Scalar H_{∞} autopilot synthesis for control systems and evaluation via HIL simulation
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Abstract
The improvement in calculation capabilities conciliate the design and performing of advanced robust control. Among the critical applications are the antitank missiles systems. The present work is concerned with enhancing the performance of an antitank guided missile system via robust synthesis of autopilot and guidance systems. This paper is concerned to the derivation of the missile airframe transfer functions. The obtained transfer functions are expanded with formulation for the system uncertainty to be considered during the robust design. The next objective for this paper is to design an autopilot using robust technique with complete justification against related work putting into consideration the overall requirements of flight tactics characteristics. The proposed design is implemented within the simulation model. The obtained results explain the capability of the designed controller to maintain the stability of the system in the presence of unmodeled dynamics. These autopilots proved its robustness to thrust uncertainties within 30% degradation, and about ± 30% of nominal aerodynamic coefficients. It is limited to wind speed of about 20 m/s in both directions. It proved its capability of less sensitivity to measurement noise and reject disturbance of 50% within 0.09 s and 95% within 0.22 s. Then, the actuator hardware is implemented within the simulation via interfacing cards using the designed robust autopilot in presence of the prescribed uncertainty. The results show the accuracy via using the hardwareinthe loop simulation. The obtained results revealing that the developed autopilot has the capability to stabilize the system in presence of noise and disturbance.
Keywords
Command guidance systems CLOS Robust control H_{∞} UncertaintiesList of symbols
Symbols
 X_{1}, Y_{1}, and Z_{1}
Vectors components along the board reference axes
 X_{g}, Yg, and Z_{g}
Vectors components along the ground reference axes
 X, Y, and Z
Vectors components along the velocity reference axes
 δ_{jp} and δ_{jy}
Thrust jetivator angles in pitch and yaw planes
 \(F_{{TX_{1} }} , F_{TY1} {\text{ and }}F_{{TZ_{1} }} \,\)
Thrust forces along the board reference axes
 F_{AX}, F_{AY} and F_{AZ}
Drag, lateral, and lift forces along the velocity axes
 S
Characteristic area
 q
Dynamic pressure given by q = 0.5ρ (Vm)^{2} (kg/m/s^{2})
 ρ
Air density (kg/m^{3})
 V_{M}
Missile velocity
 C_{x}, C_{y}, and C_{z}
Dimensionless aerodynamic coefficients
 \(\overline{g}\)
Vector of gravity acceleration
 M
Mach number and given by M = V_{m}/V_{a}
 V_{a}
Sound velocity at missile position
 l_{T}
Perpendicular distance between the missile C.G. and the point of lateral thrust forces action
 l_{TX}
Perpendicular distance between longitudinal axis and thrust force line
 l_{x}, l_{y}, l_{z}
Characteristic linear dimensions of missile
 \(m_{{x_{1} }}\), \(m_{{y_{1} }}\) and \(m_{{z_{1} }}\)
Dimensionless aerodynamic coefficients
 \(\omega_{{x_{1} }} ,\omega_{{y_{1} }} and\omega_{{z_{1} }}\)
Airframeturn rates along board coordinate axes
 \(\overline{J}\)
Acceleration of missile
 Ω
Angular velocity of VCS w.r.t GCS
 I_{XX}, I_{YY}, and I_{ZZ}
Moments of inertia components along the BCS
 α
Angle of attack [angle of incidence] (°)
 β
Sideslip angle [angle of drift] (°)
 U, V, and W
Velocities Along board coordinate axis
 U_{d}, V_{d}, and W_{d}
Derivative of velocities along board coordinate axis
 g_{x}, g_{y}, and g_{z}
Gravity acceleration along board coordinate axis
Abbreviations
 CLOS
Commanded to line of sight
 AP
Autopilot
 ATGM
Antitank guided missile
 6DOF
Six degrees of freedom
 BCS
Board coordinate system
 LFT
Linear fractional transformation
 BTT
Bank to turn
 STT
Skid to turn
 C.G.
Centre of gravity
 VCS
Velocity coordinate system
 BCS
Body coordinate system
1 Introduction
The everincreasing development of tanks capabilities necessitates the design of accurate control and guidance system for an antitank missile in presence of unmodelled dynamics, fin disturbances and circumstances noise. To achieve this objective, (Ouda, shady, Eslam, El Banna and El gabri) extracted a nonlinear mathematical model representing the dynamical behavior of the underlying missile for different flight phases with uncertainty quantification [1, 2, 3, 4, 5]. The system uncertainties included thrust variation due to different causes, variation in aerodynamic coefficients and parameters, wind velocity in different directions and different trim conditions. To overcome different sources of uncertainty, robust control is used to design the autopilot such that the system is stable with the ability to overcome unmodelled dynamics, to reject the disturbances and minimize the effects of measurement noises overall the missile flight envelope as discussed by (Ouda, hassan, Islam and Abdallah) [6, 7, 8, 9]. The performance specifications include overshoot, speed of response, steady state error, and system stability in addition to flight paths with different engagement scenarios. To overcome the effects due to uncertainties and achieve the performance requirements this paper is devoted to design a robust guidance and control for the underlying missile system using the H_{∞} with evaluation. Amr and El Sheikh verified that this control system is said to be robust when it maintains a satisfactory level of stability and performance over a range of plant parameters, suddenly disturbances and circumstances noises [10, 11]. Thus, the objective is to investigate the robustness of the designed autopilot against uncertainties due to different sources. On the other hand, the recent advances in robust control theory discussed by (Zhu, Gabrel and Rastegar) offer all requirements of next generation missiles [12, 13, 14]. There is different goals obtained from using robust control approach such as better flexibility in the choice of airframe geometry, more tolerance to uncertainty in the underlying system autopilot design. The designed controller is implemented within the missile control system and should be insensitive to model uncertainties and be able to suppress disturbances and noise over the whole envelope of operation to prove its robustness. This paper is devoted to the autopilot design including the jetivator control using the H_{∞} in state space form and its implementation for guidance and control performance analysis.
