The planar ducted fan is a fun nonlinear control problem whose plant is a non-minimum phase system. Non-minimum phase systems have a right half plane zero, which impacts the achievable closed-loop response of the system. The reason that closed-loop response has constraints is that non-minimum phase systems tend to go in the wrong direction initially before going in the right direction (to go left, the system will initially go right then come around). Many other real-world systems exhibit this behavior.

As a system, the planar ducted fan is an example of a flying machine employing vectored thrust. A fan or other thrust generator sends air out of an orifice past to flaps. These flaps, moving identically, redirect the air. Defining $q = (x, y)^T$ to be the center of mass of the ducted fan, and $\theta$ to be the orientation of the ducted fan, \begin{equation} \begin{split} m \ddot q & = -d \dot q + R(\theta) f - m \vec g \\ J \ddot \theta & = r\, \pi_1(f) \end{split} \end{equation} where the force vector $f$ is in the body frame of the ducted fan, generated from the fan's thrust. Adjustable flaps control the decomposition of the thrust vector into its two components as per \begin{equation} f = R(\psi) \begin{bmatrix} 0 \\ \tau \end{bmatrix} \end{equation} where $R$ is a planar rotation matrix \begin{equation} R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} \end{equation} The mapping $\pi_2:\mathbb{R}^2 \rightarrow \mathbb{R}$ is the projection of the 2-vector $f$ onto its second coordinate value. The control inputs are the thrust $\tau$ and the flap angle $\psi$. From these equations, it is clear that the system is an interesting nonlinear problem (as even the control inputs enter nonlinearly).

Another formulation of the equations of motion is \begin{equation} \begin{split} m \ddot q & = -d \dot q + R(\theta) R(\psi) \begin{bmatrix} 0 \\ \tau \end{bmatrix} - m \vec g \\ J \ddot \theta & = r\, \pi_1 ( e_1(\psi) \tau ) = r \sin(\psi) \tau \end{split} \end{equation}

These activities sketch what should be done, but do not necessarily indicate what should be turned in. By now you should have seen enough solution postings and possibly also read enough papers on control that you should have an understanding of what should be turned in. This would include the mathematics or derivations, the synthesized controller, and sufficient plots to demonstrate that the task objective was met. Discussion of outcomes should be included.

Linearize the equations of motion about hover at $\theta = 0$ (radians), so that the linearized state and control inputs have an equiblirium at the origin with zero linearized control input. Establish performance specifications for the system and design a linear feedback controller that will stabilize the system and meet the performance specifications.

Modify by 10-20% some of the model parameters of the system and compare the outcomes under a traditional linear controller. Incorporate linear adaptive control and show the resulting outcomes. Confirm how well the adaptive system meets the performance specifications versus the static controller.

Here, you should consider two cases. One is the initial case, where tracking a particular reference signal will lead to an adaptation transient. Simulate as normal, however pick a time post-transient and grab the adaptive gains from the output signals. Prepare a second simulation that starts with these gains. The second simulation would act like a second or subsequent deployment post-adaptation. Show that the system better meets the performance objectives.

Rearrange the equations to see if they can be put into a form that looks more like a linear system plus a non-linear defect. Design a controller with nonlinear cancelling terms. Can they all be cancelled or are there constraints on what can be done? How were those constraints, if any, managed by you, the designer? Repeat the same procedure with parameter mismatch and demonstrate improved performance relative to the static nonlinear controller design.

There are some references below whose equations might differ from the ones above. There are a few models for this fan. The model chosen depends on what the authors wish to demonstrate.

• Yu, Jadbabaie, Primbs, and Huang. “Comparison of Nonlinear Control Design Techniques on a Model of the Caltech Ducted Fan.” Automatica, 37:1971-1978, 2001. Article

Back – Main