🛩️  An MDO Framework for Relaxed Static Stability Constraint in Tail Sizing of High Performance Airplanes.

Overview

 Historically, airplane designers have relied on conservative static stability margins, leading to oversized tail areas that increase trim drag, despite modern flight control systems capable of mitigating stability issues. By relaxing these static stability constraints, it becomes possible to reduce tail size, decrease drag, and enhance maneuverability. Our study explores the potential benefits of relaxed static stability (RSS) in high-performance aircraft, supported by case studies of advanced designs like the X-29 and F-117, which achieved superior performance with significantly unstable static margins. A focus is placed on the interplay between horizontal tail design, center of gravity (CG) placement, and control feedback systems, addressing critical balance, stability, and flying quality requirements. The study seeks to quantify the extent to which feedback control can compensate for reduced static stability, enabling optimized designs with smaller tail areas without compromising performance.

1. 🎯Project Goal

The primary goal of this project is to quantify the extent to which feedback control can compensate for reduced static stability in high-performance aircraft. By doing so, we aim to enable optimized designs with smaller horizontal tail areas without compromising overall performance or flying qualities. This involves developing a framework to determine the maximum allowable instability an airframe can possess while still being effectively stabilized by a feedback controller, considering critical balance, trim, and control limitations.

2. Background 

The traditional airplane conceptual design process often prioritizes wing optimization, with tail design typically following based on assumed tail volume ratios. Flight dynamists then size the empennage and control surfaces, adjusting the center of gravity (CG) margin to meet static stability requirements (e.g., 5-10% static margin). For aircraft with flight control technology, control engineers design feedback laws late in the process to meet flying qualities.

This conventional approach is inefficient for modern, high-performance aircraft equipped with advanced flight control systems. The typical static stability requirement is often unnecessarily stringent, leading to larger tail areas and increased trim drag. Since feedback controllers can rectify stability characteristics, relaxing this constraint allows for smaller tails, reduced drag, and improved maneuverability. It's well-established that relaxed static stability (i.e., a more aft CG) positively impacts trim drag and maneuverability.

Numerous studies highlight the advantages of RSS in modern high-performance aircraft. For instance, Kehrer reported that stability augmentation in the Boeing 2707-300 Supersonic Transport (SST) led to a 150-inch fuselage reduction, decreasing vertical tail size and gear length, saving 6,000 lbs, and increasing range by 225 nautical miles. However, this required a more complex flight control system. Swortzel and Barfield observed increased sustained turn rates and load factors for the YF-16 as its airframe became more unstable. The YB-49 was flight-demonstrated with a 10% unstable static margin. The X-29, designed for performance benefits from its forward-swept-wing-canard configuration, required a 35% unstable static margin. The F-117 and B-2 are also examples of unstable aircraft.

The horizontal tail serves three primary functions: (i) Balance/Trim, (ii) Stability, and (iii) Control. Its sizing, along with CG adjustment, must satisfy several requirements:

1. Sufficient pitching-up moment at rotation speed (take-off).

2. Sufficient pitching-up moment for trim during landing.

3. Sufficient pitching-down moment for trim at Flap-Extended-Design Speed (tail stall at V_FE = 1.8Vs).

4. Stability and flying qualities requirements.

The first three requirements (plus tip-back) define the forward CG limit, while the last (plus nose gear tip-up) defines the aft CG limit. These are typically visualized in a "scissors plot," as shown in Figure 1. Other constraints include pitch recovery from high angles of attack (sufficient pitching-down moment to recover while counteracting nose-up pitching moment from roll-inertial cross-coupling).

Figure 1 is used to size horizontal tail areas (and adjust wing location) for a given CG range. Designers reduce the horizontal tail area until the specified CG range violates a constraint. This minimum area can be further reduced by relaxing constraints. While feedback control cannot handle balance/trim constraints, it can significantly relax the stability constraint. If the static stability constraint is relaxed (shifted right in Figure 1), a smaller horizontal tail area can accommodate the CG margin. The core question of this report is: how much can feedback control relax an open-loop stability constraint? In other words, how unstable can an airframe be while the feedback controller still stabilizes it and ensures good flying qualities?

