سید حمید جلالی نایینی

The miss distance analysis of the first-order explicit guidance law (EGL) is carried out using linearized equation of motion in the normalized form in order to obtain normalized miss distance curves. The initial heading error, constant target, acceleration limit, radome refraction error, and fifth-order binomial control system are considered. Moreover, body rate feedback is added to the explicit guidance law as a well-known classical compensation method of the radome effect as in proportional navigation. The analysis is performed for different values of the power of the alpha function, defined as the time decrease rate of the zero-effort miss to unit control input. As a special case, the EGL with unit power gives the first-order optimal guidance strategy for minimizing the integral of the square of the commanded acceleration during the total flight time. For the performance/stability analysis, the rms miss distance versus turning rate time constant and radome slope can be plotted for different values of the power of alpha function.

This paper presents a modification to a type of Pulse-width Pulse-Frequency (PWPF) Modulator utilized an integrator block. In this modulator that called here as "Integral Pulse-Width Pulse-Frequency (IPWPF)," an integrator is used instead of the first-order low-pass filter. To improve the performance of the control system, the modulator is modified by using a logical circuit in order to reset the output of the integrator. In this logical circuit, if the error signal becomes less than a specified small value, the integrator will be reset, that is, "Small Error-Reset Integrator (SE-RI)." The modification is applied to the stabilization and pointing modes. In stabilization mode, the control gain is obtained analytically such that the angular rate of the satellite becomes zero or less than a specific percentage of its initial value by a single pulse. Simulation results show that the performance of the modified IPWPF is comparable with that of PWPF in pointing mode.

In this paper, a modified proportional navigation (PN) with weighted combination of linear acceleration and line-of-sight (LOS) acceleration feedback is suggested. For this purpose, a comprehensive miss distance analysis is carried out for PN with linear acceletation feedback and PN with LOS acceleration feedback using a fifth-order binomial guidance and control system. The miss distance (MD) due to initial heading error, target acceleration, and seeker noise is separately analysed. As a special case, a modified PN with acceleration feedback using variable gains is suggested based on MD analysis for infra red seekers. The comparison of PN strategies is carried out using an equivalent effective navigation ratio, defined by using LOS rate profile solution. In addition, the first-order optimal guidance law is converted into PN with PD block with variable gains.

In the present work, the performance of True Proportional Navigation (TPN) with different profiles for effective navigation ratio in terms of the relative velocity angle with respect to line-of-sight is investigated due to initial heading errors and target maneuvers. Since an appropriate profile of effective navigation ratio for heading error is almost in contrary with the appropriate profile for maneuvering targets, a variable effective navigation ratio in terms of the relative velocity angle and its rate is introduced. In this way, the miss distance (MD) for the first two peaks of the normalized MD graph versus normalized final time, for example under an effective navigation ratio of 4, is reduced together. The analysis is carried out using normalized equations in polar coordinates for perfect and binomial fifth-order control systems without acceleration limit. Due to practical consideration, the effective navigation ratio is tuned and limited between 3 and 5.

In this paper, the effect of system time delay on the miss distance of proportional navigation guidance law is studied for high-order binomial guidance and control systems. For this purpose, a linearized model is utilized in one dimension. The normalized miss distance due to heading error (HE), step, ramp, and parabolic target maneuvers, and seeker noise is computed using normalized adjoint equations. The glint, range-independent, and (semi-) active range-dependent noises are considered for the seeker, and the results are verified by straight-forward method. Moreover, the effects of system time lag, system time constant, effective navigation ratio, system order up to 30 in proportional navigation miss distance are also investigated. An extremum effective navigation ratio is also obtained to minimize the worst case miss distance with respect to the final time. Furthermore, approximate miss distance formulas are presented based on worst case analysis on final time using curve fitting for mentioned miss distance sources. Finally, the approximate formulas of steady state nondimensional coefficients due to seeker noise in terms of effective navigation coefficient, for example for system orders of 5 and 10, are obtained.

