LI Zongxing and ZHANG Rui

Beijing Institute of Electronic System Engineering, Beijing 100854, China

Abstract: This paper proposes a time-varying sliding mode control method to address nonlinear missile body kinematics based on the suboptimal control theory. The analytical solution of suboptimal time-varying sliding surface and the corresponding suboptimal control law are obtained by solving the state-dependent Riccati equation analytically. Then, the Lyapunov method is used to analyze the motion trend in sliding surface and the asymptotic stability of the closed-loop system is validated. The suboptimal control law is transformed to the form of pseudo-angle-ofattack feedback. The simulation results indicate that the satisfactory performance can be obtained and the control law can overcome the influence of parameter errors.

Keywords: suboptimal control, Riccati equation, analytic solution, sliding mode control, nonlinear control.

1. Introduction

The optimal control method is a well-known method in the design of the linear time-invariant system. The optimal solution can be obtained by designing the target function and solving the corresponding Riccati equation [1-5].The traditional design of the nonlinear control system is to select the specific operating points and linearize the original system, and then design the control law based on the approximate linear system [6,7]. The suboptimal control method which evolved from traditional optimal control has been widely used in the nonlinear control system design [8-14]. The application of suboptimal control is based on the pseudo-linear structure constructed from the original system model. The crux of system design and analysis is the stability analysis of the closed-loop system in the operational range [15].

The traditional way of getting the suboptimal control law is based on the numerical solution of the state-dependent Riccati equation, which hinders the stability analysis and prompts researchers to find the analytical solution of Riccati equation. In [16,17], the analytical solution of Riccati equation in the specific form has been used to design and analyze the nonlinear system with the particular structure. In [18], the analytical solution of the two-dimensional state-dependent Riccati equation was proposed to design the suboptimal control law and analyze the stability of the closed-loop system. The suboptimal control law proposed in [18] is composed of the feedback of acceleration and the pitch rate, ensuring good performance under the nominal model. However, when the actual value of the model parameters deviates from the nominal value, the control effect will quickly deteriorate. It is known that in actual application, it is hard to get the accurate model, indicating that the model bias always exists [19]. Hence, for the design of missile autopilot, the robustness of the control system must be considered.

Sliding mode control is a nonlinear control method widely used in engineering design [20-24]. This control method shows good robustness by designing appropriate sliding surface, ensuring satisfactory response performance in the presence of model parameter deviation [25].To obtain better control effect, the traditional linear timeinvariant sliding surface has been improved to the timevarying sliding surface [26-28]. In [29], a kind of sliding mode control method based on state transition was proposed aimed at the multivariable system, which was combined with the optimization of the indicator function.

In actual engineering, we often encounter time-varying systems in which the model parameters change with states, and the design of time-invariant sliding surfaces may not guarantee the control effect under different system states. By using the measurable system state information to adjust the parameters of the sliding surface, the design of the time-varying sliding surface can ensure that the control effect will not change significantly under different conditions. In [27] and [28], the time-varying sliding mode control law was designed for the inverted pendulum and the robotic manipulator, and achieved good control effects. In [30], the design using time-varying sliding surface overcame the influence caused by the nonlinear aerodynamic parameters.

Aimed at the nonlinear system, designing sliding surface with the suboptimal theory is an approach to improve sliding mode control. In [30], the linear approximate approach was used to acquire the suboptimal sliding mode control law based on the missile nonlinear model. However, it is difficult to analyze the stability in the sliding mode designed in [30], because only the numerical solution of the sliding surface can be obtained. Although the suboptimal sliding mode control method has good robustness, further study is needed to analyze the stability in the nonlinear sliding surface.

In this paper, aimed at the missile nonlinear model, the state transition is utilized to design the suboptimal sliding surface, and its analytical expression is solved to design the control law. On the basis of analytical solution,the stability in the nonlinear time-varying sliding surface can be acquired, and the control law can be adjusted to the form of pseudo-angle-of-attack feedback.

2. Missile model

In this paper, the object of study is the missile longitudinal model, and the coordinate system is shown in Fig.1 [18].

Fig. 1 Missile longitudinal model

Consider the following model:

where α,ωy,δrepresenttheangle of attack,thepitch rate and thefindeflection angle,respectively.Ωisthe operational range of α , i.e., α∈Ω .q,S,m,d,Iy,Vrepresent the dynamic pressure, the characteristic area, the mass, the characteristic length, the moment of inertia about the pitchaxisandthetotalvelocity,respectively.CZα,CZδ,CMα,s ystem,CZα,CZδ,CMδ,CMωyare negative,andthesign of when themissileisstaticallystable,CMαhasanegative value, otherwiseCMαispositive.CMδ,CMωyareaerodynamiccoefficients. InthiscoordinateCMαis dependentonthestaticstabilityofthe missile, i.e.,

Becauseb1,b2are constants, (4) can be represented as follows:

The system matrix of (6) is denoted asA(α), andaij(i=1,2,3,j=1,2,3)represents the element of the matrix.

3. Suboptimal sliding mode control law design

To facilitate the design of the control law, assume that the acceleration command is 0, i.e., the last term of (6) can be neglected, and the system response will tend to be 0.

