Discussion on “Stability, $l _{2}$-Gain and Asynchronous $H _{\infty }$ Control of Discrete-Time Switched Systems With Average Dwell Time”

This note addresses a defect occurring in , in which the desired $l _{2}$-gain should be replaced by weighted $l _{2}$-gain performance. The corrected results and proof are presented.

The recent article [1] provided a class of Lyapunov-like functions, which are allowed to increase during the running time of subsystems, to investigate the stability and $l _{2}$-gain problems for following discrete-time switched system:

$$\eqalignno{x(k + 1) =&\, {f_{\sigma} }(x(k),u(k)) &{\hbox{(1a)}} \cr y(k) =&\, {h_{\sigma} }(x(k)) &{\hbox{(1b)}}}$$ in which the switching signal $\sigma$ belongs to average dwell time (ADT) switching. Then the results were applied into studying the issue of asynchronous control. However, a defect occurs in Theorem 2 which played a key role in $l _{2}$-gain analysis and asynchronous $H _{\infty }$ controller design.

In [1, Theorem 2], sufficient conditions were presented to ensure the switched system globally uniformly asymptotically stable (GUAS) for any switching signal satisfying ADT condition below

$${\tau _{a}} > {\tau _{a}^{\ast}} = {{- \{{{\cal T}_{M}}[\ln (1 + \beta) - \ln \overline \alpha ] + \ln \mu \} } \over {\ln \overline \alpha}} \eqno{\hbox{(2)}}$$ and has an $l _{2}$-gain no greater than ${\gamma _{s}} = \max \{\sqrt {{\theta ^{{{\cal T}_{M}} - 1}}} {\gamma _{i}}\}$. However, although the GUAS can be established by ADT condition (2), the required $l _{2}$-gain performance with index ${\gamma _{s}} = \max \{\sqrt {{\theta ^{{{\cal T}_{M}} - 1}}} {\gamma _{i}}\}$ cannot be guaranteed. In fact, the $l _{2}$-gain problem for switched system under ADT switching has been considered as an open and unsolved problem so far [2]. Thus, as other reported results [3]–​[7], the $l _{2}$-gain performance should be replaced by weighted $l _{2}$-gain performance under the conditions in Theorem 2. Inspired by the guidelines in [3] and with the aid of introduced notations ${{\cal T}_{\uparrow} }(k - s)$ and ${{\cal T}_{\downarrow} }(k - s)$ in [1], which denote the length of union of the Lyapunov function increasing and decreasing intervals, respectively, the corrected result is presented by following theorem.

Theorem 1

Consider switched system (1) and let $0 < \alpha < 1$, $\beta \ge 0$ and ${\gamma _{i}} \ge 0$, $\forall i \in {\cal I}$ be given constants. Suppose that there exist positive definite ${{\cal C}^{1}}$ functions ${V_{\sigma (k)}}$: ${{\BBR }^{n}} \to {\BBR }$, $\sigma (k) \in {\cal I}$, with ${V_{\sigma ({k_{0}})}}({x_{{k_{0}}}}) \equiv 0$ such that $\forall (i,j) \in {\cal I} \times {\cal I}$, $i \ne j$, ${V_{i}}({x_{{k_{l}}}}) \le \mu {V_{j}}({x_{{k_{l}}}})$ and $\forall i \in {\cal I}$, denoting $\Gamma (k) = {y_{k}^{T}}{y_{k}} - {\gamma _{i}^{2}}{u_{k}^{T}}{u_{k}}$

$$\Delta {V_{i}}({x_{k}}) \le \cases{{- \alpha {V_{i}}(k) - \Gamma (k),\forall k \in {{\cal T}_{\downarrow} }({k_{l}},{k_{l + 1}})} \cr {\beta {V_{i}}(k) - \Gamma (k),\forall k \in {{\cal T}_{\uparrow} }({k_{l}},{k_{l + 1}})} } .\eqno{\hbox{(3)}}$$

