Research Article | Open Access
Shang Jiang, Fuqing Tian, Shiyan Sun, "Integrated Guidance and Control Design of Rolling-Guided Projectile Based on Adaptive Fuzzy Control with Multiple Constraints", Mathematical Problems in Engineering, vol. 2019, Article ID 6309462, 17 pages, 2019. https://doi.org/10.1155/2019/6309462
Integrated Guidance and Control Design of Rolling-Guided Projectile Based on Adaptive Fuzzy Control with Multiple Constraints
In the terminal guidance section of large caliber naval gun-guided projectile while striking nearshore maneuvering target, an integrated guidance and control (IGC) method based on an adaptive fuzzy and block dynamic surface sliding mode (AFCBDSM) was proposed with multiple constraints, including the impact angle, control limitations, and limited measurement of the line of sight (LOS) angle rate. The strict feedback cascade model of rolling-guided projectile IGC in space was constructed, and the extended state observer (ESO) was used to estimate the LOS angle rate and uncertain disturbances inside and outside the system, such as target maneuvering, model errors, and wind. A nonsingular terminal sliding mode (NTSM) was designed to zero the LOS angle tracking errors and LOS angle rate in finite time, with the adaptive exponential reaching law. The cascade system was effectively stabilized by the block dynamic surface sliding mode, which prevented differential explosions. To compensate for the saturated nonlinearity of canard control constraints, an adaptive Nussbaum gain function was adopted. The switching chatter of the block dynamic surface sliding mode was reduced through adaptive fuzzy control. Proven by Lyapunov theory, the LOS angle tracking error and LOS angle rate were convergent in finite time, the closed-loop system was uniformly ultimately bounded (UUB), and the system states could be made arbitrarily small at the steady state. Hardware-in-the-loop simulation (HILS) experiments showed that the AFCBDSM provided the guided projectile with good guidance performance while striking targets with different maneuvering forms.
In recent years, the rapid development of high-tech and advanced naval warfare theory has required naval gun weapons to be capable of continuous naval surface fire support and accurate striking of maritime and coastal targets. The naval gun-guided projectile, rolling at low speeds during flight, not only possesses a higher firing speed, more ammunition carrying capacity, and superior cost effectiveness than missiles, but it also has a longer range and higher accuracy compared with traditional ammunition, thereby providing reliable naval surface fire supports for amphibious forces .
Terminal guidance and control play a core role in achieving accurate strikes of naval gun-guided projectile. As attack and defense equipment systems are upgraded, the relative movement speed between the projectile and target increases, which makes the frequency of the guidance loop close to that of the control loop. Consequently, the frequency assumption of spectral separation might be invalid . Due to the limited space of guided projectile control modules, it is necessary for guidance and control systems to share sensors, such as gyroscopes and accelerometers, to promote economic efficiency and reliability. Furthermore, to achieve a better damage effect, the multiple constraints of actual combat should be fully considered, such as the impact angle , nonlinear saturation of canard deflection , and limited measurements of the LOS angle rate . The traditional time-scale separation design method that merely meets the miss-distance constraint has difficulty meeting the aforementioned requirements, which has attracted experts, scholars, and engineers to the IGC design method with multiple constraints.
IGC, first proposed by Williams et al. , utilizes the coupling relationship between the guidance and control system to construct a direct connection through aerodynamic angles and form a cascade closed-loop system. Based on the relative motion between the projectile and target, the control law of canard deflection can be directly calculated by IGC algorithm. Scholars have subsequently achieved results in IGC, combining it with robust control [7, 8], dynamic surface control [9, 10], adaptive control [7–9], sliding mode control (SMC) [11–14], fuzzy control , and other modern control theories. Yang et al.  proposed a robust IGC design method for guided projectile based on a SMC observer. Nevertheless, its high performance depended on precise measurements of the LOS angle rate, which was too demanding for rolling-guided projectile.
