Trajectory tracking problem for Markov jump systems with Itô stochastic disturbance and its application in orbit manoeuvring

Trajectory tracking problem for Markov jump systems with Itô stochastic disturbance and its... Abstract This paper deals with the trajectory tracking problem for jump systems with Itô stochastic disturbance. The main contribution is that the form of the controller is complete parametric. The controller consists of the additive contribution of two terms: a feedback term and a feedforward term. The feedback term is presented using the linear matrix inequality method for the state feedback control law to make the system stochastically stable and can guarantee the input constraints. The feedforward term is the complete parametric feedforward tracking compensator by utilizing the parametric solution of the generalized Sylvester matrix equation for the trajectory tracking problem. The simulation about spacecrafts hovering task is carried out in relative orbit manoeuvring control. The simulation results show the effectiveness of the proposed approach. 1. Introduction In the past few decades, Markov jump systems (MJS) have been widely investigated and many useful results have been obtained. At present, numerous research results have been obtained (see Souza & Fragoso 2003; Li et al. 2010; Wang et al. 2015). The motivation for the study is that many practical systems subject to random abrupt variations can be modelled by MJS (see Zhang et al., 2011). Recently, much work has been done for the systems (see Li et al., 2002; Dragan et al., 2004; Hou et al., 2006; Dong et al., 2008; Lin et al., 2009; Chan, 2015,Shi et al., 2015; Kang et al., 2016). It was presented in a survey on recent developments of modelling, analysis and design of MJS (see Shi & Li, 2015). Kang et al. (2016) investigated the input-to-state stability problem of a class of switched stochastic non-linear retarded systems under extended asynchronous switching. The linear quadratic optimization problem for linear Markovian jump stochastic systems subject to multiplicative white noise was investigated in Dragan & Morozan (2004). Chan (2015) studied the market share and forecasting problem using the Markov model. The stochastic stability problem was investigated in Hou et al. (2006) for Itô differential equations with jump parameters. Based on the solvability of a set of coupled Hamilton–Jacobi inequalities, Lin et al. (2009) studied a unified design method for both state and dynamic output $$H_\infty$$ feedback control for the stochastic systems. Li et al. (2015) concerned with the state estimation and sliding mode control problems for phase-type semi-MJS. It was concerned with the problem of designing robust $$H_2$$-state feedback controllers for continuous-time Markov jump linear systems subject to polytopic-type parameter uncertainty in Dong et al. (2008). Almost all the control problems for these systems have been tackled, and we can find the results in the literature. The demand for trajectory tracking is that the closed-loop system is stable and can track the given command, which can be conversed into a model reference tracking control problem. Model reference tracking control plays an important role in control system design, and it has many applications arising in areas such as the control of mobile robots, motor drive, network, mechanical systems and aircraft (see Shyu et al., 1998, Wang & Yang, 2009, Liu & Tao, 2010, Wilcox et al., 2010, Heusden et al., 2011, Li & Wang, 2013, Joshi, 2014). Li & Wang (2013) investigated the finite time-tracking control problem of the mobile robots via visual servoing. Shyu et al. (1998) researched the neural networks-adaptive speed controller for the induction motor to track a reference model. Wang & Yang (2009) investigated the $$H_\infty$$ model reference tracking control for continuous-time networked control systems with communication constraints. A data-driven controller was presented in tuning method that includes a set of constraints for ensuring the system stability in Heusden et al. (2011). Multivariable output tracking adaptive control approach has been studied in Liu & Tao (2010) for the aircraft dynamics with damage. Model reference adaptive control method was studied in Joshi (2014), which considered the actuator failures and sensor bias faults for the aircraft. Wilcox et al. (2010) represented a robust model reference control strategy for a hypersonic vehicle system with additive bounded disturbances and aerothermoelastic effects. Boukas (2009) synthesized a state feedback controller that could make the state vector of the MJS track precisely a given state vector of a reference model. In this paper, the trajectory tracking problem for MJS with Itô stochastic disturbance is considered. Just as it was said in Boukas (2009), this problem has not been fully researched, not to mention the application in the spacecrafts relative orbit manoeuvring control. Thus, this paper will develop a controller, the aim of which is to make the output of the system track the output of the given reference model. Different from Boukas (2009), the controller in this paper is composed of two parts: the first one is a state feedback controller that can stabilize the system and the second one is a complete parametric feedforward tracking compensator. A linear matrix inequality (LMI) method is presented to design the state feedback controller to stabilize the system and two additional LMI conditions are proposed to guarantee the control input constraints. Based on the parametric solution of the generalized Sylvester matrix equation, the complete parametric solution of the feedforward compensator is established for the trajectory tracking problem. Our main contribution is that we obtained the complete parametric solution of the controller for the system. The derived results that the feedforward compensator contain have extra degrees of freedom in parametric design and it can be used to cope with some additional performance. The outline of the content of this paper is as follows. Section 2 presents the problem formulation. Section 3 obtains the main results. In Section 4, the designed controller is applied to the spacecrafts hovering task and the simulation results show the validity of the proposed methodology to handle the tracking problem. Finally, we end this paper in Section 5 with a brief concluding remark. 