Optimal Sliding Mode Controllers for Attitude Stabilization of Flexible Spacecraft

The robust optimal attitude control problem for a flexible spacecraft is considered. Two optimal sliding mode control laws that ensure the exponential convergence of the attitude control system are developed. Integral sliding mode control ISMC is applied to combine the first-order sliding modewith optimal control and is used to control quaternion-based spacecraft attitudemanoeuvres with external disturbances and an uncertainty inertia matrix. For the optimal control part the state-dependent Riccati equation SDRE and optimal Lyapunov techniques are employed to solve the infinite-time nonlinear optimal control problem. The second method of Lyapunov is used to guarantee the stability of the attitude control system under the action of the proposed control laws. An example of multiaxial attitude manoeuvres is presented and simulation results are included to verify the usefulness of the developed controllers.


Introduction
In recent years considerable attention has been focused on the optimal control problems of a spacecraft.Various optimal control methods have been proposed for solving the attitude control problems of a rigid spacecraft.Nonlinear H ∞ control was used in 1 to design a stabilizing feedback control for the spacecraft tracking problem.Sharma and Tewari 2 devised a Hamilton-Jacobi formulation for tracking attitude manoeuvres of spacecraft to derive a nonlinear optimal control law.An H ∞ inverse optimal adaptive controller was applied to attitude tracking of spacecraft by Luo et al. 3 .An adaptive control and nonlinear H ∞ control were merged to design robust optimal controllers.An alternative way to design a robust optimal controller is to use an optimal sliding mode controller design scheme.Sliding mode control SMC is a very effective approach when applied to a system with disturbances which satisfy the matched uncertainty condition.In recent decade, SMC has widely been extended to incorporate new techniques, such as adaptive sliding mode control, higher-order develop an optimal control law and the ISMC concept is used to merge the resulting optimal controller and first-order sliding control.The control law obtained could be called an optimal Lyapunov-based SM controller.
This paper is organized as follows.In Section 2 the dynamic equations of a fexible spacecraft and the attitude kinematics 33, 34 are described.Section 3 presents the integral sliding mode 16 technique.In Section 4 we discuss an optimal sliding mode controller and study the sliding manifold and control law in the stability proof of this controller.Section 5 describes the SDRE technique and includes a systematic procedure for the design of the SDRE controller.In Section 6 the optimal Lyapunov technique is used to produce an optimal feedback controller that yields global asymptotic stability.A controller design procedure is proposed and the stability proof of this controller is investigated.In Section 7 an example of spacecraft attitude manoeuvres is presented in order to make comparisons between the SDRE-based SM controller and the optimal Lyapunov-based SM controller.In Section 8 we present conclusions.

Mathematical Model of Flexible Spacecraft
A flexible spacecraft is composed of a rigid main body and some flexible appendages.The kinematics of the spacecraft determine the attitude of the main body and are described by the four unitary quaternions 34 where e ∈ R 3 and β denote the Euler axis and Euler angle, respectively.We define here the quaternion q q 0 q T T with q ∈ R 3 .Then the kinematic equations are described in terms of the attitude quaternion 34 and are given by q 1 2 where ω ∈ R 3 denotes the angular velocity vector and I 3×3 is the 3 × 3 identity matrix.The skew-symmetric matrix q× is and the elements of q are restricted by q 1.Note that a quaternion consists of the scalar q 0 and the three-dimensional vector q, so it has four components.The scalar term is used for avoidance of singular points in the attitude representation 34 .The quaternion kinematics equation is required to be solved for all four components.However, to indicate the orientation of the spacecraft or a rotational motion, it is sufficient to use only the vector q because this vector completely shows rotation axis and angle.Furthermore, the scalar q 0 can be calculated easily using the vector q and the condition q 1.More details of quaternion and other attitude representations can be found in 34, 35 .The equation governing a flexible spacecraft is expressed as 33 where J J T is the total inertia matrix of the spacecraft, η is the modal displacement, and δ is the coupling matrix between the central rigid body and the flexible attachments.u ∈ R 3 denotes the control input, d ∈ R 3 represents the external disturbance torque, and K and C denote the stiffness and damping matrices, respectively, which are defined as where

2.8
Note that because of the constraint relation among the four unitary quaternions q 2 0 q T q 1, and q 0 are not independent of the other components of the quaternion.

