Copter Control - Hardware Synthesis Approaches

The goal of this lab is to design a controller for the plant using various synthesis approaches for the desired Quanser Aero 2 system. The Quanser Aero is an aerospace control system that has two degrees of freedom. Rotation around the lateral axis is called pitch, and rotation around the vertical axis is called yaw. The plant is a DIDO system that takes a specified pitch and yaw voltage as input, and provides the angular position of the pitch and yaw DC motors as output. Performance and control weights are used to specify the objectives for synthesis design. The performance weight specifies target response characteristics and is the inverse of the desired sensitivity function. The control weight influences the magnitude of control input that the controller can specify to the system. Limiting this value is important both when systems can saturate, as is the case with the Aero system in this lab, but also in general because of practical limitations on how much power can be supplied to a plant.

The performance weight is specified to be first order and diagonal with the properties:

● DC disturbance rejection by a factor of 1000

● Sensitivity peak of no more than 3 

Nominal Hardware Evaluation

The response and controller usage from hardware implementation are plotted. The pitch and yaw outputs have excellent tracking when compared to the reference amplitudes of 0.5 rad for pitch and 0.79 rad for yaw. Because we specified a crossover frequency of 5 rad/s in the loop shaping process, the simulated system had no problem tracking the reference signal at 0.02 Hz or 0.13 rad/s.

The control usage for both pitch and yaw spike every 25 seconds. The timing corresponds to when the square wave changes sign. This makes sense because a large control signal is necessary to make the response follow the reference signal when the direction changes. Although our controller outputs control signals of beyond ±1000V, our Simulink model has saturation blocks that keep the actual control usage within ±25V.

The process was repeated for all desired controllers and can be found more in depth in the paper below.

Uncertain Hardware Evaluation

The results of the uncertain hardware implementation of each optimal control synthesis method can be found in Table 1 below. Based on the values of 𝜇 for robust stability, we expected all controllers to stabilize the plant. Indeed, the hardware response was stable for all uncertainty samples, although they were more oscillatory than the simulated response. Although we only ran 5 uncertain samples, robust stability was not violated in the experiment, so it is likely that our controller is robustly stable to all uncertainties. 

Conclusion

In summary, the controllers used in the experiment were expected to stabilize the plant based on the values of μ for robust stability. The hardware response was indeed stable for all uncertainty samples, although it exhibited more oscillation compared to the simulated response. The experiment confirmed that robust stability was not violated, indicating that the controller is likely robustly stable to all uncertainties.

On the other hand, the values of μ for robust performance suggested that none of the controllers would meet the performance specifications. Surprisingly, during hardware tests, the H∞ Loop Shaping controller performed the best among the controllers, providing accurate tracking to the reference signals with the least variance across responses to uncertain samples. This unexpected outcome can be attributed to the fact that the H∞ Loop Shaping controller was not subjected to a control input limit, unlike the H∞ and H2 controllers. The latter controllers relied on moderate control inputs that often fell within the large deadzone of the physical system, resulting in little or no impact on the output. In contrast, the H∞ Loop Shaping controller used larger control inputs, pushing against the saturation limit and avoiding the deadzone.

While all controllers were able to track the pitch reference signal, the H2 and H∞ optimal controllers failed to track the yaw reference signal. These controllers could not generate control signals of sufficient magnitude to track changes in the yaw reference, partially due to the control weight that limits control usage. In contrast, the μ-synthesis controller sacrificed robust stability to achieve robust performance, as indicated by its larger value of μ for robust stability. Additionally, the sinusoidal nature of the yaw reference signal, as opposed to a square wave, may have contributed to the controllers' inability to track it effectively. The gradual change in the yaw reference resulted in smaller error magnitudes and subsequently smaller control responses, which may have encountered the deadzone or simply been insufficient to bring the output signal to the reference peak before the reference changed again.

Report