Gunther Reissig

Computing Abstractions of Nonlinear Systems

Sufficiently accurate finite state models, also called symbolic models or discrete abstractions, allow one to apply fully automated methods, originally developed for purely discrete systems, to formally reason about continuous and hybrid systems and to design finite state controllers that provably enforce predefined specifications. We present a novel algorithm to compute such finite state models for nonlinear discrete-time and sampled systems which depends on quantizing the state space using polyhedral cells, embedding these cells into suitable supersets whose attainable sets are convex, and over-approximating attainable sets by intersections of supporting half-spaces. We prove a novel recursive description of these half-spaces and propose an iterative procedure to compute them efficiently. We also provide new sufficient conditions for the convexity of attainable sets which imply the existence of the aforementioned embeddings of quantizer cells. Our method yields highly accurate abstractions and applies to nonlinear systems under mild assumptions, which reduce to sufficient smoothness in the case of sampled systems. Its practicability in the design of discrete controllers for nonlinear continuous plants under state and control constraints is demonstrated by an example.

Feedback Refinement Relations for the Synthesis of Symbolic Controllers

We present an abstraction and refinement methodology for the automated controller synthesis to enforce general predefined specifications. The designed controllers require quantized (or symbolic) state information only and can be interfaced with the system via a static quantizer. Both features are particularly important with regard to any practical implementation of the designed controllers and, as we prove, are characterized by the existence of a feedback refinement relation between plant and abstraction. Feedback refinement relations are a novel concept introduced in this paper. Our work builds on a general notion of system with set-valued dynamics and possibly non-deterministic quantizers to permit the synthesis of controllers that robustly, and provably, enforce the specification in the presence of various types of uncertainties and disturbances. We identify a class of abstractions that is canonical in a well-defined sense, and provide a method to efficiently compute canonical abstractions. We demonstrate the practicality of our approach on two examples.

Strong Structural Controllability and Observability of Linear Time-Varying Systems

In this note we consider continuous-time systems ẋ(t)= A(t)x(t)+B(t)u(t), y(t)=C(t)x(t)+D(t)u(t) as well as discrete-time systems ẋ(t+1)=A(t)x(t)+B(t)u(t), y(t)= C(t)x(t)+D(t)u(t) whose coefficient matrices A, B, C and D are not exactly known. More precisely, all that is known about the systems is their nonzero pattern, i.e., the locations of the nonzero entries in the coefficient matrices. We characterize the patterns that guarantee controllability and observability, respectively, for all choices of nonzero time functions at the matrix positions defined by the pattern, which extends a result by Mayeda and Yamada for time-invariant systems. As it turns out, the conditions on the patterns for time-invariant and for time-varying discrete-time systems coincide, provided that the underlying time interval is sufficiently long. In contrast, the conditions for time-varying continuous-time systems are more restrictive than in the time-invariant case.

Human arm motion modeling and long-term prediction for safe and efficient Human-Robot-Interaction

Modeling and predicting human behavior is indispensable when industrial robots interacting with human operators are to be manipulated safely and efficiently. One challenge is that human operators tend to follow different motion patterns, depending on their intention and the structure of the environment. This precludes the use of classical estimation techniques based on kinematic or dynamic models, especially for the purpose of long-term prediction. In this paper, we propose a method based on Hidden Markov Models to predict the region of the workspace that is possibly occupied by the human within a prediction horizon. In contrast to predictions in the form of single points such as most likely human positions as obtained from previous approaches, the regions obtained here may serve as safety constraints when the robot motion is planned or optimized. This way one avoids collisions with a probability not less than a predefined threshold. The practicability of our method is demonstrated by successfully and accurately predicting the motion of a human arm in two scenarios involving multiple motion patterns.

Feedback refinement relations for symbolic controller synthesis

A common issue with existing approaches to symbolic controller synthesis lies in the huge complexity of the resulting controllers. In particular, the controllers usually need full plant state information and contain an abstraction of the plant as a building block. In this note, we present an extension which helps reduce that complexity. Our technique is based on the novel concept of feedback refinement relations to compare plants with their finite-state approximations. As an additional feature, our approach builds on infinitary completed trace semantics and allows for the synthesis of controllers for arbitrary, not necessarily prefix-closed specifications. We also reveal if and how existing symbolic controller synthesis procedures should be extended to benefit from the advantages of our technique.

