This paper considers a flexible cantilever beam, which is actuated by piezoelectric macro-fiber composite (MFC) patch actuators. For accurate positioning tasks, special attention has to be paid to the inherent nonlinear hysteresis and creep behavior of these actuators. A detailed analysis of the MFC-actuated cantilever verifies that these nonlinearities can be efficiently captured by an operator-based model using Prandtl-Ishlinskii's theory. Based on a Hammerstein-like model with the nonlinearities at the input connected in series with a linear infinite-dimensional beam model it follows that hysteresis and creep effects can be compensated by application of the inverse operator. Experimental results prove the feasibility of this approach. With this result, the tracking accuracy of the combination of the compensator with the flatness-based feedforward control design as proposed in the companion paper (Schröck et al 2011 Smart Mater. Struct. 20 015015) can be verified. Measurements demonstrate the applicability of this approach for the realization of highly dynamic trajectories for the beam's tip deflection.

This paper considers a systematic approach for motion planning and feedforward control design for a flexible cantilever actuated by piezoelectric macro-fiber composite (MFC) patches. For accurate feedforward tracking control, special attention has to be paid to the inherent nonlinear hysteresis and creep behavior of these actuators. In order to account for these effects an appropriate compensator is applied which allows to perform the tracking controller design on the basis of a linear infinitedimensional model. A detailed analysis of the nonlinear actuator behavior as well as the compensator design and the overall experimental validation is presented in the companion paper [1]. The governing equations of motion of the hysteresis and creep compensated cantilever are determined by means of the extended Hamilton's principle. This allows to consider the influence of the bonded patch actuators on the mechanical properties of the underlying beam structure in a straightforward manner and results in a model with spatially varying system parameters. For the solution of the motion planning and feedforward control problem a flatness-based methodology is proposed. In a first step, the infinite-dimensional system of the MFC-actuated flexible cantilever is approximated by a finite-dimensional model, where all system variables, i.e. the states, input, and output, can be parametrized in terms of a so-called flat output. In a second step, it is shown by numerical simulations that these parametrizations converge with increasing system order of the finite-dimensional model such that the feedforward control input can be directly calculated in order to realize prescribed output trajectories.

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The motion planning problem is considered for a Euler-Bernoulli beam with viscous damping. For its solution, a systematic spectral approach is proposed, which is based on the Riesz spectral properties of the system operator. This enables to analyze both boundary and in-domain control in a common framework.

Die Trajektorienplanungsaufgabe fü r verteilt-parametrische Systeme wird anhand des Beispiels einer einseitig eingespannten orthotropen Kirchhoff-Platte mit ö rtlich verteilten Patch-Aktoren analysiert. Hierzu wird auf Basis der spektralen Darstellung der Bewegungsgleichung ein systematischer Ansatz zur flachheitsbasierten Parametrierung des Auslenkungsprofils und der Stellgrö ßen vorgestellt. Dies ermö glicht einen intuitiven Zugang zur Trajektorienplanung und zur Realisierung hochdynamischer Ü bergä nge zwischen stationä ren Auslenkungsprofilen. Die Verwendung gewichteter Residuenverfahren ermö glicht zudem eine effiziente Computer-gestü tzte Umsetzung des Entwurfsverfahrens. Simulationsergebnisse bestä tigen die Anwendbarkeit der entwickelten Methodik und illustrieren das erzielbare Folgeverhalten. Schlü sselwö rter: verteilt-parametrisches System; Trajektorienplanung; Kirchhoff-Platte; differenzielle Flachheit; Steuerungsentwurf; gewichtete Residuenverfahren; adaptive Flü gelstruktur Motion planning for a piezo-actuated elastic Kirchhoff plate.The motion planning problem is considered for a cantilevered orthotropic Kirchhoff plate with spatially varying coefficients and distributed piezoelectric patch actuators. For this, the spectral representation of the corresponding equations of motion is utilized to systematically construct a flatness-based parametrization of state and inputs. These enable a very intuitive motion planning to realize prescribed high-speed rest-to-rest motions. Moreover, the incorporation of weighted residuals approaches yields a very efficient computational implementation. Simulation results confirm the applicability of the design approach and the achievable tracking performance.