Robust variable structure observer design for non-linear large-scale systems with non-linear interconnections

Robust variable structure observer design for non-linear large-scale systems with non-linear... Abstract In this paper, a variable structure observer is designed for a class of non-linear large-scale interconnected systems in the presence of uncertainties and non-linear interconnections. The modern geometric approach is used to explore system structure and a transformation is employed to facilitate the observer design. Based on the Lyapunov direct method, a set of conditions are developed such that the proposed variable structure systems can be used to estimate the states of the original interconnected systems asymptotically. The internal dynamical structure of the isolated nominal subsystems as well as the structure of the uncertainties are employed to reduce the conservatism. The bounds on the uncertainties are non-linear and are employed in the observer design to reject the effect of the uncertainties. A numerical example is presented to illustrate the approach and the simulation results showthat the proposed approach is effective. 1. Introduction The development of advanced technologies has produced many complex systems. An important class of complex systems, which is frequently called a system of systems or large-scale system, can usually be expressed by sets of lower-order ordinary differential equations which are linked through interconnections. Such models are typically called large-scale interconnected systems (see, e.g. Yan et al., 1999; Bakule, 2008; Mahmoud, 2011; Yan et al., 2013;). Large-scale interconnected systems widely exist in practice, for example, power networks, ecological systems, transportation networks, biological systems and information technology networks (Lunze, 1992; Mahmoud, 2011). Increasing requirements for system performance have resulted in increasing complexity within system modelling and it becomes of interest to consider non-linear large-scale interconnected systems. Such models are then used for controller design. In order to obtain good performance levels, a controller may benefit from knowledge of all the system states. This state information may be difficult or expensive to obtain and it becomes of interest to design an observer to estimate all the system states using only the subset of information available from the measured and known input and output of the system. Large-scale interconnected systems have been studied since the 1970s (Sandell et al., 1978). Early work focussed on linear systems. Subsequent results used decentralized control frameworks for non-linear large-scale interconnected systems. In much of this work, however, it is assumed that all the system state variables are available for use by the controller (Wu, 2005; Bakule, 2008; Mahmoud, 2011; Zhao et al., 2014). However, this may be limiting in practice as only a subset of state variables may be available/measureable. It becomes of interest to establish observers to estimate the system states and then use the estimated states to replace the true system states in order to implement state feedback decentralized controllers. It is also the case that observer design has been heavily applied for fault detection and isolation (Diao & Yan, 2008; Yan & Edwards, 2008; Reppa et al., 2014). This further motivates the study of observer design for non-linear large scale interconnected systems. The concept of an observer was first introduced by Luenberger (1964) where the difference between the output measurements from the actual plant and the output measurements of a corresponding dynamical model were used to develop an injection signal to force the resulting output error to zero. Later the approach was extended to non-linear systems and an extended Luenberger observer for non-linear systems is proposed in Zeitz (1987) where uncertainties are not considered. It should be noted that many approaches have been developed for observer design such as the sliding mode observer approach in Yan et al. (2013), the adaptive observer in Wu (2009) and an error linearization approach in Xia & Gao (1989). However, results concerning observer design for interconnected systems are very few when compared with the corresponding results available on controller design for interconnected systems. Sliding mode techniques have been used to design observers for non-linear interconnected power systems in Modarres et al. (2012). In Li et al. (2015) state estimation and sliding mode control for a special class of stochastic dynamic systems which is semi-Markovian jump systems is presented. The authors designed a state observer to generate the estimate of unmeasured state components, and then synthesize a sliding mode control law based on the state estimates. Wang & Fei (2015) discussed the position regulation problem of permanent magnet synchronous motor servo system based on adaptive fuzzy sliding mode control method. They used adaptive method to estimate the upper bound of the approximation error between the equivalent control law and the fuzzy controller are utilized in the paper. An adaptive observer is designed for a class of interconnected systems in Wu (2009) in which it is required that the isolated nominal subsystems are linear. Observer schemes for interconnected systems are proposed in Keliris et al. (2015), Reppa et al. (2014), Sharma & Aldeen (2011) and Yan & Edwards (2008) where the obtained results are unavoidably conservative as it is required that the designed observer can be used for certain fault detection and isolation problems. For example, it is required that the uncertainty can be decoupled with faults in Yan & Edwards (2008) and the considered system is not interconnected systems. Robust observer design is considered in Mohmoud (2012) for a class of linear large scale dynamical systems where it is required that the interconnections satisfy quadratic constraints. In Swarnakar et al. (2007) a new decentralized control scheme which uses estimated states from a decentralized observer within a feedback controller is proposed. This uses a design framework based on linear matrix inequalities and is thus applicable for linear systems. A robust observer for non-linear interconnected systems based on a constrained Lyapunov equation has been developed in Yan et al. (2003). A proportional integral observer is utilized for non-linear interconnected systems for disturbance attenuation in Ghadami & Shafai (2011) and interconnected non-linear dynamical systems are considered in Dashkovskiy & Naujok (2015) where the authors combine the advantages of input-to-state dynamical stability and use reduced order observers to obtain quantitative information about the state estimation error. This work does not, however, consider uncertainties. It should be noted that in all the existing work relating to observer design for large scale interconnected systems, it is required that either the isolated subsystems are linear or the interconnections are linear. Moreover, most of the designed observers are used for special purposes such as fault detection or stabilization and thus they impose specific requirements on the class of interconnected systems considered. This paper is an extension and modification of the authors’ conference paper in Mohamed et al. (2016). In this paper, a class of non-linear interconnected systems with disturbances is considered where both the nominal isolated subsystems and interconnections are non-linear. It is not required that either the nominal isolated subsystems or the interconnections are linearizable. A robust variable structure observer is established based on a simplified system structure by using Lyapunov analysis methodology. The structure of the internal dynamics, the structure of uncertainties and the bounds on uncertainties are fully used in the observer design to reduce the conservatism. These bounds are allowed to have a general non-linear form. The observer states converge to the system states asymptotically. An example with simulation is given to demonstrate the proposed approach. 