Remark 2The assumption that acceleration command is 0 is based on the transition from the reachability problem to the controllability problem. Although reachability is not equivalent to controllability for the nonlinear system, the approximate treatment is reasonable in the engineering design.

Noting that input control term δ only exists in the third line of the matrix of (6), the sliding surface can be defined as

Under sliding mode control, the system motion state can be divided into two parts: the state variables move from outside the sliding surface to inside and then converge to origin in the sliding surface. Defineu=δ. The control termuonly directly acts on the third line of the matrix of (6) to prompt the system states to converge to the sliding surface. When states arrive, σ(z,t)=0, and the following equation holds:twolines ofthe matrixof (6).

Atthe moment,z3worksas a control term for the first

First, design sliding mode control lawuto realize the state movement in the first part. Take the derivative of the sliding mode variable σ and obtain the following equation:

The control law can be designed as

wherek＞0 is a constant parameter designed. Substituting (10) into (9), the following equation can be obtained:

After the states arrive at the sliding surface, the system can be denoted as

The key to obtain the analytical expression of the suboptimal sliding surface is to solve matrixPby the analytic method.

Choose

ProofObviously, the Riccati equation (19) satisfies Condition 1 and Condition 2 in Lemma 1. Verify Condition 3 as follows.

The controllability matrix is as follows:

Noting thata13a22-a12a23=0 , the determinant ofMccan be derived as follows:

Becausea3＜0 , under Assumption 2,a13＞0 .xcf,represent the distances from the nose of the missile to the aerodynamic center, the center of pressure and the center of gravity, as shown in Fig.1. Then the following equations hold [18]:

In this paper, we only study the missile with normal configuration, i.e.,xcp＜xcf. Using (25) and (26), the following relations hold:

The observability matrix is as follows:

4. Analysis of stability in sliding surface

Substituting (33) and (34) into (8) and (12), the following relations hold:

where ΔM=(a11a22-a12a21)2.

ProofConsider the following Lyapunov candidate function:

Obviously,VLis positive and bounded. Take the derivative ofVLand the following equation can be obtained:

The characteristic polynomial ofAκis as follows:

Analyze the second term in the right-hand side of (48),and the following relation holds:

Analyze the first term in the right-hand side of (51),and the following equation can be derived:

Analyze the second term in the right-hand side of (51),and the following relation can be acquired:

Noting the first term in the right-hand side of (53), the following inequality holds:

5. Control law in the form of pseudo-angle-ofattack feedback

In the third section, the suboptimal control law has been designed. In this section, the control law (10) will be transformed into the form of pseudo-angle-of-attack feedback as shown in Fig. 2.

Fig. 2 Closed-loop system with pseudo-angle-of-attack feedback control law

Theorem 3Under Assumption 1, the control law(10) can be rewritten as follows:

where

ProofUsing (5) and (10), the following equation can be obtained:

The proof of Theorem 3 is completed. □

6. Simulation results

The model of the missile at the altitude of 6 000 m is as follows:

whereMais the mach number and Ω={α∈R|π/3＜α＜π/3}. In this section, the simulation results for the missile at 2Ma(static stable) and 3Ma(static unstable) are given. At 2Ma, the following relation holds:

At 3Ma, the following relation holds:

The parameter deviation is as follows:

To avoid the overlarge rudder deflection angle, step inputs are processed with the transition function.

The simulation results at 2Maare shown in Fig. 3-Fig. 6. The simulation results at 3Maare shown in Fig. 7-Fig. 10. As shown in Fig. 3 and Fig. 7, the control laws in[18] and this paper all have good performance under nominal models. However, as shown in Fig. 5 and Fig. 8,under real models with parameter deviation, the control law in [18] will result in steady state error and the control law proposed in this paper still keeps good tracking performance. Hence, the suboptimal sliding mode control law is more robust.

Fig. 3 Simulation results under nominal model at 2Ma

Fig. 4 Rudder deflection angle under nominal model at 2Ma

Fig. 5 Simulation results under real model at 2Ma

Fig. 6 Rudder deflection angle under real model at 2Ma

Fig. 7 Simulation results under nominal model at 3Ma

Fig. 8 Rudder deflection angle under nominal model at 3Ma

Fig. 9 Simulation results under real model at 3Ma

Fig. 10 Rudder deflection angle under real model at 3Ma

7. Conclusions

In this paper, the analytical solution of the suboptimal sliding surface has been proposed and the stability in the sliding surface has been proved. The suboptimal sliding mode control law shows good performance and can overcome the effect of parameter error.

The control law designed in this paper can be written in the form of pseudo-angle-of-attack feedback, and each parameter in the control law is the function of the dynamic coefficient. Given the speed and altitude, the parameters are mainly determined by the angle of attack. The purpose of designing this control law is to overcome the influence of aerodynamic nonlinearity by using the angle of attack information, and to provide the functional relationship of the control parameters with the angle of attack (dynamic coefficient). In practical applications, the control parameters can be calculated by substituting the information of speed, altitude and angle of attack into the corresponding functions, and the control law in the form of the continuous function replaces the original control law in the form of the interpolation table.