Sensitivity Analysis of Uncertain Dynamic Systems Using Set-Valued Integration with Application to Complete-Search Optimization

Set-valued integration methods for uncertain dynamic systems find applications in many research areas, including reachability and invariance analysis, set-membership estimation, global optimization, and robust optimization and control [1,2]. Existing methods can be classified as either continuous-time or discretized approaches. The former considers an auxiliary set of differential equations, which propagate a parameterization describing an enclosure of the system's trajectories, pointwise in time. The latter relies on a predictor-validation approach, which constructs parameterized tubes using time-series expansion and determines time intervals whereby these tubes yield a guaranteed enclosure of the trajectories. Both methods can propagate a variety of tube parameterizations describing either convex enclosures (e.g., interval boxes, ellipsoids or zonotopes) or nonconvex enclosures (e.g., image sets of multivariate polynomials); see [3,4].

The main contribution of this paper is an extension of existing set-valued integrators to enable sensitivity analysis. Given an uncertain dynamic system in the form of a nonlinear parametric ODE,
dx/dt = f(t,x(t),p),
with initial condition x(0)=h(p) and uncertain parameters p∈P⊂IRⁿ, we consider the problem to compute bounds on the gradients of state-dependent functions such as Φ(p):=φ(x(T),p), with T>0. Our focus is on continuous-time set-valued integration, and we consider both forward and reverse (adjoint) sensitivity analysis. A derivation of auxiliary ODEs describing enclosures of the sensitivity or adjoint trajectories of the parametric ODE is presented, and we discuss efficient numerical integration procedures for these ODEs based on SUNDIALS-CVODES [5] in order to exploit their underlying structures.

An application of this new sensitivity/adjoint bounding capability for parametric ODEs is presented in the context of parameter estimation in dynamic systems using experimental data. Numerical experiments are carried out by introducing cuts derived from the first-order optimality conditions in order to tighten the relaxations and reduce clustering in branch-and-bound search for global optimization.

References:

1. B. Chachuat, B. Houska, R. Paulen, N. Peric, J. Rajyaguru, M.E. Villanueva, "Set-theoretic approaches in analysis, estimation and control of nonlinear systems," IFAC-PapersOnLine 48(8):981-995, 2015.
2. S. Streif, K.K. Kim, P. Rumschinski, M. Kishida, D.E. Shen, R. Findeisen, and R.D. Braatz, "Robustness analysis, prediction and estimation for uncertain biochemical networks," Journal of Process Control 42:14-34, 2016.
3. M.E. Villanueva, B. Houska, B. Chachuat, "Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs," Journal of Global Optimization 62(3):575-613, 2015.
4. B. Houska, M.E. Villanueva, and B. Chachuat, "Stable set-valued integration of nonlinear dynamic systems using affine set-parameterizations," SIAM Journal on Numerical Analysis 53(5):2307-2328, 2015.
5. A.C. Hindmarsh, P.N. Brown, K.E. Grant, S.L. Lee, R. Serban, D.E. Shumaker, and C.S. Woodward, "SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers," ACM Transactions on Mathematical Software, 31(3), pp. 363-396, 2005.