Greek symbol

$${\rm \rrho }$$

density of the air

$${\rm \ralpha }$$

aerodynamic angle-of-attack

$$\rphi ,{\rm \rtheta },{\rm \rpsi }$$

Euler angles (roll, pitch and azimuth angles) defining body frame w.r.t. inertial frame

$${\rm \rgamma }$$

flight path angle

$${\rm \rbeta }$$

side slip angle

2.1 Smart adaptation kit (SAK) model

The SAK developed in this study consists of a central main body to which the aerial munition is attached. The SAK itself is attached to the belly of the aircraft via a bolting mechanism. Upon release from the aircraft, SAK wings pop-out from their initial folded position to the fully extended position. This enables the SAK to glide towards the target. The geometrical view of the designed SAK obtained through USAF Datcom⁽ Reference Finck ^{27 )} is depicted in Fig. 1.

Figure 1 SAK used in this research.

The GCM developed for the SAK provides real-time guidance commands to achieve stabilisation and directional control. To reduce the cost of the modification kit, the number of control surfaces has been kept to a bare minimum. Utilising a single pair of control fins located at the rear end of the fuselage, both the roll and yaw movements are controlled. Like ailerons, these control surfaces can be asymmetrically deflected to control the roll movement. For directional control, they can be symmetrically deflected similar to the rudder functions. This modification resulted in a highly non-linear coupled motion of the vehicle in response to control surface deflection. This design, therefore, made the control architecture more challenging. The dedicated GCM designed for the SAK constitutes (a) a scheduler, for near real-time trajectory generation; (b) a stability controller, for platform stabilisation; and (c) a tracking controller for tracking the optimised trajectory. Presently, 1,000 lb class Mk-82 series free-fall munitions can be retrofit to smart munition through this kit. When the munitions reach the vicinity of the target, the explosive charge is activated by a suitable control permitting SAK to jettison the munition without interference. Complete geometrical description and design parameters of SAK are shown in Table 1.

Table 1 SAK geometric and mass properties

2.2 Flight dynamics model

In order to evaluate performance, as well as to develop a guidance and control system, it has to create a suitable model that meets the specific properties of the guided bomb. In this research, 6-DOF ⁽ Reference Mir, Maqsood and Akhtar ^{30 )} model which is typically utilised in flight dynamic modelling, is used to model the vehicle motion in the 3D space. A flat Earth approximation is used to define the inertial co-ordinate system. The body frame is defined by the conventional manner. The dynamic equations are written with respect to body co-ordinates with appropriate kinematic equations relating body translational and rotational rates and with inertial translational and Euler angle rotational rates, respectively. The governing equations of motion representing (a) dynamics of translation (Equations (1)), (b) dynamics of rotation (Equations (2)), (c) kinematics (Equations (3)) and (d) navigation (Equation (4)), assuming non-rotating Earth, are defined as

(…1) $$\matrix{ {\dot{U}} \hfill & {{\equals}RV{\minus}QW{\minus}g\sin \rtheta {\plus}{{X_{A} {\plus}X_{T} } \over m}} \hfill \cr {\dot{V}} \hfill & {{\equals}{\minus}RU{\plus}PW{\plus}g\sin \rphi \cos \rtheta {\plus}{{Y_{A} {\plus}Y_{T} } \over m}} \hfill \cr {\dot{U}} \hfill & {{\equals}QU{\minus}PV{\plus}g\cos \rphi \cos \rtheta {\plus}{{Z_{A} {\plus}Z_{T} } \over m}} \hfill \cr {} \hfill & {} \hfill \cr } $$

(…2) $$\matrix{ {\Gamma \dot{P}} \hfill & {{\equals}J_{{XZ}} (J_{X} {\minus}J_{Y} {\plus}J_{Z} )PQ{\minus}[J_{Z} (J_{Z} {\minus}J_{Y} ){\plus}J_{{XZ}}^{2} ]QR{\plus}J_{Z} l{\plus}J_{{XZ}} n} \hfill \cr {\Gamma \dot{Q}} \hfill & {{\equals}(J_{Z} {\minus}J_{X} )PR{\minus}J_{{XZ}} (P^{2} {\minus}R^{2} ){\plus}m} \hfill \cr {\Gamma \dot{P}} \hfill & {{\equals}[J_{X} (J_{X} {\minus}J_{Y} ){\plus}J_{{XZ}}^{2} ]PQ{\minus}J_{{XZ}} (J_{X} {\minus}J_{Y} {\plus}J_{Z} )QR{\plus}J_{{XZ}} l{\plus}J_{X} n} \hfill \cr {} \hfill & {} \hfill \cr } $$