2. Problem formulation Let us consider a continuous-time MJS with Itô stochastic disturbance, the model is defined by   {x˙(t)=Ax(t)+B(γ(t))u(t)+Mx(t)dω(t),y(t)=Cx(t),  (2.1) where $$x(t)\in\mathbb{R}^{n}$$, $$u(t)\in\mathbb{R}^{r}$$ and $$y(t)\in\mathbb{R}^{d}$$ are the state vector, the input vector and the output vector respectively, and $$\mathbb{R}$$ represents the real space. The matrices $$A\in\mathbb{R}^{n\times n}$$, $$B(\gamma(t))\in\mathbb{R}^{n\times r}$$ and $$C\in\mathbb{R}^{d\times n}$$ are the system coefficient matrices. $$\omega (t)\in\mathbb{R}$$ is a standard Wiener process that is supposed to be independent of the Markov process $$\{r(t),t>0\}$$. $$M$$ is system matrix with appropriate dimensions. The Markov process $$\{\gamma(t),t>0\}$$ describes the jump mode of the system (2.1) and takes values in a finite set $$\mathbb{S}=\{1,2,\ldots,N\}$$, with transition probability matrix $$\Lambda \triangleq \{\lambda_{ij}\}$$ given by   λij=Pr{γ(t+h)=j|γ(t)=i}={λijh+o(h),i≠j,1+λiih+o(h),i=j,  (2.2) where $$h>0$$, $$\lim\limits_{h\rightarrow 0^+}\frac{o(h)}{h}=0$$; here $$\lambda_{ij}$$ is the transition rate from mode $$i$$ to mode $$j(i\neq j)$$, with $$\lambda_{ij} \geq 0$$ and $$\lambda_{ii}=-\sum\limits_{j=1,i\neq j}^{N}\lambda_{ij}$$. For each possible value of the process $$\gamma(t)\in\mathbb{S}$$, we write $$B(\gamma(t))=B_i$$, which are known matrices that represent various types of the stochastic fault of the actuator. Introduce a set of matrices $$F_i\in\mathbb{R}^{r\times r}$$ to describe the actuator fault, $$ F_i$$ is a diagonal matrix with diagonal elements $$f_{il}$$ subject to $$0 \leq f_{il}\leq 1$$, $$l=1,2,\ldots,r$$. When the system (1) is in mode $$\gamma(t)=i$$ , defining $$f_{il}=0 $$ means the $$l$$th actuator has failed completely without any control input; if $$f_{il}=1$$, the $$l$$th actuator works normally; if $$ 0<f_{il}<1 $$, the $$l$$th actuator has partially lost its effectiveness, but it still works. Thus, the matrix $$B_i$$ can be written as   Bi=BFi, (2.3) where $$B\in\mathbb{R}^{n\times r}$$ represents all the actuators work normally. Definition 1 The closed-loop system $$\dot{x}(t)=[A+B_iK_i]x(t)+Mx(t)\texttt{d}\omega (t)$$ is said to be stochastically stable (SS) if for every initial condition $$x(0)=x_0$$ and $$\gamma (0)=\gamma_0$$, the following holds   E{∫t0∞∥x(t)∥2dt|x0,r0}<∞, (2.4) where $$K_i$$ is the state feedback gain matrix and the symbol ‘E’ represents the mathematical expectation operator. This paper mainly researches the model reference tracking control problem for the continuous-time MJS with Itô stochastic disturbance. Without any loss of generality, the reference signal can be generated by the following linear continuous-time reference model   {x˙m(t)=Amxm(t),ym(t)=Cmxm(t),  (2.5) where $$x_m(t)\in\mathbb{R}^{q_1}$$ and $$y_m(t)\in\mathbb{R}^{q_2}$$ are the state vector and the output vector of the reference model. $$A_m$$ and $$C_m$$ are given matrices with appropriate dimension. The purpose is to design a controller to make the output $$y(t)$$ track the output $$y_m(t)$$ of the reference model in mean square sense, which is to satisfy the following tracking requirement   E{∫t0∞∥{y(t)−ym(t)}∥2dt|x0,r0}<∞ (2.6) for arbitrary initial values $$x(0)$$, $$x_m(0)$$ and $$\gamma (0)$$ . The following problem needs to be done to realize the purpose of this paper. Problem 1 Given the system (2.1) and the linear constant reference model (2.5), design a controller in form of   u(t)=Kix(t)+Kmixm(t), (2.7) where $$K_{mi}=K_m(\gamma (t)=i)$$ is the gain matrix of feedforward compensator, such that (a) The closed-loop system is SS. (b) The output $$y(t)$$ can track the output $$y_m(t)$$ in mean square sence (i.e. satisfy (2.6)). Lemma 1 (Boukas, 2005; Schur Complement Lemma): The LMI   [A11A12A21A22]<0 is equivalent to   A22<0,A11−A12A22−1A21<0, where $$A_{11}=A_{11}^{\rm T}$$, $$A_{22}=A_{22}^{\rm T}$$ and $$A_{12}=A_{21}^{\rm T}$$. 3. The main results According to Problem 1, we need to design the feedback controller and the feedforward compensator. The following theorem gives the controller existence conditions. Theorem 1 Problem 1 has a solution if the closed-loop system is SS, and there exist matrices $$G_i\in\mathbb{R}^{n\times q}$$ and $$H_i\in\mathbb{R}^{r\times q}$$ satisfying the following matrix equations   {AGi+BiHi−GiAm=0CGi−Cm=0.  (3.1) In this case, the feedforward gain matrices $$K_{mi}$$ is   Kmi=Hi−KiGi, (3.2) where $$K_i$$ is the state feedback gain matrix such that the closed-loop system is SS. Proof. Let   {δx(t)=x(t)−Gixm(t)δu(t)=u(t)−Hixm(t)δy(t)=y(t)−ym(t)O(t)=Mx(t)dω(t)  (3.3) after calculation, we can obtain   δx˙(t) =x˙(t)−Gix˙m(t)+O(t) =Ax(t)+Biu(t)+O(t)−GiAmxm(t) =[AGi+BiHi−GiAm]xm(t)  +Aδx(t)+Biδu(t)+O(t) (3.4) and   δy(t) =Cδx(t)+(CGi−Cm)xm(t) (3.5) If the conditions (3.1) is satisfied, it is able to show that   {δx˙(t)=Aδx(t)+Biδu(t)+O(t)y(t)=Cδx(t).  (3.6) Notice system (3.6) and system (2.1) have the same structure, hence for arbitrary state feedback control law $$u(t)=K_ix(t)$$, which can stabilize the system (2.1), the state feedback control law   δu(t)=Kiδx(t) (3.7) can also stabilize the system (3.6). That is to say the system   {δx˙(t)=[A+BiKi]δx(t)+O(t)y(t)=Cδx(t)  (3.8) is SS, thus equation (2.6) holds. Combined with the equation (3.3) and equation (3.7), the controller can be written as   u(t)=Kix(t)+[Hi−KiGi]xm(t). (3.9) Then $$K_{mi}=H_i-K_iG_i$$ is obtained. The proof is finished. □ Remark 1 It is not difficult to understand $$ \texttt{E}[O(t)]=\texttt{E}[Mx(t)\texttt{d}\omega (t)]=0$$, as $$\omega (t)\in\mathbb{R}$$ is a standard Wiener process. Thus, the controller in equation (2.7) can ensure equation (2.6) set-up. The next is the design of feedback control law and feedforward tracking compensator. 3.1. State feedback controller design Lemma 2 (Li et al., 2011) The closed-loop system is SS if and only if there exists a set of symmetric positive definite matrices $$P=(P_1, P_2,\ldots,P_N)>0$$, such that the following continuous-time stochastic Lyapunov equations hold   (A+BiKi)TPi+Pi(A+BiKi)+∑k=1NλikPk+MTPiM=−Qi (3.10) for arbitrarily given symmetric positive definite matrices $$Q=(Q_1, Q_2,\ldots,Q_N)$$. Corollary 1 The closed-loop system is SS if there exists a set of symmetric positive definite matrices $$P=(P_1, P_2,\ldots,P_N)>0$$, such that the following holds for each $$i\in\mathbb{S}$$,   (A+BiKi)TPi+Pi(A+BiKi)+∑k=1NλikPk+MTPiM<0. (3.11) Consider the control input constraints in project, there is   ∥u(t)∥2≤umax, (3.12) where $$u_{max}$$ represents the maximum control amplitude. Using the Schur Complement Lemma, the following theorem is proposed to show the conditions that can guarantee the stability and the constraints of the closed-loop system. Theorem 2 The closed-loop system is SS that satisfies the control constraints described in equation (3.12), if there exists a given positive number $$\varrho$$, a set of matrices $$W_i$$ and symmetric positive definite matrices $$X_i$$, $$i\in\mathbb{S}$$, meets the following conditions:   [ϱIxT(0)x(0)Xi]≥0 (3.13)  [umax2IWiWiT1ϱXi]≥0 (3.14)  [ΦiΓiΓiTΔi]<0 (3.15) where   Φi=[AXi+XiAT+BiWi+WiTBiT+λiiXiXiMTMXi−Xi]Γi=[λi1Xi...