Finite-Time Control and Filtering

Abstract

In this chapter, finite-time control and filtering problems are investigated for discrete-time switched systems with MDPDT. First of all, sufficient conditions are deduced to guarantee the finite-time boundedness and $\mathscr {H}_{\infty }$ performance of switched systems. Then, control and filtering schemes are developed. The time-scheduled technology is adopted to achieve less conservative disturbance attenuation performance index. Finally, numerical examples are presented to verify the effectiveness of developed methods.

1 Finite-Time Performance Analysis

It is known that the main concern in many practical problems is the performance of systems over a fixed time interval [1, 2, 4, 5]. Thus, considering the finite-time performance is necessary. In Chap. 8, finite-time control and filtering issues for discrete-time switched systems are addressed with MDADT switching. In this chapter, the MDPDT switching mechanism is adopted to deduce the finite-time performance.

Consider a class of switched discrete-time systems given by

$$\begin{aligned} x(k+1)= & {} A_{\sigma (k)}x(k) + E_{\sigma (k)} \omega (k), \end{aligned}$$

(12.1)

$$\begin{aligned} z(k)= & {} G_{\sigma (k)}x(k) + L_{\sigma (k)} \omega (k), \end{aligned}$$

(12.2)

where $x(k)\in \mathbb {R}^{n_{x}}$ is the discrete state vector of the system, $z(k)\in \mathbb {R}^{n_{z}}$ is the output, $\omega (k)\in \mathbb {R}^{r}$ is the exogenous disturbance that satisfies

$$\begin{aligned} \sum _{k=0}^{N}\omega ^{\top }(k)\omega (k)\le d,d>0 , \end{aligned}$$

(12.3)

where N is a given constant.

In this section, the QTD technique is adopted to deduce sufficient conditions which can guarantee that the system (12.1)–(12.2) is FTB with a prescribed $\mathscr {H}_{\infty }$ performance. Here, the following QTD multiple Lyapunov-like function is adopted:

$$\begin{aligned} V_{\sigma (k)}(x(k),k)=V_{\sigma (k)}(x(k),q_{k}), \end{aligned}$$

(12.4)

where $q_{k}$ is the time scheduler and can be easily computed by

• (i) in the $\tau$-portion,

  $$\begin{aligned} q_{k}=\left\{ \begin{array}{cl} k-k_{s_{p}},&{} \quad k\in [ k_{s_{p}},k_{s_{p}}+\tau _{m}), \\ \tau _{m},&{} \quad k\in [ k_{s_{p}}+\tau _{m},k_{s_{p}+1}), \end{array} \right. \end{aligned}$$

  (12.5)

• (ii) in the T-portion,

  $$\begin{aligned} q_{k}=\min \{k-H_{r}, \tau _{m}\}, k\in [k_{s_{p}+1},k_{s_{p+1}}), \end{aligned}$$

  (12.6)

where the positive integer $\tau _{m}$ is the pre-chosen maximum step, and

$$\begin{aligned} H_{r}\triangleq \arg \left\{ \max \{k_{s_{p}+r}, r\in \mathbb {Z}_{[1,N(k_{s_{p}+1}, k_{s_{p+1}})]} | k_{s_{p}+r} \le k, k_{s_{p}+r} \in [k_{s_{p}+1}, k_{s_{p+1}}) \}\right\} . \end{aligned}$$

Firstly, we provided the following lemma, which can guarantee the finite-time boundedness of the system (12.1).

Lemma 12.1

Consider the discrete-time switched linear system (12.1), for given scalars $\alpha _{i}>1$, $\mu _{i}>1$, $i \in \mathscr {M}$, integer $\tau _{m}>0$, and a prescribed period of persistence T, if there exist matrices $P_{i}(\phi )>0$, $i \in \mathscr {M}$, $\phi \in \mathbb {Z}_{[ 0,\tau _{m}]}$, such that $\forall (i,j)\in \mathscr {M}\times \mathscr {M}$, $i \ne j$, $\varphi \in \mathbb {Z}_{[0,\tau _{m}-1]}$,

$$\begin{aligned} \left[ \begin{array}{cc} A_{i}^{\top }P_{i}(\varphi +1)A_{i}-\alpha _{i}P_{i}(\varphi ) &{} A_{i}^{\top }P_{i}(\varphi +1)E_{i} \\ *&{} E_{i}^{\top }P_{i}(\varphi +1)E_{i}-\gamma ^{2}I \end{array}\right] <0, \end{aligned}$$

(12.7)

$$\begin{aligned} \left[ \begin{array}{cc} A_{i}^{\top }P_{i}(\tau _{m})A_{i}-\alpha _{i}P_{i}(\tau _{mi}) &{} A_{i}^{\top }P_{i}(\tau _{m})E_{i} \\ *&{} E_{i}^{\top }P_{i}(\tau _{m})E_{i}-\gamma ^{2}I \end{array}\right] <0, \end{aligned}$$

(12.8)

$$\begin{aligned} P_{i}(0)-\mu _{i}P_{j}(\varphi +1) \le 0, \end{aligned}$$

(12.9)

$$\begin{aligned} (c_{1}\kappa _{1}+\gamma ^{2}d)e^{\varepsilon }<c_{2}\kappa _{2}, \end{aligned}$$

(12.10)

where $\kappa _{1}=\max _{i\in \mathscr {M}, \phi \in \mathbb {Z}_{[ 0,\tau _{m}]}}\{(\lambda _{\max }(\bar{P}_{i}(\phi ))\}$, $\kappa _{2}=\min _{i\in \mathscr {M}, \phi \in \mathbb {Z}_{[ 0,\tau _{mi}]} }\{\lambda _{\min }(\bar{P}_{i}(\phi ))\}$, $\bar{P}_{i}(\phi ))=R^{-1/2}P_{i}(\phi )R^{-1/2}$, $\varepsilon = N\ln \alpha _{\max } +(T+1)\ln \mu _{\max }$, $\alpha _{\max }=\max _{i\in \mathscr {M}} \{\alpha _{i}\}$, and $\mu _{\max }=\max _{i\in \mathscr {M}} \{\mu _{i}\}$.

Then, for any MDPDT switching signal satisfying

$$\begin{aligned} \tau _{a\min }>\tau _{a\min }^{*}=\frac{N(T+1)\ln \mu _{i}}{\ln (c_{2}\kappa _{2})-\ln (c_{1}\kappa _{1}+\gamma ^{2}d)-\varepsilon }-T, \end{aligned}$$

(12.11)

the system (12.1) is FTB with respect to $(c_{1},c_{2},R,d,N,\sigma )$, where $\tau _{a\min }= \min _{i\in \mathscr {M}} \{\tau _{ai}\}$.

Proof

Choose the following Lyapunov-like function:

$$\begin{aligned} V_{\sigma (k)}(x(k),k)=V_{\sigma (k)}(x(k),q_{k})=x^{\top }(k)P_{\sigma (k)}(q_{k})x(k), \end{aligned}$$

(12.12)

where $P_{\sigma (k)}(q_{k})>0$ for $\sigma (k) \in \mathscr {M}$, and $q_{k}$ is the time scheduler defined by (12.5) and (12.6).

