Robust ISS stabilization on disturbance for uncertain singularly perturbed systems

Robust ISS stabilization on disturbance for uncertain singularly perturbed systems Abstract This article considers the robust input-to-state stability (ISS) control for a class of uncertain singularly perturbed systems with disturbances. By using the fixed-point principle, we first provide a linear matrix inequality (LMI) sufficient condition to guarantee that the original systems are in a standard form. Secondly, the two-time scale decomposition technique is applied to make the corresponding slow and fast subsystems ISS. Based on the established results, a sufficient condition is presented to guarantee that original systems are also ISS for all sufficiently small values of the perturbation parameter. Finally, a numerical example is given to show the effectiveness of the obtained theoretical results. 1. Introduction Singularly perturbed systems (SPSs) have been an emerging topic that attracted the attention of many researchers due to its wide applications in control engineering. A key to handle SPS is related to the construction of the two subsystems (slow and fast) throughout the so-called reduction technique (O’Malley, 1974; Kokotovic, 1984; Kokotovic et al., 1986; Naidu, 2002). It can be described that for an adequately small $$\varepsilon$$, if the two subsystems of linear SPSs are both stable, then the stability of the original system can be guaranteed. In the recent decades, many stability and stabilization results for SPSs have been obtained (Kahalil, 1989; Chen & Lin, 1990; Chen & Chen, 1995; Christofides & Teel, 1996; Shi et al., 1998; Singh et al., 2001; Teel et al., 2003; Shao, 2004; Shao & Sawan, 2005). For example, by the Riccati equations approach, Shi et al. (1998) investigate robust disturbance attenuation with stability for singularly perturbed linear systems with matched condition. In Shao & Sawan (2005), the robust stabilization problem of SPSs with nonlinear uncertainties is studied, where a control law is presented by the solutions of two independent Lyapunov equations. In addition, the bound of stability is also derived via a state transformation and the constructing Lyapunov function. However, from the view of application, the proposed method is shown to be very complex and difficult. Recently, the linear matrix inequality (LMI) technique has also been proposed to solve different kinds of SPSs (Liu et al., 2004; Lin & Li, 2006; Lu & Ho, 2006; Xu & Lam, 2006; Xu & Feng, 2009; Gao et al., 2011; Yang et al., 2011; Zhou & Lu, 2011; Chen et al., 2013; Yang et al., 2013). It is worth pointing out that the reduced technique is not adopted in these results, where the singular perturbation parameter $$\varepsilon >0$$ is viewed as a static scalar. Unfortunately, there is no clear-cut answer to the question if the control property of the full-order system will remain unchanged as $$\varepsilon \to 0$$. Motivated by the work above, this article considers the robust input-to-state stability (ISS) stabilization problem for a class of SPSs with time-varying uncertainties. We will use the traditional reduced technique to decompose the original system into slow and fast subsystems. However, as a precondition, we will first investigate the existence and uniqueness of the isolate root by the fixed-point principle, thus the given system is in a standard form. Furthermore, the ISS properties for the corresponding reduced-order subsystems are established by the two-time scale decomposition technique. Based on these, sufficient condition for the existence of the isolate root and ISS property of the system is obtained simultaneously via a unified LMI. After that, a design method of the state feedback controller is given to make the closed-loop system ISS. Finally, the upper bound of the parameter $$\varepsilon^{\ast }$$ for ISS is also explicitly estimated in a workable computation way. Finally, a new condition on searching for the allowable upper bound $$\varepsilon^{\ast }$$ is proposed in a workable computation way. It is worth pointing out that the upper bound $$\varepsilon^{\ast }$$ is not prescribed and fewer matrix variables are used, while such a requirement is needed in Liu et al. (2004) and Xu & Feng (2009). Thus, the effectiveness of the proposed method is clearly shown. The rest of the article is organized as follows. Section 2 gives the problem formulation. The main results are given in Section 3. Section 4 gives an example to show the effectiveness of the proposed methods. Finally, the conclusion is drawn in Section 5. Notation: throughout this paper, $$R^{n}$$ denotes de n-dimensional Euclidian space; $$I$$ denotes the identity matrix; 0 represents the zero scalar, the zero vector or zero matrix, which can be determined from context; the notation $$P>0$$ (or $$P<0)$$ means that $$P$$ is positive definite (or negative definite);$${\kern 1pt}{\kern 1pt}{\kern 1pt}\left\| {\cdot {\kern 1pt}{\kern 1pt}} \right\|$$ stands for the standards Euclidean norm in $$R^{n}$$; and the superscript $$T$$ denotes the transpose of a matrix. 2. Problem formulation Consider the following linear uncertain SPS with disturbances in compact form   Eεx˙(t)=(A+ΔA(t))x(t)+B1w(t)+(B2+ΔB2(t))u(t), (1) where $$x=(x_{1}^{T} ,x_{2}^{T} )^{T}\in R^{n}$$ is the system state, $$u\in R^{m_{1} }$$ is the control input; $$w\in R^{m_{2} }$$ is the bounded disturbance input; $$\varepsilon >0$$ is a perturbation parameter; $$x_{1} (t_{0} )=x_{10}$$ and $$x_{2} (t_{0} )=x_{20}$$ are initial conditions. are time-varying uncertainties and norm bounded as follows   ΔA(t)=HF(t)E,ΔB2=HF(t)E3 (2) where $$E=\left({{{E_{1}}\ {E_{2}}}}\right)$$ and $$E_{3}$$ are constant matrices with appropriate dimensions, $$F(t)$$ is an unknown time-varying function with appropriate dimension satisfying   FT(t)F(t)⩽I,t∈[0,∞), (3) Definition 1 (Khalil, 2000) Consider the system   x˙=f(t,x,w), (4) where state $$x(t)$$ is in $$R^{n}$$, and the control input $$w(t)$$ in $$R^{m_{2} }$$,$$f:\left[ {0,\infty } \right)\times R^{n}\times R^{m_{2} }\to {\kern 1pt}R^{n}$$ is continuous and locally Lipchitz in $$x$$ and $$w$$. The input $$w$$ is a bounded function for all $$t\geqslant 0$$. Then the system is said to be input-to-state stable (ISS) if there exist a class $$KL$$ function $$\beta$$ and a class $$K$$ function $$\gamma$$ such that for any initial state $$x(t_{0} )$$, the solution $$x(t)$$ exists for all $$t\geqslant t_{0}$$ and satisfies:   x(t)⩽β(‖x(t0)‖,t−t0)+γ(supt0⩽τ⩽t⁡‖w(τ)‖). Remark 1 The last inequality guarantees that for any bounded input $$w(t)$$, the state $$x(t)$$ will be bounded, and as $$t$$ increases, the state $$x(t)$$ will be ultimately bounded by a class $$K$$ function of $$\left\| w \right\|$$. Furthermore, the inequality also shows that if $$w(t)$$ converges to zero as $$t\to \infty$$, so does $$x(t)$$, which will be verified later. Lemma 1 (Khalil, 2000) Let $$V:[ {0,\infty} )\times R^{n}\to R$$ be a continuously differentiable function such that   α1(‖x‖)⩽V(t,x)⩽α2(‖x‖),∂V∂t+∂V∂xf(t,x,w)⩽−W(x),∀‖x‖⩾ρ(‖w‖)>0, where $$\alpha_{1} ,\alpha_{2}$$ are class $$K_{\infty }$$ functions, $$\rho$$ is a class $$K$$ function, and $$W(x)$$ is a continuous positive definite function on $$R^{n}$$. Then, the system (4) is input-to-state stable with $$\gamma =\alpha_{1}^{-1} \circ \alpha_{2} \circ \rho$$. Lemma 2 (Khargonekar et al., 1990) Let $${\it{\Sigma}}_{1}$$ and $${\it{\Sigma}}_{2}$$ be real matrices of appropriate dimensions. Then for any matrix $$F(t)$$ satisfying $$F^{T}(t)F(t)\leqslant I$$ and a scalar $$\sigma >0$$,   Σ1F(t)Σ2+(Σ1F(t)Σ2)T⩽σ−1Σ1Σ1T+σΣ2TΣ2. 