Ⅰ. INTRODUCTION

IN recent years,unmanned aerial vehicles (UAVs),such as hypersonic aircraft,highly maneuverable technology (HiMAT) vehicle and agile missile,have received considerable amounts of attention^{[1, 2, 3, 4]}. The control of UAVs is recognized to be extraordinarily challenging,due to the high nonlinearity and complexity of their motion equations,which stem from the strong couplings between dramatic parameter variations within the full-envelope,wide velocity range and the large unknown environmental disturbances^{[5, 6, 7, 8]}.

A traditional approach to handle the nonlinearity of full-envelope UAVs is by means of gain-scheduling^{[9, 10]}. Its design principle is to design local controllers for each member of a set of operating points that cover the whole flight envelope firstly and then to obtain a global controller in flight by using interpolation method according to the current value of the scheduling parameters^{[11, 12]}.

However,some poorly developed but important theoretical issues still exist in the gain-scheduled method,such as no guarantees on the robustness,performance,or even nominal stability of the overall gain-scheduled design,especially when the flight envelope becomes much wider^{[13, 14]}. Therefore,it is quite attractive to improve the gain-scheduled method by combining other approach which has appeared in the extant literatures^{[15, 16]}.

Considering the advantages of switched system theory and the computational complexity of the switched linear parameter-varying (LPV) approach^{[17, 18, 19, 20]},an improved gain-scheduled controller design approach is proposed in this work based on the switched polytopic systems which have been paid much attention in recent years^{[21, 22, 23, 24, 25, 26]}. To describe the full-envelope vehicle$'$s nonlinear model,a locally overlapped switched polytopic system is established evolving on the locally overlapped switching laws which are constructed by considering that the flight dynamics can only switch from one region to another adjacent region,where each polytopic subsystem presents the system dynamics in a part of the full flight envelope,and its vertices describe the linear dynamics on given characteristic operating points within this part of flight envelope. For each polytopic subsystem,a gain-scheduled subcontroller is obtained by interpolating between the controllers on vertices according to the weighted coefficients,and the switched parameter-dependent controllers of the full-envelope are synthesized using these polytopic subsystems$'$ gain-scheduled subcontrollers.

On the other hand,the implicit assumption that the measurements of the system states always contain useful signals is not actually possible^{[27, 28, 29, 30, 31]}. It can be proved that many measurements consist of noise alone,or involve data missing which can be caused by many practical reasons,such as,measurement failure,intermittent sensor failure,imperfect communication,some of the data being disturbed or coming from a very noisy environment^[27]. In many engineering applications,such as flight control^{[29, 30, 31]},due to sensor temporal failure or transmission disturbance at certain time instants,the system measurement may only contain noise,which means the real signal is missing^[28].

Moreover,because switching signals and weighted coefficients of the switched polytopic systems are dependent on the current Mach number and altitude which are measured online in flight,missing measurements will lead to the asynchronous switching problem which means there are updating lags between the system mode switching and the controller switching^{[32, 33]}. To the best of the authors$'$ knowledge,the UAVs$'$ control with missing measurements based on switched polytopic system approach is challenging,and has not been fully investigated yet.

This paper investigates the $H_\infty $ controller design for UAVs with missing measurements. Firstly,to reduce designing conservatisms and solving complexities,a locally overlapped switched polytopic system is established to describe the full-envelope flight dynamics evolving on the locally overlapped switching laws with the aid of the proposition of a locally overlapped region division method. Secondly,to overcome the effects of the asynchronous switching problems induced by the missing measurements,an asynchronous $H_{\infty }$ control method is proposed to guarantee that the system is stable and satisfies a prescribed attenuation level against the external disturbances. Then,the stability and $l_{2}$ performance analysis is performed by combining the switched parameter-dependent Lyapunov function method and the average dwell time method,and the sufficient existing conditions of the switched parameter-dependent $H_{\infty}$ controller are established in the form of LMIs. Finally,a full-envelope HiMAT flight example is given to demonstrate the effectiveness of the proposed approach.

Ⅱ. MODEL DESCRIPTION

The UAV investigated in this paper is a HiMAT vehicle,a full-envelope and open-loop unstable UAV,which is sponsored by NASA and the U.S. Air Force. It is studied to incorporate technological advances in many fields,such as a close-coupled canard configuration,aero elastic tailing with composite structures,relaxed static stability,and advanced transonic aerodynamics^{[34, 35]}.

To derive the linear system model from nonlinear dynamics of the full-envelope vehicle^{[14, 15, 16]},Jacobian linearization is performed on the 20 characteristic operating points within the full envelope presented in [34],which are chosen based on the altitude $H$ and Mach number $Ma$,as depicted in Fig. 1. As each linear model of the chosen operating point can describe the dynamics in its vicinity,the 20 models can cover the nonlinear dynamic behaviors of the HiMAT vehicle within the full envelope^[34].

Considering that the longitudinal motion stability and maneuverability primarily depend on the short period motion,with the aid of 20 continuous-time longitudinal linear models in [34],the longitudinal short period model of the $i$-th operating point can be modeled as the discrete time model (1) by setting the system$'$s sampling period to be $T$,which is

\begin{align} \left\{ {\begin{array}{*{20}l} \pmb x(k+1)=A_i \pmb x(k)+B_i \pmb u(k)+B_{d,i} \pmb d(k),\\ \pmb y(k)=C_i \pmb x(k)+D_{d,i} \pmb d(k). \nonumber\\ \end{array}} \right. \end{align}

(1)

Here,${\pmb x}=(\alpha ,q)^{\rm T}$ is the state vector,and $\alpha,q$ denote angle of attack,pitch rate,respectively. ${\pmb u}=(\xi _e ,\xi _v,\xi _c )^{\rm T}$ is the control input,$\xi _e ,\xi _v ,\xi _c $ denote elevator,elevon and canard deflection,respectively. ${\pmb y}$ is the output,and ${\pmb d}\in L_2 \left[{0,\infty } \right)$ is the unknown external disturbances. The real system matrices $A_i ,B_i ,C_i ,B_{d,i} ,D_{d,i} $ for the $i$-th characteristic operating point,$i\in \Omega=\{1,2,\cdots ,20\}$,are of appropriate dimensions with the index set $\Omega$ denoting the collection of all characteristic operating points within the envelope,and the trim condition for every operating point is illustrated in Table Ⅰ.

To reduce designing conservatisms and solving complexities,the large-scale flight envelope shown in Fig. 1 is divided into $N$ regions referring to the switched LPV approach. The index set $\Omega $ is accordingly divided into $N$ index subsets which are defined as $\Omega _1 ,\Omega _2 ,\cdots ,\Omega _N $ to represent the collection of the characteristic operating points within each region.

In our work,a polytopic system is adopted to represent the system dynamics in each region,and its vertices are those characteristic operating points within this region. Inspired by the gain-scheduled method,the system dynamics of current operating point within this region is obtained by interpolating among the system dynamics on vertices.

Considering the flight dynamics can only switch from the current region to another adjacent region in the full-envelope flight,a locally overlapped switched polytopic system is furthermore established as follows:

\begin{align} \left\{\!\!\! {\begin{array}{*{20}l} \pmb x(k+1)\!=\!A_{\sigma (k)} (\pmb a_k )\pmb x(k)\!\!+\!\!B_{\sigma (k)} (\pmb a_k )\pmb u(k)\!\!+\!\!B_{d,\sigma (k)}(\pmb a_k )\pmb d(k),\\ \pmb y(k)\!=\!C_{\sigma (k)} (\pmb a_k )\pmb x(k)\!\!+\!\!D_{d,\sigma (k)} (\pmb a_k )\pmb d(k),\end{array}} \right. \end{align}

(2)

where $k \in {\bf N}$ is time instant,and the switching laws $\sigma (k) \in \Gamma $ stands for variation of the flight region according to $k$,where the set $\Gamma = \left\{ {1,2,\cdots ,N} \right\}$ is the collection of the flight region indices,with $N$ being a finite integer denoting the number of flight regions. The system matrices of (2) satisfy that

\begin{align} &\left[{{\begin{array}{*{20}c} {A_m (\pmb a_k )} & {B_m (\pmb a_k )} & {B_{d,m} (\pmb a_k )} \\ {C_m (\pmb a_k )} & 0 & {D_{d,m} (\pmb a_k )} \\ \end{array} }} \right]=\notag \end{align} \begin{align} &\qquad \sum\limits_{i\in \Omega _m } {a_{i,k} \left[ {{\begin{array}{*{20}c} {A_i } & {B_i } & {B_{d,i} } \\ {C_i } & 0 & {D_{d,i} } \nonumber\\ \end{array} }} \right]},\ \end{align}

(3)

where $\sum\nolimits_{i\in \Omega _m } {a_{i,k} =1} ,~a_{i,k} \ge 0,~\forall \sigma (k)=m\in \Gamma $.

