Advances in Reachability Analysis for Nonlinear Dynamic Systems

Systems of nonlinear ordinary differential equations (ODEs) are used to model an incredible variety of dynamic phenomena in chemical, oil and gas, and pharmaceutical industries. In reality, such models are nearly always subject to significant uncertainties in their initial conditions, parameters, and inputs. This dissertation provides new theoretical and numerical techniques for rigorously enclosing the set of solutions reachable by a given systems of nonlinear ODEs subject to uncertain initial conditions, parameters, and time-varying inputs. Such sets are often referred to as reachable sets, and methods for enclosing them are critical for designing systems that are passively robust to uncertainty, as well as for optimal real-time decision-making. Such enclosure methods are used extensively for uncertainty propagation, robust control, system verification, and optimization of dynamic systems arising in a wide variety of applications. Unfortunately, existing methods for computing such enclosures often provide an unworkable compromise between cost and accuracy. For example, interval methods based on differential inequalities (DI) can produce bounds very efficiently but are often too conservative to be of any practical use. In contrast, methods based on more complex sets can achieve sharp bounds, but are far too expensive for real-time decision-making and scale poorly with problem size. Recently, it has been shown that bounds computed via differential inequalities can often be made much less conservative while maintaining high efficiency by exploiting redundant model equations that are known to hold for all trajectories of interest (e.g., linear relationships among chemical species in a reaction network that hold due to the conservation of mass or elements). These linear relationships are implied by the governing ODEs, and can thus be considered redundant. However, these advances are only applicable to a limited class of system in which pre-existing linear redundant model equations are available. Moreover, the theoretical results underlying these algorithms do not apply to redundant equations that depend on time-varying inputs and rely on assumptions that prove to be very restrictive for nonlinear redundant equations, etc. This dissertation continues a line of research that has recently achieved very promising bounding results using methods based on differential inequalities. In brief, the major contributions can be divided into three categories: (1) In regard to algorithms, this dissertation significantly improves existing algorithms that exploit linear redundant model equations to achieve more accurate and efficient enclosures. It also develops new fast and accurate bounding algorithms that can exploit nonlinear redundant model equations. (2) Considering theoretical contributions, it develops a novel theoretical framework for the introduction of redundant model equations into arbitrary dynamic models to effectively reduce conservatism. The newly developed theories have more generality in terms of application. For example, complex nonlinear constraints that involve states, time derivatives of the system states, and time- varying inputs are allowed to be exploited. (3) A new differential inequalities method called Mean Value Differential Inequalities (MVDI) is developed that can automatically introduce redundant model equations for arbitrary dynamic systems and has a second-order convergence rate reported the first time among DI-based methods

Open- and Closed-Loop Neural Network Verification using Polynomial Zonotopes

We present a novel approach to efficiently compute tight non-convex enclosures of the image through neural networks with ReLU, sigmoid, or hyperbolic tangent activation functions. In particular, we abstract the input-output relation of each neuron by a polynomial approximation, which is evaluated in a set-based manner using polynomial zonotopes. While our approach can also can be beneficial for open-loop neural network verification, our main application is reachability analysis of neural network controlled systems, where polynomial zonotopes are able to capture the non-convexity caused by the neural network as well as the system dynamics. This results in a superior performance compared to other methods, as we demonstrate on various benchmarks

Sensitivity analysis of uncertain dynamic systems using set-valued integration

We present an extension of set-valued integration to enable efficient sensitivity analysis of parameter-dependent ordinary differential equation (ODE) systems, using both the forward and adjoint methods. The focus is on continuous-time set-valued integration, whereby auxiliary ODE systems are derived whose solutions describe high-order inclusions of the parametric trajectories in the form of polynomial models. The forward and adjoint auxiliary ODE systems treat the parameterization error of the original differential variables as a time-varying uncertainty, and propagate the sensitivity bounds forward and backward in time, respectively. This construction enables building on the sensitivity analysis capabilities of state-of-the-art solvers, such as CVODES in the SUNDIALS suite. Several numerical case studies are presented to assess the performance and accuracy of these set-valued sensitivity integrators

Branch-and-lift algorithm for deterministic global optimization in nonlinear optimal control

This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced, which refines the control parameterization via a Gram-Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example. © 2013 Springer Science+Business Media New York