Backstepping control requires calculating high-order derivatives of virtual control laws, which easily leads to differential expansion. Therefore, Seyedipour et al.  designed a dynamic surface and adopted a low-pass filter to avoid differential expansion, which simplified the design process by guaranteeing the guidance performance and system stability. Combining it with ESO, Shao and Wang  designed a dynamic surface backstepping controller. The model error, target maneuvering, and other uncertain disturbances were observed by the ESO, which significantly reduced the switching range of the dynamic surface. The ESO, initially proposed by Han , is a feasible scheme to determine the time-varying nonlinear IGC problem with uncertain disturbances of which a simple algorithm can accurately observe the system states and internal and external disturbances without a precise model of the research object. As the sole actuator of the guided projectile, a canard usually exhibits deflection saturation phenomenon, which allows the available overload to be less than the required overload and easily leads to the system loss of control or even instability. Hence, Wen et al.  combined an adaptive Nussbaum gain function with the dynamic surface sliding mode, which effectively solved the nonlinear saturation of the actuator and stabilized the IGC cascade system.
Possessing strong robustness against instability factors, such as system parameter perturbations and external disturbances, SMC has been widely applied to the IGC design. Shtessel and Tournes  regarded the normal acceleration generated by target maneuver as a disturbance and designed the IGC method based on the high order sliding mode, which is robust against uncertainty disturbances of the target maneuver. To meet the terminal impact angle constraint, Wu and Yang  defined the impact angle as the angle between the projectile velocity and horizontal plane and proposed an effective IGC design scheme. However, it was pointed out that it is more universal to define the impact angle as the angle between the projectile velocity and target velocity at the moment of impact . Furthermore, combining a block dynamic surface with ESO, an IGC method was proposed by Guo and Liang , which allowed the nonrolling near-space interceptor to obtain better guidance performance. It is well known that chatter is a difficult and urgent problem in SMC, which can be partially improved by adopting a continuous saturation function. Resulting from the difficulty of determining switching gains, the effect of buffeting reduction must be further improved. Therefore, Ran et al.  designed an adaptive fuzzy system using the LOS angle rate and distance between the projectile and target, which not only effectively weakened chattering but it also improved the robustness of the whole system.
The previous research on IGC basically focused on nonrolling vehicles in a plane or space with a single constraint. However, there have been few IGC designs for rolling aircraft. A continuous roll character significantly enhances the couple relationship between the pitch and yaw channel of naval gun-guided projectiles, and multiple constraints introduce challenges to the finite-time convergence and stability of the system. With the above constraints and disturbances comprehensively considered in this paper, the AFCBDSM is proposed. It was proven using Lyapunov theory that the LOS angle tracking errors and LOS angle rate could converge in finite time, the closed-loop system was UUB, and the system states could be made arbitrarily small at the steady state. The main innovations were as follows: (1) with multiple constraints and various disturbances comprehensively considered, the IGC strict feedback cascade model for rolling naval gun-guided projectile was constructed, (2) the switching chatter of the block dynamic surface sliding mode was effectively weakened through adaptive fuzzy control, (3) the finite-time convergence of the LOS angle-tracking errors, LOS angle rate, and UUB of the closed-loop system were strictly proven, and (4) the effectiveness and feasibility of the AFCBDSM were verified by the designed HILS, which could satisfy multiple constraints.
2. Model Establishment
2.1. Motion and Mechanical Model
The relative motion relationship between the projectile and target in space is shown in Figure 1, where , , , , and represent the projectile reference coordinate (), projectile trajectory coordinate (), LOS coordinate (), target reference coordinate (), and target trajectory coordinate (), respectively. Furthermore, , , , , and denote the projectile, target, distance between the projectile and target, LOS inclination angle, and LOS azimuth angle, respectively. , , , , , and denote the trajectory inclination angle, trajectory azimuth angle, and speed of the guided projectile and target, respectively.
In addition, , , , , , , and represent the pitch angle, yaw angle, roll angle, attack angle, sideslip angle, pitch canard angle, and yaw canard angle, respectively. Because , , , and change periodically with the roll of the projectile, it is necessary to establish nonrolling coordinates , including quasi-projectile coordinates and quasi-velocity coordinates . , , , and represent the quasi attack angle, quasi sideslip angle, equal pitch canard angle, and equal yaw canard angle, respectively, which are utilized to calculate forces and moments of the rolling-guided projectile. The transformation relations are given by
Assumption 1 (see ). The guided projectile can be regarded as a rigid body, and the target can be regarded as a particle. , , , , , , , , and can be measured easily. P and T only possess acceleration in the normal direction of the respective velocity, which always satisfies .