(…3) $$\matrix{ {\dot{\rphi }} \hfill & {{\equals}P{\plus}\tan \rtheta (Q\sin \rphi {\plus}R\cos \rphi )} \hfill \cr {\dot{\rtheta }} \hfill & {{\equals}Q\cos \rphi {\minus}R\sin \rphi } \hfill \cr {\dot{\rpsi }} \hfill & {{\equals}{{Q\sin \rphi {\plus}R\cos \rphi } \over {\cos \rtheta }}} \hfill \cr {} \hfill & {} \hfill \cr } $$

(…4) $$\matrix{ {\dot{P}_{E} } \hfill & {{\equals}U\cos \rtheta \cos \rpsi {\plus}V({\minus}\cos \rphi \sin \rpsi {\plus}\sin \rphi \sin \rtheta \cos \rpsi )} \hfill \cr {} \hfill & {{\plus}W(\sin \rphi \sin \rpsi {\plus}\cos \rphi \sin \rtheta \cos \rpsi )} \hfill \cr {\dot{P}_{N} } \hfill & {{\equals}U\cos \rtheta \sin \rpsi {\plus}V(\cos \rphi \cos \rpsi {\plus}\sin \rphi \sin \rtheta \sin \rpsi )} \hfill \cr {} \hfill & {{\plus}W({\minus}\sin \rphi \cos \rpsi {\plus}\cos \rphi \sin \rtheta \sin \rpsi )} \hfill \cr {\dot{h}} \hfill & {{\equals}U\sin \rtheta {\minus}V\sin \rphi \cos \rtheta {\minus}W\cos \rphi \cos \rtheta } \hfill \cr {} \hfill & {} \hfill \cr } $$

where U, V and W are the linear velocities along body x-, y- and z-axis, respectively, $$\rphi ,{\rm \rtheta }\,\,{\rm and}\,\,{\rm \rpsi }$$ are the Euler angles defining the orientation of body frame with respect to inertial frame, P, Q and R are the angular velocities along body x-, y- and z-axis, respectively, $$P_{n} $$ and $$P_{e} $$ are the position co-ordinates along the inertial north and east directions, h is the vehicle altitude, $$X_{A} ,Y_{A} ,Z_{A} $$ are the axial, side and tangential forces defining components of the aerodynamic force in the body frame, m is the mass, g is the acceleration due to gravity, J is the moment of inertia matrix, l, m and n are the angular velocity components (roll, pitch and yaw moments) in the body axis, α and β are the aerodynamic angles representing angle-of-attack and side slip angle, respectively.

2.3 Aerodynamic parameters estimation

The aerodynamic forces and moments acting on the aerial vehicle during different stages of the flight are governed by Equations (5) and (6) respectively:

(…5) $$\matrix{ {L{\equals}q_{\infty} SC_{L} ,D{\equals}q_{\infty} SC_{D} ,Y{\equals}q_{\infty} SC_{Y} } \hfill \cr {} \hfill \cr } $$

where L, D and Y are the aerodynamic lift, drag and side force, respectively, in the wind axis. $$q_{\infty} $$ is the dynamic pressure, S is the wing area and $$C_{L} ,C_{D} \,{\rm and}\,C_{Y} $$ are the dimensionless aerodynamic coefficients for lift, drag and side forces, respectively:

(…6) $$l_{w} {\equals}q_{\infty} bSC_{l} ,\quad m_{W} {\equals}q_{\infty} cSC_{m} ,\quad n_{W} {\equals}q_{\infty} bSC_{n} $$

where l _w , m _w and n _w are the roll, pitch and yaw moment in the wind axis, respectively, b is the wing span, c is the wing chord and $$C_{l} ,C_{m} \,{\rm and}\,C_{n} $$ are the dimensionless aerodynamic coefficients for roll, pitch and yaw moments, respectively.

The body aerodynamic force and moment coefficients in Equations (5) and (6) vary with the flight conditions and control settings. A high fidelity aerodynamic model is necessary to accurately determine these aerodynamic coefficients. In this research, both empirical⁽ Reference Roskam ^{31 )} and non-empirical techniques (such as CFD⁽ Reference Petterson ^{29 )} and USAF Datcom⁽ Reference Finck ^{27 )}) are utilised to determine these coefficients. The high fidelity model employed for aerodynamic parameter estimation is elaborated in the following equation:

(…7) $$\matrix{ {C_{i} {\equals}C_{{i,{\rm static}}} ({\rm \ralpha ,\rbeta ,\delta }_{{{\rm control}}} ,M){\plus}C_{{i,{\rm dynamic}}} ({\dot{\ralpha }, \dot{\rbeta }},p,q,r)} \hfill \cr {} \hfill \cr } $$

where C _i =C _L , C _D , C _Y , C _l , C _m , C _n represent the coefficient of lift, drag, side force, rolling moment, pitch moment and yawing moment, respectively.