ISMC
The concept of integral sliding mode 16 concentrates on the robustness of the motion in the whole state space.It is constructed without the reaching phase and it ensures insensitivity of the desired trajectory with respect to matched uncertainties, starting at time zero.However, since this controller design requires a first-order SMC, the main drawback of sliding control, namely, the chatter problem, is encountered.
Integral sliding control is applied to obtain robustness against external disturbances during the sliding mode.Next the concepts of ISMC 16 are discussed.
Consider a nonlinear dynamic system ẋ F x G x u, 3.1 where x ∈ R n , u ∈ R m .Suppose there exists a feedback control law u u 0 x such that system 3.1 can be stabilized to follow a desired trajectory.
For system 3.1 the control law 16 is designed as where u 0 is the ideal control and u 1 is designed to be discontinuous to reject the disturbances.The switching function 16 is designed as The switching function consists of two parts; the first part s 0 x may be designed as the linear combination of the system states similar to the conventional sliding mode design ; and the second part γ induces the integral term and will be determined as where γ 0 is determined based on the requirement s 0 0 since the sliding mode occurs at time zero.To derive the sliding mode equation, the time derivative of s on the system trajectories should be made equal to zero.The equivalent control u eq is determined by solving the algebraic equation ṡ 0 with respect to the control input and then substituting into the equation for u.

Optimal Sliding Mode Controller Design
We continue the controller design by using the ISMC technique.The way to obtain the optimal performance is to design an ideal control in 3.2 using nonlinear optimal control techniques.So our designs will consider an ideal control as the optimal control law for nonlinear attitude regulation systems.In this paper the SDRE and the optimal Lyapunov approaches are used to produce optimal control laws and the details will be given in subsequent sections. .We use the integral sliding mode concept to obtain the sliding manifold and a new robust optimal control law is then developed.The second method of Lyapunov is used to show that reaching and sliding on the manifold are guaranteed.
Using 3.3 and letting s 0 x ω κq, the switching function is designed as where κ is a 3 × 3 symmetric positive-definite constant matrix and φ is an auxiliary variable that is the solution of the differential equation 16 with υ * being the optimal controller for the system 3.1 .Using the ISMC 3.2 we obtain an optimal sliding mode controller where μ is a m × m positive definite diagonal matrix, and τ ∈ R m the ith component of τ is given by

4.4
Now we show that the control law 4.3 is designed such that the reaching and sliding mode conditions are satisfied.The candidate Lyapunov is selected as 18 and the time derivative of V is Vs s T ṡ.

4.6
With the substitution of ṡ and 4.2 we obtain With external disturbances and the optimal control υ * , the control law 3.2 can be written as Substituting 4.2 and 4.8 in 4.7 , the time derivative of V s can be written as

4.9
Let the discontinuous control input u 1 have the following form where μ ∈ R m×m is a positive definite diagonal matrix.Letting Ψ ∂s 0 /∂x G x , we obtain Vs s T Ψ ξ − μ sign s .

4.11
We choose s 0 such that Ψ is positive definite and then 4.11 becomes Vs |s| Ψ ξ sign s − μ .

4.12
Obviously if μ is chosen such that μ > sup |ξ|, then Vs < 0. This guarantees reaching and sliding on the manifold.

SDRE Controller
In this section we mention briefly the optimal controller design scheme using the SDRE approach.The regulation motion of a flexible spacecraft is considered.The Xu method 15 and the integral sliding mode 16 are merged to design a new controller which consists of two parts; the sliding mode and optimal control.The first-order sliding mode is used for the sliding mode controller design while the optimal control law is designed using the SDRE method 20 to solve the infinite-time optimal quadratic problem.
The SDRE method requires factorization of the nonlinear dynamics into the state vector and the product of a matrix valued function which depends the state itself.This matrix is so called the state-dependent coefficient SDC matrix.For the optimal controller design, the difficulty of using the SDRE approach is how choose the appropriate SDC matrix.Using the basic concepts described in 32 we can rewrite the spacecraft dynamics equation in a more suitable form and the appropriate SDC matrix is then selected.After we obtain the optimal control law, a new optimal sliding mode controller will be designed by merging the optimal control and first-order sliding mode controller.
This controller is designed such that it minimizes the performance index