                     Figure 1: Typical scissors plot for sizing horizontal tails (for B 737-800). SM stands for static margin. 

3. Mathematical Model

   3.1 Aerodynamic Model

The aerodynamics group at a major aerospace organization performs RANS CFD simulations for the entire aircraft. Their setup is favorable because: (i) it provides contributions from individual components (wing, tail, fuselage), and (ii) all contributions are normalized by the same reference values and summed at the quarter chord of the wing MAC (mean aerodynamic chord), not the CG. This allows studying the effect of varying the CG.

For this study, the following derivatives are obtained from simulations:

C_L_alpha, C_M_alpha, C_L_q, C_M_q, C_L_delta, C_M_delta

The first four derivatives are decomposed into wing, tail, and fuselage contributions. For example, C_L_alpha is:

C_L_alpha = C_L_alpha_w + C_L_alpha_t + C_L_alpha_f

Since simulations are at a particular horizontal tail area S_H_ref, and this study requires analysis at different S_H values, all tail contributions are normalized by area:

C_t = C_t_CFD * (S / S_H)_ref * (S_H / S)

where C_t is any tail coefficient (e.g., C_L_alpha_t), and C_t_CFD is the CFD value (normalized by wing area and calculated at S_H_ref). We define C_t_eff = C_t_CFD * (S / S_H)_ref. So, C_L_alpha_t for any S_H/S is:

C_L_alpha_t = C_L_alpha_t_eff * (S_H / S)....................................................................................................          (1)

where C_L_alpha_t_eff = C_L_alpha_t_CFD * (S / S_H)_ref.

Furthermore, since CFD provides moment at c_bar/4 (quarter chord of wing MAC), the moment at a CG location x_cg is:

C_M_cg = C_M_c_bar/4 + C_L * (x_cg / c_bar - 0.25).....................................................................................          (2)

Based on this, the following equations compute the required derivatives:

C_L_alpha = C_L_alpha_w + C_L_alpha_t_eff * (S_H / S) + C_L_alpha_f 

C_L_q = C_L_q_w + C_L_q_t_eff * (S_H / S) + C_L_q_f 

C_L_delta = C_L_delta_eff * (S_H / S)..............................................................................................................          (3)

C_M_alpha_c_bar/4 = C_M_alpha_w + C_M_alpha_t_eff * (S_H / S) + C_M_alpha_f ..........................

 C_M_q_c_bar/4 = C_M_q_w + C_M_q_t_eff * (S_H / S) + C_M_q_f ..........................................................

C_M_delta_c_bar/4 = C_M_delta_eff * (S_H / S) ...............................................................................................          (4)

C_L_delta and C_M_delta are assumed to be solely from the horizontal tail. Q-derivatives can also be assumed to be solely from the horizontal tail. Equation (2) then transfers these moment coefficients to a given CG location.

Summary: 14 derivatives are provided from CFD simulations at S_H_ref: C_L_alpha_w, C_L_alpha_t, C_L_alpha_f; C_L_q_w, C_L_q_t, C_L_q_f; C_L_delta; C_M_alpha_w, C_M_alpha_t, C_M_alpha_f; C_M_q_w, C_M_q_t, C_M_q_f; C_M_delta;

Then, for given S_H/S and x_cg/c_bar, Equations (2-4) yield the five coefficients: C_L_alpha, C_M_alpha_cg, C_M_q_cg, C_L_delta, C_M_delta_cg, which are used for flight dynamics and handling qualities analysis.