Consider $\sigma (k_{s_{p}})=i\in \mathscr {M}$. For $k \in [k_{s_{p}}, k_{s_{p}}+\tau _{m})$, it holds that

$$\begin{aligned}&V_{i}(x(k+1),k+1)-\alpha _{i}V_{i}(x(k),k)-\gamma ^{2} \omega ^{\top }(k)\omega (k) \\= & {} x^{\top }(k+1)P_{i}(k+1-k_{s_{p}})x(k+1) - \alpha _{i}x^{\top }(k)P_{i}(k-k_{s_{p}})x(k) -\gamma ^{2} \omega ^{\top }(k)\omega (k) \\= & {} \xi ^{\top }(k) \left[ \begin{array}{cc} A_{i}^{\top }P_{i}(k+1-k_{s_{p}})A_{i}-\alpha _{i}P_{i}(k-k_{s_{p}}) &{} A_{i}^{\top }P_{i}(k+1-k_{s_{p}})E_{i} \\ *&{} E_{i}^{\top }P_{i}(k+1-k_{s_{p}})E_{i}-\gamma ^{2}I \end{array}\right] \xi (k), \end{aligned}$$

where $\xi (k)= \left[ \begin{array}{cc} x^{\top }(k)&\omega ^{\top }(k) \end{array} \right] ^{\top }$.

Since $0\le k-k_{s_{p}} < \tau _{m}$ for $k \in [k_{s_{p}}, k_{s_{p}}+\tau _{m})$. Therefore, (12.7) can guarantee that

$$\begin{aligned} V_{i}(x(k+1),k+1)-\alpha _{i}V_{i}(x(k),k)-\gamma ^{2} \omega ^{\top }(k)\omega (k)<0. \end{aligned}$$

(12.13)

By the similar manipulation, (12.8) can guarantee (12.13) for $k \in [k_{s_{p}}+\tau _{m}, k_{s_{p}+1})$.

Therefore, we can conclude that for $k \in [k_{s_{p}}, k_{s_{p}+1})$, i.e., in the $\tau$-portion, it holds that

$$\begin{aligned} V_{\sigma (k)}(x(k+1),k+1)<\alpha _{i}V_{\sigma (k)}(x(k),k)+\gamma ^{2} \omega ^{\top }(k)\omega (k). \end{aligned}$$

(12.14)

According to the definition of $q_{k}$, (12.14) also holds in the T-portion. Therefore, (12.14) holds for each switching interval $[k_{s}, k_{s+1})$.

By summating (12.14) for $k \in [k_{s}, k_{s+1})$, one can achieve that

$$\begin{aligned} V_{\sigma (k)}(x(k),k)<\alpha _{\sigma (k_{s})}^{k-k_{s}} V_{\sigma (k_{s})}(x(k_{s}),k_{s})+\sum _{l=k_{s}}^{k-1} \alpha _{k_{s}}^{k-1-s} \gamma ^{2} \omega ^{\top }(l)\omega (l). \end{aligned}$$

(12.15)

On the other hand, $\sigma (k_{s})=i \in \mathscr {M}$, $\sigma (k_{s}-1) =j \in \mathscr {M}$, $i\ne j$, (12.9) indicates that

$$\begin{aligned} V_{\sigma (k_{s})}(x(k_{s}), k_{s}) \le \mu _{\sigma (k_{s})} V_{\sigma (k_{s}-1)}(x(k_{s}), k_{s}). \end{aligned}$$

(12.16)

Combining (12.15) and (12.16), one can obtain that

$$\begin{aligned} V_{\sigma (k)}(x(k),k)< & {} \alpha _{\sigma (k_{N_{\sigma }(0,k)})}^{k-k_{N_{\sigma }(0,k)}} V_{\sigma (k_{N_{\sigma }(0,k)})}(x(k_{N_{\sigma }(0,k)}),k_{N_{\sigma }(0,k)}) \\&+\sum _{l=k_{N_{\sigma }(0,k)}}^{k-1}\alpha _{k_{N_{\sigma }(0,k)}}^{k-1-l} \gamma ^{2} \omega ^{\top }(l)\omega (l) \\\le & {} \mu _{\sigma (k_{N_{\sigma }(0,k)})}\alpha _{\sigma (k_{N_{\sigma }(0,k)})}^{k-k_{N_{\sigma }(0,k)}} V_{\sigma (k_{N_{\sigma }(0,k)-1})}(x(k_{N_{\sigma }(0,k)}),k_{N_{\sigma }(0,k)}) \\&+\sum _{l=k_{N_{\sigma }(0,k)}}^{k-1}\alpha _{k_{N_{\sigma }(0,k)}}^{k-1-l} \gamma ^{2} \omega ^{\top }(l)\omega (l) \\\le & {} \cdots \\\le & {} \mu _{\sigma (k_{N_{\sigma }(0,k)})} \mu _{\sigma (k_{N_{\sigma }(0,k)-1})} \ldots \mu _{\sigma (0)} \\&\times \alpha _{\sigma (k_{N_{\sigma }(0,k)})}^{k-k_{N_{\sigma }(0,k)}} \alpha _{\sigma (k_{N_{\sigma }(0,k)-1})}^{k_{N_{\sigma }(0,k)}-k_{N_{\sigma }(0,k)-1}} \ldots \alpha _{\sigma (0)}^{k_{1}-k_{0}} V_{\sigma (0)}(x(0), 0) \\&+ \mu _{\sigma (k_{N_{\sigma }(0,k)})} \mu _{\sigma (k_{N_{\sigma }(0,k)-1})} \ldots \mu _{\sigma (0)} \\&\times \alpha _{\sigma (k_{N_{\sigma }(0,k)})}^{k-k_{N_{\sigma }(0,k)}} \alpha _{\sigma (k_{N_{\sigma }(0,k)-1})}^{k_{N_{\sigma }(0,k)}-k_{N_{\sigma }(0,k)-1}} \ldots \alpha _{\sigma (k_{1})}^{k_{2}-k_{1}} \\&\times \sum _{l=0}^{k_{1}-1} \alpha _{\sigma (0)}^{k_{1}-1-l}\gamma ^{2} \omega ^{\top }(l)\omega (l) + \cdots \\&+ \sum _{l=k_{N_{\sigma }(0,k)}}^{k-1}\alpha _{k_{N_{\sigma }(0,k)}}^{k-1-l} \gamma ^{2} \omega ^{\top }(l)\omega (l) \\\le & {} \mu _{\max }^{N_{\sigma }(0,N)} \prod _{i=1}^{M} \alpha _{i}^{H_{i}(0,N)} V_{\sigma (0)}(x(0), 0) \\&+\gamma ^{2} \sum _{l=0}^{k-1}\left\{ \mu _{\max }^{N_{\sigma }(l,k)} \prod _{i=1}^{M} \alpha _{i}^{H_{i}(l,k)} \omega ^{\top }(l)\omega (l)\right\} \\\le & {} \mu _{\max }^{N_{\sigma }(0,N)} \prod _{i=1}^{M} \alpha _{i}^{H_{i}(0,N)} (V_{\sigma (0)}(x(0),0)+\gamma ^{2}d). \end{aligned}$$

