Optimal PID Control of Spatial Inverted Pendulum With Big Bang–Big Crunch Optimization

Jia-Jun Wang and Tufan Kumbasar,,

Abstract—As the extension of the linear inverted pendulum(LIP)and planar inverted pendulum(PIP),this paper proposes a novel spatial inverted pendulum(SIP).The SIP is the most general inverted pendulum(IP)than any existing IP.The model of the SIP is presented for the first time.The SIP inherits all the characteristics of the LIP and the PIP,which is a nonlinear,unstable and underactuated system.The SIP has five degrees of motion freedom and three control forces.Thus,it is a multipleinput and multiple-output(MIMO)system with nonlinear dynamics.To realize the spatial trajectory tracking of the SIP,the control structure with five PID controllers will be designed.The parameter tuning of the multiple PIDs is a challenging work for the proposed SIP model.To alleviate the difficulties of the parameter tuning for the multiple PID controllers,optimal PIDs can be achieved with the help of Big Bang–Big Crunch(BBBC) optimization. The BBBC algorithm can successfully optimize the parameters of the multiple PID controllers with high convergence speed.The optimization performance index of the BBBC algorithm is compared with that of the particle swarm optimization(PSO).Simulation results certify the rightness and effectiveness of the proposed control and optimization methods.

I.INTRODUCTION

INVERTED pendulums(IPs)are one kind of the most important tools to test the control algorithms[1]?[4].The typical IP has a pendulum above the pivot point.According to the motion of the pivot point,the IPs can be mainly classified into three types of IPs.When the pivot point moves along a direct line,we call this type of IP as linear IP(LIP).When the pivot point moves in a horizontal or vertical plane,we call this type of IP as planar IP(PLP).The last one is defined when the pivot point moves in three dimensional space,and this type of IP can be called as spatial IP(SIP).Almost all the presented IPs can be classified as the above three types of IPs.The Furuta’s IP[5]and Kapitza’s IP[6]can be seen as the PIPs.The wheeled IPs,such like the Segway PUMA[7],can be seen as the PIPs that are restricted by nonholonomic(nonintegrable)constraints.The inverted 3-D pendulum proposed in[2],[8]was fully actuated by control forces.So it can be seen as a simplified SIP.As far as we know,at present,there does not exist any research literature on the general SIP.

IP models have their pedagogical and research meanings.The models can represent the simplified robotics, rocket,spacecraft or some mechanical systems[9].They can be used to demonstrate the foundations of nonlinear dynamics and control.At the same time,they also motivate the research in nonlinear dynamics and nonlinear control[2].To the best of our knowledge,we can not find a model that can wholly describe the dynamics of balancing a thin rod or stick on the end of one’s finger at present.The dynamics of the rod or the stick have five degrees of motion freedom,which include three degrees of moving freedom and two degrees of rotating freedom.The LIP or PIP has not enough degrees of freedom to describe these dynamics of balancing the rod or the stick,and the high buildings in the earthquake.It is necessary to extend the LIP or PIP to SIP.The first objective of this paper is to find a model of the SIP,which can wholly describe the dynamics of balancing the rod or the stick.

As the most general IP,the SIP inherits all the characteristics of the LIP and the PIP,which is a nonlinear,unstable and underactuated system.It is also a complex multiple-input and multiple-output(MIMO)system.The LIP and PIP can be seen as the special case of the SIP.The control design of the SIP is a challenging work.The PID controllers are the most popular controllers in the industrial control design[10].To design the controllers for the SIP,the first control strategy we can consider is the PID controller.In[11],a two-loop robust PID controller was designed for an IP system via pole placement technique.The model of the IP was linearized to reduce the difficulty of the design.In[12],the position tracking control of the PIP was designed with PID and neural networks.The neural network was used to decouple the system and compensate for dynamic coupling uncertainties.In[13],the PID control schemes were designed for the LIP and PIP.Although the proposed PID methods in[11]?[13]can not directly applied to the PID control design for the SIP,they give us some very useful hints.The second objective of this paper is to solve multiple PID controller design and the parameter tuning of the PIDs for the SIP.