3. Main results In this section we will present a sufficient condition of LMI such that the system (1) is in the standard form and ISS with respect to $$w$$. 3.1 Standard singularly perturbed model Definition 2 System (1) with $$u=0$$ is said to be a standard SPS, if the algebraic equation   0=(A21+ΔA21(t))x1(t)+(A22+ΔA22(t))x2(t)+B21w(t), (5) has a unique isolate root $$x_{2} =\phi (t,x_{1} ,w)$$ for any given $$(x_{1}, w)$$. From the Definition 2, we can see that the existence of the isolate root ensures that the $$n_{1}$$-dimensional reduced model is well defined. Thus it becomes a standard requirement for most singularly perturbed control systems, see Kokotovic et al. (1986), Naidu (2002) and O’Malley (1974). The following result presents a sufficient condition in terms of LMI to guarantee the existence of an isolate root for the system (1). Lemma 3 If there exist a scalar $$\sigma >0$$, symmetric matrices $$P_{11}$$, $$P_{22}$$ and matrix $$P_{21}$$ such that the following LMI holds   Φ0=(ATP+PTA+σETEPTH∗−σI)<0, (6) where Then the uncertain system (1) is a standard SPS. Proof. Let $$\varepsilon =0$$, we obtain the reduced-order system from (1)   E0x˙(t)=(A+ΔA(t))x(t)+B1w(t), where $$E_{0} =diag\{I_{n} ,O\}$$. Condition (6) implies that $$A^{T}P+P^{T}A+\sigma E^{T}E<0$$. Noticing that $$\sigma E^{T}E$$ is non-negative, which means   ATP+PTA<0 (7) Now, making a partition for (7) gives $$A_{22}^{T} P_{22} +P_{22}^{T} A_{22} <0$$. So $$A_{22}^{T} P_{22}$$ is non-singular, which implies $$A_{22}$$ is non-singular too. According to Xu & Lam (2006), it is easy to obtain that the pair $$(E_{0} ,A)$$ is regular and impulse free, thus there exist matrices $$M_{1} \in R^{n_{1} \times n},\;M_{2} \in R^{n_{2} \times n},\;N_{1} \in R^{n\times n_{1} },\;N_{2} \in R^{n\times n_{2} }$$ such that $$M=(M_{1}^{T} ,M_{2}^{T} )^{T}$$ and $$N=(N_{1} ,N_{2} )$$ are non-singular upper and lower triangular matrices, respectively, and the following decomposition holds:   ME0N=diag(In1,O),MAN=diag(A1,In2), where $$A_{1} \in R^{n_{1} \times n_{1} }$$. Noticing that $$M_{2} HH^{T}M_{2}^{T}$$ is positive semi definite, thus one has $$Q_{\zeta } =(M_{2} HH^{T}M_{2}^{T} +\zeta I)^{-\frac{1}{2}}$$ is positive definite for any $$\zeta >0$$. Let $$T_{0} =diag(I_{n_{1} } ,Q_{\zeta } ),\;\bar{{M}}=T_{0} M,\;\bar{{N}}=NT_{0}^{-1}$$, then we have $$\bar{{M}}E_{0} \bar{{N}}=diag(I_{n_{1} } ,O)$$, $$\bar{{M}}A\bar{{N}}=diag(A_{1} ,I_{n_{2} } )$$ and   QζM2HHTM2TQζ =(M2HHTM2T+ζI)−12M2HHTM2T(M2HHTM2T+I)−12 <(M2HHTM2T+ζI)−12(M2HHTM2T+ζI)(M2HHTM2T+I)−12=I, which implies that $$\left\| {Q_{\zeta } M_{2} H} \right\|<1$$. By the Schur’s complement lemma, it is obtained from LMI (6) that   ATP+PTA+σETE+σ−1PTHHTP<0. (8) Pre- and post-multiplying both sides of (8) with $$\bar{{N}}^{T}$$ and $$\bar{{N}}$$ respectively, we obtain that   (M¯AN¯)TM¯−TPN¯ +(M¯−TPN¯)TM¯AN¯+σN¯TETEN¯+σ−1(M¯−TPN¯)TM¯HHTM¯−T(M¯−TPN¯)<0. (9) Let Then it is easy to know that the structure of $$\bar{{M}}^{-T},\;P$$ and $$\bar{{N}}$$ implies that $$P_{2} =0$$. By decomposing (9) and further calculation, we have that the block matrix at the second block row and the second block column of the left-hand side of (9) is negative definite, which is   P4+P4T+σQζ−TN2TETEN2Qζ−1+σ−1P4TQζM2HHTM2TQζT<0. It implies that there exists a sufficiently small $$\zeta >0$$ such that   P4+P4T+σQζ−TN2TETEN2Qζ−1+σ−1P4TQζ(M2HHTM2T+QζI)QζT<0, that is   P4+P4T+σQζ−TN2TETEN2Qζ−1+σ−1P4TP4<0, which is equivalent to   σ−1(P4+σI)T(P4+σI)−σI−σQζ−TN2TETEN2Qζ−T<0. Then, it implies that $$Q_{\zeta }^{-T} N_{2}^{T} E^{T}EN_{2} Q_{\zeta }^{-1} <I$$. Thus there exists a sufficiently small scalar $$\eta >0$$ such that   ‖EN2Qζ−1‖<11+η. In order to show the existence of the isolate root, we introduce a change of coordinates   N¯−1x=(x11T  x12T)T, (10) where $$x_{11} \in R^{n_{1} },\;x_{12} \in R^{n_{2} }$$. Then the reduced-order system can be rewritten equivalently as follows:   x˙11(t)=A1x11+M1HF(t)E(N1x11+N2Qζ−1x12)+M1B11w, (11)  0=x12+QζM2HF(t)E(N1x11+N2Qζ−1x12)+QζM2B12w. (12) For any given $$x_{12} ,\;\bar{{x}}_{12} \in R^{n}$$, we have   ‖QζM2HF(t)EN2Qζ−1(x12−x¯12)‖ ⩽‖QζM2H‖‖F(t)‖‖EN2Qζ−1‖‖x12−x¯12‖ ⩽11+η‖x12−x¯12‖. (13) According to the fixed-point principle, we get that there exists a unique solution $$x_{12} =\phi (x_{11} ,w)$$ for any given $$(x_{11} ,w)$$ from (12). Thus, the existence of isolate root $$x_{2} =\phi (t,x_{1} ,w)$$ is obtained by (10), that is, the system (1) is in the standard form, which completes the proof. □ Remark 2 From the proof of Lemma 3, it is noticed that the existence of the isolate root for system (1) is inherited from the reduced-order system, which is a key step for the two-time scale decomposition technique. However, it is shown in Shao (2004) and Shao & Sawan (2005) that the corresponding problem has not been investigated. Thus, this can be viewed as an extension of Shao (2004) and Shao & Sawan (2005). Furthermore, we can also show that $$x_{2} =\phi (t,x_{1} ,w)$$ is Lipchitz with respect to $$(x_{1} ,w)$$, that is, there exist two scalars $$\alpha_{1} >0$$ and $$\alpha_{2} >0$$ satisfying the following constraint   ‖ϕ(t,x1,w)‖⩽α1‖x1‖+α2‖w‖. (14) The mathematical derivation is similar to that of (13), thus the detail is omitted here. 3.2 ISS investigation Attention is now focused on how the ISS property of the original system (1) can be deduced from the reduced order systems in separate time scales. In fact the slow and fast subsystems (15) and (16) of system (1) can be described by setting $$\varepsilon =0$$, we have:   E0x¯˙=(A+ΔA)x¯+B1ws, (15) where $$\bar{{x}}=\left( {x_{s}^{T} \;} \right.\left. {\bar{{x}}_{2}^{T} } \right)^{T}$$ and $$\bar{{x}}_{2} =\phi (t,x_{s} ,w_{s} )$$;   x˙f=(A22+ΔA22)xf+B12wf,xf(0)=x2−ϕ, (16) where $$x_{f} =x_{2} -\phi ,\;w_{f} =w-w_{s}$$. Based on the reduced technique, next we will give sufficient conditions under which the full-order system is ISS. First, we have following results for the slow and fast subsystems. Theorem 1 Under the condition of Lemma 3, if the matrix $$P_{11}$$ is symmetric and positive definite, then the slow subsystem (15) is made ISS with respect to the disturbance $$w_{s}$$. Theorem 2 Under the condition of Lemma 3, if the matrix $$P_{22}$$ is symmetric and positive definite, then the fast subsystem (16) is made ISS with respect to the disturbance $$w_{f}$$. Using the traditional Lyapunov direct method, we can easily prove the two theorems by choosing $$S_{0} (x_{s} )=x_{s}^{T} P_{11} x_{s} ,\;S_{0} (x_{f} )=x_{f}^{T} P_{22} x_{f}$$ as Lyapunov candidate functions for the slow and fast subsystems, respectively, and computing the corresponding derivatives along trajectories (15) and (16). Based on Theorems 1 and 2, we can now state the main result in the following theorem: Theorem 3 If the condition of Theorems 1 and 2 holds, then there exists an $$\varepsilon^{\ast }>0$$, such that the following results hold: 1. System (1) is a standard SPSs; 2. System (1) is made ISS