Robust adaptive fuzzy control design for nearspace vehicle

1Introduction

It is well known that the fuzzy control technique provides a means of collecting knowledge and expertise. Over the past decade, it has proved to be very useful in many applications [8, 10, 13, 14, 19]. It is not surprising that T-S fuzzy models have become one of the most useful control approaches for complex nonlinear systems. Many nonlinear systems can be represented by T-S fuzzy systems, allowing designers to take advantage of conventional linear system methods to for design and analysis [2, 4, 6, 7, 18, 21, 22]. The fuzzy adaptive control can not only automatically adjust to control rules in the face of change in performance and parameters of the controlled object, but also enhance the adaptive ability to deal with environmental changes and realize the purpose of control.

The control design of NSVs has attracted increasing attention in recent years. The primary reason is that they have potential and promising applications in both military and civilian fields. Since a Nearspace vehicle is a complex dynamic system, it is difficult to study according to traditional control methods. To overcome this limitation, a T-S fuzzy control scheme has been considered to cope with such problems. The major advantage of this scheme is that an accurate mathematical model is not necessary, and consequently, the T-S fuzzy control theory is suitable for the design of the flight control system of an NSV [5, 15]. However, the Nearspace hypersonic vehicle dynamics are severely nonlinear, time-varying, highly uncertain and strongly coupled. It also suffers from different external disturbances and uncertainties due to changes in the flight environment. Therefore, ensuring the robust stability of the NSV flight is challenging. To date, this subject has not been fully investigated.

This paper proposes the design of a feedback and feedforward control for T-S fuzzy systems, which has been applied to the tracking control of attitude angle of the NSV. The organization of the paper is as follows: Section 2 describes design formulation. Section 3 describes the tracking controller design of the NSV, and presents the design of an anti-interference composite controller, as well as the calculation method of feedforward gain and feedback gain by LMI. The simulation results which demonstrate the effectiveness of the proposed approaches are presented in Section 4, followed by conclusions in Section 5.

2Problem formulation

The mathematic model of the NSV developed at NASA (National Aeronautics and Space Administration) Langley Research Center is given asfollows:

where α is the angle of attack, β is the sideslip angle, μ is the bank angle, p is the roll rate, q is the pitch rate and r is the yaw rate.

According to singular perturbation theory, the six equations can be divided into the fast loop and the slow loop, respectively. The above attitude motion equations can thus be rewritten as follows:

According to the following: x(t) = [ω(t)^T, Ω(t)^T], u(t) = T_C , y(t) = ω(t) system (2) can also be represented as follows:

3The tracking controller design for NSV

4Simulation

In this section, simulation results are presented in order to illustrate the effectiveness of the proposed robust control scheme. Considering the nonlinearity of NSV dynamics, the aerodynamic coefficients are taken as the nominal cruising flight; the nominal flight of an NSV occurs at a trimmed cruise condition (V = 2500 m/s, H = 45 km); the initial attitude angle conditions are chosen as α = 1°, β = -1°, μ = -1°; the body-axis angular rate is assumed to be p = q = r = 0°/s; the reference commands are α_c = 3 .4°, β_c = 0°, and μ_c = 1 .2°, respectively. The external disturbance of the NSV system is 2 × 10⁵ · [sin(2t) , sin(2t) , sin(2t)] ^T Nm, and the system exists – 20% aerodynamic parameter perturbation.

According to Theorem 2, the feedback control gains K_j can be obtained as follows:

The reference model is given as follows:

In Theorem 3, K_cj =  F_j -  K_j D were obtained.

The simulation results indicate the following conclusions. Figure 1 depicts the tracking curve of the angle of attack α; Fig. 2 represents the tracking curve of the sideslip angle β; Fig. 3 depicts the tracking curve of the bank angle μ; and Fig. 4 shows the p, q, r state response curve (p is the roll rate, q is the pitch rate and r is the yaw rate.). The variables α_c, β_c, μ_c are the reference commands of attitude angle α, βμ, respectively.

As shown in Figs. 1 through 4, the closed-loop system is asymptotically stable under the proposed robust compound controller. Thus, the proposed fuzzy control scheme based on feedback control and feedforward control is validated.

5Conclusion

T-S fuzzy control schemes have been developed for the NSV system with functional uncertainty and external disturbance. Feedforward and feedback composite control is adopted to eliminate the external disturbance of the NSV. The controller design has been implemented in a unified manner in which gains are solved according to a set of LMIs. Simulation results have demonstrated the effectiveness of the proposed model. NSV is a complex dynamic system; future work will take random factors and stochastic noises in developments of NVS models into accountance [11, 12, 16], and will study attitude tracking control and accommodation approaches to NSVs with functional uncertainty and external disturbance.