Several researches for missile autopilot design and guidance computer implementation was reviewed in this section. Yang and Li developed a control method using decoupling technique based on robust state feedback control scheme and also, developed a disturbance observer for the (BTT) missile system [15]. The (BTT) missile dynamic systems categorized into three subchannels (roll, yaw, and pitch). Then disturbance observer was developed for each channel in order to estimate both the external disturbances and the nonlinear couplings. The robust state feedback controller was introduced to stabilize linear parts of each channel. Mattei investigated a nonlinear robust controller for a very highly maneuverable missile in the presence of a wide range of uncertainties in the circumstances [16]. The designed backstepping robust controller was applied to the MIMO model in order to achieve both (BTT) and (STT) maneuvers. Liu emphasized the problem related to composite antidisturbance autopilot design for missile systems with multiple disturbances [17]. The first part is developing an observer in order to estimate the disturbances. The finite time integral sliding mode control method was investigated to develop a controller in a feedback path. Lee and Singh developed an adaptive longitudinal autopilot to control the angle of attack of the missile in the presence external disturbance [18].
The main contribution of this work is the derivation of the missile airframe transfer functions representing the dynamics of the underlying system in pitch plane. The obtained transfer functions are expanded with a mathematical formulation for the system uncertainty to be considered during the robust (H_{∞}) design. Furthermore, the autopilot design using H_{∞} technique putting into consideration the requirements of flight path characteristics. The designed autopilot is implemented within the simulation from which the obtained results revealing that the designed autopilot capable to stabilize the system with the validation via HIL experiment and in presence of noise and disturbance.
This paper is organized in seven sections ended by section seven that presents a conclusion of the paper. Section 1 presents the introduction of underlying missile and the objective for using robust (H_{∞}) controller. Section 2 is devoted to the types of uncertainties, controller performance evaluation and used model order reduction algorithms. Section 3 presents the problem of underlying missile and airframe dynamics with uncertainty modelling. Section 4 is devoted to nominal pitch channel modeling, autopilot design using H_{∞} loop shaping and autopilot robustness evaluation. Section 5 is devoted to flight performance evaluation concerning thrust variation with measurement noise, effect of yaw separation angle (ψ_{s}) and effect of target motion. Section 6 is devoted to flight performance evaluation concerning HIL experiment.
2 Robust control
One way to describe the performance of a control system can be considered as the size of interested signals. For example, the performance of a tracking system can be considered as the size of the error signal. There are several ways of defining a signal’s size (i.e. several norms for signals), among these norms is the ∞Norm. The ∞Norm of a signal u(t) is the least upper bound of its absolute value \(\left\ u \right\_{\infty } : = \mathop {\sup }\limits_{t} \left {u(t)} \right\).There are several robust techniques, among them is the H_{∞} where a quantitative measure for the size of the system uncertainty is considered. Da, Ouda, Bahaa and Selim illustrated that the infinity norm of the transfer function relating the input to the output is the worstcase gain between the two, where both the input and output are measured either by their energy or peak value [21, 22, 23, 24]. Other measures of gain can also characterize worst–case amplifications, but in ways which seem to be less useful in practice.
The set of all stable transfer functions whose infinity norms are finite forms a Hardy space as illustrated by Bryant and Nandakumar and denoted by \(\left\ H \right\_{\infty }\) [25, 26]. Moreover, it is the approach which gives much of recent robust control theory its name. The theory is of great interest because it gives solutions to realistic robust control problems known as \(\left\ H \right\_{\infty }\) optimization problems. One would expect it to be harder than LQG theory, because min–max optimization problems are usually harder than quadratic ones, but in fact recent developments have shown the theory to have remarkable similarities with the LQG theory, and LQG problems can even be seen as special cases of \(\left\ H \right\_{\infty }\) problems. In addition to the theoretical advances, one should add that a major reason why this theory is of practical interest is the availability of low–cost interactive software, like MATLAB, which makes it possible to perform all the necessary computations quickly and easily.
2.1 Types of uncertainties
Unstructured uncertainties assume less knowledge of the system and it is only assumed that the frequency response of the system lies between two bounds. This type of uncertainty comprises two types: additive and multiplicative uncertainty. Suppose that the actual system (P) is modelled by \(\tilde{P}\), then the additive perturbation represents the absolute error in the model, whereas the multiplicative perturbation represents the relative error. The multiplicative perturbation is more frequently used because it is easier to manipulate.

