How can I control the flight path angle (or the ROC) with throttle in Simulink?

Question

I need to implement an autopilot for my 3DoF (longitudinal motion) aircraft model in Simulink. My model is nonlinear, but linearizing it in Simulink hasn’t been an issue. I’m trying to develop a control logic that makes my model track a specific flight path angle $\gamma$ using the throttle $\delta_T$ while adjusting speed $V_{\infty}$ using pitch $\theta$ (elevator deflection, $\delta_e$) along the trajectory. I’m struggling to understand how to connect the flight path angle with the throttle. Should I perhaps use this relationship

$$ ROC = V_{\infty}\sin\gamma = \frac{TV_{\infty}-DV_{\infty}}{W} $$ and linearize it, starting from a trim condition?

Thank you.

Answer 1

Instead of using two separate loops for controlling the elevator and thrust, I would recommend to use the TECS (Total Energy Control System) algorithm which simultaneously controls the elevator and the throttle of the aircraft to manipulate the energy state of the aircraft.

The idea behind this algorithm is to control the energy state of the aircraft. One has to realize that the elevator only changes the energy distribution between the kinetic energy (speed) and potential energy (height), while only the throttle introduces or deletes energy! With this in mind, one can easily calculate the energy required for a given airspeed and climb command, calculate the energy distribution between potential and kinetic energy, and then control the elevator and throttle to realize the demanded energy state and distribution.

The energy of an aircraft can be easily described by

$E = m \cdot g \cdot h + \frac{1}{2} \cdot m \cdot v^2$

Dividing by the weight of the aircraft $m\cdot g$ and differentiating once, yields the energy rate of the aircraft per weight:

$\dot{E} = \dot{h} + V\cdot \dot{V}/ g$

And further dividing by $V$ yields the specific Energy rate

$\dot{E}_s = \gamma + \dot{V}/ g$

Note that $\dot{h}/V\approx\gamma$ if the small angle approximation is made.

The classical TECS algorithm is displayed below. The TECS algorithm calculates an specific Energy error (how much energy is lacking) $\dot{E}_{s,c} - \dot{E}_s$ with $\dot{E}_{s,c}$ indicating the commanded specific Energy rate to achieve the commanded descent/climb rate and target acceleration ($\dot{V}$). To do this, a classical PI Controller is used (these are the KTI/s and KTP blocks in the diagram). The result is then given to the throttle command to change the throttle such that the desired energy rate is achieved.

Similarly for the elevator the energy distribution term $L = \gamma - \frac{\dot{V}}{g}$ is formed. Analogously to the throttle command, the pitch command is formed by first calculating a commanded energy distribution $L_c$, forming the energy distribution error $L_e = L_c - L$ and using a PI controller to achieve the desired value.

It should be noted that another control loop for controlling pitch attitude and dampening short period motion should be introduced in the "airplane dependent design" side of the algorithm

This established algorithm was first developed by Lambregts and has since been applied quite successfully in a number of applications. For example by NASA on their Boeing 737 test aircraft or the popular UAV open source control software ArduPilot and Pixhawk PX4.

Answer 2

can I still get some physics advice?

With aircraft it is far easier to control pitch to the horizon (therefor the aircraft flight path descending, level, or climbing) with throttle control.

The elevator or stabiliator trim is used to set a given airspeed. This is why we are taught "pitch controls speed, power controls altitude".

In order for this all to work, the aircraft must be staticly stable. It must have the property of pitching up when it goes faster than its trim speed, and pitching down if it goes slower than it's trim speed.

Once this aerodynamic requirement is satisfied, it is only a matter of power setting to determine angle to the horizon. Using throttle to do this allows for "fine tuning" of flight path because engine RPM can be adjusted to a high degree of accuracy, instead of struggling to adjust drag with only 3 flap settings. Power on landings are therefor much easier$^1$.

What should become apparent is that gravity will play a role in aircraft power requirement depending on angle to the horizon, when airspeed and trim setting are constant.

Angle to the horizon is determined by power setting. If power is insufficient (or none) for level fight, flight path drops below the horizon and gravity assists by the formula W × sin (angle below the horizon).

In a climb, it is opposite, now gravity resists forward airspeed by the same formula.

The amount of Net Thrust to fly the airplane at a given airspeed is always "Gravity thrust" (positive, 0, or negative depending on angle to the horizon) + Propulsive thrust (0 to maximum available thrust).

Gliding at a given airspeed with no power provides the quantity of Net Thrust required for linear steady state flight as W × sin (Glide Angle).

$^1$ except, with a single engine, if the engine is lost you don't make the runway

Answer 3

Commercial solutions to this problem are often pretty complicated but the core is similar to what you're suggesting with thrust target = sin(flight path angle target)*W - predicted drag.

Some catches:

• Some ability to account for modelling errors like incorrect weight or drag is essential for speed control. For example the integral part of a PI controller provides this functionality
• since engines can have a slow response time, you may want to have an outer loop for thrust or EPR targets vs fpa, and an inner loop for thrust vs throttle
• flight path acceleration can be added to the equation so you don't overestimate the extra thrust needed while you're already speeding up