Considering the locally overlapped property of $\sigma (k)$ within the full-envelope flight,the flight regions must be divided under the following conditions:

Condition 1. $\bigcup\nolimits_{m\in \Gamma } {\Omega _m } =\Omega.$

Condition 2. $\Omega _{\sigma (k)} \cap \Omega _{\sigma (k+1)} \ne \emptyset,~\forall k\in {\bf N}. $

Taking the flight trajectory depicted in Fig. 1 for example,the flight envelope can be partitioned into three flight regions,and one has

\begin{align} \left\{ {\begin{array}{l} \Gamma =\left\{ {1,2,3} \right\},\Omega _1 =\left\{ {\mbox{1,2,3,4,5,6,7}} \right\},\\ \Omega _2 =\left\{ {\mbox{6,7,8,9,11,12,16}} \right\},\\ \Omega _3 =\left\{ {\mbox{10,12,13,14,15,16,17,18,19,20}} \right\}. \nonumber\\ \end{array}} \right. \end{align}

(4)

Remark 1. Condition 1 guarantees the index subsets must involve all the characteristic operating points within the flight envelope. Condition 2 guarantees the switching occurs between two adjacent regions that share common characteristic operating points. Thus,the switching on the boundary of two adjacent regions will not induce non-smooth change of the weighted coefficient $\pmb a_k $,provided $\pmb a_k $ is only calculated by using these common vertices of the adjacent polytopic subsystems. For example,the controller gains on the boundary of the Region 1 and Region 2 are only interpolated by the gains on characteristic operating Points 6 and 7.

Ⅲ. PROBLEM FORMULATION

In this section,a switched parameter-dependent state feedback $H_\infty $ controller is designed to guarantee that the system is stable and satisfies a prescribed performance level. Firstly,the following assumptions are made.

Assumption 1. System state $\pmb x$ is measurable,and the missing measurements of $\pmb x$ are described by a Bernoulli distributed stochastic variable $\theta (k)\in \{0,1\}$,where $\mbox{Pr\{}\theta (k)=1\mbox{\}}=\rho,~ \mbox{Pr\{}\theta (k)=0\mbox{\}}=1-\rho$. $\theta (k)=1$ stands for the normal case,while $\theta (k)=0$ stands for the missing measurement case,accordingly^[27].

Thus,the state feedback controller$'$s input $\pmb z(k)$ is given by

\begin{align} \pmb z(k)=\theta (k)\pmb x(k)+(1-\theta (k))\pmb z(k-1). \end{align}

(5)

With the locally overlapped switching laws $\forall \sigma (k)\in \Gamma $,the discrete state feedback switched parameter-dependent controller (6) is designed as

\begin{align} \pmb u(k)=K_{\sigma (k)} (\pmb a_k )\pmb z(k), \end{align}

(6)

where subcontroller $K_m (\pmb a_k ),~ \forall \sigma (k)=m \in \Gamma $ of polytopic subsystem $m$ is synthesized as follows:

\begin{align} K_m (\pmb a_k )=\sum\limits_{i\in \Omega _m } {a_{i,k} K_i},~ \sum\limits_{i\in \Omega _m } {a_{i,k} =1} ,~a_{i,k} \ge 0, \end{align}

(7)

where $K_i $ is the controller gain to be determined for the $i$-th characteristic operating point.

Remark 2. Because the partition information of the flight regions is stored in controller unit previously,$\sigma (k)$ can be determined online by judging which region the current Mach number and altitude belong to,and $\pmb a_k$ could also be calculated referring to the finite element theory^[23].

Remark 3. Considering the measured Mach number and altitude data will be missing at some instants,there will be updating lags for $\sigma (k)$ and $\pmb a_k $,which will further induce asynchronous switching phenomena between the system modes and controllers. Moreover,the asynchronous switching here will be more complex than that in [32,33],because neither $\sigma (k)$ nor $\pmb a_k $ can be obtained in time.

Due to the asynchronous switching,the Lyapunov function value may moderately increase even during the active time of certain subsystems except at the switching instants,but the increase rate must be bounded^[32]. For concise notation,let $k_v $ and $k_{v+1},v\in {\bf N}$ denote the starting time and ending time of some active subsystem,while ${\cal T}_\uparrow (k_v,k_{v+1} )$ and ${\cal T}_\downarrow (k_v,k_{v+1} )$ represent the unions value of the dispersed intervals during which Lyapunov function is increasing and decreasing within the interval,where ${\cal T}_\downarrow (k_{v+1}-k_v )$ and ${\cal T}_\uparrow (k_{v+1}-k_v )$ denote the length of ${\cal T}_\uparrow (k_v,k_{v+1} )$ and ${\cal T}_\downarrow (k_v,k_{v+1} )$,respectively^[33].

As a result of the updating lags of $\sigma (k)$,the Lyapunov function value of the polytopic subsystem may moderately increase, and the corresponding ${\cal T}_\uparrow (k_v,k_{v+1} )$ will be only the interval close to the switching instants. As ${\cal T}_\uparrow (k_{v+1}-k_v )$ may vary at different instants, without loss of generality,the maximal lag of the Mach number and altitude data is assumed to be $\delta _mT$,where $\delta _m $ is known a priori here^[33].

Thus,the controller in (6) will be transformed to (8) under the asynchronous switching,i.e.,

\begin{align} \pmb u(k)=K_{\sigma (k-\delta_m )} (\pmb a_{k-\delta_m} )\pmb z(k). \end{align}

(8)

Denote $\pmb \xi (k)=\left( {{\begin{array}{*{20}c} {\pmb x^{\rm T}(k)} ,& {\pmb z^{\rm T}(k-1)} \\ \end{array} }} \right)^{\rm T}$,then,by combining (2),(5)-(8), the augmented locally overlapped switched polytopic stochastic system (9) is given as

\begin{align} &\left\{ {\begin{array}{lll} \pmb \xi (k+1)\!=\!\hat {A}_n (\pmb a_{\bar {k}})\pmb \xi (k)\!\!+\!\!\bar {\theta }(k)\hat {A}_{1,n} (\pmb a_{\bar {k}} )\pmb \xi (k)\!\!+\!\!\hat {B}_n (\pmb a_{\bar {k}} )\pmb d(k),\\ \pmb y(k)\!=\!\hat {C}_n (\pmb a_{\bar {k}} )\pmb \xi (k)\!+\!\hat {D}_n (\pmb a_{\bar {k}})\pmb d(k),\forall k\in \left[{k_v ,k_v +\delta _m \;} \right), \end{array}} \right. \notag\\ &\left\{ {\begin{array}{l} \pmb \xi (k+1)\!=\!\bar {A}_m (\pmb a_{\bar {k}} )\pmb \xi (k)\!\!+\!\!\bar {\theta } (k)\bar {A}_{1,m}(\pmb a_{\bar {k}} )\pmb \xi (k)\!\!+\!\!\bar {B}_m (\pmb a_{\bar {k}} )\pmb d(k),\\ \pmb y(k)\!=\!\bar {C}_m (\pmb a_{\bar {k}} )\pmb \xi (k)\!+\!\bar {D}_m (\pmb a_{\bar {k}})\pmb d(k),\forall k\in \left[{k_v +\delta _m \;,k_{v+1} } \right), \end{array}} \right. \end{align}