Then switched system is GUAS for any switching signal satisfying (2) and has a weighted $l _{2}$-gain

$$\sum\limits_{s = {k_{0}}}^{\infty} {{{\overline \alpha }^{(s - {k_{0}})}}{y_{s}^{T}}{y_{s}}} < \sum\limits_{s = {k_{0}}}^{\infty} {{\gamma _{s}^{2}}{u_{s}^{T}}{u_{s}}}\eqno{\hbox{(4)}}$$ where $\overline \alpha = 1 - \alpha$, ${\gamma _{s}} = {\max _{i \in {\cal I}}}\{\sqrt {{\mu ^{{N_{0}}}}{\theta ^{{{\cal T}_{M}}{N_{0}}}}} {\gamma _{i}}\}, \theta \!\!=\!\! (1 \!+\! \beta)/(1 \!-\!\alpha)$ and ${{\cal T}_{M}} = {\max _{l \in {\BBN}}}{{\cal T}_{\uparrow} }({k_{l + 1}} - {k_{l}})$.

Proof

By (3) and choosing the Lyapunov-like function ${V_{\sigma (k)}}({x_{k}})$, we have that

$$\displaylines{{V_{\sigma (k)}}({x_{k}}) \le {\overline \alpha ^{(k - {k_{l}})}}{\theta ^{{{\cal T}_{\uparrow} }(k - {k_{l}})}}{V_{\sigma (k)}}({x_{{k_{l}}}}) \hfill\cr\hfill- \sum\limits_{s = {k_{l}}}^{k - 1} {{{\overline \alpha }^{k - s - 1}}{\theta ^{{{\cal T}_{\uparrow} }(k - s - 1)}}\Gamma (s)}}$$ where $\overline \alpha = 1 - \alpha$ and $\theta = {{(1 + \beta)} / {\overline \alpha}}$. Then, by similar guidelines of [3, Theorem 5], we get $$\displaylines{\sum\limits_{s = {k_{0}}}^{k - 1} {{\mu ^{{N_{\sigma} }(s,k)}}{{\overline \alpha }^{k - s - 1}}{\theta ^{{{\cal T}_{\uparrow} }(k - s - 1)}}{y_{s}^{T}}{y_{s}}} \hfill\cr \hfill \le \sum\limits_{s = {k_{0}}}^{k - 1} {{\mu ^{{N_{\sigma} }(s,k)}}{{\overline \alpha }^{k - s - 1}}{\theta ^{{{\cal T}_{\uparrow} }(k - s - 1)}}{\gamma _{\sigma (s)}^{2}}{u_{s}^{T}}{u_{s}}} .\quad{\hbox{(5)}}}$$

Multiplying both sides of (5) by ${\mu ^{- {N_\sigma }({k_0},k)}}{\theta ^{- {{\cal T}_ \uparrow }(k - {k_0} - 1)}}$ and further following the guidelines in [3], one has

$$\displaylines{\sum\limits_{s = {k_{0}}}^{k - 1} {{\mu ^{- {N_{\sigma} }({k_{0}},s)}}{\theta ^{- {{\cal T}_{M}}{N_{\sigma} }({k_{0}},s)}}{{\overline \alpha }^{k - s - 1}}{y_{s}^{T}}{y_{s}}} \le \sum\limits_{s = {k_{0}}}^{k - 1} {{{\overline \alpha }^{k - s - 1}}{\gamma _{\sigma (s)}^{2}}{u_{s}^{T}}{u_{s}}} .\hfill\cr\hfill{\hbox{(6)}}}$$

By definition of ADT, we see

$$\displaylines{{\mu ^{- {N_{\sigma} }({k_{0}},s)}}{\theta ^{- {{\cal T}_{M}}{N_{\sigma} }({k_{0}},s)}} \hfill\cr\hfill > {\mu ^{- {N_{0}}}}{\theta ^{- {{\cal T}_{M}}{N_{0}}}}{\mu ^{{{- (s - {k_{0}})}/ {{\tau _{a}}}}}}{\theta ^{{{- {{\cal T}_{M}}(s - {k_{0}})}/ {{\tau _{a}}}}}} .}$$

