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Robust Control for Uncertain Vertical Electric Stabilization System With Flexible Nonlinearity Using Backstepping Idea
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Peng Liu, Tan Lu, He Zhang
International Journal of Mechanical System Dynamics | 2025, 5(3) : 443 - 462
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International Journal of Mechanical System Dynamics | 2025, 5(3): 443-462
• RESEARCH ARTICLE •
Robust Control for Uncertain Vertical Electric Stabilization System With Flexible Nonlinearity Using Backstepping Idea
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Peng Liu, Tan Lu, He Zhang
Affiliations
  • School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, China
doi: 10.1002/msd2.70029
Outline
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A robust control method for the uncertain vertical electric stabilization system (VESS) with flexible nonlinearity is proposed, and the mismatched uncertainty is considered and compensated based on the backstepping idea. First, based on evaluating the coupling effects of the flexible nonlinearity, the analytical dynamics model of the VESS is established. Second, the tracking error is defined as the evaluation of the system's pitch-pointing tracking control, and then the mismatched state space model with two interconnected subsystems is established as the controlled system. Third, the original mismatched system is converted to the locally matched system using the backstepping design to transform the system state variables. The robust control is proposed to handle the flexible nonlinearity and mismatched uncertainty, which can make both the original system and the reconfigured system present practical stability. Finally, the effectiveness of the proposed control is verified by numerical simulation experiments. This study should be the first to consider flexible nonlinearity coupling and two different uncertainties (matched and mismatched uncertainty) in the design of pitch-pointing tracking control for the vertical electric stabilization system (VESS).

