Hybridní systémy

Temporal regularization of the water tank problem to avoid Zeno behaviour.

• Modify the provided code for the water tank problem to avoid Zeno behaviour by introducing some time delay into the system. You can do it by introducing a new state variable that will play the role of a timer. Using this timer you can enforce some minimum time that the system must stay in the current discrete state before transitioning to the other. 
• Alternatively, you can introduce another discrete state that will model the transition of the hose. While in this mode (that is, during the transition), there is no inflow to either tank. 

Plotting the state portrait and some estimate of the region of attraction (ROA) for a second-order nonlinear system

• This is ultimately motivated by the problem of swinging up and stabilizing a pendulum on a cart.
• Use the quiver function in Matlab to plot the state portrait of the system \(f(x) = \begin{bmatrix}-x_2\\ x_1 + (x_1^2-1)x_2\end{bmatrix}\). Set the ranges for the two axes [-3,3].
• Using the plot, try to delineate the region of attraction just visually (pointing a finger).
• Should this model represent a feedback controlled nonlinear system, the motivation for characterizing the ROA is that we needed to know when to switch from a global controller to the local one (the latter perhaps being just an LQR).
• In this particular example, a convenient way to draw the boundary of the ROA is to solve the system equations backwards in time, starting somewhere close to the origin. Solving an ODE \(\dot x = f(x)\) backwards amounts to changing the sign on the RHS, that is, solving \(\dot x = -f(x)\).
• We get the boundary or the ROA, but for practical purposes, it must be characterised in a computationally tractable way. How about inscribing a circle?
• How about inscribing an ellipse? Generate an ellipse as a sublevel set for \(V(x) = x^T P x\) for \(P = \begin{bmatrix}3/2 & -1/2\\ -1/2 & 1\end{bmatrix}\), that is, a solution set for \(V(x)\leq c\), where c is set to a few values, say \(c = 0.25, 1, 4\). It seems that for this particular matrix P by choosing an appropriate c we can cover a large part (albeit not the whole) of the ROA.  

Implementation of Selector control (aka Override control) as a preparation for later analysis of stability

• Implement the following feedback control scheme. The y variable is the primary one. We design a feedback controller C with which a reference value (aka set point) for y is tracked. 
• At the same time, the secondary output z should not leave the interval [zmin, zmax] (in fact, these boundary values must be set a bit more conservative than their true bounds because with the considered control scheme we are going to keep the z variable "around" the zmin and zmax values).

Figure 4.8 from [1]

• Consider the plant modelled by \(P_1 = \frac{40}{0.05s^2 + 2s^2 + 22s + 40}\) and \(P_2=\frac{5}{s^2 + 7s + 5}\). 
  
• Design a first- or second-order linear controller C as a lead-lag controller (avoid having a pure integrator in the controller this time, so that we do not have to handle the windup issues, just for simplicity of exposition) so that the rise time is about 1s and the overshoot not larger than 20%.
• Restrict yourself just to proportional controllers Cmin and Cmax.
• Set zmin and zmax to -1.2 and 1.2, respectively.