PD-like finite time controller with simple structure for satellite attitude control

Li You ,Ye Dong ,Xio Bing

a Xidian University,Xi'an,710126,China

b Harbin Institute of Technology,Harbin 150090,China

c Northwestern Polytechnical University,Xi'an,710072,China

Keywords:Finite time control PD-like Simple structure Attitude control

ABSTRACT A finite time controller with PD-like structure for satellite attitude control is proposed in this paper.The controller is constructed with simple structure based on standard PD controller.The fractional order term is designed hence system could both have strong robustness and finite time convergence rate,and the advantage of finite time control and PD control is combined in this paper.System convergence rate is discussed by Lyapunov method,and the constraint on control parameters is given by implementing the coupled term of angular velocity and attitude quaternion.Moreover,the accuracy at steady stage depending on control parameters is given hence system could converge to this field within finite time.System stability and performance is demonstrated by numerical simulation results.

1.Introduction

In satellite attitude control issue,PD control is the most mature and widely used control algorithm,and much work has been done on this field.PD controller has the advantage of simple structure and definite physical meaning,also,it is robust to disturbance and model uncertainty.Hu [1,2]has done some fundamental work on control algorithms with PD/PID like structure for satellite attitude control and some typical methods to analyze system performance were proposed in his work.System global stability is proved and the basic structure of standard PD controller is constructed in his work.Recent years researchers also have done some work on PD controllers.In order to improve the convergence rate of standard PD controller,Li[3,4]proposed a modified PD controller with better convergence rate.In his work,the PD controller is treated as a special form of sliding mode controller,and a novel sliding mode with three-stage structure is proposed.System convergence rate is improved by constructing the maneuver stage with constant angular velocity.Li [5]designed an optimal PID controller for spacecraft attitude stabilization.The main focus of his work is to optimize convergence time and fuel cost.Generally,PD/PID control algorithm is quite mature,but its main disadvantage is its convergence rate.Linear system governed by linear controller generally could have exponential convergence rate,and how to improve the terminal convergence rate of linear system is a main focus of current research (see Table 1).

In order to improve the exponential convergence rate of linear system,finite time controller based on fractional order feedback is discussed.The terminal sliding mode constructed by angular velocity and attitude quaternion is firstly proposed.Wu [6,7]and Liang [8,9]designed terminal finite time controller for satellite attitude stabilization issue,the basic structure of finite time sliding mode is constructed and finite time controllers robust to model uncertainty and disturbance torque are proposed in their work.The mainly focus of their work is to solve the singularity issue caused by fractional order feedback and chattering issue caused by sliding mode surface.Xiao[10-12]also designed finite time controllers for satellite attitude control and robot manipulator maneuver issue.The main idea of his work is also to construct fractional order feedback sliding mode to achieve finite time stability,the main contribution of his work is the fault tolerant algorithm.Finite time controller under sensor failure such as angular velocity missing is proposed based on state observer and fault diagnose algorithm.System stability is discussed under the worst condition hence system performance is ensured.Generally,finite time controllers are mostly constructed based on finite time sliding mode,and the convergence process could be concluded as two steps: (1) System could converge to desired sliding mode within finite time;(2)System could converge to its equilibrium point within finite time along the sliding mode surface.

In satellite attitude control issue,the existence of perturbations makes it more difficult to design attitude controllers.Hu [13-15]discussed the sensor failure in engineering practice and several state observers are proposed in his work.Li [16-18]discussed the model uncertainty in finite time controller design.Generally in sliding mode control design,the accurate inertia matrix is needed and this could not be achieved in engineering practice.In order to solve this issue,error inertia matrix is assumed to be norm upper bounded and sign function is brought into controller design to offset the effect of model uncertainty.Ye [19,20]discussed the structure robustness of satellite attitude controller.In his work it is pointed out that controller with simple structure and definite physical meaning is helpful for system robustness.Some researchers [21-24]have discussed the overall robustness to disturbance,sensor failure and model uncertainty,however its cost is the drop of system performance.Gui [32]designed finite time controllers with simple structure,but accurate system model is needed in the design process.