Although in[13],many PID adjusting methods were given for the LIP and the PIP,the PID parameters were still adjusted based on the operator’s experience.It is very difficult to achieve an optimal result for different control error,overshoot and settling time indexes.To alleviate the difficulties of the parameter tuning for the PID controllers,many evolutionary algorithms have been used to solve the problems,such as genetic algorithm(GA),particle swarm optimization(PSO)and differential evolution algorithm (DE) [14]?[16]. Big Bang–Big Crunch(BBBC)algorithm is a novel evolutionary computing algorithm proposed by Erol and Eksin in 2006[17].The BBBC algorithm has several advantages over other evolutionary methods,such as the inherent simplicity with few parameters,easy implementation,and quick convergence.In[18],[19],the BBBC algorithm was successfully applied to the optimization of the interval type-2 fuzzy PID controller and the zSlices-based general type-2 fuzzy PI controller in the mobile robot.Based on the application of the BBBC algorithm,the third objective of this paper is to realize the optimization of the multiple PID controllers with the BBBC algorithm for the SIP.

This paper is organized as following six sections.Section II introduces the modelling procedure of the SIP step by step.Section III gives the control structure with multiple PID controllers for the SIP.Section IV shows the optimization of the multiple PIDs with BBBC algorithm.Section V gives some comparisons and discussions.And Section VI concludes the work presented in this paper.

II.MODELLING PROCEDURE OF THE SIP

A.The Structure of the SIP

The structure of the SIP is given in Fig.1.The pivot of the SIP is activated by three control forces,which areFx,Fy,andFz.Three control forces are along the direction of thex-axis,y-axis andz-axis in the spatial coordinate system,respectively.We assume the pivot point is atin thexyzspatial coordinate system,and the origin point ofspatial coordinate system is set at pivot pointTheandaxes are parallel withx,yandzaxes,respectively.The center of mass of the pendulum is atp(xp,yp,zp)in thespatial coordinate system.The angleθis the angle between the pendulum with the projection of the pendulum in thevertical plane.And the angleφis the angle between the-axis with the projection of the pendulum in thevertical plane.The mass of the pivot isM.And the mass of the pendulum ism.The length of the pendulum is 2l.We assume that the center of mass of the pendulum is at the middle point of the pendulum.And the distance between the center of mass of the pendulum and pivot point isl.

B.Modelling of the SIP

In this subsection,we will present the derivation of the SIP model with six steps.

Step 1:is the computation of the total kinetic energyKand potential energyPof the SIP.The total kinetic energyKcan be computed with the following expression:

Fig.1. The structure of the SIP.

The potential energyPcan be computed with the expression

wheregrepresents the acceleration constant due to gravity.

Step 2:is the computation of the Lagrangian formulationLof the SIP.The Lagrangian formulationLcan be computed with the difference between the total kinetic energyKand potential energyP,that isL=K ?P

The expression of Lagrangian formulationLinxyzcoordinate is given in(39)of Appendix A.

Step 3:is the computation of the Lagrangian equations of the SIP.The Lagrangian equations of the SIP can be expressed as the following six equations:

The detailed computation of each formulation is given in(40)?(54)of Appendix A.

Step 4:is the computation of the kinematic equations for the SIP.Based on the equation in(7)?(11)and(40)?(54),the kinematic equations of the SIP can be obtained as the following equations:

Step 5:is the solvation of the system kinematic equations.In the equations(12)?(16),andare considered as the primary variables.The following equations can be defined:

Step 6:is the transformation of the state equations.We define thatandx10= ˙φ.Then the state equations of the SIP can be easily obtained,which are given in Appendix B.From the state equation of the SIP in(55)of Appendix B,we can observe that the SIP has the following characteristics.

1)Through the approximate linearization of the state equations of the SIP,we can find that the upright point is open-loop unstable.Without external forces,the SIP can not be stabilized at the vertical position.

2)It can be concluded that the SIP model is a nonlinear MIMO system.The SIP has five degrees of motion freedom with only three control forces.Thus,the resulting system is a typical underactuated control system.

3)Every state variable of the SIP is affected by at least two control forces.There exists strong coupling between the state variables.The SIP is a high dimensional,strong coupled nonlinear system.

III.PID CONTROL DESIGN FOR THE SIP

A.Control Structure Design With Multiple PID Controllers

In Section II,the modelling problem of the SIP is solved.In this part,we will use and employ the PID controllers to solve the spatial trajectory tracking problem of the SIP.From the model of the SIP,we know that the SIP has five variables that should be controlled.According to the general idea,we should use five PID controllers to control these five variables.However,the main challenge is how to arrange/match the PID controllers with the system variables.If the kinematic equations in(17)?(21)are simplified near the origin point,we can obtain the following equations:

From the above simplified kinematic equations,we can find the following three rules near the stable point.