In this study, the preferred regions of Pulse-Width Pulse-Frequency Modulator (PWPFM) are obtained analytically for the static analysis. For this purpose, a comprehensive parametric study is carried out based on the two performance indices of fuel consumption and the number of thruster firings. The preferred regions are presented by normalized relations and curves. Moreover, the exact analytical solutions of the two performance indices are obtained for a class of modulators with the assumption of constant inputs. The advantages of the present study are non dimensional analysis and obtaining the preferred regions in terms of each others, resulting in more accurate regions as opposed to inequality relations using constant values for a specified input signal. In addition, in the case of specified minimum pulse width (having the update frequency and thruster time constant), determining the preferred regions becomes more limited. In this regard, useful relations and curves based on the maximum possible value for the number of the thruster firings are derived and presented.

his paper suggests arobust optimization algorithm for the design of the satellite attitude control system in order to increase the robustness of the performance under uncertainties. A single-axis on-off attitude control with rigid dynamics is considered using Schmitt-Trigger and PID controller. The model uncertainties include the moment of inertia, thrust level, thruster delay and theexternal disturbance amplitude.A weighted combination of expected value and standard deviation of pointing error is considered as an objective function for the robust optimization. The numerical solutions show that the robust optimization reduces the variations of the objective function, i.e. it increases the robustness of the system performance compared to the deterministic optimization.

In this study, stabilization attitude control of a rigid satellite with on-off thrusters using pulse-width pulse-frequency (PWPF) modulator is investigated in presence of sensor noise. The preferred regions of the PWPF modulator parameters and stabilization control gain are obtained based on the two performance indices of the fuel consumption and the total number of thruster firings. The analyses include tumbling, detumbling, and stabilization block as an internal loop of the satellite pointing mode. The design parameters are reduced by using the quasi-normalized equations of PWPF modulator. Therefore, the preferred regions are extracted based on search method in terms of grouped parameters, regardless of the value of each parameter, separately. In quasi-normalized form, the computational burden is considerably decreased, especially in the statistical analysis in the presence of sensor noise. The parametric study is carried out with/without sensor noise. The parameters are also tuned using multi-objective optimization with genetic algorithm for stabilization mode without sensor noise. In the presence of sensor noise, the behaviors of the parameters are plotted versus the noise power spectral density. In order to better specify the preferred regions, each quasi-normalized design curve is plotted for a specified value of the input noise power spectral density. The parameters of the satellite attitude control system are suggested to be tuned/optimized within the preferred regions of the parameters in the stabilization loop as an internal loop.

In this paper¡ the preferred regions of pulse-width pulse-frequency (PWPF) modulator parameters are obtained based on zero-input¡ static¡ and dynamic analysis in the presence of sensor noise as an input noise to PWPF modulator. The design parameters are reduced to 3 by using the quasi-normalized equations of PWPF modulator. Therefore¡ the results are applicable for grouped parameters¡ regardless of the value of each parameter¡ separately. Moreover¡ the computational burden is highly decreased¡ especially in a statistical analysis. The input noise of the modulator is constructed by a low pass filter driven by a white Gaussian noise. The fuel consumption and number of thruster firings are considered as performance indices. The modulator output frequency is also limited to 50 Hz. The preferred regions of quasi-normalized system are extracted based on eliminating the upper 30% (and 10%) of the plotted graphs for the above-mentioned performance indices. Finally¡ the preferred regions can simply be viewed in our resulting curves¡ i.e.¡ normalized hysteresis plotted versus normalized PWPF on-threshold for different values of modulator time constant. Each of these curves is plotted for a specified value of input noise power spectral density.

In this paper, an approximate solution of required velocity with final position constraint is derived using a piecewise linear gravity assumption for elliptical earth model. In this approach, the total flight time is divided into several time intervals and the gravitational acceleration is assumed to be linear at each interval. The solution gives an explicit relation in terms of the current position vector, desired final position and flight time in three dimensions. The accuracy and computational burden of the method are obtained numerically in terms of the number of time intervals, and compared with linearized solution and Zarchan's iterative algorithm. Numerical solution shows that the present method has better accuracy than the two mentioned approaches with the same computational burden up to a range angle of 18 deg for minimum energy trajectory in an elliptical earth model. The presented method can be extended for two or multi-body problem and also for the computation of sensitivity matrix of required velocity.

In this paper, the performance of a single-axis attitude control with pulse-width pulse-frequency (PWPF) modulation is enhanced using a modified proportional-integral-derivative (PID) controller for a rigid satellite with on-off thruster actuators. For this purpose, the well-known observer-based PID approach is utilized. The on-off thruster actuator is modeled with a constant delay followed by a second-order binomial transfer function. The modulator update frequency is limited to 40 Hz as an input to the on-off thruster actuators. In this study, the design criteria of pointing accuracy, overshoot of the attitude response, fuel consumption, and the number of thruster firings are considered for a step external disturbance (with different values). The parameters of the observer-based PID controller are tuned using parametric search method. Simulation results show that the fuel consumption and settling time of the observer-based approach are considerably decreased with respect to those of PID controller with PWPF modulator. Moreover, the overshoot of the observer-based approach is omitted. Finally, the robustness of the observer-based modified PID controller is investigated in presence of uncertainties in satellite moment of inertia and thrust level of on-off actuators.