It is noted that

$$\begin{aligned} 0\le N_{\sigma }(0,N) \le \left( \frac{N}{\tau _{a\min }+T}+1\right) (T+1). \end{aligned}$$

Thus, we have

$$\begin{aligned} V_{\sigma (k)}(x(k),k)\le & {} \exp \left\{ \left( \frac{N}{\tau _{a\min }+T}+1\right) (T+1)\ln \mu _{\max } +\sum _{i=1}^{M}H_{i}(0,N) \ln \alpha _{i} \right\} \\&\times (V_{\sigma (0)}(x(0),0)+\gamma ^{2}d) \\\le & {} \exp \left\{ \left( \frac{N}{\tau _{a\min }+T}+1\right) (T+1)\ln \mu _{\max } + N \ln \alpha _{\max } \right\} \\&\times (V_{\sigma (0)}(x(0),0)+\gamma ^{2}d). \end{aligned}$$

Considering $\bar{P}_{i}(\phi )=R^{-1/2}P_{i}(\phi ) R^{-1/2}$ for $\sigma (k)=i \in \mathscr {M}$, one can obtain that

$$\begin{aligned} V_{\sigma (0)}(x(0), 0)= & {} x^{\top }(0)P_{\sigma (0)}(0)x(0) = x^{\top }(0)R^{1/2}\bar{P}_{\sigma (0)}(0)R^{1/2}x(0) \\\le & {} \max _{i\in \mathscr {M}} \{\lambda _{\max }(\bar{P}_{i}(0))\} x^{\top }(0)R x(0)\le c_{1} \kappa _{1}, \end{aligned}$$

and

$$\begin{aligned} V_{\sigma (k)}(x(k),k)= & {} x^{\top }(k)P_{\sigma (k)}(k)x(k) = x^{\top }(k)R^{1/2}\bar{P}_{\sigma (k)}(q_{k})R^{1/2}x(k) \\\ge & {} \min _{i\in \mathscr {M}, \phi \in \mathbb {Z}_{[ 0,\tau _{m}]} } \{\lambda _{\min }(\bar{P}_{i}(\phi ))\} x^{\top }(k)R x(k)\ge \kappa _{2}x^{\top }(k)R x(k). \end{aligned}$$

It is noted that (12.10) indicates

$$\begin{aligned} \ln (c_{2}\kappa _{2})-\ln (c_{1}\kappa _{1}+\gamma ^{2}d)-\varepsilon >0. \end{aligned}$$

Notice that $\varepsilon = N\ln \alpha _{\max } +(T+1) \ln \mu _{\max }$. Combining with (12.11), for $\forall i \in \mathscr {M}$, it holds that

$$\begin{aligned} \left( \frac{N}{\tau _{a\min }+T}+1\right) (T+1)\ln \mu _{\max } +\max _{i\in \mathscr {M}}\left\{ N \ln \alpha _{i}\right\} < \ln (c_{2}\kappa _{2})-\ln (c_{1}\kappa _{1}+\gamma ^{2}d). \end{aligned}$$

Therefore, we can arrive at

$$\begin{aligned} x^{\top }(k)R x(k)\le & {} \frac{1}{\kappa _{2}}V_{\sigma (k)}(x(k),k) \\\le & {} \frac{c_{1}\kappa _{1}+\gamma ^{2}d}{\kappa _{2}} \exp \left\{ \left( \frac{N}{\tau _{a\min }+T}+1\right) (T+1)\ln \mu _{\max } + N \ln \alpha _{\max } \right\} \\< & {} \frac{c_{1}\kappa _{1}+\gamma ^{2}d}{\kappa _{2}} \exp \left\{ \ln (c_{2}\kappa _{2})-\ln (c_{1}\kappa _{1}+\gamma ^{2}d) \right\} \\< & {} c_{2}. \end{aligned}$$

According to Definition 8.2, the switched linear system (12.1) is FTB with respect to $(c_{1},c_{2},R,d,N,\sigma )$, which ends the proof.

Remark 12.1

It can be seen that condition (12.10) in Lemma 12.1 is not a strict LMI constraint. We can firstly solve (12.7)–(12.9) to gain $P_{i}(\phi )$, $i \in \mathscr {M}$, $\phi \in \mathbb {Z}_{[0,\tau _{m}]}$. Then we can further obtain the values of $\kappa _{1}$ and $\kappa _{2}$. Substituting $\kappa _{1}$, $\kappa _{2}$ into (12.10) and adjusting parameters $\alpha _{i}$, $\mu _{i}$, $c_{1}$ and $c_{2}$ properly, we can manage to find an admissible solution such that (12.10) holds.

Based on Lemma 12.1, we further deduce the criterion to guarantee that the system is finite-time boundedness with $\mathscr {H}_{\infty }$ performance of the switched system (12.1)–(12.2).

Lemma 12.2

Consider the discrete-time switched linear system (12.1)–(12.2), for given scalars $\alpha _{i}>1$, $\mu _{i}>1$, $i \in \mathscr {M}$, integer $\tau _{m}>0$, and a prescribed period of persistence T, if there exist matrices $P_{i}(\phi )>0$, $i \in \mathscr {M}$, $\phi \in \mathbb {Z}_{[ 0,\tau _{m}]}$, such that $\forall (i,j)\in \mathscr {M}\times \mathscr {M}$, $i \ne j$, $\varphi \in \mathbb {Z}_{[0,\tau _{m}-1]}$, (12.9) holds, and

$$\begin{aligned} \left[ \begin{array}{cc} A_{i}^{\top }P_{i}(\varphi +1)A_{i}+G_{i}^{\top }G_{i}-\alpha _{i}P_{i}(\varphi ) &{} A_{i}^{\top }P_{i}(\varphi +1)E_{i}+G_{i}^{\top }L_{i} \\ *&{} E_{i}^{\top }P_{i}(\varphi +1)E_{i}+L_{i}^{\top }L_{i}-\gamma ^{2}I \end{array}\right] <0, \end{aligned}$$

(12.17)

$$\begin{aligned} \left[ \begin{array}{cc} A_{i}^{\top }P_{i}(\tau _{m})A_{i}+G_{i}^{\top }G_{i}-\alpha _{i}P_{i}(\tau _{m}) &{} A_{i}^{\top }P_{i}(\tau _{m})E_{i}+G_{i}^{\top }L_{i} \\ *&{} E_{i}^{\top }P_{i}(\tau _{m})E_{i}+L_{i}^{\top }L_{i}-\gamma ^{2}I \end{array}\right] <0, \end{aligned}$$

(12.18)

$$\begin{aligned} \gamma ^{2}de^{\varepsilon }<c_{2}\kappa _{2}, \end{aligned}$$

(12.19)

where $\kappa _{2}=\min _{i\in \mathscr {M}, \phi \in \mathbb {Z}_{[ 0,\tau _{m}]} }\{\lambda _{\min }(\bar{P}_{i}(\phi ))\}$, $\bar{P}_{i}(\phi ))=R^{-1/2}P_{i}(\phi )R^{-1/2}$, $\varepsilon = N\ln \alpha _{\max } +(T+1)\ln \mu _{\max }$, $\alpha _{\max }=\max _{i\in \mathscr {M}} \{\alpha _{i}\}$, and $\mu _{\max }=\max _{i\in \mathscr {M}} \{\mu _{i}\}$.