1)Thex-axis motion andθare mainly manipulated byFx.

2)They-axis motion andφare mainly manipulated byFy.

3)Thez-axis motion is mainly manipulated byFz.

According to the above observations,the control structure of the SIP with five PID controllers can be designed as shown in Fig.2.In Fig.2,xd,ydandzdrepresent the desired position of the pivot.The control structure in Fig.2 has the following three characteristics.

Fig.2. The control structure for the SIP.

1)The PID1 to PID5 controllers are the controllers for the five variables of the SIP.PID1 and PID2 are combined to produce the control forceFx.PID3 and PID4 are combined to produce the control forceFy.And the PID5 is used to produce the control forceFz.

2)This structure realizes the decoupling for the control forces in different directions of the SIP.This is very important for the control of the SIP.

3)This structure make the control of the SIP become very simple and clear.Note that,other control strategies can be easily integrated into the PID control for the SIP.

B.Parameter Tuning for the Multiple PID Controllers

The control structure of the SIP is given in above subsection.There exist five PID controllers,therefore we need tune fifteen parameters.If the researcher has little experience in the tuning of the PID controllers,the task is a big challenge even for an experienced control engineer.Referenced from the idea in[13],the five PID controllers can be tuned with the following five steps.

Step 1:Before tuning the PID1,we first disconnect four pointsA,B,CandDin Fig.2.At the same time,we makeFz=(M+m)gandFy=0.Then it is easy to tune the PID1 to make the angleθconverge to zero.

Step 2:Based on the Step 1,other conditions are unchanged,we only connect the pointA.The PID1 need not change any more.We can only tune the PID2 to make the variablexof the SIP trackxd.When the tracking performance is acceptable.Then the parameters of PID2 is ok.

Step 3:Based on the Step 2,other conditions are unchanged,we connect the pointDand cancel the conditionFy=0.According to the symmetrical kinematics ofx-axis andy-axis of the SIP,the parameters of PID1 can be used as the reference parameters of the PID3.And the parameters of the PID3 can be easily obtained by fine-tuning to make the angleφconverge to zero.

Step 4:Based on the Step 3,other conditions are unchanged,we connect the pointC.Similar with the Step 3,the parameters of PID2 can be used as the reference parameters of the PID4.The parameters of the PID4 can be easily obtained by fine-tuning to make the variableyof the SIP trackyd.

Step 5:Based on the Step 4,other conditions are unchanged,we connect the pointBand cancel the conditionFz=(M+m)g.Without changing the parameters of PID1 to PID4,we only need to tune the parameters of the PID5 to make the variablezof the SIP trackzd.

C.Simulation Results With PID Controllers

In above subsection,a systematic design method for tuning the five PID controllers was presented.Next,we will test the control performance of the five PID controllers for the SIP with MATLAB/Simulink.The parameters of the SIP and five PID controllers are given in Tables I and II,respectively.

The spatial moving trajectory of the pivot are given as two spatial curves.The first spatial curve is a spatial spiral curve,which is shown as following expressions:

The second spatial curve is a spatial periodic waveform curve,which is given as the following expressions:

TABLE I THE PARAMETERS OF ISP

TABLE II THE PARAMETERS OF THE FIVE PID CONTROLLERS

Assumption 1:In the simulation,we assume that the operation space of the SIP is confined with the following conditions:

In the simulation,we define thatex=xd ?x,ey=yd ?yandez=zd ?z. The initial state variables are set asθ(0)=π/6,φ(0)=?π/6,x(0)=?0.2,y(0)=0,andz(0)=0.Fig.3 shows the simulation results when the SIP tracks the spatial spiral curve given in(27)with five PID controllers.And Fig.4 demonstrates the simulation results when the SIP tracks the spatial periodic waveform curve given in(28).In Figs.3 and 4,(a)represents the anglesθandφ,(b)is the tracking errorex,eyandez,(c)is the three control forcesFx,FyandFz,and(d)shows the tracking of the desired space trajectory.From the simulation results in Figs.3 and 4,we can obtain the following three conclusions.

1)The proposed control structure with five PID controllers can solve the tracking control problem of the SIP with a satisfactory control performance.

2)The simulation is based on the nonlinear model of the SIP.It need not do any simplification for the SIP model.The proposed control structure with five PID controllers are effective for the SIP with strong nonlinearity.