5.2
Here, the state x ∈ R n is defined as x ω T q T ϑ T T .The weighting matrices Q x , N x , and R x are positive semidefinite and positive definite, respectively.f x and g x are all sufficiently smooth functions of the state vector x t , and x 0 is the initial conditions of the process.It is assumed that f 0 0 and g x / 0 for all x.To apply the SDRE method f x must be decomposed as f x a x x.Using the theorem in 32 the SDC matrix a x can be obtained by writing f x in a more suitable form as To use the SDRE approach the SDC matrix a x is chosen as Thus, the optimal control υ * 15 is given as where Π x is the solution to the generalized SDRE 15

5.6
Note that the parameterization in 5.3 is not unique.It has explicitly been selected to make the stability problem analytically tractable.It is convenient to use this parameterization, since the state-dependent Riccati equation 5.6 will have a simple form to be solved for Π x .Next, it should be checked whether the parametrization in 5.3 is pointwise controllable for all x.
The controllability matrix is where

5.8
Clearly the condition for pointwise controllability is where |M C | denotes the determinant of the matrix M C .Letting T q q× q 0 I 3 , one obtains

5.10
Using 5.10 we can ensure that |M C | / 0. This implies that M C has full rank.In Cimen 36 it was stated that the observability matrix has full rank by choosing the Q x positive definite for all x ∈ R n .If our design selects Q x to be positive definite, the observability matrix will be a full rank matrix and the sufficient condition for observability is satisfied.Thus, the SDRE method generates a closed loop solution, which is locally asymptotically stable.However, the success of the SDRE approach depends on a good choice of the matrix a x .It is difficult to obtain global stability because of the limitations of this technique.
After we obtain the matrix a x , an optimal controller υ * can be designed using 5.5 .Substituting this control into 4.3 the SDRE-based SM controller is obtained.

Optimal Lyapunov Controller
This section presents an optimal control moment which stabilizes the zero solution and minimizes the selected performance index 5.1 .The basic principles in Krasovskii 37 with a Lyapunov function are applied to develop an optimal controller.The stability proof of this control law is also performed by using the optimal Lyapunov technique see 28-31 .

Mathematical Problems in Engineering
The optimal Lyapunov function, defined as the minimum value of the performance index 5.1 , is given by The function V x represents the value of the integral performance index when evaluated along the optimal trajectory.Let the positive definite symmetric matrix Q ∈ R n×n and the matrix N ∈ R m×n be partitioned as respectively.We next show that V x can be treated as a Lyapunov function.Based on Krasovskii's theorem, the function V x satisfies the following equation: We assume that u o is the optimal control law.From the optimality conditions of Krasovskii's theorem, function V must satisfy the following partial differential equation: which is known as the Hamilton-Jacobi equation.Clearly, the resulting partial differential equation 6.4 contains the optimal control u o .In particular if the optimal control u o is known, we can solve this equation for the optimal Lyapunov function V .On the other hand if the function V is known, this equation can be solved for the optimal control u o .In fact both solutions of this equation and control torques are dependent on each other.Substituting 2.2 and 2.7 into 6.4 , one obtains

Mathematical Problems in Engineering 11
Let a Lyapunov function be chosen as where P is a positive-definite matrix that is a solution of the Lyapunov equation A T P PA −Q A with a positive-definite matrix Q A .The gradient of V with respect to ω, q and ϑ is given by This function consists of the sum of quadratic terms, so it is a positive definite function with respect to stabilizing variables.Substituting 6.7 into 6.5 , one obtains