Then, for any MDPDT switching signal satisfying

$$\begin{aligned} \tau _{a\min }>\tau _{a\min }^{*}=\max \left\{ \frac{N(T+1)\ln \mu _{\max }}{\ln (c_{2}\kappa _{2})-\ln (\gamma ^{2}d)-\varepsilon }-T, \frac{(T+1)\ln \mu _{\max }}{\ln \alpha _{\max }}-T \right\} , \end{aligned}$$

(12.20)

the system (12.1)–(12.2) is FTB and has a prescribed $\mathscr {H}_{\infty }$ performance with respect to $(0,c_{2},R,d,\gamma _{s},N,\sigma )$, where $\tau _{a\min }= \min _{i\in \mathscr {M}} \{\tau _{ai}\}$, and

$$\begin{aligned} \gamma _{s}=\sqrt{e^{(T+1)\ln \mu _{\max }+2N\ln \alpha _{\max }}}\gamma . \end{aligned}$$

(12.21)

Proof

It is noted that (12.17) can be rewritten as

$$\begin{aligned} \left[ \begin{array}{cc} A_{i}^{\top }P_{i}(\varphi +1)A_{i}-\alpha _{i}P_{i}(\varphi ) &{} A_{i}^{\top }P_{i}(\varphi +1)E_{i} \\ *&{} E_{i}^{\top }P_{i}(\varphi +1)E_{i}-\gamma ^{2}I \end{array}\right] +\left[ \begin{array}{c} G_{i}^{\top } \\ L_{i}^{\top } \end{array}\right] \left[ \begin{array}{cc} G_{i}&L_{i} \end{array}\right] <0. \end{aligned}$$

Since

$$\begin{aligned} \left[ \begin{array}{c} G_{i}^{\top } \\ L_{i}^{\top } \end{array}\right] \left[ \begin{array}{cc} G_{i}&L_{i} \end{array}\right] \ge 0, \end{aligned}$$

then one can obtain that

$$\begin{aligned} \left[ \begin{array}{cc} A_{i}^{\top }P_{i}(\varphi +1)A_{i}-\alpha _{i}P_{i}(\varphi _{i}) &{} A_{i}^{\top }P_{i}(\varphi +1)E_{i} \\ *&{} E_{i}^{\top }P_{i}(\varphi +1)E_{i}-\gamma ^{2}I \end{array}\right] <0. \end{aligned}$$

That is to say, (12.7) can be ensured by (12.17). By the similar manipulation, (12.18) can guarantee (12.8). Therefore, according to Lemma 12.1, Lemma 12.2 can guarantee that the switched system (12.1) is FTB with respect to $(c_{1},c_{2},R,d,N,\sigma )$ by setting $c_{1}=0$.

Select (12.12) as the Lyapunov-like function. Denote $\varGamma (k)=\gamma ^{2}\omega ^{\top }(k)\omega (k) - z^{\top }(k)z(k)$. Consider $\sigma (k_{s_{p}})=i\in \mathscr {M}$. For $k \in [k_{s_{p}}, k_{s_{p}}+\tau _{m})$, it holds that

$$\begin{aligned}&V_{i}(x(k+1),k+1)-\alpha _{i}V_{i}(x(k),k)-\varGamma (k) \\= & {} \xi ^{\top }(k) \left[ \begin{array}{cc} A_{i}^{\top }P_{i}(k+1-k_{s})A_{i}-\alpha _{i}P_{i}(k-k_{s}) &{} A_{i}^{\top }P_{i}(k+1-k_{s})E_{i} \\ *&{} E_{i}^{\top }P_{i}(k+1-k_{s})E_{i} \end{array}\right] \xi (k)-\varGamma (k) , \end{aligned}$$

where $\xi (k)= \left[ \begin{array}{cc} x^{\top }(k)&\omega ^{\top }(k) \end{array} \right] ^{\top }$.

It is noted that $0\le k-k_{s_{p}} < \tau _{m}$ for $k \in [k_{s_{p}}, k_{s_{p}}+\tau _{m})$. Thus, (12.17) indicates that

$$\begin{aligned} V_{i}(x(k+1),k+1)-\alpha _{i}V_{i}(x(k),k)-\varGamma (k)<0. \end{aligned}$$

(12.22)

Similarly, for $k \in [k_{s}+\tau _{mi}, k_{s+1})$, (12.22) can also be ensured by (12.18). That is to say, in the $\tau$-portion, (12.18) always holds. Obviously, in the T-portion, (12.18) can also be ensured. Therefore, one can conclude that for each switching interval $k \in [k_{s}, k_{s+1})$,

$$\begin{aligned} V_{\sigma (k)}(x(k+1),k+1)<\alpha _{i}V_{\sigma (k)}(x(k),k)+\varGamma (k). \end{aligned}$$

(12.23)

By summating (12.23) for $k \in [k_{s}, k_{s+1})$, one can obtain

$$\begin{aligned} V_{\sigma (k)}(\xi (k)) < \alpha _{\sigma (k_{s})}^{k-k_{s}} V_{\sigma (k_{s})}(x(k_{s}), k_{s}) + \sum ^{k-1}_{l=k_{s}} \alpha _{\sigma (k_{s})}^{k-1-l} \varGamma (k). \end{aligned}$$

(12.24)

Combining (12.16) and (12.24), it holds that

$$\begin{aligned} V_{\sigma (k)}(x(k), k)< & {} \mu _{\sigma (k_{N_{\sigma }(0,k)})} \alpha _{\sigma (k_{N_{\sigma }(0,k)})}^{k-k_{N_{\sigma }(0,k)}} V_{\sigma (k_{N_{\sigma }(0,k)})}(x(k_{N_{\sigma }(0,k)}), k_{N_{\sigma }(0,k)}) \\&+ \sum ^{k-1}_{l=k_{N_{\sigma }(0,k)}} \alpha _{\sigma (k_{N_{\sigma }(0,k)})}^{k-1-l} \varGamma (k) \\\le & {} \prod _{i=1}^{M} \mu _{i}^{N_{\sigma i}(0,k)} \alpha _{i}^{H_{i}(0,k)} V_{\sigma (0)}(x(0), 0) + \sum _{l=0}^{k-1}\left\{ \prod _{i=1}^{M} \mu _{i}^{N_{\sigma i}(l,k)} \alpha _{i}^{H_{i}(l,k)}\varGamma (l) \right\} . \end{aligned}$$

