(257c) Generalized Chaos Expansions with Arbitrary Multivariate Probability Measures: Applications in Closed-Loop Performance Verification for Stochastic Dynamic Systems

Modern control design methods, such as model predictive control (MPC) and its many variants including robust & stochastic MPC, often use a control-relevant model and a control-relevant uncertainty description (e.g., finite number of scenarios) that is amenable to online computations [1]. However, in many practical applications, there can be significant mismatch between the “real system” behavior and the model internally used in the controller. A complex high-fidelity model of the true system can sometimes be developed using first principles models, which more accurately predicts the system response over a significantly larger range of length- and time-scales. High-fidelity models can then be used for performance verification as well as for shaping closed-loop performance via tuning the many different changeable parameters (e.g., objective function, uncertainty description, prediction horizon, etc.) of the “robust” model-based controller [2].

In many engineering applications, however, even the high-fidelity models will have significant sources of uncertainty. This fact has prompted the development of broad range of tools for uncertainty quantification (UQ), which attempts to systematically understand the impact of these errors on the model predictions [3]. Therefore, the goal of this work is to develop an efficient approach for UQ in the context of closed-loop performance verification of nonlinear systems with arbitrary probability uncertainties. In particular, we focus on solving the “forward problem” of propagating uncertainty in unknown parameters on key quantities of interests (QoIs) that are directly related to closed-loop performance metrics (e.g., objective or constraint violations).

A huge variety of methods have been developed for stochastic UQ each with their own set of advantages and disadvantages. Our particular choice of methods is motived by two key features that are important in the context of closed-loop performance verification: (1) the closed-loop system is expensive to simulate for every realization of the uncertain model parameters due to the fact that an embedded optimization problem must be solved at sample time and (2) the uncertain parameters can have an arbitrary probability distribution. This latter point is motivated by the fact that the parameter distribution will often be defined as the solution to a Bayesian inversion problem, which is known to lead to highly correlated/dependent random variables. Monte Carlo Sampling (MCS) is the standard approach for solving this type of problem and has been applied for tuning backoff constraints in MPC [4]; however, the estimated statistics of the QoIs are known to converge very slowly with respect to number of parameter realizations meaning a large number of realizations is required for accurate predictions. A recently developed alternative to MCS, broadly termed generalized polynomial chaos (gPC) [5], has become one of the most widely used methods for stochastic UQ. The gPC framework can be thought of as a spectral representation (i.e., basis function expansion) in random space wherein the basis is chosen to be a set of polynomials that are orthogonal with respect to a user-specified germ distribution to simplify computations. However, gPC and its variants require the germ to have independent random elements. Thus, in order to apply gPC, the dependent random parameters must be transformed to any chosen independent germ. Not only is defining this transformation difficult (or impossible) in many practical cases, but the use of this transformation can introduce additional nonlinearities that result in a big drop in the expansion’s rate of convergence

The main contribution of this work is to further generalize the concept of gPC so that the germ can have a truly arbitrary multivariate distribution. This notion is very similar to that of so-called arbitrary PC (aPC) [6]; however, it differs in the sense that the basis functions do not have to be polynomials and the independence assumption is completely removed. This proposed concept of a generalized chaos expansion (GCE) involves two key steps: (1) defining an orthonormal basis (ONB) with respect to arbitrary probability measures and (2) determining the expansion coefficients. Thus, this work also presents an efficient algorithm for numerically solving these two problems. The proposed method only requires moments of the germ distribution to be known so that the full distribution does not need to be known in closed-form. Owing to the orthogonal projection theorem, the latter problem of determining the expansion coefficients can be defined exactly as a multivariate integral with respect to the chosen arbitrary measure. This represents the computational bottleneck in this setting due to the expensiveness of the closed-loop simulations. Utilizing concepts from moment-matching optimization [7], we can develop a “quasi-optimal” cubature rule to accurately approximate these integrals with a small number of simulations, which substantially reduces the computational cost of the method.

The proposed tractable stochastic UQ method is demonstrated in simulations on a benchmark continuously-stirred tank reactor (CSTR) that is run in closed-loop with a robust “scenario-tree” (or multi-stage) MPC algorithm. It is shown that the proposed method can guarantee satisfaction of distributional state chance constraints through proper determination of controller parameters (e.g., the MPC constraint backoff in this case). Additionally, the proposed method is compared to gPC wherein we show that it can achieve higher accuracy with lower-order expansions while requiring an order-of-magnitude fewer closed-loop simulations (resulting in huge computational savings), owing to the fact that the germ transformation has been avoided.

References

[1] D. Q. Mayne, “MPC: Recent developments and future promise,” Automatica, vol. 50, pp. 2967–2986, 2014.

[2] J. A. Paulson and A. Mesbah. “Arbitrary Polynomial Chaos for Quantification of General Probabilistic Uncertainties: Shaping Closed-loop Behavior of Nonlinear Systems,” submitted to the 57^th IEEE Conference on Decision and Control (under review).

[3] D. Xiu. "Fast numerical methods for stochastic computations: a review." Communications in computational physics 5.2-4 (2009): 242-272.

[4] H. Jang, J. H. Lee, and L. T. Biegler. "A robust NMPC scheme for semi-batch polymerization reactors." IFAC-PapersOnLine 49.7 (2016): 37-42.

[5] D. Xiu and G. E. Karniadakis, “The Wiener-Askey polynomial chaos for stochastic differential equations,” SIAM Journal of Scientific Computation, vol. 24, pp. 619–644, 2002.

[6] J. A. Paulson, E. A. Buehler, and A. Mesbah, “Arbitrary polynomial chaos for uncertainty propagation of correlated random variables in dynamic systems,” IFAC-PapersOnLine, vol. 50, pp. 3548–3553, 2017.

[7] E. K. Ryu and S. P. Boyd, “Extensions of Gauss quadrature via linear programming,” Foundations of Computational Mathematics, vol. 15, pp. 953–971, 2015.