As an enthusiast of Control Engineering, I would like to share with you a very interesting study that I’ve done of Control Aircraft Pitch Angle, using Lead Compensator on Frequency Domain. For this analysis I used Altair Compose (integrated with Python language) and Altair Activate.

It was discussed and applied some very important concepts of control engineering in a case study, which although simplified, will translate very well how to use Control Engineering in real life! (Something that some of my undergraduate / engineering colleagues question me).

The starting point of this study is understand how the pitch angle is changed on an aircraft. The component which changes it, is the Elevator [1] changing its deflection angle.

To start this analysis is necessary obtain the Transfer Function (which will be used to analyze the Response on Time and Frequency Domain, Stability etc), which will have as input the elevator deflection angle, and output the aircraft pitch angle. For that, the free body diagram of the Aircraft is shown as below [2]:

The scope of this analysis is not valuate each force component and its effects on the aircraft flight, if you have more interesting on that, please read the second reference article. It was assumed the aircraft is in steady-cruise (constant altitude and velocity) providing forces equilibrium, and finally that a change in pitch angle would not change the aircraft speed (unrealistic but simplifies the problem a bit) [2]. Finally the Plant Transfer Function, commonly called G(s), is obtained:

Where: Θ(s) is the Pitch Angle; Δ(s) is the Elevator Deflection (Frequency Domain).

The Frequency Response is the representation of the system’s response to sinusoidal inputs at variable frequencies. The output from a linear system to a sinusoidal input is a sinusoid with the same frequency, but with different magnitude and phase. The Frequency Response is thus defined as the Magnitude and Phase differences between the input and output signals.

Now the system Stability will be analyzed to be controlled. A BIBO (bounded-input bounded-output) stable system is a system for which the outputs will remain bounded for all time, for any finite initial condition and input [3].

One of the most used techniques to analyze the Stability in Time Domain is simply apply a Step input (0.2 rad of amplitude in this analysis) to the initial Open Loop System and analyze some parameters like Overshoot, Rise time, Settling time, Steady-state (described on the table as below).

In addition, already thinking about the Frequency Domain, it is usually analyzed the Poles and Zeros of the Transfer Function and the Bode Diagram, once the bode plot of the open-loop system indicates behavior of the closed-loop system. For a system to be stable, poles must be located on the left semiplane of the plane [4], it is clear once that poles are the denominators of the Transfer Function system.

Some usual parameter values were used (described below) and as output of both described analysis (Time and Frequency Domain) above, it was obtained:

It is clear that the Open Loop system is unstable! Its output grows unbounded with step input and the pole is located at the origin, contradicting the condition of stability mentioned above. Otherwise, the Closed Loop (unitary feedback) is stable once the gain margin is infinite (phase never crosses -180°), doesn’t matter what gain (K) you put.

The most reasonable step is to analyze the time response in a closed-loop system with a Controller Gain equal to 1, and analyze its required parameters (described on the table bellow) once we know that it is stable (by bode plot), and then implement a more robust control. (You can re-check this stability just plotting Root Locus and see that the pole is not on the origin anymore).

As expected, just turning the system into a Closed-Loop would not be enough to achieve the described parameters, although As expected, just turning the system into a Closed-Loop would not be enough to achieve the described parameters, although the system is now stable! For example, settling time of 10 seconds is not even close (now it is around 50s). The simplest way to change this, would be to get a system with faster response, but it will affect overshoot…

The simplest way to decrease settling time and reduce overtshoot is adding a Lead Compensator C(s), that adds positive phase to the system increasing the phase margin and damping, and compensates the excessive phase delay associated with the components of a fixed system.

The following parameters K, α and T need to be calculated! The simplest one is the gain K once it has relation with the steady-state error. You can analyze that we are working with a Type 1 control system, and following a table to step input [4], the steady-state error is 0! By increasing the Gain value K, the system magnitude at all frequencies will be increased, it is a Proportional Effect! However, this increase will decrease the phase margin and consequently, increase the overshoot. Hence the Lead Compensator will be used to add damping to the system, reducing the overshoot, considering that the gain K will be set as 10 on this analysis.

People that already studied Compensators know that its parameters can be obtained by a step-by-step, depending on what is known and what is required. In this problem, the natural way will be: Calculate Damping Ratio (ξ), once Overshoot threshold was given; with it, calculate Phase Margin (ϕm) considering Phase Margin (PM) Compensations (5-15°) and then calculate α. Later will be calculated the Amount of Magnitude Increase (bode plote) that will be supplied by the Lead Compensator at the bump in phase location. Finally T can be calculated centering the maximum bump in phase at the new gain crossover frequency (ωcg). Therefore, it is obtained the Lead Compensator C(s):

Finally, with Lead Compensator Transfer Function it can be obtained the final system results:

It is important to notice that sometimes the parameters calculate steps consist of an iterative process mainly due extra added angle to compensate the phase lag. Also sometimes the minimum phase compensation is divided and specified each item on the requirements.

Please feel free to contact me if you have some note about that study or even interest in Control Engineering! Thank you for your attention!

References:

1. Huang, Zhen & Tang, Chengcheng & Dinavahi, Venkata. (2019). Unified Solver Based Real-Time Multi-Domain Simulation of Aircraft Electro-Mechanical-Actuator. IEEE Transactions on Energy Conversion. PP. 1-1. 10.1109/TEC.2019.2932381.
2. http://ctms.engin.umich.edu/CTMS/index.php?example=AircraftPitch&section=SystemModeling
3. https://academic.csuohio.edu/simond/linearsystems/stability/bibo/
4. Ogata, K. (2010) Modern Control Engineering. 5th Edition, Pearson, Upper Saddle River.