(9)

where $ \bar {k}=k-\delta_m,~ \bar {\theta }(k)=\theta (k)-\rho,~ \forall \{ \sigma (k_v )=m,~ \sigma (\bar {k})= n\}\in \Gamma \times \Gamma,~ m\ne n,~ \Omega _m \cap \Omega_n \ne \emptyset$,and one has

\begin{align*} &\mbox{E}\left\{ {\bar {\theta }(k)} \right\}=0,~\mbox{E}\left\{ {\bar {\theta }(k)\bar {\theta }(k)} \right\}=\rho (1-\rho ),\notag\\ &\left[{{\begin{array}{*{20}c} {\hat {A}_n (\pmb a_{\bar {k}} )} & {\hat {A}_{1,n} (\pmb a_{\bar {k}} )} & {\hat {B}_n (\pmb a_{\bar {k}} )} \\ {\hat {C}_n (\pmb a_{\bar {k}} )} & 0 & {\hat {D}_n (\pmb a_{\bar {k}} )} \\ \end{array} }} \right] =\notag\\ &\qquad \sum\limits_{l\in \Omega _n } {a_{l,\bar {k}} } \sum\limits_{i\in \Omega _m } {a_{i,k} \left[ {{\begin{array}{*{20}c} {\hat {A}_{il} } & {\hat {A}_{1,il} } & {B_{d,i} } \\ {C_i } & 0 & {D_{d,i} } \\ \end{array} }} \right]},\\ &\left[{{\begin{array}{*{20}c} {\bar {A}_m (\pmb a_{\bar {k}} )} & {\bar {A}_{1,m} (\pmb a_{\bar {k}} )} & {\bar {B}_m (\pmb a_{\bar {k}} )} \\ {\bar {C}_m (\pmb a_{\bar {k}} )} & 0 & {\bar {D}_m (\pmb a_{\bar {k}} )} \\ \end{array} }} \right] = \end{align*} \begin{align*} &\qquad \sum\limits_{l\in \Omega _m } {a_{l,\bar {k}} } \sum\limits_{i\in \Omega _m } {a_{i,k} \left[ {{\begin{array}{*{20}c} {\hat {A}_{il} } & {\hat {A}_{1,il} } & {B_{d,i} } \\ {C_i } & 0 & {D_{d,i} } \\ \end{array} }} \right]},\\ &\hat {A}_{il} =\left[{{\begin{array}{*{20}c} {A_i +\rho B_i K_l } & {(1-\rho )B_i K_l } \\ \rho & {(1-\rho )} \\ \end{array} }} \right],\\ &\hat {A}_{1,il} =\left[{{\begin{array}{*{20}c} {B_i K_l } & {-B_i K_l } \\ I & {-I} \\ \end{array} }} \right]. \end{align*}

Then,the $H_\infty $ controller design for the HiMAT vehicle with missing measurements is summarized as follows. Considering the locally overlapped switched polytopic system (2) with missing measurements described in (5),for a positive constant $\gamma $, determine the matrices $K_i~(i\in \Omega) $ of asynchronous controller in (8),such that system (9) is stable and has a weighted $l_2\mbox{-}$gain performance (10) under zero initial conditions, that is,

\begin{align} \mbox{E}\left\{ {\sum\limits_{s=0}^\infty {(1-}\lambda )^s \pmb y^{\rm T}(s) \pmb y(s)} \right\}\le \sum\limits_{s=0}^\infty {\gamma^2} \pmb d^{\rm T}(s)\pmb d(s), \end{align}

(10)

where $0<\lambda <1,$ and $\gamma >0$.

Remark 4. In the presence of missing measurements and the locally overlapped switching laws,a Bernoulli distributed stochastic variable is introduced to describe the missing measurements,and an augmented locally overlapped switching polytopic stochastic system (9) is further established. Obviously, the asynchronous $H_\infty $ control problem for (9) in our work is more complicated than the existent $H_\infty $ control problem for the switched polytopic system.

Ⅳ. MAIN RESULTS

In this section,the sufficient existing conditions of the desired controller will be derived in the form of LMIs. Firstly,we present the following definition and lemmas on the stability and $l_2\mbox{-}$gain analysis here for later use.

Definition 1^[32]. For switching signal $\sigma (\cdot )$ and any $k\ge 1$,let $N_\sigma [0,k)$ be the switching numbers of $\sigma (\cdot )$ over the interval $[0,k)$. If for any given $N_0 \ge 0$ and $\tau _a >0$,we have ${N_\sigma }[0,k) \le {N_0} + k/{\tau _a}$,then $\tau_a$ and $N_0$ are called average dwell time and the chatter bound, respectively. Without loss of generality,we choose $N_0 =0$ for convenience as commonly used in the literature.

Lemma 1. Consider system (9),and let $0<\lambda <1$,$\beta \ge 0$ and $\mu \ge 1$ be given constants. Suppose that there exist $C^1$ functions $V_{\sigma (k)} $,$\sigma (k)=m\in \Gamma $,and two class $\kappa_\infty$ functions $\kappa_1,\kappa_2$ such that

\begin{align} \kappa _1 \left( {\left\| {\pmb \xi (k)} \right\|} \right)\le V_m (\pmb \xi (k))\le \kappa _2 \left( {\left\| {\pmb \xi (k)} \right\|} \right), \end{align}

(11)

\begin{align} \mbox{E}\left( {\Delta V_m (\pmb \xi (k))} \right)\le \left\{ {\begin{array}{l} -\lambda V_m (\pmb \xi (k)),\;\;\;\;\forall k\in \left[{k_v +\delta _m,k_{v+1} } \right),\\ \beta V_m (\pmb \xi (k)),\;\;\;\;\;\;\forall k\in \left[{k_v ,k_v +\delta _m } \right),\\ \end{array}} \right. \end{align}

(12)

\begin{align} V_m (\pmb \xi (k))\le \mu V_n (\pmb \xi (k)),~~ \forall (m,n)\in \Gamma \times \Gamma,~~ m\ne n. \end{align}

(13)

Thus,the system is globally uniformly asymptotically stable in the mean (GUAS-M) for any switching signal with average dwell time $\tau_a$ satisfying

\begin{align} {\tau _a} < \tau _a^ * = - \{ {\delta _m}[\ln \tilde \beta - \ln \tilde \lambda] + \ln \mu \} {\rm{ /}}\ln \tilde \lambda , \end{align}

(14)

where $\tilde {\lambda }=1-\lambda$,$\tilde {\beta }=1+\beta $, and $\hbar = \tilde \beta {\rm{ }}/\tilde \lambda$.

Proof. Referring to the method in [33],$\forall k\in\left[ {k_v ,k_{v+1} } \right)$,it holds from (12) that

\begin{array}{l} {\rm{E}}\left( {{V_{\sigma (k)}}({\rm{\backslash pmb}}\xi (k))} \right) \le {{\tilde \lambda }^{{T_{ \downarrow (}}k - {k_v})}}{{\tilde \beta }^{{T_{ \uparrow (}}k - {k_v})}}{\rm{E}}\left( {{V_{\sigma ({k_v})}}({\rm{\backslash pmb}}\xi ({k_v}))} \right) \le \\ \quad {{\tilde \lambda }^{(k - {k_v})}}{\hbar ^{{\delta _m}}}{\rm{E}}\left( {{V_{\sigma ({k_v})}}({\rm{\backslash pmb}}\xi ({k_v}))} \right) \le \\ \quad {{\tilde \lambda }^{(k - {k_v})}}{\hbar ^{{\delta _m}}}\mu {\rm{E}}\left( {{V_{\sigma ({k_v} - 1)}}({\rm{\backslash pmb}}\xi ({k_v}))} \right) \le \cdots \le \\ \quad {{\tilde \lambda }^{(k - 0)}}{\left( {{\hbar ^{{\delta _m}}}\mu } \right)^{{N_\sigma }[0,k)}}{V_{\sigma (0)}}({\rm{\backslash pmb}}\xi (0)) \le \\ \quad {\mu ^{{N_0}}}{\hbar ^{{N_0}{\delta _m}}}{\left( {\tilde \lambda {\hbar ^{{\delta _m}/{\tau _a}}}{\mu ^{1/{\tau _a}}}} \right)^k}{V_{\sigma (0)}}({\rm{\backslash pmb}}\xi (0)).{\rm{ }} \end{array}

(15)

If average dwell time $\tau _a $ satisfies (14),define $\kappa : = - \ln \tilde \lambda / - \ln \tilde \lambda \left( {{\delta _m}\ln \hbar + \ln \mu } \right){\rm{ }} - \left( {{\delta _m}\ln \hbar + \ln \mu } \right)$,then

\begin{array}{l} \tilde \lambda \hbar {\rm{ }}{{\rm{ }}^{{\delta _m}{\rm{ }}/{\tau _a}}}{\rm{ }}{\mu ^{1{\rm{ }}/{\tau _a}}} < \tilde \lambda {\rm{ }}{\hbar ^{\frac{{ - \delta {\rm{ }}{{\rm{ }}_m}{\rm{ }}\ln \tilde \lambda }}{{{\delta _m}\ln {\rm{ }}\hbar {\rm{ }} + {\rm{ }}\ln {\rm{ }}\mu }}}}{\mu ^{\frac{{ - \ln {\rm{ }}\tilde \lambda }}{{\delta {\rm{ }}{{\rm{ }}_m}{\rm{ }}\ln {\rm{ }}\hbar {\rm{ }} + {\rm{ }}{\mathop{\rm l}\nolimits} {\rm{ }}n{\rm{ }}\mu }}}} = \\ \quad \tilde \lambda \left( {\hbar {{\rm{ }}^{{\delta _{{\rm{ }}m}}}}\mu {\rm{ }}} \right){\rm{ }}{{\rm{ }}^\kappa } = \tilde \lambda \left( {{{\rm{e}}^{\delta {{\rm{ }}_m}{\rm{ }}\ln {\rm{ }}\hbar {\rm{ }} + {\rm{ }}\ln {\rm{ }}\mu {\rm{ }}}}} \right){\rm{ }}{{\rm{ }}^\kappa } = \tilde \lambda /\tilde \lambda = 1. \end{array}

(16)

Therefore,we can get that $\mbox{E}\left( {V_{\sigma (k)} (\pmb \xi(k))} \right)\to 0$ as $k\to \infty $,and system (9) is GUAS-M with the aid of (11) by using the same method as in [36].

