### 3.1 Stability analysis

In this section, two necessary lemmas are given firstly for the following non-switched positive system:

\{\begin{array}{c}\dot{x}(t)=Ax(t)+{A}_{d}x(t-d(t)),\hfill \\ x({t}_{0}+\theta )=\phi (\theta ),\phantom{\rule{1em}{0ex}}\theta \in [-\tau ,0],\hfill \end{array}

(6)

where *A* is a Metzler constant matrix and {A}_{d}\u2ab00 is a constant matrix; d(t) denotes the time-varying delay satisfying 0\le d(t)\le \tau, \dot{d}(t)\le d; \phi (\theta )\u2ab00.

Choose the co-positive type Lyapunov-Krasovskii functional candidate for system (6) as follows:

V(t,x(t))={V}_{1}(t,x(t))+{V}_{2}(t,x(t))+{V}_{3}(t,x(t)),

(7)

where

and v,\upsilon ,\vartheta \in {R}_{+}^{n}, \alpha >0.

For the sake of simplicity, V(t,x(t)) is written as V(t) in this paper.

**Lemma 2** *For a given positive constant* *α*, *if there exist vectors*v,\upsilon ,\vartheta \in {R}_{+}^{n}*and*\varsigma \in {R}^{n}*such that*

*where*

*with*{a}_{r} ({a}_{dr}) *represents the* *rth column vector of matrix* *A* ({A}_{d}), *and*v={[{v}_{1},{v}_{2},\dots ,{v}_{n}]}^{T}, \upsilon ={[{\upsilon}_{1},{\upsilon}_{2},\dots ,{\upsilon}_{n}]}^{T}, \vartheta ={[{\vartheta}_{1},{\vartheta}_{2},\dots ,{\vartheta}_{n}]}^{T}, \varsigma ={[{\varsigma}_{1},{\varsigma}_{2},\dots ,{\varsigma}_{n}]}^{T}, {v}_{r}({\upsilon}_{r},{\vartheta}_{r},{\varsigma}_{r})*represents the* *rth element of the vector*v(\upsilon ,\vartheta ,\varsigma ). *Then along the trajectory of system* (6), *we have*

V(t)\le {e}^{-\alpha (t-{t}_{0})}V({t}_{\mathrm{0}}).

*Proof* Along the trajectory of system (6), for the co-positive type Lyapunov-Krasovskii functional (7), we have

Using the Leibniz-Newton formula, one has

{\int}_{t-d(t)}^{t}\dot{x}(s)\phantom{\rule{0.2em}{0ex}}ds=x(t)-x(t-d(t)).

(11)

Considering that

{\int}_{t-d(t)}^{t}\dot{x}(s)\phantom{\rule{0.2em}{0ex}}ds={\int}_{t-d(t)}^{t}(Ax(s)+{A}_{d}x(s-d(s)))\phantom{\rule{0.2em}{0ex}}ds,

(12)

the following relationship can be obtained for any vector \varsigma \in {R}^{n}:

{(x(t)-x(t-d(t))-{\int}_{t-d(t)}^{t}(Ax(s)+{A}_{d}x(s-d(s)))\phantom{\rule{0.2em}{0ex}}ds)}^{T}\varsigma =0.

(13)

From (10) and (13), we have

\begin{array}{rcl}\dot{V}(t)+\alpha V(t)& \le & {x}^{T}(t)({A}^{T}v+\alpha v+\upsilon +\tau \vartheta )+\varsigma \\ +{x}^{T}(t-d(t))({A}_{d}^{T}v-(1-d){e}^{-\alpha \tau}\upsilon -\varsigma )\\ -{\int}_{t-d(t)}^{t}{\left[\begin{array}{c}x(s)\\ x(s-d(s))\end{array}\right]}^{T}(\left[\begin{array}{c}{A}^{T}\varsigma \\ {A}_{d}^{T}\varsigma \end{array}\right]+\left[\begin{array}{c}{e}^{-\alpha \tau}\vartheta \\ 0\end{array}\right])\phantom{\rule{0.2em}{0ex}}ds.\end{array}

(14)

One can obtain from (8) and (9) that

It follows that

\dot{V}(t)\le -\alpha V(t).

Then, along the trajectory of system (6), we have

V(t)\le {e}^{-\alpha (t-{t}_{0})}V({t}_{\mathrm{0}}).