6.8
After lengthy calculus manipulation, one can obtain the optimal control u o as u o −αq − Λω. 6.9 We can conclude that 6.5 can be solved by using the Lyapunov function V x and the controller u o with the following relations:

6.10
We also prove that the optimal control 6.9 with the chosen Lyapunov function 6.6 yields global asymptotic stability.For the purpose of the stability analysis, the first time derivative of V x is considered and it takes the form V x ω T J mb ω 2α q 0 − 1 q0 2αq T q ϑ T P θ. 6.11 Substituting 2.7 into 6.11 , we obtain  which can be further expressed as which can be further expressed as where We know that there exists an appropriate controller parameter Λ such that Φ is positive definite.Therefore V x is negative definite and global asymptotic stability has been proved.After obtaining the resulting optimal controller, a new optimal sliding mode controller will be developed by combining this controller with a first-order SMC.The optimal Lyapunov-based SM controller can be obtained by letting υ * u o and substituting 6.9 into 4.3 .

Simulation Results
An example of attitude control of flexible spacecraft 33 is presented with numerical simulations to validate and compare both controllers; SDRE-based SM controller and optimal , 1 and R diag 1, 1, 1 .The initial states of the rotation motion are given by q 0 0.173648 − 0.263201 0.789603 − 0.526402 T , ω 0 0 0 0 T rad/sec, and ϑ 0 0 0 0 0 0 0 0 0 T .For the SDRE-based SM controller the control vector is designed by using 4.3 with optimal control 5.5 .The optimal Lyapunov-based SM controller is obtained by using 4.3 with optimal control 6.9 .For both controllers the switching function 4.1 is chosen using the same constant matrix κ defined as κ λI 3×3 with λ 1.2.To obtain s 0 0 the initial φ is chosen to be φ 0 − ω 0 κq 0 .The attitude control problem is considered in the presence of external disturbance d t .The disturbance model 33 is  Simulation studies have been performed to test both controllers.Figures 1 and 2 clearly show the performance of the SDRE-based SM controller.The responses of quaternion and angular velocity components reach zero after 70 seconds.Obviously the effect of external disturbances on quaternion and angular velocity is totally removed.From Figure 4 it can be seen that the SDRE-based SM controller stabilizes the closed-loop system of flexible spacecraft and provides quite smooth control torque responses.As shown in Figures 5 and 6 the modal displacements μ 1 -μ 1 converge to the neighborhood of zero.
On the other hand Figure 7 shows that the optimal Lyapunov-based SM controller provides good trajectories of the quaternions and they reach zero in about 80 seconds.Similarly, from Figure 8 it can be seen that the angular velocities reach zero after 100 seconds.Making comparisons between the simulation results obtained by the SDRE-based SM controller and optimal Lyapunov-based SM controller, it can be seen that the SDREbased SM controller provides smoother attitude responses and achieves the desired attitude faster.In view of these simulation results, the SDRE-based SM controller seems to be a more useful approach for general cases of attitude regulation problems.Different models and manoeuvres may yield different behaviour.However, the success of the SDRE approach depends on a good choice of the SDC matrix a x .It is difficult to obtain global stability because of the limitations of this technique.On the other hand the difficulty of using the optimal Lyapunov approach is how to find a suitable Lyapunov function.Once this function is known, the optimal Lyapunov approach can be used for the optimal controller design that yields global asymptotic stability.

Conclusion
We have studied two controller designs for attitude stabilization of a flexible spacecraft.Both the SDRE-based SM controller and optimal Lyapunov-based SM controller have been successfully applied to the spacecraft attitude manoeuvres.To obtain both controller designs, ISMC combined with first-order sliding mode and optimal control has been applied to quaternion-based spacecraft attitude manoeuvres with external disturbances and an uncertain inertia matrix.The SDRE and the optimal Lyapunov approaches are used to solve the infinite-time nonlinear optimal control problem.The second method of Lyapunov is used to guarantee the stability of the attitude control system under the action of both controllers.An example of multiaxial attitude manoeuvres is presented and simulation results are given and compared to verify the usefulness of the developed controllers.