Additive errors
The additive uncertainties for the missile trajectory system can be considered as the unmodelled dynamics which is the uncertainties in determining some values such as the weight of missile which is vary during flight because of fuel consumption.

Multiplicative errors
Where W_{m} is the weight defined as a fixed stable transfer function and Δ_{m} is a variable stable transfer function satisfying \(\left\ {\Delta_{m} } \right\_{\infty } < 1\).
The multiplicative uncertainties for the missile trajectory system can be considered as the applied noise to measuring devices (sensors) such as gyros and accelerometers or the applied disturbances to mechanical actuators (fins).
Considering the realistic frequency range to analyze the system, the frequency response of the extracted pitch airframe transfer function (θ/δ_{jp}) at different operating conditions is clarifies that the max gain variation is 52.8 dB at low frequency and 26.1 dB at high frequency. In addition, the maximum phase variation is 12.7° at low frequency and 80° at high frequency. Also, the frequency response of the yaw airframe jetivator is examined for the ten operating points, where the maximum gain variation is 52.2 dB and the maximum phase variation is 359.8°.
2.2 H _{∞} control theory
Given that γ > γ_{min}. This problem can be solved efficiently using the algorithm of Doyle et al. (1989), by reducing γ iteratively to yield the optimal solution [19].
2.3 Controller performance evaluation
Considering the feedback control system shown in Fig. 1, the stability margins and performance of such systems can be quantified using the singular values of the closedloop transfer function matrices from r to each of the three outputs e, u, and y as defined in Eq. 1 and 5.
The singular value bode plot of each of the three transfer functions represent sensitivity, control sensitivity, and complementary sensitivity has an important influence in the design.
The singular value Bode plot of R(s) and of T(s) are used to quantify the stability margins of feedback designs in the face of additive plant perturbations Δ_{A} and multiplicative plant perturbations Δ_{M}, respectively.
The smaller is \(\bar{\sigma }\{ T(j\omega )\}\), the larger will be the size of the smallest destabilizing multiplicative perturbation and consequently the larger will be the stability margin.

Good command following ⇐ L large.

Good disturbance rejection ⇐ L large

Good noise attenuation ⇐ L small.

Small input u ⇐ C, L small.

Nominal stability (stable plant) ⇐ L small

Robust stability (stable plant) ⇐ L small
Fortunately, these objectives are in different frequency ranges and consequently most of them can be met with a large gain (L > 1) for low frequencies (below crossover), and a small gain (L < 1) at high frequencies (above crossover). An alternative approach to the loop shaping is the H_{∞} mixedsensitivity loop shaping, where the performance and stability specifications, Eqs. (8, 9) are combined into a single infinity norm specification of the form \(\left\ {T_{y1u1} } \right\ \le 1\) in which \(T_{y1u1} = \left[ {\begin{array}{*{20}c} {W_{p} S} \\ {W_{t} T} \\ \end{array} } \right]\).
The term \(\left\ {T_{y1u1} } \right\_{\infty }\) is called a mixedsensitivity cost function because it penalizes both sensitivity S(s) and the complementary sensitivity T(s). Loop shaping is achieved via choosing W_{p} to have the target loop shape for frequencies ω < ω_{c} and choosing 1/W_{t} to be the target for ω > ω_{c}. In choosing design specifications W_{p} and W_{t} for a controller design, it is required to ensure that the 0 dB crossover frequency for the Bode plot of W_{p} is below the 0 dB crossover frequency of 1/W_{t} as shown in Fig. 7. This is to allow a gap for the desired loop shape Gd to pass between the performance bound W_{p} and the robustness bound 1/W_{t}; otherwise the performance and robustness requirements will not be achievable.
In addition, the following tradeoffs should be exercised: good command following and disturbance rejection necessitate L to be large while good noise attenuation and robust stability necessitates L to be small. That is, compromise between conflicting requirements should be experienced.
2.4 Model order reduction
The robust controller’s design for complex systems necessitates model reduction to simplify the obtained controller to a reasonable order. The order reduction can be carried to the original system, to the obtained controller or to the system as a whole.
In this work a model reduction algorithm was applied to the control designed controller to reduce its complexity.
Glover proofed that eigenvalues define system stability whereas Hankel singular values define the energy of each state in the system [30].
 1.
It is desirable to simplify the available model for the purpose of reduced order controller design.
 2.
In using certain design methods (including the H_{∞} method), fictitious unobservable/uncontrollable states are generated by the algorithms which must be stripped away by a reliable model reduction algorithm.
 3.
Finally, if a modern control method such as H_{∞} is employed for which the complexity of the control law is not explicitly constrained, the order of the resultant control law is likely to be considerably greater than is truly needed. Consequently, a good model reduction algorithm is applied to the control law to reduce its complexity with little change in control system performance.
Hankel singular values

Additive error method in which the reducedorder model Gred has an additive error bounded by an error criterion \(\left\ {G  G_{red} } \right\_{\infty }^{{}}\).