□

**Lemma 3** *Consider system* (6), *for a given positive constant* *β*, *if there exist vectors*v,\upsilon ,\vartheta \in {R}_{+}^{n}*and*\varsigma \in {R}^{n}*such that*

*where*

{a}_{r} ({a}_{dr}) *represents the* *rth column vector of matrix* *A* ({A}_{d}); *and*v={[{v}_{1},{v}_{2},\dots ,{v}_{n}]}^{T}, \upsilon ={[{\upsilon}_{1},{\upsilon}_{2},\dots ,{\upsilon}_{n}]}^{T}, \vartheta ={[{\vartheta}_{1},{\vartheta}_{2},\dots ,{\vartheta}_{n}]}^{T}, \varsigma ={[{\varsigma}_{1},{\varsigma}_{2},\dots ,{\varsigma}_{n}]}^{T}. *Then*, *along the trajectory of system* (6), *we have*

V(t)\le {e}^{\beta (t-{t}_{0})}V({t}_{\mathrm{0}}).

*Proof* Choose the following co-positive type Lyapunov-Krasovskii functional candidate for system (6):

V(t)={V}_{1}(t)+{V}_{2}(t)+{V}_{3}(t),

where

and v,\upsilon ,\vartheta \in {R}_{+}^{n}, \beta >0.

The rest of the proof of this lemma is similar to that of Lemma 2, and thus is omitted here. □

Now we are in a position to provide the stability conditions for the following positive switched system:

\{\begin{array}{c}\dot{x}(t)={A}_{\sigma (t)}x(t)+{A}_{d\sigma (t)}x(t-d(t)),\hfill \\ x({t}_{0}+\theta )=\phi (\theta ),\phantom{\rule{1em}{0ex}}\theta \in [-\tau ,0],\hfill \end{array}

(21)

where {A}_{i}, i\in \underline{N}, are Metzler constant matrices and {A}_{di}\u2ab00, i\in \underline{N}, are constant matrices; d(t) denotes the time-varying delay satisfying 0\le d(t)\le \tau, \dot{d}(t)\le d; \phi (\theta )\u2ab00.

Let *Q* denote the index set of all stable subsystems, which is a nonempty subset of \underline{N}, and \overline{Q} denote the index set of all unstable subsystems. Let {T}^{+}({t}_{0},t) denote the total activation time of the unstable subsystems during [{t}_{0},t), let {T}^{-}({t}_{0},t) denote the total activation time of the stable subsystems during [{t}_{0},t), then we have the following result.

**Theorem 1** *Given positive constants* *α* *and* *β*, *if there exist*{v}_{i},{\upsilon}_{i},{\vartheta}_{i}\in {R}_{+}^{n}*and*{\varsigma}_{i}\in {R}^{n}, i\in \underline{N}, *such that*

*where*

{a}_{ir} ({a}_{dir}) *represents the* *rth column vector of matrix*{A}_{i} ({A}_{di}), i\in \underline{N}; *and*{v}_{i}={[{v}_{i1},{v}_{i2},\dots ,{v}_{in}]}^{T}, {\upsilon}_{i}={[{\upsilon}_{i1},{\upsilon}_{i2},\dots ,{\upsilon}_{in}]}^{T}, {\vartheta}_{i}={[{\vartheta}_{i1},{\vartheta}_{i2},\dots ,{\vartheta}_{in}]}^{T}, {\varsigma}_{i}={[{\varsigma}_{i1},{\varsigma}_{i2},\dots ,{\varsigma}_{in}]}^{T}. *Then system* (21) *is exponentially stable for any switching signals*\sigma (t)*with the average dwell time*

\underset{t>{t}_{0}}{inf}\frac{{T}^{-}({t}_{0},t)}{{T}^{+}({t}_{0},t)}\ge \frac{\beta +\lambda}{\alpha -\lambda},\phantom{\rule{2em}{0ex}}{T}_{a}>{T}_{a}^{\ast}=\frac{ln(\mu \eta )}{\lambda},

(26)

*where*\eta ={e}^{(\alpha +\beta )\tau}, 0<\lambda <\alpha*and*\mu \ge 1*satisfies*

{v}_{i}\u2aaf\mu {v}_{j},\phantom{\rule{2em}{0ex}}{\upsilon}_{i}\u2aaf\mu {\upsilon}_{j},\phantom{\rule{2em}{0ex}}{\vartheta}_{i}\u2aaf\mu {\vartheta}_{j},\phantom{\rule{1em}{0ex}}\mathrm{\forall}(i,j)\in \underline{N}\times \underline{N}.