Although system convergence rate,model uncertainty,disturbance,actuator failure and singularity issues are considered in these papers,the proposed method are relatively complex comparing with standard PID controller and inconvenient for engineering practice.In order to simplify the structure of finite time controller,researchers also have done a lot of work.Jose [25]designed PID like controller for mobile robot control and his main focus is to simplify the structure of controller,however in his work system stability is not strict finite time.Javier [26]designed finite time controller with simple structure for Lagrange system based on hyperbolic tangent function.In this work,the structure is simple but the sliding mode surface is relatively complex,and there are some constraints on system model.Shakil[27]designed finite time controller with simple structure for electronic system,but sign function which would caused chattering issue is used.Martin[28],Yang [29],Liu [30]and Mohammad [31]designed finite time controller based on terminal sliding mode surface,although the controller structure is simpler than existing methods but the structure of sliding mode is complex and system accurate model is used.In conclusion,it could be found that although there are some finite time controllers with simple structure,but the cost is the complex structure of sliding mode surface or relying on system accurate model,and these two properties are both inconvenient for engineering practice.

Above all,for engineering practice,it is necessary to develop a controller both have simple structure(non sliding mode controller)and better convergence rate which is robustness to model uncertainty and disturbance.In this paper,a finite time PD-like controller with simple structure will be proposed.The advantage of PD controller and finite time controller would be combined and system performance would be largely improved comparing with standard PD controller.The structure of this paper would be concluded as follows:the 2nd section would give the models used in this paper;the 3rd section would give the relative assumptions and lemmas;the 4th section will show the main contribution of this paper and give the finite time controller;the performance of proposed controller would be demonstrated by numerical simulations in 5th section and 6th section would conclude the paper.

2.Dynamic and kinetic model

The dynamic model of satellite could be written as follow:

where ω is a three-dimensional vector which describes system angular velocity,J is inertia matrix of satellite which is a 3×3 symmetric matrix,d is three-dimensional unknown disturbance torque.Product matrix r×of vector r is defined as

Generally inertia matrix J could not be accurate known and it could be described as follow:

Product matrix has an important property which will be used in the later part that the eigenvalues of r×satisfies

The kinetic model based on attitude quaternion could be written as follow:

In order to simplify the text,the maximum and minimum eigenvalue of matrix A is described as λM(A) and λm(A).

3.Assumptions and lemmas

Assumption 1.In this paper,the norm of angular velocity is assumed to be norm upper bounded i.e.following inequality holds.In fact,this assumption is reasonable since the satellite angular momentum is constant and the angular velocity of actuator is limited.

Remark 1.In satellite attitude control system,the actuator is usually thruster and momentum exchanging devices,and thruster consumes propellant (usually liquefied gas) which is non renewable,and using thruster to achieve attitude control would shorten satellite life (when carried fuel is depleted,the satellite loses its orbit maintain capability and its life will end).Hence most satellite control objects except for some emergency conditions are achieved by momentum exchanging devices such as reaction wheel (RW)and momentum gyro,and this means the momentum of whole satellite is constant.Noting that the maximum momentum of RW and momentum gyro is limited,the maximum angular velocity of satellite is limited.Also it is worth noting that although angular velocity is limited,it is much larger than that of satellite attitude stabilization and satellite tracking object.

Assumption 2.The norm of disturbance torque is assumed to be norm upper bounded and following inequality holds wheredω,dq,are all positive scalars.

Assumption 3.For system Eq.(9) and positive definite scalar functionV(x)≥0,when inequality Eq.(10) holds system is asymptotically stable and its convergence rate is exponential,when inequality Eq.(11)holds system is stable and its convergence rate is finite time.Noting the existence of random disturbance in satellite attitude system,system could not stay at its equilibrium point forever,hence the finite time stability actually means system could converge to a range around the equilibrium point with certain accuracy within finite time.

Assumption 4.In this paper,the description for a matrix A ＞0means matrix A is positive definite.

Lemma 1.For systemEq.(1)andEq.(5)governed by following controller with any positive scalar kpand kdis globally stable and any three-dimensional vectorr.

Proof: Select Lyapunov function as follows:

Calculate its derivative and substitute controller Eq.(14)

Based on system stability proof it could be found that any positive control gain factorkpandkdcould stabilize system,which means enlarging or narrowing the control torque proportional and differential terms does not affect system stability.