2. Preliminaries Consider the single input–single output non-linear system x˙(t)=f(x)+g(x)u (2.1) y(t)=h(x), (2.2) where $$x\in \Omega\subset R^{n}$$ ($$\Omega$$ is a neighbourhood of the origin), $$y\in R$$ and $$u\in U \subset R$$ ($$ U $$ is an admissible control set) are the state, output and input, respectively, $$f(x)$$, $$g(x) \in R^{n}$$ are smooth vector fields defined in the domain $$\Omega$$, and $$h(x)\in R^{m}$$ is a smooth vector in the domain $$\Omega$$. Firstly, recall some key elements of the geometric approach in Isidori (1995) which will be used in the later analysis. The notation used in this paper is the same as Isidori (1995) unless it is specifically defined. Definition 1 From Isidori (1995), system (2.1)–(2.2) is said to have uniform relative degree $$r$$ in the domain $$\Omega$$ if for any $$x\in\Omega$$, (i)$$L_{g} L^k_f h(x)=0,\quad\mbox{for}\quad k=1,2,\ldots,r-1,$$ (ii)$$L_{g} L^{r-1}_f h(x)\neq 0.$$ Now consider system (2.1)–(2.2). It is assumed that system (2.1)–(2.2) has uniform relative degree $$r$$ in domain $$\Omega$$. Construct a mapping $$\phi: x \rightarrow z$$ as follows: ϕ(⋅):{z1=h(x)z2=Lfh(x)⋮zr=Lfr−1h(x)zr+1=ϕr+1⋮zn=ϕn(x), (2.3) where $$\phi(\cdot)=\mbox{col}(\phi_1(x),\phi_2(x),\ldots,\phi_n(x)),$$$$\phi_1(x)=h(x),$$$$\phi_2(x)=L_f h(x),\ldots,\phi_r(x)=L^{r-1}_f h(x)$$ and the functions $$\phi_{r+1}(x),$$$$\ldots,$$$$\phi_{n}(x)$$ need to be selected such that Lgϕi(x)=0,i=r+1,r+2,…,N and the Jacobian matrix Jϕ:=∂ϕ(x)∂x is non-singular in domain $$\Omega$$. Then the mapping $$\phi\hspace{0mm}: x \rightarrow z$$ forms a diffeomorphism in the domain $$\Omega$$. For the sake of simplicity, let ζ=[ζ1ζ2⋯ζr]T:=[z1z2⋯zr]Tη=[ζr+1ζr+2⋯ζn]T:=[zr+1zr+2⋯zn]T. Then, from Isidori (1995), it follows that in the new coordinates $$z$$, system (2.1)–(2.2) can be described by ζ˙1=ζ2ζ˙2=ζ3⋮ζ˙r−1=ζrζ˙r=a(ζ,η)+b(ζ,η)uη˙=q(ζ,η), (2.4) where a(ζ,η)=Lfrh(ϕ−1(ζ,η))b(ζ,η)=LgLfr−1h(ϕ−1(ζ,η)) and q(ζ,η)=[qr+1(ζ,η)qr+2(ζ,η)⋮qn(ζ,η)]=[Lfϕr+1(ϕ−1(ζ,η))Lfϕr+2(ϕ−1(ζ,η))⋮Lfϕn(ϕ−1(ζ,η))]. It should be noted that the coordinate transformation (2.3) will be available if $$\phi_i(x)$$ are available for $$i=r+1,\ldots,N$$, and in this case, the system (2.4) can be obtained directly. 3. Large-scale system description and problem statement Consider the non-linear interconnected systems x˙i(t)=fi(xi)+gi(xi)ui+Δfi(xi)+∑j≠ij=1NDij(xj), (3.1) yi(t)=hi(xi),i=1,2,…,N, (3.2) where $$x_i\in \Omega_i\subset R^{n_i}$$ ($$\Omega_i$$ is a neighbourhood of the origin), $$y_i\in R$$ and $$u_i\in U_i \subset R$$ ($$ U_i $$ is an admissible control set) are the state, output and input of the $$i$$th subsystem, respectively, $$f_i(x_i)\in R^{n_i}$$ and $$g_i(x_i) \in R^{n_i}$$ are smooth vector fields defined in the domain $$\Omega_i$$, and $$h_i(x_i)\in R^{m_i}$$ are smooth in the domain $$\Omega_i$$ for $$i=1,2,\ldots,N$$. The term $$\Delta {f_i} (x_i)$$ includes all the uncertainties experienced by the $$i$$th subsystem. The term $$\sum_{\stackrel{j=1}{j\not=i}}^N D_{ij}(x_j)$$ is the non-linear interconnection of the $$i$$th subsystem. Definition 2 The systems x˙i(t)=fi(xi)+gi(xi)ui+Δfi(xi) (3.3) yi(t)=hi(xi),i=1,2,…,N (3.4) are called the isolated subsystems of the systems (3.1)–(3.2), and the systems x˙i(t)=fi(xi)+gi(xi)ui (3.5) yi(t)=hi(xi),i=1,2,…,N (3.6) are called the nominal isolated subsytems of the systems (3.1)–(3.2). In this paper, under the assumption that the isolated subsystems (3.5)–(3.6) have uniform relative degree $$r_i$$ in the considered domain $$\Omega_i$$, the interconnected systems (3.1)–(3.2) are to be analysed. The objective is to explore the system structure based on a geometric transformation to design a robust asymptotic observer for the interconnected system (3.1)–(3.2). It should be noted that the following results can be extended to the case where the isolated subsystems are multi-input and multi-output using the corresponding framework to Section 2 for the multi-input and multi-output case provided in Isidori (1995). 4. System analysis and assumptions In this section, some assumptions are imposed on the system (3.1)–(3.2) to facilitate the observer design. Assumption 1 The nominal isolated subsystem (3.5)–(3.6) has uniform relative degree $$r_i$$ in domain $$x_i\in \Omega_i$$ for $$i=1,2,\ldots,N$$. Under Assumption 1, it follows from Section 2 that there exists a coordinate transformation Ti:xi→col(ζi,ηi), (4.1) where ζi=[ζi1ζi2⋮ζiri]=[hi(xi)Lfhi(xi)⋮Lfri−1hi(xi)]∈Rri (4.2) and $$\eta_i\in R^{n_i-r_i}$$ is defined by ηi=[ηi1ηi2⋮ηni−ri]=[ϕi(ri+1)(xi)ϕi(ri+2)(xi)⋮ϕini(xi)]∈Rni−ri (4.3) for $$i=1,2,\ldots,N$$. The functions $$ \phi_{i(r_i+1)}(x_i),$$$$\phi_{i(r_i+2)}(x_i),\ldots,\phi_{in_i}(x_i)$$ can be obtained by solving the following partial differential equations: Lgiϕi(xi)=0,xi∈Ωi,i=1,2,…,N. (4.4) From Section 2, it follows that in the new coordinate system $$(\zeta_i,\eta_i)$$, the nominal isolated subsystem (3.5)–(3.6) can be described by ζ˙i=Aiζi+βi(ζi,ηi,ui), (4.5) η˙i=qi(ζi,ηi) (4.6) yi=Ciζi, (4.7) where Ai=[010⋯0001⋯0⋮⋮⋮⋮⋮000⋯1000⋯0]∈Rri×ri,Ci=[10⋯0]∈R1×ri (4.8) βi(ζi,ηi,ui)=[0⋮0Lfirihi(Ti−1(ζi,ηi))+LgiLfiri−1hi(Ti−1(ζi,ηi))ui]. (4.9) It is clear to see that the pair $$(A_i,C_i)$$ is observable. Thus, there exists a matrix $$L_i$$ such that $$A_i-L_iC_i$$ is Hurwitz stable. This implies that, for any positive definite matrix $$Q_i\in R^{r_i\times r_i}$$, the Lyapunov equation (Ai−LiCi)TPi+Pi(Ai−LiCi)=−Qi (4.10) has a unique positive-definite solution $$P_i\in R^{r_i\times r_i}$$ for $$i=1,2,\ldots,N$$. Assumption 2 The uncertainty $$\Delta {f_i}(x_i)$$ in (3.1) satisfies ∂Ti∂xiΔfi(xi)=[EiΔΨ(xi)0] , (4.11) where $$T_i(\cdot)$$ is defined in (4.1), $$E_i \in R^{r_i \times r_i}$$ is a constant matrix satisfying EiTPi=HiCi (4.12) for some matrix $$H_i$$, with $$P_i$$ satisfying (4.10), and $$\|\Delta \Psi_i(x_i)\|\leq\kappa_i(x_i)$$, where $$\kappa_i(x_i)$$ is continuous and Lipschitz about $$x_i$$ in the domain $$\Omega_i$$ for $$i=1,2,\ldots,N$$. Remark 1 Solving the Lyapunov equation (4.10) in the presence of the constraint (4.12) is the well-known constrained Lyapunov problem (Galimidi & Barmish, 1986). Although there is no general solution available for this problem, associated discussion and an algorithm can be found in Edwards et al. (2007). Remark 2 Assumption 2 is a limitation on the uncertainty $$\Delta f_i(x_i)$$, and this is necessary to guarantee the existence of asymptotic observers. Denote the non-linear uncertain term $$\Delta \Psi_i(x_i)$$ in (4.11) in the new coordinate frame $$(\zeta_i,\eta_i)$$ by $$\Delta\Phi_i(\zeta_i,\eta_i),$$ i.e. ΔΦi(ζi,ηi)=[ΔΨi(ζi,ηi)]xi=Ti−1(ζi,ηi). (4.13) From Assumption 2, there exists a function $$\rho_i(\zeta_i,\eta_i)$$ such that ‖ΔΦi(ζi,ηi)‖≤ρi(ζi,ηi) (4.14) and $$\rho_i(\zeta_i,\eta_i)$$ satisfies the Lipschitz condition in $$T_i(\Omega_i)$$. Thus for any $$(\zeta_i,\eta_i)$$ and $$(\hat{\zeta_i},\hat{\eta_i}) \in T_i(\Omega_i)$$, ‖ρi(ζi,ηi)−ρi(ζ^i,η^i)‖≤lia‖ζi−ζ^i‖+lib‖ηi−η^i‖, (4.15) where both $$l_i^a$$ and $$l_i^b$$ are non-negative constants. Consider the interconnections $$D_{ij}(x_j)$$ in system (3.1). Partition the term $$\frac{\partial T_i}{\partial