with respect to disturbance $$w$$ for any given $$\varepsilon \in \left( {0\quad \varepsilon^{\ast }} \right]$$. Proof. The proof of Lemma 1 has shown that system (1) is in standard form, which completes the proof of part 1). We now show the ISS property of system (1). Under the condition of Theorems 1 and 2, it is shown that both $$P_{11}$$ and $$P_{22}$$ are symmetric and positive definite matrices, then there exists a sufficiently small scalar $$\varepsilon_{1} >0$$ such that $$P_{11} -\varepsilon P_{12}^{T} P_{22}^{-1} {\kern 1pt}P_{21} >0$$ for all $$\varepsilon \in \left( {0\quad \varepsilon _{1} } \right]$$. By the Schur’s Complement Lemma, it yields   EεTPε=PεTEε=(P11εP21TεP21εP22)>0, where Define a Lyapunov function candidate for system (1) as follows   S(x)=xTEεTPεx, (17) then, for any a constant $$\sigma >0$$, the derivative of $$S$$ satisfies   S˙(x)⩽xTΥεx+2xTPεTB1w, where $$\Upsilon_{\varepsilon } =A^{T}P_{\varepsilon } +P_{\varepsilon }^{T} A+\sigma E^{T}E+\sigma^{-1}P_{\varepsilon }^{T} HH^{T}P_{\varepsilon }$$. For the above inequality, by the Schur’s complement Lemma, we have that $$\Upsilon_{\varepsilon } <0$$ is equivalent to   Φ0+εΦ<0, where   Φ=(ATP0+P0TAP0TH∗O),P0=(OP21TOO). (18) It follows from (6) that there exists a sufficiently small scalar $$\varepsilon_{2} >0$$ such that $${\it{\Phi}}_{0} +\varepsilon {\it{\Phi}} <0$$ for any given $$\varepsilon \in \left( {0\quad \varepsilon_{2} } \right]$$. Thus $$\Upsilon_{\varepsilon } <0$$ for $$\varepsilon \in \left( {0\ \varepsilon_{2} } \right]$$ is guaranteed. Let $$\bar{{a}}=\lambda_{\min } (-\Upsilon_{\varepsilon } )$$, then $$\bar{{a}}>0$$ for $$\varepsilon \in \left( {0\quad \varepsilon_{2} } \right]$$. Further, let $$\varepsilon^{\ast }=\min \{\varepsilon_{1} ,\varepsilon_{2} \}$$, then we have $$E_{\varepsilon }^{T} P_{\varepsilon } >0$$ and   S˙(x)⩽−a¯‖x‖2+2‖xT‖‖PεTB1‖‖w‖⩽−a¯(1−θ¯)‖x‖2, for 0<θ¯<1,‖x‖>maxε∈[0ε∗]⁡‖PεTB1‖a¯θ¯‖w‖, (19) where $$0<\bar{{\theta }}<1$$. So the conditions of Lemma 1 are satisfied and we say that the system (1) is ISS with respect to $$w(t)$$. This completes the proof. □ Remark 3 Theorem 3 presents a unified sufficient for the existence of the isolate root and ISS property for system (1) by the two-scale decomposition technique. By combining this and LMIs, not only the high dimensionality and the ill condition are alleviated, but also the regularity restrictions attached to the Riccati-based solutions are avoided. Moreover, compared with one-step methodology in Kahalil (1989) and Christofides & Teel (1996), it can be seen from Theorems 1–3 that the ISS property for the corresponding limit systems (i.e. slow subsystems and fast subsystems) of the original system still be guaranteed. As a special case, when the disturbance $$w(t)$$ satisfies the condition $$w(t)\to 0$$ as $$t\to \infty$$, we have the following result. Corollary 1 Under the condition of Theorem 3, if the disturbance $$w(t)\to 0$$ as $$t\to \infty$$ then system (1) is asymptotically stable for all $$\varepsilon \in (0\ \varepsilon^{\ast }]$$. For SPSs, the robustness with respect to the perturbation parameter $$\varepsilon$$ is often referred to as the problem of determining the stability bound; here we will give without proof the following result Theorem 4 If there exist a constant $$\lambda >0$$, positive definite matrices $${\it{\Pi}} >0,\;P_{11} >0,\;P_{22}$$ and matrix $$P_{21}$$, satisfying the following linear matrix inequalities:   Π<λP11,(ΠP21TP21P22)>0,Φ0<0,Φ<−λΦ0, (20) where $${\it{\Phi}}_{0}$$ and $${\it{\Phi}}$$ are defined in (6) and (18), respectively. Then system (1) is in the standard form and ISS with respect to $$w(t)$$ for all $$\varepsilon \in (0\quad \varepsilon^{\ast }]$$ with $$\varepsilon^{\ast }=\lambda^{-1}$$. It follows from Theorem 4 that the upper bound $$\varepsilon^{\ast }$$ can be obtained by solving the following generalized eigenvalue problem (Boyd et al., 1994)   min λ subject to (20), which can be solved effectively by applying GEVP solver in LMI Control Toolbox. In many cases, when the unforced system is not stable, we include feedback transformation to make the system stabilizable and achieve ISS. Therefore, we need to find a state feedback transformation   u=K1x1+K2x2, (21) where $$K=\left( {K_{1} \;K_{2} } \right)$$ is a constant matrix, such that the resulting closed-loop system is ISS with respect to the disturbance $$w(t)$$ when feedback implementation.. Substituting the above control function (21) into (1), we obtain the closed-loop system as follows:   Eεx˙(t)=(A+B2K+ΔA(t)+ΔB2(t)K)x(t)+B1w(t). (22) Applying Theorem 3 to the closed-loop system (22), we have the following result. Theorem 5 If there exists a constant $$\sigma >0$$, matrix $$Y$$ and a lower triangular matrix with $$0<X_{11} \in R^{n_{1} \times n_{1} }$$ and $$X_{22} \in R^{n_{2} \times n_{2} }$$, satisfying the following LMI   Ω0=(AX+XTAT+B2Y+YTB2TXTET+YTE3TEX+E3Y−σI)<0, (23) then there exists $$\varepsilon^{\ast }>0$$ such that the resulting closed-loop system (22) is in the standard form and ISS with respect to $$w(t)$$ for any $$\varepsilon \in (0\ \varepsilon^{\ast }]$$. Moreover, state feedback gain matrix can be chosen as   K=YX−1, (24) Proof. By substituting (24) into (23) and using the Schur’s Complement Lemma, we obtain that inequality (23) is equivalent to   ((A+B2K)X+XT(A+B2K)T+σXT(E+E3K)T(E+E3K)HHT−σI)<0. (25) Pre- and post-multiplying inequality (25) by $$diag\,(X^{-T},I)$$ and $$diag\,(X^{-1},I)$$, respectively and letting $$X=\bar{{P}}^{-1},\;Y=K\bar{{P}}^{-1}$$, then (25) is equivalent to   Ω¯0=((A+B2K)TP¯+P¯T(A+B2K)+σ(E+E3K)T(E+E3K)P¯THHTP¯−σI)<0. (26) We choose a Lyapunov function candidate as follows   S(x)=xTEεTP¯εx, where Then, for any a constant $$\sigma >0$$, the derivative of $$S(x)$$ along (22) yields   S˙(x)=(Eεx˙)TP¯εx+xTP¯εT(Eεx˙)⩽xTΥ¯εx+2xTP¯εTB1w, where $$\bar{{Y}}_{\varepsilon } =(A+B_{2} K)^{T}\bar{{P}}_{\varepsilon } +\bar{{P}}_{\varepsilon }^{T} (A+B_{2} K)+\sigma (E+E_{3} K)^{T}(E+E_{3} K)+\sigma^{-1}\bar{{P}}_{\varepsilon }^{T} HH^{T}\bar{{P}}_{\varepsilon }$$ For the above inequality, by the Schur’s complement Lemma, we have that $$\bar{{Y}}_{\varepsilon } <0$$ is equivalent to   Ω¯0+εΩ¯<0, where   Ω¯=((A+B2K)TP¯0+P¯0T(A+B2K)P¯0THHTP¯0O). (27) The proof is similar to that of Theorem 3, thus there exists a scalar $$\varepsilon^{\ast }>0$$ such that the closed-loop system (22) is in the standard form and ISS with respect to $$w(t)$$ for any $$\varepsilon \in (0\quad \varepsilon^{\ast }]$$. This completes the proof. □ According to Theorem 5, we have the following direct result which gives the method for solving the upper bound for the ISS property of closed-loop system. Theorem 6 After the control gain matrix $$K$$ has been obtained from (22) and (26), if there exists a scalar $$\bar{{\lambda }}>0$$, positive definite matrices $$\bar{{{\it{\Pi}} }}>0,\;\bar{{P}}_{11} >0,\;\bar{{P}}_{22}$$ and matrix $$\bar{{P}}_{21}$$, satisfying the following linear matrix inequalities:   Π¯<λ¯P¯11,(Π¯P¯21TP¯21P¯22)>0,Ω0<0,Ω<−λ¯Ω0, (28) where   Ω=((A+B2K)TP¯0+P¯0T(A+B2K)P¯0THHTP¯0O), then the resulting closed-loop system (22) is in the standard form and ISS with respect to $$w(t)$$ for any $$\varepsilon \in (0\quad \varepsilon^{\ast }]$$ with $$\varepsilon^{\ast }=\bar{{\lambda }}^{-1}$$. Remark 4 It can be seen that the obtained stability bound conditions in Theorem 6 do have dependencies on the control gain matrix $$K$$. However, from Theorem 5 we know that solution pair $$\left( {X,\;Y} \right)$$ derived from LMI (23) is not unique, thus this leads to the non-uniqueness of control gain matrix. Obviously, different stability bounds may be caused by using different control gain matrices; this situation can be an opportunity for future works. 