Then, we have the equation shown at the top of the page.

$$\eqalignno{\sum\limits_{k = {k_{0}}}^{\infty}& {\sum\limits_{s = {k_{0}}}^{k - 1} {{\mu ^{- {N_{0}}}}{\theta ^{- {{\cal T}_{M}}{N_{0}}}}{\mu ^{{{- (s - {k_{0}})} / {{\tau _{a}}}}}}{\theta ^{{{- {{\cal T}_{M}}(s - {k_{0}})} / {{\tau _{a}}}}}}{{\overline \alpha }^{k - s - 1}}{y_{s}^{T}}{y_{s}}} } \cr <&\, \sum\limits_{k = {k_{0}}}^{\infty} {\sum\limits_{s = {k_{0}}}^{k - 1} {{{\overline \alpha }^{k - s - 1}}{\gamma _{\sigma (s)}^{2}}{u_{s}^{T}}{u_{s}}} } \cr \Rightarrow&\, \sum\limits_{s = {k_{0}}}^{\infty} {\sum\limits_{k = s}^{\infty} {{\mu ^{- {N_{0}}}}{\theta ^{- {{\cal T}_{M}}{N_{0}}}}{\mu ^{{{- (s - {k_{0}})} / {{\tau _{a}}}}}}{\theta ^{{{- {{\cal T}_{M}}(s - {k_{0}})} / {{\tau _{a}}}}}}{{\overline \alpha }^{k - s - 1}}{y_{s}^{T}}{y_{s}}} } \cr <&\, \sum\limits_{s = {k_{0}}}^{\infty} {\sum\limits_{k = s}^{\infty} {{{\overline \alpha }^{k - s - 1}}{\gamma _{\sigma (s)}^{2}}{u_{s}^{T}}{u_{s}}} } \cr \Rightarrow&\, \sum\limits_{s = {k_{0}}}^{\infty} {{\mu ^{- {N_{0}}}}{\theta ^{- {{\cal T}_{M}}{N_{0}}}}{\mu ^{{{- s(s - {k_{0}})} / {{\tau _{a}}}}}}{\theta ^{{{- {{\cal T}_{M}}(s - {k_{0}})} / {{\tau _{a}}}}}}{y_{s}^{T}}{y_{s}} \sum\limits_{k = s}^{\infty} {{{\overline \alpha }^{k - s - 1}}} } \cr <&\, \sum\limits_{s = {k_{0}}}^{\infty} {{\gamma _{\sigma (s)}^{2}}{u_{s}^{T}}{u_{s}} \sum\limits_{k = s}^{\infty} {{{\overline \alpha }^{k - s - 1}}} } \cr \Rightarrow&\, \sum\limits_{s = {k_{0}}}^{\infty} {{\mu ^{- {N_{0}}}}{\theta ^{- {{\cal T}_{M}}{N_{0}}}}{\mu ^{{{- (s - {k_{0}})} / {{\tau _{a}}}}}}{\theta ^{{{- {{\cal T}_{M}}(s - {k_{0}})} /{{\tau _{a}}}}}}{y_{s}^{T}}{y_{s}}} < \sum\limits_{s = {k_{0}}}^{\infty} {{\gamma _{\sigma (s)}^{2}}{u_{s}^{T}}{u_{s}}} }$$

Thus, by the ADT Condition (2) which implies ${\mu ^{{{- (s - {k_{0}})} /{{\tau _{a}}}}}}{\theta ^{{{- {{\cal T}_{M}}(s - {k_{0}})}/{{\tau _{a}}}}}} > {\overline \alpha ^{(s - {k_{0}})}}$, the weighted $l _{2}$-gain performance (4) can be obtained. $\hfill\square$