(27)

*Proof*

Choose the following piecewise co-positive type Lyapunov-Krasovskii functional candidate:

V(t)={V}_{\sigma (t)}(t).

Let {t}_{1}<\cdots <{t}_{l} denote the switching instants of \sigma (t) over the interval [{t}_{0},t). By Lemmas 2 and 3, one can obtain from (22)-(25) that

{V}_{\sigma (t)}(t)\le \{\begin{array}{cc}{e}^{-\alpha (t-{t}_{k})}{V}_{\sigma ({t}_{k})}({t}_{k})\hfill & \text{if}\sigma (t)\in Q,t\in [{t}_{k},{t}_{k+1}),\hfill \\ {e}^{\beta (t-{t}_{k})}{V}_{\sigma ({t}_{k})}({t}_{k})\hfill & \text{if}\sigma (t)\in \overline{Q},t\in [{t}_{k},{t}_{k+1}).\hfill \end{array}

(28)

From (27) and the co-positive type Lyapunov-Krasovskii functional, at the switching instants {t}_{k}, k=1,2,\dots ,l, it is obtained that

{V}_{i}({t}_{k})\le \mu \eta {V}_{j}\left({t}_{k}^{-}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall}(i,j)\in \underline{N}\times \underline{N},

(29)

where \eta ={e}^{(\alpha +\beta )\tau}.

By (26), (28), (29) and Definition 4, for t\in [{t}_{l},{t}_{l+1}), it is not hard to get

\begin{array}{rcl}{V}_{\sigma (t)}(t)& \le & {e}^{-\alpha {T}^{-}({t}_{l},t)+\beta {T}^{+}({t}_{l},t)}{V}_{\sigma ({t}_{l})}({t}_{l})\\ \le & \mu \eta {e}^{-\alpha {T}^{-}({t}_{l},t)+\beta {T}^{+}({t}_{l},t)}{V}_{\sigma ({t}_{l}^{-})}\left({t}_{l}^{-}\right)\\ \le & \mu \eta {e}^{-\alpha {T}^{-}({t}_{l-1},t)+\beta {T}^{+}({t}_{l-1},t)}{V}_{\sigma ({t}_{l-1})}({t}_{l-1})\\ \le & \cdots \\ \le & {(\mu \eta )}^{{N}_{\sigma}({t}_{0},t)}{e}^{-\alpha {T}^{-}({t}_{0},t)+\beta {T}^{+}({t}_{0},t)}{V}_{\sigma ({t}_{0})}({t}_{0})\\ \le & {e}^{-\alpha {T}^{-}({t}_{0},t)+\beta {T}^{+}({t}_{0},t)}{e}^{(t-{t}_{0})ln(\mu \eta )/{T}_{a}}{V}_{\sigma ({t}_{0})}({t}_{0})\\ \le & {e}^{-(\lambda -ln(\mu \eta )/{T}_{a})(t-{t}_{0})}{V}_{\sigma ({t}_{0})}({t}_{0}).\end{array}

(30)

Denoting {\epsilon}_{1}={min}_{(r,i)\in \underline{n}\times \underline{N}}\{{v}_{ir}\} yields

{V}_{\sigma (t)}(t)\ge {\epsilon}_{1}\parallel x(t)\parallel .

(31)

Denote

and a=max\{{a}_{1},{a}_{2}\}, then

{V}_{\sigma ({t}_{0})}({t}_{0})\le a\underset{{t}_{0}-\tau \le \delta \le {t}_{0}}{sup}\parallel x(\delta )\parallel .