3.2 Flight Dynamics Model

The ultimate objective is to meet flying qualities requirements, specifically for the short-period mode in horizontal tail sizing. The short-period approximation is accurate for short-period dynamic characteristics (damping ratio zeta_sp and natural frequency omega_sp). It is written as:

[alpha_dot(t); q_dot(t)] = [-L_alpha/u_0 1; M_alpha M_q] * [alpha(t); q(t)] + [-L_delta/u_0; M_delta] * delta(t)      (5)

where u_0 is forward speed, and the state vector x includes angle of attack (alpha) and pitch rate (q). System parameters (stability derivatives) are defined using stability coefficients:

L_alpha = (1/m) * (0.5 * rho * u_0^2 * S * C_L_alpha)

L_delta = (1/m) * (0.5 * rho * u_0^2 * S * C_L_delta) 

M_alpha = (1/I_y) * (0.5 * rho * u_0^2 * S * c_bar * C_M_alpha_cg)

M_delta = (1/I_y) * (0.5 * rho * u_0^2 * S * c_bar * C_M_delta_cg) 

M_q = (1/I_y) * (0.25 * rho * u_0 * S * c_bar^2 * C_M_q_cg)

where the definition of C_M_q = (2 * u_0 / c_bar) * (d C_M / d q) is used. The five coefficients (C_L_alpha, C_M_alpha_cg, C_M_q_cg, C_L_delta, C_M_delta_cg) from the aerodynamic model determine the five derivatives (L_alpha, L_delta, M_alpha, M_delta, M_q), which populate the matrices in the short-period approximation (5).

The short-period approximation can be written as:

x_dot(t) = [A(x_cg, S_H)] x(t) + [B(x_cg, S_H)] delta(t)

where matrices A and B are functions of horizontal tail area S_H and CG location x_cg.

3.3 Flight Control

The control group at a major aerospace organization uses an effective control strategy with four gains: pitch rate feedback, normal acceleration (az) feedback, integrator gain, and feedforward gain, plus a 3rd-order horizontal tail actuator model. For preliminary sizing, we use the simple short-period approximation (5) with a standard state-feedback control law:

delta = K x = [k_alpha k_q] * [alpha; q] .................................................................................................................             (6)

Including actuator dynamics with this simple strategy might lead to an unstable closed-loop system, but other gains (acceleration feedback, integrator, feedforward) can be tuned later. We use a simple model and controller for preliminary sizing, leaving advanced controller adjustments for later design stages.

Using control law (6) with system (5), the closed-loop system is:

x_dot(t) = [A_cl(x_cg, S_H, k_alpha, k_q)] x(t) + [B(x_cg, S_H)] delta(t)

where the closed-loop system matrix A_cl = A + B K determines short-period characteristics. Specifically, zeta_sp, omega_sp, and CAP = omega_sp^2 / (g L_alpha) can be obtained, which are functions of S_H, x_cg, k_alpha, and k_q. Once these are fixed, the closed-loop response x(t) to a disturbance x_0 is:

x(t) = exp(A_cl * t) * x_0

where exp(A_cl * t) is the matrix exponential. The required deflection delta and deflection rate delta_dot for recovery from x_0 are:

delta(t) = K x(t)

 delta_dot(t) = K A_cl x(t)

4. Flying Qualities Requirements

While the control group at a major aerospace organization detailed flying qualities requirements, we consider only the following for preliminary sizing from MIL-F-8785C for short-period longitudinal dynamics:

0.35 <= zeta_sp <= 1.3 0.28 <= CAP <= 3.6

When the simple state-feedback controller (6) is optimized for these requirements on model (5), many other requirements are naturally satisfied. For instance, minimum 6 dB gain margin and 45 degree phase margin are always met. Overshoot requirements (less than 5% for normal acceleration and 50% for pitch rate responses to a 1g step command) are also satisfied, as the optimizer yields an overdamped system (zeta_sp > 1) with no overshoot. We removed the maximum CAP constraint as it had an adverse effect. The main ignored consideration is the actuator model, leaving freedom for the control group later.

It's important to note that DEF-STAN 00-970 and MIL-F-8785C may not yield completely satisfactory responses for heavily stability-augmented airplanes. Some advanced aircraft designed to these standards have received negative pilot opinions, prompting research since the 1970s for highly augmented aircraft. The Hoh et al. proposal in 1982 evolved into MIL-STD-1797A, but it's not publicly available. Even MIL-STD-1797A is not a "cookbook"; its requirements are often left blank for the procuring agency to fill. Thus, publicly available knowledge on flying qualities may not fully satisfy highly augmented aircraft, and criteria will likely be revised during design and testing. This is another reason to simplify requirements in preliminary sizing.