Multiplicative error method where the reducedorder model has a multiplicative or relative error bounded by an error criterion \(\left\ {G^{  1} (G  G_{red} )} \right\_{\infty }\).
The error is measured in terms of peak gain across frequency (H_{∞} norm), where the error bounds are functions of the neglected Hankel singular values.
Additive model reduction

Squareroot balanced model truncation

Schur balanced model truncation

Hankel minimum degree approximation
Multiplicative Model Reduction
 1.Find the controllability grammian P and observability grammian Q of the left spectral factor \(\varPhi = \varGamma (\sigma )\varGamma * (  \sigma ) = \varOmega * (  \sigma )\varOmega (\sigma )\) by solving the following Lyapunov and Riccati equations;$$\begin{aligned} & AP + PA^{T} + BB^{T} = 0 \\ & B_{W} = PC^{T} + BD^{T} \\ & QA + A^{T} Q + (QB_{W}  C^{T} )(  DD^{T} )(QB_{W}  C^{T} )^{T} = 0 \\ \end{aligned}$$
 2.Find the Schur decomposition for PQ in both ascending and descending order, respectively;$$V_{A}^{T} PQV_{A} = \left[ {\begin{array}{*{20}c} {\lambda_{1} } & \ldots & \ldots \\ 0 & \ldots & \ldots \\ 0 & 0 & {\lambda_{n} } \\ \end{array} } \right]\quad V_{D}^{T} PQV_{D} = \left[ {\begin{array}{*{20}c} {\lambda_{n} } & \ldots & \ldots \\ 0 & \ldots & \ldots \\ 0 & 0 & {\lambda_{1} } \\ \end{array} } \right]$$
 3.Find the left/right orthonormal Eigenbases of PQ associated with the kth big Hankel singular values of the allpass phase matrix \(\{ W^{ * } (S)\}^{  1} G(S)\)$$\begin{aligned} V_{A} = [V_{R,SMALL} ,\overbrace {{V_{L,BIG} }}^{K}] \hfill \\ V_{D} = [\overbrace {{V_{R,BIG} }}^{K},V_{L,SMALL} ] \hfill \\ \end{aligned}$$
 4.
Find the SVD of \((V_{L,BIG}^{T} \, V_{R,BIG} ) = U\sum {\zeta T}\)
 5.Form the left/right transformation for the final kth order reduced model;$$\begin{aligned} S_{L,BIG} = V_{L,BIG} \, U\sum {(1:K,1:K)^{  1/2} } \hfill \\ S_{R,BIG} = V_{R,BIG} \, V\sum {(1:K,1:K)^{  1/2} } \hfill \\ \end{aligned}$$
 6.Finally, the reduced order model is obtained as:$$\left[ {\begin{array}{*{20}c} {\tilde{A}} & {\tilde{B}} \\ {\tilde{C}} & {\tilde{D}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {S_{L,BIG}^{T} \, A \, S_{R,BIG} } & {S_{L,BIG}^{T} \, B} \\ {C \, S_{R,BIG} } & D \\ \end{array} } \right]$$
3 Problem formulation
This section briefly describes the basic subsystems that form a guidance system which steers the missile towards its target according to a certain guidance law.
Missile equations of motion
Geometrical relations

VCS w.r.t. GCS by angles (γ, ϑ, χ)

VCS w.r.t. BCS by angles (α, β)

