NOMENCLATURE

ARI

aileron to rudder interconnect

$C{a_b} = \frac{{\partial Ca}}{{\partial b}}$

partial derivative of parameter Ca with respect to parameter b in 1/radian

${\text{Ix}}, {\text{Iy}}, {\text{Iz}}$

moment of inertia along X-axis, Y-axis and Z-axis, respectively, in kg m²

${\text{Ixy}}, {\text{Iyz}}, {\text{Ixz}}$

moment of inertia in XY-plane, YZ-plane and XZ-plane, respectively, in kg m²

$K_{ARI}$

aileron to rudder interconnect gain

$K_{p}, K_{\beta}$

roll rate feedback gain and side-slip feedback gain, respectively

${K_{{B_w}}}$

wash-out side-slip rate feedback gain

$L_{\beta}, L_{\delta a}, L_{\delta r}$

roll angular acceleration per unit β, δa and δr, respectively, in °/s²/°

$L_{p}, L_{r}$

roll angular acceleration per unit p and r, respectively, in 1/s

${\text{loe}}, {\text{lie}}$

port-side left outboard and inboard elevon position, respectively, in °.

$N_\beta, N_{\delta a}, N_{\delta r}$

yaw angular acceleration per unit β, δa and δr, respectively, in °/s²/°

$N_p, N_r$

yaw angular acceleration per unit p and r, respectively, in 1/s

$N_{y}$

lateral acceleration in g

$\dot p, \dot r$

roll acceleration and yaw acceleration, respectively, in °/s²

$q, p, r$

body-fixed axis pitch rate, roll rate and yaw rate, respectively, in °/s

${\text{roe}}, {\text{rie}}$

starboard-side right outboard and inboard elevon position, respectively, in °

${\text{rud}}$

rudder surface position in °

$V_{T}$

true airspeed in m/s

$Y_{p}, Y_{r}$

lateral acceleration per unit p and r, respectively, in m/s/°

$Y_\beta, Y_{\delta a}, Y_{\delta r}$

lateral acceleration per unit β, δa and δr, respectively, in m/s²/°

α

angle-of-attack (AoA) in °

β

angle of side slip (AoSS) or beta in °

$\delta a, \delta r$

aileron and rudder surface deflection, respectively, in °

$\phi $

roll angle in °

${{\dot{\beta}}}$

side-slip rate in °/s

$\dot \phi $

rate of roll angle or bank angle in °/s

α₀

initial angle-of-attack in °

τ_w

wash-out filter time constant in s

2.0 AIRCRAFT MODEL AND CONVENTIONAL ARI DESIGN

The non-linear rigid-body equations of motion^{(Reference Stevens and Lewis5–Reference Roskam7)} form the basis for simulation and controller design. The aircraft dynamics is linearised by using a small perturbation technique, and a lateral/directional linear model, suitable for control law design, is obtained with β, p, r and $\phi $ as the states, δa and δr as the inputs, and $\phi $, β, p, r and Ny as observations. By ignoring the $\dot \phi $ equation and the dependence on $\phi $, the lateral and directional model for the aircraft in state space is given by Equation (1). The spiral mode, being a low-frequency mode because of its long time constant, is not significant from a handling quality point of view and is thus excluded from the state-space model. Hence, a third-order simplified state variable model is considered with the state vector defined as x = [p, r, β] and u = [δa, δr].

$\dot x = A\;x + B\;u$, where A and B are as defined in Equation (1)

(1) \begin{align}\left[ \begin{array}{c} {\dot p}\\[3pt] {\dot r}\\[3pt] {\dot \beta} \end{array} \right] = \left[ \begin{array}{c@{\quad}c@{\quad}c} {{L_p}} & {{L_r}} & {{L_\beta }}\\[3pt] {{N_p}} & {{N_r}} & {{N_\beta }}\\[3pt] {{Y^{\prime}_p}} & {{Y^{\prime}_r}} & {{Y^{\prime}_\beta}}\end{array} \right] \left[ {\begin{array}{c} p\\[3pt] r\\[3pt] \beta \end{array}} \right] + \left[ {\begin{array}{c@{\quad}c} {{L_{\delta a}}} & {{L_{\delta r}}}\\[3pt] {{N_{\delta a}}} & {{N_{\delta r}}}\\[3pt] {{Y_{\delta a}}} & {{Y_{\delta r}}}\end{array}} \right] \left[ \begin{array}{c} {\delta a}\\[3pt] {\delta r} \end{array} \right] \end{align}