5. Optimization Problem Formulation

The objective is to determine how unstable an airframe can be while a feedback controller (like (6)) ensures closed-loop flying qualities and handles severe gusts without exceeding control surface limitations (delta_max and delta_max_dot). This requires optimizing controller gains to find the "best" controller for the most unstable airplane. We seek an alternative to the right boundary in the scissors plot (Figure 1): the most relaxed boundary, relying on the best controller in the assumed form (6).

The solution is similar to Kaminer et al., but we move away from their LMI formulation for greater flexibility in allowing non-LMI constraints. Their LMI approach also doesn't allow general optimization (only linear costs), requiring an outer search loop for maximum CG. Our nonlinear optimization approach allows arbitrary nonlinear (smooth) constraints and objective functions simultaneously. For a given horizontal tail area ratio S_H/S, the following optimization problem is solved:

maximize (chi) x_cg / c_bar

subject to: (i) 0.3 <= zeta_sp <= 1.3 (ii) 0.28 <= CAP (iii) |delta(t)| <= delta_max = 25 degrees and |delta_dot(t)| <= delta_max_dot = 60 degrees/s for the closed-loop response simulation due to a severe vertical gust of 66 ft/sec (i.e., x_0 = [66(ft/sec)/u_0; 0]), where the vector chi of design variables includes the CG location and feedback gains: chi = [x_cg/c_bar; k_alpha; k_q].

The Matlab program solves this optimization for a given S_H/S. It uses "fmincon" (nonlinear constrained optimization). Two user-defined Matlab functions are needed: "xcg_max" (takes chi, returns objective function to minimize (-x_cg/c_bar)) and "Tail_Sizing_Constraint_CFD" (takes chi, returns constraints to be less than zero, e.g., 0.35 - zeta_sp). The constraint function calls "Parameters_CFD" (flight condition, geometry, 14 CFD coefficients). The "interior-point" algorithm is used, and six random initial guesses check for local minima. The main program "CG Maximization CFD" yields the maximum CG and corresponding gains, showing closed-loop response to gust and g-step command, along with required deflection and rate.

6. Results

Solving the optimization problem using representative geometrical parameters and aerodynamic data at M = 0.8 and sea level for

 S_H/S = 0.35, the optimizer yielded the following optimum solution consistently across six random guesses:

x_cg / c_bar = 0.91 

k_alpha = 4.52 

k_q = 0.32

This results in:

h_n = -0.61 

T_2 = 0.06s 

zeta_sp = 1.3

 omega_sp = 6.5 rad/s

 T_theta_2 = 0.23s

Figure 2 shows satisfactory closed-loop response. Horizontal tail deflection is below 25 degrees, but initial deflection rate hits 60 degrees/s. Thus, deflection rate and maximum damping ratio constraints drive the optimization. This satisfactory response is achieved at an open-loop time-to-double T_2 = 0.06s, significantly lower than the AGARD report's limit, allowing a more unstable airframe without sacrificing flying qualities.

To construct an alternative for the right (stability) boundary in the scissors diagram (Figure 1), we solve the optimization problem at different S_H/S values, each yielding a maximum CG location. We tested S_H/S from 0.05 to 0.5.

For S_H/S = 0.05, the optimizer yielded:

x_cg / c_bar = 0.44

 k_alpha = 6.25 

k_q = 0.69

This results in:

h_n = -0.21 ,T_2 = 0.11s, zeta_sp = 1.3 ,omega_sp = 3.52 rad/s, T_theta_2 = 0.37s

Figure 3 shows satisfactory response despite the small area. Initial deflection hits 25 degrees, while initial deflection rate is below 60 degrees/s. Maximum deflection and damping ratio constraints are active here.