In this paper, the miss distance analysis of proportional navigation for high-order binomial guidance and control system is presented in presence of seeker noise using normalized adjoint method. In this analysis, the effects of effective navigation ratio and system order up to 30 are studied due to heading error, target maneuvers, and seeker noise. Numerical solutions for the cases of initial heading error, target with constant, linear, and parabolic maneuvers show a quasi-asymptotic behavior with respect to system order. In addition, the quasi-asymptotic behavior occurs for higher system order due to seeker noise. The values for the infinity's order system are obtained numerically and analytically. Based on the analysis and its assumptions, an order of 8 to 10 is suggested for miss distance analysis of a binomial guidance and control system.

In this paper, an explicit formulation of optimal line-of-sight strategy is derived in closed-loop for integrated guidance and control (IGC) system without consideration of fin deflection limit. The airframe dynamics is modeled by a second-order nonminimum phase transfer function, describing short period approximation. In the derivation of our optimal control problem, the actuator is assumed to be perfect and without limitation on fin deflection, whereas fin deflection limit is applied for the performance analysis of the presented optimal IGC solution. The problem geometry is assumed in one dimension and the final position and final time are fixed. The formulation is obtained in four different normalized forms to give more insight into the design and performance analysis of the optimal IGC strategy. In addition, guidance gains are obtained analytically in explicit form for steady-state solution. In overall, the performance of IGC is better than that of IGC with steady-state gains, but have more computational burden; however, it is reasonable for now-a-day microprocessors. Curve fitting or look-up table may be used instead for an implementation of optimal IGC strategy. Moreover, parametric study of nondimensional IGC parameters is carried out, such as weighing factor, dc gain, and short period frequency. Finally, the performance of the both IGC strategies is evaluated with airframe model uncertainties.

In this paper, an approximate solution of zero-effort-miss distance in spherical earth model is obtained in Earth-centered inertial (ECI) coordinates. In this approach, the gravitational acceleration is assumed to be inversely proportional to the cubic distance of space vehicle from the earth center. The present solution gives an explicit formula in terms of initial conditions and flight time. Moreover, the piecewise solution, i.e., the connection of solutions, is utilized in order to increase the accuracy of the algorithm. For this propose, the total flight time is divided into several interval using middle points. In each interval, the gravitational acceleration is taken to be inversely proportional to the cubic distance, but the correction constant is updated for each interval. The accuracy of the proposed method and its computational burden are calculated for various numbers of time intervals. For comparison purpose, the piecewise solution is also applied to the linear gravity approximation. The simulation results and calculation of computational burden show that the presented method has better accuracy than linear solution in ECI reference and linearized solution for relative motion with the same computational burden.

In this paper, effect of order of the thruster transfer function is investigated for a single-axis attitude control of a rigid satellite using two sets of opposing thrusters. The thruster dynamics is modeled by a pure delay in series with an arbitrary-order transfer function (TF) consisting of similar first-order lags. For this purpose, bang-bang controllers, bang-bang with dead zone, Schmitt-trigger and PWPF modulator are used in the analysis. The thruster dynamics order is selected using existing empirical thruster profiles. Considering the empirical profiles, usually a TF’s order lower than 2 is not recommended. Simulation results show that the TF’s order may be considered as an uncertainty applied to the attitude control problem and its sensitivity analysis.

In this paper, an explicit optimal line-of-sight guidance law for second-order binomial control systems is derived in closed-loop without acceleration limit. The problem geometry is assumed in one dimension and the final time and final position are fixed. The formulation is normalized in three forms to give more insight into the design and performance analysis of the guidance law. The computational burdun of the guidance law is reasonable for now-a-day microprocessors; however curve fitting or look-up table may be used for the implementation of the second-order optimal guidance law. The performance of the second-order optimal guidance law is compared in normalized forms with zero-lag and first-order optimal guidance laws using third-, fourth-, and sixth-order binomial control systems with/without acceleration limit. Moreover, the effect of the final time, the equivalent time constant of the vehicle control system, the vehicle-to-target line-of-sight weighting factor in cost function, and acceleration limit are investigated. Normalized miss distance analysis shows that the miss distance of the second-order guidance law is smaller than the two mentioned schemes for small total flight times, especially with large maneuvering capability.