The non-dimensional coefficients are usually obtained through linear interpolations using data obtained from various sources. In this research, different methodologies were adopted to ascertain the static and dynamic components of these coefficients⁽ Reference Le Roy, Morgand and Farcy ^{32 )}. Evaluation of static (basic) coefficient data (see Equation (8)) was performed employing CFD⁽ Reference Buning, Gomez and Scallion ^{28 ,} Reference Petterson ^{29 )} technique. All the parameters were obtained as a function of control ( $${\rm \delta }_{{{\rm control}}} )$$ , angle-of-attack (α), side slip (β) and Mach number (M)

(…8) $$\matrix{ {C_{{i,{\rm static}}} ({\rm \ralpha ,\rbeta ,\delta }_{{{\rm control}}} {\rm ,M})} \hfill & {\Rightarrow C_{{D_{b} }} ({\rm \ralpha ,\rbeta ,\delta }_{{{\rm control}}} {\rm ,M}),C_{{L_{b} }} ({\rm \ralpha ,\rbeta ,\delta }_{{{\rm control}}} {\rm ,M}),C_{{Y_{b} }} ({\rm \ralpha ,\rbeta ,\delta }_{{{\rm control}}} {\rm ,M}),} \hfill \cr {} \hfill & \vskip8pt {C_{{l_{b} }} ({\rm \ralpha ,\rbeta ,\delta }_{{{\rm control}}} {\rm ,M}),C_{{m_{b} }} ({\rm \ralpha ,\rbeta ,\delta }_{{{\rm control}}} {\rm ,M}),C_{{n_{b} }} ({\rm \ralpha ,\rbeta ,\delta }_{{{\rm control}}} {\rm ,M}),} \hfill \cr {} \hfill & {} \hfill \cr } $$

where $$C_{{D_{b} }} $$ , $$C_{{L_{b} }} $$ , $$C_{{Y_{b} }} $$ , $$C_{{l_{b} }} $$ $$C_{{m_{b} }} $$ and $$C_{{n_{b} }} $$ represents the basic components of the aerodynamic forces and moments as a function of ( $${\rm \delta }_{{{\rm control}}} )$$ , angle-of-attack (α), side slip (β) and Mach number (M).

Dynamic component (Equation (9)) consists of rate and acceleration derivatives. These are evaluated utilising empirical⁽ Reference Roskam ^{31 )} and non-empirical ('USAF Stability and Control DATCOM' ⁽ Reference Finck ^{27 )}) techniques:

(…9) $$\matrix{ {C_{{i,{\rm dynamic}}} ({\dot{\ralpha },\dot{\rbeta }},p,q,r){\equals}{\rm Rate}\,{\rm derivatives}{\plus}{\rm Acceleration}\,{\rm derivatives}} \hfill \cr {} \hfill \cr } $$

where the rate derivatives are the derivatives due to roll (p) rate, pitch rate (q) and yaw rate (r) as shown in the following equation:

(…10) $$\matrix{ {{\rm Rate}\,{\rm derivatives}{\equals}(C_{{L_{q} }} ,C_{{D_{q} }} ,C_{{m_{q} }} ){\plus}(C_{{Y_{p} }} ,C_{{l_{p} }} ,C_{{n_{p} }} ){\plus}(C_{{Y_{r} }} ,C_{{l_{r} }} ,C_{{n_{r} }} )} \hfill \cr {} \hfill \cr } $$

Acceleration derivatives are the derivatives due to change in the aerodynamic angles $$({\rm \dot{\ralpha },\dot{\rbeta }})$$ and are defined in the following equation:

(…11) $$\matrix{ {{\rm Acceleration}\,{\rm derivatives}{\equals}(C_{{L_{{\dot{\ralpha }}} }} {\plus}C_{{D_{{\dot{\ralpha }}} }} {\plus}C_{{m_{{\dot{\ralpha }}} }} ){\plus}(C_{{Y_{{\dot{\rbeta }}} }} {\plus}C_{{l_{{\dot{\rbeta }}} }} {\plus}C_{{m_{{\dot{\rbeta }}} }} ).} \hfill \cr {} \hfill \cr } $$