backstepping idea  /  flexible nonlinearity  /  mismatched uncertainty  /  robust control  /  vertical electric stabilization system
Peng Liu, Tan Lu, He Zhang. Robust Control for Uncertain Vertical Electric Stabilization System With Flexible Nonlinearity Using Backstepping Idea[J]. International Journal of Mechanical System Dynamics, 2025 , 5 (3) : 443 -462 . DOI: 10.1002/msd2.70029
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1 | Introduction
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To enhance survival and combat effectiveness on the battlefield, modern tanks are designed with an emphasis on lightness and speed. However, this design approach often leads to increased hull vibration [1], which exacerbates the impact of flexible nonlinearity on the tank pitch-pointing tracking control [2, 3]. Given that the mechanical structure of the tank vertical electric stabilization system (VESS) primarily consists of the actuator (electric cylinder) and the load mechanism (barrel), a comprehensive analysis of the flexible characteristics of both the electric cylinder and the barrel is essential for improving the pitch-pointing tracking control performance of modern tanks.
As a critical component of modern tank fire control systems, the control performance of the VESS directly influences the accuracy and hit rate of tank firing [4, 5]. In recent years, significant research has been conducted to analyze the coupling relationships among the servo motor, gearbox, ball screw system, and control system of the VESS. Novel control algorithms have been proposed to enhance pitch-pointing tracking control [6-8]. However, these studies primarily focus on addressing friction nonlinearity, gear backlash nonlinearity, and damping nonlinearity, while largely neglecting the flexible nonlinearity. This oversight limits the ability to meet control requirements under strong vibration conditions, which are common in battlefield scenarios. To address this gap, we constructed an analytical dynamics model of the VESS that incorporates flexible nonlinearity by developing an axial stiffness model for the electric cylinder and a modal solution for the flexible barrel. Through the fine modeling of the VESS, the adverse effects of flexible nonlinearity can be suppressed.
In fact, for the controller design of the system, strong nonlinear factors and complex uncertainties inevitably increase the complexity of the control process. In a general electromechanical servo system, uncertainty includes matched and mismatched uncertainty. The matched uncertainty is located in the parameter space of the control matrix and can be compensated for directly by adjusting the control input. In contrast, the mismatched uncertainty exists outside the parameter space of the control matrix and cannot be solved directly by adjusting the control input [9, 10]. Most of the pitch-point tracking control strategies used for VESS focus on compensating for the matched uncertainty [11-13], but the research on the mismatched uncertainty is limited, which can reduce the difficulty of controller design. However, the unmodeled dynamics of the VESS are easily excited by the strong robustness of the controller to suppress the mismatched uncertainty, which leads to the instability of the VESS. Given the inherent complex and flexible nonlinearity of the system, it is important to solve the problem of mismatch uncertainty to meet the requirements of control design. To address this challenge, we construct a state-space model of controlled systems with mismatched uncertainties based on the analytical dynamics model of VESS.
Many advanced control algorithms have been developed to address system uncertainties and nonlinearities, including adaptive robust control [14-16], sliding mode control [17, 18], adaptive state-constrained control [19, 20], event-triggered control [21-23], Lyapunov-based control [24-26], and intelligent control [27-29]. The above controller provides a systematic reference for the control algorithm design of high-end equipment, and can be integrated and optimized according to specific working conditions in practical applications. Moreover, in dealing with flexible nonlinearity, Xu [30] proposed a singular perturbation theory-based composite learning control for flexible-link manipulators, while Yang et al. [31] investigated trajectory tracking and vibration reduction in flexible manipulators using an adaptive control method with an iterative learning scheme. For mismatched uncertainty, Chen [32] introduced a novel control algorithm that characterizes the structure of uncertainty in mismatched systems, and Xu et al. [33] developed a fuzzy-based optimal approach for robust control design in interconnected uncertain systems with mismatched conditions. Additionally, Sun et al. [34] designed a robust controller to handle both matched and mismatched uncertainties in vertical electrohydraulic stabilization systems. Despite the advancements in control strategies for VESS, existing methods often fail to simultaneously address flexible nonlinearity and mismatched uncertainty, which are critical challenges in tank gun control systems. This gap motivates the development of a novel control approach that can effectively handle both issues while ensuring practical stability and high tracking accuracy. Building on these advancements, this paper adopts the backstepping design approach to transform the system state variables, converting the original uncertain system into a locally matched uncertainty system. Subsequently, a robust control method is proposed to address the flexible nonlinearity and mismatched uncertainty, ensuring practical stability for both the original and reconfigured systems.
In this paper, the flexible nonlinearity of the system is characterized through nonlinear mechanism functions, which are integrated into the analytical dynamics model. The mismatched state space model is developed to address the mismatched uncertainty, providing a foundation for subsequent controller design. Meanwhile, the original mismatched uncertain system is transformed into a locally matched uncertain system by redefining the system state variables. Furthermore, a robust control method is proposed to effectively handle both mismatched uncertainty and flexible nonlinearity, ensuring practical stability for both the original and reconfigured systems. Finally, simulation and experimental results demonstrate that the proposed control method outperforms existing approaches in terms of accuracy, robustness, and stability. The primary innovation of this study lies in the simultaneous consideration of flexible nonlinear coupling and two types of uncertainties (matched and mismatched) in the pitch-pointing tracking control design for the vertical electric stabilization system (VESS). Unlike existing methods that typically focus on either matched uncertainties or specific types of nonlinearities (e.g., friction or gear backlash), our approach addresses the complex interplay between flexible nonlinearity and mismatched uncertainty, which is particularly critical in applications such as tank gun control systems. This comprehensive treatment enables superior control performance under strong vibration conditions, where traditional methods often fall short.