λi,i−1Xiλi,i+1Xi...λiNXi0...00...0]Δi=Diag[X1...Xi−1Xi+1...Xn] In this case, the feedback gain matrices are given by   Ki=WiXi−1. (3.16) Proof. It follows from Corollary 1 that the closed-loop system is SS if there exist symmetric positive definite matrices $$P_i$$, such that the following holds for each $$i\in\mathbb{S}$$,   (A+BiKi)TPi+Pi(A+BiKi)+∑k=1NλikPk+MTPiM<0. (3.17) As $$P_i$$ is symmetric positive definite matrix, there exists inverse matrix $$P_i^{-1}$$, satisfying $$P_i^{-1}=(P_i^{-1})^{\rm T}$$. Pre-multiplying and post-multiplying by $$P_i^{-1}$$ on both sides of the aforementioned inequality gives   Pi−1AT+Pi−1KiTBiT+APi−1+BiKiPi−1+Pi−1(∑k=1NλikPk)Pi−1+MTPiM<0. By defining $$X_i=P_i^{-1}$$ and $$W_i=K_iP_i^{-1}$$, the above inequality is turned into the following form   XiAT+WiTBiT+AXi+BiWi+Xi(∑k=1NλikXj−1)Xi+MTPiM<0. (3.18) Using Lemma 1, the inequality equation (3.18) can be equivalently arranged into the form of inequality equation (3.15). There exists a positive number $$\varrho$$ satisfying the initial condition $$x^{\rm T}(0)P_ix(0)\leq \varrho$$ for every $$i\in\mathbb{S}$$. Then, while $$t>0$$, it follows that   xT(t)Pix(t)≤xT(0)Pix(0)≤ϱ. (3.19) Using the Schur Complement Lemma, the inequality equation (3.19) can be equivalently arranged into the form of inequality equation (3.13). In view of equation (3.12), there is   uT(t)u(t)=xT(t)KiTKix(t)≤umax2. (3.20) Combining (3.19) and (3.20) and using the Schur Complement Lemma once again, it can be obtained that   [umax2IKiKiT1ϱPi]≥0. (3.21) Pre- and post-multiplying by diag $$(I,X_i)$$ on both sides of the aforementioned inequality, the inequality equation (3.14) can be obtained. The proof is then completed. □ 3.2. Feedforward compensator design The purpose of feedforward compensator is to satisfy the tracking requirement in Problem 1. In order to solve the compensator, the first is to find two matrices $$G_i$$ and $$H_i$$ satisfying equation (3.1). Lemma 3 (Bhatia, 1997) The second equation of equation (3.1) has a solution if and only if there exists a generalized inverse matrix $$C^{-}$$of matrix $$C$$ such that   CC−Cm=Cm, (3.22) then the general complete solution of it can be written as   Gi=C−Cm+[I−C−C]Yi, (3.23) where $$Y_i\in\mathbb{R}^{n\times p}$$ is arbitrary matrix. Substituting (3.23) into the first equation of equation (3.1), gives   BiHi+AC−Cm−C−CmAm= [I−C−C]YiAm−AYi+AC−CYi. (3.24) Let   W=AC−Cm−C−CmAm,E=I−C−C,A^=AE, (3.25) then equation (3.1) can be written in the following form:   BiHi+W=−A^Yi+EYiAm. (3.26) The next gives the equation (3.26) a complete parametric solution. Lemma 1 (Wu & Duan 2008) If the matrix pair [$$sE-\hat{A}$$, $$B_i$$] is controllable, there exists a unimodular matrix $$V_i(s)\in\mathbb{R}^{(n+r)\times (n+r)}$$[s] satisfying   [sE−A^Bi]Vi(s)=[△0] (3.27) where $$\triangle=\Sigma_i(s)$$ is usually a diagonal matrix and satisfies $$\texttt{det}\triangle\neq0$$. Partition the matrix $$V_i(s)\in\mathbb{R}^{(n+r)\times (n+r)}$$[s] as   Vi(s)=[∑a=0αUiasa∑a=0αLiasa∑a=0αQiasa∑a=0αDiasa] (3.28) where $$U_{ia}(s)\in\mathbb{R}^{n\times n}$$, $$L_{ia}(s)\in\mathbb{R}^{n\times r}$$, $$Q_{ia}(s)\in\mathbb{R}^{r\times n}$$ and $$D_{ia}(s)\in\mathbb{R}^{r\times r}$$. Using the Theorem 4.7 and Theorem 5.2 in Duan (2015), the general complete solution of ($$Y_i$$, $$H_i$$) to the non-homogeneous generalized Sylvester equation (3.26) is given by   [YiHi] =∑a=0α[LiaUiaDiaQia][ZiW]Ama. (3.29) Further, in combination with equation (3.29) and equation (3.23), a complete parametric solution of equation (3.1) can be obtained as follows   {Gi=C−Cm+E∑a=0α[LiaZi+UiaW]Ama,Hi=∑a=0α[DiaZi+QiaW]Ama,  (3.30) where $$Z_i\in\mathbb{R}^{r\times q}$$ is arbitrary parameter matrix that represents the degree of freedom in the solution. 3.3. Algorithm for solving the controller Given the reference model coefficient matrices $$A_m$$, $$C_m$$, the transition probability matrix $$\Lambda$$ and the failure matrices $$B_i$$ for every $$i=1,2,\ldots,N$$, then the following algorithm for solving Problem 1 can be presented. Algorithm 1 Step 1. Compute $$W_i$$ and $$X_i$$ according to the equations (3.13), (3.14) and (3.15). Based on equation (3.16), solve the feedback gain matrix $$K_i$$. Step 2. If the matrix pair [$$\hat{A}$$, $$B_i$$]($$\hat{A}$$ is defined in equation (3.25)) is controllable, solve the equation (3.27) and go to next step; otherwise, the feedforward compensation does not exist or the algorithm is failed. Step 3. Computing $$H_i$$ and $$G_i$$ by equation (3.30), utilize the equation (3.2) to compute the gain matrix of the feedforward compensation and the parametric form of the controller (2.7) is obtained. From the above analysis, the following remarks can be deduced. Remark 2 If one or more actuators fail completely, the system will become under-actuated system (Benosman, 2009), then the designed controller in equation (2.7) will not guarantee the system stability. Here, all the actuators are assumed to be active even if there exist faults in some actuators. The under-actuated system is not considered further in this paper. Remark 3 Given the matrices $$A, B_i, C$$ in system (2.1) and $$A_m$$, $$C_m$$ in the reference model (2.5), assume that there exists a generalized inverse matrix $$C^{-}$$ of the matrix $$C$$ that satisfying $$CC^{-}C_m=C_m$$ as well as the matrix pair [$$\hat{A}$$, $$B_i$$] is controllable. 4. The Application to the spacecraft relative orbit manoeuvre In this section, we will apply the proposed model reference tracking control approach to the application for the spacecrafts relative orbit manoeuvring in an circular orbit. Suppose that the target spacecraft is in the geosynchronous orbit, a circular orbit on the Earth’s equator plane with the same period as Earth rotation. The orbital parameters are as follows: the radius $$R=4.2241\times 10^7$$ m, the period $$T=86,164$$ s and the angular rate $$w=7.2921\times 10^{-5}$$ rad/s. The linearized equations of relative motion can be expressed as   {x˙(t)=Ax(t)+Bu(t)+Mx(t)dω(t),y(t)=Cx(t),  (4.1) where the system parameter matrices are given respectively in Zhao & Jia (2014),   A=[0001000000100000013w20002w0000−2w0000−w2000],B=[000100000010000001]T,C=[100000010000001000] The coefficient matrices $$M$$ is randomly generated by Matlab:   