Lemma 2.If proper control parameters kpand kdare selected and there exists positive scalars k1,k2and c2,c2,αto satisfy inequalitiesEq.(16),systemEq.(1)andEq.(5)governed by controllerEq.(15)is globally asymptotically stable and system convergence rate is exponential.

Proof: Select Lyapunov function as follow:

Hence Lyapunov function Eq.(16) is positive definite.Calculate its derivative it could be got that

Hence system is asymptotically stable and its convergence rate is exponential.Lemma 2 has been proved.

Lemma 3.For positive scalars a1,a2,…,an and p∈(0,2),following inequality holds.

Proof: Ref.[33].

4.PD-like finite-time controller

4.1.PD-like controller

In satellite attitude control issue,standard sliding mode could be written as follows:

withkdandkpare all positive scalars.

The structure of standard PD controller is very simple and the control element could be measured by gyroscope and star sensor directly,hence control loop structure is simple and reliable.Moreover,the controller is globally asymptotic stable for any positive control parameterkdandkp,which means that once the direction of controller Eq.(21) is determined,the norm of control torque does not affect system stability,and this property could be used to solve control torque saturation issue and improve system robustness.Since standard PD control has so many advantages,it is the most mature and widely used control algorithm in the field of satellite attitude control.

Although standard PD controller has such advantages,the exponential convergence rate is the main drawback.When approaching the equilibrium point,angular velocity descends drastically and the convergence rate of attitude quaternion also slows down.System capability and control torque is not fully utilized around the equilibrium point.In fact,linear control feedback brings determines system has the character of standard two-stage system,and in order to improve system convergence rate,fractional feedback could be implemented.

The PD-like finite-time controller with simple structure could be written as follows

wherekd,kdr,kp,kprare all positive scalars,ris a positive scalar which satisfies 0 ＜r＜1,vector function sigr(·)is defined as follows.

The first two terms in controller Eq.(22) are totally same as standard PD controller,and the last two terms with fractional feedback are the key to improve system convergence rate.Also it could be found that controller Eq.(22)does not need inertia matrix hence it is robust to model uncertainty.

In order to achieve finite time stability,control parameters should satisfies Eq.(24),and this would be explained in later text.

Based on Eq.(23) it could be found that controller Eq.(22) is essentially a PD controller with variable gain factor.Also,based on the definition of vector function sigr(·),control torque tends to zero when approaching system equilibrium point and does not have the singularity issue.The definition of unclaimed positive variable γpwould be given in later text.

PD-like controller Eq.(22) has following properties: (1)System Eq.(1),Eq.(5) governed by controller Eq.(22) would converge to the field ‖ω‖＜ε1,‖qv‖＜ε2within finite time with small positive scalars ε1and ε2,this property is the main contribution of this paper since controller Eq.(22) combines the advantage of PD controller and finite time controller;and this property would be proved in later text;(2)This controllerdis constructed by the same element as standard PD controller i.e.angular velocity and attitude quaternion and no extra state is needed,moreover,the accurate inertia matrix is also not needed in this controller which means controller Eq.(22)is robustness to model uncertainty;(3)Based on the definition it could easily be found controller Eq.(22) does not have singularity issue.Above all discussion,controller Eq.(22)could combine the advantage of standard PD controller and finite time controller.

4.2.System stability analysis

The stability analysis could be concluded as following two steps:(1) System Eq.(1) and Eq.(5) governed by controller Eq.(22) is globally stable with any positive control parameters;(2)System Eq.(1)and Eq.(5)governed by controller Eq.(22)could converge to the field ‖ω‖＜ε1,‖qv‖＜ε2within finite time.

First the globally stability will be discussed.

Noting the two fractional terms in controller Eq.(22) could be treated as special differential term and proportional term since following transformation holds

Hence controller Eq.(22) could be re-written as follow:

Comparing the structure of controller Eq.(22) with controller Eq.(12) in Lemma 1,it could be found that the corresponding parameters could be written as follows:

When system is away from its equilibrium point,the disturbance torque and the relevant terms in controller could be ignored,and controller Eq.(22)could be treated as a standard PD controller with time-variant parameters,hence based on Lemma 1 attitude system governed by controller Eq.(22) is stable.