3)The PID controllers for the SIP can realize the stabilization of the pendulum in a relatively large range.And the PID controllers have robustness to the disturbances or uncertainties.

IV.PID OPTIMIZATION WITH BBBC

In the above section,five PID controllers are successfully applied to the trajectory tracking control for the SIP.There are fifteen parameters that need to be tuned.It is very difficulty to obtain the optimal PID parameters only according to the research’s experience.In this section,the BBBC algorithm will be adopted to optimize the parameters of the five PID controllers designed for the SIP.

Fig.3. Tracking of spatial spiral curve with PIDs.(a)θ and φ.(b)ex,ey and ez.(c)Fx,Fy and Fz.(d)Spatial trajectory tracking of the SIP.

A.The Design of the BBBC

The working principle of the BBBC algorithm can transform a convergent solution to a chaotic or disorder state and then pullback the state to a single tentative solution point.The flowchart of the BBBC algorithm is given in Fig.5[20].According to the flowchart,the design of PID controller via the BBBC algorithm can be realized with following six steps.

Step 1:This step is used to specify the parameters for the BBBC algorithm.In BBBC,there are normally three parameters that need to be specified,which are iteration parameterNi,population size parameterNp,and space limiting parameterα.

Fig.4. Tracking of spatial periodic waveform curve with PIDs.(a)θ and φ.(b)ex,ey and ez.(c)Fx,Fy and Fz.(d)Spatial periodic trajectory tracking of the SIP.

Step 2:This step is used to initialize the first population for the BBBC algorithm.To optimize the five PID controllers,there are fifteen parameters that need to be optimized.The optimized variable for the BBBC can be defined as

wherek,P,IandDrepresent population number,proportion parameter,integral parameter and derivative parameter,respectively.And the subscripts ofP,IandDrepresent five different PIDs.The first time populations can be initialized with the following expression:

wherexmaxandxminrepresent are the upper and lower limits defined with in a 15-dimensional search space.Andrand(Np,1)representsNprandom numbers in the interval[0,1].

Fig.5. The flowchart of the BBBC algorithm.

Step 3:This step is the design of the fitness function or objective function for the BBBC algorithm.Different fitness functions can be used in the BBBC optimization,such as integral absolute error(IAE),integral squared error(ISE),integral time-weighted squared error(ITSE)or integral timeweighted absolute error(ITAE).The IAE performance index is selected to optimize the five PID controllers for the SIP,which is defined as the following expression:

wherek=1,2,...,Nkis thekth population number.

Step 4:This step is the Big Crunch phase.Big Crunch phase is a contraction procedure.The contraction operation takes the current position of each candidate solution in the population and its associated fitness function value,and then computes the centroid of the population.The centroid of the population can be computed with the following expression:

wherexcis the centroid,xkis the position of the candidate,fkis the fitness function value of thekth candidate.

Step 5:This step is the Big Bang phase.The new generation for the next iteration Big Bang phase is normally distributed aroundxcand can be computed as the following expression:

whereis the new generated candidates,ris random number generated in the interval[?1,1],αis space limiting parameter,andiis the current iteration step.

Step 6:This step is the design of the stopping criteria for the BBBC optimization.The stopping criteria of the BBBC can use the maximal iteration criteria or minimal error criteria.In this optimization of the five PID controllers,the maximum number of iteration is applied as the stopping criteria.

B.Simulation Results With BBBC Optimized PID Controllers

In the BBBC optimization for the five PID controllers,the tracking trajectories and the initial value of the SIP are the same as in Section III.The initial value of the BBBC are set as:the iteration parameterNi=40,the population size parameterNp=40,and the space limiting parameterα=0.15.

Because the SIP is strongly nonlinear system,the change of the PID parameters may make the system unstable.The optimization range for the BBBC can not be given arbitrarily.In the BBBC optimization,for the positive values,we confine the maximal value is the 180%of the PID parameter values that we have obtained in Table II,and the minimal value is the 20% of the PID parameter values. While for the negative values,the maximal value is the 20%of the PID parameter values,and the minimal value is the 180%of the PID parameter values.

Figs.6 and 7 show the simulation results for the optimization of the five PID controllers with the desired spatial spiral curve and spatial periodic waveform curve,respectively.In Figs.6 and 7,(a),(b)and(c)shows the optimizedP/Pb,I/IbandD/Dbof the five PID controllers respectively,and(d)demonstrates the tracking performance with BBBC optimization.