M=[0.31050.03170.25670.15070.42110.08860.28690.43020.08880.14780.27950.33140.02600.46720.19930.16650.42700.16540.46560.49220.06700.23350.17390.44920.36430.42950.01540.32410.22300.05910.36890.39280.46960.01260.02710.4942]×10−7 Now, we consider the actuator occur the stochastic fault, the matrices $$F_i$$ describing the failure channel of the thrust, suppose in three modes, mode 1: $$f_{11}=0.5, f_{12}=f_{13}=1$$ means the first actuator supplies 50% of the thrust and the other two works normally; mode 2: $$f_{22}=0.5, f_{21}=f_{23}=1$$ means the second actuator supplies 50% of the thrust and the other two works normally; mode 3: $$f_{33}=0.5, f_{31}=f_{32}=1$$ means the third actuator supplies 50% of the thrust and the other two works normally; thus, the fault matrices $$B_i$$ are given as   B1=[0000000000.500010001],B2=[00000000010000.50001],B3=[000000000100010000.5], the transition rates matrix is   Λ=[−0.60.30.30.3−0.30.00.20.5−0.7] (4.2) In this paper, the interest is put on the chaser spacecraft hovering the target spacecraft. Assume the maximum acceleration that the spacecraft thruster can provide is $$u_{max}=0.2{\rm{m/s^2}}$$, the hovering state $x_f=\left[ \begin{array}{@{}cccccc@{}} 1000{\rm{m}} & 1000{\rm{m}} & 1000{\rm{m}} & 0{\rm{m/s}} & 0{\rm{m/s}} & 0{\rm{m/s}}\\ \end{array}\right]^ \mathrm{ T }$, the reference model matrices $$A_m=\textbf{0}_{2\times 2}$$, $$C_m=I_2$$ and the initial state $x_0=\left[ \begin{array}{@{}cccccc@{}} 500{\rm{m}} & 2000{\rm{m}} & 1500{\rm{m}} & 0{\rm{m/s}} & 0{\rm{m/s}} & 0{\rm{m/s}}\\ \end{array}\right]^ \mathrm{T}$. Set the simulation time $$T_{simu}=6000$$ s and a random mode changing curve based on the transition probability rate matrix in equation (4.2) is shown in Fig. 1. Fig. 1. View largeDownload slide Jumping mode of the chased spacecraft. Fig. 1. View largeDownload slide Jumping mode of the chased spacecraft. The state trajectories of the closed-loop system and the model reference tracking controller described by equation (2.7) are abtained. The position and velocity change relatively to the target spacecraft are shown in Figs 2 and 3, respectively. The controller output is shown in Fig. 4 and the tracking errors are recorded in 5. The simulation results from Figs 1 to 5 show that although the actuator occurs fault and the Itô disturbance exist simultaneously, the controlled system can track the reference signal. It follows that the closed-loop system is asymptotically stable. In this numerical simulation example, the task of hovering is accomplished at about Ts = 5000 s. Fig. 2. View largeDownload slide Relative position. Fig. 2. View largeDownload slide Relative position. Fig. 3. View largeDownload slide Relative velocity. Fig. 3. View largeDownload slide Relative velocity. Fig. 4. View largeDownload slide Output of control. Fig. 4. View largeDownload slide Output of control. Fig. 5. View largeDownload slide Tracking errors. Fig. 5. View largeDownload slide Tracking errors. The magnitude of the control is recorded in Fig. 4, which shows the jumping characteristics of the controller. From Fig. 1, it is clear that the time of occurrence of jumping is consistent with the jumping mode of the chased spacecraft, which illustrates the validity of the designed controller. But the output of the controller is not asymptotically convergent to zero may be due to the following two major aspects, one is that there exist both the disturbance and the actuator fault, another is that for the realization of hovering, the orbit of the chased spacecraft is not the Kepler orbit. Thus, for a long time to realize hovering, the output of the controller should be simultaneously acting on the system. It is also clear to see that the desired thrust constraint is satisfied. The steady tracking error is less than 2 m from Fig. 5. The error is not zero, because there exists the Itô stochastic disturbance in the system, but the mean value of the tracking error is zero because the stochastic disturbance is the white noise in this paper. Figure 5 illustrates this point just right from the theoretical viewpoint. This suggests that the chased spacecraft can accurately achieve the desired trajectory tracking that can realize the hovering task. In order to better validate the performance of the controller proposed in the paper, we use the conventional proportional—integral—derivative (PID) controller with saturation constraint for comparison. The calculation process is omitted and the simulation parameters are the same. The results are given directly in Figs 6 and 7. Compared with Fig. 2, the dynamic process of Fig. 6 is more slow for the same simulation time and the tracking error is about 20 m from Fig. 5, which shows the effectiveness of the proposed method in this paper. Fig. 6. View largeDownload slide Relative position using PID method. Fig. 6. View largeDownload slide Relative position using PID method. Fig. 7. View largeDownload slide Tracking error using PID method. Fig. 7. View largeDownload slide Tracking error using PID method. Remark 4 The problem of spacecrafts relative orbit manoeuvring control based on circular orbit describing can be abstracted as the trajectory tracking control problem. In particular, two spacecrafts relative orbit manoeuvring control problem such as space rendezvous, space interception, space hovering and collision avoidance can be simplified as the proposed model reference tracking control problem in this paper. Remark 5 In this point, the dynamic characteristics that the relative motion of chased spacecraft and target spacecraft is not obvious, so the tracking trajectory can be treated as a step signal. Thus, the core problem is how to select the parameter matrices of the reference model. This paper only considers the hovering task, and therefore, how to select the reference model matrices in other circumstances needs to be dealt with in the further research. Remark 6 The tracking error is $$y(t)-y_m(t)=C\int Mx(t)\texttt{d}\omega (t)$$, which relates with the matrices $$C$$, $$M$$ and Wiener process $$\omega (t)$$, thereby assuming that $$\Vert M\Vert^{2}\leq \hat{M}$$, $$\hat{M}$$ is a bounded number. 5. Conclusions In this paper, we have studied the model reference tracking control problem of jump system with Itô stochastic disturbance. The purpose is to make the closed-loop system stable and track the given command. An LMI method is presented to make the stochastic disturbance system asymptotically mean square stable and guarantees the input constraints. Based on the theory of the generalized Sylvester equations, a parametric method is established for the model reference tracking problem. 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Trajectory tracking problem for Markov jump systems with Itô stochastic disturbance and its application in orbit manoeuvring