Utilising Equation (9), dynamic derivatives are evaluated:

(…12) $$\matrix{ {C_{{D({\rm Dyn})}} } \hfill & {{\equals}C_{{D_{b} }} {\plus}C_{{D_{{\dot{\ralpha }}} }} {\rm \dot{\ralpha }}\left( {{{\bar{c}} \over {2V_{t} }}} \right){\plus}C_{{D_{q} }} q\left( {{{\bar{c}} \over {2V_{t} }}} \right),} \hfill \vskip10pt \cr {C_{{L({\rm Dyn})}} } \hfill & {{\equals}C_{{L_{b} }} {\plus}C_{{L_{{\dot{\ralpha }}} }} {\rm \dot{\ralpha }}\left( {{{\bar{c}} \over {2V_{t} }}} \right){\plus}C_{{L_{q} }} q\left( {{{\bar{c}} \over {2V_{t} }}} \right),} \hfill \vskip10pt \cr {C_{{Y({\rm Dyn})}} } \hfill & {{\equals}C_{{Y_{b} }} {\plus}C_{{D_{{\dot{\rbeta }}} }} {\rm \dot{\rbeta }}\left( {{{\bar{c}} \over {2V_{t} }}} \right){\plus}(C_{{Y_{p} }} p{\plus}C_{{Y_{r} }} r)\left( {{{\bar{c}} \over {2V_{t} }}} \right),} \hfill \cr } $$

(…13) $$\matrix{ {C_{{l(Dyn)}} } \hfill & {{\equals}C_{{l_{b} }} {\plus}C_{{l_{{\dot{\rbeta }}} }} \dot{\rbeta }({{\bar{c}} \over {2V_{t} }}){\plus}(C_{{l_{p} }} p{\plus}C_{{l_{r} }} r)({{\bar{c}} \over {2V_{t} }}),} \hfill \vskip10pt \cr {C_{{m(Dyn)}} } \hfill & {{\equals}C_{{m_{b} }} {\plus}C_{{m_{{\dot{\ralpha }}} }} \dot{\ralpha }({{\bar{c}} \over {2V_{t} }}){\plus}C_{{m_{q} }} q({{\bar{c}} \over {2V_{t} }}),} \hfill \vskip10pt \cr {C_{{n(Dyn)}} } \hfill & {{\equals}C_{{n_{b} }} {\plus}C_{{n_{{\dot{\rbeta }}} }} \dot{\rbeta }({{\bar{c}} \over {2V_{t} }}){\plus}(C_{{n_{p} }} p{\plus}C_{{n_{r} }} r)({{\bar{c}} \over {2V_{t} }}),} \hfill \cr {} \hfill & {} \hfill \cr } $$

where $$C_{{L(Dyn)}} $$ , $$C_{{D(Dyn)}} $$ , $$C_{{Y(Dyn)}} $$ , $$C_{{l(Dyn)}} $$ , $$C_{{m(Dyn)}} $$ and $$C_{{n(Dyn)}} $$ represent the dynamic derivatives for the aerodynamic forces (lift, drag and side force) and moments (roll, pitch and yaw moments) coefficients, respectively.

The resultant aerodynamic coefficients are then calculated utilising Equation (7) and are then utilised to compute the values of the aerodynamic load during different stages of the flight.

2.4 Generation of aerodynamic flight profile

Generation of an optimal flight profile, in which parameters such as velocity, dynamic pressure, and Mach number are optimised, forms an essential part of the design of the GCM⁽ Reference Jiang, Li, Li, Huang, Yang and Jiang ^{33 –} Reference Jain and Tsiotras ^{36 )}. This is important since the use of ad hoc profile or control policies to evaluate competing configurations may inappropriately penalise the performance of one configuration over another. Moreover, to avoid any undesirable aero-elastic phenomena, maximum dynamic pressure must be constrained. Thus, to guarantee optimised trajectory, it is important to optimise the aerodynamic flight profile and develop control policy for each configuration early in the design process. Two different types of optimised flight profiles (based on the dynamic pressure) are created to acquire ranges between 100 and 120 km, once SAK is air launched from an altitude of 35,000 ft. Table 2 depicts the flight profile developed for acquiring 100 km range.

Table 2 SAK geometric and mass properties

Graphical illustration of the optimised flight profile (for both 100 km and 120 km ranges) is depicted in Fig. 2. The velocity, dynamic pressure and Mach number are optimised for the entire flight regime (ranging from 35,000 ft to the ground target).

Figure 2 Optimal trajectories: aerodynamic flight profile.