Remark 1:To show the BBBC optimization procedure for the parametersP,I,Dmore clearly in one figure,the optimized valueP,IandDare divided by their base valuesPb,Ib,Dbthat are given in Table II.

From the simulation results in Figs.6 and 7,we can obtain the following three conclusions.

1) The BBBC algorithm can successfully optimize the parameters of the five PID controllers for the SIP in the spatial trajectory tracking.And the optimized PID parameters can be obtained with fast convergence.

2)The BBBC algorithm not only can alleviate the difficulty of the parameter tuning for the five PID controllers,but also can enhances the control performance for the spatial trajectory tracking control.The adoption of the BBBC algorithm in the PID controller optimization simplify the parameter tuning for five PID controllers.

3)The BBBC algorithm can realize the optimization of strong nonlinear and high dimensional control problems with few population and iteration.

Fig.6. PID parameter optimization with BBBC algorithm for the spatial curve tracking.(a)P/Pb.(b)I/Ib.(c)D/Db.(d)Spatial trajectory tracking of the SIP.

V.COMPARISONS AND DISCUSSIONS

A.Comparisons for Different Control Method

In the above sections,we have solved the control problems of the SIP.In this section,we will give some comparisons among the PID control,PID control with BBBC optimization and PID control with PSO optimization.

The PSO algorithm are given as following expressions:

Fig.7. PID parameter optimization with BBBC algorithm for the spatial periodic waveform curve.(a)P/Pb.(b)I/Ib.(c)D/Db.(d)Spatial periodic trajectory tracking of the SIP.

wherek,V(k+1),X(k+1),Vk,Xk,XBkandXGkare population number(k=1,2,...,Np ?1),new velocity,new position,present velocity,present position,local best value and global best value,respectively.λis the weighted(inertia)factor which has a range from 0.1 to 0.9,c1andc2are acceleration coefficients,andr1andr2are random numbers uniformly distributed in[0,1].

In the simulation,the population number and the iteration number for the PSO are set as 40,which are the same as in the BBBC algorithm.And the control parameters are set as:ω=0.5,c1=1.5 andc2=0.25.

Firstly,we compare the optimization speed between the BBBC and PSO in the optimization of the five PID controllers for the SIP.The desired tracking trajectories are the same.And the initial state variables are given as the following two cases.

1)Initial state variables of the first case:θ(0)=π/6,φ(0)=0,x(0)=0,y(0)=0,andz(0)=0.

2)Initial state variables of the second case:θ(0)=0,φ(0)=π/6,x(0)=0,y(0)=0,andz(0)=0.

The comparison results are given in Figs.8 and 9.From the simulation results in Figs.8 and 9,we can find that the BBBC optimization not only has faster convergence speed,but also has less fitness value than the PSO.

Fig.8. The comparison of the BBBC and PSO for the first case.(a)Fitness function of spatial spiral curve.(b)Fitness function of spatial periodic waveform curve.

Secondly,the IAE performance of the SIP are compared between the PID controllers with and without optimization.The initial state variables are the same as the above first comparison.Figs.10 and 11 show the simulation results of the IAE performance indexes.It can be seen from the simulation results in Figs.10 and 11 that the PID controllers without optimization has larger IAE.The optimization with BBBC and PSO can greatly reduce the IAE performance indexes for trajectory tracking control of the SIP.

Thirdly,to further prove the effectiveness of the proposed method for the SIP,the IAE for different control methods with different initial values are given in Table III.In Table III,the initial values of the SIP for Cases 1 and 2 are the same as given in the first part in this section.And Case 3 is given asθ(0)=π/6,φ(0)=?π/6,x(0)=?0.2,y(0)=0,andz(0)=0.

The parameters of the SIP,such as the mass of the pivotM,the massmand the lengthlof the pendulum,can not be given very accurately,or they may change at some conditions.The sensitivity analysis of the SIP for the parameter variations are given in Tables IV and V for spatial spiral curve and spatial periodic waveform curve,respectively.The initial data are given asθ(0)=π/6,φ(0)=0,x(0)=0,y(0)=0,andz(0)=0.In the sensitivity analysis,we assume thatMandmcan only be increased.Whereas,the lengthl,representing the distance between the mass of the center of the pendulum and the pivot,can become longer or shorter.When one of the parameter is changed,the other parameters are assumed to be unchanged.From Tables IV and V,it can be seen that the BBBC still can maintain perfect optimization performance under the variation of the system parameters.