The next step is to discuss the effect on system accuracy of disturbance torque.In this condition,system state is near its equilibrium point,disturbance could no longer be ignored and system state is relatively small.Noting Assumption 2,the norm of disturbance torque consists of three parts: quaternion term,velocity term and constant part.The former two terms has the same structure as PD controller hence the condition to consider these two terms is not relevant to system state.The third term i.e.constant termis the term effects when the.disturbance should be taken into consideration.Assuming when inequality Eq.(28)holds,disturbance should be considered with all positive scalars.

Similar as the discussion in Lemma 2,select Lyapunov function as follows

and select control parameters to satisfy inequality Eq.(16).Substitute controller Eq.(22) into the dynamic model and ignore the term ω×(since angular velocity error inertia matrix are all relatively small under this condition),it could be got that

Noting Assumption 2 and Assumption 6,by implementing following property

It could be got that

Hence when following inequality is satisfied,1≤0 could be ensured.

In other words,when disturbance torque is taken into consideration,following inequality describes the steady accuracy under the worst condition.

Moreover,based on the discussion above and Lemma 2,system convergence rate is exponential when disturbance could be ignored,and convergence time satisfies following inequality.

4.3.Finite time stability analysis

In previous section,the fractional order feedback is treated as another form of PD controller with variable parameters,and it has been proved that system could converge to certain accuracy i.e.with exponential convergence rate at least.In this section,the main focus is to prove that when reaching this field,system could converge to a higher level accuracy within finite time by selecting proper control parameters.

Noting that system state is near its equilibrium point,hence the following approximation holds.

Select Lyapunov function as follows:

Calculate its derivative

In other words,system Eq.(1) and system Eq.(5) governed by controller Eq.(22) could converge to the field ‖ω‖≤γ2,‖qv‖≤γ2within finite time,and convergence time satisfies following inequality.

And the overall convergence time from any initial condition to the field ‖ω‖≤γ2,‖qv‖≤γ2satisfies

whereT1andT2are defined in Eq.(36) and Eq.(51) respectively.Hence system finite time stability under worst condition has been proved.

Moreover,based on the discussion above it could be found that following equations:

Comparing with controller Eq.(53) with Eq.(22) it could be found that the basic structure is maintained,and system finite time convergence rate still holds based on the stability proof.The main difference is the fractional feedback,the former controller needs to implement the direction vector of system state,and when approaching equilibrium point,the measurement error would be aggravated and harm system performance.Later controller would release this issue,however,the inherent PD controller structure is changed and system stability may not be ensured under some extreme condition.

4.4.Global stability analysis

In previous section,system stability is discussed for two stages and this brings much inconvenient for controller design.In this condition,the global condition will be considered,the model error term would be ignored and norm upper bound of system state would be used to derive system stability constraint.The loose condition is considered and a typical constraint for control parameters would be given.

Similar as previous section,select Lyapunov function as follows and calculate its derivative

Hence system could converge to the field descried by Eq.(59)within finite time,and the convergence time satisfies Eq.(60).

The constraint on control parameters could be concluded as follows

Comparing with control parameter constraints Eq.(24) and Eq.(61) it could be found that,the principles to select control parameters are basically the same and the later constraint is slightly easier to satisfy,the main difference between non-global stability analysis and global stability analysis is the steady accuracy.The matrixD3is much larger thanD2in section 4.3 and this expands the steady field.In conclusion,if high accuracy is not requested and control parameters are hard to select,the constraint Eq.(61) could be implemented;and if high accuracy is needed,the stability analysis method in section 4.2 and section 4.3 should be implemented.

Remark 2.It is worth noting that the Lyapunov stability analysis is sufficient but not necessary for system stability,and the proposed constraints on control parameters is strict.Noting that the proposed controller could be treated as special PD controller,and when these constraints is not satisfied,system could still converge to its equilibrium point (since the stability of PD controller proved in Lemma 1) but the finite time stability could not be guaranteed.