Under zero initial conditions, we have

$$\begin{aligned}&\sum _{l=0}^{k-1}\left\{ \prod _{i=1}^{M} \mu _{i}^{N_{\sigma i}(l,k)} \alpha _{i}^{H_{i}(l,k)}z^{\top }(l)z(l) \right\} \nonumber \\\le & {} \gamma ^{2} \sum _{l=0}^{k-1}\left\{ \prod _{i=1}^{M} \mu _{i}^{N_{\sigma i}(l,k)} \alpha _{i}^{H_{i}(l,k)}\omega ^{\top }(l)\omega (l) \right\} \end{aligned}$$

(12.25)

Furthermore, one can arrive at

$$\begin{aligned}&\sum _{l=0}^{k-1}\left\{ \prod _{i=1}^{M} \mu _{i}^{N_{\sigma i}(l,k)} \alpha _{i}^{H_{i}(l,k)}z^{\top }(l)z(l) \right\} \\\le & {} \gamma ^{2} \sum _{l=0}^{k-1}\left\{ \mu _{\max }^{N_{\sigma }(l,k)}\alpha _{\max }^{k-l}\omega ^{\top }(l)\omega (l) \right\} . \end{aligned}$$

Since

$$\begin{aligned} 0\le N_{\sigma }(l,k) \le \left( \frac{k-l}{\tau _{a\min }+T}+1\right) (T+1). \end{aligned}$$

It holds that

$$\begin{aligned}&\sum _{l=0}^{k-1}z^{\top }(l)z(l) \nonumber \\\le & {} \sum _{l=0}^{k-1}\left\{ \left( \frac{k-l}{\tau _{a\min }+T}+1\right) (T+1)\ln \mu _{\max }+(k-l)\ln \alpha _{\max } \right\} \omega ^{\top }(l)\omega (l). \end{aligned}$$

(12.26)

On the other hand, if (12.20) holds, one can obtain that

$$\begin{aligned} \frac{\ln \mu _{\max }}{\tau _{a\min }+T} \le \ln \alpha _{\max }. \end{aligned}$$

Then, we can derive

$$\begin{aligned}&\sum _{l=0}^{k-1}\left\{ \left( \frac{k-l}{\tau _{a\min }+T}+1\right) (T+1)\ln \mu _{\max }+(k-l)\ln \alpha _{\max } \right\} \omega ^{\top }(l)\omega (l) \nonumber \\\le & {} \sum _{l=0}^{k-1} \exp \left\{ (T+1)\ln \mu _{\max } +2(k-l)\ln \alpha _{\max } \right\} \omega ^{\top }(l)\omega (l). \end{aligned}$$

(12.27)

By setting $k-1=N$, it can be concluded from (12.25)–(12.27) that

$$\begin{aligned} \sum _{l=0}^{N}z^{\top }(l)z(l) \le \gamma _{s}^{2}\omega ^{\top }(l)\omega (l), \end{aligned}$$

where $\gamma _{s}=\sqrt{\mathrm {exp}((T+1)\ln \mu _{\max }+2N\ln \alpha _{\max })}\gamma$.

According to Definition 8.3, one can conclude that the system (12.1)–(12.2) is FTB and has a prescribed $\mathscr {H}_{\infty }$ performance with respect to $(0,c_{2},R,d,\gamma _{s},N,\sigma )$, which ends the proof.

In Lemma 12.2, sufficient conditions are deduced to guarantee the finite-time $\mathscr {H}_{\infty }$ performance for discrete-time switched systems. On this basis, the control and filtering schemes are developed in the subsequent sections.

2 Finite-Time Control

In this section, we consider the finite-time $\mathscr {H}_{\infty }$ control problem. Consider a class of discrete-time switched linear systems:

$$\begin{aligned} x(k+1)= & {} A_{\sigma (k)}x(k)+B_{\sigma (k)}u(k)+E_{\sigma (k)}\omega (k), \end{aligned}$$

(12.28)

$$\begin{aligned} z(k)= & {} G_{\sigma (k)}x(k)+H_{\sigma (k)}u(k)+L_{\sigma (k)}\omega (k), \end{aligned}$$

(12.29)

where $x(k) \in \mathbb {R}^{n_{x}}$ is the state vector, $u(k) \in \mathbb {R}^{n_{u}}$ is the control input, $z(k)\in \mathbb {R}^{n_{z}}$ is the controlled output, and $\omega (k)\in \mathbb {R}^{n_{\omega }}$ is the exogenous disturbance satisfying (12.3). $\sigma (k):[0,\infty )\rightarrow \mathscr {M}$ is the switching signal. $A_{i}$, $B_{i}$, $E_{i}$, $G_{i}$, $H_{i}$ and $L_{i}$ are real constant matrices with appropriate dimensions for $\sigma (k)=i\in \mathscr {M}$.

The considered controller is with the following form:

$$\begin{aligned} u(k)=K_{\sigma (k)}(q_{k})x(k), \end{aligned}$$

(12.30)

where $K_{\sigma (k)}(q_{k})$ is the controller gains to be determined, and $q_{k}$ can be computed by (12.5) and (12.6).

For $\sigma (k)=i \in \mathscr {M}$, the corresponding closed-systems are

$$\begin{aligned} x(k+1)= & {} \mathscr {A}_{i}(q_{k})x(k)+\mathscr {B}_{i}\omega (k), \end{aligned}$$

(12.31)

$$\begin{aligned} z(k)= & {} \mathscr {C}_{i}(q_{k})x(k)+\mathscr {D}_{i}\omega (k), \end{aligned}$$

(12.32)

where $\mathscr {A}_{i}(q_{k})=A_{i}+B_{i}K_{i}(q_{k})$, $\mathscr {B}_{i}=E_{i}$, $\mathscr {C}_{i}(q_{k})=G_{i}+H_{i}K_{i}(q_{k})$, and $\mathscr {D}_{i}=L_{i}$.

Here, we are aimed to design a set of QTD state feedback controllers (12.30) for the system (12.28)–(12.29), such that the closed-loop system (12.31)–(12.32) is FTB and has a prescribed $\mathscr {H}_{\infty }$ performance.

Theorem 12.1

Consider the discrete-time switched linear system (12.28)–(12.29), for given scalars $\alpha _{i}>1$, $\mu _{i}>1$, $i \in \mathscr {M}$, integer $\tau _{m}>0$, and a prescribed period of persistence T, if there exist matrices $\tilde{P}_{i}(\phi )>0$, $U_{i}(\phi )$, $i \in \mathscr {M}$, $\phi \in \mathbb {Z}_{[ 0,\tau _{m}]}$, such that $\forall (i,j)\in \mathscr {M}\times \mathscr {M}$, $i \ne j$, $\varphi \in \mathbb {Z}_{[ 0,\tau _{m}-1]}$, (12.19) holds, and

$$\begin{aligned} \varXi _{i}(\varphi +1, \varphi ) < 0, \end{aligned}$$

(12.33)

$$\begin{aligned} \varXi _{i}(\tau _{m}, \tau _{m}) < 0, \end{aligned}$$

(12.34)

$$\begin{aligned} \tilde{P}_{j}(\varphi +1)-\mu _{i}\tilde{P}_{i}(0) \le 0, \end{aligned}$$