The main contributions of this study are threefold: (1) the first integration of flexible nonlinear coupling and mismatched uncertainty into the pitch-pointing tracking control design for VESS; (2) the development of a mismatched state space model and its transformation into a locally matched system using backstepping design; and (3) the proposal of a robust control method that ensures practical stability for both the original and reconfigured systems. These innovations provide a comprehensive solution to the challenges posed by flexible nonlinearity and mismatched uncertainty in high-vibration environments.
2 | System Modeling: Flexible Nonlinearity and Dynamics of VESS
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2.1 | Axial Stiffness Model of the Electric Cylinder
As an essential part of the VESS, the flexible deformation of the electric cylinder has a significant impact on the system stability and control performance. Considering that axial deformation is the main influencing factor, the focus of this paper is to construct the axial stiffness model of the electric cylinder. As shown in Figure 1, the servo electric cylinder in VESS is mainly composed of servo motor, gearbox, bearing set, ball screw, and pushrod.
The total axial stiffness of the electric cylinder is the series sum of the stiffness of each component, among which the axial stiffness of the bearing set, screw, nut, and pushrod has the most significant influence, and the influence of other parts is small and negligible. From the axial stiffness model of the bearing set, screw, nut, and pushrod [35], it is known that the axial stiffness of the servo-electric cylinder is not only related to the structural size and material performance parameters of the electric cylinder, but also proportional to the one-third power of the axial load size. Considering the small change of the electric cylinder output force size when the tank VESS is working, the axial stiffness of the electric cylinder can be regarded as the system uncertainty, the nominal part of which is constant value and independent of the axial load size, the uncertainty part may change rapidly with time, but bounded.
2.2 | Modal Solution of Flexible Barrel
The tank barrel is a kind of tubular flexible structure using gun steel as the material, and its Schematic diagram is shown in Figure 2. The tail end of the barrel is connected with the cradle and the gun breech to form the load unit of the VESS. The length of the restrained section of the barrel is the distance between the gun breech and the trunnion center point, and the length of the unrestrained section of the barrel is the distance between the muzzle and the trunnion center point. Considering that the bending deformation of the flexible barrel is much larger than its shear deformation and axial deformation, it is simplified as the Euler-Bernoulli beam in Figure 3, only considering its bending deformation and ignoring its shear deformation and axial deformation, and the structure stiffness of the gun breech and cradle is larger, so the flexible deformation of the restrained section of the barrel is smaller and can be ignored. In Figure 3, is the inertial coordinate system, is the floating coordinate system fixed on the barrel, is the center point of the trunnion, is the mass of the load unit, is the density of the flexible barrel, is the cross-sectional area of the barrel, is the flexural stiffness of the flexible barrel, where is the modulus of elasticity, is the moment of inertia of the cross-section, the barrel does pitching motion around the center point of the trunnion under the action of the moment is the pitch angle at the cradle measured by the gyroscope at the trunnion, is the pitch angle of the barrel, is the flexible deformation in the process of motion due to the flexibility of the barrel at . By geometric relationship, we have
where is the displacement of the barrel.
According to the hypothetical modal method, the deformation of the barrel at moment during the motion can be described as
where denotes the th order vibration function, and denotes the corresponding modal coordinates. Considering the small vibration amplitude of the higher-order modals of the flexible barrel, this paper mainly studies the influence of the bending deformation of the flexible barrel on the control performance of the system, to simplify the research process, only the first-order modal of the barrel is considered, and the influence of the remaining modals on the system can be neglected, that is
Considering the small change in the cross-sectional area of the barrel and the overall use of the same material, the unrestrained section of the barrel can be regarded as a homogeneous beam of equal cross-section with one end fixed and one end free (cantilever beam), from which the boundary conditions can be determined, and the specific expression of the vibration function is [36]
where can be obtained from the following equation:
According to the initial vibration conditions of the flexible barrel, the expressions for the inherent frequency and modal coordinates of the flexible barrel are
With the flexible nonlinearity of the system characterized, the next step is to develop a dynamical model that incorporates both matched and mismatched uncertainties. In Section 2.3, we construct a mismatched state space model of the VESS, which serves as the foundation for the subsequent controller design.
2.3 | Dynamical Model of the Uncertain VESS
The tank VESS is mainly composed of a motor, a flexible electric cylinder, and a flexible barrel, and its multibody model is shown in Figure 4, in which the finite element mesh cell is taken to characterize the flexible characteristics of the uncertain system.
The output torque of the motor is amplified by the gearbox to drive the ball screw of the electric cylinder to rotate, the screw nut, which is limited in the radial direction, is driven by the rotating force of the screw to do linear motion together with the pushrod, and the pushrod of the electric cylinder applies thrust at point of the barrel to drive the load to rotate around the center point of the trunnion for meeting the angle adjustment command of the fire control system. The simplified schematic diagram of the system mechanics is shown in Figure 5, where is the center of gravity of the load, is the initial angle of the electric cylinder, is the initial length of the electric cylinder, is the theoretical working length of the electric cylinder, is the axial flexible deformation of the electric cylinder, and are the lengths of the segment and segment , respectively, is the distance from the upper pivot of the electric cylinder to the center point of the trunnion, and is the thrust of the electric cylinder. Based on the working principle of motor-gearbox system, the dynamics model can be expressed as
where is the output torque of motor-gearbox system, is the gear ratio, is the control input voltage, and are the motor torque constant and the electromotive force constant, respectively. and are the moments of inertia and viscous damping coefficients of the motor, is the angular displacement of the motor-gearbox system, and is the total resistance of the armature circuit.
According to the working principle of electric cylinder system, the dynamics model of ball screw can be expressed as
