Inner-Approximating Reachable Sets for Polynomial Systems With Time-Varying Uncertainties

Abstract

In this paper we propose a convex programming based method to address a long-standing problem of inner-approximating backward reachable sets of state-constrained polynomial systems subject to time-varying uncertainties. The backward reachable set is a set of states, from which all trajectories starting will surely enter a target region at the end of a given time horizon without violating a set of state constraints in spite of the actions of uncertainties. It is equal to the zero sub-level set of the unique Lipschitz viscosity solution to a Hamilton-Jacobi partial differential equation (HJE). We show that inner-approximations of the backward reachable set can be formed by zero sub-level sets of its viscosity super-solutions. Consequently, we reduce the inner-approximation problem to a problem of synthesizing polynomial viscosity super-solutions to this HJE. Such a polynomial solution in our method is synthesized by solving a single semi-definite program. We also prove that polynomial solutions to the formulated semi-definite program exist and can produce a convergent sequence of inner-approximations to the interior of the backward reachable set in measure under appropriate assumptions. This is the main contribution of this work. Several illustrative examples demonstrate the merits of our approach. Related WorkAs mentioned above, inner-approximate reachability analysis of ordinary differential equations subject to time-varying uncertainties and state constraints, is still in its infancy and thus provides an open area of research.For ordinary differential equations free of time-varying uncertainties and state constraints, [17], [18] proposed a method based on modal intervals with affine forms to inner-approximate reachable sets using intervals. By making use of the homeomorphism property of the solution mapping, a boundary based reachability analysis method was proposed to inner-approximate reachable sets with polytopes in [44], and it was extended to a class of delay differential equations in [43]. Since reachable sets of nonlinear systems tend to be non-convex, the above mentioned methods based on convex set representations may result in poor approximations. As accuracy is also an important factor in performing reachability analysis (e.g.,[37], [28]), more complex shapes of representations such as Taylor models and semi-algebraic sets are desirable.[7] proposed a Taylor model backward flowpipe method to compute inner-approximations. [41] proposed an iterative method, with each iteration relying on solving semi-definite programming problems, to compute semi-algebraic inner-approximations for polynomial systems using the advection map of the given dynamical system. [45] extended the method in [44] to compute semi-algebraic inner-approximations of reachable sets for polynomial systems and beyond by solving semi-definite programming problems. Recently, [42] formulated the problem of solving HJEs as a semi-definite program to compute inner-approximations for polynomial systems. For state-constrained polynomial systems ...

Under-Approximate Reachability Analysis for a Class of Linear Uncertain Systems

“…Using the triangular inequality, d(I1, I3) ≤ d (I1, I2) + d (I2, I3) . The term d (I1, I2) can be bounded above using the triangular inequality, the definition of I given in Equation (10), and Lemma 3(a),(c), as follows:…”

“…For example, when the input set is associated with control constraints, it is often preferred to under-estimate the set of attainable states [9]; hence, under-approximations can be useful in such contexts. Moreover, under-approximations of backward reachable sets starting from target sets can be used to detect subsets of the initial states from which all trajectories fulfill safety specifications with final values laying inside the target sets [10]. Furthermore, under-approximations of forward reachable sets can be used in solving falsification problems: verifying if the reachable sets/tubes intersect with unsafe sets [11].…”

Online Inner Approximation of Reachable Sets of Nonlinear Systems with Diminished Control Authority

“…Another application is flight envelope estimation for aircraft; the penalty of overconfidence in flight envelope estimation is often severe, and over-preparedness as the cost of underconfidence is much preferred in such a context [26]. Methods for determining inner approximations of reachable sets have been based on various principles, including relying on polynomial inner approximation of the nonlinear system dynamics using interval calculus [9], ellipsoid calculus [8], and viscosity solutions to HJB equations [29].…”

Finite Horizon Backward Reachability Analysis and Control Synthesis for Uncertain Nonlinear Systems

“…In contrast, the exact BRS is computed in [5]- [8] as the sublevel set of the solution to Hamilton-Jacobi (HJ) partial differential equations (PDEs). Other results provide BRS inner-approximations using relaxed HJ equations [9]- [11] and Lyapunov-based methods [12].…”