x_i} D_{ij}(x_j)$$ as follows ∂Ti∂xiDij(xj)|xj=Tj−1(ζj,ηj)=[Γija(ζj,ηj)Γijb(ζj,ηj)], (4.16) where $$\Gamma^a_{ij}{(\zeta_j,\eta_j)} \in R^{r_i}$$, $$\Gamma^b_{ij}{(\zeta_j,\eta_j)} \in R^{n_i-r_i}$$ for $$i=1,2,\ldots,N$$ and $$i\neq j$$. Assumption 3 The non-linear terms $$\Gamma^a_{ij}{(\zeta_j,\eta_j)} \in R^{r_i}$$ and $$\Gamma^b_{ij}{(\zeta_j,\eta_j)} \in R^{n_i-r_i}$$ in (4.16) satisfy the Lipschitz condition in $$T_i(\Omega_i)$$. Assumption 3 implies that there exist non-negative constants $$\alpha_{ij}^a$$, $$\alpha_{ij}^b $$, $$\mu_{ij}^a $$ and $$\mu_{ij}^b$$ such that ‖Γija(ζi,ηi)−Γija(ζ^i,η^i)‖≤αija‖ζj−ζ^j‖+αijb∥ηj−η^j‖ (4.17) ‖Γijb(ζi,ηi)−Γijb(ζ^i,η^i)‖≤μija‖ζj−ζ^j‖+μijb∥ηj−η^j‖ (4.18) for $$i=1,2,\ldots,N$$ and $$i \neq j$$. From (4.5)–(4.7) and the analysis above, it follows that under Assumption 2, in the new coordinate system $$(\zeta_i,\eta_i)$$, the system (3.1)–(3.2) can be described by ζ˙i=Aiζi+βi(ζi,ηi,ui)+EiΔΨi(ζi,ηi)+∑j≠ij=1NΓija(ζj,ηj), (4.19) η˙i=qi(ζi,ηi)+∑j≠ij=1NΓijb(ζj,ηj), (4.20) yi=Ciζi, (4.21) where $$A_i$$ and $$ C_i$$ are given in (4.8), $$\beta_i(\cdot)$$ is defined in (4.9) and $$\Gamma^a_{ij}(\cdot)$$ and $$\Gamma^b_{ij}(\cdot)$$ are defined in (4.16). Remark 3 Since $$\beta_i(\cdot)$$ is continuous in the domain $$ T_i(\Omega_i)$$, it is straightforward to see that there exists a subset in the domain $$T_i(\Omega_i)$$ such that the function $$\beta_i(\cdot)$$ is Lipschitz in the subset ∥βi(ζi,ηi,ui)−βi(ζi^,ηi^,ui)∥via(ui)∥ζi−ζ^i∥+vib(ui)∥ηi−η^i∥, (4.22) where $$v_i^a(u_i)$$ and $$v_i^b(u_i)$$ are non-negative functions of $$u_i$$ for $$i=1,2,\ldots,N$$. Assumption 4 The function $$q_i(\zeta_i,\eta_i)$$ in equation (4.20) has the following decomposition qi(ζi,ηi)=Miηi+θi(ζi,ηi), (4.23) where $$ M_i \in R^{(n_i-r_i)\times (n_i-r_i)}$$ is a Hurwitz matrix and $$\theta_i(\zeta_i,\eta_i)$$ are Lipschitz in domain $$T_i(\Omega_i)$$. Under Assumption 4, there exist non-negative constants $$\tau_i^a$$ and $$\tau_i^b$$ such that. ∥θi(ζi,ηi)−θi(ζ^i,η^i)∥τia∥ζi−ζ^i∥+τib∥ηi−η^i∥ (4.24) for $$i= 1,2,\ldots,N.$$ Further, from the fact that $$M_i$$ is Hurwitz stable for $$\Lambda_i>0$$, the following Lyapunov equation has a unique solution $$\Pi_i>0$$ MiTΠi+ΠiMi=−Λi,i=1,2,…,N. (4.25) 5. Non-linear observer synthesis In this section, an observer is designed for the transformed systems (4.19)–(4.21) which provides asymptotic estimation of the states of the interconnected systems (4.19)–(4.21). For system (4.19)–(4.21), construct dynamical systems ζ^˙i=Aiζ^i+Li(yi−Ciζ^i)+βi(ζ^i,η^i,ui)+Ki(y,ζ^i,η^i)+∑j≠ij=1NΓija(ζ^j,η^j), (5.1) η^˙i=Miη^i+θi(ζ^i,η^i)+∑j≠ij=1NΓijb(ζ^j,η^j), (5.2) where the term $$K_i(y_i,\hat{\zeta}_i,\hat{\eta}_i)$$ is defined by Ki(yi,ζ^i,η^i)={Pi−1CiT(yi−Ciζ^i)‖yi−Ciζ^i‖∥Hi∥ρi(ζ^i,η^i),yi−Ciζ^i≠00,yi−Ciζ^i=0, (5.3) where $$P_i$$ and $$H_i$$ satisfy (4.10) and (4.12), respectively. Remark 4 System (5.1)–(5.2) is called a variable structure system throughout this paper due to the discontinuous terms defined in (5.3). It should be noted that this observer is different from a sliding mode observer as the proposed observer (5.1)–(5.2) may not produce a sliding motion. The following results are ready to be presented. Theorem 1 Suppose Assumptions 1–4 hold. Then, the dynamical system (5.1)–(5.2) is a robust asymptotic observer of system (4.19)–(4.21), if the function matrix $$W^T(\cdot)+W(\cdot)$$ is positive definite in the domain $$T(\Omega) \times U:=T(\Omega_1) \times U_1 \times T(\Omega_2)\times U_2 \times\cdots\times T(\Omega_N)\times U_N$$, where the matrix $$W(\cdot)=\left[w_{ij}(\cdot)\right]_{2N \times 2N}$$, and its entries $$w_{ij}(\cdot)$$ are defined by wij={λmin(Qi)−2λmax(Pi)via(.)−2lia‖Ci‖‖Hi‖,i=j,1≤i≤N−2λmax(Pi)αija,i≠j,1≤i≤N,1≤j≤Nλmin(Λi−N)−2λmax(Πi−N)τi−Nb,i=j,N+1≤i≤2N−2λmax(Π(i−N))μ(i−N)(j−N)b,i≠j,N+1≤i≤2N,N+1≤j≤2N−2[λmax(Pi)vib(.)+lib‖Ci‖‖Hi‖+λmax(Πi)τia],j−i=N,1≤i≤N,N+1≤j≤2N−2λmax(Pi)αi(j−N)b,j−i≠N,1≤i≤N,N+1≤j≤2N0,i−j=N,N+1≤i≤2N,1≤j≤N−2λmax(Πi−N)μ(i−N)ja,i−j≠N,N+1≤i≤2N,1≤j≤N. Proof Let $$e_{\zeta_i}=\zeta_i-\hat{\zeta}_i $$ and $$ e_{\eta_i}=\eta_i-\hat{\eta}_i$$ for $$i= 1,2,\ldots, N$$. Compare systems (4.19)–(4.20) and (5.1)–(5.2). It follows that the error dynamical systems are described by e˙ζi=(Ai−LiCi)eζi+βi(ζi,ηi,ui)−βi(ζi^,ηi^,ui)+EiΔΨi(ζi,ηi)−Ki(yi,ζ^i,η^i)+∑j=ij=1NΓija(ζj,ηj)−∑j=ij=1NΓija(ζ^j,η^j), (5.4) e˙ηi=Mieηi+θi(ζi,ηi)−θi(ζ^i,η^i)+∑j=ij=1NΓijb(ζj,ηj)−∑j=ij=1NΓijb(ζ^j,η^j). (5.5) Now, for the system (5.4) and (5.5) consider the following candidate Lyapunov function V=∑i=1NeζiTPieζi+∑i=1NeηiTΠieηi. (5.6) Then, the time derivative of the candidate Lyapunov function can be described by V˙=∑i=1N[(e˙ζiTPieζi+eζiTPieζi˙)+(e˙ηiTΠieηi+eηiTΠie˙ηi)]. (5.7) Substituting both $$\dot{e}_{\zeta_i}$$ in (5.4) and $$\dot{e}_{\eta_i}$$ in (5.5) into equation (5.7), it follows by direct computation that the time derivative of the function $$V$$ in (5.6) can be described by V˙=∑i=1N{eζiT[(Ai−LiCi)TPi+Pi(Ai−LiCi)]eζi+2eζiTPi[βi(ζi,ηi,ui)−βi(ζi^,ηi^,ui)]+2[eζiTPiEiΔΨi(ζi,ηi)−eζiTPiKi(yi,ζ^i,η^i)]+2eζiTPi∑j≠ij=1N[Γija(ζj,ηj)−Γija(ζ^j,η^j)]+eηiT(MiTΠi+ΠiMi)eηi+2eηiTΠi[θi(ζi,ηi)−θi(ζ^i,η^i)]+2eηiTΠi∑j≠ij=1N[Γijb(ζj,ηj)−Γijb(ζ^j,η^j)]}. (5.8) From (4.12), (4.14), (4.15) and (5.3), it follows that: (i) If $$y_i-C_i\hat{\zeta}_i=0$$, then from (4.12) and $${e}^T_{\zeta_i}C_i^T=(y_i-C_i\hat{\zeta}_i)^T$$ eζiTPiEiΔΦi(ζi,ηi)−eζiTPiKi(yi,ζ^i,η^i)=eζiTCiTHiTΔΦi(ζi,ηi)=(Hi(yi−Ciζ^i))TΔΦi(ζi,ηi)=0. (ii) If $$y_i-C_i\hat{\zeta}_i\neq 0$$, then from (4.12), (4.14), (4.15) and (5.3) eζiTPiEiΔΦi(ζi,ηi)−eζiTPiKi(yi,ζ^i,η^i)=eζiTCiTHiTΔΦi(ζi,ηi)−eζiTPiPi−1CiT(yi−Ciζ^i)‖yi−Ciζ^i‖∥Hi∥ρi(ζ^i,η^i)=(Cieζi)THiTΔΦi(ζi,ηi)−eζiTCiTCieζi‖Cieζi‖‖Hi‖ρi(ζ^i,η^i)≤‖Cieζi‖‖Hi‖{ρi(ζi,ηi)−ρi(ζ^i,η^i)}≤‖Cieζi‖‖Hi‖{lia‖ζi−ζ^i‖+lib‖ηi−η^i‖}. Then, from (i) and (ii) above, it follows that eζiTPiEiΔΦi(ζi,ηi)−eζiTPiKi(yi,ζ^i,η^i)≤‖Cieζi‖‖Hi‖(lia‖eζi‖+lib‖eηi‖). (5.9) Substituting (4.17), (4.18), (4.22), (4.24) and (5.9) into (5.8) yields V˙≤∑i=1N{−eζiTQieζi+2‖eζi‖‖Pi‖[via‖eζi‖+vib‖eηi‖]+2‖eζi‖‖Ci‖‖Hi‖[lia‖eζi‖+lib‖eηi‖]+2‖eζi‖‖Pi‖∑j≠ij=1N[αija‖eζj‖+αijb‖eηj‖]−eηiTΛieηi+2eηiT‖Πi‖[τia‖eζi‖+τib‖eηi‖]+2eηiT‖Πi‖∑j≠ij=1N[μija‖eζj‖+μijb‖eηj‖]}≤∑i=1N{−eζiTQieζi+2via‖eζi‖2‖Pi‖+2vib‖eζi‖‖eηi‖‖Pi‖+2lia‖eζi‖2‖Ci‖‖Hi‖+2lib‖eζi‖‖eηi‖Ci‖‖Hi‖+∑j≠ij=1N[2αija‖eζi‖‖eζj‖‖Pi‖+2αijb‖eζi‖‖eηj‖‖Pi‖]−eηiTΛieηi+2τia‖Πi‖‖eζi‖‖eηi‖+2τib‖Πi‖‖eηi‖2+∑j≠ij=1N[2μija‖Πi‖‖eζj‖‖eηi‖+2μijb‖Πi‖‖eηi‖‖eηj‖]}≤∑i=1N{−[λmin(Qi)−2λmax(Pi)via−2lia‖Ci‖‖Hi‖]‖eζi‖2+[∑j≠ij=1N2λmax(Pi)αija]‖eζi‖‖eζj‖+[2λmax(Pi)vib+2lib‖Ci‖‖Hi‖+2λmax(Πi)τia]‖eζi‖‖eηi‖+2∑j≠ij=1Nλmax(Pi)αijb‖eζi‖‖eηj‖+∑j≠ij=1N2λmax(Πi)μija‖eζj‖‖eηi‖−[λmin(Λi)−2λmax(Πi)τib]‖eηi‖2+[∑j≠ij=1N2λmax(Πi)μijb]‖eηi‖‖eηj‖}≤−∑i=1N{[λmin(Qi)−2λmax(Pi)via−2lia‖Ci‖‖Hi‖]‖eζi‖2−[∑j≠ij=1N2λmax(Pi)αija]‖eζi‖‖eζj‖−[2λmax(Pi)vib+2lib‖Ci‖‖Hi‖+2λmax(Πi)τia]‖eζi‖‖eηi]‖−2∑j≠ij=1Nλmax(Pi)αijb‖eζi‖‖eηj‖]−∑j≠ij=1N2λmax(Πi)μija‖eζj‖‖eηi‖+[λmin(Λi)−2λmax(Πi)τib]‖eηi‖2−[∑j≠ij=1N2λmax(Πi)μijb]‖eηi‖‖eηj‖}. Then, from the definition of the matrix $$W(\cdot)$$ and the inequality above, it follows that V˙≤−12XT[WT(⋅)+W(⋅)]X, where $$X= [\|e_{\zeta_1}\|,\|e_{\zeta_2}\|, \ldots, \|e_{\zeta_N}\|,\|e_{\eta_1}\|,\|e_{\eta_2}\|,\ldots,$$$$\|e_{\eta_N}\|]^T$$. Since $$W^T(\cdot)+W(\cdot)$$ is positive definite in the domain $$ T(\Omega) \times U$$, it is clear that $$\dot V|_{(\text{5.1})-(\text{5.2})}$$ is negative definite. Therefore, the error system (5.4)–(5.5) is asymptotically stable, that is, limt→∞‖ζi(t)−ζ^i(t)‖=0andlimt→∞‖ηi(t)−η^i(t)‖=0. (5.10) Hence, the conclusion follows. Remark 5 Theorem 1 shows that variable structure system (5.1)–(5.2) is an asymptotic observer of the interconnected system (4.19)–(4.21). It is called a variable