The transformation matrix from to is as follows:The relative motion relationship between the projectile and target  is given bywhere and are unknown disturbances caused by the target maneuver and and are components of the projectile acceleration and target acceleration in , respectively. is the projection of on the yaw plane. The components of in and the components of in are defined as and , respectively. The transformation relation for from to can be obtained as follows:where and are coordinate transformation errors. The combined external force acting on the projectile mainly consists of a gravitational force , lift force , Magnus force , equal operating force , and wind force . The components of in the and axes are the normal force and lateral force , respectively. The external moment mainly consists of a static moment , equatorial damping moment , Magnus moment , equal operating moment , and wind moment . The components of in the and axes are the pitch moment and yaw moment , respectively. Their formulas can be written as follows:where and could be obtained through sensors of the projectile, such as microinertial navigation device and , , , , , and are the mass, dynamic pressure, reference area, reference diameter, reference length, and distance from the canard to the pressure center of the guided projectile, respectively. , , and represent the lift coefficient derivative, Magnus force coefficient joint partial derivative, and canard lift coefficient derivative, respectively. , , and are the static moment coefficient derivative, equatorial damping moment coefficient derivative, and Magnus moment coefficient joint partial derivative, respectively. , , , and represent unknown model errors of the force and moment. and are the additional attack angle caused by the vertical wind and plumb wind , respectively, and is the additional sideslip angle from the cross wind . These are related as follows:where is a continuous nondifferentiable saturation function defined as follows:where is the maximum canard deflection angle and has the same form. For the convenience of the IGC design, the following reasonable assumption is made.
Assumption 2 (see ). and are mainly generated by and , respectively. The forces generated by and account for a small proportion of the lift force, which can be regarded as bounded uncertain disturbances.
The kinetics and kinematic equations of a projectile body rotating around mass center yield the following:Considering the delay characteristic of the canard as a first-order inertial link with time constant , the IGC model of a guided projectile can be obtained as follows:
2.2. Nearshore Maneuvering Target Model
A maritime and coastal maneuvering target can be approximately expressed by the following first-order term:where and are time constants and and are normal and lateral acceleration commands of the target, respectively, which are unknown and bounded.
2.3. Impact Angle Model
In the pitch plane, the impact angle is defined as the angle between and at the final impact moment. Zeroing the relative normal velocity of the projectile and target leads to where and are the final LOS inclination angle and final target trajectory inclination angle, respectively, which can be acquired by unmanned reconnaissance. For any specified , there exists a unique corresponding to it, and the impact angle constraint can be converted into a LOS angle constraint.
2.4. System States Space
The system state variables, input variables, and output variables are defined as follows:
For the convenience of the IGC design, a continuous differentiable hyperbolic tangent function vector is introduced to describe the canard deflection saturation:
The derivative function vector of is and
Moreover, the IGC strict feedback state space of a continuous rolling-guided projectile can be obtained as follows:where . For the convenience of the IGC design, the following reasonable assumption is made.
Assumption 3 (see ). The disturbances and their first derivative are bounded, i.e., there are unknown normal numbers and that satisfy and , respectively.
Remark 1. According to the definition of , one could be obtained that is a continuous function in its definition domain and there is no discontinuous point. As result of the left derivative is not equal to the right derivative at , and is only nonderivative at in its definition domain. However, is derivable in its definition domain except these two points, and the derivative is bounded.
3. AFCBDSM Design
The purpose of the AFCBDSM design is to generate an appropriate control law for equation (14) that can allow and to converge to any small neighborhood of zero in finite time and guarantee the closed-loop system is UUB , and the system states converge to a small neighborhood of the origin, with limited measurements of , the control saturation of , and the unknown bounded disturbances .
3.1. ESO Design
To acquire an accurate observation value of , the following third-order ESO is designed, with the observational variables defined as , , and :where , , , , , , and is the function vector .
In a previous report , the stability of the second- and third-order ESO designed in cascade systems were deduced and proven. By choosing appropriate parameters values, in particular, letting be much larger than and , the ESO given by equation (16) can observe accurately and quickly. Similarly, for observing , the observational variables are defined as and and the second order ESO is designed as follows:
3.2. Nonsingular Terminal Sliding Mode Design
Lemma 1 (see ). Considering the following system:where is the system state vector, : is a nonlinear continuous function defined on , and is an open neighborhood of the origin . Assuming is a smooth positive definite function defined on and is negative semidefinite on for and , there exists an area such that any that starts from can reach in finite time. Furthermore, if is the time needed to reach , then , where is the initial value of .