where , and are the lead, the moments of inertia, and the viscous damping coefficients of ball-screw, is the theoretical displacement of the screw.
For the load unit of the VESS rotating around the center point of the trunnion, the total kinetic energy can be expressed as
The total potential energy can be expressed as
The Euler-Lagrange equations of the flexible barrel are
where is the Lagrangian, is the generalized coordinates, and is the generalized force. The computation of Equation (15) results in the following equations:
where is the external disturbance torque mainly caused by the road excitation. Equation (16) can be further rewritten as
where and . Here, and are coefficients related to the flexible barrel's modal characteristics. represents the influence of the barrel's bending deformation on the system dynamics, while captures the coupling effect between the barrel vibration and the system's motion.
According to the geometric relationship in Figure 5, then
with Equations (11), (12), and (18):
where
Based on the dynamical model of the uncertain VESS, the next step is to formulate the pitch-pointing tracking problem. In Section 3, we define the tracking error and establish the control objectives, laying the groundwork for the design of the robust backstepping controller in Section 4.
3 | Formulation of Pointing Tracking Problem for the VESS
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The main function of the VESS is to achieve real-time pitch-pointing tracking of the tank to the enemy target, so the goal of the tank vertical stabilization control is to design the controller so that the pitch angle of the barrel is maintained at the target angle. Thus, the tank pitch-pointing tracking control can be converted into a class of problems that induce the pitch angle of barrel to track the desired reference signal approximately, while enabling the controlled system to suppress uncertainty interference during the signal tracking process autonomously. Assume that is a second-order differentiable signal function, and are uniformly bounded. The tracking error is thus defined as
Taking Equations (2) and (4) into Equation (22):
Then
Let and , substitute it into Equation (24):
substitute Equation (25) into Equation (20):
with the definitions
Consider the axial stiffness of the electric cylinder and the external disturbance torque as the uncertainties in the VESS (26) with flexible nonlinearity, and decompose them into nominal and uncertain portions as
where and are the nominal portions, and are the uncertain portions, which are possibly fast time varying but bounded, and the bounds can be described as
Remark 1. Considering that unbounded uncertainty requires infinite energy to be maintained, any parameter with physical significance and its uncertainty part is bounded. Besides, the nominal part of the parameter with uncertainty is usually chosen based on engineering experience, so it is always known as well.
Let be the uncertain portions of the system; by introducing the decompositions of and into the uncertain system (26) and classifying the system as the nominal and uncertain portions, Equation (26) can be rewritten as
with the definitions of
where is a function of and , and is a function of and .
As a result, the VESS with flexible nonlinearity can be described as a coupling system in a state-space form as
where is the time, are the system state variable, is the servo motor control input, is the system uncertain parameter, is unknown compact, which stands for the possible boundary of . Besides, and are matrices of appropriate dimensions; , and are continuous functions, which can be generalized to be Lebesgue measurable in .
Remark 2. From Equation (32), the dynamic model of the VESS of the tank consisting of two subsystems is nonlinear, and its state variables x1 and x2 are coupled to each other, providing the dynamical model for the subsequent design of a robust backstepping controller.
Remark 3. There exists the matrix L(x1, x2) such that PT = BL, that is, the input matrix satisfies the adjacency matched condition, and thus the input matrix P of subsystem N1 can be described by the input matrix B of subsystem N2. In other words, the control input u(t) can make the dynamic link between the state variable x1 of subsystem N1 and the state variable x2 of subsystem N2, so that the control input u(t) can realize the dynamic regulation of the whole system (including subsystems N1 and N2) although it enters the system from subsystem N2.
Having established the formulation of the pitch-pointing tracking problem for the VESS, the next step is to design a robust control strategy that can effectively handle the system's flexible nonlinearity and mismatched uncertainty. In Section 5, we propose a robust backstepping control method that transforms the original mismatched uncertain system into a locally matched system. This transformation enables the design of a control law that ensures practical stability for both the original and reconfigured systems. The detailed design process, stability analysis, and simulation results are presented in the following section.
4 | Robust Backstepping Control Strategy for the VESS
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Since the VESS considers flexible nonlinearity to be a mismatched uncertain system, the control method under conventional logic is no longer applicable. The backstepping control idea is used to transform the state of the original controlled system, specifically from to , so as to reconstruct the controlled system and make the reconstructed system satisfy the matched condition, and further design the robust controller to make the reconstructed system present practical stability while the original controlled system presents the same characteristics, so as to realize the barrel vertical stability control.
4.1 | Design of Robust Backstepping Controller
At first, choose the appropriate functions and to make the uncontrolled nominal systems and to be uniformly asymptotically stable at the origin , then, choose the second-order differential functions and continuous, strictly increasing functions are constructed to satisfy
besides, choose a strictly positive constant , such that for all
In addition, exist a continuous function , which satisfies
which means that for the uncontrolled nominal systems and , there exists the legal Lyapunov functions and .
Then, the robust controller is designed to induce the following system to present practical stability
Considering that the system state variable and the matrix can be described as
For discussing the boundary value of the uncertain term , according to Equation (31), can be decomposed into the form of multiplying the “certain” and “uncertain” portions.
substituting it into Equation (32), we have
where is a continuous function of and .
In preparation for the subsequent controller design, the boundary conditions of are now further analyzed
According , we have
then, the controller is designed as , where
where
where is the design parameter and is a continuous differentiable smooth function and .
Theorem 1. When the controller (42) is applied to the controlled system (36), the solution of this controlled system will exhibit practical stability as follows:
A