BCS w.r.t GCS by angles (ψ, θ, φ)
3.1 Pitch jetivator and airframe dynamics

pitch and/or yaw motion

piecewise constant velocity

Neglect g and consider small firing angles (α, β) and small thrust jetivator angles δ_{jp} and δ_{jy} such that \(F_{{Tx_{1} }} = F_{{T_{1} }}\), \(F_{{Ty_{1} }} =  F_{{T_{1} }} \delta_{jy}\), and \(F_{{Tz_{1} }} =  F_{{T_{1} }} \delta_{jp}\) [31, 32, 33].
Body rate transfer function
The perturbed variables are n_{α}, \(n_{{\delta_{jp} }}\), \(M_{y1\alpha }\), \(M_{{\delta_{jp} }}\) and \(M_{{y1\dot{\theta }}}\) in pitch plane and n_{β}, \(n_{{\delta_{jy} }}\), \(M_{y1\beta }\), \(M_{{\delta_{jy} }}\) and \(M_{{y1\dot{\psi }}}\) in yaw plane.
3.2 Pitch airframejetivator with uncertainty modelling
4 Robust autopilot synthesis
4.1 Nominal pitch channel modeling
 CaseA: The 6th pitch airframe transfer function is considered and thus the overall plant transfer function is$$\frac{\theta }{{\Delta_{{p_{command} }} }} = \frac{{1.024 \times 10^{  8} s^{2}  1.362 \times 10^{5} s  5.385 \times 10^{4} }}{{s^{5} + 392.6s^{4} + 4.934 \times 10^{4} s^{3} + 4.246 \times 10^{6} s^{2} + 1.959 \times 10^{6} s}}$$(34)
 CaseB: The 4th pitch airframe transfer function is considered and yields the overall plant transfer function is$$\frac{\theta }{{\Delta_{{p_{command} }} }} = \frac{{2.559 \times 10^{  9} s^{2}  1.387 \times 10^{5} s  3.342 \times 10^{4} }}{{s^{5} + 234.1s^{4} + 2.947 \times 10^{4} s^{3} + 1.731 \times 10^{6} s^{2} + 5.179 \times 10^{5} s}}$$(35)
4.2 Autopilot design using H _{∞} loop shaping
This approach is utilized for the autopilot design without uncertainty modelling i.e. there is no certain structure for the uncertainty during the design.
4.2.1 Autopilot design trials
 Trial1: Considering the nominal plant CaseA, the weights (36) and the optimal H_{∞} robust control yield the controller CA1 as:$${\text{C}}_{\text{A1}} = \frac{{2.867 \times 10^{5} {\text{s}}^{6} + 1.133 \times 10^{8} {\text{s}}^{5} + 1.444 \times 10^{10} {\text{s}}^{4} + 1.255 \times 10^{12} {\text{s}}^{3} + 3.765 \times 10^{12} {\text{s}}^{2} + 1.478 \times 10^{12} {\text{s }} + 143.3}}{{{\text{s}}^{7} + 612.4{\text{s}}^{6} + 1.553 \times 10^{5} {\text{s}}^{5} + 2.327 \times 10^{7} {\text{s}}^{4} + 1.994 \times 10^{9} {\text{s}}^{3} + 3.272 \times 10^{10} {\text{s}}^{2} + 1.311 \times 10^{10} {\text{s}} + 1.892 \times 10^{8} }}$$
 Trial2: Considering the nominal plant CaseA, the weights (37) and the optimal H_{∞} robust control yield the controller CA2 as:$${\text{C}}_{\text{A2}} = \frac{{ 8. 6 2 8\times 1 0^{ 6} {\text{s}}^{ 6} \, + 3.41 \times 10^{9} {\text{s}}^{ 5} \, + 4.346 \times 10^{11} {\text{s}}^{ 4} \, + 3.776 \times 10^{13} {\text{s}}^{ 3} \, + { 1} . 1 3 3\times 1 0^{ 1 4} {\text{s}}^{ 2} + 4.448 \times 10^{13} {\text{s }} + { 2189}}}{{{\text{s}}^{ 7} \, + {\text{ 1063 s}}^{ 6} + { 5} . 1 7 8\times 1 0^{ 5} {\text{s}}^{ 5} + { 1} . 4 9 4\times 1 0^{ 8} {\text{s}}^{ 4} \, + { 2} . 5 4 4\times 1 0^{ 1 0} {\text{s}}^{ 3} \, + { 6} . 9 2 5\times 1 0^{ 1 1} {\text{s}}^{ 2} + { 2} . 8 0 1\times 1 0^{ 1 1} {\text{s }} + { 4} . 0 4 5\times 1 0^{ 9} }}$$
 Trial3: Considering the nominal plant CaseA, the weights (38) and the optimal H_{∞} robust control yield the controller CA3 as:$${\text{C}}_{\text{A3}} = \frac{{1.237 \times 10^{7} {\text{s}}^{6} + 4.888 \times 10^{9} {\text{s}}^{5} + 6.23 \times 10^{11} {\text{s}}^{4} + 5.412 \times 10^{13} {\text{s}}^{3} + 1.624 \times 10^{14} {\text{s}}^{2} + 6.376 \times 10^{13} {\text{s}} + 5103}}{{{\text{s}}^{7} + 1200{\text{s}}^{6} + 6.681 \times 10^{5} {\text{s}}^{5} + 2.211 \times 10^{8} {\text{s}}^{4} + 4.314 \times 10^{10} {\text{s}}^{3} + 1.932 \times 10^{12} {\text{s}}^{2} + 7.86 \times 10^{11} {\text{s}} + 1.136 \times 10^{10} }}$$
 Trial4: Considering the nominal plant CaseB, the weights (36) and the optimal H_{∞} robust control yield the controller CB1 as:$$C_{B1} = \frac{{ 2. 4 0 3\times 1 0^{ 5} s^{6} + 5.69 \times 10^{7} s^{5} + 7.231 \times 10^{9} s^{4} + 4.347 \times 10^{11} s^{3} + 1.219 \times 10^{12} s^{2} + 3.276 \times 10^{11} s + 31.43}}{{s^{7} + 507.7 \, s^{6} + 1. 2 7 8\times 1 0^{ 5} s^{5} + 1. 9 4\times 1 0^{ 7} s^{4} + 1. 7 0 1\times 1 0^{ 9} s^{3} + 2. 7 6 7\times 1 0^{ 1 0} s^{2} + 6.978 \times 10^{9} s + 9.842 \times 10^{7} \, }}$$
 Trial5: Considering the nominal plant CaseB, the weights (37) and the optimal H_{∞} robust control yield the controller CB2 as:$${\text{C}}_{\text{B2}} = \frac{{ 1. 9 0 6\times 1 0^{ 6} {\text{s}}^{ 6} \, + 4.513 \times 10^{8} {\text{s}}^{ 5} \, + 5.735 \times 10^{10} {\text{s}}^{ 4} \, + 3.448 \times 10^{12} {\text{s}}^{ 3} \, + { 9} . 6 7 1\times 1 0^{ 1 2} {\text{s}}^{ 2} + 2.598 \times 10^{12} {\text{s}} + { 187} . 7}}{{{\text{s}}^{ 7} \, + { 833} . 3 {\text{ s}}^{ 6} + { 2} . 8 7 4\times 1 0^{ 5} {\text{s}}^{ 5} + { 5} . 8 4\times 1 0^{ 7} {\text{s}}^{ 4} \, + { 7} . 1 5 1\times 1 0^{ 9} {\text{s}}^{ 3} \, + { 1} . 5 6 8\times 1 0^{ 1 1} {\text{s}}^{ 2} + { 3} . 9 6 9\times 1 0^{ 1 0} {\text{s }} + { 5} . 6 0 1\times 1 0^{ 8} }}$$
Obtained controllers characteristics
Trial number  Obtained controller  Rise time  Settling time  \(\left\ {\omega_{p} S} \right\_{\infty }\)  \(\left\ {\omega_{t} T} \right\_{\infty }\) 