Remark 3.There are four control parameters in the proposed controller,noting that satellite attitude system governed by PD-like controller could be treated as two-stage system,the differential term determines its convergence rate and the proportional term determines its steady accuracy.Hence if faster convergence rate is needed larger differential termkdandkdrcould be selected,and if better steady accuracy is needed larger proportional termkpandkprcould be selected.But too large control parameters would cause control saturation issue and these parameters should be selected properly in engineering practice.

Remark 4.In this section,system global stability is proved under the assumption that angular velocity is norm upper bounded,when this assumption is not satisfied(such as when satellite is during the orbit entry stage it is spinning very fast and its angular velocity would reach the level of 1 rad/s) system stability may not be guaranteed,and it is necessary to suppress its fast spinning and then implement the proposed equations.

5.Simulation

5.1.Comparing group

Set system parameters as follows

In order to demonstrate the superiority of controller Eq.(22)proposed in this paper,finite time controller proposed in Ref.[34]is firstly compared.

The simulation results of controller Eq.(63)are given as follows.

Based on Fig.1 and Fig.2 it could be found that system convergence time is about 90 s and the steady accuracy at 100 s is about of 8×10-4rad/s angular velocity and 2×10-3of attitude quaternion.The convergence rate and steady accuracy still needs improvement.

Fig.1.Curve of angular velocity.

Fig.2.Curve of attitude quaternion.

Fig.3.Curve of control torque and its norm.

Considering that the controller in Ref.[34]did not consider angular velocity norm upper bound,and this is an important assumption in this paper,the finite time controller Eq.(64) proposed in Ref.[17]which system angular velocity upper bound is taken into consideration is secondly compared.

The simulation results of controller Eq.(64) are shown as follows.

Based on Fig.4 and Fig.5 it could be found that system convergence time is about 110 s and steady accuracy at 200 s is about of 5×10-7rad/s angular velocity and 6×10-8of attitude quaternion.It is obvious that by implementing fractional order feedback,system convergence time and steady accuracy are both largely improved.However,it is also worth noting that the structure of controller is relatively complex,and in order to resist model uncertainty and disturbance,sign function term which would bring high frequency vibration is added in the controller (see Fig.6).

Fig.4.Curve of angular velocity.

Fig.5.Curve of attitude quaternion.

Fig.6.Curve of control torque and its norm.

5.2.Main results

The next step is to demonstrate system performance governed by controller Eq.(22)proposed in this paper.Noting that there are such many control parameters needs to be selected,hence it is necessary to demonstrate the principle to select control parameters.Firstly the Lyapunov function should be determined properly,hence selectk1,kd,kpas follows:

Check if the Lyapunov function is positive definite

Hence the positive definite condition is satisfied and positive scalarsc1,c2could be determined.The next step is to check if the derivative of Lyapunov function is negative definite.

After checking the stability condition,the next step is to calculate the convergence time and steady accuracy.Select control parameters as follows

And it could be calculated that the latter two inequalities in Eq.(24) are satisfied.Based on Eq.(50) it could be got that

And convergence time satisfies

Based on Eq.(65)-Eq.(71),the principle to select control parameter and the method to estimate convergence time and steady accuracy are given.

Simulation results of controller Eq.(22) proposed in this paper are shown as follows.

Based on Fig.7 and Fig.8 it could be found that system convergence time is about 60 s,and system steady accuracy at 100 s is about 1×10-6rad/s and 4×10-7of angular velocity,the convergence time and steady are both largely improved comparing with the proposed adaptive finite time controller.Moreover,when comparing with finite time controller Eq.(64),the convergence rate is also improved meanwhile the steady accuracy is maintained at the same level.Comparing Fig.9 with Fig.3,it could be found that the norm upper bound of controller Eq.(22) and standard PD controller are basically the same,and there is no singularity issue in controller Eq.(22).It is obvious that by implementing the fractional order state feedback,system convergence rate and the efficiency on control torque are both largely improved.Also,by simulation results it could be found that the convergence time condition and stability condition in Eq.(24)and Eq.(52)both hold,and there are some redundancy on the convergence time estimation since Eq.(52) considers the worst condition (see Fig.11) (see Fig.12) (see Fig.10).

Fig.7.Curve of angular velocity.

Fig.8.Curve of attitude quaternion.

Fig.9.Curve of control torque and its norm.

Fig.10.Curve of angular velocity.

