(581b) Rigorous Approach for Robust Design of Nonlinear Dynamic Systems

Robust design problems involving dynamic processes arise in many areas of engineering.  In chemical engineering, processes and control systems often must be designed to satisfy certain safety and/or quality control constraints, and to do so in a way that is robust to uncertainties in process parameters and to possible disturbances.  To address this problem, dynamic processes can be modeled as continuous-time hybrid systems [1] subject to appropriate terminal constraints and path constraints.  The design problem considered here is then to rigorously determine robust subregions of the design space; that is, subregions in which all constraints will be satisfied over a finite time horizon, and this will remain true for all possible parameter values and disturbances.  Most existing methods [2-4] for addressing this problem are restricted to linear models.  Huang et al. [5] have proposed a region-transition-model (RTM) framework for uncertain nonlinear systems.  However, they used difference equations to approximate the evolution of the continuous states, which may not realistically represent the true dynamics of a nonlinear system.  Recently, Lin and Stadtherr [6] extended the RTM framework to continuous-time systems and used it to develop a rigorous approach for safety analysis.  In general, their approach was not designed to handle uncertainties, although the approach could be used to address robust design problems by treating uncertain parameters as additional design variables, which introduces unnecessary computational cost.

In this study, we present a new, rigorous approach for the robust design of nonlinear, continuous-time dynamic systems.  This method is based on constraint propagation using Taylor-models [7-9] under a double-layer RTM framework.  In the inner layer, a possible subregion of the design space is tested over different subregions of uncertainty.  In the outer layer, the results from the inner layer are collected and summarized to determine overall feasibility and subsequent action.  A key feature of the method is the use of a validated solver (VSPODE) for parametric ODEs [10], which is used to produce guaranteed bounds on the solutions of dynamic systems with interval-valued parameters and/or initial states.  VSPODE consists of two phases applied at each integration step.  In the first phase, existence and uniqueness of the solution are proven, and a coarse enclosure of the solution for the entire integration step is computed.  In the second phase, a tighter enclosure of the solution is computed and bounded by Taylor models, which are symbolic (algebraic) functions of the uncertain parameters and initial states.

Our strategy provides four main advantages in this application.  First, the approach can be directly applied to the original nonlinear dynamic model without introducing any linearization or discrete-time approximation.  Second, the path constraint functions can be easily formulated using additional state variables and then tested for feasibility over the entire time interval of an integration step in the first phase of VSPODE, thus providing rigorous assurance that the path constraints are satisfied over the entire time horizon, and not just at discrete points.  Third, the use of Taylor models to represent bounds provides the power to obtain rigorous and very tight path enclosures with a lower computational cost, in comparison to other commonly used bounding methods.  Fourth, because the Taylor model method used provides an explicit algebraic representation of the state trajectories in terms of the design variables, constraint propagation techniques can be exploited to efficiently distinguish feasible subregions and infeasible ones.  The computational aspects of this new approach will be demonstrated through application to several example problems.

References

[1] Barton, P.; Pantelides, C.  Modeling of Combined Discrete-continuous Process. AIChE J. 1994, 40 , 966.

[2] Srinivasan, R. et al.  Safety Verification using a Hybrid Knowledge-based Mathematical Programming Framework. AIChE J. 1998, 44 , 361.

[3] Park, T.; Barton, P.  Implicit Model Checking of Logic-based Control Systems.

AIChE J. 1997, 43 , 2246.

[4] Dimitriadis, V. et al.  A Case Study in Hybrid Process Safety Verification.

Comput. Chem. Eng. 1996, 20 , S503.

[5] Huang, H et al.  Quantitative Framework for Reliable Safety Analysis. AIChE J. 2002, 48 , 78.

[6] Lin, Y.; Stadtherr, M.A.  Rigorous Model-based Safety Analysis for Nonlinear Continuous Time Systems. Comput. Chem. Eng. 2009, 32. 493.

[7] Lin, Y.; Stadtherr, M. A.  Deterministic Global Optimization of Nonlinear Dynamic Systems. AIChE J. 2007, 53 , 866.

[8] Lin, Y.; Stadtherr, M. A.  Deterministic Global Optimization for Parameter Estimation of Dynamic Systems. Ind. Eng. Chem. Res. 2006, 45, 8438.

[9] Lin, Y.; Stadtherr, M. A.  Enclosing All Solutions of Two-Point Boundary Value Problems for ODEs. Comput. Chem. Eng. 2008, 32, 1714.

[10] Lin, Y.; Stadtherr, M. A.  Validated Solutions of Initial Value Problems for Parametric ODEs. Appl. Num. Math. 2007, 57, 1145.