2.5 Formulation of optimal control problem

In this section, steady-state values for the state and control variables are determined at each point of a pre-specified trajectory by computing an equilibria point of the differential equations specified in Equations (1)–(4). The key idea includes linearising the non-linear system along the trajectory, then using the resulting time-varying linearisation to obtain a time-varying state feedback controller that locally stabilises the system along the trajectory. The optimisation techniques similar to Refs. 37–41 enhanced SAK range to about 100 km with high accuracy. The problem is configured as a constrained optimisation problem⁽ Reference Maqsood and Hiong Go ^{42 )}, with an objective to determine an open loop control that optimises the specified performance index, subject to certain constraints⁽ Reference Maqsood and Hiong Go ^{42 )}. For optimisation, Matlab^® non-linear constrained optimisation technique, based on Sequential Quadratic Programming (SQP) and quasi-Newton methods, is utilised. Steady states for the optimised trajectory were obtained for a co-ordinated turn flight at different turn rates as shown in the following equation:

(…14) $${\rm Turn}\,{\rm rate}\,({\rm \dot{\rpsi }}){\equals}[{\minus}10,{\minus}7,{\minus}5,{\minus}3,{\minus}2,1,0,1,2,3,4,5,10]^{T\circ} \,/\,{\rm s}.$$

In order to generate the optimised trajectories at the desired turn rates (as specified in Equation (14), the performance measure that is minimised is defined in the following equation:

(…15) $$J_{{min}} {\equals}w_{1} \dot{V}_{T} {\plus}w_{2} \dot{\ralpha }{\plus}w_{3} \dot{\rbeta }{\plus}w_{4} \dot{p}{\plus}w_{5} \dot{q}{\plus}w_{6} \dot{r}$$

where $$w_{1} ...w_{6} {\equals}1$$ and $${\dot{\ralpha }}$$ , $${\dot{\rbeta }}$$ , $$\dot{p}$$ , $$\dot{q}$$ , $$\dot{r}$$ are the rate derivatives of velocity, angle-of-attack, side slip angle and roll, pitch and yaw rates, respectively. This cost function is minimised by the optimisation algorithm at each equilibria point of the pre-specified trajectory governed by the differential equations specified in Equations (1)–(4).

The state and control variables utilised during the optimisation process are defined in the following equations:

(…16) $$\matrix{ {{\mib{x}} {\equals}} \hfill & {[V_{T} ,\ralpha ,\rbeta ,\rphi ,\rtheta ,\rpsi ,p,q,r,P_{n} ,P_{e} ,h]^{T} ,\quad {\bf x}\in{\Bbb R}^{{12}} } \hfill \cr {{\mib{u}} {\equals}} \hfill & {[LCF,RCF]^{T} ,\quad {\bf u}\in{\Bbb R}^{2} } \hfill \cr } $$

where left control fin (LCF) and right control fin (RCF) are the control variables.

Path constraints along with the bounds on the state and control variables are defined in the following equation:

(…17) $$\matrix{ {0\leq h\leq 35000ft,{\minus}3^{\circ} \,\lt\,\ralpha \,\lt\,6^{\circ} ,{\minus}6^{\circ} \,\lt\,\rbeta \,\lt\,6^{\circ} } \cr {Mach\leq 0.75,100km\leq Range\leq 120km.} \cr } $$

Terminal constraints are defined in the following equations:

(…18) $$\matrix{ {} \hfill & {P_{n} (te){\minus}P_{{n_{{(te)}} }} \leq \Delta P_{n} ,} \hfill \cr {} \hfill & {P_{e} (te){\minus}P_{{e_{{(te)}} }} \leq \Delta P_{e} ,} \hfill \cr {} \hfill & {h(te){\minus}h_{{te}} \leq \Delta h,} \hfill \cr {} \hfill & {} \hfill \cr } $$

where $$P_{n} (te),P_{e} (te),h(te)$$ are the SAK co-ordinates at the terminal point, $$P_{{n_{{(te)}} }} ,P_{{e_{{(te)}} }} ,h_{{te}} $$ are the target co-ordinates and $$\Delta P_{n} ,\Delta P_{e} ,\Delta h$$ are the permissible tolerances.

According to the model assumptions, the orientation of SAK at any point 'b' can be described in terms of earlier point 'a', by the following equation:

(…19) $$\matrix{ {{\mib{x}} (t_{a} ){\equals}{\b{x}} (t_{a} ){\plus}{\int}_{t_{a} }^{t_{b} } {\b{\dot{x}}} (t)dt,} \hfill \cr {} \hfill \cr } $$

where $${\mib{x}} (t_{b} )$$ and $${\mib{x}} (t_{a} )$$ represents the state variables at time t _b and t _a , respectively.