structure observer in this paper. Now, consider interconnected system (3.1)–(3.2). Assume that $$\frac {\partial T_i(\zeta_i,\eta_i)}{\partial (\zeta_i,\eta_i)}$$ is bounded in $$T_i(\Omega_i)$$ for $$i=1,2,\ldots,N$$. There exists a positive constant $$\gamma_i$$ such that ‖∂Ti(ζi,ηi)∂(ζi,ηi)‖≤γi,(ζi,ηi)∈Ti(Ωi),i=1,2,…,N. Define $$\hat{x_i}=T^{-1}_i(\hat{\zeta}_i,\hat{\eta}_i), i=1,2,\ldots,N$$. Then, ‖xi−x^i‖=‖Ti−1(ζi,ηi)−Ti−1(ζ^i,η^i)‖≤γi(‖ζi−ζ^i‖+‖ηi−η^i‖). (5.11) From (5.10) and (5.11), it follows that limt→∞‖xi(t)−x^i(t)‖=0. This implies that $$ \hat{x}_i$$ is an asymptotic estimate of $$x_i$$ for $$i=1,2,\ldots,N$$. Therefore, x^i=Ti−1(ζ^i,η^i) provide an asymptotic estimation for the states $$x_i$$ of system (3.1)–(3.2), where $$\hat{\zeta}_i$$ and $$\hat{\eta}_i$$ are given by (5.1)–(5.2) for $$i=1,2,\ldots,N$$. Remark 6 From the analysis above, it is clear to see that, in this paper, it is not required that either the nominal isolated subsystems or the interconnections are linearizable. The uncertainties are bounded by non-linear functions and are fully used in the observer design in order to reject the effects of the uncertainties, and thus robustness is enhanced. The designed observer is an asymptotic observer and the developed results can be extended to the global case if the associated conditions hold globally. 6. Numerical example Consider the non-linear interconnected systems: x˙1=[x12−0.1sin⁡x12−3x112−3.25x13−2x12]⏟f1(x1)+[010]⏟g1(x1)u1+[Δσ10.5Δσ1−2Δσ1]⏟Δf1(x1)+[0.2(x212+x22)00.1sin⁡x21]⏟D12(x2) (6.1) y1=x11⏟h1(x1) (6.2) x˙2=[−x21−x212−3x22+cos⁡(x212+x22)−1−2x23+0.2x212]⏟f2(x2)+[1−2x210]⏟g2(x2)u2+[−Δσ22x21Δσ20]⏟Δf2(x2)+[00.1sin⁡(x13+2x11)0]⏟D21(x1) (6.3) y2=x21⏟h2(x2), (6.4) where $$x_1=\mbox{col} (x_{11},x_{12},x_{13})$$ and $$x_2= \mbox{col}(x_{21},x_{22},x_{23})$$, $$h_1(x_1)$$ and $$h_2(x_2)$$ and $$u_1(t)$$ and $$u_2(t)$$ are the system state, output and input, respectively, $$D_{12}(\cdot)$$ and $$D_{21}(\cdot)$$ are interconnected terms and $$\Delta f_1(x_1)$$ and $$\Delta f_2(x_2)$$ are the uncertainties experienced by the system which satisfy ||Δf1(x1)||=0.1|x13+2x11|sin2⁡t (6.5) ||Δf2(x2)||=0.1x212|cos⁡t|. (6.6) The domain considered is Ω={(x11,x12,x13,x21,x22,x23),||x11|<3,|x21|≤1.3,x12,x13,x22,x23∈R}. (6.7) By direct computation, it follows that the first subsystem has a uniform relative degree $${2}$$, and the second subsystem has a uniform relative degree $${1}$$. The corresponding transformations are obtained as follows: T1:{ζ11=x11ζ12=x12η1=x13+2x11,T2:{ζ2=x21η21=x212+x22η22=x23. In the new coordinates, the system (6.1)–(6.4) can be described by: ζ˙1=[0100]⏟A1[ζ11ζ12]+[0−0.1sin⁡ζ11+u1]⏟β1+[Δσ1(ζ1,η1)0.5Δσ1(ζ1,η1)]⏟E1ΔΨ(ζ1,η1)+[0.2η210]⏟Γ12a (6.8) η˙1=−3.25η1+0.25ζ112⏟q1(ζ1,η1)+0.4η21+0.1sin⁡ζ2⏟Γ12b (6.9) y1=[10][ζ11ζ12] (6.10) ζ˙2=−⏟A2ζ2+u2⏟β2−Δσ2(ζ2,η2)⏟E2ΔΨ(ζ2,η2) (6.11) η˙2=[−300−2][η21η22]+[cos⁡η21−10.2ζ22]⏟q2(ζ2,η2)+[0.1sin⁡η10]⏟Γ21b (6.12) y2=ζ2, (6.13) where $$\zeta_1=(\zeta_{11}, \zeta_{12})^T, \eta_1 \in R, \zeta_2 \in R \mbox{and} \eta_2=(\eta_{21}, \eta_{22})^T$$. From (6.5) and (6.6) ‖ΔΨ1(ζ1,η1)‖≤||Δσ1(ζ1,η1)||≤0.1|η1|sin2⁡t⏟ρ1(⋅)‖ΔΨ2(ζ2,η2)‖≤||Δσ2(ζ2,η2)||≤0.1ζ22|cos⁡t|⏟ρ2(⋅). Then, for the first subsystem, choose $$L_1=\left[\ 3\ \ 2\ \right]^T$$ and $$Q=I$$. It follows that the Lyapunov equation (4.10) has a unique solution: P1=[0.5−0.5−0.51] and the solution to equation (4.12) is $$H_1=0.25$$. As $$M_1=-3.25$$, let $$\Lambda_1= 3.25$$. Thus the solution of equation (4.25) is $$\Pi_1=0.5$$. Now, for the second subsystem, choose $$L_2=0$$ and $$Q_2=2$$. It follows that the Lyapunov equation (4.10) has a unique solution $$P_2=1$$ and the solution to equation (4.12) is $$H_2=-1$$. As M2=[−300−2],letΛ2=[1001]. Then Π2=[0.1667000.25]. By direct computation, it follows that the matrix $$W^T+W$$ is positive definite in the domain $$\Omega$$ defined in (6.7). Thus, all the conditions of Theorem 1 are satisfied. This implies that the dynamical system ζ^˙1=[0100][ζ^11ζ^12]+[32](y1−C1ζ1^)+[0u1]+K1(⋅)+[0.2η^210] (6.14) η^˙1=−3.25η^1+0.25ζ^112+0.4η^21+0.1sin⁡ζ^2 (6.15) ζ^˙2=−ζ^2+u2+K2(⋅) (6.16) η^˙2=[−300−2][η^21η^22]+[cos⁡η^21−10.2ζ^22]+[0.1sin⁡η^10] (6.17) is a robust observer of the system (6.8)–(6.13) where $${\hat{\zeta}}_1=\mbox{col}({\hat{\zeta}}_{11},{\hat{\zeta}}_{12})$$, $${\hat{\eta}}_2=\mbox{col}({\hat{\eta}}_{21},{\hat{\eta}}_{22})$$ and $$K_1(\cdot)$$ and $$K_2(\cdot)$$ defined in (5.3) are as follows K1(y1,ζ^1,η^1)={[0.10.05](ζ11−ζ^11)‖ζ11−ζ^11‖|η1|sin2⁡t),ζ11−ζ^11≠00,ζ11−ζ^11=0K2(y2,ζ^2,η^2)={0.1(ζ2−ζ^2)‖ζ2−ζ^2‖ζ22|cos⁡t|,ζ2−ζ^2≠00,ζ2−ζ^2=0. Therefore, x^11=ζ^11x^12=ζ^12x^13=η^1−2ζ^11andx^21=ζ^2x^22=η^21−ζ^22x^23=η^22 with $$\hat{\zeta}_{1}=\mbox{col}(\hat{\zeta}_{11}, \hat{\zeta}_{12}), \hat{\eta}_1, \hat{\zeta}_{2}$$ and $$ \hat{\eta}_2=\mbox{col} (\hat{\eta}_{21},\hat{\eta}_{22})$$ given by system (6.14)–(6.17), provide an asymptotic estimate for $$x_1$$ and $$x_2$$ of system (6.1)–(6.4). For simulation purposes, the controllers are chosen as: u1=−ζ11−2ζ12andu2=cos⁡ζ2+5. (6.18) The initial conditions used in the simulation are chosen as $$x_{10}=[-2 2 -2]$$ and $$x_{20}=[1 4 -5]$$. The simulation results in Figs 1 and 2 show that the designed observer estimates the states of the interconnected system $$x_1=\mbox{col} ( {x_{11}, x_{12}, x_{13}})$$ and $$x_2=\mbox{col} ( {x_{21}, x_{22}, x_{23}})$$ very well in (6.1)–(6.4) even if the system is not asymptotically stable. Fig. 1. View largeDownload slide The time response of the first subsystem states $$x_1=\mbox{col} ( {x_{11},x_{12},x_{13}} )$$ and their estimates $$\hat {x}_1=\mbox{col} ( {\hat {x}_{11},\hat {x}_{12}, \hat {x}_{13}} )$$. Fig. 1. View largeDownload slide The time response of the first subsystem states $$x_1=\mbox{col} ( {x_{11},x_{12},x_{13}} )$$ and their estimates $$\hat {x}_1=\mbox{col} ( {\hat {x}_{11},\hat {x}_{12}, \hat {x}_{13}} )$$. Fig. 2. View largeDownload slide The time response of second subsystem states $$x_2=\mbox{col} ( {x_{21},x_{22},x_{23}} )$$ and their estimates $$\hat {x}_2=\mbox{col} ( {\hat {x}_{21},\hat {x}_{22}, \hat {x}_{23}} ).$$ Fig. 2. View largeDownload slide The time response of second subsystem states $$x_2=\mbox{col} ( {x_{21},x_{22},x_{23}} )$$ and their estimates $$\hat {x}_2=\mbox{col} ( {\hat {x}_{21},\hat {x}_{22}, \hat {x}_{23}} ).$$ Remark 7 The aim of this paper is to design an observer for a class of non-linear interconnected systems in the presence of uncertainties. Note that in this paper, it is not required that the considered systems are asymptotically stable. In order to guarantee the performance of the observer, it is only required that the error dynamical systems are asymptotically stable. The simulation results have shown that the errors between the estimated states and the actual states converge to zero even though the second subsystem is not asymptotically stable as shown in Fig. 2. 7. Conclusions In this paper, a class of non-linear large scale interconnected systems with uniform relative degree has been considered. An asymptotic observer has been developed for non-linear interconnected systems with uncertainties using the Lyapunov approach together with a geometric transformation which has been employed to exploit the system structure. It is not required that either the isolated nominal subsystems or the interconnections are linearizable. Robustness to uncertainties is enhanced by using the system structure and the structure of the uncertainties within the design framework. Acknowledgement The authors gratefully acknowledge the support of the National Natural Science Foundation of China (61573180) for this work. References Bakule L. (2008) Decentralized control: an overview . Annu. Rev. Control , 32 , 87 – 98 . Google Scholar CrossRef Search ADS Dashkovskiy S. & Naujok L. (2015) Quasi- ISS/ISDS observers for interconnected systems and applications . Syst. Control Lett. , 77 , 11 – 21 . Google Scholar CrossRef Search ADS Diao Z. F. & Yan X. G. 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Robust variable structure observer design for non-linear large-scale systems with non-linear interconnections