1  C_A1  0.99  2.21  0.4635  3.0008 
2  C_A2  0.756  1.79  0.3297  0.6786 
3  C_A3  1.36  2.91  0.6438  0.8713 
4  C_B1  0.99  2.21  0.4635  3.0008 
5  C_B2  0.752  1.79  0.3330  0.6810 
The results clarify that the weights (37, 38) yield designs with \(\left\ {W_{p} S} \right\_{\infty }\) and \(\left\ {W_{t} T} \right\_{\infty }\) < 1 and the closed loop step response has settling and rise times smaller than that obtained from another controller obtained using the weights (36) which yields that \(\left\ {W_{p} S} \right\_{\infty }\) < 1 and \(\left\ {W_{t} T} \right\_{\infty }\) > 1. In addition, the angle of departure between sensitivity and complementary weighting functions in the first case with \(\left\ {W_{p} S} \right\_{\infty }\) and \(\left\ {W_{t} T} \right\_{\infty }\) < 1 is greater than those obtained in the second case where \(\left\ {W_{p} S} \right\_{\infty } < 1\) and \(\left\ {W_{t} T} \right\_{\infty } > 1\).
4.2.2 Reduced order autopilot

Trial1

Trial2,3,4,5
The obtained results clarify that the multiplicative reduction method yields a better result which is consistent with the nonreduced controller and consequently it will be used forward in this work.
4.2.3 Autopilot robustness evaluation

Unmodelled dynamics

Noise attenuation

Disturbance rejection
4.3 Autopilot synthesis with uncertainty modelling
This section is devoted to the design of a robust system for attitude/horizontal stabilization of a timevarying thrust vector control. The linearized equations of the longitudinal motion are derived with the consideration of variations in the aerodynamic coefficients as parametric uncertainties in the design such that the desired closedloop performs in the presence of uncertainty, disturbances and noises.
4.3.1 Pitch plane performance requirements
This system has a reference signal r and two weighted outputs e_{p} and e_{u} which characterize performance requirements. The transfer function W_{g} represent the free gyro dynamics that measures θ. The system M is the ideal model to be matched by the designed closed loop system. The dotted box represents the perturbed plant model \(G = F_{u} (G_{mis} ,\Delta )\), where G_{mis} is the nominal model of the missile and Δ parameterizes the model uncertainty. The matrix Δ is unknown but has a diagonal structure and is norm bounded, i.e. \(\left\ \Delta \right\_{\infty }\) < 1. For robust performance, it is required that the transfer function matrix from r to e_{p} and e_{u} should be small in the sense of \(\left\ . \right\_{\infty }\), for all possible uncertain matrices Δ. The transfer function matrices W_{p} and W_{u} are employed to represent the relative significance of performance requirements over different frequency ranges. The measured output feedback signal is y = W_{g}θ and the gyro transfer function is chosen as \(W_{g} = \frac{0.044}{0.022s + 1}\).