(32)

From (30)-(32), we obtain

Thus, by denoting \kappa =a/{\epsilon}_{1}, \epsilon =\lambda -\frac{ln(\mu \eta )}{{T}_{a}}>0, it can be seen from (33) that \parallel x(t)\parallel \le \kappa {\parallel {x}_{{t}_{0}}\parallel}_{c}{e}^{-\epsilon (t-{t}_{0})}, \mathrm{\forall}t\ge {t}_{0}, where {\parallel {x}_{{t}_{0}}\parallel}_{c}={sup}_{{t}_{0}-\tau \le \delta \le {t}_{0}}\parallel x(\delta )\parallel. Therefore, we can conclude that system (21) is exponentially stable for any switching signal with average dwell time (26). □

**Remark 2** In Theorem 1, sufficient conditions for the existence of the exponential stability for positive switched system (21) with both stable and unstable subsystems are presented via the average dwell time approach. It is shown by (26) that when the average dwell time is sufficiently large and the total activation time of unstable subsystems is relatively small compared with that of stable subsystems, the stability of the system can be guaranteed.

### 3.2 {L}_{1}-gain property analysis

**Theorem 2** *Given positive constants* *α*, *β* *and* *γ*, *if there exist*{v}_{i},{\upsilon}_{i},{\vartheta}_{i}\in {R}_{+}^{n}*and*{\varsigma}_{i}\in {R}^{n}, i\in \underline{N}, *such that*

*where*

{e}_{ir}*represents the* *rth column vector of matrix*{E}_{i}, i\in \underline{N}. *Then system* (1) *is exponentially stable and has*{L}_{1}-*gain performance index* *γ* *for any switching signal*\sigma (t)*with the average dwell time* (26).

*Proof* It is easy to get that (22)-(25) can be deduced from (34)-(37). According to Theorem 1, system (1) with w(t)=0 is exponentially stable. In the sequel, we will prove that the {L}_{1}-gain performance of system (1) is guaranteed.

Let {t}_{1}<\cdots <{t}_{l} denote the switching instants of \sigma (t) over the interval [{t}_{0},t). Following the proof line of Theorem 1, one can obtain from (34)-(37) that

{V}_{\sigma (t)}(t)\le \{\begin{array}{cc}{e}^{-\alpha (t-{t}_{k})}{V}_{\sigma ({t}_{k})}({t}_{k})-{\int}_{{t}_{k}}^{t}{e}^{-\alpha (t-s)}\mathrm{\Gamma}(s)\phantom{\rule{0.2em}{0ex}}ds\hfill & \text{if}\sigma (t)\in Q,t\in [{t}_{k},{t}_{k+1}),\hfill \\ {e}^{\beta (t-{t}_{k})}{V}_{\sigma ({t}_{k})}({t}_{k})-{\int}_{{t}_{k}}^{t}{e}^{\beta (t-s)}\mathrm{\Gamma}(s)\phantom{\rule{0.2em}{0ex}}ds\hfill & \text{if}\sigma (t)\in \overline{Q},t\in [{t}_{k},{t}_{k+1}),\hfill \end{array}

(38)

where \mathrm{\Gamma}(s)=\parallel z(s)\parallel -\gamma \parallel w(s)\parallel.

