Bengt Lennartson

Perspectives and results on the stability and stabilizability of hybrid systems

This paper introduces the concept of a hybrid system and some of the challenges associated with the stability of such systems, including the issues of guaranteeing stability of switched stable systems and finding conditions for the existence of switched controllers for stabilizing switched unstable systems. In this endeavour, this paper surveys the major results in the (Lyapunov) stability of finite-dimensional hybrid systems and then discusses the stronger, more specialized results of switched linear (stable and unstable) systems. A section detailing how some of the results can be formulated as linear matrix inequalities is given. Stability analyses on the regulation of the angle of attack of an aircraft and on the PI control of a vehicle with an automatic transmission are given. Other examples are included to illustrate various results in this paper.

Stability and robustness for hybrid systems

Stability and robustness issues for hybrid systems are considered in this paper. Present stability results, that are extensions of classical Lyapunov theory, are not straightforward to apply in general due to two reasons. First, existing theory do not unveil how to find needed Lyapunov functions. Secondly, at some time instants it is necessary to know the values of the continuous trajectory. Because of these drawbacks, stronger conditions for stability are suggested. The search for Lyapunov functions can then be formulated as a linear matrix inequality problem. Additionally, it is shown how to obtain robustness properties. An example illustrates the results.

Stabilization of hybrid systems using a min-projection strategy

This paper describes a method of how to stabilize a system consisting of several subsystems. The subsystems are described by nonlinear models with different vector fields. The method is denoted the min-projection strategy, since the vector field associated with the smallest (skew) projection is selected for each state. Conditions are given guaranteeing (exponential) stability. It is also shown how these conditions can be formulated as a nonlinear optimization problem, or, for a pre-determined projection matrix, a linear matrix inequality (LMI) problem. Sliding motions may occur in the basic form of the strategy. However, it is shown how this behavior can be avoided by introducing hysteresis around the switch surfaces, still preserving the stability of the closed-loop hybrid system. Two examples are given to motivate and exemplify the strategy.

Feedback stabilization of nonaxisymmetric resistive wall modes in tokamaks. I. Electromagnetic model

Active feedback stabilization of pressure-driven modes in tokamaks is studied computationally in toroidal geometry. The stability problem is formulated in terms of open-loop transfer functions for fluxes in sensor coils resulting from currents in feedback coils. The transfer functions are computed by an extended version of the MARS stability code [A. Bondeson et al., Phys. Fluids B 4, 1889 (1992)] and can be accurately modeled by low order rational functions. In the present paper stability is analyzed for a system with an ideal amplifier (current control). It is shown that feedback with modest gain, and a single coil array poloidally, gives substantial stabilization for a range of coil shapes. Optimum design uses sensors for the poloidal field, located inside the resistive wall, in combination with rather wide feedback coils outside the wall. Typically, the feedback does not strongly modify the plasma-generated magnetic field perturbation. A future companion paper [C. M. Fransson et al., Phys. Plasmas (ac...

LMI for stability and robustness of hybrid systems

This paper presents results for stability and robustness of hybrid systems consisting of nonlinear subsystems. Present stability results that are extensions of classical Lyapunov theory are restricted to certain kinds of hybrid systems. For example, it is required that all subsystems have the same equilibrium point. In this paper these results are generalized, but more importantly, it is shown how Lyapunov functions for hybrid systems can be solved by linear matrix inequalities (LMI). Furthermore, it is shown how uncertainties around the nominal switch sets can be handled by introducing acceptable switch regions as additional stability conditions. The theory is illustrated by an example.

Hybrid systems in process control

Modeling and control of hybrid systems, with particular emphasis on process control applications, are considered in this article. Based on a number of observations on typical mixed discrete and continuous features for such applications, a fairly general model structure for hybrid systems is proposed. This model structure, which clearly separates the open-loop plant from the closed-loop system, is suitable for analysis and synthesis of hybrid control systems. To illustrate this, three different approaches for control-law synthesis based on continuous and discrete specifications are discussed. In the first one, the hybrid plant model is replaced by a purely discrete event model, related to the continuous specification, and a supervisor is synthesized applying supervisory control theory suggested by Ramadge and Wonham (1987). The other two methods directly utilize the continuous specification for determination of a control event generator, where time-optimal aspects are introduced as an option in the last approach.

Hybrid system stability and robustness verification using linear matrix inequalities

This paper addresses issues concerning exponential stability and robustness of hybrid systems. Stability conditions using Lyapunov techniques are given. The search for the Lyapunov functions is formulated as a linear matrix inequality (LMI) problem for hybrid systems with affine as well as non-linear vector fields. It is shown how the Lyapunov approach most advantageously also can be used to guarantee stability despite the presence of model uncertainties. Several examples are given to illustrate the theory.

Robust tuning of PI and PID controllers: using derivative action despite sensor noise

This article presents some easily understood and applied methods for close-to-optimal tuning PI and PID controllers. By optimal it means good midfrequency robustness and the best possible tradeoff between output performance and control activity. For plants with all poles on the negative real axis, a simple step response can provide adequate plant knowledge. For plants with integral action, an impulse response or a relay experiment can be used. In all PID cases, the controller zeros can be fixed, the control activity can be varied by the filter factor, and, finally, the integral gain can be adjusted to the required damping of a step response for the closed-loop system. with this strategy, tuning a PID controller is as easy as tuning a PI controller, the difference being that the PID solution gives additional freedom for selecting slightly higher control activity which significantly improves the output performance. When the situation demands HF rolloff of the controller, the PI or PID controller can be augmented by an additional lowpass filter. In these cases, the inclusion of derivative action is recommended. Four of the five controller parameters are then easily found, and the remaining gain can be manually tuned to obtain a desired tradeoff between output performance and damping (MF robustness).

Sequence Planning for Integrated Product, Process and Automation Design

In order to obtain a unified information flow from early product design to final production, an integrated framework for product, process and automation design is presented. The framework is based on sequences of operations and includes a formal relation between product properties and process operations. This relation includes liaisons (interfaces) and precedence relations, where the precedence relations generate preconditions for the related process operations. From this information a set of sequences of operations (SOPs) is generated. A formal graphical language for hierarchical operations and SOPs is then introduced and defined based on automata extended with variables. Since the operations are self-contained they can be grouped and viewed from different angles, e.g., from a product or a resource perspective. These multiple views increase the interoperability between different engineering disciplines. A case study is performed on a car manufacturing cell, where the suggested modeling framework is shown to give comprehensible SOPs.

Efficient supervisory synthesis of large systems

Due to the state-space explosion, many synthesis and verification problems for discrete event systems cannot be solved using traditional state-space traversal algorithms. This work presents efficient methods for reachability search based on symbolic computations using Binary Decision Diagrams. In addition, simple guidelines and quantities for separating hard and easy reachability problems are presented. Furthermore, the performance of the presented algorithms and heuristics is demonstrated on a set of industrial benchmark examples. It is also shown that such examples are much more challenging than some well-known handmade benchmark models. This is also indicated by the presented hardness quantities.