Increasing efficiency of optimization-based path planning for robotic manipulators

Path planning for robotic manipulators interacting with obstacles is considered, where an end-effector is to be driven to a goal region in minimum time, collisions are to be avoided, and kinematic and dynamic constraints are to be obeyed. The obstacles can be time-varying in their positions, but the positions should be known or estimated over the prediction horizon for planning the path. This non-convex optimization problem can be approximated by Mixed Integer Programs (MIPs), which usually leads to a large number of binary variables, and hence, to inacceptable computational time for the planning. In this paper, we present a geometric result whose application drastically reduces the number of binary decision variables in the aforementioned MIPs for 3D motion planning problems. This leads to a reduction in computational time, which is demonstrated for different scenarios.

Abstraction-based solution of optimal stopping problems under uncertainty

In this paper we present novel results on the solution of optimal control problems with the help of finite-state approximations (“symbolic models”) of infinite-state plants. We investigate optimal stopping problems in the minimax sense, with undiscounted running and terminal costs, for nonlinear discrete-time plants subject to perturbations and constraints. This problem class includes finite-horizon and exit-(entry-)time problems as well as pursuit-evasion and reach-avoid games as special cases. We utilize symbolic models of the plant to upper bound the value function, i.e., the achievable closed-loop performance, and to compute controllers realizing the bounds. The symbolic models are obtained from suitable discretizations of the state and input spaces, and we prove that the computed bounds converge to the value function as the discretization errors approach zero. The value function is in general discontinuous, and the convergence (in the hypographical sense) is uniform on every compact subset of the state space. We apply the proposed method to design an approximately optimal feedback controller that starts up a DC-DC converter and is robust against supply voltage as well as load fluctuations.

Abstraction based solution of complex attainability problems for decomposable continuous plants

The focus of the present paper is systems of nonlinear continuous sub-plants that share a common input but are otherwise coupled only through the specification of a control problem, possibly including state constraints. Examples include cart-pole systems and collision avoidance problems involving multiple vehicles. We propose a method that uses finite state models for solving highly complex continuous attainability problems. We first prove that finite state models, also called discrete abstractions, of the overall plant may be obtained as products of abstractions of sub-plants. The latter, which we call factors, may be determined quickly and concurrently. We also modify a state-of-the-art algorithm for the discrete, auxiliary attainability problems that arise, to work directly with the set of factors and prove that the asymptotic computational complexity of the modified algorithm matches that of the original one. In practice, the latter will often be much slower since the representation of the abstraction of the overall plant on a computer is likely to require an excessive amount of memory. Practicability of our method is demonstrated by successfully designing discrete controllers that globally stabilize decomposable nonlinear continuous plants whose overall finite state models would include millions of states and billions of transitions. Working with the factors instead, problem data fit into main memory of a customary personal computer, and computations take only minutes.

Sufficient conditions for strong structural controllability of uncertain linear time-varying systems

In this paper, we extend the notion of strong structural controllability of linear time-invariant systems, a property that requires the controllability of each system in a specific class given by the zero-nonzero pattern of the system matrices, to the linear time-varying case ẋ (t) = A(t) · x(t) + B(t) · u(t), where A and B are matrices of analytic functions. It is demonstrated that the requirements for strong structural controllability of linear time-invariant systems are not sufficient for strong structural controllability of linear time-varying systems. Moreover, in the main result of this paper, sufficient conditions for strong structural controllability of linear time-varying systems are given and an algorithm for verifying this property is provided. Since time-invariant systems are included in the class of time-varying system, these conditions are also new sufficient conditions for strong structural controllability of time-invariant systems. Finally, the results are illustrated by means of an example.

Mixed-integer programming for optimal path planning of robotic manipulators

One of the fundamental problems in the field of robotic motion planning is to safely and efficiently drive the end effector of a robotic manipulator to a specified goal position. Here, safety refers to the requirement that the robotic manipulator must have no collision with surrounding obstacles, and efficiency requires that some predefined cost function is minimized. In addition, kinematic and dynamic constraints have to be satisfied. These requirements lead to non-convex optimization problems, which may be approximated by mixed-integer linear programs (MILPs). The solution of the latter, however, is often intolerably complex due to a huge number of binary decision variables. In the present paper, we consider motion planning scenarios with polyhedral obstacles and velocity constraints for the joint positions of the robotic manipulator. We provide a geometric result whose application leads to MILPs with drastically reduced numbers of binary decision variables. Computational efficiency is demonstrated for two- and three-link manipulators interacting with obstacles, where the number of simplex steps during the MILP solution is reduced by a factor of roughly 200 over previous methods. We also demonstrate the application of the proposed method to an industrial robot.