4. A numerical example Consider the following DC-Motor plant with matched uncertainty in Shi et al. (1998).   diadt=−RaLaia−KbLaωm+1Laet,d2θmdt2=KiJmia−BmJmdθmdt−KiJm(θm−θl),d2θldt2=KlJl(θm−θl)+TfJl, (29) where $$J_{l} =5(1+\delta_{l} ),\;\left| {\delta_{l} } \right|\leqslant 0.5;\;T_{f} =(1+\delta_{f} )u_{1} ,\;\left| {\delta_{f} } \right|\leqslant 0.2$$ Let $$x_{1} =\theta_{m} -\theta_{l} ,\;x_{2} =\theta_{m} =\omega_{m} ,\;x_{3} =\theta_{l} =\omega_{l} ,\;x_{4} =i_{a} ,\;u_{2} =e_{t}$$ and $$\varepsilon =L_{a} >0$$. Then, we have the state space form of the uncertain DC-Motor model given by   (x˙1x˙2x˙3εx˙4)=(01−10−KlJm−BmJm0−KiJm−KlJm0000−Kb0−Ra)(x1x2x3x4)+(00001Jl001)(Tfu2). (30) Substituting the system parameters (see Table 1 in Shi et al., 1998) and noticing the uncertainties $$J_{l}$$ and $$T_{f}$$, we can transfer (30) into the following form   (x˙1x˙2x˙3εx˙4)=((01−10−2−0.20−22000052×10−50−2)+ΔA(t))(x1x2x3x4)+((00001Jl001)+ΔB(t))(u1u2), (31) where $${\it{\Delta}} A(t)$$, $${\it{\Delta}} B(t)$$ are expressed in (2), $$F(t)$$ is a time-varying function satisfying (3) and   H=(0.20.20.20.2)T,E=(1110),E3=(11),C=I4,D=(00100001)T. We find the feasible solutions of LMIs (22) as follows:   X=(3.8833−1.64411.35110−1.64415.45880.379901.35110.37994.228700.8891−3.2438−2.14684.4639),Y=(−4.498310.7055−6.9933−4.8031.1984−15.07280.86094.6366),σ=1.3619. Thus, we can obtain the control gain matrix from (24)   K=YX−1=(0.71321.7155−2.5827−1.0774−1.6030−2.73041.48841.0387). Therefore, there exists $$\varepsilon^{\ast }>0$$ such that the unforced SPS is asymptotically stable for all $$\varepsilon \in (0,\;\varepsilon^{\ast }]$$. The upper bound $$\varepsilon^{\ast }=0.6709$$ is obtained by solving the generalized eigenvalue minimization problem. We take $$w(t)=5\cos (0.5t)N$$, then the simulation results for trajectories of the closed-loop system with different parameters are shown in Figs 1 and 2. As shown in the simulations, the closed-loop system bounded when the disturbance input is bounded. Thus, the effectiveness of the proposed method is shown clearly. Fig. 1. View largeDownload slide Trajectories of the closed-loop system with $$\varepsilon =0.01$$. Fig. 1. View largeDownload slide Trajectories of the closed-loop system with $$\varepsilon =0.01$$. Fig. 2. View largeDownload slide Trajectories of the closed-loop system with $$\varepsilon =0.1$$. Fig. 2. View largeDownload slide Trajectories of the closed-loop system with $$\varepsilon =0.1$$. 5. Conclusion In this paper, we have investigated robust ISS for a class of uncertain SPS with disturbance. The uncertainty considered consists of norm-bounded parameter uncertainty (Shi et al., 1998). Based on the reduction technique, we decomposed the whole systems into slow and fast subsystems; knowing that the properties of the original system can be derived from those of slow and fast subsystems for sufficiently small $$\varepsilon$$. Unlike the work in Shi et al. (1998), we have been able to demonstrate that it does not require to set $${\it{\Delta}} A_{22} =0$$ in order to achieve the desired properties of stability. In fact the absence of $${\it{\Delta}} A_{22} =0$$ in the original system is intended to avoid the computation of $$(A_{22} +{\it{\Delta}} A_{22} )^{-1}$$ which is the inverse of an uncertain matrix. In our approach, the LMI technique is used to proof Lemma 1 and it took out the obstacle related to the term $$(A_{22} +{\it{\Delta}} A_{22} )^{-1}$$ which is ‘impossible’ to compute because of the uncertainty $${\it{\Delta}} A_{22}$$. Successively we proved the ISS for the slow subsystem, the fast subsystem and the normal system; this is the condition that made the ‘robustness’ of the ISS in this work. Comparing with Shi et al. (1998) where a complex state transformation is done for stability, it is difficult to apply the abstract theory and complex technique to practical engineers’ problems; however we cannot deny that nowadays the LMI method gives rise to many questions about its relationship with the original problem and the final purpose. But in our work, that relationship is quite clear because from the uncertain model we achieve robust ISS with careful proof and from the simulations, it can be seen that the presented approach guarantees the ISS of the closed-loop system. Funding This article is supported by the National Science Foundation of China (11171113), Science and Technology Commission of Shanghai Municipality (STCSM), grant No.13dz2260400, Shanghai Key Laboratory of PMMP and Science and Technology Planning Project of Henan Province of China (172102210608). References Boyd S., Ghaoui L. E. L., Feron E. & Balakrishnan V. ( 1994) Linear Matrix Inequalities in System and Control Theory . Philadelphia, PA: SIAM. Chen Y. H. & Chen J. S. ( 1995) Robust composite control for singularly perturbed systems with time-varying uncertainties. J. Syst. Measure. Control,  117, 445– 452. Google Scholar CrossRef Search ADS   Chen W. H., Guo Y. & Zheng W. X. ( 2013) Robust stability of singularly perturbed impulsive systems under nonlinear perturbation. IEEE Trans. Autom. Control,  58, 168– 174. Google Scholar CrossRef Search ADS   Chen B. S. & Lin C. L. ( 1990) On the stability bounds of singularly perturbed systems. IEEE Trans. Autom. Control,  35, 1265– 1270. 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IEE Proc. Control Theory Appl. , 148, 391– 396. Google Scholar CrossRef Search ADS   Teel A. R., Moreau L. & Ne¡sic D. ( 2003) A unified framework for input-to-state stability in systems with two time scales. IEEE Trans. Autom. Control,  48, 1526– 1544. Google Scholar CrossRef Search ADS   Zhou L. & Lu G. P. ( 2011) Robust stability of singularly perturbed descriptor systems with nonlinear perturbation. IEEE Trans. Autom. Control,  56, 858– 863. Google Scholar CrossRef Search ADS   Yang C. Y., Sun J. & Ma X. P. ( 2013) Stabilization bound of singularly perturbed systems subject to actuator saturation. Automatica , 49, 457– 462. Google Scholar CrossRef Search ADS   Xu S. Y. & Feng G. ( 2009) New results on $$H_{\infty }$$ control of discrete singularly perturbed systems. Automatica , 45, 2339– 2343. Google Scholar CrossRef Search ADS   Xu S. Y. & Lam J. ( 2006) Robust Control and Filtering of Singular Systems . Berlin, Germany: Springer. Yang C. Y., Zhang Q. L., Sun J. & Chai T. Y. ( 2011) Lur’e Lyapunov function and absolute stability criterion for Lur’e singularly perturbed systems. IEEE Trans. Autom. Control,  56, 2666– 2671. Google Scholar CrossRef Search ADS   © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

Robust ISS stabilization on disturbance for uncertain singularly perturbed systems

, Volume Advance Article – Apr 10, 2017
13 pages

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Publisher
Oxford University Press
ISSN
0265-0754
eISSN
1471-6887
D.O.I.