Uniform boundedness: For any , there exists a positive real number such that if , then for all ;

B

Uniform ultimate boundedness: For any and , there exists such that if , then for all , where ;

C

Uniform stability: There exists a real number such that if , then for .

Proof. With control μ1(·), the derivative function of the Lyapunov function V1(·) along the trajectory of the system (36) is
Recalling △g1 = Pq1 and with Equation (34)
Let , according to Equation (42), if
if
Consequently, for all (x1, t)
This means that the controlled system (36) exhibits practical stability under the action of control (42).
Since subsystem N1(32) actually contains x2 rather than control μ1, defining the implanted control
where is the scalar design parameter and the new term is added to compensate for the difference between and . Substituting into Equation (32), the subsystem can be redescribed as
Recalling Pr = [0, P21]T and Equation (49), v1(t) can be described as
Based on control v, the derivative function of V1(·) is
With and Equation (48), we have
The analysis of the third and fourth terms on the right-hand side of the inequality will be given later.
Then, the system state variable x2 is converted to . According to Equation (32)
according to Equation (32), defining
By introducing function , Equation (54) can be rewritten as
where
With Equations (31), (32), (55) and the definitions of z2 and g2(z2, t), Equation (57) can be rewritten as
where
Considering that Δa22, B21, ΔB21 are continuous functions and B21 is a positive scalar, Δa22 and ΔB21 can be decomposed into the form of multiplying the “certain” and “uncertain” portions:
Substituting it into Equation (32), we have
In preparation for the subsequent controller design, the boundary conditions q2 and E2 are now further analyzed, according to Equations (31) and (59), we have
where the uncertain portion of axial stiffness of the electric cylinder usually fluctuates in the range of – of the certain portion of axial stiffness of the electric cylinder , so existing constant to make . For obtaining , the following projection is made
Finally, to achieve stability tracking control of the VESS with flexible nonlinearity, the following robust backstepping controller is designed for the controlled system:
where
where is the design parameter, , thus . Similar to the analysis of , can be ensured to be non-zero by picking the appropriate function . With control , the derivative function of the Lyapunov function along the trajectory of the system (56) is
Recalling Δa2 = Bq2 and ΔB = BE2, according to Equation (34), we have
According to Equation (67), if ||β|| > ε2
Let , Equation (73) can be rewritten as
Recalling , we have
According to Equation (67), if ||β|| ≤ ε2
Consequently, for all (z2, t)
4.2 | Stability Analysis of the Reconfigured System
For efficiently designing the robust backstepping controller, the state variables of the controlled system (32) are transformed and the transformation process can be summarized as
where is the pre-transformation state variables, is the post-transformation state variables, is the implanted controller. The dynamics of the transformed state variables , that is, the reconfigured system dynamics can be described as
Remark 4. To deal with the mismatched uncertainty disturbance in the VESS, the original controlled system (32) is reconstructed based on the system state transformation, so that the reconstructed system (79) meets the matched condition, and thus a robust backstepping controller is designed. If the controller can make the reconstructed system (79) and the original VESS (32) present practical stability, the tank pitch-pointing tracking control can be realized.
Theorem 2. When the controller (67) is applied to the controlled system (79), the solution of this controlled system will exhibit practical stability as follows:
A

Uniform boundedness: For any , there exists a positive real number such that if , then for all .

B

Uniform ultimate boundedness: For any and , there exists such that if , then for all , where .

C

Uniform stability: There exists a real number such that if , then for .

Proof. Choosing the Lyapunov function candidate
The derivative function of the Lyapunov function V(·) along the trajectory of the system (79) is
With Equations (53) and (77), we have
Then, Using the inequality
for any ,
substituting it into Equation (82), we have
By choosing and , we have
where
Let
According to Equation (86), by the literature, the practical stability of the system described by the Theorem 2 is guaranteed, the details are as follows:
A

The reconfigured system presents uniform boundedness, for any , if , there exists

where
to make for all tt0.

B

The reconfigured system presents uniform ultimate boundedness: For any , satisfying

when , where

C

When is selected, the reconfigured system presents uniform stability.