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Abstract

Abstract This paper deals with the trajectory tracking problem for jump systems with Itô stochastic disturbance. The main contribution is that the form of the controller is complete parametric. The controller consists of the additive contribution of two terms: a feedback term and a feedforward term. The feedback term is presented using the linear matrix inequality method for the state feedback control law to make the system stochastically stable and can guarantee the input constraints. The feedforward term is the complete parametric feedforward tracking compensator by utilizing the parametric solution of the generalized Sylvester matrix equation for the trajectory tracking problem. The simulation about spacecrafts hovering task is carried out in relative orbit manoeuvring control. The simulation results show the effectiveness of the proposed approach. 1. Introduction In the past few decades, Markov jump systems (MJS) have been widely investigated and many useful results have been obtained. At present, numerous research results have been obtained (see Souza & Fragoso 2003; Li et al. 2010; Wang et al. 2015). The motivation for the study is that many practical systems subject to random abrupt variations can be modelled by MJS (see Zhang et al., 2011). Recently, much work has been done for the systems (see Li et al., 2002; Dragan et al., 2004; Hou et al., 2006; Dong et al., 2008; Lin et al., 2009; Chan, 2015,Shi et al., 2015; Kang et al., 2016). It was presented in a survey on recent developments of modelling, analysis and design of MJS (see Shi & Li, 2015). Kang et al. (2016) investigated the input-to-state stability problem of a class of switched stochastic non-linear retarded systems under extended asynchronous switching. The linear quadratic optimization problem for linear Markovian jump stochastic systems subject to multiplicative white noise was investigated in Dragan & Morozan (2004). Chan (2015) studied the market share and forecasting problem using the Markov model. The stochastic stability problem was investigated in Hou et al. (2006) for Itô differential equations with jump parameters. Based on the solvability of a set of coupled Hamilton–Jacobi inequalities, Lin et al. (2009) studied a unified design method for both state and dynamic output $$H_\infty$$ feedback control for the stochastic systems. Li et al. (2015) concerned with the state estimation and sliding mode control problems for phase-type semi-MJS. It was concerned with the problem of designing robust $$H_2$$-state feedback controllers for continuous-time Markov jump linear systems subject to polytopic-type parameter uncertainty in Dong et al. (2008). Almost all the control problems for these systems have been tackled, and we can find the results in the literature. The demand for trajectory tracking is that the closed-loop system is stable and can track the given command, which can be conversed into a model reference tracking control problem. Model reference tracking control plays an important role in control system design, and it has many applications arising in areas such as the control of mobile robots, motor drive, network, mechanical systems and aircraft (see Shyu et al., 1998, Wang & Yang, 2009, Liu & Tao, 2010, Wilcox et al., 2010, Heusden et al., 2011, Li & Wang, 2013, Joshi, 2014). Li & Wang (2013) investigated the finite time-tracking control problem of the mobile robots via visual servoing. Shyu et al. (1998) researched the neural networks-adaptive speed controller for the induction motor to track a reference model. Wang & Yang (2009) investigated the $$H_\infty$$ model reference tracking control for continuous-time networked control systems with communication constraints. A data-driven controller was presented in tuning method that includes a set of constraints for ensuring the system stability in Heusden et al. (2011). Multivariable output tracking adaptive control approach has been studied in Liu & Tao (2010) for the aircraft dynamics with damage. Model reference adaptive control method was studied in Joshi (2014), which considered the actuator failures and sensor bias faults for the aircraft. Wilcox et al. (2010) represented a robust model reference control strategy for a hypersonic vehicle system with additive bounded disturbances and aerothermoelastic effects. Boukas (2009) synthesized a state feedback controller that could make the state vector of the MJS track precisely a given state vector of a reference model. In this paper, the trajectory tracking problem for MJS with Itô stochastic disturbance is considered. Just as it was said in Boukas (2009), this problem has not been fully researched, not to mention the application in the spacecrafts relative orbit manoeuvring control. Thus, this paper will develop a controller, the aim of which is to make the output of the system track the output of the given reference model. Different from Boukas (2009), the controller in this paper is composed of two parts: the first one is a state feedback controller that can stabilize the system and the second one is a complete parametric feedforward tracking compensator. A linear matrix inequality (LMI) method is presented to design the state feedback controller to stabilize the system and two additional LMI conditions are proposed to guarantee the control input constraints. Based on the parametric solution of the generalized Sylvester matrix equation, the complete parametric solution of the feedforward compensator is established for the trajectory tracking problem. Our main contribution is that we obtained the complete parametric solution of the controller for the system. The derived results that the feedforward compensator contain have extra degrees of freedom in parametric design and it can be used to cope with some additional performance. The outline of the content of this paper is as follows. Section 2 presents the problem formulation. Section 3 obtains the main results. In Section 4, the designed controller is applied to the spacecrafts hovering task and the simulation results show the validity of the proposed methodology to handle the tracking problem. Finally, we end this paper in Section 5 with a brief concluding remark. 