10.1093/imamci/dnx017
Publisher site
See Article on Publisher Site

Abstract

Abstract This article considers the robust input-to-state stability (ISS) control for a class of uncertain singularly perturbed systems with disturbances. By using the fixed-point principle, we first provide a linear matrix inequality (LMI) sufficient condition to guarantee that the original systems are in a standard form. Secondly, the two-time scale decomposition technique is applied to make the corresponding slow and fast subsystems ISS. Based on the established results, a sufficient condition is presented to guarantee that original systems are also ISS for all sufficiently small values of the perturbation parameter. Finally, a numerical example is given to show the effectiveness of the obtained theoretical results. 1. Introduction Singularly perturbed systems (SPSs) have been an emerging topic that attracted the attention of many researchers due to its wide applications in control engineering. A key to handle SPS is related to the construction of the two subsystems (slow and fast) throughout the so-called reduction technique (O’Malley, 1974; Kokotovic, 1984; Kokotovic et al., 1986; Naidu, 2002). It can be described that for an adequately small $$\varepsilon$$, if the two subsystems of linear SPSs are both stable, then the stability of the original system can be guaranteed. In the recent decades, many stability and stabilization results for SPSs have been obtained (Kahalil, 1989; Chen & Lin, 1990; Chen & Chen, 1995; Christofides & Teel, 1996; Shi et al., 1998; Singh et al., 2001; Teel et al., 2003; Shao, 2004; Shao & Sawan, 2005). For example, by the Riccati equations approach, Shi et al. (1998) investigate robust disturbance attenuation with stability for singularly perturbed linear systems with matched condition. In Shao & Sawan (2005), the robust stabilization problem of SPSs with nonlinear uncertainties is studied, where a control law is presented by the solutions of two independent Lyapunov equations. In addition, the bound of stability is also derived via a state transformation and the constructing Lyapunov function. However, from the view of application, the proposed method is shown to be very complex and difficult. Recently, the linear matrix inequality (LMI) technique has also been proposed to solve different kinds of SPSs (Liu et al., 2004; Lin & Li, 2006; Lu & Ho, 2006; Xu & Lam, 2006; Xu & Feng, 2009; Gao et al., 2011; Yang et al., 2011; Zhou & Lu, 2011; Chen et al., 2013; Yang et al., 2013). It is worth pointing out that the reduced technique is not adopted in these results, where the singular perturbation parameter $$\varepsilon >0$$ is viewed as a static scalar. Unfortunately, there is no clear-cut answer to the question if the control property of the full-order system will remain unchanged as $$\varepsilon \to 0$$. Motivated by the work above, this article considers the robust input-to-state stability (ISS) stabilization problem for a class of SPSs with time-varying uncertainties. We will use the traditional reduced technique to decompose the original system into slow and fast subsystems. However, as a precondition, we will first investigate the existence and uniqueness of the isolate root by the fixed-point principle, thus the given system is in a standard form. Furthermore, the ISS properties for the corresponding reduced-order subsystems are established by the two-time scale decomposition technique. Based on these, sufficient condition for the existence of the isolate root and ISS property of the system is obtained simultaneously via a unified LMI. After that, a design method of the state feedback controller is given to make the closed-loop system ISS. Finally, the upper bound of the parameter $$\varepsilon^{\ast }$$ for ISS is also explicitly estimated in a workable computation way. Finally, a new condition on searching for the allowable upper bound $$\varepsilon^{\ast }$$ is proposed in a workable computation way. It is worth pointing out that the upper bound $$\varepsilon^{\ast }$$ is not prescribed and fewer matrix variables are used, while such a requirement is needed in Liu et al. (2004) and Xu & Feng (2009). Thus, the effectiveness of the proposed method is clearly shown. The rest of the article is organized as follows. Section 2 gives the problem formulation. The main results are given in Section 3. Section 4 gives an example to show the effectiveness of the proposed methods. Finally, the conclusion is drawn in Section 5. Notation: throughout this paper, $$R^{n}$$ denotes de n-dimensional Euclidian space; $$I$$ denotes the identity matrix; 0 represents the zero scalar, the zero vector or zero matrix, which can be determined from context; the notation $$P>0$$ (or $$P<0)$$ means that $$P$$ is positive definite (or negative definite);$${\kern 1pt}{\kern 1pt}{\kern 1pt}\left\| {\cdot {\kern 1pt}{\kern 1pt}} \right\|$$ stands for the standards Euclidean norm in $$R^{n}$$; and the superscript $$T$$ denotes the transpose of a matrix. 2. Problem formulation Consider the following linear uncertain SPS with disturbances in compact form   Eεx˙(t)=(A+ΔA(t))x(t)+B1w(t)+(B2+ΔB2(t))u(t), (1) where $$x=(x_{1}^{T} ,x_{2}^{T} )^{T}\in R^{n}$$ is the system state, $$u\in R^{m_{1} }$$ is the control input; $$w\in R^{m_{2} }$$ is the bounded disturbance input; $$\varepsilon >0$$ is a perturbation parameter; $$x_{1} (t_{0} )=x_{10}$$ and $$x_{2} (t_{0} )=x_{20}$$ are initial conditions. are time-varying uncertainties and norm bounded as follows   ΔA(t)=HF(t)E,ΔB2=HF(t)E3 (2) where $$E=\left({{{E_{1}}\ {E_{2}}}}\right)$$ and $$E_{3}$$ are constant matrices with appropriate dimensions, $$F(t)$$ is an unknown time-varying function with appropriate dimension satisfying   FT(t)F(t)⩽I,t∈[0,∞), (3) Definition 1 (Khalil, 2000) Consider the system   x˙=f(t,x,w), (4) where state $$x(t)$$ is in $$R^{n}$$, and the control input $$w(t)$$ in $$R^{m_{2} }$$,$$f:\left[ {0,\infty } \right)\times R^{n}\times R^{m_{2} }\to {\kern 1pt}R^{n}$$ is continuous and locally Lipchitz in $$x$$ and $$w$$. The input $$w$$ is a bounded function for all $$t\geqslant 0$$. Then the system is said to be input-to-state stable (ISS) if there exist a class $$KL$$ function $$\beta$$ and a class $$K$$ function $$\gamma$$ such that for any initial state $$x(t_{0} )$$, the solution $$x(t)$$ exists for all $$t\geqslant t_{0}$$ and satisfies:   x(t)⩽β(‖x(t0)‖,t−t0)+γ(supt0⩽τ⩽t⁡‖w(τ)‖). Remark 1 The last inequality guarantees that for any bounded input $$w(t)$$, the state $$x(t)$$ will be bounded, and as $$t$$ increases, the state $$x(t)$$ will be ultimately bounded by a class $$K$$ function of $$\left\| w \right\|$$. Furthermore, the inequality also shows that if $$w(t)$$ converges to zero as $$t\to \infty$$, so does $$x(t)$$, which will be verified later. Lemma 1 (Khalil, 2000) Let $$V:[ {0,\infty} )\times R^{n}\to R$$ be a continuously differentiable function such that   α1(‖x‖)⩽V(t,x)⩽α2(‖x‖),∂V∂t+∂V∂xf(t,x,w)⩽−W(x),∀‖x‖⩾ρ(‖w‖)>0, where $$\alpha_{1} ,\alpha_{2}$$ are class $$K_{\infty }$$ functions, $$\rho$$ is a class $$K$$ function, and $$W(x)$$ is a continuous positive definite function on $$R^{n}$$. Then, the system (4) is input-to-state stable with $$\gamma =\alpha_{1}^{-1} \circ \alpha_{2} \circ \rho$$. Lemma 2 (Khargonekar et al., 1990) Let $${\it{\Sigma}}_{1}$$ and $${\it{\Sigma}}_{2}$$ be real matrices of appropriate dimensions. Then for any matrix $$F(t)$$ satisfying $$F^{T}(t)F(t)\leqslant I$$ and a scalar $$\sigma >0$$,   Σ1F(t)Σ2+(Σ1F(t)Σ2)T⩽σ−1Σ1Σ1T+σΣ2TΣ2. 