Reachability analysis and deterministic global optimization of differential-algebraic systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 447-460).Systems of differential-algebraic equations (DAEs) are used to model an incredible variety of dynamic phenomena. In the chemical process industry in particular, the numerical simulation of detailed DAE models has become a cornerstone of many core activities including, process development, economic optimization, control system design and safety analysis. In such applications, one is primarily interested in the behavior of the model solution with respect variations in the model inputs or uncertainties in the model itself. This thesis addresses two computational problems of general interest in this regard. In the first, we are interested in computing a guaranteed enclosure of all solutions of a given DAE model subject to a specified set of inputs. This analysis has natural applications in uncertainty quantification and process safety verification, and is used for many important tasks in process control. However, for nonlinear dynamic systems, this task is very difficult. Existing methods apply only to ordinary differential equation (ODE) models, and either provide very conservative enclosures or require excessive computational effort. Here, we present new methods for computing interval bounds on the solutions of ODEs and DAEs. For ODEs, the focus is on efficient methods for using physical information that is often available in applications to greatly reduce the conservatism of existing methods. These methods are then extended for the first time to the class of semi-explicit index-one DAEs. The latter portion of the thesis concerns the global solution of optimization problems constrained by DAEs. Such problems arise in optimal control of batch processes, determination of optimal start-up and shut-down procedures, and parameter estimation for dynamic models. In nearly all conceivable applications, there is significant economic and/or intellectual impetus to locate a globally optimal solution. Yet again, this problem has proven to be extremely difficult for nonlinear dynamic models. A small number of practical algorithms have been proposed, all of which are limited to ODE models and require significant computational effort. Here, we present improved lower-bounding procedures for ODE constrained problems and develop a complete deterministic algorithm for problems constrained by semi-explicit index-one DAEs for the first time.by Joseph Kirk Scott.Ph.D

Set-based state estimation and fault diagnosis using constrained zonotopes and applications

This doctoral thesis develops new methods for set-based state estimation and active fault diagnosis (AFD) of (i) nonlinear discrete-time systems, (ii) discrete-time nonlinear systems whose trajectories satisfy nonlinear equality constraints (called invariants), (iii) linear descriptor systems, and (iv) joint state and parameter estimation of nonlinear descriptor systems. Set-based estimation aims to compute tight enclosures of the possible system states in each time step subject to unknown-but-bounded uncertainties. To address this issue, the present doctoral thesis proposes new methods for efficiently propagating constrained zonotopes (CZs) through nonlinear mappings. Besides, this thesis improves the standard prediction-update framework for systems with invariants using new algorithms for refining CZs based on nonlinear constraints. In addition, this thesis introduces a new approach for set-based AFD of a class of nonlinear discrete-time systems. An affine parametrization of the reachable sets is obtained for the design of an optimal input for set-based AFD. In addition, this thesis presents new methods based on CZs for set-valued state estimation and AFD of linear descriptor systems. Linear static constraints on the state variables can be directly incorporated into CZs. Moreover, this thesis proposes a new representation for unbounded sets based on zonotopes, which allows to develop methods for state estimation and AFD also of unstable linear descriptor systems, without the knowledge of an enclosure of all the trajectories of the system. This thesis also develops a new method for set-based joint state and parameter estimation of nonlinear descriptor systems using CZs in a unified framework. Lastly, this manuscript applies the proposed set-based state estimation and AFD methods using CZs to unmanned aerial vehicles, water distribution networks, and a lithium-ion cell.Comment: My PhD Thesis from Federal University of Minas Gerais, Brazil. Most of the research work has already been published in DOIs 10.1109/CDC.2018.8618678, 10.23919/ECC.2018.8550353, 10.1016/j.automatica.2019.108614, 10.1016/j.ifacol.2020.12.2484, 10.1016/j.ifacol.2021.08.308, 10.1016/j.automatica.2021.109638, 10.1109/TCST.2021.3130534, 10.1016/j.automatica.2022.11042

LNCS

Reachability analysis is difficult for hybrid automata with affine differential equations, because the reach set needs to be approximated. Promising abstraction techniques usually employ interval methods or template polyhedra. Interval methods account for dense time and guarantee soundness, and there are interval-based tools that overapproximate affine flowpipes. But interval methods impose bounded and rigid shapes, which make refinement expensive and fixpoint detection difficult. Template polyhedra, on the other hand, can be adapted flexibly and can be unbounded, but sound template refinement for unbounded reachability analysis has been implemented only for systems with piecewise constant dynamics. We capitalize on the advantages of both techniques, combining interval arithmetic and template polyhedra, using the former to abstract time and the latter to abstract space. During a CEGAR loop, whenever a spurious error trajectory is found, we compute additional space constraints and split time intervals, and use these space-time interpolants to eliminate the counterexample. Space-time interpolation offers a lazy, flexible framework for increasing precision while guaranteeing soundness, both for error avoidance and fixpoint detection. To the best of out knowledge, this is the first abstraction refinement scheme for the reachability analysis over unbounded and dense time of affine hybrid systems, which is both sound and automatic. We demonstrate the effectiveness of our algorithm with several benchmark examples, which cannot be handled by other tools