(12.35)

where

$$\begin{aligned} \varXi _{i} (\iota _{1},\iota _{2})=\left[ \begin{array}{cccc} -\alpha _{i}\tilde{P}_{i}(\iota _{2}) &{} 0 &{} \tilde{P}_{i}(\iota _{2})A_{i}^{\top }+U_{i}^{\top }(\iota _{2})B_{i}^{\top } &{} \tilde{P}_{i}(\iota _{2})G_{i}^{\top }+U_{i}^{\top }(\iota _{2})H_{i}^{\top } \\ *&{} -\gamma ^{2}I &{} E_{i}^{\top } &{} L_{i}^{\top } \\ *&{} *&{} -\tilde{P}_{i}(\iota _{1}) &{} 0 \\ *&{} *&{} *&{} -I \end{array} \right] , \end{aligned}$$

$\kappa _{2}=\min _{i\in \mathscr {M}, \phi _{i} \in \mathbb {Z}_{[ 0,\tau _{m}]} }\{\lambda _{\min }(\bar{P}_{i}(\phi ))\}$, $\bar{P}_{i}(\phi ))=R^{-1/2}\tilde{P}_{i}^{-1}(\phi )R^{-1/2}$, $\varepsilon = N\ln \alpha _{\max } +(T+1)\ln \mu _{\max }$, $\alpha _{\max }=\max _{i\in \mathscr {M}} \{\alpha _{i}\}$, and $\mu _{\max }=\max _{i\in \mathscr {M}} \{\mu _{i}\}$.

Then, for any MDPDT switching signal satisfying (12.20), the system (12.31)–(12.32) is FTB and has a prescribed $\mathscr {H}_{\infty }$ performance with respect to $(0,c_{2},R,d,\gamma _{s},N,\sigma )$, where $\gamma _{s}$ is given in (12.21). Moreover, the controller gains are given by

$$\begin{aligned} K_{i}(\phi )=U_{i}(\phi )\tilde{P}_{i}^{-1}(\phi ). \end{aligned}$$

(12.36)

Proof

Construct the following Lyapunov-like function:

$$\begin{aligned} V_{\sigma (k)}(x(k),\phi )=x^{\top }(k)P_{\sigma (k)}(\phi )x(k), \end{aligned}$$

(12.37)

where $P_{\sigma (k)}(\phi )=\tilde{P}_{\sigma (k)}(\phi )$ for $\forall \sigma (k) \in \mathscr {M}$, $\phi \in \mathbb {Z}_{[ 0,\tau _{m}]}$. In this case, (12.35) is equivalent to (12.9).

By using Schur complement and congruence transformation, (12.33) can guarantee (12.17) and (12.18) can be ensured by (12.34).

It is noted that all conditions in Lemma 12.2 are satisfied. Therefore, one can conclude that the closed-loop system (12.31)–(12.32) is FTB and has a prescribed $\mathscr {H}_{\infty }$ performance, which ends the proof.

In the following, the traditional time-independent state feedback controller is designed.

Corollary 12.1

Consider the discrete-time switched linear system (12.28)–(12.29), for given scalars $\alpha _{i}>1$, $\mu _{i}>1$, $i \in \mathscr {M}$, and a prescribed period of persistence T, if there exist matrices $\tilde{P}_{i}>0$, $U_{i}$, $i \in \mathscr {M}$, such that $\forall (i,j)\in \mathscr {M}\times \mathscr {M}$, $i \ne j$, (12.19) holds, and

$$\begin{aligned} \left[ \begin{array}{cccc} -\alpha _{i}\tilde{P}_{i} &{} 0 &{} \tilde{P}_{i}A_{i}^{\top }+U_{i}^{\top }B_{i}^{\top } &{} \tilde{P}_{i}G_{i}^{\top }+U_{i}^{\top }H_{i}^{\top } \\ *&{} -\gamma ^{2}I &{} E_{i}^{\top } &{} L_{i}^{\top } \\ *&{} *&{} -\tilde{P}_{i} &{} 0 \\ *&{} *&{} *&{} -I \end{array} \right] <0, \end{aligned}$$

(12.38)

$$\begin{aligned} \tilde{P}_{j}-\mu _{i}\tilde{P}_{i} \le 0, \end{aligned}$$

(12.39)

where $\kappa _{2}=\min _{i\in \mathscr {M}, \phi _{i} \in \mathbb {Z}_{[ 0,\tau _{m}]} }\{\lambda _{\min }(\bar{P}_{i}(\phi ))\}$, $\bar{P}_{i}(\phi ))=R^{-1/2}\tilde{P}_{i}^{-1}(\phi )R^{-1/2}$, $\varepsilon = N\ln \alpha _{\max } +(T+1)\ln \mu _{\max }$, $\alpha _{\max }=\max _{i\in \mathscr {M}} \{\alpha _{i}\}$, and $\mu _{\max }=\max _{i\in \mathscr {M}} \{\mu _{i}\}$.

Then, for any MDPDT switching signal satisfying (12.20), the system (12.31)–(12.32) is FTB and has a prescribed $\mathscr {H}_{\infty }$ performance with respect to $(0,c_{2},R,d,\gamma _{s},N,\sigma )$, where $\gamma _{s}$ is given in (12.21). Moreover, the controller gains are given by

$$\begin{aligned} K_{i}=U_{i}\tilde{P}_{i}^{-1}. \end{aligned}$$

(12.40)

Proof

Omitted.

Remark 12.2

In this section, finite-time $\mathscr {H}_{\infty }$ control for discrete-time switched systems is addressed with MDPDT switching. The designed controllers are based on state feedback. In [3], finite-time output feedback control for discrete-time switched systems is studied with MDPDT switching. One can refer to it for more details.

Example 12.1

Consider a discrete-time switched linear system (12.28)–(12.29) consisting of three subsystems, and the system parameters are given by:

$$\begin{aligned} A_{1}= & {} \left[ \begin{array}{cc} 1.1 &{} -0.4 \\ 0.6 &{} -1.2 \end{array}\right] , B_{1}=\left[ \begin{array}{c} 0.1 \\ 0.2 \end{array}\right] , E_{1}=\left[ \begin{array}{c} 0.1 \\ 0.1 \end{array}\right] , \\ G_{1}= & {} \left[ \begin{array}{cc} 0.2&-0.1 \end{array}\right] , H_{1}=0.2, L_{1}=0.1; \\ A_{2}= & {} \left[ \begin{array}{cc} 1.2 &{} 0.2 \\ 0.7 &{} 1.2 \end{array}\right] , B_{2}=\left[ \begin{array}{c} 0.1 \\ 0.1 \end{array}\right] , E_{2}=\left[ \begin{array}{c} 0.1 \\ 0.2 \end{array}\right] , \\ G_{2}= & {} \left[ \begin{array}{cc} 0.1&0.2 \end{array}\right] , H_{2}=-0.1, L_{2}=0.1;\\ A_{3}= & {} \left[ \begin{array}{cc} -1.2 &{} 0.3 \\ 0.2 &{} 1.1 \end{array}\right] , B_{3}=\left[ \begin{array}{c} 0.1 \\ 0.1 \end{array}\right] , E_{3}=\left[ \begin{array}{c} 0.2 \\ 0.1 \end{array}\right] , \\ G_{3}= & {} \left[ \begin{array}{cc} 0.1&0.1 \end{array}\right] , H_{3}=0.1, L_{3}=0.3. \end{aligned}$$