where ${Y^{\prime}_p} = ({Y_p}/{V_T}) + sin{\alpha _0}$, ${Y^{\prime}_r} = ({Y_r}/{V_T}) - \cos {\alpha _0}$, ${Y^{\prime}_\beta} = ({Y_\beta }/{V_T})$.

Consider the nominal lateral/directional control schematic^{(Reference Chetty and Deodhare12,Reference Srinathkumar13)} shown in Fig. 1. The main objective of the lateral directional control law for a fighter aircraft is to augment the directional stability and Dutch roll damping (which could be quite poor) and to improve the handling quality (e.g. a crisp roll rate response, good disturbance rejection and good turn coordination). In response to roll stick input, the aircraft at high angles of attack rolls about the velocity vector to suppress the kinematic coupling between the AoA and the side-slip angle. The stability axis angular rates ${{\rm{p}}_{\rm{s}}} = {\rm{p\;}}$ and ${{\rm{r}}_{\rm{s}}} = {\rm{r - ptan}}\alpha {\rm{\;}}$ are used instead of the body axis angular rates for feedback. Furthermore, the inertial coupling compensation term $\left( {{{\Delta \dot r}} = \left( {\frac{{{{\rm{I}}_{\rm{y}}}{\rm{ - }}{{\rm{I}}_{\rm{x}}}}}{{{{\rm{I}}_{\rm{z}}}}}} \right){\rm{pr}}} \right)$ is also added to the feedback terms of the directional axis. The lateral/directional control law^(14,15) is designed to meet the specifications given in the MIL requirements documents.

Figure 1.

Lateral directional feedback control law schematic.

Note that ${K_{ARI^\circ }}$ is replaced by ${G_{ARI^\circ }}$ obtained from method 1 for the simulations used.

The transfer functions for the roll rate (p) and side slip (β) can be derived from Equation (1), and expressed as

${G_{p\delta a}}(s) = \frac{{p(s)}}{{\delta a}}$, ${G_{p\delta r}}(s) = \frac{{p(s)}}{{\delta r}}$, ${G_{\beta \delta r}}(s) = \frac{{\beta (s)}}{{\delta r}}$, as given in Equations (2) and (3).

(2) \begin{equation}p(s) = {G_{p\delta a}}(s )\delta a( s ) + \;{G_{p\delta r}}( s )\delta r( s )\end{equation}

(3) \begin{equation}\beta ( s ) = {G_{\beta \delta a}}( s )\delta a( s ) + \;{G_{\beta \delta r}}( s )\delta r( s )\end{equation}

The ARI gain (${K_{ARI^\circ }}$) is synthesised, after the primary feedback loops have been appropriately designed, to decouple the Dutch roll mode from the roll rate response to the aileron input. This (${K_{ARI^\circ }}$) gain is conventionally implemented as shown in Equation (4)

(4) \begin{equation}\delta r = {K_{ARI}}\delta a\end{equation}

With the ARI gain as given in Equation (4), Equation (2) can be rewritten as

(5) \begin{equation}p(s) = \left( {{G_{p\delta a}}(s) + {K_{ARI}}{G_{p\delta r}}(s)} \right)\delta a(s)\end{equation}

Thus, the effect of increasing the ARI gain is to move the closed-loop $\left({G_{p\delta a}}(s) + \right.$ $\left. {K_{ARI}}{G_{p\delta r}}(s)\right)$ zeros, as given in Equation (5), towards the open-loop (${{G_{p\delta r}}}$) poles. The gain is then selected to place a zero close to the augmented Dutch roll poles, to achieve approximate pole-zero cancellation.