Redesign the robust controller using the following performance weighting functions \(W_{p} (s) = \frac{0.1s + 5}{27.2522s + 0.015}\) and \(W_{u} (s) = \frac{0.4s + 0.9}{5.9s + 5}\) at the same operating point which results in a controller of 10th order. The frequency response of the structured singular value for the case of robust stability analysis is considered, where µmax = 0.47705, from the frequency response of the nominal performance it is seen that the obtained peak value of γ is 1.001 and less than 1 in the high frequency range which shows that the nominal performance has achieved. In addition, the peak value of µ is 1.6, and less than 1 in the high frequency range which shows that the robust performance has achieved. That is, the system does preserve performance under all relative parameter changes. The transient response of the closed loop system with the designed H_{∞} controller for a step command is obtained, where the system is under damped with accepted characteristics. The singular value plot of closed loop poles reveals that all singular values < 1 which show that the H_{∞} norm of the closed loop system is less than 1, and the condition \(\left\ {W_{p} (I + GC)^{  1} } \right\_{\infty } < 1\) is satisfied in this case.
4.3.2 Yaw plane performance requirements
The performance weighting functions are \(W_{p} (s) = \frac{0.1s + 4.261}{18.5s + 0.02}\) and \(W_{u} (s) = \frac{0.4s + 0.9}{5.9s + 5}\) from which the designed controller obtained is of 10th order. For the case of robust stability analysis, the maximum value of structured singular value µmax = 0.40335, which means that the stability of the closedloop system is preserved under all perturbations that satisfy \(\left\ \Delta \right\_{\infty } < \frac{1}{0.40335}\). The nominal performance of the closed loop system transfer matrix is tested via the frequency response. The obtained peak value of γ is 1.0009 and less than 1 at high frequency range which shows that the nominal performance has been achieved. For the case of robust performance analysis, the peak value of µ is 1.5821, and less than 1 at high frequency which shows that the robust performance has been achieved. In other words, the system does preserve performance under all relative parameter changes with uncertainty range (25%). The simulation shows the transient responses of the closed loop system with the designed H_{∞} controller for step command where the system is underdamped with accepted characteristics. The singular value plot of closed loop poles is considered, which reveals that all singular values < 1 and that the H_{∞} norm of the closed loop system is less than 1. Note that, the model order reduction techniques are applied to the obtained controller using the multiplicative method and yields 4th order autopilots.
4.3.3 Autopilot robustness evaluation

Unmodeled dynamics

Noise attenuation

Disturbance rejection
The same evaluation of yaw plane clarifies that the designed robust controllers with uncertainty modeling have faster response a lowest steady state control effort than classical one and the obtained robust controller without modeling the uncertainty. In addition, applying noise clarify that designed robust controllers with uncertainty modeling is less sensitive to additive noise compared to others. Also the response results to disturbance on the jetivator output clarify its capability reject 50% within 0.09 s and 95% within 0.22 s.

Flight path evaluation
The robust autopilot proved its robustness to thrust uncertainties to about 30% degradation with little oscillation at the gathering phase compared to nominal thrust case.
Variation in aerodynamic coefficients
The results obtained with varying aerodynamics clarify that the robustness of designed autopilots without uncertainty modeling is limited to about 5% to − 20% of nominal values after which the miss distance will be large or missile ground impact occurs. While the robustness of designed autopilot, with uncertainty modeling, is limited to about ± 30% of nominal value.
Wind speed in X direction

Figures 51 and 52 clarify that all flight path trajectories have an acceptable trajectory and miss distance when the wind speed along the X1axis reaches 30 m/s.

Its clear from Fig. 53 that at maximum tactical range when the wind speed along the X1axis reaches − 5 m/s a ground impact occurred by the missile using the classical controller at 2580.7 m, C_A1 at 2704 m, CA2 at 2619.2 m, CA3 at 2401 m, CB1 At 2494 m, CB2 at 2545 m.

Its clear from Fig. 54 that at maximum tactical range when the wind speed along the X1axis reaches − 10 m/s a ground impact occurred by the missile using the controller C_A1 at 2453 m, CA2 at 2405 m, CA3 at 2261 m, CB1 at 2271 m, CB2 at 2303 m. While, a ground impact occurred by the missile using the classical controller at 734.7 m.

The results obtained with considering wind velocity along the Xaxis clarify that the robustness of designed autopilots without uncertainty modeling is limited to about − 5:30 m/s, while the robustness of designed autopilots with uncertainty modeling is limited to about − 19:30 m/s.

Its clear from Fig. 55 that at maximum tactical range when the wind speed along the Y1axis reaches ± 20 m/s a ground impact occurred by the missile using the controller CA1 at 2784.5 m, CA2 at 2753.5 m, CA3 at 2525.5 m, CB1 at 2656.5 m, CB2 at 2714.5 m.

It’s clear from Fig. 56 that at maximum tactical range when the wind speed along the Y1axis reaches ± 40 m/s a ground impact occurred by the missile using the controller CA1 at 2445 m, CA2 at 2450 m, CA3 at 2351 m, CB1 at 2337.5 m, CB2 at 2395 m. While, a ground impact occurred by the missile using the classical controller at 800 m.