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Abstract

Abstract In this paper, a variable structure observer is designed for a class of non-linear large-scale interconnected systems in the presence of uncertainties and non-linear interconnections. The modern geometric approach is used to explore system structure and a transformation is employed to facilitate the observer design. Based on the Lyapunov direct method, a set of conditions are developed such that the proposed variable structure systems can be used to estimate the states of the original interconnected systems asymptotically. The internal dynamical structure of the isolated nominal subsystems as well as the structure of the uncertainties are employed to reduce the conservatism. The bounds on the uncertainties are non-linear and are employed in the observer design to reject the effect of the uncertainties. A numerical example is presented to illustrate the approach and the simulation results showthat the proposed approach is effective. 1. Introduction The development of advanced technologies has produced many complex systems. An important class of complex systems, which is frequently called a system of systems or large-scale system, can usually be expressed by sets of lower-order ordinary differential equations which are linked through interconnections. Such models are typically called large-scale interconnected systems (see, e.g. Yan et al., 1999; Bakule, 2008; Mahmoud, 2011; Yan et al., 2013;). Large-scale interconnected systems widely exist in practice, for example, power networks, ecological systems, transportation networks, biological systems and information technology networks (Lunze, 1992; Mahmoud, 2011). Increasing requirements for system performance have resulted in increasing complexity within system modelling and it becomes of interest to consider non-linear large-scale interconnected systems. Such models are then used for controller design. In order to obtain good performance levels, a controller may benefit from knowledge of all the system states. This state information may be difficult or expensive to obtain and it becomes of interest to design an observer to estimate all the system states using only the subset of information available from the measured and known input and output of the system. Large-scale interconnected systems have been studied since the 1970s (Sandell et al., 1978). Early work focussed on linear systems. Subsequent results used decentralized control frameworks for non-linear large-scale interconnected systems. In much of this work, however, it is assumed that all the system state variables are available for use by the controller (Wu, 2005; Bakule, 2008; Mahmoud, 2011; Zhao et al., 2014). However, this may be limiting in practice as only a subset of state variables may be available/measureable. It becomes of interest to establish observers to estimate the system states and then use the estimated states to replace the true system states in order to implement state feedback decentralized controllers. It is also the case that observer design has been heavily applied for fault detection and isolation (Diao & Yan, 2008; Yan & Edwards, 2008; Reppa et al., 2014). This further motivates the study of observer design for non-linear large scale interconnected systems. The concept of an observer was first introduced by Luenberger (1964) where the difference between the output measurements from the actual plant and the output measurements of a corresponding dynamical model were used to develop an injection signal to force the resulting output error to zero. Later the approach was extended to non-linear systems and an extended Luenberger observer for non-linear systems is proposed in Zeitz (1987) where uncertainties are not considered. It should be noted that many approaches have been developed for observer design such as the sliding mode observer approach in Yan et al. (2013), the adaptive observer in Wu (2009) and an error linearization approach in Xia & Gao (1989). However, results concerning observer design for interconnected systems are very few when compared with the corresponding results available on controller design for interconnected systems. Sliding mode techniques have been used to design observers for non-linear interconnected power systems in Modarres et al. (2012). In Li et al. (2015) state estimation and sliding mode control for a special class of stochastic dynamic systems which is semi-Markovian jump systems is presented. The authors designed a state observer to generate the estimate of unmeasured state components, and then synthesize a sliding mode control law based on the state estimates. Wang & Fei (2015) discussed the position regulation problem of permanent magnet synchronous motor servo system based on adaptive fuzzy sliding mode control method. They used adaptive method to estimate the upper bound of the approximation error between the equivalent control law and the fuzzy controller are utilized in the paper. An adaptive observer is designed for a class of interconnected systems in Wu (2009) in which it is required that the isolated nominal subsystems are linear. Observer schemes for interconnected systems are proposed in Keliris et al. (2015), Reppa et al. (2014), Sharma & Aldeen (2011) and Yan & Edwards (2008) where the obtained results are unavoidably conservative as it is required that the designed observer can be used for certain fault detection and isolation problems. For example, it is required that the uncertainty can be decoupled with faults in Yan & Edwards (2008) and the considered system is not interconnected systems. Robust observer design is considered in Mohmoud (2012) for a class of linear large scale dynamical systems where it is required that the interconnections satisfy quadratic constraints. In Swarnakar et al. (2007) a new decentralized control scheme which uses estimated states from a decentralized observer within a feedback controller is proposed. This uses a design framework based on linear matrix inequalities and is thus applicable for linear systems. A robust observer for non-linear interconnected systems based on a constrained Lyapunov equation has been developed in Yan et al. (2003). A proportional integral observer is utilized for non-linear interconnected systems for disturbance attenuation in Ghadami & Shafai (2011) and interconnected non-linear dynamical systems are considered in Dashkovskiy & Naujok (2015) where the authors combine the advantages of input-to-state dynamical stability and use reduced order observers to obtain quantitative information about the state estimation error. This work does not, however, consider uncertainties. It should be noted that in all the existing work relating to observer design for large scale interconnected systems, it is required that either the isolated subsystems are linear or the interconnections are linear. Moreover, most of the designed observers are used for special purposes such as fault detection or stabilization and thus they impose specific requirements on the class of interconnected systems considered. This paper is an extension and modification of the authors’ conference paper in Mohamed et al. (2016). In this paper, a class of non-linear interconnected systems with disturbances is considered where both the nominal isolated subsystems and interconnections are non-linear. It is not required that either the nominal isolated subsystems or the interconnections are linearizable. A robust variable structure observer is established based on a simplified system structure by using Lyapunov analysis methodology. The structure of the internal dynamics, the structure of uncertainties and the bounds on uncertainties are fully used in the observer design to reduce the conservatism. These bounds are allowed to have a general non-linear form. The observer states converge to the system states asymptotically. An example with simulation is given to demonstrate the proposed approach. 