Hybridní systémy

Nonlinear harmonic oscilaltor

Consider a nonlinear oscillator modelled by \(\ddot x = -\text{sign} x\). Draw its state portrait (you may find the quiver function in Matlab useful) and a state trajectory for some nonzero initial condition. Argue if the oscillator admits a classical or extended solution.

Box sliding on a tilted plane subject to dry friction

Consider a box sliding on a tilted plane. Besides the projection of the gravitational force to the direction of motion, the only other force that affects the motion along the plane is the dry friction. Recall that dry friction force depends only on the sign of the velocity, the normal force (here given by the projection of the gravitational force into the normal direction) and some material/friction coefficient \(\mu\). 

Consider the velocity the only state variable (we are not interested in the position). Write down the state equation (indeed, it will be just a single equation). Plot the state portrait. For a given coefficient \(\mu\) it would be just one-dimensional, but consider a range of values of the coefficient \(\mu\), say \(\mu\in [0,2]\) and produce a 2D plot using the quiver function similarly as in the previous example. Using this plot, determine the value of the material coefficient \(\mu\) for which, for the given initial velocity, the state portrait would predict the box coming to a stop (on the chosen time interval). Solve the corresponding ODE, either in Simulink or directly in Matlab. What is you conclusion about the type of the solution? Does the problem admit a classical solution? The extended (Carathéodory)? How about Filipppov? Can you find the coefficients of the convex combination that would describe the motion (actually its absence) after the velocity reduces to zero?

Swingup of a pendulum on a cart

Consider a pendulum on a cart controlled directly through acceleration of the cart. Design a hybrid controller for swinging the pendulum up. The hybrid controller will contain two sub-controllers: a global one and a local one. The former will be based on energy shaping, the latter will be just an LQR controller. For the former, a Simulink implementation is made available to you. The latter you must design and implement on your own, as well as the logic for switching between the two controllers. Do not forget to implement some hysteresis in the controller.