Efficient polyhedral enclosures for the reachable set of nonlinear control systems

This work presents a general theory for the construction of a polyhedral outer approximation of the reachable set (“polyhedral bounds”) of a dynamic system subject to time-varying inputs and uncertain initial conditions. This theory is inspired by the efficient methods for the construction of interval bounds based on comparison theorems. A numerically implementable instance of this theory leads to an auxiliary system of differential equations which can be solved with standard numerical integration methods. Meanwhile, the use of polyhedra provides greater flexibility in defining tight enclosures on the reachable set. These advantages are demonstrated with a few examples, which show that tight bounds can be efficiently computed for general, nonlinear systems. Further, it is demonstrated that the ability to use polyhedra provides a means to meaningfully distinguish between time-varying and constant, but uncertain, inputs.Novartis-MIT Center for Continuous Manufacturin

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

Optimization Methods and Algorithms for Classes of Black-Box and Grey-Box Problems

There are many optimization problems in physics, chemistry, finance, computer science, engineering and operations research for which the analytical expressions of the objective and/or the constraints are unavailable. These are black-box problems where the derivative information are often not available or too expensive to approximate numerically. When the derivative information is absent, it becomes challenging to optimize and guarantee optimality of the solution. The objective of this Ph.D. work is to propose methods and algorithms to address some of the challenges of blackbox optimization (BBO). A top-down approach is taken by first addressing an easier class of black-box and then the difficulty and complexity of the problems is gradually increased. In the first part of the dissertation, a class of grey-box problems is considered for which the closed form of the objective and/or constraints are unknown, but it is possible to obtain a global upper bound on the diagonal Hessian elements. This allows the construction of an edge-concave underestimator with vertex polyhedral solution. This lower bounding technique is implemented within a branch-and-bound framework with guaranteed convergence to global optimality. The technique is applied for the optimization of problems with embedded system of ordinary differential equations (ODEs). Time dependent bounds on the state variables and the diagonal elements of the Hessian are computed by solving auxiliary set of ODEs that are derived using differential inequalities. In the second part of the dissertation, general box-constrained black-box problems are addressed for which only simulations can be performed. A novel optimization method, UNIPOPT (Univariate Projection-based Optimization) based on projection onto a univariate space is proposed. A special function is identified in this space that also contains the global minima of the original function. Computational experiments suggest that UNIPOPT often have better space exploration features compared to other approaches. The third part of the dissertation addresses general black-box problems with constraints of both known and unknown algebraic forms. An efficient two-phase algorithm based on trust-region framework is proposed for problems particularly involving high function evaluation cost. The performance of the approach is illustrated through computational experiments which evaluate its ability to reduce a merit function and find the optima

Optimization Methods and Algorithms for Classes of Black-Box and Grey-Box Problems

There are many optimization problems in physics, chemistry, finance, computer science, engineering and operations research for which the analytical expressions of the objective and/or the constraints are unavailable. These are black-box problems where the derivative information are often not available or too expensive to approximate numerically. When the derivative information is absent, it becomes challenging to optimize and guarantee optimality of the solution. The objective of this Ph.D. work is to propose methods and algorithms to address some of the challenges of blackbox optimization (BBO). A top-down approach is taken by first addressing an easier class of black-box and then the difficulty and complexity of the problems is gradually increased. In the first part of the dissertation, a class of grey-box problems is considered for which the closed form of the objective and/or constraints are unknown, but it is possible to obtain a global upper bound on the diagonal Hessian elements. This allows the construction of an edge-concave underestimator with vertex polyhedral solution. This lower bounding technique is implemented within a branch-and-bound framework with guaranteed convergence to global optimality. The technique is applied for the optimization of problems with embedded system of ordinary differential equations (ODEs). Time dependent bounds on the state variables and the diagonal elements of the Hessian are computed by solving auxiliary set of ODEs that are derived using differential inequalities. In the second part of the dissertation, general box-constrained black-box problems are addressed for which only simulations can be performed. A novel optimization method, UNIPOPT (Univariate Projection-based Optimization) based on projection onto a univariate space is proposed. A special function is identified in this space that also contains the global minima of the original function. Computational experiments suggest that UNIPOPT often have better space exploration features compared to other approaches. The third part of the dissertation addresses general black-box problems with constraints of both known and unknown algebraic forms. An efficient two-phase algorithm based on trust-region framework is proposed for problems particularly involving high function evaluation cost. The performance of the approach is illustrated through computational experiments which evaluate its ability to reduce a merit function and find the optima