2. Preliminaries Consider the single input–single output non-linear system x˙(t)=f(x)+g(x)u (2.1) y(t)=h(x), (2.2) where $$x\in \Omega\subset R^{n}$$ ($$\Omega$$ is a neighbourhood of the origin), $$y\in R$$ and $$u\in U \subset R$$ ($$ U $$ is an admissible control set) are the state, output and input, respectively, $$f(x)$$, $$g(x) \in R^{n}$$ are smooth vector fields defined in the domain $$\Omega$$, and $$h(x)\in R^{m}$$ is a smooth vector in the domain $$\Omega$$. Firstly, recall some key elements of the geometric approach in Isidori (1995) which will be used in the later analysis. The notation used in this paper is the same as Isidori (1995) unless it is specifically defined. Definition 1 From Isidori (1995), system (2.1)–(2.2) is said to have uniform relative degree $$r$$ in the domain $$\Omega$$ if for any $$x\in\Omega$$, (i)$$L_{g} L^k_f h(x)=0,\quad\mbox{for}\quad k=1,2,\ldots,r-1,$$ (ii)$$L_{g} L^{r-1}_f h(x)\neq 0.$$ Now consider system (2.1)–(2.2). It is assumed that system (2.1)–(2.2) has uniform relative degree $$r$$ in domain $$\Omega$$. Construct a mapping $$\phi: x \rightarrow z$$ as follows: ϕ(⋅):{z1=h(x)z2=Lfh(x)⋮zr=Lfr−1h(x)zr+1=ϕr+1⋮zn=ϕn(x), (2.3) where $$\phi(\cdot)=\mbox{col}(\phi_1(x),\phi_2(x),\ldots,\phi_n(x)),$$$$\phi_1(x)=h(x),$$$$\phi_2(x)=L_f h(x),\ldots,\phi_r(x)=L^{r-1}_f h(x)$$ and the functions $$\phi_{r+1}(x),$$$$\ldots,$$$$\phi_{n}(x)$$ need to be selected such that Lgϕi(x)=0,i=r+1,r+2,…,N and the Jacobian matrix Jϕ:=∂ϕ(x)∂x is non-singular in domain $$\Omega$$. Then the mapping $$\phi\hspace{0mm}: x \rightarrow z$$ forms a diffeomorphism in the domain $$\Omega$$. For the sake of simplicity, let ζ=[ζ1ζ2⋯ζr]T:=[z1z2⋯zr]Tη=[ζr+1ζr+2⋯ζn]T:=[zr+1zr+2⋯zn]T. Then, from Isidori (1995), it follows that in the new coordinates $$z$$, system (2.1)–(2.2) can be described by ζ˙1=ζ2ζ˙2=ζ3⋮ζ˙r−1=ζrζ˙r=a(ζ,η)+b(ζ,η)uη˙=q(ζ,η), (2.4) where a(ζ,η)=Lfrh(ϕ−1(ζ,η))b(ζ,η)=LgLfr−1h(ϕ−1(ζ,η)) and q(ζ,η)=[qr+1(ζ,η)qr+2(ζ,η)⋮qn(ζ,η)]=[Lfϕr+1(ϕ−1(ζ,η))Lfϕr+2(ϕ−1(ζ,η))⋮Lfϕn(ϕ−1(ζ,η))]. It should be noted that the coordinate transformation (2.3) will be available if $$\phi_i(x)$$ are available for $$i=r+1,\ldots,N$$, and in this case, the system (2.4) can be obtained directly. 3. Large-scale system description and problem statement Consider the non-linear interconnected systems x˙i(t)=fi(xi)+gi(xi)ui+Δfi(xi)+∑j≠ij=1NDij(xj), (3.1) yi(t)=hi(xi),i=1,2,…,N, (3.2) where $$x_i\in \Omega_i\subset R^{n_i}$$ ($$\Omega_i$$ is a neighbourhood of the origin), $$y_i\in R$$ and $$u_i\in U_i \subset R$$ ($$ U_i $$ is an admissible control set) are the state, output and input of the $$i$$th subsystem, respectively, $$f_i(x_i)\in R^{n_i}$$ and $$g_i(x_i) \in R^{n_i}$$ are smooth vector fields defined in the domain $$\Omega_i$$, and $$h_i(x_i)\in R^{m_i}$$ are smooth in the domain $$\Omega_i$$ for $$i=1,2,\ldots,N$$. The term $$\Delta {f_i} (x_i)$$ includes all the uncertainties experienced by the $$i$$th subsystem. The term $$\sum_{\stackrel{j=1}{j\not=i}}^N D_{ij}(x_j)$$ is the non-linear interconnection of the $$i$$th subsystem. Definition 2 The systems x˙i(t)=fi(xi)+gi(xi)ui+Δfi(xi) (3.3) yi(t)=hi(xi),i=1,2,…,N (3.4) are called the isolated subsystems of the systems (3.1)–(3.2), and the systems x˙i(t)=fi(xi)+gi(xi)ui (3.5) yi(t)=hi(xi),i=1,2,…,N (3.6) are called the nominal isolated subsytems of the systems (3.1)–(3.2). In this paper, under the assumption that the isolated subsystems (3.5)–(3.6) have uniform relative degree $$r_i$$ in the considered domain $$\Omega_i$$, the interconnected systems (3.1)–(3.2) are to be analysed. The objective is to explore the system structure based on a geometric transformation to design a robust asymptotic observer for the interconnected system (3.1)–(3.2). It should be noted that the following results can be extended to the case where the isolated subsystems are multi-input and multi-output using the corresponding framework to Section 2 for the multi-input and multi-output case provided in Isidori (1995). 4. System analysis and assumptions In this section, some assumptions are imposed on the system (3.1)–(3.2) to facilitate the observer design. Assumption 1 The nominal isolated subsystem (3.5)–(3.6) has uniform relative degree $$r_i$$ in domain $$x_i\in \Omega_i$$ for $$i=1,2,\ldots,N$$. Under Assumption 1, it follows from Section 2 that there exists a coordinate transformation Ti:xi→col(ζi,ηi), (4.1) where ζi=[ζi1ζi2⋮ζiri]=[hi(xi)Lfhi(xi)⋮Lfri−1hi(xi)]∈Rri (4.2) and $$\eta_i\in R^{n_i-r_i}$$ is defined by ηi=[ηi1ηi2⋮ηni−ri]=[ϕi(ri+1)(xi)ϕi(ri+2)(xi)⋮ϕini(xi)]∈Rni−ri (4.3) for $$i=1,2,\ldots,N$$. The functions $$ \phi_{i(r_i+1)}(x_i),$$$$\phi_{i(r_i+2)}(x_i),\ldots,\phi_{in_i}(x_i)$$ can be obtained by solving the following partial differential equations: Lgiϕi(xi)=0,xi∈Ωi,i=1,2,…,N. (4.4) From Section 2, it follows that in the new coordinate system $$(\zeta_i,\eta_i)$$, the nominal isolated subsystem (3.5)–(3.6) can be described by ζ˙i=Aiζi+βi(ζi,ηi,ui), (4.5) η˙i=qi(ζi,ηi) (4.6) yi=Ciζi, (4.7) where Ai=[010⋯0001⋯0⋮⋮⋮⋮⋮000⋯1000⋯0]∈Rri×ri,Ci=[10⋯0]∈R1×ri (4.8) βi(ζi,ηi,ui)=[0⋮0Lfirihi(Ti−1(ζi,ηi))+LgiLfiri−1hi(Ti−1(ζi,ηi))ui]. (4.9) It is clear to see that the pair $$(A_i,C_i)$$ is observable. Thus, there exists a matrix $$L_i$$ such that $$A_i-L_iC_i$$ is Hurwitz stable. This implies that, for any positive definite matrix $$Q_i\in R^{r_i\times r_i}$$, the Lyapunov equation (Ai−LiCi)TPi+Pi(Ai−LiCi)=−Qi (4.10) has a unique positive-definite solution $$P_i\in R^{r_i\times r_i}$$ for $$i=1,2,\ldots,N$$. Assumption 2 The uncertainty $$\Delta {f_i}(x_i)$$ in (3.1) satisfies ∂Ti∂xiΔfi(xi)=[EiΔΨ(xi)0] , (4.11) where $$T_i(\cdot)$$ is defined in (4.1), $$E_i \in R^{r_i \times r_i}$$ is a constant matrix satisfying EiTPi=HiCi (4.12) for some matrix $$H_i$$, with $$P_i$$ satisfying (4.10), and $$\|\Delta \Psi_i(x_i)\|\leq\kappa_i(x_i)$$, where $$\kappa_i(x_i)$$ is continuous and Lipschitz about $$x_i$$ in the domain $$\Omega_i$$ for $$i=1,2,\ldots,N$$. Remark 1 Solving the Lyapunov equation (4.10) in the presence of the constraint (4.12) is the well-known constrained Lyapunov problem (Galimidi & Barmish, 1986). Although there is no general solution available for this problem, associated discussion and an algorithm can be found in Edwards et al. (2007). Remark 2 Assumption 2 is a limitation on the uncertainty $$\Delta f_i(x_i)$$, and this is necessary to guarantee the existence of asymptotic observers. Denote the non-linear uncertain term $$\Delta \Psi_i(x_i)$$ in (4.11) in the new coordinate frame $$(\zeta_i,\eta_i)$$ by $$\Delta\Phi_i(\zeta_i,\eta_i),$$ i.e. ΔΦi(ζi,ηi)=[ΔΨi(ζi,ηi)]xi=Ti−1(ζi,ηi). (4.13) From Assumption 2, there exists a function $$\rho_i(\zeta_i,\eta_i)$$ such that ‖ΔΦi(ζi,ηi)‖≤ρi(ζi,ηi) (4.14) and $$\rho_i(\zeta_i,\eta_i)$$ satisfies the Lipschitz condition in $$T_i(\Omega_i)$$. Thus