Andreas B. Hempel

Software engineering meets control theory

The software engineering community has proposed numerous approaches for making software self-adaptive. These approaches take inspiration from machine learning and control theory, constructing software that monitors and modifies its own behavior to meet goals. Control theory, in particular, has received considerable attention as it represents a general methodology for creating adaptive systems. Control-theoretical software implementations, however, tend to be ad hoc. While such solutions often work in practice, it is difficult to understand and reason about the desired properties and behavior of the resulting adaptive software and its controller. This paper discusses a control design process for software systems which enables automatic analysis and synthesis of a controller that is guaranteed to have the desired properties and behavior. The paper documents the process and illustrates its use in an example that walks through all necessary steps for self-adaptive controller synthesis.

Inverse Parametric Optimization With an Application to Hybrid System Control

We present a number of results on inverse parametric optimization and its application to hybrid system control. We show that any function that can be written as the difference of two convex functions can also be written as a linear mapping of the solution to a convex parametric optimization problem. We exploit these results in application to the control of systems with piecewise affine dynamics, and show that it is possible to model such systems as optimizing processes. Optimal control problems for such systems can be remodeled as bilevel optimization problems and solved with existing techniques.

Inverse Parametric Quadratic Programming and an Application to Hybrid Control

Abstract We present the complete solution to the inverse parametric quadratic programming problem: from a given continuous piecewise affine function we construct both the constraints and objective function of a parametric quadratic program, such that the supplied function is the unique parametric minimizer for the constructed problem data. In contrast to past approaches to this problem, our method does not rely on prior knowledge of the constraint set or sufficient sampling of the optimizer function, and is guaranteed to solve the inverse optimization problem exactly if a solution exists. We then apply this inverse optimization technique to the control of piecewise affine systems. By recasting the hybrid system dynamics as the parametric solution to a quadratic program obtained from our inverse optimization technique, we derive an equivalent linear complementarity model via the Karush-Kuhn-Tucker conditions of the identified optimization problem. This approach allows one to solve an optimal control problem for a piecewise affine system by solving a mathematical program with equilibrium constraints. Simulation results suggest that the computational effort required to solve such problems can be significantly smaller than that required for conventional mixed-integer quadratic programming approaches for systems with piecewise affine dynamics. We demonstrate via two numerical examples that globally optimal points can be identified using this approach.

Robust stability of a class of interconnected nonlinear positive systems

We present conditions for robust stability of a class of linear systems interconnected by uncertain nonlinear, norm-bounded functions. We show that such conditions can be reformulated as classical small gain like conditions for a related linear system. Under further assumptions that render the related linear system positive, we show that we can achieve sharp tractable conditions for robust stability of the original nonlinear system.

Control Strategies for Self-Adaptive Software Systems

The pervasiveness and growing complexity of software systems are challenging software engineering to design systems that can adapt their behavior to withstand unpredictable, uncertain, and continuously changing execution environments. Control theoretical adaptation mechanisms have received growing interest from the software engineering community in the last few years for their mathematical grounding, allowing formal guarantees on the behavior of the controlled systems. However, most of these mechanisms are tailored to specific applications and can hardly be generalized into broadly applicable software design and development processes. This article discusses a reference control design process, from goal identification to the verification and validation of the controlled system. A taxonomy of the main control strategies is introduced, analyzing their applicability to software adaptation for both functional and nonfunctional goals. A brief extract on how to deal with uncertainty complements the discussion. Finally, the article highlights a set of open challenges, both for the software engineering and the control theory research communities.

A novel method for modelling cardinality and rank constraints

Constraints on the cardinality or rank of decision variables in optimization problems are generally modelled separately from algebraic constraints. In this paper we show that cardinality constraints on vectors and rank constraints on matrices can be represented using purely algebraic constraints on continuous variables, by exploiting classical results on the Ky Fan norm for matrices and its analogous norm for vectors. Using this technique, a vector cardinality constraint can be modelled via introduction of a small number of additional variables and linear constraints, in conjunction with a single bilinear inequality. Analogously, a matrix rank constraint can be modelled via introduction of additional matrix variables and linear matrix inequalities, in conjunction with a single bilinear matrix inequality. We discuss a number of variations on cardinality and rank constraints that can be modelled in a similar way.

Strong Stationarity Conditions for Optimal Control of Hybrid Systems

We present necessary and sufficient optimality conditions for finite-time optimal control problems for a class of hybrid systems described by linear complementarity models. Although these optimal control problems are difficult in general due to the presence of complementarity constraints, we provide a set of structural assumptions ensuring that the tangent cone of the constraints possesses geometric regularity properties. These imply that the classical Karush–Kuhn–Tucker conditions of nonlinear programming theory are both necessary and sufficient for local optimality, which is not the case for general mathematical programs with complementarity constraints. We also present sufficient conditions for global optimality. We proceed to show that the dynamics of every continuous piecewise affine (PWA) system can be written as the optimizer of a mathematical program, which results in a linear complementarity model satisfying our structural assumptions. Hence, our stationarity results apply to a large class of hybrid systems with PWA dynamics. We present simulation results showing the substantial benefits possible from using a nonlinear programming approach to the optimal control problem with complementarity constraints instead of a more traditional mixed-integer formulation.

A necessary optimality condition for constrained optimal control of hybrid systems

Using the fact that continuous piecewise affine systems can be written as special inverse optimization models, we present necessary optimality conditions for constrained optimal control problems for hybrid dynamical systems. The modeling approach is based on the fact that piecewise affine functions can be written as the difference of two convex functions and has been described in previous publications. The inverse optimization model resulting from this approach can be replaced by its Karush-Kuhn-Tucker conditions to yield a linear complementarity model. An optimal control problem for this model class is an instance of a mathematical program with complementarity constraints for which classical Karush-Kuhn-Tucker optimality conditions may not hold. Exploiting the regularity properties of the inverse optimization model, we show why for the class of control problems under consideration this is not the case and the classical optimality conditions also characterize optimal input trajectories.

Output-feedback controlled-invariant sets for systems with linear parameter-varying state transition matrix

The notion of output-feedback controlled-invariant sets is extended from LTI systems to systems with linear parameter-varying state transition matrix. A theorem is presented that can be used to verify whether a given polytope can be made invariant under output-feedback. The theorem also provides the constraints a control input has to fulfill to make the candidate set invariant. Predictive output-feedback controllers based on such a set can satisfy hard constraints on both the plant state and the control inputs in the presence of process disturbances and measurement noise. Simulation results demonstrate the strength of such a controller that can guarantee constraints for a subset of the state space without requiring state information or estimation.

Model-based current limiting for traction control of an electric four-wheel drive race car

This paper describes a novel traction control method and its application to an electric four-wheel driven race car. The proposed control method is based on a detailed model of tire dynamics and is designed for hardware with limited memory and computational power. We derive a linear parameter-varying model from first principles and validate it against a full nonlinear vehicle model. We then use the model to design a gain-scheduled LQRI controller, parametric on the measured vehicle velocity and lateral acceleration. We show that when incorporating additional information about the tire state, a gain scheduled LQRI controller is capable of minimizing excessive wheel spin by limiting the maximum torque available to the driver. This leads to a performance gain in acceleration while improving the handling characteristics of the race car. The proposed controller is thoroughly tested for its sensitivity to sensor noise and changes in system parameters in simulation and then implemented on a prototype race car competing in Formula Student. Experiments indicate satisfactory experimental performance from the initial control design without additional tuning of the controller parameters. This illustrates the simplicity of the design and ease of implementation.