2. Problem formulation Let us consider a continuous-time MJS with Itô stochastic disturbance, the model is defined by   {x˙(t)=Ax(t)+B(γ(t))u(t)+Mx(t)dω(t),y(t)=Cx(t),  (2.1) where $$x(t)\in\mathbb{R}^{n}$$, $$u(t)\in\mathbb{R}^{r}$$ and $$y(t)\in\mathbb{R}^{d}$$ are the state vector, the input vector and the output vector respectively, and $$\mathbb{R}$$ represents the real space. The matrices $$A\in\mathbb{R}^{n\times n}$$, $$B(\gamma(t))\in\mathbb{R}^{n\times r}$$ and $$C\in\mathbb{R}^{d\times n}$$ are the system coefficient matrices. $$\omega (t)\in\mathbb{R}$$ is a standard Wiener process that is supposed to be independent of the Markov process $$\{r(t),t>0\}$$. $$M$$ is system matrix with appropriate dimensions. The Markov process $$\{\gamma(t),t>0\}$$ describes the jump mode of the system (2.1) and takes values in a finite set $$\mathbb{S}=\{1,2,\ldots,N\}$$, with transition probability matrix $$\Lambda \triangleq \{\lambda_{ij}\}$$ given by   λij=Pr{γ(t+h)=j|γ(t)=i}={λijh+o(h),i≠j,1+λiih+o(h),i=j,  (2.2) where $$h>0$$, $$\lim\limits_{h\rightarrow 0^+}\frac{o(h)}{h}=0$$; here $$\lambda_{ij}$$ is the transition rate from mode $$i$$ to mode $$j(i\neq j)$$, with $$\lambda_{ij} \geq 0$$ and $$\lambda_{ii}=-\sum\limits_{j=1,i\neq j}^{N}\lambda_{ij}$$. For each possible value of the process $$\gamma(t)\in\mathbb{S}$$, we write $$B(\gamma(t))=B_i$$, which are known matrices that represent various types of the stochastic fault of the actuator. Introduce a set of matrices $$F_i\in\mathbb{R}^{r\times r}$$ to describe the actuator fault, $$ F_i$$ is a diagonal matrix with diagonal elements $$f_{il}$$ subject to $$0 \leq f_{il}\leq 1$$, $$l=1,2,\ldots,r$$. When the system (1) is in mode $$\gamma(t)=i$$ , defining $$f_{il}=0 $$ means the $$l$$th actuator has failed completely without any control input; if $$f_{il}=1$$, the $$l$$th actuator works normally; if $$ 0<f_{il}<1 $$, the $$l$$th actuator has partially lost its effectiveness, but it still works. Thus, the matrix $$B_i$$ can be written as   Bi=BFi, (2.3) where $$B\in\mathbb{R}^{n\times r}$$ represents all the actuators work normally. Definition 1 The closed-loop system $$\dot{x}(t)=[A+B_iK_i]x(t)+Mx(t)\texttt{d}\omega (t)$$ is said to be stochastically stable (SS) if for every initial condition $$x(0)=x_0$$ and $$\gamma (0)=\gamma_0$$, the following holds   E{∫t0∞∥x(t)∥2dt|x0,r0}<∞, (2.4) where $$K_i$$ is the state feedback gain matrix and the symbol ‘E’ represents the mathematical expectation operator. This paper mainly researches the model reference tracking control problem for the continuous-time MJS with Itô stochastic disturbance. Without any loss of generality, the reference signal can be generated by the following linear continuous-time reference model   {x˙m(t)=Amxm(t),ym(t)=Cmxm(t),  (2.5) where $$x_m(t)\in\mathbb{R}^{q_1}$$ and $$y_m(t)\in\mathbb{R}^{q_2}$$ are the state vector and the output vector of the reference model. $$A_m$$ and $$C_m$$ are given matrices with appropriate dimension. The purpose is to design a controller to make the output $$y(t)$$ track the output $$y_m(t)$$ of the reference model in mean square sense, which is to satisfy the following tracking requirement   E{∫t0∞∥{y(t)−ym(t)}∥2dt|x0,r0}<∞ (2.6) for arbitrary initial values $$x(0)$$, $$x_m(0)$$ and $$\gamma (0)$$ . The following problem needs to be done to realize the purpose of this paper. Problem 1 Given the system (2.1) and the linear constant reference model (2.5), design a controller in form of   u(t)=Kix(t)+Kmixm(t), (2.7) where $$K_{mi}=K_m(\gamma (t)=i)$$ is the gain matrix of feedforward compensator, such that (a) The closed-loop system is SS. (b) The output $$y(t)$$ can track the output $$y_m(t)$$ in mean square sence (i.e. satisfy (2.6)). Lemma 1 (Boukas, 2005; Schur Complement Lemma): The LMI   [A11A12A21A22]<0 is equivalent to   A22<0,A11−A12A22−1A21<0, where $$A_{11}=A_{11}^{\rm T}$$, $$A_{22}=A_{22}^{\rm T}$$ and $$A_{12}=A_{21}^{\rm T}$$. 3. The main results According to Problem 1, we need to design the feedback controller and the feedforward compensator. The following theorem gives the controller existence conditions. Theorem 1 Problem 1 has a solution if the closed-loop system is SS, and there exist matrices $$G_i\in\mathbb{R}^{n\times q}$$ and $$H_i\in\mathbb{R}^{r\times q}$$ satisfying the following matrix equations   {AGi+BiHi−GiAm=0CGi−Cm=0.  (3.1) In this case, the feedforward gain matrices $$K_{mi}$$ is   Kmi=Hi−KiGi, (3.2) where $$K_i$$ is the state feedback gain matrix such that the closed-loop system is SS. Proof. Let   {δx(t)=x(t)−Gixm(t)δu(t)=u(t)−Hixm(t)δy(t)=y(t)−ym(t)O(t)=Mx(t)dω(t)  (3.3) after calculation, we can obtain   δx˙(t) =x˙(t)−Gix˙m(t)+O(t) =Ax(t)+Biu(t)+O(t)−GiAmxm(t) =[AGi+BiHi−GiAm]xm(t)  +Aδx(t)+Biδu(t)+O(t) (3.4) and   δy(t) =Cδx(t)+(CGi−Cm)xm(t) (3.5) If the conditions (3.1) is satisfied, it is able to show that   {δx˙(t)=Aδx(t)+Biδu(t)+O(t)y(t)=Cδx(t).  (3.6) Notice system (3.6) and system (2.1) have the same structure, hence for arbitrary state feedback control law $$u(t)=K_ix(t)$$, which can stabilize the system (2.1), the state feedback control law   δu(t)=Kiδx(t) (3.7) can also stabilize the system (3.6). That is to say the system   {δx˙(t)=[A+BiKi]δx(t)+O(t)y(t)=Cδx(t)  (3.8) is SS, thus equation (2.6) holds. Combined with the equation (3.3) and equation (3.7), the controller can be written as   u(t)=Kix(t)+[Hi−KiGi]xm(t). (3.9) Then $$K_{mi}=H_i-K_iG_i$$ is obtained. The proof is finished. □ Remark 1 It is not difficult to understand $$ \texttt{E}[O(t)]=\texttt{E}[Mx(t)\texttt{d}\omega (t)]=0$$, as $$\omega (t)\in\mathbb{R}$$ is a standard Wiener process. Thus, the controller in equation (2.7) can ensure equation (2.6) set-up. The next is the design of feedback control law and feedforward tracking compensator. 3.1. State feedback controller design Lemma 2 (Li et al., 2011) The closed-loop system is SS if and only if there exists a set of symmetric positive definite matrices $$P=(P_1, P_2,\ldots,P_N)>0$$, such that the following continuous-time stochastic Lyapunov equations hold   (A+BiKi)TPi+Pi(A+BiKi)+∑k=1NλikPk+MTPiM=−Qi (3.10) for arbitrarily given symmetric positive definite matrices $$Q=(Q_1, Q_2,\ldots,Q_N)$$. Corollary 1 The closed-loop system is SS if there exists a set of symmetric positive definite matrices $$P=(P_1, P_2,\ldots,P_N)>0$$, such that the following holds for each $$i\in\mathbb{S}$$,   (A+BiKi)TPi+Pi(A+BiKi)+∑k=1NλikPk+MTPiM<0. (3.11) Consider the control input constraints in project, there is   ∥u(t)∥2≤umax, (3.12) where $$u_{max}$$ represents the maximum control amplitude. Using the Schur Complement Lemma, the following theorem is proposed to show the conditions that can guarantee the stability and the constraints of the closed-loop system. Theorem 2 The closed-loop system is SS that satisfies the control constraints described in equation (3.12), if there exists a given positive number $$\varrho$$, a set of matrices $$W_i$$ and symmetric positive definite matrices $$X_i$$, $$i\in\mathbb{S}$$, meets the following conditions:   [ϱIxT(0)x(0)Xi]≥0 (3.13)  [umax2IWiWiT1ϱXi]≥0 (3.14)  [ΦiΓiΓiTΔi]<0 (3.15) where   Φi=[AXi+XiAT+BiWi+WiTBiT+λiiXiXiMTMXi−Xi]Γi=[λi1Xi...