for any $$(\zeta_i,\eta_i)$$ and $$(\hat{\zeta_i},\hat{\eta_i}) \in T_i(\Omega_i)$$, ‖ρi(ζi,ηi)−ρi(ζ^i,η^i)‖≤lia‖ζi−ζ^i‖+lib‖ηi−η^i‖, (4.15) where both $$l_i^a$$ and $$l_i^b$$ are non-negative constants. Consider the interconnections $$D_{ij}(x_j)$$ in system (3.1). Partition the term $$\frac{\partial T_i}{\partial x_i} D_{ij}(x_j)$$ as follows ∂Ti∂xiDij(xj)|xj=Tj−1(ζj,ηj)=[Γija(ζj,ηj)Γijb(ζj,ηj)], (4.16) where $$\Gamma^a_{ij}{(\zeta_j,\eta_j)} \in R^{r_i}$$, $$\Gamma^b_{ij}{(\zeta_j,\eta_j)} \in R^{n_i-r_i}$$ for $$i=1,2,\ldots,N$$ and $$i\neq j$$. Assumption 3 The non-linear terms $$\Gamma^a_{ij}{(\zeta_j,\eta_j)} \in R^{r_i}$$ and $$\Gamma^b_{ij}{(\zeta_j,\eta_j)} \in R^{n_i-r_i}$$ in (4.16) satisfy the Lipschitz condition in $$T_i(\Omega_i)$$. Assumption 3 implies that there exist non-negative constants $$\alpha_{ij}^a$$, $$\alpha_{ij}^b $$, $$\mu_{ij}^a $$ and $$\mu_{ij}^b$$ such that ‖Γija(ζi,ηi)−Γija(ζ^i,η^i)‖≤αija‖ζj−ζ^j‖+αijb∥ηj−η^j‖ (4.17) ‖Γijb(ζi,ηi)−Γijb(ζ^i,η^i)‖≤μija‖ζj−ζ^j‖+μijb∥ηj−η^j‖ (4.18) for $$i=1,2,\ldots,N$$ and $$i \neq j$$. From (4.5)–(4.7) and the analysis above, it follows that under Assumption 2, in the new coordinate system $$(\zeta_i,\eta_i)$$, the system (3.1)–(3.2) can be described by ζ˙i=Aiζi+βi(ζi,ηi,ui)+EiΔΨi(ζi,ηi)+∑j≠ij=1NΓija(ζj,ηj), (4.19) η˙i=qi(ζi,ηi)+∑j≠ij=1NΓijb(ζj,ηj), (4.20) yi=Ciζi, (4.21) where $$A_i$$ and $$ C_i$$ are given in (4.8), $$\beta_i(\cdot)$$ is defined in (4.9) and $$\Gamma^a_{ij}(\cdot)$$ and $$\Gamma^b_{ij}(\cdot)$$ are defined in (4.16). Remark 3 Since $$\beta_i(\cdot)$$ is continuous in the domain $$ T_i(\Omega_i)$$, it is straightforward to see that there exists a subset in the domain $$T_i(\Omega_i)$$ such that the function $$\beta_i(\cdot)$$ is Lipschitz in the subset ∥βi(ζi,ηi,ui)−βi(ζi^,ηi^,ui)∥via(ui)∥ζi−ζ^i∥+vib(ui)∥ηi−η^i∥, (4.22) where $$v_i^a(u_i)$$ and $$v_i^b(u_i)$$ are non-negative functions of $$u_i$$ for $$i=1,2,\ldots,N$$. Assumption 4 The function $$q_i(\zeta_i,\eta_i)$$ in equation (4.20) has the following decomposition qi(ζi,ηi)=Miηi+θi(ζi,ηi), (4.23) where $$ M_i \in R^{(n_i-r_i)\times (n_i-r_i)}$$ is a Hurwitz matrix and $$\theta_i(\zeta_i,\eta_i)$$ are Lipschitz in domain $$T_i(\Omega_i)$$. Under Assumption 4, there exist non-negative constants $$\tau_i^a$$ and $$\tau_i^b$$ such that. ∥θi(ζi,ηi)−θi(ζ^i,η^i)∥τia∥ζi−ζ^i∥+τib∥ηi−η^i∥ (4.24) for $$i= 1,2,\ldots,N.$$ Further, from the fact that $$M_i$$ is Hurwitz stable for $$\Lambda_i>0$$, the following Lyapunov equation has a unique solution $$\Pi_i>0$$ MiTΠi+ΠiMi=−Λi,i=1,2,…,N. (4.25) 5. Non-linear observer synthesis In this section, an observer is designed for the transformed systems (4.19)–(4.21) which provides asymptotic estimation of the states of the interconnected systems (4.19)–(4.21). For system (4.19)–(4.21), construct dynamical systems ζ^˙i=Aiζ^i+Li(yi−Ciζ^i)+βi(ζ^i,η^i,ui)+Ki(y,ζ^i,η^i)+∑j≠ij=1NΓija(ζ^j,η^j), (5.1) η^˙i=Miη^i+θi(ζ^i,η^i)+∑j≠ij=1NΓijb(ζ^j,η^j), (5.2) where the term $$K_i(y_i,\hat{\zeta}_i,\hat{\eta}_i)$$ is defined by Ki(yi,ζ^i,η^i)={Pi−1CiT(yi−Ciζ^i)‖yi−Ciζ^i‖∥Hi∥ρi(ζ^i,η^i),yi−Ciζ^i≠00,yi−Ciζ^i=0, (5.3) where $$P_i$$ and $$H_i$$ satisfy (4.10) and (4.12), respectively. Remark 4 System (5.1)–(5.2) is called a variable structure system throughout this paper due to the discontinuous terms defined in (5.3). It should be noted that this observer is different from a sliding mode observer as the proposed observer (5.1)–(5.2) may not produce a sliding motion. The following results are ready to be presented. Theorem 1 Suppose Assumptions 1–4 hold. Then, the dynamical system (5.1)–(5.2) is a robust asymptotic observer of system (4.19)–(4.21), if the function matrix $$W^T(\cdot)+W(\cdot)$$ is positive definite in the domain $$T(\Omega) \times U:=T(\Omega_1) \times U_1 \times T(\Omega_2)\times U_2 \times\cdots\times T(\Omega_N)\times U_N$$, where the matrix $$W(\cdot)=\left[w_{ij}(\cdot)\right]_{2N \times 2N}$$, and its entries $$w_{ij}(\cdot)$$ are defined by wij={λmin(Qi)−2λmax(Pi)via(.)−2lia‖Ci‖‖Hi‖,i=j,1≤i≤N−2λmax(Pi)αija,i≠j,1≤i≤N,1≤j≤Nλmin(Λi−N)−2λmax(Πi−N)τi−Nb,i=j,N+1≤i≤2N−2λmax(Π(i−N))μ(i−N)(j−N)b,i≠j,N+1≤i≤2N,N+1≤j≤2N−2[λmax(Pi)vib(.)+lib‖Ci‖‖Hi‖+λmax(Πi)τia],j−i=N,1≤i≤N,N+1≤j≤2N−2λmax(Pi)αi(j−N)b,j−i≠N,1≤i≤N,N+1≤j≤2N0,i−j=N,N+1≤i≤2N,1≤j≤N−2λmax(Πi−N)μ(i−N)ja,i−j≠N,N+1≤i≤2N,1≤j≤N. Proof Let $$e_{\zeta_i}=\zeta_i-\hat{\zeta}_i $$ and $$ e_{\eta_i}=\eta_i-\hat{\eta}_i$$ for $$i= 1,2,\ldots, N$$. Compare systems (4.19)–(4.20) and (5.1)–(5.2). It follows that the error dynamical systems are described by e˙ζi=(Ai−LiCi)eζi+βi(ζi,ηi,ui)−βi(ζi^,ηi^,ui)+EiΔΨi(ζi,ηi)−Ki(yi,ζ^i,η^i)+∑j=ij=1NΓija(ζj,ηj)−∑j=ij=1NΓija(ζ^j,η^j), (5.4) e˙ηi=Mieηi+θi(ζi,ηi)−θi(ζ^i,η^i)+∑j=ij=1NΓijb(ζj,ηj)−∑j=ij=1NΓijb(ζ^j,η^j). (5.5) Now, for the system (5.4) and (5.5) consider the following candidate Lyapunov function V=∑i=1NeζiTPieζi+∑i=1NeηiTΠieηi. (5.6) Then, the time derivative of the candidate Lyapunov function can be described by V˙=∑i=1N[(e˙ζiTPieζi+eζiTPieζi˙)+(e˙ηiTΠieηi+eηiTΠie˙ηi)]. (5.7) Substituting both $$\dot{e}_{\zeta_i}$$ in (5.4) and $$\dot{e}_{\eta_i}$$ in (5.5) into equation (5.7), it follows by direct computation that the time derivative of the function $$V$$ in (5.6) can be described by V˙=∑i=1N{eζiT[(Ai−LiCi)TPi+Pi(Ai−LiCi)]eζi+2eζiTPi[βi(ζi,ηi,ui)−βi(ζi^,ηi^,ui)]+2[eζiTPiEiΔΨi(ζi,ηi)−eζiTPiKi(yi,ζ^i,η^i)]+2eζiTPi∑j≠ij=1N[Γija(ζj,ηj)−Γija(ζ^j,η^j)]+eηiT(MiTΠi+ΠiMi)eηi+2eηiTΠi[θi(ζi,ηi)−θi(ζ^i,η^i)]+2eηiTΠi∑j≠ij=1N[Γijb(ζj,ηj)−Γijb(ζ^j,η^j)]}. (5.8) From (4.12), (4.14), (4.15) and (5.3), it follows that: (i) If $$y_i-C_i\hat{\zeta}_i=0$$, then from (4.12) and $${e}^T_{\zeta_i}C_i^T=(y_i-C_i\hat{\zeta}_i)^T$$ eζiTPiEiΔΦi(ζi,ηi)−eζiTPiKi(yi,ζ^i,η^i)=eζiTCiTHiTΔΦi(ζi,ηi)=(Hi(yi−Ciζ^i))TΔΦi(ζi,ηi)=0. (ii) If $$y_i-C_i\hat{\zeta}_i\neq 0$$, then from (4.12), (4.14), (4.15) and (5.3) eζiTPiEiΔΦi(ζi,ηi)−eζiTPiKi(yi,ζ^i,η^i)=eζiTCiTHiTΔΦi(ζi,ηi)−eζiTPiPi−1CiT(yi−Ciζ^i)‖yi−Ciζ^i‖∥Hi∥ρi(ζ^i,η^i)=(Cieζi)THiTΔΦi(ζi,ηi)−eζiTCiTCieζi‖Cieζi‖‖Hi‖ρi(ζ^i,η^i)≤‖Cieζi‖‖Hi‖{ρi(ζi,ηi)−ρi(ζ^i,η^i)}≤‖Cieζi‖‖Hi‖{lia‖ζi−ζ^i‖+lib‖ηi−η^i‖}. Then, from (i) and (ii) above, it follows that eζiTPiEiΔΦi(ζi,ηi)−eζiTPiKi(yi,ζ^i,η^i)≤‖Cieζi‖‖Hi‖(lia‖eζi‖+lib‖eηi‖). (5.9) Substituting (4.17), (4.18), (4.22), (4.24) and (5.9) into (5.8) yields V˙≤∑i=1N{−eζiTQieζi+2‖eζi‖‖Pi‖[via‖eζi‖+vib‖eηi‖]+2‖eζi‖‖Ci‖‖Hi‖[lia‖eζi‖+lib‖eηi‖]+2‖eζi‖‖Pi‖∑j≠ij=1N[αija‖eζj‖+αijb‖eηj‖]−eηiTΛieηi+2eηiT‖Πi‖[τia‖eζi‖+τib‖eηi‖]+2eηiT‖Πi‖∑j≠ij=1N[μija‖eζj‖+μijb‖eηj‖]}≤∑i=1N{−eζiTQieζi+2via‖eζi‖2‖Pi‖+2vib‖eζi‖‖eηi‖‖Pi‖+2lia‖eζi‖2‖Ci‖‖Hi‖+2lib‖eζi‖‖eηi‖Ci‖‖Hi‖+∑j≠ij=1N[2αija‖eζi‖‖eζj‖‖Pi‖+2αijb‖eζi‖‖eηj‖‖Pi‖]−eηiTΛieηi+2τia‖Πi‖‖eζi‖‖eηi‖+2τib‖Πi‖‖eηi‖2+∑j≠ij=1N[2μija‖Πi‖‖eζj‖‖eηi‖+2μijb‖Πi‖‖eηi‖‖eηj‖]}≤∑i=1N{−[λmin(Qi)−2λmax(Pi)via−2lia‖Ci‖‖Hi‖]‖eζi‖2+[∑j≠ij=1N2λmax(Pi)αija]‖eζi‖‖eζj‖+[2λmax(Pi)vib+2lib‖Ci‖‖Hi‖+2λmax(Πi)τia]‖eζi‖‖eηi‖+2∑j≠ij=1Nλmax(Pi)αijb‖eζi‖‖eηj‖+∑j≠ij=1N2λmax(Πi)μija‖eζj‖‖eηi‖−[λmin(Λi)−2λmax(Πi)τib]‖eηi‖2+[∑j≠ij=1N2λmax(Πi)μijb]‖eηi‖‖eηj‖}≤−∑i=1N{[λmin(Qi)−2λmax(Pi)via−2lia‖Ci‖‖Hi‖]‖eζi‖2−[∑j≠ij=1N2λmax(Pi)αija]‖eζi‖‖eζj‖−[2λmax(Pi)vib+2lib‖Ci‖‖Hi‖+2λmax(Πi)τia]‖eζi‖‖eηi]‖−2∑j≠ij=1Nλmax(Pi)αijb‖eζi‖‖eηj‖]−∑j≠ij=1N2λmax(Πi)μija‖eζj‖‖eηi‖+[λmin(Λi)−2λmax(Πi)τib]‖eηi‖2−[∑j≠ij=1N2λmax(Πi)μijb]‖eηi‖‖eηj‖}. Then, from the definition of the matrix $$W(\cdot)$$ and the inequality above, it follows that V˙≤−12XT[WT(⋅)+W(⋅)]X, where $$X= [\|e_{\zeta_1}\|,\|e_{\zeta_2}\|, \ldots, \|e_{\zeta_N}\|,\|e_{\eta_1}\|,\|e_{\eta_2}\|,\ldots,$$$$\|e_{\eta_N}\|]^T$$. Since $$W^T(\cdot)+W(\cdot)$$ is positive definite in the domain $$ T(\Omega) \times U$$, it is clear that $$\dot V|_{(\text{5.1})-(\text{5.2})}$$ is negative