This paper designed a smooth fixed-time-convergent sliding mode controller for a missile flight system considering aerodynamic uncertainties. Fixed-time convergence theory is incorporated with the sliding mode control technique to ensure that the system tracks desired commands within uniform bounded time under different initial conditions. Unlike previous terminal sliding mode approaches, not only is the bound of settling time independent of initial state, indicating that performance metrics like convergence rate can be predicted beforehand, but the control input is designed to be smooth based on adaptive estimations and some mathematical results without introducing any discontinuous items like the signum function, which avoids the problem of chattering consequently. A cascade control structure is employed with the derived control algorithm, and therein, the control input signal is obtained. Finally, a number of simulations are carried out and demonstrate the effectiveness of the designed controller.

Increasing demand for high accuracy and system reliability has stimulated the development of control techniques in the past decades. For the missile flight system, a well-behaved controller can not only track the command signal swiftly but also exhibit appropriate robustness against disturbances and uncertainties which exist throughout the fickle flight regime. From the perspective of overall design, it is also preferable to evaluate the control performances as much as possible once the preliminary controller design is accomplished.

The conventional missile autopilot focused on improving the poor underdamped dynamics of the missile airframe. By adding feedback loops, the PI regulator [

Among the myriad control approaches for missile controller design, the sliding mode control (SMC) technique is widely used and performs remarkably against uncertainties and disturbances without exact modeling of the uncertainties or disturbance estimation [

However, in the abovementioned works, where finite-time convergence characteristic is incorporated, exhaustive estimations of settling time are not addressed. Only theoretical results are derived in the citation of lemmas. From the expression of theoretical settling time, prior knowledge of the initial system state must be known to estimate the convergence rate. Different initial conditions correspond to various settling time, of which the maximum value depends on the maximum initial errors related to the Lyapunov function. In the preliminary design of a missile control system, it would be very helpful if the settling time can be predicted without information of the initial conditions. Thus, in contrast with existing finite-time controllers, the upper bound of the settling time could be estimated in the sense of the fixed-time convergence concept [

Further, variable structure items, which comprise discontinuous parts like the signum function, are normally introduced in the SMC control methodologies mentioned above to suppress the disturbances. The discontinuous signum function induces the well-known chattering problem as the switching gain must be chosen larger than the bound of uncertainty, which results in a degradation of system performance to some considerable extent under specific occasions. Some works replaced the discontinuous item with saturation function or sigmoid function [

Inspired by the previous discussion, this paper designed a new robust smooth sliding-mode-based controller with fixed-time convergence. An adaptive estimation and some primary mathematical results are utilized to alleviate the effect of uncertainties, as well as deducing the control algorithm. Unlike existing works, the designed smooth fixed-time-convergent sliding mode controller motivates missile control variables to converge to the equilibrium point before the uniform bounded settling time in the presence of aerodynamic uncertainties with its input inherently continuous without using any discrete items, like the signum function. The upper bound of settling time here is simply a function of designed parameters, and therefore, prior knowledge of the convergence rate can be evaluated in advance without any information of system initial conditions. With a uniform bounded convergence time, the controller can track the desired command in the presence of aerodynamic uncertainties under different initial state situations, which is very helpful for both preliminary design and performance evaluation. The smooth input avoids the problem of singularity and chattering, which ensures better performance of the missile control system and exhibits superior availability for practical application. Comprehensive simulations considering a nominal missile dynamic model with aerodynamic uncertainties are carried out to demonstrate the effectiveness of the designed attitude controller. The fixed-time convergence characteristic is fully reflected under the cases of various controller gains and initial states, and the smooth input exhibits nice continuity through comparison with the conventional terminal sliding mode method.

The rest of this paper is organized as follows. In the forthcoming section, a missile dynamic model with aerodynamic uncertainties is presented. Next, some preliminaries involving primary mathematical results and the fixed-time convergence theory are introduced, and then a smooth fixed-time-convergent sliding mode controller is designed in Section

Typical missile control embodies the tracking of commanded attitude angle. To begin with, some fundamental assumptions are introduced.

The variation of static parameters like mass and moment of inertial through entire flight is not taken into consideration.

Trustworthy measurements of attitude angles and angular velocities are available with high precision sensors.

Aerodynamic uncertainties and their derivatives are bounded.

In the controller design, missile velocity and flight altitude are assumed to be constant.

Regardless of gravity, the longitudinal model of the missile in the presence of aerodynamic uncertainties is considered here:

In the above equation, _{z}_{z}_{α}_{δz}_{α}_{δz}_{ωz}

For our convenience, (

System (_{1} and _{2}, items like

The dynamic model developed in Section

Two fundamental equations are presented here. For any

Equations (

Another important synthesis in this paper is the concept of fixed-time stability, which can be assumed as the extension of finite-time stability.