Lemma 2. Consider system (9),and let $0<\lambda<1,~\beta \ge 0,~ \mu \ge 1$ and $\gamma_m >0~(\forall m\in \Gamma)$ be given constants. Suppose that there exist $C^1$ functions $V_{\sigma (k)} ,~\sigma (k)=m\in \Gamma $,such that (13) is satisfied and

\begin{align} &\mbox{E}\left( {\Delta V_m (\pmb \xi (k))} \right)\le\notag\\ &\quad \left\{ {\begin{array}{l} -\lambda V_m (\pmb \xi (k))-\phi (k),\;\; \forall k\in \left[{k_v +\delta _m,k_{v+1} } \right),\\ \beta V_m (\pmb \xi (k))-\phi (k),\;\;\;\;\;\forall k\in \left[{k_v ,k_v +\delta_m } \right). \end{array}} \right. \end{align}

(17)

Then the system has a weighted $l_2\mbox{-}$gain no greater than $\gamma =\mathop {\max }\nolimits_{m\in \Gamma } \{\sqrt {\mu ^{N_0}\hbar ^{\delta _m N_0 }} \gamma _m \}$ like (10),where $\phi (k)=\mbox{E}\left( {\pmb y^{\rm T}(k)\pmb y(k)} \right)-\gamma _m \pmb d^{\rm T}(k)\pmb d(k)$.

Proof. By replacing $V_m (\pmb x(k))$ and $\Delta V_m (\pmb x(k))$ with $\mbox{E}\left( {V_{\sigma (k)} (\pmb \xi (k))} \right)$ and $\mbox{E}\left( {\Delta V_{\sigma (k)} (\pmb \xi (k))} \right)$ in [33],Lemma 2 for the switched polytopic system with missing measurement can be directly obtained with the aid of the theoretical analysis for the stochastic switched system in [37],and the proofs are omitted here.

A. Stability and $l_2\mbox{-}$Gain Performance Analysis

Theorem 1. Given scalars $0<\lambda<1,~~ \beta \ge 0,~~ \vartheta =\sqrt {\rho (1-\rho )}$ and $\gamma _m >0,~~ \forall m\in \Gamma $,if there exist matrices $P_i>0,~~i\in \Omega _m,~~ \forall m\in \Gamma $,such that the following LMIs (18) and (19) hold for $\forall \left( {m,n}\right)\in \Gamma \times \Gamma $:

\begin{align} & \Upsilon _{i,j,l}\!\! =\!\!\left[\!\!\!{{\begin{array}{*{20}c} {P_j } & 0 & 0 & {P_j \hat {A}_{il} } & {P_j B_{d,i} } \\ \ast & {P_j } & 0 & {\vartheta P_j \hat {A}_{1,il} } & 0 \\ \ast & \ast & {-I} & {C_i } & {D_{d,i} } \\ \ast & \ast & \ast & {-(1-\lambda )P_i } & 0 \\ \ast & \ast & \ast & \ast & {-\gamma _m^2 I} \\ \end{array} }} \!\!\!\right]<0,\notag\\ &\qquad \qquad \qquad \qquad \forall i,j,l\in \Omega _m, \end{align}

(18)

\begin{align} & \bar {\Upsilon }_{i,j,l} \!=\!\left[\!\!\! {{\begin{array}{*{20}c} {P_j } & 0 & 0 & {P_j \hat {A}_{il} } & {P_j B_{d,i} } \\ \ast & {P_j } & 0 & {\vartheta P_j \hat {A}_{1,il} } & 0 \\ \ast & \ast & {-I} & {C_i } & {D_{d,i} } \\ \ast & \ast & \ast & {-(1+\beta )P_i } & 0 \\ \ast & \ast & \ast & \ast & {-\gamma _m^2 I} \\ \end{array} }}\!\!\! \right]\!\!<0,\notag\\ &\qquad \forall i,j\in \Omega _m,~ l\in \Omega _n,~ \Omega _m \cap \Omega _n \ne \emptyset,~ m\ne n. \end{align}

(19)

Then system (9) is GUAS-M when $\pmb d(k)=0$ for any switching signal with average dwell time $\tau _a $ satisfying

\begin{align} \begin{array}{*{35}{l}} {{\tau }_{a}} >\tau _{a}^{*}=-\{{{\delta }_{m}}[\ln \tilde{\beta }-\ln \tilde{\lambda }]\}/\ln \tilde{\lambda },\\ \end{array} \end{align}

(20)

and has the weighted $l_2\mbox{-}$gain performance index (10) under zero initial conditions,where $\gamma =\max\{\gamma _m \},~~ \forall m\in \Gamma,~~ \tilde {\lambda }=1-\lambda,$ and $ \tilde {\beta }=1+\beta $.

Proof. Firstly,we choose the following Lyapunov-like function (21) based on the switched parameter-dependent Lyapunov function method^[38]:

\begin{align} V_m (\pmb \xi (k))=V_m (\pmb a_k,\pmb \xi (k))=\pmb \xi ^{\rm T}(k)P_m (\pmb a_k )\pmb \xi (k), \end{align}

(21)

where the weighting matrices for the polytopic subsystem $\left({\bar {A}_m (\pmb a_k ),\bar {A}_{1,m} (\pmb a_k ),\bar {C}_m (\pmb a_k ),\bar{D}_m (\pmb a_k )} \right)$ are $P_m (\pmb a_k )=\sum\nolimits_{i\in \Omega _m} {a_{i,k} P_i } ,~~\forall \sigma (k)=m\in \Gamma $.

From Remark 1,it is clear that switchings on the boundary of two adjacent regions will not induce non-smooth change of the weighted coefficient $\pmb a_k$,because $\pmb a_k$ is calculated by using only these common vertices of the adjacent polytopic subsystems. So,when the switching occurs,we have

\begin{align} \begin{array}{l} \begin{aligned} &V_m (\pmb a_k,\pmb \xi (k_v ))=\pmb \xi ^{\rm T}(k_v )P_m (\pmb a_{k_v } )\pmb \xi (k_v ) =\\ &\quad \pmb \xi ^{\rm T}(k_v )\left( {\sum\limits_{s\in \{\Omega _m \cap \Omega _n\}} {a_{s,k_v } P_s } } \right)\pmb \xi (k_v ),\\ &V_n (\pmb a_k,\pmb \xi (k_v ))=\pmb \xi ^{\rm T}(k_v )P_n (\pmb a_{k_v} )\pmb \xi (k_v ) =\\ &\quad \pmb \xi ^{\rm T}(k_v )\left( {\sum\limits_{s\in \{\Omega _m \cap \Omega _n\}} {a_{s,k_v} P_s } } \right)\pmb \xi (k_v ). \\ \end{aligned}\end{array} \end{align}

(22)

Thus,we get $\mu =1$ at the switching instant from (13),and it can be derived from (14) that the dwell time $\tau _a $ of system (9) must satisfy (20).