Remark 5. The robust backstepping controller proposed in this paper enables the reconfigured systems (i.e., systems and ) to exhibit practical stability (including uniform boundedness, uniform ultimate boundedness, and uniform stability) under uncertainty disturbances.
4.3 | Stability Analysis of the Original System
In Sections 5.1 and 5.2, the robust backstepping controller (67) for the reconfigured system (79) is proposed, and the stability of the reconfigured system is proved theoretically. However, the actual controlled system is the tank VESS (32), and the goal is to design an appropriate controller to make it exhibit the expected characteristics, so the stability of the tank VESS under the action of the proposed robust backstepping controller is further analyzed.
Theorem 3. Under the action of the controller u(t), the original tank VESS presents practical stability (including uniform boundedness, uniform ultimate boundedness, and uniform stability). At the same time, by adjusting the control design parameters εi, i = 1, 2, the size of the region of uniform ultimate boundedness and uniform stability can be made infinitesimal.
Proof. At first, supposing the boundary of z is κ, that is, ||z|| ≤ κ. According to z = [z1, z2]T, we have
with Equation (78)
Since
we have
recalling and Equation (35), we have
since and are the continuous functions, and , if and , existing the constants and to make (where )
where
The boundedness of x1,2 can be described as
where
It can be seen that if is continuous, then is bounded. This implies that if and are bounded, then is bounded. Furthermore, if , then there are and .
For any , the following definition is made
According to Theorem 2, when is uniformly bounded, for any . By replacing with , Equation (101) can be rewritten as
So
From the above analysis, it can be seen that when the reconfigured system state variable is uniformly bounded, the original system state variable also presents uniform boundedness. For any , there exists where , it means that . Let presents uniform ultimate boundedness: For any , there exists , then for all . Finally, when choosing
the original system presents uniform stability.
Furthermore, according to Equations (89) and (93), when choose the control parameters , then , thus . In summary, when the reconfigured system presents practical stability, the original system equally presents practical stability. At the same time, by adjusting the control design parameters , the size of the region of uniform ultimate boundedness and uniform stability can be made infinitesimal.
Remark 6. For the problem of tank pitch-pointing tracking control, the original controlled system is reconstructed by system state transformation to make the reconfigured system a matched uncertain system to deal with the mismatched uncertainty of the VESS. Thus, the robust backstepping controller is designed. Under the action of this controller, both the reconfigured system and the original VESS can present practical stability, which makes the pitch angle of barrel φr(t) track approximately the desired reference signal, thus realizing the tank pitch-pointing tracking control.
5 | Controller Design Ideas and Processes
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To effectively suppress the coupling influence of the complex flexible nonlinearity and two types of uncertainty (matched uncertainty and mismatched uncertainty) of the VESS, a novel robust backstepping control strategy is proposed. As shown in Figure 6, the design flow of the proposed robust backstepping control method can be summarized as follows:
a

Analyze the complex, flexible nonlinearity of tank VESS and consider the dynamics of the control actuator under fully electric drive. Based on the axial stiffness model of the electric cylinder and the modal solution of the flexible barrel, the coupling dynamics model of VESS is established (20).

b

Study the mathematical description method of tank pitch-pointing tracking control, the tracking error of the barrel pitch angle is defined as the control tracking object; describe the coupling dynamics model in the state space of tank VESS (20), considering the matched and mismatched uncertainty; construct the mismatched state space model (32) as the controlled system.

c

Choose the appropriate functions and to make the uncontrolled nominal system present uniform asymptotically stable at the origins; meanwhile, choose the appropriate functions to satisfy Equations (33) and (34). Analyze the boundary conditions of the uncertain part of the VESS to obtain , as shown in Equation (41), and select the control parameter , which leads to the design of the implanted robust controller , as shown in Equation (49).

d

Analyze the boundary conditions of uncertain part of the reconfigured system to obtain , as shown in Equation (66) and select the control parameters , which leads to the actual robust backstepping controller , as shown in Equation (67). According to the practical stability theory described in Theorems 2 and 3, the practical stability of the reconfigured system and the original system is proved.