Then, for t\in [{t}_{l},{t}_{l+1}), we have

\begin{array}{rcl}{V}_{\sigma (t)}(t)& \le & {e}^{-\alpha {T}^{-}({t}_{l},t)+\beta {T}^{+}({t}_{l},t)}{V}_{\sigma ({t}_{l})}({t}_{l})-{\int}_{{t}_{l}}^{t}{e}^{-\alpha {T}^{-}(s,t)+\beta {T}^{+}(s,t)}\mathrm{\Gamma}(s)\phantom{\rule{0.2em}{0ex}}ds\\ \le & \mu \eta {e}^{-\alpha {T}^{-}({t}_{l},t)+\beta {T}^{+}({t}_{l},t)}{V}_{\sigma ({t}_{l}^{-})}\left({t}_{l}^{-}\right)-{\int}_{{t}_{l}}^{t}{e}^{-\alpha {T}^{-}(s,t)+\beta {T}^{+}(s,t)}\mathrm{\Gamma}(s)\phantom{\rule{0.2em}{0ex}}ds\\ \le & \mu \eta {e}^{-\alpha {T}^{-}({t}_{l-1},t)+\beta {T}^{+}({t}_{l-1},t)}{V}_{\sigma ({t}_{l-1})}({t}_{l-1})-{\int}_{{t}_{l}}^{t}{e}^{-\alpha {T}^{-}(s,t)+\beta {T}^{+}(s,t)}\mathrm{\Gamma}(s)\phantom{\rule{0.2em}{0ex}}ds\\ -\mu \eta {\int}_{{t}_{l-1}}^{{t}_{l}}{e}^{-\alpha {T}^{-}(s,{t}_{l-1})+\beta {T}^{+}(s,{t}_{l-1})}\mathrm{\Gamma}(s)\phantom{\rule{0.2em}{0ex}}ds\\ \le & \cdots \\ \le & {(\mu \eta )}^{{N}_{\sigma}({t}_{0},t)}{e}^{-\alpha {T}^{-}({t}_{0},t)+\beta {T}^{+}({t}_{0},t)}{V}_{\sigma ({t}_{0})}({t}_{0})-{\int}_{{t}_{0}}^{t}{(\mu \eta )}^{{N}_{\sigma}(s,t)}{e}^{-\alpha {T}^{-}(s,t)+\beta {T}^{+}(s,t)}\mathrm{\Gamma}(s)\phantom{\rule{0.2em}{0ex}}ds\\ \le & {e}^{-\alpha {T}^{-}({t}_{0},t)+\beta {T}^{+}({t}_{0},t)}{e}^{(t-{t}_{0})ln(\mu \eta )/{T}_{a}}{V}_{\sigma ({t}_{0})}({t}_{0})\\ -{\int}_{{t}_{0}}^{t}{(\mu \eta )}^{{N}_{\sigma}(s,t)}{e}^{-\alpha {T}^{-}(s,t)+\beta {T}^{+}(s,t)}\mathrm{\Gamma}(s)\phantom{\rule{0.2em}{0ex}}ds.\end{array}

(39)

Under the zero initial condition, we have {V}_{\sigma ({t}_{0})}({t}_{0})=0, then (39) becomes

0\le -{\int}_{{t}_{0}}^{t}{(\mu \eta )}^{{N}_{\sigma}(s,t)}{e}^{-\alpha {T}^{-}(s,t)+\beta {T}^{+}(s,t)}\mathrm{\Gamma}(s)\phantom{\rule{0.2em}{0ex}}ds.

From the condition (26), it is obvious that

-\alpha {T}^{-}({t}_{0},t)+\beta {T}^{+}({t}_{0},t)\le -\lambda (t-{t}_{0}),

then

{\int}_{{t}_{0}}^{t}{e}^{-\lambda (t-s)}{(\mu \eta )}^{{N}_{\sigma}(s,t)}\mathrm{\Gamma}(s)\phantom{\rule{0.2em}{0ex}}ds\le 0.

That is,

{\int}_{{t}_{0}}^{t}{e}^{-\lambda (t-s)}{(\mu \eta )}^{{N}_{\sigma}(s,t)}\parallel z(s)\parallel \phantom{\rule{0.2em}{0ex}}ds\le \gamma {\int}_{{t}_{0}}^{t}{e}^{-\lambda (t-s)}{(\mu \eta )}^{{N}_{\sigma}(s,t)}\parallel w(s)\parallel \phantom{\rule{0.2em}{0ex}}ds.

(40)

Multiplying both sides of (40) by {e}^{-{N}_{\sigma}({t}_{0},t)ln(\mu \eta )} yields

{\int}_{{t}_{0}}^{t}{e}^{-\lambda (t-s)}{e}^{-{N}_{\sigma}({t}_{0},s)ln(\mu \eta )}\parallel z(s)\parallel \phantom{\rule{0.2em}{0ex}}ds\le \gamma {\int}_{{t}_{0}}^{t}{e}^{-\lambda (t-s)}{e}^{-{N}_{\sigma}({t}_{0},s)ln(\mu \eta )}\parallel w(s)\parallel \phantom{\rule{0.2em}{0ex}}ds.

(41)

By Definition 4 and condition (26), one can obtain

{\int}_{{t}_{0}}^{t}{e}^{-\lambda (t-s)}{e}^{-\lambda (s-{t}_{0})}\parallel z(s)\parallel \phantom{\rule{0.2em}{0ex}}ds\le \gamma {\int}_{{t}_{0}}^{t}{e}^{-\lambda (t-s)}\parallel w(s)\parallel \phantom{\rule{0.2em}{0ex}}ds.