Remark 1

From the definition of weighted $l _{2}$-gain given in [4], the disturbance attenuation performance (4) is a typical weighted $l _{2}$-gain performance, not the constant $l _{2}$-gain performance. The proof procedure of Theorem 2 in [1] considered that “For the $i$th subsystem, we know the $l _{2}$-gain is not greater than $\max \{\sqrt {{\theta ^{{{\cal T}_{M}} - 1}}} {\gamma _{i}}\}$. Therefore, we can conclude that system (1) can have the $l _{2}$-gain as ${\gamma _{s}} = \max \{\sqrt {{\theta ^{{{\cal T}_{M}} - 1}}} {\gamma _{i}}\}$”, which is actually not correct, since the switching among subsystems can affect $l _{2}$-gain performance even though all subsystems have a desired $l _{2}$-gain. In fact, some other reported literatures about $l _{2}$-gain analysis for switched system by ADT approach are mostly based on the conception of weighted $l _{2}$-gain performance, in which the conventional Lyapunov functions are employed and the synchronous switching case has been thoroughly discussed [3]–​[7]. Thus, the corrected results in this note can be viewed as the extended results for weighted $l _{2}$-gain analysis and can be readily applied into asynchronous switching.

Remark 2

Moreover, it has to be pointed out that similar errors occur in [8, Theorem 2] and [9, Theorem 3] which are concerned about $l _{2}$-gain under synchronous switching. The constant $l _{2}$-gain should also be replaced by weighted $l _{2}$-gain performance. In the case related to [8] and [9], we have ${\cal T} _{M}=0$, hence the ADT condition becomes

$${\tau _{a}} > {\tau _{a}^{\ast}} =- {{\ln \mu } \over {\ln \overline \alpha}} .\eqno{\hbox{(7)}}$$

By commonly choosing chatter bound ${N_{0}} = 0$, Theorem 1 turns into following corollary usually related to synchronous switching, which is exactly reduced to the results presented in [3]–​[7].

Corollary 1

Consider switched system (1) and let $0 < \alpha < 1$ and ${\gamma _{i}} \ge 0$, $\forall i \in {\cal I}$ be given constants. Suppose that there exist positive definite ${{\cal C}^{1}}$ functions ${V_{\sigma (k)}}$: ${{\BBR }^{n}} \to {\BBR }$, $\sigma (k) \in {\cal I}$, with ${V_{\sigma ({k_{0}})}}({x_{{k_{0}}}}) \equiv 0$ such that $\forall (i,j) \in {\cal I} \times {\cal I}$, $i \ne j$, ${V_{i}}({x_{{k_{l}}}}) \le \mu {V_{j}}({x_{{k_{l}}}})$ and $\forall i \in {\cal I}$, denoting $\Gamma (k) = {y_{k}^{T}}{y_{k}} - {\gamma _{i}^{2}}{u_{k}^{T}}{u_{k}}$, $\Delta {V_{i}}({x_{k}}) \le - \alpha {V_{i}}(k) - \Gamma (k)$. Then switched system is GUAS for any switching signal satisfying (7) and has a weighted $l _{2}$-gain $\sum_{s = {k_{0}}}^{\infty} {{{\overline \alpha }^{(s - {k_{0}})}}{y_{s}^{T}}{y_{s}}} < \sum_{s = {k_{0}}}^{\infty} {{\gamma _{s}^{2}}{u_{s}^{T}}{u_{s}}}$, where $\overline \alpha = 1 - \alpha$ and ${\gamma _{s}} = {\max _{i \in {\cal I}}}\{{\gamma _{i}}\}$.

Since the results of [1] in $H _{\infty }$ control problem were primarily based on the incorrect Theorem 2, the Theorem 3 for bounded real lemma and Theorem 4 for $H _{\infty }$ controller design should also be corrected by replacing the $l _{2}$-gain with weighted $l _{2}$-gain performance.