3. Main results In this section we will present a sufficient condition of LMI such that the system (1) is in the standard form and ISS with respect to $$w$$. 3.1 Standard singularly perturbed model Definition 2 System (1) with $$u=0$$ is said to be a standard SPS, if the algebraic equation   0=(A21+ΔA21(t))x1(t)+(A22+ΔA22(t))x2(t)+B21w(t), (5) has a unique isolate root $$x_{2} =\phi (t,x_{1} ,w)$$ for any given $$(x_{1}, w)$$. From the Definition 2, we can see that the existence of the isolate root ensures that the $$n_{1}$$-dimensional reduced model is well defined. Thus it becomes a standard requirement for most singularly perturbed control systems, see Kokotovic et al. (1986), Naidu (2002) and O’Malley (1974). The following result presents a sufficient condition in terms of LMI to guarantee the existence of an isolate root for the system (1). Lemma 3 If there exist a scalar $$\sigma >0$$, symmetric matrices $$P_{11}$$, $$P_{22}$$ and matrix $$P_{21}$$ such that the following LMI holds   Φ0=(ATP+PTA+σETEPTH∗−σI)<0, (6) where Then the uncertain system (1) is a standard SPS. Proof. Let $$\varepsilon =0$$, we obtain the reduced-order system from (1)   E0x˙(t)=(A+ΔA(t))x(t)+B1w(t), where $$E_{0} =diag\{I_{n} ,O\}$$. Condition (6) implies that $$A^{T}P+P^{T}A+\sigma E^{T}E<0$$. Noticing that $$\sigma E^{T}E$$ is non-negative, which means   ATP+PTA<0 (7) Now, making a partition for (7) gives $$A_{22}^{T} P_{22} +P_{22}^{T} A_{22} <0$$. So $$A_{22}^{T} P_{22}$$ is non-singular, which implies $$A_{22}$$ is non-singular too. According to Xu & Lam (2006), it is easy to obtain that the pair $$(E_{0} ,A)$$ is regular and impulse free, thus there exist matrices $$M_{1} \in R^{n_{1} \times n},\;M_{2} \in R^{n_{2} \times n},\;N_{1} \in R^{n\times n_{1} },\;N_{2} \in R^{n\times n_{2} }$$ such that $$M=(M_{1}^{T} ,M_{2}^{T} )^{T}$$ and $$N=(N_{1} ,N_{2} )$$ are non-singular upper and lower triangular matrices, respectively, and the following decomposition holds:   ME0N=diag(In1,O),MAN=diag(A1,In2), where $$A_{1} \in R^{n_{1} \times n_{1} }$$. Noticing that $$M_{2} HH^{T}M_{2}^{T}$$ is positive semi definite, thus one has $$Q_{\zeta } =(M_{2} HH^{T}M_{2}^{T} +\zeta I)^{-\frac{1}{2}}$$ is positive definite for any $$\zeta >0$$. Let $$T_{0} =diag(I_{n_{1} } ,Q_{\zeta } ),\;\bar{{M}}=T_{0} M,\;\bar{{N}}=NT_{0}^{-1}$$, then we have $$\bar{{M}}E_{0} \bar{{N}}=diag(I_{n_{1} } ,O)$$, $$\bar{{M}}A\bar{{N}}=diag(A_{1} ,I_{n_{2} } )$$ and   QζM2HHTM2TQζ =(M2HHTM2T+ζI)−12M2HHTM2T(M2HHTM2T+I)−12 <(M2HHTM2T+ζI)−12(M2HHTM2T+ζI)(M2HHTM2T+I)−12=I, which implies that $$\left\| {Q_{\zeta } M_{2} H} \right\|<1$$. By the Schur’s complement lemma, it is obtained from LMI (6) that   ATP+PTA+σETE+σ−1PTHHTP<0. (8) Pre- and post-multiplying both sides of (8) with $$\bar{{N}}^{T}$$ and $$\bar{{N}}$$ respectively, we obtain that   (M¯AN¯)TM¯−TPN¯ +(M¯−TPN¯)TM¯AN¯+σN¯TETEN¯+σ−1(M¯−TPN¯)TM¯HHTM¯−T(M¯−TPN¯)<0. (9) Let Then it is easy to know that the structure of $$\bar{{M}}^{-T},\;P$$ and $$\bar{{N}}$$ implies that $$P_{2} =0$$. By decomposing (9) and further calculation, we have that the block matrix at the second block row and the second block column of the left-hand side of (9) is negative definite, which is   P4+P4T+σQζ−TN2TETEN2Qζ−1+σ−1P4TQζM2HHTM2TQζT<0. It implies that there exists a sufficiently small $$\zeta >0$$ such that   P4+P4T+σQζ−TN2TETEN2Qζ−1+σ−1P4TQζ(M2HHTM2T+QζI)QζT<0, that is   P4+P4T+σQζ−TN2TETEN2Qζ−1+σ−1P4TP4<0, which is equivalent to   σ−1(P4+σI)T(P4+σI)−σI−σQζ−TN2TETEN2Qζ−T<0. Then, it implies that $$Q_{\zeta }^{-T} N_{2}^{T} E^{T}EN_{2} Q_{\zeta }^{-1} <I$$. Thus there exists a sufficiently small scalar $$\eta >0$$ such that   ‖EN2Qζ−1‖<11+η. In order to show the existence of the isolate root, we introduce a change of coordinates   N¯−1x=(x11T  x12T)T, (10) where $$x_{11} \in R^{n_{1} },\;x_{12} \in R^{n_{2} }$$. Then the reduced-order system can be rewritten equivalently as follows:   x˙11(t)=A1x11+M1HF(t)E(N1x11+N2Qζ−1x12)+M1B11w, (11)  0=x12+QζM2HF(t)E(N1x11+N2Qζ−1x12)+QζM2B12w. (12) For any given $$x_{12} ,\;\bar{{x}}_{12} \in R^{n}$$, we have   ‖QζM2HF(t)EN2Qζ−1(x12−x¯12)‖ ⩽‖QζM2H‖‖F(t)‖‖EN2Qζ−1‖‖x12−x¯12‖ ⩽11+η‖x12−x¯12‖. (13) According to the fixed-point principle, we get that there exists a unique solution $$x_{12} =\phi (x_{11} ,w)$$ for any given $$(x_{11} ,w)$$ from (12). Thus, the existence of isolate root $$x_{2} =\phi (t,x_{1} ,w)$$ is obtained by (10), that is, the system (1) is in the standard form, which completes the proof. □ Remark 2 From the proof of Lemma 3, it is noticed that the existence of the isolate root for system (1) is inherited from the reduced-order system, which is a key step for the two-time scale decomposition technique. However, it is shown in Shao (2004) and Shao & Sawan (2005) that the corresponding problem has not been investigated. Thus, this can be viewed as an extension of Shao (2004) and Shao & Sawan (2005). Furthermore, we can also show that $$x_{2} =\phi (t,x_{1} ,w)$$ is Lipchitz with respect to $$(x_{1} ,w)$$, that is, there exist two scalars $$\alpha_{1} >0$$ and $$\alpha_{2} >0$$ satisfying the following constraint   ‖ϕ(t,x1,w)‖⩽α1‖x1‖+α2‖w‖. (14) The mathematical derivation is similar to that of (13), thus the detail is omitted here. 3.2 ISS investigation Attention is now focused on how the ISS property of the original system (1) can be deduced from the reduced order systems in separate time scales. In fact the slow and fast subsystems (15) and (16) of system (1) can be described by setting $$\varepsilon =0$$, we have:   E0x¯˙=(A+ΔA)x¯+B1ws, (15) where $$\bar{{x}}=\left( {x_{s}^{T} \;} \right.