6 | Simulation Analysis
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6.1 | Parameter Selections
In the simulation, two different desired reference signals are considered for validating the steady-state tracking performance and stability, as well as the dynamic tracking performance and stability of the designed controller. In Simulation I, the pitch angle of barrel is set to adjust from to for validating the steady-state comprehensive performance of the controller, so the desired reference signal is set as , thus, the initial tracking errors (i.e., ) and the initial thrust of the electric cylinder (i.e., ). Meanwhile, the other initial states are set as and . In Simulation II, the pitch angle of barrel is set to adjust from to for validating the dynamic comprehensive performance of the controller, so the desired reference signal is set as , thus, the initial tracking errors (i.e., ) and the initial thrust of the electric cylinder (i.e., ). Meanwhile, the other initial states are set as and .
Then, the system modeling error and external disturbance are considered as system uncertainty to validate the suppression effect of the proposed control method on the complex time-varying uncertainty. For system modeling error, according to the electric cylinder model parameters and past experimental results, choose the axial stiffness of the electric cylinder , for external disturbance. Based on the vibration data of the tank traveling on the complex road, approximately choose the external disturbance torque .
Next, recalling the design process of the proposed controller, it is necessary to choose appropriate functions and to make the uncontrolled nominal system and present uniform asymptotically stable at the origins (i.e., ). Assuming , thus , according to Equation (32), the uncontrolled nominal system can be expressed as
where , according to Equations (33)–(35), choose and . Choose and , so , choose , through Lyapunov equation
can be obtained.
Finally, in the simulation, the parameters of the flexible barrel are set as , the parameters of the servo motor and the electric cylinder are set as , and the installation parameters of the electric cylinder are set as .
To better verify the effectiveness of the proposed pitch-pointing tracking control method of the VESS, the PID controller and sliding mode controller (SMC controller) are selected for comparisons with the proposed controller in the simulation process. For a PID controller, the control input can be designed as
For SMC controller, the control input can be designed as
where
where are the design parameters.
Remark 7. The reasons for selecting PID control and SMC control are: (1) at this stage, the traditional tank fire control system adopts PID control, which is of more practical significance for comparison; (2) SMC control is a class of advanced nonlinear control methods, which has the advantages of fast response, insensitivity to parameter changes and perturbations, no requirement for online identification of the system, and simple physical realization, and so forth, which can further verify the effectiveness of the proposed control algorithms by selecting it for comparison.
In Simulation I, the control design parameters of the proposed controller are set as , for the PID controller, , for the SMC controller, . In Simulation II, the control design parameters of the proposed controller are set as , for the PID controller, , and for the SMC controller, .
The determination of control parameters in this study was guided by a combination of theoretical analysis and simulation-based optimization. The backstepping design provides theoretical guidelines for selecting parameter ranges, while simulation experiments were conducted to fine-tune these parameters for optimal performance. Specifically, the parameters , and were adjusted to balance the trade-off between control accuracy, robustness, and stability. For example, and were tuned to minimize the tracking error while ensuring robustness to uncertainties, and was optimized to guarantee stability during the state variable transformation process. This two-step approach ensures that the control parameters are both theoretically sound and practically effective.
6.2 | Simulation Results
The simulation results are presented in two distinct scenarios, each with different desired signals, as shown in Figures 7-10 and 11-14, respectively.
In Simulation I, Figures 7 and 8 illustrate the comparative tracking error and the comparative pitch angle of the barrel . It is evident that under the proposed control method, the tracking error e converges from −0.1 rad to 0 rad within approximately 5 s and stabilizes near 0 rad. Similarly, the pitch angle of the barrel converges and stabilizes near the desired angle rad after 5 s. In contrast, both PID control and SMC exhibit longer convergence times and significant fluctuations around zero in the steady state. Furthermore, the pitch angle of the barrel takes considerably longer to converge and stabilize near the desired angle, with a maximum steady-state error of up to 0.08 rad. These results highlight the superior accuracy and stability of the proposed control method compared to PID and SMC.
Figures 9 and 10 depict the comparative thrust of the electric cylinder and the comparative control input voltage , respectively. With the proposed control, once the pitch angle of the barrel stabilizes around the desired angle (after approximately 5 seconds), both the thrust of the electric cylinder and the control input voltage reach a stable state, indicating that the entire VESS achieves stability. In contrast, under PID and SMC control, the thrust of the electric cylinder and the control input voltage exhibit continuous fluctuations, making it difficult for the system to reach a steady state.
In Simulation II, Figures 11 and 12 present the comparative tracking error and the comparative pitch angle of the barrel , respectively, while Figures 13 and 14 show the comparative thrust of the electric cylinder and the comparative control input voltage . The results are consistent with those of Simulation I, further validating the accuracy and stability of the proposed control method in dynamic tracking processes. Notably, the influence of system uncertainty and nonlinearity becomes more pronounced when the desired signal is dynamic. Traditional PID and SMC controls fail to effectively suppress nonlinear disturbances, whereas the proposed control method demonstrates superior performance in handling system uncertainty and nonlinearity.
