Zero dynamics and funnel control for linear electrical circuits

Abstract

Abstract. We consider electrical circuits containing linear resistances, capacitances, inductances. The circuits can be described by differential-algebraic input-output systems, where the input consists of voltages of voltage sources and currents of current sources and the output consists of currents of voltage sources and voltages of current sources. We generalize a characterization of asymptotic stability of the circuit and give sufficient topological criteria for its invariant zeros being located in the open left half-plane. We show that asymptotic stability of the zero dynamics can be characterized by means of the interconnectivity of the circuit and that it implies that the circuit is high-gain stabilizable with any positive high-gain factor. Thereafter we consider the output regulation problem for electrical circuits by funnel control. We show that for circuits with asymptotically stable zero dynamics, the funnel controller achieves tracking of a class of reference signals within a pre-specified funnel; this means in particular that the transient behaviour of the output error can be prescribed and the funnel controller does neither incorporate any internal model for the reference signals nor any identification mechanism, it is simple in its design. The results are illustrated by a simulation of a discretized transmission line.

Fault tolerant funnel control

“…The funnel controller has been successfully applied e.g. in control of industrial servo-systems [3] and voltage and current control of electrical circuits [4]. A certain redundancy of the actuators is necessary in (1.1), thus m is usually much larger than p. A typical situation is that rk B = p, i.e., the number of linearly independent actuators equals the number of outputs of the system; one may think of p groups of actuators, where actuators in the same group perform the same control task.…”

Funnel control in the presence of infinite-dimensional internal dynamics

“…Note that no asymptotic tracking is pursued, but a prescribed tracking performance is guaranteed over the whole time interval. The funnel controller proved to be the appropriate tool for tracking problems in various applications, such as temperature control of chemical reactor models [21], control of industrial servo-systems [12] and underactuated multibody systems [2], speed control of wind turbine systems [10,11], DC-link power flow control [27], voltage and current control of electrical circuits [6], oxygenation control during artificial ventilation therapy [24] and adaptive cruise control [4,5].…”

Funnel control for nonlinear systems with higher relative degree

“…The funnel controller has been successfully applied e.g. in control of industrial servosystems [12] and voltage and current control of electrical circuits [3]. Funnel control is not restricted to systems of ordinary differential equations, but can also be used for infinite-dimensional systems (see e.g.…”