A kind of NTSM is used to construct the sliding mode vectors:where and . The time derivative of equation (19) can be obtained as follows:To ensure good dynamic properties in the sliding mode approach process, the sliding mode adaptive approach law is designed as follows:where , , , , and is a symbolic function vector . Combining equations (20) and (21) and removing the singular factors and yields,
Theorem 1. For the subsystem composed of the first two equations of equation (15), adopting the ESO given by equation (16) and the control law given by equation (22), the system state variables and will converge to any small neighborhood of zero in finite time.
Proof. A Lyapunov function is chosen as , and the time derivative of can be obtained:According to Lemma 1, can converge to in finite time while , which is also satisfied if . Thus, the following can be obtained:A Lyapunov function is chosen as , and the time derivative of can be obtained:According to Lemma 1, one can obtain , and can converge to any small neighborhood of in finite time. The proof ends.
3.3. Block Dynamic Surface Sliding Mode Design
is defined as the second block dynamic surface sliding mode, and the third, fourth, and fifth block dynamic surface sliding modes are defined as follows:where is the abbreviated notation for the function . Virtual control laws and can be obtained  as follows:
To avoid direct differentiation with respect to , , and , the filtered virtual control laws , , and are obtained using the following first-order filter:where , , and are positive constants.
Definition 1 (see ). If the continuous function satisfies the following property and , then is a Nussbaum function.
For handling the nonlinear saturation of the canard deflection effectively, control laws are designed as follows:where and are positive parameters whose form and scope are similar to and , respectively. According to Definition 1, the following Nussbaum function is implemented:where and are adaptive variables of the Nussbaum function with the following adaptive law:where and are positive constants.
3.4. Adaptive Fuzzy System Design
Resulting from the time variability and nonlinearity of uncertain disturbances, it is difficult to determine appropriate switching gains, which causes serious chatter of the sliding mode. Possessing the universal approximation ability, the adaptive fuzzy system, composed of product inference, singleton fuzzification, center-average defuzzification, and Gauss membership functions , will be used to approximate observation error of disturbance. Thus, the high frequency chatter of the control law could be effectively weakened.
The input variable vector is defined as , then there are fuzzy rules in common as follows:where are fuzzy sets associated with the fuzzy membership functions, and is the output variable as follows:where is an adaptive fuzzy parameter vector in the dimension of , is the Gauss membership function, is the fuzzy basis vector in the dimension of , and the element is as follows:
Lemma 2 ([see 22]). For a continuous function defined in a closed set and any precision , there must exist an adaptive fuzzy system composed of equations (33) and (34) such that it satisfies .
To guarantee the finite time convergence of and , the adaptive fuzzy systems are designed for the virtual control law given by equation (27) and the control law given by equation (29). The optimal adaptive fuzzy parameter approximation vectors are defined as follows:where . According to Lemma 2, the following inequalities hold for any given small constant :The approximation errors of the fuzzy adaptive parameter vectors were defined as , of which the adaptive approach laws are as follows:where . From equations (27) and (29), the virtual control laws and control law are, respectively, modified as follows:where .
4. System Stability Analysis
Lemma 3 ([see 23]). Let and be smooth functions, which are defined on with . , and is a Nussbaum gain function. If the following inequality holds, then and must be bounded on :where and . The errors of the virtual control laws are defined as follows:Differentiating and yieldsThrough some simple simplifications, we obtain the following:According to a previous report , there are positive real numbers and such that the inequalities and hold. Through some simple calculations, we obtainWe obtain the following Lyapunov function of the whole system state:
Theorem 2. For system (15), which satisfies Assumptions 1–3, the closed-loop system is UUB, and the system states could be made arbitrarily small at the steady state, with the application of the ESO given by equations (16) and (17), the block dynamic surface sliding mode given by equations (18) and (26), the control law given by equations (22) and (37), and the appropriate parameter selections.
Proof. For convenience, taking the third dynamic surface as an example, the simplification process of equalities and inequalities related to is deduced as follows. From equations (27) and (28), one can obtain