\left. {\bar{{x}}_{2}^{T} } \right)^{T}$$ and $$\bar{{x}}_{2} =\phi (t,x_{s} ,w_{s} )$$;   x˙f=(A22+ΔA22)xf+B12wf,xf(0)=x2−ϕ, (16) where $$x_{f} =x_{2} -\phi ,\;w_{f} =w-w_{s}$$. Based on the reduced technique, next we will give sufficient conditions under which the full-order system is ISS. First, we have following results for the slow and fast subsystems. Theorem 1 Under the condition of Lemma 3, if the matrix $$P_{11}$$ is symmetric and positive definite, then the slow subsystem (15) is made ISS with respect to the disturbance $$w_{s}$$. Theorem 2 Under the condition of Lemma 3, if the matrix $$P_{22}$$ is symmetric and positive definite, then the fast subsystem (16) is made ISS with respect to the disturbance $$w_{f}$$. Using the traditional Lyapunov direct method, we can easily prove the two theorems by choosing $$S_{0} (x_{s} )=x_{s}^{T} P_{11} x_{s} ,\;S_{0} (x_{f} )=x_{f}^{T} P_{22} x_{f}$$ as Lyapunov candidate functions for the slow and fast subsystems, respectively, and computing the corresponding derivatives along trajectories (15) and (16). Based on Theorems 1 and 2, we can now state the main result in the following theorem: Theorem 3 If the condition of Theorems 1 and 2 holds, then there exists an $$\varepsilon^{\ast }>0$$, such that the following results hold: 1. System (1) is a standard SPSs; 2. System (1) is made ISS with respect to disturbance $$w$$ for any given $$\varepsilon \in \left( {0\quad \varepsilon^{\ast }} \right]$$. Proof. The proof of Lemma 1 has shown that system (1) is in standard form, which completes the proof of part 1). We now show the ISS property of system (1). Under the condition of Theorems 1 and 2, it is shown that both $$P_{11}$$ and $$P_{22}$$ are symmetric and positive definite matrices, then there exists a sufficiently small scalar $$\varepsilon_{1} >0$$ such that $$P_{11} -\varepsilon P_{12}^{T} P_{22}^{-1} {\kern 1pt}P_{21} >0$$ for all $$\varepsilon \in \left( {0\quad \varepsilon _{1} } \right]$$. By the Schur’s Complement Lemma, it yields   EεTPε=PεTEε=(P11εP21TεP21εP22)>0, where Define a Lyapunov function candidate for system (1) as follows   S(x)=xTEεTPεx, (17) then, for any a constant $$\sigma >0$$, the derivative of $$S$$ satisfies   S˙(x)⩽xTΥεx+2xTPεTB1w, where $$\Upsilon_{\varepsilon } =A^{T}P_{\varepsilon } +P_{\varepsilon }^{T} A+\sigma E^{T}E+\sigma^{-1}P_{\varepsilon }^{T} HH^{T}P_{\varepsilon }$$. For the above inequality, by the Schur’s complement Lemma, we have that $$\Upsilon_{\varepsilon } <0$$ is equivalent to   Φ0+εΦ<0, where   Φ=(ATP0+P0TAP0TH∗O),P0=(OP21TOO). (18) It follows from (6) that there exists a sufficiently small scalar $$\varepsilon_{2} >0$$ such that $${\it{\Phi}}_{0} +\varepsilon {\it{\Phi}} <0$$ for any given $$\varepsilon \in \left( {0\quad \varepsilon_{2} } \right]$$. Thus $$\Upsilon_{\varepsilon } <0$$ for $$\varepsilon \in \left( {0\ \varepsilon_{2} } \right]$$ is guaranteed. Let $$\bar{{a}}=\lambda_{\min } (-\Upsilon_{\varepsilon } )$$, then $$\bar{{a}}>0$$ for $$\varepsilon \in \left( {0\quad \varepsilon_{2} } \right]$$. Further, let $$\varepsilon^{\ast }=\min \{\varepsilon_{1} ,\varepsilon_{2} \}$$, then we have $$E_{\varepsilon }^{T} P_{\varepsilon } >0$$ and   S˙(x)⩽−a¯‖x‖2+2‖xT‖‖PεTB1‖‖w‖⩽−a¯(1−θ¯)‖x‖2, for 0<θ¯<1,‖x‖>maxε∈[0ε∗]⁡‖PεTB1‖a¯θ¯‖w‖, (19) where $$0<\bar{{\theta }}<1$$. So the conditions of Lemma 1 are satisfied and we say that the system (1) is ISS with respect to $$w(t)$$. This completes the proof. □ Remark 3 Theorem 3 presents a unified sufficient for the existence of the isolate root and ISS property for system (1) by the two-scale decomposition technique. By combining this and LMIs, not only the high dimensionality and the ill condition are alleviated, but also the regularity restrictions attached to the Riccati-based solutions are avoided. Moreover, compared with one-step methodology in Kahalil (1989) and Christofides & Teel (1996), it can be seen from Theorems 1–3 that the ISS property for the corresponding limit systems (i.e. slow subsystems and fast subsystems) of the original system still be guaranteed. As a special case, when the disturbance $$w(t)$$ satisfies the condition $$w(t)\to 0$$ as $$t\to \infty$$, we have the following result. Corollary 1 Under the condition of Theorem 3, if the disturbance $$w(t)\to 0$$ as $$t\to \infty$$ then system (1) is asymptotically stable for all $$\varepsilon \in (0\ \varepsilon^{\ast }]$$. For SPSs, the robustness with respect to the perturbation parameter $$\varepsilon$$ is often referred to as the problem of determining the stability bound; here we will give without proof the following result Theorem 4 If there exist a constant $$\lambda >0$$, positive definite matrices $${\it{\Pi}} >0,\;P_{11} >0,\;P_{22}$$ and matrix $$P_{21}$$, satisfying the following linear matrix inequalities:   Π<λP11,(ΠP21TP21P22)>0,Φ0<0,Φ<−λΦ0, (20) where $${\it{\Phi}}_{0}$$ and $${\it{\Phi}}$$ are defined in (6) and (18), respectively. Then system (1) is in the standard form and ISS with respect to $$w(t)$$ for all $$\varepsilon \in (0\quad \varepsilon^{\ast }]$$ with $$\varepsilon^{\ast }=\lambda^{-1}$$. It follows from Theorem 4 that the upper bound $$\varepsilon^{\ast }$$ can be obtained by solving the following generalized eigenvalue problem (Boyd et al., 1994)   min λ subject to (20), which can be solved effectively by applying GEVP solver in LMI Control Toolbox. In many cases, when the unforced system is not stable, we include feedback transformation to make the system stabilizable and achieve ISS. Therefore, we need to find a state feedback transformation   u=K1x1+K2x2, (21) where $$K=\left( {K_{1} \;K_{2} } \right)$$ is a constant matrix, such that the resulting closed-loop system is ISS with respect to the disturbance $$w(t)$$ when feedback implementation.. Substituting the above control function (21) into (1), we obtain the closed-loop system as follows:   Eεx˙(t)=(A+B2K+ΔA(t)+ΔB2(t)K)x(t)+B1w(t). (22) Applying Theorem 3 to the closed-loop system (22), we have the following result. Theorem 5 If there exists a constant $$\sigma >0$$, matrix $$Y$$ and a lower triangular matrix with $$0<X_{11} \in R^{n_{1} \times n_{1} }$$ and $$X_{22} \in R^{n_{2} \times n_{2} }$$, satisfying the following LMI   Ω0=(AX+XTAT+B2Y+YTB2TXTET+YTE3TEX+E3Y−σI)<0, (23) then there exists $$\varepsilon^{\ast }>0$$ such that the resulting closed-loop system (22) is in the standard form and ISS with respect to $$w(t)$$ for any $$\varepsilon \in (0\ \varepsilon^{\ast }]$$. Moreover, state feedback gain matrix can be chosen as   K=YX−1, (24) Proof. By substituting (24) into (23) and using the Schur’s Complement Lemma, we obtain that inequality (23) is equivalent to   ((A+B2K)X+XT(A+B2K)T+σXT(E+E3K)T(E+E3K)HHT−σI)<0. (25) Pre- and post-multiplying inequality (25) by $$diag\,(X^{-T},I)$$ and $$diag\,(X^{-1},I)$$, respectively and letting $$X=\bar{{P}}^{-1},\;Y=K\bar{{P}}^{-1}$$, then (25) is equivalent to   Ω¯0=((A+B2K)TP¯+P¯T(A+B2K)+σ(E+E3K)T(E+E3K)P¯THHTP¯−σI)<0. (26) We choose a Lyapunov function candidate as follows   S(x)=xTEεTP¯εx, where Then, for any a constant $$\sigma >0$$, the derivative of $$S(x)$$ along (22) yields   S˙(x)=(Eεx˙)TP¯εx+xTP¯εT(Eεx˙)⩽xTΥ¯εx+2xTP¯εTB1w, where $$\bar{{Y}}_{\varepsilon } =(A+B_{2} K)^{T}\bar{{P}}_{\varepsilon } +\bar{{P}}_{\varepsilon }^{T} (A+B_{2} K)+\sigma (E+E_{3} K)^{T}(E+E_{3} K)+\sigma^{-1}\bar{{P}}_{\varepsilon }^{T} HH^{T}\bar{{P}}_{\varepsilon }$$ For the above inequality, by the Schur’s complement Lemma, we have that $$\bar{{Y}}_{\varepsilon } <0$$ is equivalent to   Ω¯0+εΩ¯<0, where   Ω¯=((A+B2K)TP¯0+P¯0T(A+B2K)P¯0THHTP¯0O). (27) The proof is similar to that of Theorem 3, thus there exists a