An LQR control-based optimisation framework was, therefore, formulated which utilises a set of dynamic constraints represented by Equations (1)–(4), path constraints shown in Equations (17) and terminal constraints of Equations (18), while minimising the performance measure represented by Equation (15). As a result of the optimisation process, steady-state values for the state and control variables along various optimal trajectories (governed by Equation (14)) were obtained. The optimisation process utilised the optimal flight profile parameters (velocity, dynamic pressure and Mach number) evaluated in Section. 2 2.4.

During optimisation process, certain states and variables were kept fixed along the trim points of the optimal ballistic trajectories while other states and control variables were kept free for optimisation (Equation (20)) within the permissible ranges:

(…20) $$\matrix{ {{\rm Fixed\_states}} {{\equals}[V_{T} ,h,\dot{\rphi },{\rm \dot{\rtheta },\dot{\rpsi }}]} \hfill \cr {{\rm Free\_states\,\, \hbox {\&}\;Controls}} \hfill {{\equals}[{\rm \ralpha ,\rbeta ,\rgamma },\rphi ,{\rm \rtheta },p,q,r,LCF,RCF]} \hfill \cr {} \hfill & {} \hfill \cr } $$

The numeric values of the V _T and h utilised along different trim points of the trajectory are specified in Table 2. Also numeric values of $$\dot{\rphi }{\equals}0,{\rm \dot{\rtheta }}{\equals}0\,\&\,{\rm \dot{\rpsi }}{\equals}0$$ are used during optimisation. The optimal trim state value for $${\rm \rgamma },{\rm \ralpha },{\rm \rbeta },\rphi ,{\rm \rtheta },p,q,r,LCF,RCF$$ within the permissible ranges was numerically ascertained along different trim points by the optimisation algorithm.

As a result of the optimisation, optimal trim values of the state and control variables were ascertained along trim points for the entire flight envelope (35,000-ground level). This included determining optimal trajectories for different turn rates governed by Equation (14). The trim values of the state and control variables along different trim points for one such optimal trajectory $$({\rm \dot{\rpsi }}{\equals}0)$$ are depicted in Table.3.

Table 3 Optimal state and control variables along equilibria points of the optimal trajectory ( $$\dot{\psi }{\equals}0$$ )

Figure 3 graphically depicts the values of steady-state control needed to guide the vehicle towards an intended target at 100 km range from the launching platform. System state response is also depicted in the same figure for the optimal $$\dot{\rpsi }{\equals}0$$ trajectory.

Figure 3 Optimal trajectories for the state and control variable for $$\dot{\psi }{\equals}0$$ trajectory.

3.1 Stability analysis

In this subsection, stability analysis of the SAK is performed utilising linear analysis tools. The stability aspects are determined for the system linearised about the equilibria point ascertained in Section 2.5. Eigen-value analysis revealed the system as unstable, as it contained a number of eigenvalues having strictly positive real part(s). The eigen-values are shown in Fig. 4.

Figure 4 Linearised system eigenvalues.

Due to this intrinsic instability, the non-linear system (governed by Equations (1)–(4)), when simulated for launch from an altitude of 35,000 ft, showed a complete unstable response (see Fig. 5). The SAK instead of following the optimised trajectory (Fig. 5(a)), exhibited an unstable response which resulted in the ballistic range of just 15 km (see Fig. 5(b)). This instability⁽ Reference Christophersen, Pickell, Neidhoefer, Koller, Kannan and Johnson ^{43 )} caused the SAK to move in an uncontrolled manner immediately after launch. Resultantly, the SAK instead of following a gradual flight path angle showed a steep vertical descent.

Figure 5 Ballistic munition range.

