Modeling for control of an inflatable space reflector, the linear 1-D case

In this paper we develop a mathematical model of the dynamics for an inflatable space reflector, which can be used to design a controller for the shape of the inflatable structure. Inflatable structures have very nice properties, suitable for aerospace applications. We can construct e.g. a huge light weight reflector for a satellite which consumes very little space in the rocket because it can be inflated when the satellite is in the orbit. So with this technology we can build inflatable reflectors which are about 100 times bigger than solid ones. But to be useful for telescopes we have to actively control the surface of the inflatable to achieve the desired surface accuracy. The starting point of the control design is modeling for control, in our case port-Hamiltonian (pH) modeling. We will show how to derive an infinite and also finite dimensional port-Hamiltonian model of a 1-D Euler-Bernoulli beam with piezo actuation. In the future we will also focus on 2-D models.