Figure 2 show the locus of ${{{\rm{G}}_{{\rm{p\delta a}}}}}$ zero with different ${{{\rm{K}}_{{\rm{ARI}}}}}$ gains and augmented Dutch roll pole. The side-slip (beta) time response for different ${{{\rm{K}}_{{\rm{ARI}}}}}$ gains is also plotted in Fig. 2 with roll stick doublet (at 1 s) input. Note from Fig. 2 that the pole-zero distance is lower for ARI gains of approximately 2.2 to 2.4. However, the ARI gain that generates the minimum side slip, as seen from the time response plots, is seen to be ${{{\rm{K}}_{{\rm{ARI}}}}}$ = 1.7. This example reveals the issues with the traditional design procedure.

Figure 2.

Plot showing the location of the Dutch roll pole and zero of ${G_{p\delta a}}(s) + {K_{ARI}}\;{G_{p\delta r}}( {\rm{s}} )$ to study the sensitivity of ARI gain for 0.4M, 1km and also the plot of the angle of side-slip response with roll stick input for different ARI gains.

ARI gain selection cannot be automated if it is based on the placement of the root location close to the Dutch roll pole, or based on the Hankel norm^{(Reference Jee, Sharma, Rao, Zachariah, Brinda, Lalithambika, Dhekane, Deodhare and Chetty16)}, or verifying it with closed-loop time response simulations or HQ assessment using real-time simulators. The methods currently used for ${{K_{ARI}}}$ design are, therefore, based on an iterative process. For online reconfiguration, a non-iterative method for ${{K_{ARI}}}$ gain design is essential.

3.0 PROPOSED METHODS OF ARI DESIGN

In this section, two alternative methods for ARI design are proposed. The basic requirement is to nullify the side slip caused by an aileron input by applying an appropriate rudder input. The side-slip response, as derived in Equation (3), is equated to zero to nullify the side slip, as shown below:

(6) \begin{equation}\beta (s) = {{\rm{G}}_{\beta {\rm{\delta a}}}}(s)\delta a(s) + {{\rm{G}}_{\beta {\rm{\delta a}}}}(s)\delta r(s) = 0\end{equation}

The required rudder $\delta r$ can be derived from Equation (6) as

(7) \begin{equation}\delta r = - \frac{{{G_{\beta \delta a}}(s)}}{{{G_{\beta \delta r}}(s)}}\delta a = {G_{ARI}}(s)\delta a\end{equation}

where ${{\rm{G}}_{{\rm{ARI}}}}\left( {\rm{s}} \right)$ is the ARI transfer function. Note that Equation (3), used in the above derivation, is the open-loop transfer function. Hence, it does not consider the feedback loops discussed in Section II, as shown in Fig. 1. It can be shown that the above derivation of ${G_{ARI}}(s)$ holds in the closed-loop case also.

1. A. Proposed method 1 for the ARI design

In this method, the ARI is directly taken as the transfer function, as in Equation (7) above, instead of the static ${K_{ARI}}$ gain. ${G_{ARI}}(s)$ can be derived analytically from the third-order lateral directional model given in Equation (1), and is given in Equation (8). For real-time implementation of the ARI transfer function, real-time estimation of the lateral and directional parameters, as listed in Equation (8), is required. All the elements or parameters required to calculate ${G_{ARI}}(s)$ can be obtained using real-time estimation of the parameters (and possibly from aero database tables, which can also be carried onboard to make parameter estimation more reliable).

(8) \begin{equation}{G_{ARI}}(s) = - \frac{{\frac{{\beta (s)}}{{\delta a(s)}}}}{{\frac{{\beta (s)}}{{\delta r(s)}}}} = - \frac{{{G_{\beta \delta a}}(s)}}{{{G_{\beta \delta r}}(s)}} = \frac{{A1\,{s^2} + A2\,s + A3}}{{B1\,{s^2} + B2\,s + B3}}\end{equation}

Where $A1 = {Y_{\delta a}}$, $B1 = - {Y_{\delta r}}$,

$A2 = - ( {{Y_{\delta a}}{L_p} + {Y_{\delta a}}{N_r} - {L_{\delta a}}Y^{\prime}_p - {N_{\delta a}}{Y^{\prime}_r}} )$