3.2 Controllability analysis

Having shown in previous subsection that the system is intrinsically unstable, in this section, we determine the controllability aspects of the SAK. This is to ascertain that through suitable control, whether the unstable system can be made linearly/non-linearly controllable. The non-linear dynamic system governed by Equations (1)–(4) can be represented in the general form shown in the following:

(…21) $${{\dot{\mib x}}} {\equals}{\bf F}(x(t),u(t))$$

A sufficient condition for the local controllability of the system (21) at $${\mib{x}} _{0} $$ is that the linearised system about $${\mib{x}} _{0} $$ as defined in the following equation to be controllable:

(…22) $${\mib{\dot{x}}} {\equals}{\bf Ax}{\plus}{\bf Bu}.$$

By setting $${\mib{A}} {\equals}\left[ {{{\partial f({\mib{x}} ,{\mib{u}} )} \over {\partial {\mib{x}} }}} \right]_{{({\mib{x}} _{0} ,{\mib{u}} _{0} )}} $$ ( $${\mib{A}} $$ $$\in{\Bbb R}^{{n{\times}n}} $$ ) and $${\mib{B}} {\equals}\left[ {{{\partial {\mib{f}} ({\mib{x}} ,{\mib{u}} )} \over {\partial {\mib{u}} }}} \right]_{{({\mib{x}} _{0} ,{\mib{u}} _{0} )}} $$ ( $${\mib{B}} $$ $$\in{\Bbb R}^{{n{\times}m}} $$ ), a controllability matrix $${\mib{C}} $$ can be constructed as in Equation (23) to determine local controllability:

(…23) $${\mib{C}} {\equals}\left[ {\matrix{ {{\mib{B}} ,{\mib{AB}} ,...,{\mib{A}} ^{n} {\mib{B}} } \cr } } \right],\quad {\mib{C}} \in{\Bbb R}^{{n{\times}nm}} .$$

A sufficient condition for the controllability of the linearised system is that the $${\Bbb R}^{{n{\times}nm}} $$ controllability matrix defined by Equation (23) has full row rank (i.e. $${\rm rank}(C){\equals}n$$ ).

In this study, the non-linear system, governed by Equations (1)–(4) was linearised about the equilibria (obtained in Section 2.5). To perform model linearisation, mathematical framework was developed in Matlab Simulink environment. In this model, the non-linear dynamics governed by Equations (1)–(4) along with all dependencies were incorporated. Resultantly, a linearised state space representation of the form specified in Equation (22) was ascertained about the equilibrium point.

The linearised system dynamics was concluded to be controllable as it had full rank Kalman matrix. This satisfied a sufficient condition for concluding the controllability of the original non-linear system, about the equilibria. This guaranteed that through a suitable control architecture, the unstable states of the system can be forced to follow the reference trajectory.

4.1 Stabilisation controller

In this section, control law synthesis⁽ Reference Dadkhah and Mettler ^{49 –} Reference Sujit, Saripalli and Sousa ^{52 )} is performed for the SAK to meet the stabilisation issues after launch from the aerial platform (as discussed in Section 3.1). To achieve optimality with guaranteed degree of robustness, LQR⁽ Reference Hespanha ^{48 )}-based control architecture is designed for stability purpose. The controller utilises the state information for providing the desired control. To simply the architecture, system non-effective states were excluded from the control loop. States mentioned in Equation (24) which includes (a) navigation state variables (x,y,h) and (b) heading state ( $${\rm \rpsi }$$ ) were neglected in the control design. This was because the system dynamics was independent of the variations in parameters of these states:

(…24) $$\matrix{ {{\mib{x}} _{{{\rm not\hbox{-}included}}} {\equals}[x,y,h,{\rm \rpsi }]^{T} .} \hfill \cr {} \hfill \cr } $$

The states utilised in the control architecture is represented in the following equation:

(…25) $$\matrix{ {{\mib{x}} _{{{\rm utilized}}} {\equals}[V_{T} ,{\rm \ralpha ,\rbeta },\rphi ,{\rm \rtheta },p,q,r]^{T} .\quad {\mib{x}} _{{red}} \in{\Bbb {\rm R}}^{8} } \hfill \cr {} \hfill \cr } $$

The longitudinal dynamics of the reduced system is elaborated in the following equation:

(…26) $$\matrix{ {{\mib{x}} _{{{\rm utilized}_{{{\rm (long)}}} }} {\equals}[V_{T} ,{\rm \ralpha },\rtheta ,q]^{T} ,\quad {\mib{x}} _{{{\rm red}_{{{\rm (long)}}} }} \in{\Bbb {\rm R}}^{4} } \hfill \cr {} \hfill \cr } $$

whereas the lateral dynamics are defined in the following equation:

(…27) $$\matrix{ {{\mib{x}} _{{{\rm utilized}_{{{\rm (lat)}}} }} {\equals}[{\rm \rbeta },\rphi ,p,r]^{T} .\quad {\mib{x}} _{{{\rm red}_{{{\rm (lat)}}} }} \in{\Bbb {\rm R}}^{4} } \hfill \cr {} \hfill \cr } $$

The optimised values of the state (Q) and controller weighting matrices are shown below in the following equation:

(…28) $$\matrix{ {} \hfill & {{\mib{Q}} {\equals}{\rm diag}[10,13,5,30,56,70,13,40]^{T} } \hfill \cr {} \hfill & {{\mib{R}} {\equals}{\rm diag}[10,4]^{T} } \hfill \cr } $$

Eigenvalue analysis of the linearised system with LQR controller was then performed to determine close loop stability. The system was identified to be stable as all the eigenvalues along the trim points of the optimised trajectory have strictly negative real parts. Eigenvalues analysis of the linearised system is depicted in Fig. 6.