In this paper, implicit guidance equations are derived in polar coordinates. Depending on applications, implicit guidance equations in polar coordinates may be preferred over cartesian coordinates. Moreover, depending on the type of guidance problem, analytical solutions for sensitivity matrices may be simplified using polar coordinates. Therefore, transformation of implicit guidance equation into polar coordinates can be useful in guidance problems. In addition, the resulting equations are extended to cylindrical coordinates.

In this paper, a fuzzy logic guidance algorithm is presented for the ascending phase of satellite launch vehicles in the presence of wind effects. In this algorithm, the midcourse constraints including maximum allowable angle of attack at the maximum dynamic pressure and the product of the dynamic pressure and angle of attack, as well as constraints on the final altitude and flight-path angle are considered. The algorithm uses a Mamdani-type fuzzy controller with centroid defuzzification.Maximizing and minimizing set methods to reduce wind effect, while satisfying the midcourse and final constraints. Simulation results show that the presented algorithm improves the performance of the satellite launch vehicle, satisfying the constraints within the maximum allowable estimation error on wind speed.

In this paper, miss distance analysis of a modified proportional navigation with lateral acceleration feedback is presented using normalized equations of adjoint method in the presence of radome effect and seeker noise. In this analysis, linear model is utilized. Also, radome error slope is assumed to be constant. Approximate formulas of miss distance are presented for second-order control system with two real distinct poles, using obtained nondimensional miss distance curves. These curves and formulas are useful for guidance system analysis and preliminary design. Moreover, using the approximate formulas, the guidance gain and time constant can be designed as variables to enhance the stability and performance of the guidance and control system in low and high altitudes.

In this paper, the effect of non-minimum phase control system on the miss distance of a class of one-dimensional guidance law due to initial heading error and also constant and time-varying target maneuvers (including ramp and sinusoidal) is presented. The guidance and control system dynamics is modeled by a fifth-order transfer function with acceleration limit and radome effect. More precisely, each of the seeker and noise filter is taken as a first-order transfer function. The control system and airframe are modeled by a third-order non-minimum phase one. Miss distance analysis is made by using normalized governing equations and plotting normalized curves, resulting in useful curves for parametric study specially the effect of zero location of airframe dynamics for analysis, modification, and design of a class of linear and nonlinear guidance laws.

This paper presents analytical solution of miss distance of a proportional navigation (PN) modified with body rate and lateral acceleration feedback using adjoint method in the presence of radome effect and seeker noise. The control system dynamics is modeled by a first-order transfer function. Therefore, the transfer function from line-of-sight rate to lateral acceleration is derived as a first-order transfer function, for which the equivalent time constant and gain are obtained in terms of radome slope and guidance coefficients. Consequently, the miss distance relations due to radome slope and seeker noise are obtained in terms of the radome slope and the acceleration and its rate feedback. The resulting relations are verified using statistical analysis. Moreover, the guidance law coefficients are designed as variables to enhance the PN performance, thereby increasing the stability region and reducing the miss distance.

In this paper، the linearized 2-D equations of proportional navigation guidance (PNG) are normalized in order to carry out the missdistance analysis considering acceleration saturation and radome effect. The control system dynamics is modeled using the first-order transfer function. To reduce the missdistance due to radome effect، the body rate feedback is added to the commanded acceleration in the analysis. Using this approach، useful normalized curves are obtained for parametric study and modification of PNG. Finally، the normalized equations of a modified PNG، having nonlinear terms consisting of line-of-sight rate with different powers، are derived.

In this paper، a three-dimensional midcourse guidance law is presented for interception of moving targets using moving waypoints. The waypoints are selected such that the interceptor ascends initially at higher altitudes to fly its trajectory in the thin layers of atmosphere. The interceptor uses proportional navigation guidance with gravity compensation to fly towards the next waypoint. The waypoint positions are modified in case the target changes its course. Simulation results show that the proposed midcourse guidance has better performance for the long range in comparison with classical midcourse schemes.

In this research, an approximate solution of the required velocity with final velocity constraint is derived using linear gravity assumption along the path of current position to the final position vectors. Moreover, an approximate solution for sensitivity matrix of the required velocity with respect to the position vector is obtained. The presented solutions are given for a predetermined final time. In this case, for the free final position, the analytical solution is rather more difficult than its fixed final position counterpart. Therefore, three approaches are utilized for approximation of the final position vector. Finally, the obtained solutions are compared to exact numerical one for the spherical- Earth model.