$B2 = ( {{Y_{\delta r}}{L_p} + {Y_{\delta r}}{N_r} - {L_{\delta r}}Y^{\prime}_p - {N_{\delta r}}{Y^{\prime}_r}} )$

$A3 = - {Y_{\delta a}}( {{L_r}{N_p} - {L_p}{N_r}} ) + {L_{\delta a}}( {{N_p}Y^{\prime}_r - {N_r}{Y^{\prime}_p}} ) + {N_{\delta a}}( {{L_r}Y^{\prime}_p - {L_p}{Y^{\prime}_r}} )$

$B3 = {Y_{\delta r}}( {{L_r}{N_p} - {L_p}{N_r}} ) - {L_{\delta r}}( {{N_p}Y^{\prime}_r - {N_r}{Y^{\prime}_p}} ) - {N_{\delta r}}( {{L_r}Y^{\prime}_p - {L_p}{Y^{\prime}_r}} )$

1. B. Proposed method 2 for the ARI design

Traditionally, ARI is taken to be a flight-condition scheduled gain, instead of a transfer function as obtained above. As seen from Fig. 2, the primary effect of the ${K_{ARI}}$ gain is visible in the transient response when a roll manoeuvre is initiated. ${K_{ARI}}$ is a static interconnect gain which plays an important role in shaping the transient response. The roll mode is typically the fastest among the roll, spiral and Dutch roll modes. Therefore, the transient response is primarily influenced by the roll mode, which is defined by the roll damping (${L_p}$). The Dutch roll mode and the spiral mode are relatively long-term responses. Hence, intuitively, the magnitude of ${G_{ARI}}(s)$ at the roll mode (${L_p}$) frequency can be considered to be the ${K_{ARI}}$ gain. To confirm this, six degree of freedom simulations were carried out for a generic fighter aircraft in various flight conditions. It was found that ${K_{ARI}}$ computed as the magnitude of ${G_{ARI}}(s)$ at the roll damping (${L_p}$) frequency gives the minimum side slip, and therefore, the same is proposed in this paper. The sign of ${K_{ARI}}$ is selected based on the sign of the real part of ${G_{ARI}}(s)$, as shown in Equation (9).

(9) \begin{equation}{K_{ARI}} = sign\left( {real\left( {{G_{ARI}}\left(\, {j{L_p}} \right)} \right)} \right){\rm{*}}\left| {{G_{ARI}}\left(\, {j{L_p}} \right)} \right|\end{equation}

The proposed method for computing ${K_{ARI}}$ is simple and non-iterative, and can be used for gain reconfiguration in near real time in case of faults. It is computationally more efficient than method 1, as it does not require a transfer function computation in each control cycle.

1. C. Comparison of the nonlinear simulation responses with the proposed methods

Nonlinear 6DoF simulation is carried out with all 12 states^{(Reference Stevens and Lewis5)} using flight-validated simulation software. The step size used in the simulation is 0.0125 s, and simulations were carried out for a fixed mass (~9000kg), moments of inertia (Iyy = 66,558.9, Ixx = 14,087.9, Izz = 77,875.2, Ixy = 162.5, Iyz = 32.5, Izx = 2490.5), and centre of gravity (34% of the mean aerodynamic chord). In this section, simulation results for a typical fighter aircraft are presented for the three flight conditions of wings level trim (1g) as listed in Table 1. The stability and control derivatives of the three flight conditions considered are presented in Table 2.

The ARI obtained using the proposed methods is presented in Table 1 for these flight conditions. Note that the ARI gain values as obtained using the conventional method and method 2 are in fairly good agreement.