Here, we are aimed to design a set of state feedback controllers and find admissible MDPDT switching signals such that the corresponding closed-loop system (12.31)–(12.32) is FTB and has a prescribed $\ell _{2}$-gain performance. Set $\alpha _{1}=1.009$, $\alpha _{2}=1.008$, $\alpha _{3}=1.007$, $\mu _{1}=\mu _{2}=\mu _{3}=1.01$, and $\tau _{m1}=\tau _{m2}=\tau _{m3}=1$. Moreover, consider $c_{2}=0.4$, $d=0.4$, $N=80$, $T=10$ and $R=I$. By using Theorem 12.1, feasible solutions can be found with $\tau _{a\min }^{*}=2.2162$, and $\gamma _{s}=0.8641$. Using Corollary 12.1 can also obtain feasible solutions with $\gamma _{s}=1.0490$. It can be seen that Theorem 12.1 can achieve less conservative performance index compared with Corollary 12.1.

Fig. 12.1

A designed switching signal $\sigma (k)$ with $\tau _{a\min }\ge 3$ and $T=10$

By Theorem 12.1, a set of both mode-dependent and QTD state feedback controllers with

$$\begin{aligned} K_{1}(0)= & {} \left[ \begin{array}{cc} -3.6520&4.5658 \end{array}\right] , K_{1}(1)=\left[ \begin{array}{cc} -3.6611&4.5895 \end{array}\right] ; \\ K_{2}(0)= & {} \left[ \begin{array}{cc} -10.9575&-7.2757 \end{array}\right] , K_{2}(1)=\left[ \begin{array}{cc} -10.9574&-7.2757 \end{array}\right] ;\\ K_{3}(0)= & {} \left[ \begin{array}{cc} 8.2843 -6.3012 \end{array}\right] , K_{3}(1)=\left[ \begin{array}{cc} 8.2855 -6.2970 \end{array}\right] . \end{aligned}$$

Assume that the exogenous disturbance input is $\omega (k)=0.1\sin (0.4k)$. The switching signal is given in Fig. 12.1. Under zero initial condition, the state responses of the closed-loop system are shown in Fig. 12.2. The history of $x^{\top }(k)Rx(k)$ is displayed in Fig. 12.3, which is far less than the given bound $c_{2}$.

Fig. 12.2

State responses

Fig. 12.3

The history $x^{\top }(k)Rx(k)$

Fig. 12.4

Output responses

Fig. 12.5

The actual disturbance attenuation performance $\gamma _{l}(k)$

The output responses of the closed-loop system are shown in Fig. 12.4. Introduce

$$\begin{aligned} \gamma _{l}(k)=\sqrt{\frac{\sum _{l=0}^{k}z^{\top }(l)z(l)}{\sum _{l=0}^{k}\omega ^{\top }(l)\omega (l)}}, \end{aligned}$$

which indicates the influence of the disturbance input $\omega (k)$ to the controlled output z(k). The evolution of $\gamma _{l}(k)$ is shown in Fig. 12.5. It can be seen that $\gamma _{l}(k)$ is no greater than 0.2962 in the concerned interval, which is less than the prescribed value 0.8641. The simulation results show the effectiveness of the designed controllers.

3 Finite-Time Filtering

In this section, the finite-time $\mathscr {H}_{\infty }$ filtering problem is investigated. Consider the following discrete-time switched linear system:

$$\begin{aligned} x(k+1)= & {} A_{\sigma (k)}x(k)+E_{\sigma (k)}\omega (k), \end{aligned}$$

(12.41)

$$\begin{aligned} y(k)= & {} C_{\sigma (k)}x(k)+D_{\sigma (k)}\omega (k), \end{aligned}$$

(12.42)

$$\begin{aligned} z(k)= & {} G_{\sigma (k)}x(k)+L_{\sigma (k)}\omega (k), \end{aligned}$$

(12.43)

where $x(k) \in \mathbb {R}^{n_{x}}$ is the state vector, $y(k) \in \mathbb {R}^{n_{y}}$ is the output vector, $z(k)\in \mathbb {R}^{n_{z}}$ is the objective signal to be estimated, and $\omega (k)\in \mathbb {R}^{n_{\omega }}$ is the exogenous disturbance satisfying (12.3). $\sigma (k):[0,\infty )\rightarrow \mathscr {M}$ is the switching signal. $A_{i}$, $E_{i}$, $C_{i}$, $D_{i}$, $G_{i}$ and $L_{i}$ are real constant matrices with appropriate dimensions for $\sigma (k)=i\in \mathscr {M}$.

The following full-order filters are designed for the system (12.41)–(12.43):

$$\begin{aligned} x_{f}(k+1)= & {} A_{f\sigma (k)}(q_{k})x_{f}(k)+B_{f\sigma (k)}(q_{k})y(k), \end{aligned}$$

(12.44)

$$\begin{aligned} z_{f}(k)= & {} C_{f\sigma (k)}(q_{k})x_{f}(k)+D_{f\sigma (k)}(q_{k})y(k), \end{aligned}$$

(12.45)

where $x_{f}(k) \in \mathbb {R}^{n_{x}}$ is the state of the filter, $z_{f}(k) \in \mathbb {R}^{n_{z}}$ is the estimation of z(k). $A_{fi}(q_{k})$, $B_{fi}(q_{k})$, $C_{fi}(q_{k})$ and $D_{fi}(q_{k})$ are filter gains to be determined for $i\in \mathscr {M}$, and $q_{k}$ is defined in (12.5) and (12.6).

For $\sigma (k)=i\in \mathscr {M}$, the filtering error systems are with the following structure:

$$\begin{aligned} \tilde{x}(k+1)= & {} \mathscr {A}_{i}(q_{k})\tilde{x}(k)+\mathscr {B}_{i}(q_{k})\omega (k), \end{aligned}$$

(12.46)

$$\begin{aligned} e(k)= & {} \mathscr {C}_{i}(q_{k})\tilde{x}(k)+\mathscr {D}_{i}(q_{k})\omega (k), \end{aligned}$$

(12.47)

where $\tilde{x}(k)=\left[ \begin{array}{cc} x^{\top }(k)&X_{f}^{\top }(k)\end{array}\right]$, $e(k)=z(k)-z_{f}(k)$, and

$$\begin{aligned} \mathscr {A}_{i}(q_{k})= & {} \left[ \begin{array}{cc} A_{i} &{} 0 \\ B_{fi}(q_{k})C_{i} &{} A_{fi}(q_{k}) \end{array}\right] , \mathscr {B}_{i}(q_{k})=\left[ \begin{array}{c} E_{i} \\ B_{fi}(q_{k})D_{i} \end{array}\right] ,\\ \mathscr {C}_{i}(q_{k})= & {} \left[ \begin{array}{cc} G_{i}-D_{fi}(q_{k})C_{i}&-C_{fi}(q_{k}) \end{array}\right] , \mathscr {D}_{i}(q_{k})= L_{i}-D_{fi}(q_{k})D_{i}. \end{aligned}$$

Here, the objective is to design a set of filters (12.44)–(12.45) for the discrete-time switched linear system (12.41)–(12.43), such that the corresponding filtering error system (12.46)–(12.47) is FTB and has a prescribed $\mathscr {H}_{\infty }$ performance index.