Fig.11.Curve of attitude quaternion.

Fig.12.Curve of control torque and its norm.

Fig.13.Curve of angular velocity.

Fig.14.Curve of attitude quaternion.

This group of simulation demonstrates the superiority of the controller proposed in this paper which could concluded as following properties: (1) Controller Eq.(22) does not change the basic structure of standard PD controller,hence the strong robustness to disturbance and model uncertainty is maintained;(2)No more extra state such as sliding mode surface or auxiliary state is needed in the controller hence it is very convenient for engineering practice;(3) The fractional order feedback could largely improve system convergence time without causing the singularity issue;(4) System steady accuracy could be maintained at a high level by selecting proper control parameters.

In order to demonstrate the superiority of the proposed controller,next the simulation results of controller Eq.(53)will be given.Select control parameters same as Eq.(65)and Eq.(69),and the simulation results are given as follows.

Similar as discussed in previous text,the convergence time and steady accuracy are both similar to controller Eq.(22),and this proves the effectiveness of controller Eq.(53).The main difference between controller Eq.(22) and controller Eq.(53) is that the fractional terms in Eq.(53)do not maintain the direction of angular velocity and attitude quaternion.This change could release the measurement error when approaching the equilibrium point,however the strong stability may not hold in some extreme conditions.

Also,in order to demonstrate the robustness of proposed controller,system measurement error is considered as Gauss white noise shown as follows.

The simulation results are shown as follows.

Based on Figs.13 and 14 it could be easily found that under measurement error,system convergence rate is at the same level as previous results.System steady accuracy is also at the same level as measurement error,and this demonstrates that system accuracy is determined by controller,disturbance and measurement accuracy(input accuracy).This group also proves that the proposed method is robustness to measurement error and it is useful for engineering practice.

5.3.Summarize of simulation

In this section,the performance of proposed is compared with standard PD controller and finite time controller.The numerical simulation results could be concluded as following table.

Based on the simulation in this section,it could be found that the proposed controller maintained most of the advantages of PD controller and largely improves its convergence rate.Only two additional terms with simple structure are added in standard PD controller and system performance near the equilibrium point is much improved.It could be concluded that the proposed controller combined the advantages of standard PD controller and finite time controller.

6.Conclusions

In this paper,a PD-like finite time controller is proposed and system performance is largely improved comparing with standard PD controller.It is pointed out that by implementing some typical fractional order state feedback,system convergence time could be largely improved.This method uses the property that the fractional order of small scalars could enlarge themselves.In worst condition this could be treated as a linear PD controller with time variant parameters.Generally this method changes the linearity of attitude system and the inherent exponential convergence rate is improved.The expected convergence rate could be got by implementing proper nonlinear system feedback meanwhile the strong robustness could be maintained,and this is one of the main contributions of this paper.

In fact,it is relatively easy to add some typical fractional order terms in PD controller,and the difficult part is to analyze system stability.In this paper,Lyapunov method is used to prove system stability.A Lyapunov function with coupled term is developed,although the absolute positive definite property does not hold,it brings much convenient for stability analysis.By using matrix inequality and discussing the range of system state,the relationship between the derivative of V function and system state is constructed and system convergence time could be got.Also,it is also pointed out that by matrix inequality,the influence of disturbance on system accuracy could be estimated by calculating the lower bound of V function.Moreover,a loose and a strict stability are both discussed and two stability conditions are given,generally a loose constraint is convenient to select control parameters but the accuracy is sacrificed somehow,and a strict constraint makes it harder to select control parameters but the accuracy could be maintained.The method to analyze system stability is another main contribution of this paper.

Although the finite time controller with simple structure is proposed in this paper,the finite time stability is not strict condition which means system state could hardly converge to its equilibrium point strictly,but could only convergence to a field with lower and upper bound near the equilibrium point,and this could be focused in later work.Also,system stability is proved under some constraints on control parameters,and system stability is not global,and this should be improved in later work.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported partially by National Natural Science Foundation of China (Project Nos.61903289 and 62073102).The authors greatly appreciate the above financial support.The authors would also like to thank the associate editor and reviewers for their valuable comments and constructive suggestions that helped to improve the paper significantly.