Secondly,we prove the stability of system (9). To this end, assume that $\pmb d(k)=0$. For any non-zero $\pmb \xi (k),~~\forall k\in \left[{k_v +\delta _m ,k_{v+1} } \right)$,and any switching signal $\sigma (k)=m\in \Gamma $,one has (23) holds.

\begin{align} &\mbox{E}\left\{ {V_m (\pmb a_{k+1},\pmb \xi (k+1))} \right\}-V_m (\pmb a_k,\pmb \xi (k))+\lambda V_m (\pmb a_k,\pmb \xi (k)) =\notag\\ &\quad \mbox{E}\left\{ {\pmb \xi ^{\rm T}(k+1)P_m (\pmb a_{k+1} )\pmb \xi (k+1)} \right\}-\pmb \xi^{\rm T}(k)(1-\lambda )P_m (\pmb a_k )\pmb \xi (k) =\notag\\ &\quad \mbox{E}\left\{ {\pmb \xi ^{\rm T}(k)\left( {\bar {A}_m (\pmb a_{\bar {k}} ) +\bar {\theta }(k)\bar {A}_{1,m} (\pmb a_{\bar {k}} )} \right)^{\rm T}P_m(\pmb a_{k+1} )} {\left( {\bar {A}_m (\pmb a_{\bar {k}} )+\bar {\theta }(k)\bar {A}_{1,m} (\pmb a_{\bar {k}} )} \right)\pmb \xi (k)} \right\}-\pmb \xi^{\rm T}(k)(1-\lambda )P_m (\pmb a_k )\pmb \xi (k) =\notag\\ &\quad \pmb \xi ^{\rm T}(k)\Lambda _{1,m} (\pmb a_k ,\pmb a_{k+1} )\pmb \xi (k), \end{align}

(23)

where $\begin{array}{l} \Lambda _{1,m} (\pmb a_k ,\pmb a_{k+1} )=\bar {A}_m^{\rm T} (\pmb a_{\bar {k}} )P_m (\pmb a_{k+1} )\bar {A}_m (\pmb a_{\bar {k}} ) +\vartheta ^2\bar {A}_{1,m}^{\rm T} (\pmb a_{\bar {k}} )P_m (\pmb a_{k+1} )\bar {A}_{1,m} (\pmb a_{\bar {k}} )-(1-\lambda )P_m (\pmb a_k ). \\ \end{array}$

Similarly,for any non-zero $\pmb \xi (k),~~\forall k\in \left[ {k_v,k_v +\delta _m } \right)$,and any switching signal $\left\{{\sigma (k)=m,~\sigma (\bar {k})=n} \right\}\in \Gamma \times \Gamma,~\Omega _m \cap \Omega _n \ne \emptyset,~ m\ne n$, one has (24) holds.

\begin{align} &\mbox{E}\left\{ {V_m (\pmb a_{k+1},\pmb \xi (k+1))} \right\}-V_m (\pmb a_k,\pmb \xi(k))-\beta V_m (\pmb a_k,\pmb \xi (k))= \notag\\ &\quad \mbox{E}\left\{ {\pmb \xi ^{\rm T}(k+1)P_m (\pmb a_{k+1} )\pmb \xi (k+1)} \right\}-\pmb \xi^{\rm T}(k)(1+\beta )P_m (\pmb a_k )\pmb \xi (k) =\pmb \xi ^{\rm T}(k)\Lambda _{2,m} (\pmb a_k ,\pmb a_{k+1} )\pmb \xi (k), \end{align}

(24)

where $\begin{array}{l} \Lambda _{2,m} (\pmb a_k ,\pmb a_{k+1} )=\hat {A}_n^{\rm T} (\pmb a_{\bar {k}} )P_m (\pmb a_{k+1} )\hat {A}_n (\pmb a_{\bar {k}} ) +\vartheta ^2\hat {A}_{1,n}^{\rm T} (\pmb a_{\bar {k}} )P_m (\pmb a_{k+1} )\hat{A}_{1,n} (\pmb a_{\bar {k}} )-(1+\beta )P_m (\pmb a_k ) \\ \end{array}$.

As $P_i \ge 0,~i\in \Omega _m,~\forall m\in \Gamma $,(25) and (26) will be obtained by performing a congruence transformation to (18) and (19) by $\mbox{diag}\left\{ {P_j^{-\rm T},P_j^{-\rm T},I,I,I} \right\}$ and the Schur complement.

\begin{align} &\Theta _{i,j,l} =\left[\!\!{{\begin{array}{*{20}c} {\hat {A}_{il} } & {B_{d,i} } \\ {\vartheta \hat {A}_{1,il} } & 0 \\ {C_i } & {D_{d,i} } \\ \end{array} }} \!\!\!\right]^{\rm T}\!\!\left[\!\!\!{{\begin{array}{*{20}c} {P_j } & 0 & 0 \\ 0 & {P_j } & 0 \\ 0 & 0 & I \\ \end{array} }} \!\!\!\right]\!\!\left[\!\!\! {{\begin{array}{*{20}c} {\hat {A}_{il} } & {B_{d,i} } \\ {\vartheta \hat {A}_{1,il} } & 0 \\ {C_i } & {D_{d,i} } \\ \end{array} }} \!\!\!\right] -\left[\!\!\!{{\begin{array}{*{20}c} {(1-\lambda )P_i } & 0 \\ 0 & {\gamma _m^2 I} \\ \end{array} }}\!\!\! \right]<0,\notag\\ &\qquad \forall i,j,l\in \Omega _m. \\ \end{align}

(25)

\begin{align} &\bar {\Theta }_{i,j,l} =\left[\!\!\! {{\begin{array}{*{20}c} {\hat {A}_{il} } & {B_{d,i} } \\ {\vartheta \hat {A}_{1,il} } & 0 \\ {C_i } & {D_{d,i} } \\ \end{array} }}\!\!\! \right]^{\rm T}\!\!\!\left[\!\!\! {{\begin{array}{*{20}c} {P_j } & 0 & 0 \\ 0 & {P_j } & 0 \\ 0 & 0 & I \\ \end{array} }}\!\!\! \right]\!\!\!\left[\!\!\! {{\begin{array}{*{20}c} {\hat {A}_{il} } & {B_{d,i} } \\ {\vartheta \hat {A}_{1,il} } & 0 \\ {C_i } & {D_{d,i} } \\ \end{array} }} \!\!\!\right] -\left[\!\!\! {{\begin{array}{*{20}c} {(1+\beta )P_i } & 0 \\ 0 & {\gamma _m^2 I} \\ \end{array} }}\!\!\! \right]<0,\notag\\ &\qquad \forall i,j\in \Omega _m ,~~l\in \Omega _n ,~~\Omega _m \cap\Omega _n \ne \emptyset,~~ m\ne n.\\ \end{align}

(26)

Noticing the (1,1) block of $\Theta _{i,j,l} $ and $\bar {\Theta}_{i,j,l} $ can imply

\begin{align} \begin{array}{l} \Lambda _{i,j,l} =\hat {A}_{il}^{\rm T} P_j \hat {A}_{il} +\vartheta^2 \hat {A}_{1,il}^{\rm T} P_j \hat {A}_{1,il} -(1-\lambda )P_i <0,\\ \hat {\Lambda }_{i,j,l} =\hat {A}_{il}^{\rm T} P_j \hat {A}_{il} +\vartheta ^2\hat {A}_{1,il}^{\rm T} P_j \hat {A}_{1,il} -(1+\beta)P_i <0,\\ \end{array} \end{align}

(27)

we can have^[38]

\begin{align} \begin{array}{l} \Lambda _{1,m} (\pmb a_k ,\pmb a_{k+1} )\!\!=\!\!\sum\limits_{l\in \Omega _m } {a_{l,\bar{k}} } \sum\limits_{i\in \Omega _m } {a_{i,k} } \sum\limits_{j\in \Omega _m} {a_{j,k+1} } \left( {\Lambda _{i,j,l} } \right)\!\!<\!\!0,\\ \Lambda _{2,m} (\pmb a_k ,\pmb a_{k+1} )\!\!=\!\!\sum\limits_{l\in \Omega _n } {a_{l,\bar{k}} } \sum\limits_{i\in \Omega _m } {a_{i,k} } \sum\limits_{j\in\Omega _m} {a_{j,k+1} } \left( {\hat {\Lambda }_{i,j,l} } \right)\!\!<\!\!0. \\ \end{array} \end{align}

(28)

Clearly,(29) holds according to (23),(24) and (28).

\begin{align} \mbox{E}\left( {\Delta V_m (\pmb \xi (k))} \right)\le \left\{ {\begin{array}{l}-\lambda V_m (\pmb \xi (k)), \;\;\;\forall k\in \left[{k_v +\delta _m ,k_{v+1} }\right),\\ \beta V_m (\pmb \xi (k)), \;\;\;\;\;\;\forall k\in \left[{k_v ,k_v +\delta_m } \right). \\ \end{array}} \right. \end{align}

(29)

Combining the conditions in (20) and (29),one can get that system (9) is GUAS-M according to Lemma 1.