Consider the nonlinear system [

The origin is a finite-time stable equilibrium if it is Lyapunov stable and for any given initial time _{0}, such that for every solution of system (

Moreover, if the origin is finite-time stable with

Consider the differential equation of system (

In addition, if

From Lemma

Considering system (

Consider a scalar system

As denoted in (

This part derived a robust smooth sliding-mode-based controller with fixed-time convergence for system (

Suppose the desired command is

Define a sliding manifold as

Differentiating

To achieve fixed-time convergence, the reaching law is designed in the form

Substituting (

Next, with the obtained virtual control input, the control law for the inner loop can be designed under the same baseline.

Similar procedures are conducted in the inner loop. The objective is to deduce the actual control input such that tracking error between the virtual control signal _{2c} and the actual angular rate variable _{2} converges to zero within a fixed bounded time. Rewrite the second equation of system (

Another sliding manifold is defined as

Taking the derivative of

In a similar way, the reaching law for inner loop is designed as

Considering (

It can be observed from (

In this section, closed form stability analysis of system (

For the dynamic system (

For the outer loop, consider the Lyapunov function

Differentiating _{1} with respect to time and substituting (

Considering (

According to the boundedness theorem,

Similarly, define another Lyapunov function of inner loop as

Taking the derivative of _{2}, it follows from (_{2} and

For the integral system, consider the Lyapunov function

Combined with (

Denote

Equation (

To ensure the stability of the integral system, designed parameters are chosen to satisfy _{31}^{∗} ≥ _{1}/_{1}^{2} and _{32}^{∗} ≥ _{2}/_{2}^{2}. To simplify the design process and keep its uniformity, other parameters can be chosen conveniently in the form _{11} = _{12} = _{1,} _{21} = _{22} = _{2,} _{1} _{2} _{1} _{2} _{1} _{2} _{1} _{2}

Considering Lemma _{max}. The fixed settling time is bounded and can be expressed as follows:

For the control variable

Further, (

It is noticed in (_{2c} should be calculated. Because of the fact that _{x1} is a function of control input, it is difficult to deduce

Suppose the estimate error

Substituting (

Similar with the previous process, for the integral system, the derivative of Lyapunov function _{3} can be deduced in the same form with (

In this section, a missile control problem in the longitudinal plane in the presence of aerodynamic uncertainties is considered. The dynamic model of (

The flight speed and altitude are Mach 5 and 25 km, respectively. Parameters of a generic tail fin controlled missile are given in Table

The parameters for a dual-control missile.

Variable | Value |
---|---|

2.63 | |

0.45 | |

0.86 | |

10.98 | |

7.47 | |

Control parameters are chosen to be

This case simply takes the tracking ability of fixed constant command into consideration. The coefficients of aerodynamic forces and moments are all increased by 30% of their respective nominal values. First, the initial system state is set to be

Response of the angle of attack.

Time history of elevator deflection.

As visualized in Figure

Next, Monte Carlo simulations are carried out to verify the fixed-time convergence property of the designed controller, which is independent of system initial conditions. In this occasion,

Figure

System response with the designed controller.

Response of the angle of attack when

Response of the angle of attack when

Response of the angle of attack when

As calculated above, theoretical bounds of settling time with the designed controller are 1.1 s, 0.88 s, and 0.73 s, respectively. Performance diagrams in Figure

A terminal sliding mode controller (TSMC) is used here with identical designed parameters for comparison. The expressions are given in (

System response with TSMC controller.

Figure

In this case, it is assumed that aerodynamic uncertainty coefficients are increased by 20%, 40%, and 60%. A continuous signal is generated as turn commands for the missile to maneuver, of which

Time history of the angle of attack.

Response of the angle of attack

Details of response in the pitch plane

Elevator deflection.

Time history of elevator deflection

Details of elevator deflection

It can be seen from Figure

To show the effectiveness of the proposed smooth controller further, the TSMC controller is simulated with identical designed parameters for comparison.

On this occasion, aerodynamic uncertainty is assumed to increase by 20%. Figures

Time history of the angle of attack.

Response of the angle of attack

Details of response in the pitch plane

Elevator deflection.

Although it seems both controllers can track the continuous command, the control quality of the designed controller is superior to that of the TSMC method. It can be observed from the above figures that the chattering problem exists in the performance of the TSMC controller. With the designed smooth controller, the input signal is smooth without singularity and chattering, which illustrates better tracking performance than the TSMC approach.

This paper designed a smooth fixed-time-convergent sliding mode controller for a missile flight system in the presence of aerodynamic uncertainties. Based on fixed-time stability theorem, some primary mathematical results, and adaptive estimations, the derived control algorithm guarantees that control variables converge to the desired command within fixed uniform bounded time regardless of initial conditions with a smooth control input. The bounded convergence rate can be predicted in advance without knowledge of system original states, which is of significant value in system preliminary design, as well as performance evaluation. Meanwhile, the input signal is designed to be continuous without introducing any discrete items like the signum function, thus eliminating the chattering phenomenon and facilitating practical application significantly. In the end, extensive simulations are carried out and validate the effectiveness and robustness of the designed controller.

The authors declare that there is no conflict of interest.