Figure 6 Eigenvalues of closed loop stable system.

In order to validate the viability and effectiveness of the proposed control architecture, a six DOF Simulink model incorporating vehicle, actuator and controller dynamics was developed in Matlab environment. Actuator dynamics was ascertained from the desired performance specification, as specified in Table 4. Stated parameters such as settling time, damping ratio and percentage overshoot, governs the dynamics of the actuator.

Table 4 Actuator performance parameters.

Based on these specifications, the second-order approximation of the actuator dynamics was made. This is elaborated in the following equation:

(…29) $$\matrix{ {{\rm Actuator}\,{\rm dynamics}{\equals}G_{{DC}} \cdot {{w_{n}^{2} } \over {s^{2} {\plus}2{\rm \zeta }w_{n} s{\plus}w_{n}^{2} }},} \hfill \cr {} \hfill \cr } $$

where $$G_{{DC}} $$ is the DC gain , $$w_{n} {\equals}\sqrt {{k \over m}} $$ is the natural frequency and $${\rm \rzeta }$$ is the damping ratio.

Simulation results (refer Fig. 7) showed a stable system response with the vehicle glide range extended to about 100 km. In comparison with the SAK open loop range of about 15 km, this was considerable improvement.

Figure 7 Close loop response with stabilisation control.

This considerable range enhancement was attained as the SAK followed the optimised trajectory under the control of the LQR based controller architecture. During simulation, system states exhibited a stable response. This concluded the effectiveness of the designed control architecture. However, it remains to highlight that actual implementation with navigation might result in range deterioration.

4.2 Tracking error

In the designed state feed-back LQR-based control architecture, problem of deviation from the desired course was encountered. In spite of having stable system response with theoretical range enhancement to about 100 km, the SAK has a considerable off shoot from the target location. This is shown in Fig. 8. The deviation from desired course was encountered as the designed control law did not utilise the directional information ( $${\rm \rpsi })$$ provided by navigational equation (see Equation (4)). Both the longitudinal and lateral dynamics were independent from directional orientation provided by navigational equation (see Equation (4)). Accurate guidance to the target, therefore, requires incorporation of an additional control loop which minimises the drift in the heading.

Figure 8 SAK response with stabilisation controller: range enhanced to 100 km but tracking problem encountered.

4.3 Implementation of tracking controller

Guidance is the very important component to accurately commanding a guided bomb to a target. In order to guide the SAK towards the target location, a tracking controller was added in the control architecture. Utilising the information of the present heading and the desired heading, estimates for the drift are made. Steering commands are then generated to precisely guide the SAK towards the target location. With the addition of the guidance controller⁽ Reference Jamilnia and Naghash ^{53 –} Reference Asadi, Sabzehparvar, Atkins and Talebi ^{55 )} in the GCM module, the framework for the control architecture was completed. Control law⁽ Reference Fresconi, Cooper and Costello ^{56 )} based on PD controller for tracking is specified in the following equation:

(…30) $$\matrix{ {{\rm Control}\,{\rm law}{\equals}A{\rm \rpsi }_{{{\rm error}}} {\plus}B{\rm \dot{\rpsi }}_{{{\rm error}}} } \hfill \cr {} \hfill \cr } $$

where $${\rm \rpsi }_{{{\rm error}}} $$ = $${\rm \rpsi }_{{{\rm actual}}} $$ − $${\rm \rpsi }_{{{\rm error}}} $$ . Non-linear simulations carried out to evaluate the closed loop system response having both stability and guidance controllers exhibited a stable system response. Besides range enhanced to 100 km, the vehicle was accurately guided to the desired target location, with reduced CEP. This is shown in Fig. 9.

Figure 9 SAK trajectory with stabilisation and guidance control.

State and control variable response of the closed loop system is depicted in Fig. 10. It is clear from the figure that all the system states follow a trajectory to ensure that SAK after launch from the aircraft precisely hits the target within a range of about 100 km.

Figure 10 Close loop response with stabilisation and guidance controllers.