Finally,we prove that the weighted $l_2\mbox{-}$gain performance index (10) holds. For any non-zero $\pmb w(k)\in L_2 \left[ {0,\infty }\right),~~ \forall k\in \left[{k_v +\delta _m ,k_{v+1} } \right)$,and any $\sigma (k)=m\in \Gamma $,define $\pmb \zeta (k)=\left( {{\begin{array}{*{20}c} {\pmb \xi ^{\rm T}(k)},& {\pmb d^{\rm T}(k)} \\ \end{array} }} \right)^{\rm T}$,then one has (30) holds.

\begin{align} &\mbox{E}\left\{ {V_m (\pmb a_{k+1},\pmb \xi (k+1))} \right\}-V_m (\pmb a_k,\pmb \xi(k))+\lambda V_m (\pmb a_k,\pmb \xi (k)) +\mbox{E}\left\{ {\pmb y^{\rm T}(k)\pmb y(k)} \right\}-\gamma _m^2 \pmb d^{\rm T}(k)\pmb d(k)=\notag\\ &\quad \mbox{E}\left\{ {\pmb \xi ^{\rm T}(k+1)P_m (\pmb a_{k+1} )\pmb \xi (k+1)} \right\}-\pmb \xi^{\rm T}(k)(1-\lambda )P_m (\pmb a_k )\pmb \xi (k)-\gamma _m^2 \pmb d^{\rm T}(k)\pmb d(k)+\notag\\ &\quad \mbox{E}\left\{ {\left[{{\begin{array}{*{20}c} {\pmb \xi (k)}\\ {\pmb w(k)} \\ \end{array} }} \right]^{\rm T}\!\!\left[{{\begin{array}{*{20}c} {\bar {C}_m^{\rm T} (\pmb a_{\bar {k}} )} \\ {\bar {D}_m^{\rm T} (\pmb a_{\bar {k}} )} \\ \end{array} }} \right]\!\!} \right.\left. {\left[{{\begin{array}{*{20}c} {\bar {C}_m^{\rm T} (\pmb a_{\bar {k}} )} \\ {\bar {D}_m^{\rm T} (\pmb a_{\bar {k}} )} \\ \end{array} }} \right]^{\rm T}\!\!\left[{{\begin{array}{*{20}c} {\pmb \xi (k)} \\ {\pmb w(k)} \\ \end{array} }} \right]} \right\} =\notag\\ &\quad \left[{{\begin{array}{*{20}c} {\pmb \xi (k)} \\ {\pmb w(k)} \\ \end{array} }} \right]^{\rm T}\Bigg\{ {\left[{{\begin{array}{*{20}c} {\bar {A}_m^{\rm T} (\pmb a_{\bar {k}} )} \\ {\bar {B}_m^{\rm T} (\pmb a_{\bar {k}} )} \\ \end{array} }} \right]P_m (\pmb a_{k+1} )\left[{{\begin{array}{*{20}c} {\bar {A}_m^{\rm T} (\pmb a_{\bar {k}} )} \\ {\bar {B}_m^{\rm T} (\pmb a_{\bar {k}} )} \\ \end{array} }} \right]^{\rm T}} \Bigg. +\vartheta ^2\left[{{\begin{array}{*{20}c} {\bar {A}_{1,m}^{\rm T} (\pmb a_{\bar {k}} )} \\ 0 \\ \end{array} }} \right]P_m (\pmb a_{k+1} )\left[{{\begin{array}{*{20}c} {\bar {A}_{1,m}^{\rm T} (\pmb a_{\bar {k}} )} \\ 0 \\ \end{array} }} \right]^{\rm T}+\notag\\ &\quad \vartheta ^2\left[{{\begin{array}{*{20}c} {\bar {A}_{1,m}^{\rm T} (\pmb a_{\bar {k}} )} \\ 0 \\ \end{array} }} \right]P_m (\pmb a_{k+1} )\left[{{\begin{array}{*{20}c} {\bar {A}_{1,m}^{\rm T} (\pmb a_{\bar {k}} )} \\ 0 \\ \end{array} }} \right]^{\rm T} +\left[{{\begin{array}{*{20}c} {\bar {C}_m^{\rm T} (\pmb a_{\bar {k}} )} \\ {\bar {D}_m^{\rm T} (\pmb a_{\bar {k}} )} \\ \end{array} }} \right]\left[{{\begin{array}{*{20}c} {\bar {C}_m^{\rm T} (\pmb a_{\bar {k}} )} \\ {\bar {D}_m^{\rm T} (\pmb a_{\bar {k}} )} \\ \end{array} }} \right]^{\rm T} -\Bigg. {\left[{{\begin{array}{*{20}c} {(1-\lambda )P_m (\pmb a_k )} & 0 \\ 0 & {\gamma _m^2 I} \\ \end{array} }} \right]} \Bigg\} \left[{{\begin{array}{*{20}c} {\pmb \xi (k)} \\ {\pmb w(k)} \\ \end{array} }} \right] =\notag\\ &\quad \pmb \zeta ^{\rm T}(k)\Gamma _{1,m} (\pmb a_k ,\pmb a_{k+1} ,\pmb a_{\bar {k}} )\pmb \zeta (k), \end{align}

(30)

where $\Gamma _{1,m} (\pmb a_k,\pmb a_{k+1},\pmb a_{\bar {k}} )=$

Similarly,for any non-zero $\pmb w(k)\in L_2 \left[{0,\infty } \right),~ \forall k\in \left[{k_v ,k_v +\delta _m } \right)$,and any $\left\{ {\sigma (k)=m,~~ \sigma (\bar {k})=n} \right\}\in \Gamma \times \Gamma,~~ \Omega _m \cap \Omega _n \ne \emptyset, ~~m\ne n$,one has (31) holds.

\begin{align} &\left[\!\! {{\begin{array}{*{20}c} {\bar {A}_m (\pmb a_{\bar {k}} )} & {\bar {B}_m (\pmb a_{\bar {k}} )}\!\!\\ {\vartheta \bar {A}_{1,m} (\pmb a_{\bar {k}} )} & 0 \\ {\bar {C}_m (\pmb a_{\bar {k}} )} & {\bar {D}_m (\pmb a_{\bar {k}} )}\\ \end{array} }}\!\! \right]^{\rm T}\!\!\left[\!\!{{\begin{array}{*{20}c} {P_m (\pmb a_{k+1} )} & 0 & 0 \\ 0 & {P_m (\pmb a_{k+1} )} & 0 \\ 0 & 0 & I \\ \end{array} }} \right] \left[\!\!{{\begin{array}{*{20}c} {\bar {A}_m (\pmb a_{\bar {k}} )} & {\bar {B}_m (\pmb a_{\bar {k}} )} \\ {\vartheta \bar {A}_{1,m} (\pmb a_{\bar {k}} )} & 0 \\ {\bar {C}_m (\pmb a_{\bar {k}} )} & {\bar {D}_m (\pmb a_{\bar {k}} )} \\ \end{array} }}\!\! \right]-\left[{{\begin{array}{*{20}c} {(1-\lambda )P_m (\pmb a_k )} & 0 \\ 0 & {\gamma _m^2 I} \\ \end{array} }} \right].\notag\\ &\mbox{E}\left\{ {V_m (\pmb a_{k+1},\pmb \xi (k+1))} \right\}-V_m (\pmb a_k,\pmb \xi(k))-\beta V_m (\pmb a_k,\pmb \xi (k)) +\mbox{E}\left\{ {\pmb y^{\rm T}(k)\pmb y(k)} \right\}-\gamma _m^2 \pmb d^{\rm T}(k)\pmb d(k)=\notag\\ &\quad \mbox{E}\left\{ {\pmb \xi ^{\rm T}(k+1)P_m (\pmb a_{k+1} )\pmb \xi (k+1)} \right\} -\pmb \xi^{\rm T}(k)(1+\beta )P_m (\pmb a_k )\pmb \xi (k) -\gamma _m^2 \pmb d^{\rm T}(k)\pmb d(k)+\notag\\ &\quad \mbox{E}\left\{ {\left[{{\begin{array}{*{20}c} {\pmb \xi (k)} \\ {\pmb w(k)} \\ \end{array} }} \right]^{\rm T}\left[\!\!{{\begin{array}{*{20}c} {\hat {C}_n^{\rm T} (\pmb a_{\bar {k}} )} \\ {\hat {D}_n^{\rm T} (\pmb a_{\bar {k}} )} \\ \end{array} }} \!\!\right]} \right.\left. {\left[\!\! {{\begin{array}{*{20}c} {\hat {C}_n^{\rm T} (\pmb a_{\bar {k}} )} \\ {\hat {D}_n^{\rm T} (\pmb a_{\bar {k}} )} \\ \end{array} }} \!\!\right]^{\rm T}\left[\!\!{{\begin{array}{*{20}c} {\pmb \xi (k)} \\ {\pmb w(k)} \\ \end{array} }} \!\!\right]} \right\} =\pmb \zeta ^{\rm T}(k)\Gamma _{2,mn} (\pmb a_k ,\pmb a_{k+1} ,\pmb a_{\bar {k}} )\pmb \zeta (k) , \end{align}

