optimal Lyapunov function (with respect to a properly defined control cost) that guarantees this rate of convergence in this domain?

In this talk, we point out the conceptual analogy between Lyapunov functions and process design, between state domain and process uncertainties, and between asymptotic convergence and process operational feasibility. Hence, we establish that the above questions in Lyapunov stability analysis are analogous to the three aspects of process flexibility analysis. We discuss how this conceptual relation can be leveraged to:

1. Develop a systematic framework of Lyapunov stability analysis [4, 5]. We employ bilevel programming and two-stage stochastic programming formulations, which have been well established in process flexibility analysis, to the problems of Lyapunov stability analysis. Compared to the existing approaches, the proposed framework applies to more general forms of control systems under operational constraints, and scales better to higher-dimensional systems.
2. Integrate process design and control decisions [6]. Specifically, the effect of uncertainties on the operational feasibility is captured by a gain from the uncertainties to the value of Lyapunov function, i.e., the Lyapunov stability analysis offers a characterization of dynamic flexibility for process systems. By considering the Lyapunov function as a part of the process design variables, process design and control decisions are integrated into a single optimization problem, where the operating cost and operational feasibility are related to the Lyapunov function value. Compared to the traditional methods of integrating design and control, this Lyapunov flexibility framework explicit accounts for process nonlinearity, while avoiding the complexity of mixed-integer dynamic optimization.

References

[1] Grossmann, I. E., Calfa, B. A., & Garcia-Herreros, P. (2014). Evolution of concepts and models for quantifying resiliency and flexibility of chemical processes. Computers & Chemical Engineering, 70, 22-34.

[2] Zhang, Q., Grossmann, I. E., & Lima, R. M. (2016). On the relation between flexibility analysis and robust optimization for linear systems. AIChE Journal, 62(9), 3109-3123.

[3] Haddad, W. M., & Chellaboina, V. (2011). Nonlinear dynamical systems and control: a Lyapunov-based approach. Princeton University Press.

[4] Tang, W., & Daoutidis, P. (2019). A bilevel programming approach to the convergence analysis of control-Lyapunov functions. IEEE Transactions on Automatic Control, 64(10), in press.

[5] Tang, W., & Daoutidis, P. (2019). A stochastic programming approach to the optimal design of control-Lyapunov functions. Automatica, under review.

[6] Tang, W., & Daoutidis, P. (2019). Lyapunov dynamic flexibility of nonlinear processes. In Foundations of Computer-Aided Process Design. Copper Mountain Resort, Colorado.