scalar $$\varepsilon^{\ast }>0$$ such that the closed-loop system (22) is in the standard form and ISS with respect to $$w(t)$$ for any $$\varepsilon \in (0\quad \varepsilon^{\ast }]$$. This completes the proof. □ According to Theorem 5, we have the following direct result which gives the method for solving the upper bound for the ISS property of closed-loop system. Theorem 6 After the control gain matrix $$K$$ has been obtained from (22) and (26), if there exists a scalar $$\bar{{\lambda }}>0$$, positive definite matrices $$\bar{{{\it{\Pi}} }}>0,\;\bar{{P}}_{11} >0,\;\bar{{P}}_{22}$$ and matrix $$\bar{{P}}_{21}$$, satisfying the following linear matrix inequalities:   Π¯<λ¯P¯11,(Π¯P¯21TP¯21P¯22)>0,Ω0<0,Ω<−λ¯Ω0, (28) where   Ω=((A+B2K)TP¯0+P¯0T(A+B2K)P¯0THHTP¯0O), then the resulting closed-loop system (22) is in the standard form and ISS with respect to $$w(t)$$ for any $$\varepsilon \in (0\quad \varepsilon^{\ast }]$$ with $$\varepsilon^{\ast }=\bar{{\lambda }}^{-1}$$. Remark 4 It can be seen that the obtained stability bound conditions in Theorem 6 do have dependencies on the control gain matrix $$K$$. However, from Theorem 5 we know that solution pair $$\left( {X,\;Y} \right)$$ derived from LMI (23) is not unique, thus this leads to the non-uniqueness of control gain matrix. Obviously, different stability bounds may be caused by using different control gain matrices; this situation can be an opportunity for future works. 4. A numerical example Consider the following DC-Motor plant with matched uncertainty in Shi et al. (1998).   diadt=−RaLaia−KbLaωm+1Laet,d2θmdt2=KiJmia−BmJmdθmdt−KiJm(θm−θl),d2θldt2=KlJl(θm−θl)+TfJl, (29) where $$J_{l} =5(1+\delta_{l} ),\;\left| {\delta_{l} } \right|\leqslant 0.5;\;T_{f} =(1+\delta_{f} )u_{1} ,\;\left| {\delta_{f} } \right|\leqslant 0.2$$ Let $$x_{1} =\theta_{m} -\theta_{l} ,\;x_{2} =\theta_{m} =\omega_{m} ,\;x_{3} =\theta_{l} =\omega_{l} ,\;x_{4} =i_{a} ,\;u_{2} =e_{t}$$ and $$\varepsilon =L_{a} >0$$. Then, we have the state space form of the uncertain DC-Motor model given by   (x˙1x˙2x˙3εx˙4)=(01−10−KlJm−BmJm0−KiJm−KlJm0000−Kb0−Ra)(x1x2x3x4)+(00001Jl001)(Tfu2). (30) Substituting the system parameters (see Table 1 in Shi et al., 1998) and noticing the uncertainties $$J_{l}$$ and $$T_{f}$$, we can transfer (30) into the following form   (x˙1x˙2x˙3εx˙4)=((01−10−2−0.20−22000052×10−50−2)+ΔA(t))(x1x2x3x4)+((00001Jl001)+ΔB(t))(u1u2), (31) where $${\it{\Delta}} A(t)$$, $${\it{\Delta}} B(t)$$ are expressed in (2), $$F(t)$$ is a time-varying function satisfying (3) and   H=(0.20.20.20.2)T,E=(1110),E3=(11),C=I4,D=(00100001)T. We find the feasible solutions of LMIs (22) as follows:   X=(3.8833−1.64411.35110−1.64415.45880.379901.35110.37994.228700.8891−3.2438−2.14684.4639),Y=(−4.498310.7055−6.9933−4.8031.1984−15.07280.86094.6366),σ=1.3619. Thus, we can obtain the control gain matrix from (24)   K=YX−1=(0.71321.7155−2.5827−1.0774−1.6030−2.73041.48841.0387). Therefore, there exists $$\varepsilon^{\ast }>0$$ such that the unforced SPS is asymptotically stable for all $$\varepsilon \in (0,\;\varepsilon^{\ast }]$$. The upper bound $$\varepsilon^{\ast }=0.6709$$ is obtained by solving the generalized eigenvalue minimization problem. We take $$w(t)=5\cos (0.5t)N$$, then the simulation results for trajectories of the closed-loop system with different parameters are shown in Figs 1 and 2. As shown in the simulations, the closed-loop system bounded when the disturbance input is bounded. Thus, the effectiveness of the proposed method is shown clearly. Fig. 1. View largeDownload slide Trajectories of the closed-loop system with $$\varepsilon =0.01$$. Fig. 1. View largeDownload slide Trajectories of the closed-loop system with $$\varepsilon =0.01$$. Fig. 2. View largeDownload slide Trajectories of the closed-loop system with $$\varepsilon =0.1$$. Fig. 2. View largeDownload slide Trajectories of the closed-loop system with $$\varepsilon =0.1$$. 5. Conclusion In this paper, we have investigated robust ISS for a class of uncertain SPS with disturbance. The uncertainty considered consists of norm-bounded parameter uncertainty (Shi et al., 1998). Based on the reduction technique, we decomposed the whole systems into slow and fast subsystems; knowing that the properties of the original system can be derived from those of slow and fast subsystems for sufficiently small $$\varepsilon$$. Unlike the work in Shi et al. (1998), we have been able to demonstrate that it does not require to set $${\it{\Delta}} A_{22} =0$$ in order to achieve the desired properties of stability. In fact the absence of $${\it{\Delta}} A_{22} =0$$ in the original system is intended to avoid the computation of $$(A_{22} +{\it{\Delta}} A_{22} )^{-1}$$ which is the inverse of an uncertain matrix. In our approach, the LMI technique is used to proof Lemma 1 and it took out the obstacle related to the term $$(A_{22} +{\it{\Delta}} A_{22} )^{-1}$$ which is ‘impossible’ to compute because of the uncertainty $${\it{\Delta}} A_{22}$$. Successively we proved the ISS for the slow subsystem, the fast subsystem and the normal system; this is the condition that made the ‘robustness’ of the ISS in this work. Comparing with Shi et al. (1998) where a complex state transformation is done for stability, it is difficult to apply the abstract theory and complex technique to practical engineers’ problems; however we cannot deny that nowadays the LMI method gives rise to many questions about its relationship with the original problem and the final purpose. But in our work, that relationship is quite clear because from the uncertain model we achieve robust ISS with careful proof and from the simulations, it can be seen that the presented approach guarantees the ISS of the closed-loop system. Funding This article is supported by the National Science Foundation of China (11171113), Science and Technology Commission of Shanghai Municipality (STCSM), grant No.13dz2260400, Shanghai Key Laboratory of PMMP and Science and Technology Planning Project of Henan Province of China (172102210608). References Boyd S., Ghaoui L. E. L., Feron E. & Balakrishnan V. ( 1994) Linear Matrix Inequalities in System and Control Theory . Philadelphia, PA: SIAM. Chen Y. H. & Chen J. S. ( 1995) Robust composite control for singularly perturbed systems with time-varying uncertainties. J. Syst. Measure. Control,  117, 445– 452. Google Scholar CrossRef Search ADS   Chen W. H., Guo Y. & Zheng W. X. ( 2013) Robust stability of singularly perturbed impulsive systems under nonlinear perturbation. IEEE Trans. Autom. Control,  58, 168– 174. Google Scholar CrossRef Search ADS   Chen B. S. & Lin C. L. ( 1990) On the stability bounds of singularly perturbed systems. IEEE Trans. Autom. Control,  35, 1265– 1270. 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IMA Journal of Mathematical Control and InformationOxford University Press

Published: Apr 10, 2017

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