The results obtained with considering wind velocity along the Yaxis clarify that the robustness of designed autopilots with uncertainty modeling is limited to about ± 30 m/s, while the designed robust autopilots without uncertainty modeling is limited to about ± 15 m/s.
The results obtained reveal that the designed robust controller with uncertainty modelling which satisfy the equality \(\left\ {W_{p} (I + GC)^{  1} } \right\_{\infty } < 1\) (suboptimal2) is the best design. This controller proves its robustness against unmodelled dynamics, stable flight path with the consideration of different sources of uncertainties (thrust degradation, aerodynamic coefficient variation and wind speed), low missdistance, low control effort, and less sensitivity to additive noise and disturbance. The remainder designed autopilots are sorted in descending order from the best performance to lowest one as: suboptimal1, CA2, CA1, CB2, CB1, CA3 and finally the classical autopilot.
5 Flight performance evaluation
The obtained suboptimal controllers in both pitch and yaw planes are evaluated via the flight path trajectory at different flight scenarios with existence of additive random noise applying on the measuring devices (gyros).
5.1 Thrust variation with measurement noise
Obtained controllers missdistances with thrust variation
Range (m)  Thrust (%)  Conventionalpitch Conventionalyaw  Optimalpitch Conventionalyaw  Optimalpitch Optimalyaw 

500  100  2.3174  3.329  2.2893 
90  2.1433  3.414  1.2436  
85  Ground impact  2.733  1.2086  
2800  90  0.9317  1.596  0.8417 
85  Ground impact  1.78  0.4254 
5.2 Effect of yaw separation angle (ψ_{s})
Obtained controllers missdistances with yaw separation angle variation
ψ_{s} (°)  Conventionalpitch Conventionalyaw  Optimalpitch Conventionalyaw  Optimalpitch Optimalyaw 

1.5°  2.0569  2.9406  1.1713 
1°  1.9040  2.7252  1.1600 
0.5°  1.9520  2.9014  1.7157 
5.3 Effect of target motion
Obtained controllers missdistances with moving targets
Range (m)  Target  Conventionalpitch Conventionalyaw  Optimalpitch Conventionalyaw  Optimalpitch Optimalyaw 

3000  Outgoing  ground impact  1.038  1.1782 
Incoming  0.4180  0.754  0.6466  
500  Outgoing  1.6183  1.3756  1.4593 
Incoming  3.6435  2.04  2.2389 
6 Flight path performance evaluation within HIL experiment
The evaluation of the conventional autopilot and designed suboptimal robust autopilot against different uncertainties in HIL environment is illustrated in Figs. 62, 63, 64 and 65.
6.1 Nominal thrust value—nominal aerodynamic coefficient—zero wind
6.2 (75%) thrust value—nominal aerodynamic coefficient—zero wind speed
6.3 Nominal thrust value—(+ 20%) aerodynamic coefficient variation—zero wind speed
6.4 Nominal thrust value—nominal aerodynamic coefficient—V_{wx} = − 30 m/s
 (1)
At nominal flight and low uncertainties level the flight path trajectory shows stable and successful engagements for both classical and robust autopilots. In addition, the control effort obtained with the classical controller at this low level of uncertainties is lower than that obtained with a robust controller.
 (2)
Increasing the margin of uncertainties (lower thrust values—change of aerodynamic coefficients—wind speed) the classical autopilot has unstable flight path trajectory and ground impact, while the robust autopilot yields successful engagements. The control effort obtained using robust autopilots has the same level as that obtained at low uncertainties level, while the classical controller has a higher control effort.
 (3)
The missdistance obtained via robust autopilot especially at low tactical target range is lower than that obtained via classical autopilot and within an acceptable margin.
7 Conclusion
This paper presented the robust control theory in the form of two approaches; the H_{∞} and suboptimal H_{∞} designs with different sensitivities and norms. In addition, it presented the model reduction techniques that can be utilized for reducing the controller order. Then, the underlying system is formulated in structures appropriate for utilizing these design techniques. The autopilots designed using the two techniques are evaluated against stability, unmodeled dynamics, disturbance rejection, noise attenuation and flight path. The obtained results clarify that the suboptimal H_{∞} controller, where the uncertainties are modeled during the design process, is more robust than the other H_{∞} technique, where the design process is carried out without modeling the uncertainties, and the classical one. These autopilots proved its robustness to thrust uncertainties within ± 30% degradation, and about ± 30% of nominal aerodynamic coefficients. In addition, it is limited to wind speed of about V_{wx} = − 19:30 m/s, V_{wy} = ±30 m/s, and V_{wz} = ±20 m/s. It proved its capability of faster response with the lowest steady state control effort, less sensitivity to measurement noise and reject disturbance of 50% within 0.09 s and 95% within 0.22 s.
The HIL experiment is described in detail. Then, the system performance is evaluated using different types of models’ structures and at different flight phases. The results reveal the robustness of the designed the optimal H_{∞} controller via fewer excursions in the flight path which leads to less flight time, possible increased range and less possibility to ground hit.
Notes
Compliance with ethical standards
Conflict of interest
The author(s) declare that they have no competing interests.
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