(31)

where $\Gamma _{2,mn} (\pmb a_k ,\pmb a_{k+1} ,\pmb a_{\bar {k}} )=$

\begin{align} \left[\!\! {{\begin{array}{*{20}c} {\hat {A}_n (\pmb a_{\bar {k}} )} & {\hat {B}_n (\pmb a_{\bar {k}} )}\\ {\vartheta \hat {A}_{1,n} (\pmb a_{\bar {k}} )} & 0 \\ {\hat {C}_n (\pmb a_{\bar {k}} )} & {\hat {D}_n (\pmb a_{\bar {k}} )}\\ \end{array} }}\!\! \right]^{\rm T}\left[\!\!{{\begin{array}{*{20}c} {P_m (\pmb a_{k+1} )} & 0 & 0 \\ 0 & {P_m (\pmb a_{k+1} )} & 0 \\ 0 & 0 & I \\ \end{array} }}\right] \left[{{\begin{array}{*{20}c} {\hat {A}_n (\pmb a_{\bar {k}} )} & {\hat {B}_n (\pmb a_{\bar {k}} )}\\ {\vartheta \hat {A}_{1,n} (\pmb a_{\bar {k}} )} & 0 \\ {\hat {C}_n (\pmb a_{\bar {k}} )} & {\hat {D}_n (\pmb a_{\bar {k}} )}\\ \end{array} }} \!\!\right]-\left[\!\! {{\begin{array}{*{20}c} {(1+\beta )P_m (\pmb a_k )} & 0 \\ 0 & {\gamma _m^2 I} \\ \end{array} }}\!\! \right].\notag \end{align}

Clearly,(25) and (26) can imply that (32) holds.

\begin{align} \begin{array}{l} \begin{aligned} &\Gamma _{1,m} (\pmb a_k ,\pmb a_{k+1} ,\pmb a_{\bar {k}} )=\\ &\quad \sum\limits_{l\in \Omega _m }{a_{l,\bar {k}} } \sum\limits_{i\in \Omega _m } {a_{i,k} } \sum\limits_{j\in \Omega _m } {a_{j,k+1} } \left( {\Theta _{i,j,l} } \right)<0,\\ &\Gamma _{2,mn} (\pmb a_k ,\pmb a_{k+1} ,\pmb a_{\bar {k}} )=\\ &\quad \sum\limits_{l\in \Omega _n }{a_{l,\bar {k}} } \sum\limits_{i\in \Omega _m } {a_{i,k} } \sum\limits_{j\in \Omega _m } {a_{j,k+1} } \left( {\bar {\Theta }_{i,j,l} } \right)<0. \\ \end{aligned} \end{array} \end{align}

(32)

Thus,we can get

\begin{align} \begin{array}{l} \mbox{E}\left( {\Delta V_m (\pmb \xi (k))} \right)\le\\ \quad \left\{\!\!{\begin{array}{l} -\lambda V_m (\pmb \xi (k))+\mbox{E}\left\{ {\pmb y^{\rm T}(k)\pmb y(k)} \right\}-\gamma_m^2 \pmb d^{\rm T}(k)\pmb d(k),\\ \quad \forall k\in \left[{k_v +\delta _m ,k_{v+1} } \right),\\ \beta V_m (\pmb \xi (k))+\mbox{E}\left\{ {\pmb y^{\rm T}(k)\pmb y(k)} \right\}-\gamma_m^2 \pmb d^{\rm T}(k)\pmb d(k),\\ \quad \forall k\in \left[{k_v ,k_v +\delta _m } \right). \\ \end{array}} \right. \end{array} \end{align}

(33)

Combining (20),(33) and Lemma 2,it can be derived that system (9) is GUAS-M for any switching signal with average dell time (20). As a result of $\mu =1$ and $N_0 =0$,the weighted $l_2\mbox{-}$gain performance index in (10) is degraded to be $\gamma =\max \{\gamma _m \},~~\forall m\in \Gamma $.

B. $H_\infty $ Controller Design

Based on Theorem 1,the sufficient existing conditions of the state feedback $H_\infty $ controller in (8) are derived.

Theorem 2. Given scalars $0<\lambda <1,~~\beta \ge 0$ and $\gamma _m >0,~~ \forall m\in \Gamma $,let $\vartheta =\sqrt {\rho(1-\rho )} $,if there exist matrices $S_i =\left[ {{\begin{array}{*{20}c} {\bar {S}_i } & 0 \\ 0 & {\bar {S}_i } \\ \end{array} }} \right]>0$ and $U_i,$ $i\in \Omega _m,$ $\forall m\in \Gamma $, such that the LMIs hold for $\forall \left( {m,n} \right)\in \Gamma \times \Gamma $,i.e.,

\begin{align} & \left[\!\!\!\! {{\begin{array}{*{20}c} {-S_j } & 0 & 0 & {\bar {\phi }_{14} } & {B_{d,i} } \\ \ast & {-S_j } & 0 & {\bar {\phi }_{24} } & 0 \\ \ast & \ast & {-I} & {C_i S_l } & {D_{d,i} } \\ \ast & \ast & \ast & {-(1-\lambda )(S_i -S_l-S_l^{\rm T} )} & 0 \\ \ast & \ast & \ast & \ast & {-\gamma _m^2 I} \\ \end{array} }} \!\!\!\!\right]<0,\notag\\ &\qquad \qquad \qquad \qquad \forall i,j,l\in \Omega _m, \end{align}

(34)

\begin{align} & \left[\!\!\!\! {{\begin{array}{*{20}c} {-S_j } & 0 & 0 & {\bar {\phi }_{14} } & {B_{d,i} } \\ \ast & {-S_j } & 0 & {\bar {\phi }_{24} } & 0 \\ \ast & \ast & {-I} & {C_i S_l } & {D_{d,i} } \\ \ast & \ast & \ast & {-(1+\beta )(S_i -S_l-S_l^{\rm T} )} & 0 \\ \ast & \ast & \ast & \ast & {-\gamma _m^2 I} \\ \end{array} }} \!\!\!\!\right]<0,\notag\\ &\qquad \quad \forall i,j\in \Omega _m,~~l\in \Omega _n ,~~\Omega _m \cap \Omega _n \ne\emptyset,~~m\ne n. \end{align}

(35)

Then under the asynchronous delay $\delta _m $,system (9) is GUAS-M for any switching signals with $\tau _a $ satisfying (20), and has a weighted $l_2\mbox{-}$gain performance index $\gamma=\max \{\gamma _m \},~~\forall m\in \Gamma $. Moreover,if a feasible solution exists,the desired controller gains in (8) are given by

\begin{align} K_i =U_i \bar {S}_i^{-1} ,~~i\in \Omega _m ,~~\forall m,\in\Gamma, \end{align}

(36)

where

\[ \begin{array}{l} \bar {\phi }_{14} =\left[{{\begin{array}{*{20}c} {A_i \bar {S}_l +\rho B_i U_l } & {(1-\rho )B_i U_l } \\ {\rho \bar {S}_l } & {(1-\rho )\bar {S}_l } \\ \end{array} }} \right],\\ \bar {\phi }_{24} =\vartheta \left[{{\begin{array}{*{20}c} {B_i U_l } & {-B_i U_l } \\ {\bar {S}_l } & {-\bar {S}_l } \\ \end{array} }} \right]. \\ \end{array} \]

Proof. Suppose that there exist matrices $S_i $ and $U_i,~~i\in \Omega _m ,~~\forall m\in \Gamma $ satisfying (34) and (35). Now suppose matrices $P_i $ in Theorem 1 have the form of $P_i:= \left[{{\begin{array}{*{20}c} {\bar {P}_i } & 0 \\ 0 & {\bar {P}_i } \\ \end{array} }} \right]$,let $S_i =\left[{{\begin{array}{*{20}c} {\bar {S}_i } & 0 \\ 0 & {\bar {S}_i } \\ \end{array} }} \right],~~ \bar {S}_i := \bar {P}_i^{-1} $, and define $U_i:= K_i \bar {S}_i,~~ i\in \Omega _m,~~ \forall m\in \Gamma $. Considering the fact that $\left( {S_i -S_l } \right)^{\rm T}S_l^{-1} \left( {S_i -S_l } \right)\ge 0$,we have $S_i -S_l -S_l^{\rm T} \ge -S_l^{\rm T} S_i^{-1} S_l $. Then replace the (4,4) block of (34) and (35) by $-S_l^{\rm T} S_i^{-1} S_l $,and do a congruence transformation via $\mbox{diag}\left\{ {S_j ^{-{\rm T}},S_j ^{-{\rm T}},I,S_l ^{-{\rm T}},I} \right\}$, which will lead to (18) and (19).

