Altitude & Airspeed Hold

$\theta ,\gamma ,\alpha$

Pitch attitude, flight-path angle and angle of attack. Angles are radians internally; α = θ − γ.

$\Delta$

Change from the initial trim state. It does not indicate a different aircraft or an absolute angle.

$\Delta T$

Change in body-axis thrust, N. The first two equations show its contributions to the complete aircraft equations.

${\alpha }_0,m,V_0$

Trim angle of attack, rad; fixed mass 1800 kg; trim speed 60 m/s.

The same wing–tail aircraft is linearised at level-flight trim: V₀ = 60 m/s, h₀ = 1,500 m, mass 1,800 kg. Thrust acts along the body through the CG. Its first-order contributions are ΔV̇ += cos(α₀)ΔT/m and Δγ̇ += sin(α₀)ΔT/(mV₀). Height follows Δḣ = V₀Δγ.

$e_h,e_v$

eₕ is m; eᵥ is m/s. Integrated height error is m s; integrated speed error is m.

$R_h$

Dimensionless altitude-response multiplier. The three outer gains are 0.003 rad/m, 0.005 rad s/m and 0.00005 rad/(m s).

${\mathrm{\Delta \delta }}_{\text{e,c}},q$

Command is rad; q is rad/s. The rate gain is 0.5 s; actual elevator follows a 0.25 s actuator.

eh = Δhc − Δh; ev = ΔVc − ΔV. The altitude controller requests Δθc = Rh(0.003eh − 0.005Δḣ + 0.00005∫eh dt). The inner loop commands Δδe,c = −1.2(Δθc − Δθ) + 0.5q, followed by a 0.25 s actuator.

$R_v$

Dimensionless airspeed-response multiplier; gains are 400 N s/m and 40 N/m.

$\Delta T,{\Delta T}_c,{\tau }_T$

Actual and demanded thrust changes, N; engine time constant, s.

Autothrottle commands ΔTc = Rv(400ev + 40∫ev dt). Actual thrust follows τTΔṪ = ΔTc − ΔT. With autothrottle OFF, ΔTc = 0 and the speed integrator is frozen at zero. Sensors are ideal. Both integrators start at zero.