Projects

Model-based control requires the model of the system to be controlled. In general, a model is an approximation of a system and therefore contains errors. For systems described by Partial Differential Equations (PDEs) with actuators and sensors located at the spatial boundaries of the domain, one can design controllers based on a discretized model of the system. The discretized model is described by Ordinary Differential Equations (ODEs) and the controller can be designed more easily than for PDEs. However, since the discretized model is an approximation of the system described by PDEs, the closed-loop stability is not always guaranteed when using classic control tools of Linear Time-Invariant (LTI) systems. We exemplify this in the following example. We consider the vibrating string with force actuators and velocity sensors at both sides of the string. The design is based on a discretized motel that contains 59 state variables (open loop eigenvalues in yellow). We design the state feedback (eigenvalues in blue) and the Luenberger observer (eigenvalues in orange). Then, we analyze the closed-loop eigenvalues (in violet) when applying the designed observer-based state feedback (OBSF) controller to a discretized model that is more precise than the one used for the design (the idea is to approach the original system described by PDEs). Some eigenvalues that were not considered during the design are destabilized in the closed-loop. In a numerical simulation, for a short time interval, the system seems stable. However, when time goes to infinity, the system blows up.

In this thesis, we propose two methodologies for the design of the state feedback and the Luenberger observer such that we can guarantee closed-loop stability between the OBSF controller (described by ODEs) and the system described by PDEs. In the following figures, we show that the closed-loop eigenvalues (in violet) remain at the left side of the imaginary axis. In the simulation, we show that the observed-displacement approaches the real displacement while both converge to zero.

In this experiment, we control the turbidity in a sedimentation column. To this end, we model the turbidity dynamic of a fixed point of the sedimentation column. We consider a nonlinear model that contains two different dynamics: the one given by the sedimentation phenomena and the one given by the actuator. We consider the inflow as an actuator and the outflow as a perturbation (and vice versa). The inflow is a mixture of water and calcium with an unknown calcium concentration. Similarly, the outflow is a mixture of water and calcium with a higher calcium concentration (because of sedimentation phenomena). The outflow calcium concentration is also not known. The only measurement is the turbidity (related to the calcium concentration) of a fixed point of the column. The sensor is located between the inflow and the top of the column.

We show the closed-loop stability between the nonlinear model and a Proportional Integral (PI) controller. In this process, one can keep a desired set point of turbidity while the water is recovery by an overflow of the sedimentation column. In the figures, we show three different levels of turbidity. The figure at the left shows a low level of turbidity, in which we are even able to see the inlet pipe. In the other two cases, we can obtain a bigger overflow (more water recovered), but with a higher level of turbidity. I let you attached the slides related to this project.

In this project, we implemented a control system composed of a Programmable Logic Controller (PLC) as shown in the figure at the left. This controller is able to command around 60 valves along with the center pivot. The controller can be configured with an irrigation matrix that allows fixing a desired percentage of irrigation depending on the position of the center pivot. The position is measured by a Global Positioning System (GPS) connected to a device that allows communicating to the controller system remotely (middle figure). Finally, a Human Interface Machine (HMI) was implemented in a device located locally at the center pivot. The HMI allows to configure the irrigation matrix manually and visualize the operation variables.