Meanwhile,the switched parameter-dependent controller gains are given by (36) according to the definition that $U_i := K_i \bar {S}_i,~i\in \Omega _m ,~~\forall m\in \Gamma $.

Remark 5. Referring to Theorem 2,a similar conclusion can be obtained for the input-to-state stability of the homologous systems which have reference inputs. This corollary can be directly obtained from Lemma 1 in [18] which will be validated later in the numerical example section.

Ⅴ. NUMBERICAL EXAMPLE

In this section,a numerical example is provided to illustrate the effectiveness of the proposed method.

Suppose that the sampling period is $T=0.025$,s,then corresponding system matrices $A_i ,B_i $ in (1) can be obtained with the aid of the continuous-time system matrix parameters in [34]. Moreover, other system matrices are given below,where $i\in \Omega _m ,~~\forall m\in \Gamma $:

\[ C_i =\left[{{\begin{array}{*{20}c} 1 & 0 \\ \end{array} }} \right],~~B_{d,i} =\left[{{\begin{array}{*{20}c} {0.01} & {0.01} \\ \end{array} }} \right]^{\rm T},~~D_{d,i} =0.05. \]

The external disturbance $\pmb d$ is supposed to be a random sequence uniformly distributed over $\left[{-0.1,0.1} \right]$. Moreover,it is assumed that $\mbox{E\{}\theta (k)\mbox{\}}=\rho =0.95$,which means that the probability of the measured system state $\pmb x$ being successfully transmitted to the controller is 0.95,and Fig. 2 shows the actual response of $\theta (k)$.

The performance of the proposed switched parameter-dependent controller for the full-envelope HiMAT vehicle is validated by a large-scale flight trajectory,i.e.,2-4-5-7-8-9-12-14-15-17, which is depicted in Fig. 1. The variation of altitude and Mach number are depicted in Fig. 3.

The flight envelope can be partitioned into three regions which are given in (4),and the flight trajectory includes two switchings between the three polytopic subsystems. First switching occurs when the HiMAT vehicle enters characteristic operating Point 7,and the second switching occurs when the HiMAT vehicle enters characteristic operating Point 12 which can give the fact that the average dwell time $\tau_a =1\,600\,(40\,{\rm s})$ from Definition 1. Supposed that the maximal lag $\delta _m =5$,it can be calculated that $\tau_a^\ast =5\,(0.1250\,{\rm s})$ which implies the constraint condition in (20) can be easily satisfied.

To analyze the input-to-state stability^[18],the controller in (6) is modified into the following form:

\begin{align} \pmb u(k)=K_{\sigma (k)} (\pmb a_k )\left( {\pmb z(k)-\pmb r(k)} \right), \end{align}

(37)

where $\pmb r(k)=[\alpha_c~~ q_c]^{\rm T}$ is the command input, including angle of attack command and pitch rate command. By setting $\gamma_m $ as the optimization variable at the same time,the YALMIP toolbox in Matlab is used to solve the optimization problem in Theorem 2,where $\lambda =0.01$ and $\beta =0.01$ are given. Then,we can get $\gamma ^\ast =\mbox{1.9237}$,and the controller gains can be obtained,which are shown as follows:

$$ \begin{array}{l} K_2 =\left[{{\begin{array}{*{20}c} {\mbox{3.0284}} & {\mbox{0.3215}} \\ {\mbox{2.0639}} & {\mbox{0.2191}} \\ {\mbox{-0.3264}} & {\mbox{-0.0347}} \\ \end{array} }} \right],K_4 =\left[{{\begin{array}{*{20}c} {\mbox{3.4061}} & {\mbox{0.3054}} \\ {\mbox{2.2868}} & {\mbox{0.2050}} \\ {\mbox{-0.0136}} & {\mbox{-0.0012}} \\ \end{array} }} \right],\\ K_5 =\left[{{\begin{array}{*{20}c} {\mbox{3.8160}} & {\mbox{0.4595}} \\ {\mbox{2.6204}} & {\mbox{0.3156}} \\ {\mbox{-0.3823}} & {\mbox{-0.0460}} \\ \end{array} }} \right],K_7 =\left[{{\begin{array}{*{20}c} {\mbox{4.7516}} & {\mbox{0.6742}} \\ {\mbox{3.2841}} & {\mbox{0.4660}} \\ {\mbox{-0.4047}} & {\mbox{-0.0574}} \\ \end{array} }} \right],\\ K_8 =\left[{{\begin{array}{*{20}c} {\mbox{2.4448}} & {\mbox{0.2869}} \\ {\mbox{1.5827}} & {\mbox{0.1857}} \\ {\mbox{0.2334}} & {\mbox{0.0274}} \\ \end{array} }} \right],K_9 =\left[{{\begin{array}{*{20}c} {\mbox{1.6363}} & {\mbox{0.1859}} \\ {\mbox{1.1127}} & {\mbox{0.1264}} \\ {\mbox{0.0504}} & {\mbox{0.0057}} \\ \end{array} }} \right],\\ K_{12} =\left[{{\begin{array}{*{20}c} {\mbox{0.8494}} & {\mbox{0.1017}} \\ {\mbox{0.5943}} & {\mbox{0.0712}} \\ {\mbox{-0.0202}} & {\mbox{-0.0024}} \\ \end{array} }}\right],K_{14} =\left[{{\begin{array}{*{20}c} {\mbox{0.9897}} & {\mbox{0.1341}} \\ {\mbox{0.7026}} & {\mbox{0.0952}} \\ {\mbox{-0.0271}} & {\mbox{-0.0037}} \\ \end{array} }} \right],\\ K_{15} =\left[{{\begin{array}{*{20}c} {\mbox{1.9732}} & {\mbox{0.2757}} \\ {\mbox{1.4165}} & {\mbox{0.1979}} \\ {\mbox{-0.1031}} & {\mbox{-0.0144}} \\ \end{array} }} \right],K_{17} =\left[{{\begin{array}{*{20}c} {\mbox{1.5329}} & {\mbox{0.2341}} \\ {\mbox{1.1137}} & {\mbox{0.1701}} \\ {\mbox{-0.0482}} & {\mbox{-0.0074}} \\ \end{array} }} \right]. \\ \end{array} $$

The controller gains for the characteristic operating points are given above,so the gain-scheduled subcontroller for each polytopic subsystem is interpolated by (7). In this work,the gain-scheduled subcontrollers are interpolated in triangular regions,namely,three characteristic operating points,which are the closest to the current one,are chosen.

For example,for the triangular regions which are constructed by the characteristic operating Points 2,4 and 5 in Region 1,the gain-scheduled subcontroller is interpolated by three gains $K_2 $,$K_4 $ and $K_5 $. Given the flight condition in this flight region with altitude and Mach number $\left( {h_a ,M_a } \right)$ at instant $k$,then the interpolated gain-scheduled subcontroller gain can be calculated as

\[ K_1 (\pmb a_k )=a_{2,k} K_2 +a_{4,k} K_4+a_{5,k} K_5, \]

and the weighted coefficient $\pmb a_k $ satisfies

\begin{equation*} \left\{ {\begin{array}{l} a_{2,k} +a_{4,k} +a_{5,k} =1,\\ a_{2,k} h_{2} +a_{4,k} h_{4} +a_{5,k} h_{5} =h_a,\\ a_{2,k} M_{a,2} +a_{4,k} M_{a,4} +a_{5,k} M_{a,5} =M_a,\\ \end{array}} \right. \end{equation*}

where $h_i,$ $M_{a,i} ,$ $i\in \Omega $ are altitude and Mach number for the $i$-th characteristic operating point.

Fig. 4 demonstrates the response of angle of attack,and pitch rate is shown in Fig. 5 where the pitch rate command is set to be zero. We can conclude from Figs. 4 and 5 that the tracking performance of angle of attack is acceptable in the full flight envelope,and the pitch rate response is satisfying.

The control surface deflections are shown in Figs. 6$\sim$ 8, which are achievable and practical.

From the simulation results above,it can be concluded that the HiMAT vehicle is stable in the full-envelope flight with robustness against the external disturbances,and can also restrain the effect of asynchronous switching to guarantee a desired performance.

Ⅵ. CONCLUSION

This study proposed an improved gain-scheduled controller design approach for the full-envelope UAVs with missing measurements and external disturbances. A locally overlapped switched polytopic system is established to describe flight dynamics within full flight envelope,and an asynchronous robust $H_\infty $ synthesis approach is utilized to obtain a prescribed disturbance attenuation level. The proposed approach could also be expanded to other applications where the locally overlapped switched polytopic system model could be established,and the missing measurements and asynchronous switching phenomena exist. Future works will analyze the impact of system states that cannot be measured,and the output feedback control problem will be concerned.