λi,i−1Xiλi,i+1Xi...λiNXi0...00...0]Δi=Diag[X1...Xi−1Xi+1...Xn] In this case, the feedback gain matrices are given by   Ki=WiXi−1. (3.16) Proof. It follows from Corollary 1 that the closed-loop system is SS if there exist symmetric positive definite matrices $$P_i$$, such that the following holds for each $$i\in\mathbb{S}$$,   (A+BiKi)TPi+Pi(A+BiKi)+∑k=1NλikPk+MTPiM<0. (3.17) As $$P_i$$ is symmetric positive definite matrix, there exists inverse matrix $$P_i^{-1}$$, satisfying $$P_i^{-1}=(P_i^{-1})^{\rm T}$$. Pre-multiplying and post-multiplying by $$P_i^{-1}$$ on both sides of the aforementioned inequality gives   Pi−1AT+Pi−1KiTBiT+APi−1+BiKiPi−1+Pi−1(∑k=1NλikPk)Pi−1+MTPiM<0. By defining $$X_i=P_i^{-1}$$ and $$W_i=K_iP_i^{-1}$$, the above inequality is turned into the following form   XiAT+WiTBiT+AXi+BiWi+Xi(∑k=1NλikXj−1)Xi+MTPiM<0. (3.18) Using Lemma 1, the inequality equation (3.18) can be equivalently arranged into the form of inequality equation (3.15). There exists a positive number $$\varrho$$ satisfying the initial condition $$x^{\rm T}(0)P_ix(0)\leq \varrho$$ for every $$i\in\mathbb{S}$$. Then, while $$t>0$$, it follows that   xT(t)Pix(t)≤xT(0)Pix(0)≤ϱ. (3.19) Using the Schur Complement Lemma, the inequality equation (3.19) can be equivalently arranged into the form of inequality equation (3.13). In view of equation (3.12), there is   uT(t)u(t)=xT(t)KiTKix(t)≤umax2. (3.20) Combining (3.19) and (3.20) and using the Schur Complement Lemma once again, it can be obtained that   [umax2IKiKiT1ϱPi]≥0. (3.21) Pre- and post-multiplying by diag $$(I,X_i)$$ on both sides of the aforementioned inequality, the inequality equation (3.14) can be obtained. The proof is then completed. □ 3.2. Feedforward compensator design The purpose of feedforward compensator is to satisfy the tracking requirement in Problem 1. In order to solve the compensator, the first is to find two matrices $$G_i$$ and $$H_i$$ satisfying equation (3.1). Lemma 3 (Bhatia, 1997) The second equation of equation (3.1) has a solution if and only if there exists a generalized inverse matrix $$C^{-}$$of matrix $$C$$ such that   CC−Cm=Cm, (3.22) then the general complete solution of it can be written as   Gi=C−Cm+[I−C−C]Yi, (3.23) where $$Y_i\in\mathbb{R}^{n\times p}$$ is arbitrary matrix. Substituting (3.23) into the first equation of equation (3.1), gives   BiHi+AC−Cm−C−CmAm= [I−C−C]YiAm−AYi+AC−CYi. (3.24) Let   W=AC−Cm−C−CmAm,E=I−C−C,A^=AE, (3.25) then equation (3.1) can be written in the following form:   BiHi+W=−A^Yi+EYiAm. (3.26) The next gives the equation (3.26) a complete parametric solution. Lemma 1 (Wu & Duan 2008) If the matrix pair [$$sE-\hat{A}$$, $$B_i$$] is controllable, there exists a unimodular matrix $$V_i(s)\in\mathbb{R}^{(n+r)\times (n+r)}$$[s] satisfying   [sE−A^Bi]Vi(s)=[△0] (3.27) where $$\triangle=\Sigma_i(s)$$ is usually a diagonal matrix and satisfies $$\texttt{det}\triangle\neq0$$. Partition the matrix $$V_i(s)\in\mathbb{R}^{(n+r)\times (n+r)}$$[s] as   Vi(s)=[∑a=0αUiasa∑a=0αLiasa∑a=0αQiasa∑a=0αDiasa] (3.28) where $$U_{ia}(s)\in\mathbb{R}^{n\times n}$$, $$L_{ia}(s)\in\mathbb{R}^{n\times r}$$, $$Q_{ia}(s)\in\mathbb{R}^{r\times n}$$ and $$D_{ia}(s)\in\mathbb{R}^{r\times r}$$. Using the Theorem 4.7 and Theorem 5.2 in Duan (2015), the general complete solution of ($$Y_i$$, $$H_i$$) to the non-homogeneous generalized Sylvester equation (3.26) is given by   [YiHi] =∑a=0α[LiaUiaDiaQia][ZiW]Ama. (3.29) Further, in combination with equation (3.29) and equation (3.23), a complete parametric solution of equation (3.1) can be obtained as follows   {Gi=C−Cm+E∑a=0α[LiaZi+UiaW]Ama,Hi=∑a=0α[DiaZi+QiaW]Ama,  (3.30) where $$Z_i\in\mathbb{R}^{r\times q}$$ is arbitrary parameter matrix that represents the degree of freedom in the solution. 3.3. Algorithm for solving the controller Given the reference model coefficient matrices $$A_m$$, $$C_m$$, the transition probability matrix $$\Lambda$$ and the failure matrices $$B_i$$ for every $$i=1,2,\ldots,N$$, then the following algorithm for solving Problem 1 can be presented. Algorithm 1 Step 1. Compute $$W_i$$ and $$X_i$$ according to the equations (3.13), (3.14) and (3.15). Based on equation (3.16), solve the feedback gain matrix $$K_i$$. Step 2. If the matrix pair [$$\hat{A}$$, $$B_i$$]($$\hat{A}$$ is defined in equation (3.25)) is controllable, solve the equation (3.27) and go to next step; otherwise, the feedforward compensation does not exist or the algorithm is failed. Step 3. Computing $$H_i$$ and $$G_i$$ by equation (3.30), utilize the equation (3.2) to compute the gain matrix of the feedforward compensation and the parametric form of the controller (2.7) is obtained. From the above analysis, the following remarks can be deduced. Remark 2 If one or more actuators fail completely, the system will become under-actuated system (Benosman, 2009), then the designed controller in equation (2.7) will not guarantee the system stability. Here, all the actuators are assumed to be active even if there exist faults in some actuators. The under-actuated system is not considered further in this paper. Remark 3 Given the matrices $$A, B_i, C$$ in system (2.1) and $$A_m$$, $$C_m$$ in the reference model (2.5), assume that there exists a generalized inverse matrix $$C^{-}$$ of the matrix $$C$$ that satisfying $$CC^{-}C_m=C_m$$ as well as the matrix pair [$$\hat{A}$$, $$B_i$$] is controllable. 4. The Application to the spacecraft relative orbit manoeuvre In this section, we will apply the proposed model reference tracking control approach to the application for the spacecrafts relative orbit manoeuvring in an circular orbit. Suppose that the target spacecraft is in the geosynchronous orbit, a circular orbit on the Earth’s equator plane with the same period as Earth rotation. The orbital parameters are as follows: the radius $$R=4.2241\times 10^7$$ m, the period $$T=86,164$$ s and the angular rate $$w=7.2921\times 10^{-5}$$ rad/s. The linearized equations of relative motion can be expressed as   {x˙(t)=Ax(t)+Bu(t)+Mx(t)dω(t),y(t)=Cx(t),  (4.1) where the system parameter matrices are given respectively in Zhao & Jia (2014),   A=[0001000000100000013w20002w0000−2w0000−w2000],B=[000100000010000001]T,C=[100000010000001000] The coefficient matrices $$M$$ is randomly generated by Matlab:   M=[0.31050.03170.25670.15070.42110.08860.28690.43020.08880.14780.27950.33140.02600.46720.19930.16650.42700.16540.46560.49220.06700.23350.17390.44920.36430.42950.01540.32410.22300.05910.36890.39280.46960.01260.02710.4942]×10−7 