(42)

Integrating both sides of (42) from t={t}_{0} to ∞ leads to

{\int}_{{t}_{0}}^{\mathrm{\infty}}{e}^{-\lambda (t-{t}_{0})}\parallel z(s)\parallel \phantom{\rule{0.2em}{0ex}}ds\le \gamma {\int}_{{t}_{0}}^{\mathrm{\infty}}\parallel w(s)\parallel \phantom{\rule{0.2em}{0ex}}ds.

This means that system (1) achieves {L}_{1}-gain performance index *γ*.

The proof is completed. □

### 3.3 Reliable {L}_{1} control

In what follows, we design a state feedback controller for positive switched system (3) such that resulting closed-loop system (5) is exponentially stable with {L}_{1}-gain performance index *γ*.

**Theorem 3** *Consider system* (3), *for given positive constants* *α*, *β* *and* *γ*, *if there exist*{v}_{i},{\upsilon}_{i},{\vartheta}_{i}\in {R}_{+}^{n}*and*{\varsigma}_{i},{h}_{i}\in {R}^{n}, i\in \underline{N}, *such that*

*where*

{c}_{ir} ({d}_{ir}) *represents the* *rth column vector of matrix*{C}_{i} ({D}_{i}), i\in \underline{N}; {b}_{i\overline{M}r} ({b}_{iMr}) *represents the* *rth column vector of matrix*{B}_{i\overline{M}} ({B}_{i{M}_{i}}); {h}_{i}={[{h}_{i1},{h}_{i2},\dots ,{h}_{in}]}^{T}, {f}_{i}={[{f}_{i1},{f}_{i2},\dots ,{f}_{in}]}^{T}.

*Then*, *under the controller* (4), *resulting closed*-*loop system* (5) *is exponentially stable and has*{L}_{1}-*gain performance index* *γ* *for any switching signals*\sigma (t)*with the average dwell time *(26).

*Proof* Under the controller (4), the resulting closed-loop system can be written as (5).

Denote {\overline{A}}_{i}={A}_{i}+{B}_{i\overline{M}}{K}_{i\overline{M}}, {\overline{E}}_{i}=[{E}_{i}\phantom{\rule{0.25em}{0ex}}{B}_{iM}], {\overline{D}}_{i}=[{D}_{i}\phantom{\rule{0.25em}{0ex}}0] and {h}_{i}={K}_{i\overline{M}}^{T}{B}_{i\overline{M}}^{T}{v}_{i}, {f}_{i}={K}_{i\overline{M}}^{T}{B}_{i\overline{M}}^{T}{\varsigma}_{i}, then by Theorem 2, one can obtain from (43)-(46) that closed-loop system (5) is exponentially stable and has {L}_{1}-gain performance index *γ*. This completes the proof. □

We now present the following algorithm for the construction of the reliable {L}_{1} state feedback controller.

**Algorithm 1** Step 1. Input the matrices {A}_{i}, {A}_{di}, {B}_{i}, {C}_{i}, {D}_{i} and {E}_{i}, i\in \underline{N};

Step 2. By adjusting parameters *α* and *β*, we can find the solutions of {v}_{i}, {\upsilon}_{i}, {\vartheta}_{i}, {\varsigma}_{i}, {h}_{i} such that (43) and (45) hold;

Step 3. From {h}_{i}={K}_{i\overline{M}}^{T}{B}_{i\overline{M}}^{T}{v}_{i} and {h}_{i}={K}_{iM}^{T}{B}_{iM}^{T}{v}_{i}, one can obtain the gain matrices {K}_{i}={[{K}_{iM}^{T}\phantom{\rule{0.25em}{0ex}}{K}_{i\overline{M}}^{T}]}^{T}, and then substitute {K}_{i} into (44) and (46). If inequalities (44) and (46) hold and {\overline{A}}_{i}={A}_{i}+{B}_{{\overline{M}}_{i}}{K}_{{\overline{M}}_{i}} are Metzler matrices, then go to Step 4; otherwise, go back to Step 2;

Step 4. With {v}_{i}, {\upsilon}_{i}, {\vartheta}_{i}, the switching signal \sigma (t) can be obtained by (26) and (27);

Step 5. Construct the feedback controller (4), where {K}_{i} are gain matrices obtained in Step 3.