Fig.9. The comparison of the BBBC and PSO for the second case.(a)Fitness function of spatial spiral curve.(b)Fitness function of spatial periodic waveform curve.

Fig.10. The comparison of IAE for the first case.(a)IAE of spatial spiral curve.(b)IAE of spatial periodic waveform curve.

B. Discussions

In the above sections,we gave the modelling procedure for the SIP,successfully designed the control structure with five PID controllers and realized the optimization of the parameters of the five PID controllers.Furthermore,we certified the rightness of the model for the SIP,and demonstrated the effectiveness of the PID controllers and their optimization strategies.However,there are still some questions that need to be discussed and explained.

Fig.11. The comparison of IAE for the second case.(a)IAE of spatial spiral curve.(b)IAE of spatial periodic waveform curve.

TABLE III THE IAE FOR DIFFERENT CONTROL METHODS WITH DIFFERENT INITIAL VALUES

TABLE IV THE IAE FOR SPATIAL SPIRAL CURVE WITH PARAMETER VARIATIONS

TABLE V THE IAE FOR SPATIAL PERIODIC WAVEFORM CURVE WITH PARAMETER VARIATIONS

1)In modelling procedure for the SIP,we only considered the ideal conditions.We omitted lots of tiny and uncertain factors,such like the friction between the pivot and the pendulum,the moment of inertia,the measuring error and control error.We assume that the center of mass is at the middle point of the pendulum.In fact,the mass of the pendulum is distributed along the pendulum.

2)In the simulation with five PID controllers and their optimization,we did not consider the disturbances and the uncertainties.In fact,the disturbances and the uncertainties exist in all control systems.Thus,there is a need of robust or intelligent control structures that can be integrated with the PID controllers,such like the sliding mode control,neural network,fuzzy control or other advanced control strategies.

3)In the optimization of the PID controllers,we only considered the BBBC and PSO algorithms.Other optimization algorithms,such like genetic algorithm,differential evolution algorithm or other evolutionary algorithms can also be tested in the optimization of the PID controllers.

VI.CONCLUSIONS

In this paper,for the first time in literature as far as we know,a novel SIP was proposed that could wholly describe the dynamics of balancing a thin rod or stick on the end of one’s finger.We have also presented an optimization based control design strategy to solve the tracking problem of the SIP.The main contributions of this paper can be summarized as the following four points.

1)The model of the SIP was first given step by step.The SIP inherits all the characteristics of the LIP and the PIP.And almost all the present IPs can be seen as one special case of the proposed SIP.The SIP has three degrees of moving freedom and two degrees of rotating freedom.The proposed SIP can be seen as the most general IP.

2)The SIP has nonlinear,unstable and underactuated features.Thus,the control of the SIP is a challenging work.In this paper,a control structure with five PID controllers was proposed for the SIP.The proposed PID controllers could decouple three control forces and made the control structure become every clear.And the procedure of the parameter tuning for the five PIDs was given step by step.

3)The parameter tuning for five PIDs is not an easy work.To alleviate the difficulties of the parameter tuning of the PIDs,the BBBC algorithm was successfully applied to optimize the parameters of the PIDs.The optimization results of the BBBC was compared with that of the PSO.Simulation results validated that the BBBC optimization had faster convergence speed and less IAE than the PSO optimization.

4) Many simulation results are given, which not only certified the rightness of the model of the SIP, but also demonstrated the effectiveness of the proposed control and optimization methods.

This paper extended the research on the IPs and gave a new idea in the control of the IPs.The proposed method has its generality and can be directly applied in the control of the LIP,PIP and planar vertical take off and landing aircraft(PVTOL).In future,we will construct a platform to realize the control of the SIP and certify the simulation results in this paper.

APPENDIX A

1)Computation ofL

2)Computation of

3)Computation of

4)Computation of

5)Computation of

6)Computation of

7)Computation of

8)Computation of

9)Computation of

10)Computation of

11)Computation of

12)Computation of

13)Computation of

14)Computation of

15)Computation of

16)Computation of

APPENDIX B

The state equation of the SIP is given in the shape of the following expression

With the kinematic equations(17)?(21),we can know thatX,F(X),B(X),Ucan be defined as the following forms

where the parameters inF(X) can be given as the following expressions:

The parameters inB(X)can be given as the following expressions

In(55),Drepresents the disturbances or uncertainties.