Now, we consider the actuator occur the stochastic fault, the matrices $$F_i$$ describing the failure channel of the thrust, suppose in three modes, mode 1: $$f_{11}=0.5, f_{12}=f_{13}=1$$ means the first actuator supplies 50% of the thrust and the other two works normally; mode 2: $$f_{22}=0.5, f_{21}=f_{23}=1$$ means the second actuator supplies 50% of the thrust and the other two works normally; mode 3: $$f_{33}=0.5, f_{31}=f_{32}=1$$ means the third actuator supplies 50% of the thrust and the other two works normally; thus, the fault matrices $$B_i$$ are given as   B1=[0000000000.500010001],B2=[00000000010000.50001],B3=[000000000100010000.5], the transition rates matrix is   Λ=[−0.60.30.30.3−0.30.00.20.5−0.7] (4.2) In this paper, the interest is put on the chaser spacecraft hovering the target spacecraft. Assume the maximum acceleration that the spacecraft thruster can provide is $$u_{max}=0.2{\rm{m/s^2}}$$, the hovering state $x_f=\left[ \begin{array}{@{}cccccc@{}} 1000{\rm{m}} & 1000{\rm{m}} & 1000{\rm{m}} & 0{\rm{m/s}} & 0{\rm{m/s}} & 0{\rm{m/s}}\\ \end{array}\right]^ \mathrm{ T }$, the reference model matrices $$A_m=\textbf{0}_{2\times 2}$$, $$C_m=I_2$$ and the initial state $x_0=\left[ \begin{array}{@{}cccccc@{}} 500{\rm{m}} & 2000{\rm{m}} & 1500{\rm{m}} & 0{\rm{m/s}} & 0{\rm{m/s}} & 0{\rm{m/s}}\\ \end{array}\right]^ \mathrm{T}$. Set the simulation time $$T_{simu}=6000$$ s and a random mode changing curve based on the transition probability rate matrix in equation (4.2) is shown in Fig. 1. Fig. 1. View largeDownload slide Jumping mode of the chased spacecraft. Fig. 1. View largeDownload slide Jumping mode of the chased spacecraft. The state trajectories of the closed-loop system and the model reference tracking controller described by equation (2.7) are abtained. The position and velocity change relatively to the target spacecraft are shown in Figs 2 and 3, respectively. The controller output is shown in Fig. 4 and the tracking errors are recorded in 5. The simulation results from Figs 1 to 5 show that although the actuator occurs fault and the Itô disturbance exist simultaneously, the controlled system can track the reference signal. It follows that the closed-loop system is asymptotically stable. In this numerical simulation example, the task of hovering is accomplished at about Ts = 5000 s. Fig. 2. View largeDownload slide Relative position. Fig. 2. View largeDownload slide Relative position. Fig. 3. View largeDownload slide Relative velocity. Fig. 3. View largeDownload slide Relative velocity. Fig. 4. View largeDownload slide Output of control. Fig. 4. View largeDownload slide Output of control. Fig. 5. View largeDownload slide Tracking errors. Fig. 5. View largeDownload slide Tracking errors. The magnitude of the control is recorded in Fig. 4, which shows the jumping characteristics of the controller. From Fig. 1, it is clear that the time of occurrence of jumping is consistent with the jumping mode of the chased spacecraft, which illustrates the validity of the designed controller. But the output of the controller is not asymptotically convergent to zero may be due to the following two major aspects, one is that there exist both the disturbance and the actuator fault, another is that for the realization of hovering, the orbit of the chased spacecraft is not the Kepler orbit. Thus, for a long time to realize hovering, the output of the controller should be simultaneously acting on the system. It is also clear to see that the desired thrust constraint is satisfied. The steady tracking error is less than 2 m from Fig. 5. The error is not zero, because there exists the Itô stochastic disturbance in the system, but the mean value of the tracking error is zero because the stochastic disturbance is the white noise in this paper. Figure 5 illustrates this point just right from the theoretical viewpoint. This suggests that the chased spacecraft can accurately achieve the desired trajectory tracking that can realize the hovering task. In order to better validate the performance of the controller proposed in the paper, we use the conventional proportional—integral—derivative (PID) controller with saturation constraint for comparison. The calculation process is omitted and the simulation parameters are the same. The results are given directly in Figs 6 and 7. Compared with Fig. 2, the dynamic process of Fig. 6 is more slow for the same simulation time and the tracking error is about 20 m from Fig. 5, which shows the effectiveness of the proposed method in this paper. Fig. 6. View largeDownload slide Relative position using PID method. Fig. 6. View largeDownload slide Relative position using PID method. Fig. 7. View largeDownload slide Tracking error using PID method. Fig. 7. View largeDownload slide Tracking error using PID method. Remark 4 The problem of spacecrafts relative orbit manoeuvring control based on circular orbit describing can be abstracted as the trajectory tracking control problem. In particular, two spacecrafts relative orbit manoeuvring control problem such as space rendezvous, space interception, space hovering and collision avoidance can be simplified as the proposed model reference tracking control problem in this paper. Remark 5 In this point, the dynamic characteristics that the relative motion of chased spacecraft and target spacecraft is not obvious, so the tracking trajectory can be treated as a step signal. Thus, the core problem is how to select the parameter matrices of the reference model. This paper only considers the hovering task, and therefore, how to select the reference model matrices in other circumstances needs to be dealt with in the further research. Remark 6 The tracking error is $$y(t)-y_m(t)=C\int Mx(t)\texttt{d}\omega (t)$$, which relates with the matrices $$C$$, $$M$$ and Wiener process $$\omega (t)$$, thereby assuming that $$\Vert M\Vert^{2}\leq \hat{M}$$, $$\hat{M}$$ is a bounded number. 5. Conclusions In this paper, we have studied the model reference tracking control problem of jump system with Itô stochastic disturbance. The purpose is to make the closed-loop system stable and track the given command. An LMI method is presented to make the stochastic disturbance system asymptotically mean square stable and guarantees the input constraints. Based on the theory of the generalized Sylvester equations, a parametric method is established for the model reference tracking problem. 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IMA Journal of Mathematical Control and InformationOxford University Press

Published: May 3, 2017

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