Table 1

ARI gains with three proposed methods

Table 2

Stability and control derivatives for flight conditions considered in Table 1

The nonlinear simulation response to roll stick doublet input at three flight conditions, for different values of ARI (i.e., ${{G_{ARI}}}$ using method 1 and ${K_{ARI}}$ using method 2), is plotted in Fig. 3. Note that the time response is similar. To assess the computational complexity, C codes are generated for both methods and compared using the default configuration settings for a generic real-time target using Simulink MATLAB software. It is observed that the size of the code with ${K_{ARI}}$ is smaller than that with ${{G_{ARI}}}$, which can lead to a smaller footprint of embedded code. Also, the output function generated by ${{K_{ARI}}}$ is less than the output function generated by ${{G_{ARI}}}$. Hence, the single gain (${{K_{ARI}}}$) evaluation is computationally much more efficient than the transfer function (${{G_{ARI}}}$) evaluation, and thus particularly useful for real-time reconfiguration.

Figure 3.

Comparison of nonlinear simulation responses at 6km altitude with ${K_{ARI}}$ and ${G_{ARI}}$.

4.0 RECONFIGURATION OF ${{\textbf{K}}_{{\textbf{ARI}}}}$ GAIN FOR LEFT OUTBOARD CONTROL SURFACE DAMAGE SCENARIO

Some fault scenarios, such as total/partial damage to a control surface or a fault due to battle damage or a bird strike, cannot be detected either through hardware or through software redundancy management techniques. In such cases, manoeuvring by the pilot with full stick input demand could lead to departure of the aircraft, if the fault is not identified and the control law is not reconfigured in real time. To illustrate this, the case of left outboard control surface damage is considered in this paper. This is modelled as a 90% effectiveness drop, i.e., a reduction in the surface effectiveness in the aero database and simulation using the flight-validated software. For simplicity, the reduction in control effectiveness is assumed to be linear. Further work needs to be carried out based on computational fluid dynamics studies to model the exact relationship.

In this example, the value of the left elevon outboard surface (port side), used for computing the force and moment coefficients from aero database tables, is set to 10% of its value (i.e., a 90% reduction). The rolling moment coefficient^{(Reference Roskam7)}, which is modelled for the left control surface damage, is shown by Equation (10).

(10) \begin{align}{{\rm{C}}_1} & = {\rm{C}_{1\beta }}\left( \beta \right) + {\rm{C}_{1\rm{p}}}\left( {\rm{Mach}} \right)\frac{\rm{pb}}{{2\rm{v}}} + {\rm{C}_{1\rm{r}}}\left( {\rm{Mach}} \right)\frac{\rm{rb}}{{2\rm{v}}} + {\rm{C}_{1\delta \rm{r}}}\left( {\alpha,\rm{Mach},\rm{RUD}} \right)\nonumber\\ & \quad + {\rm{C}_{1\delta \rm{a}}}\left( {\alpha,\rm{Mach},\rm{LOE} * 0.1} \right) + {\rm{C}_{1\delta \rm{a}}}\left( {\alpha,\rm{Mach},\rm{LIE}} \right) + {\rm{C}_{1\delta \rm{a}}}\left( {\alpha,\rm{Mach},\rm{RIE}} \right)\nonumber\\ & \quad + {\rm{C}_{1\delta \rm{a}}}\left( {\alpha,\rm{Mach},\rm{ROE}} \right)\end{align}

where ${\rm{C}_{1\beta }}$, ${\rm{C}_{1\rm{p}}}$, ${\rm{C}_{1\rm{r}}}$, ${\rm{C}_{1\delta \rm{r}}}$, ${\rm{C}_{1\delta \rm{a}}}$ are the rolling moment coefficients of beta, roll rate, yaw rate, rudder and aileron, respectively. In the aero database tables, these coefficients are defined as a function of Mach no. (M, Mach), angle-of-attack (α, AoA), angle of side slip (β, beta), LIE, LOE, RIE and ROE (left and right inboard and outboard surface values) and RUD (rudder surface value). The coefficient ${{\rm{C}}_1}$ is the total rolling moment coefficient.

Six-DoF non-linear simulation results are presented for roll stick doublet input at a typical flight condition (0.5 Mach number, 9.6km altitude). The time responses for the following three cases are presented in Fig. 4 to demonstrate the impact of ARI gain reconfiguration in the considered surface damage fault scenario.

Figure 4.

ARI reconfiguration using method 2 for reduction of outboard surface effectiveness fault.