From the simulation results, it is clear that the proposed control strategy effectively addresses the complex flexible nonlinearity and two types of uncertainty (matched and mismatched) in the VESS. However, it is worth noting that the proposed method requires larger control inputs during the initial stage to counteract the system's nonlinearity and uncertainty. This initial demand for higher control effort is a trade-off for achieving enhanced stability and accuracy in the steady state. Overall, the proposed control method outperforms traditional approaches in terms of convergence speed, steady-state accuracy, and robustness under dynamic conditions.
7 | Experimental Validation
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7.1 | Composition Introduction
The experimental platform, constructed using the scaling theory of VESS, is mainly composed of mechanical, electrical, and testing components [13]. The platform's mechanical part mimics the upper structure of a tank, primarily consisting of the gun breech, cradle, front bush, rear bush, and barrel. In addition to the electrical part of the platform, which is electrically controlled by a servo motor and an electric cylinder, enabling the stabilized pitch motion of the tank barrel. The control equipment manages the pitch motion of the test stand as well as its parameter settings. The platform employs a TMS320F283325 DSP with a main frequency of 150 MHz. Parameters such as the working mode, desired angle, and desired angular speed are set through the HMI control interface, after which the electrical equipment must function according to the parameter settings and complete the pitch motion function. The testing part, on the other hand, consists of an inclination sensor mounted on the muzzle, an angle encoder mounted on the trunnion, and a force sensor mounted on the electric cylinder actuator, which realizes the closed-loop control of the vertical electric stabilization system by means of the data from the sensors. Besides, the platform uses a shaking table to simulate the excitation caused by road roughness, with an adjustable weight design featured in the gun breech. External interfaces all use a common interface; the console integrates a certain number of network interfaces, asynchronous serial interfaces, and image interfaces, facilitating the data exchange and information transmission of the test stand.
In the experiment, the shaking table data is set to be saddle ring vibration data of a tank moving at 30 km/h on a Class D road, so as to simulate the driving conditions of a tank moving at 30 km/h on a Class D road. The dynamic parameters of the experimental platform are given in Table 1, and the PID and the SMC are still introduced as the comparison method.
7.2 | Experimental Results
For the same purpose of the simulation, the desired reference signal is set as and in Experiments I and II, respectively.
The comparative tracking error and the comparative pitch angle of barrel in Experiment I are, respectively, shown in Figures 15 and 16. From the experimental results, under the three different control strategies, the system stabilizes in approximately the same time. However, compared to the proposed control, under SMC control, the system always vibrates after stabilization with a large amplitude, and under PID control, the system tends to stabilize at a value that is still somewhat different from the target value. The comparative tracking error and the comparative pitch angle of barrel in Experiment II are, respectively, shown in Figures 18 and 19. The experimental results similarly show that the proposed control has better dynamic tracking performance relative to the PID and SMC controls. The comparative control input voltage in Experiments I and II are, respectively, shown in Figures 17 and 20. The change rule of the control input voltage basically keeps the same trend with the corresponding error curve, characterizing the reasonableness and effectiveness of the control strategy verified by this experimental platform; moreover, the control voltage also presents a stable state after the system tends to be stabilized under the proposed control strategy.
8 | Conclusion
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To enhance the stability and accuracy of pitch-pointing tracking control, this study proposes a novel robust backstepping control strategy that explicitly incorporates flexible nonlinearity and mismatched uncertainty into the dynamics modeling process of the VESS. The research framework is structured as follows: First, a nonlinear coupled dynamics model is established by integrating the axial stiffness model of the electric cylinder with the modal solution of the flexible barrel, thereby capturing the system's flexible coupling effects. Second, to address both matched and mismatched uncertainties, a mismatched state space model, which consists of two interconnected subsystems, is developed. Third, through the application of backstepping design principles, a state variable transformation is implemented to convert the mismatched uncertainty boundary, leading to the development of an innovative robust control method. This approach guarantees practical stability for both the original and reconfigured systems. Extensive simulation and experimental results demonstrate the superior performance of the proposed method compared to traditional control approaches. Notably, this study represents the first attempt to introduce flexible nonlinearity into the pitch-pointing tracking problem of VESS, offering a groundbreaking perspective for future research in this domain.
However, the method relies on accurate modeling of flexible nonlinearities and uncertainties, which can be challenging to achieve in real-world scenarios with limited sensor data and changing environmental conditions. Secondly, the computational complexity of the control algorithm may increase the system control cost. Future research is expected to develop data-driven modeling techniques to reduce the reliance on precise analytical models. Further bridge the gap between theoretical research and practical application.
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Appendix
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Year 2025 volume 5 Issue 3
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doi: 10.1002/msd2.70029
  • Receive Date：2025-01-20
  • Online Date：2026-03-24
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  • Received：2025-01-20
  • Revised：2025-03-25
  • Accepted：2025-04-09
Affiliations
    School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, China

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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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