definite. Therefore, the error system (5.4)–(5.5) is asymptotically stable, that is, limt→∞‖ζi(t)−ζ^i(t)‖=0andlimt→∞‖ηi(t)−η^i(t)‖=0. (5.10) Hence, the conclusion follows. Remark 5 Theorem 1 shows that variable structure system (5.1)–(5.2) is an asymptotic observer of the interconnected system (4.19)–(4.21). It is called a variable structure observer in this paper. Now, consider interconnected system (3.1)–(3.2). Assume that $$\frac {\partial T_i(\zeta_i,\eta_i)}{\partial (\zeta_i,\eta_i)}$$ is bounded in $$T_i(\Omega_i)$$ for $$i=1,2,\ldots,N$$. There exists a positive constant $$\gamma_i$$ such that ‖∂Ti(ζi,ηi)∂(ζi,ηi)‖≤γi,(ζi,ηi)∈Ti(Ωi),i=1,2,…,N. Define $$\hat{x_i}=T^{-1}_i(\hat{\zeta}_i,\hat{\eta}_i), i=1,2,\ldots,N$$. Then, ‖xi−x^i‖=‖Ti−1(ζi,ηi)−Ti−1(ζ^i,η^i)‖≤γi(‖ζi−ζ^i‖+‖ηi−η^i‖). (5.11) From (5.10) and (5.11), it follows that limt→∞‖xi(t)−x^i(t)‖=0. This implies that $$ \hat{x}_i$$ is an asymptotic estimate of $$x_i$$ for $$i=1,2,\ldots,N$$. Therefore, x^i=Ti−1(ζ^i,η^i) provide an asymptotic estimation for the states $$x_i$$ of system (3.1)–(3.2), where $$\hat{\zeta}_i$$ and $$\hat{\eta}_i$$ are given by (5.1)–(5.2) for $$i=1,2,\ldots,N$$. Remark 6 From the analysis above, it is clear to see that, in this paper, it is not required that either the nominal isolated subsystems or the interconnections are linearizable. The uncertainties are bounded by non-linear functions and are fully used in the observer design in order to reject the effects of the uncertainties, and thus robustness is enhanced. The designed observer is an asymptotic observer and the developed results can be extended to the global case if the associated conditions hold globally. 6. Numerical example Consider the non-linear interconnected systems: x˙1=[x12−0.1sin⁡x12−3x112−3.25x13−2x12]⏟f1(x1)+[010]⏟g1(x1)u1+[Δσ10.5Δσ1−2Δσ1]⏟Δf1(x1)+[0.2(x212+x22)00.1sin⁡x21]⏟D12(x2) (6.1) y1=x11⏟h1(x1) (6.2) x˙2=[−x21−x212−3x22+cos⁡(x212+x22)−1−2x23+0.2x212]⏟f2(x2)+[1−2x210]⏟g2(x2)u2+[−Δσ22x21Δσ20]⏟Δf2(x2)+[00.1sin⁡(x13+2x11)0]⏟D21(x1) (6.3) y2=x21⏟h2(x2), (6.4) where $$x_1=\mbox{col} (x_{11},x_{12},x_{13})$$ and $$x_2= \mbox{col}(x_{21},x_{22},x_{23})$$, $$h_1(x_1)$$ and $$h_2(x_2)$$ and $$u_1(t)$$ and $$u_2(t)$$ are the system state, output and input, respectively, $$D_{12}(\cdot)$$ and $$D_{21}(\cdot)$$ are interconnected terms and $$\Delta f_1(x_1)$$ and $$\Delta f_2(x_2)$$ are the uncertainties experienced by the system which satisfy ||Δf1(x1)||=0.1|x13+2x11|sin2⁡t (6.5) ||Δf2(x2)||=0.1x212|cos⁡t|. (6.6) The domain considered is Ω={(x11,x12,x13,x21,x22,x23),||x11|<3,|x21|≤1.3,x12,x13,x22,x23∈R}. (6.7) By direct computation, it follows that the first subsystem has a uniform relative degree $${2}$$, and the second subsystem has a uniform relative degree $${1}$$. The corresponding transformations are obtained as follows: T1:{ζ11=x11ζ12=x12η1=x13+2x11,T2:{ζ2=x21η21=x212+x22η22=x23. In the new coordinates, the system (6.1)–(6.4) can be described by: ζ˙1=[0100]⏟A1[ζ11ζ12]+[0−0.1sin⁡ζ11+u1]⏟β1+[Δσ1(ζ1,η1)0.5Δσ1(ζ1,η1)]⏟E1ΔΨ(ζ1,η1)+[0.2η210]⏟Γ12a (6.8) η˙1=−3.25η1+0.25ζ112⏟q1(ζ1,η1)+0.4η21+0.1sin⁡ζ2⏟Γ12b (6.9) y1=[10][ζ11ζ12] (6.10) ζ˙2=−⏟A2ζ2+u2⏟β2−Δσ2(ζ2,η2)⏟E2ΔΨ(ζ2,η2) (6.11) η˙2=[−300−2][η21η22]+[cos⁡η21−10.2ζ22]⏟q2(ζ2,η2)+[0.1sin⁡η10]⏟Γ21b (6.12) y2=ζ2, (6.13) where $$\zeta_1=(\zeta_{11}, \zeta_{12})^T, \eta_1 \in R, \zeta_2 \in R \mbox{and} \eta_2=(\eta_{21}, \eta_{22})^T$$. From (6.5) and (6.6) ‖ΔΨ1(ζ1,η1)‖≤||Δσ1(ζ1,η1)||≤0.1|η1|sin2⁡t⏟ρ1(⋅)‖ΔΨ2(ζ2,η2)‖≤||Δσ2(ζ2,η2)||≤0.1ζ22|cos⁡t|⏟ρ2(⋅). Then, for the first subsystem, choose $$L_1=\left[\ 3\ \ 2\ \right]^T$$ and $$Q=I$$. It follows that the Lyapunov equation (4.10) has a unique solution: P1=[0.5−0.5−0.51] and the solution to equation (4.12) is $$H_1=0.25$$. As $$M_1=-3.25$$, let $$\Lambda_1= 3.25$$. Thus the solution of equation (4.25) is $$\Pi_1=0.5$$. Now, for the second subsystem, choose $$L_2=0$$ and $$Q_2=2$$. It follows that the Lyapunov equation (4.10) has a unique solution $$P_2=1$$ and the solution to equation (4.12) is $$H_2=-1$$. As M2=[−300−2],letΛ2=[1001]. Then Π2=[0.1667000.25]. By direct computation, it follows that the matrix $$W^T+W$$ is positive definite in the domain $$\Omega$$ defined in (6.7). Thus, all the conditions of Theorem 1 are satisfied. This implies that the dynamical system ζ^˙1=[0100][ζ^11ζ^12]+[32](y1−C1ζ1^)+[0u1]+K1(⋅)+[0.2η^210] (6.14) η^˙1=−3.25η^1+0.25ζ^112+0.4η^21+0.1sin⁡ζ^2 (6.15) ζ^˙2=−ζ^2+u2+K2(⋅) (6.16) η^˙2=[−300−2][η^21η^22]+[cos⁡η^21−10.2ζ^22]+[0.1sin⁡η^10] (6.17) is a robust observer of the system (6.8)–(6.13) where $${\hat{\zeta}}_1=\mbox{col}({\hat{\zeta}}_{11},{\hat{\zeta}}_{12})$$, $${\hat{\eta}}_2=\mbox{col}({\hat{\eta}}_{21},{\hat{\eta}}_{22})$$ and $$K_1(\cdot)$$ and $$K_2(\cdot)$$ defined in (5.3) are as follows K1(y1,ζ^1,η^1)={[0.10.05](ζ11−ζ^11)‖ζ11−ζ^11‖|η1|sin2⁡t),ζ11−ζ^11≠00,ζ11−ζ^11=0K2(y2,ζ^2,η^2)={0.1(ζ2−ζ^2)‖ζ2−ζ^2‖ζ22|cos⁡t|,ζ2−ζ^2≠00,ζ2−ζ^2=0. Therefore, x^11=ζ^11x^12=ζ^12x^13=η^1−2ζ^11andx^21=ζ^2x^22=η^21−ζ^22x^23=η^22 with $$\hat{\zeta}_{1}=\mbox{col}(\hat{\zeta}_{11}, \hat{\zeta}_{12}), \hat{\eta}_1, \hat{\zeta}_{2}$$ and $$ \hat{\eta}_2=\mbox{col} (\hat{\eta}_{21},\hat{\eta}_{22})$$ given by system (6.14)–(6.17), provide an asymptotic estimate for $$x_1$$ and $$x_2$$ of system (6.1)–(6.4). For simulation purposes, the controllers are chosen as: u1=−ζ11−2ζ12andu2=cos⁡ζ2+5. (6.18) The initial conditions used in the simulation are chosen as $$x_{10}=[-2 2 -2]$$ and $$x_{20}=[1 4 -5]$$. The simulation results in Figs 1 and 2 show that the designed observer estimates the states of the interconnected system $$x_1=\mbox{col} ( {x_{11}, x_{12}, x_{13}})$$ and $$x_2=\mbox{col} ( {x_{21}, x_{22}, x_{23}})$$ very well in (6.1)–(6.4) even if the system is not asymptotically stable. Fig. 1. View largeDownload slide The time response of the first subsystem states $$x_1=\mbox{col} ( {x_{11},x_{12},x_{13}} )$$ and their estimates $$\hat {x}_1=\mbox{col} ( {\hat {x}_{11},\hat {x}_{12}, \hat {x}_{13}} )$$. Fig. 1. View largeDownload slide The time response of the first subsystem states $$x_1=\mbox{col} ( {x_{11},x_{12},x_{13}} )$$ and their estimates $$\hat {x}_1=\mbox{col} ( {\hat {x}_{11},\hat {x}_{12}, \hat {x}_{13}} )$$. Fig. 2. View largeDownload slide The time response of second subsystem states $$x_2=\mbox{col} ( {x_{21},x_{22},x_{23}} )$$ and their estimates $$\hat {x}_2=\mbox{col} ( {\hat {x}_{21},\hat {x}_{22}, \hat {x}_{23}} ).$$ Fig. 2. View largeDownload slide The time response of second subsystem states $$x_2=\mbox{col} ( {x_{21},x_{22},x_{23}} )$$ and their estimates $$\hat {x}_2=\mbox{col} ( {\hat {x}_{21},\hat {x}_{22}, \hat {x}_{23}} ).$$ Remark 7 The aim of this paper is to design an observer for a class of non-linear interconnected systems in the presence of uncertainties. Note that in this paper, it is not required that the considered systems are asymptotically stable. In order to guarantee the performance of the observer, it is only required that the error dynamical systems are asymptotically stable. The simulation results have shown that the errors between the estimated states and the actual states converge to zero even though the second subsystem is not asymptotically stable as shown in Fig. 2. 7. Conclusions In this paper, a class of non-linear large scale interconnected systems with uniform relative degree has been considered. 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This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

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IMA Journal of Mathematical Control and InformationOxford University Press

Published: Dec 24, 2016

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