For S_H/S = 0.5, the optimizer yielded:

x_cg / c_bar = 1.02 

k_alpha = 4.18 

k_q = 0.3

This results in:

h_n = -0.69, T_2 = 0.06s, zeta_sp = 1.3, omega_sp = 7.59 rad/s, T_theta_2 = 0.2s

Figure 4 also shows satisfactory response. Similar to the nominal area case, initial deflection rate hits 60 degrees/s, while deflection remains below 25 degrees. Maximum deflection rate and damping ratio constraints are active.

In conclusion, closed-loop response is satisfactory for S_H/S values from 0.05 to 0.5. The maximum damping ratio constraint is always active. For low S_H/S, maximum deflection is active; for larger S_H/S, maximum deflection rate is active. Figures 2-4 show that larger horizontal tail areas lead to larger steady-state load factors (due to increased tail control power) and allow more instability (lower h_n and T_2).

Plotting the maximum CG location from the optimization problem at different S_H/S values constructs the modified (relaxed) right boundary in the scissors plot, as shown in Figure 5. The Matlab program "Horizontal Tail Sizing CFD" performs this task. Figure 5 also shows other stability boundaries for the aircraft at M = 0.8, sea level: 5% and -5% static margin, C_M_alpha = 0.4/rad, and the current design point at (x_cg/c_bar, S_H/S) = (0.336, 0.35). The optimized boundary significantly differs from others. At the current S_H/S = 0.35, it extends the most backward CG limit by over 200%. Alternatively, adhering to the current backward CG limit of 0.336 c_bar, it offers over 90% reduction in horizontal tail area.

 These results indicate that the static stability constraint should not be a primary concern; other requirements (balance/trim and control) will be decisive in horizontal tail sizing.

The black solid boundary in Figure 5 answers the study's core question: how unstable can an airframe be while the feedback controller maintains required flying qualities and satisfies other limits (e.g., deflection and deflection rate)?

Figure 2: Response of the closed loop short period dynamics to a severe vertical gust of 66 ft/s (equivalent to 4 degrees disturbance in the angle of attack) and to a step g-command using the representative data at M = 0.8, sea level with the maximum cg location at the nominal horizontal tail area ratio of S_H/S = 0.35.)

Figure 3: Response of the closed loop short period dynamics to a severe vertical gust of 66 ft/s (equivalent to 4 degrees disturbance in the angle of attack) and to a step g-command using the representative data at M = 0.8, sea level with the maximum cg location at a very small horizontal tail area ratio of S_H/S = 0.05. 

Figure 4: Response of the closed loop short period dynamics to a severe vertical gust of 66 ft/s (equivalent to 4 degrees disturbance in the angle of attack) and to a step g-command using the representative data at M = 0.8, sea level with the maximum cg location at a large horizontal tail area ratio of S_H/S = 0.5. 

Figure 5: Several stability boundaries for the aircraft at M = 0.8, sea level: 5% and -5% static margin, C_M_alpha = 0.4/rad, the current design point, and the optimized boundary using the proposed formulation.

7. Beyond This Study: Motivation for Future Study

Having shown that the static stability constraint is no longer active in tail sizing optimization, other considerations become decisive. Using the Aerodynamics group's CFD results, we constructed: (i) the backward limit for nose-down recovery at high alpha, (ii) the forward limit for trim during landing, and (iii) the forward limit for takeoff rotation, as shown in Figure 6.

As expected, the high-alpha pitch-down recovery backward CG constraint is more restrictive than the optimized stability constraint. Figure 6 also shows that for the aircraft, the takeoff rotation limit is more stringent than the landing trim limit. The current area ratio (0.35) and CG boundaries (23%-33.6%) violate this stringent takeoff rotation constraint, implying either the tail area must increase or the CG must move backward. Therefore, the takeoff phase needs scrutiny to determine the reason for poor takeoff performance and propose solutions other than inefficiently increasing tail area.

Figure 6: Several cg boundaries for the aircraft: trim at landing configuration, takeoff rotation, nose down recovery at high angle of attack, the optimized stability boundary using the proposed formulation, and the current cg limits.