Theorem 12.2

Consider the discrete-time switched linear system (12.41)–(12.43). For given scalars $\alpha _{i}>1$, $\mu _{i}>1$, $i \in \mathscr {M}$, integer $\tau _{m}>0$, and a prescribed period of persistence T, if there exist matrices $P_{1i}(\phi )>0$, $P_{3i}(\phi )>0$, and $P_{2i}(\phi )$, $X_{i}(\phi )$, $Y_{i}(\phi )$, $Z_{i}(\phi )$, $A_{Fi}(\phi )$, $B_{Fi}(\phi )$, $C_{Fi}(\phi )$, $D_{Fi}(\phi )$, $i \in \mathscr {M}$, $\phi \in \mathbb {Z}_{[ 0,\tau _{m}]}$, and a scalar $\gamma >0$, such that $\forall (i,j)\in \mathscr {M}\times \mathscr {M}$, $i \ne j$, $\varphi \in \mathbb {Z}_{[ 0,\tau _{m}-1]}$, (12.9) and (12.19) hold, and

$$\begin{aligned} \varOmega _{i}(\varphi +1, \varphi ) < 0, \end{aligned}$$

(12.48)

$$\begin{aligned} \varOmega _{i}(\tau _{m}, \tau _{m}) < 0, \end{aligned}$$

(12.49)

where

$$\begin{aligned} \varOmega _{i} (\iota _{1},\iota _{2})=\left[ \begin{array}{cccc} -\alpha _{i}P_{i}(\iota _{2}) &{} 0 &{} \varOmega _{i13}(\iota _{2}) &{} \varOmega _{i14}(\iota _{2}) \\ *&{} -\gamma ^{2}I &{} \varOmega _{i23}(\iota _{2}) &{} \varOmega _{i24}(\iota _{2}) \\ *&{} *&{} \varOmega _{i33}(\iota _{1},\psi _{2}) &{} 0 \\ *&{} *&{} *&{} -I \end{array} \right] , \end{aligned}$$

with

$$\begin{aligned} \varOmega _{i13}(\iota _{2})= & {} \left[ \begin{array}{cc} A_{i}^{\top }X_{i}^{\top }(\iota _{2})+C_{i}^{\top }B_{Fi}^{\top }(\iota _{2}) &{} A_{i}^{\top }Z_{i}^{\top }(\iota _{2})+C_{i}^{\top }B_{Fi}^{\top }(\iota _{2})\\ A_{Fi}^{\top }(\iota _{2}) &{} A_{Fi}^{\top }(\iota _{2}) \end{array}\right] , \\ \varOmega _{i14}(\psi _{2})= & {} \left[ \begin{array}{c} G_{i}^{\top }-C_{i}^{\top }D_{Fi}^{\top }(\iota _{2})\\ -C_{Fi}^{\top }(\iota _{2}) \end{array}\right] , \\ \varOmega _{i23}(\psi _{2})= & {} \left[ \begin{array}{cc} E_{i}^{\top }X_{i}^{\top }(\iota _{2})+D_{i}^{\top }B_{Fi}^{\top }(\iota _{2})&E_{i}^{\top }Z_{i}^{\top }(\iota _{2})+D_{i}^{\top }B_{Fi}^{\top }(\iota _{2}) \end{array}\right] , \\ \varOmega _{i24}(\iota _{2})= & {} L_{i}^{\top }-D_{i}^{\top }D_{Fi}^{\top }(\iota _{2}), \\ \varOmega _{i33}(\iota _{1},\iota _{2})= & {} P_{i}(\iota _{1})-R_{i}(\iota _{2})-R_{i}^{\top }(\iota _{2}), \\ P_{i}(\iota _{1})= & {} \left[ \begin{array}{cc} P_{1i}(\iota _{1}) &{} P_{2i}(\iota _{1}) \\ *&{} P_{3i}(\psi _{1}) \end{array}\right] \ge 0, R_{i}(\psi _{2})=\left[ \begin{array}{cc} X_{i}(\iota _{2}) &{} Y_{i}(\iota _{2}) \\ Z_{i}(\iota _{2}) &{} Y_{i}(\iota _{2}) \end{array}\right] . \end{aligned}$$

Then, for any MDPDT switching signal satisfying (12.20), the system (12.46)–(12.47) is FTB and has a prescribed $\mathscr {H}_{\infty }$ performance with respect to $(0,c_{2},R,d,\gamma _{s},N,\sigma )$, where $\gamma _{s}$ is given in (12.21). Moreover, for $\phi \in \mathbb {Z}_{[0,\tau _{m}]}$, $i \in \mathscr {M}$, the filter gains are given by

$$\begin{aligned} \left[ \begin{array}{cc} A_{fi}(\phi ) &{} B_{fi}(\phi ) \\ C_{fi}(\phi ) &{} D_{fi}(\phi ) \end{array}\right] = \left[ \begin{array}{cc} Y_{i}^{-1}(\phi ) &{} 0 \\ 0 &{} I \end{array}\right] \left[ \begin{array}{cc} A_{Fi}(\phi ) &{} B_{Fi}(\phi ) \\ C_{Fi}(\phi ) &{} D_{Fi}(\phi ) \end{array}\right] \end{aligned}$$

(12.50)

Proof

The proof procedure is similar to above previous theorems. (12.48) and (12.49) can guarantee (12.17) and (12.18), respectively. According to Lemma 12.2, one can conclude that the corresponding filtering error system is FTB with a prescribed $\mathscr {H}_{\infty }$ performance.

The follow corollary present the time-independent filtering scheme.

Corollary 12.2

Consider the discrete-time switched linear system (12.41)–(12.43). For given scalars $\alpha _{i}>1$, $\mu _{i}>1$, $i \in \mathscr {M}$, and a prescribed period of persistence T, if there exist matrices $P_{1i}>0$, $P_{3i}>0$, and $P_{2i}$, $X_{i}$, $Y_{i}$, $Z_{i}$, $A_{Fi}$, $B_{Fi}$, $C_{Fi}$, $D_{Fi}$, $i \in \mathscr {M}$, and a scalar $\gamma >0$, such that $\forall (i,j)\in \mathscr {M}\times \mathscr {M}$, $i \ne j$, (12.19) holds, and

$$\begin{aligned} \left[ \begin{array}{cccc} -\alpha _{i}P_{i} &{} 0 &{} \varOmega _{i13} &{} \varOmega _{i14} \\ *&{} -\gamma ^{2}I &{} \varOmega _{i23} &{} \varOmega _{i24} \\ *&{} *&{} P_{i}-R_{i}-R_{i}^{\top } &{} 0 \\ *&{} *&{} *&{} -I \end{array} \right] <0, \end{aligned}$$

(12.51)

$$\begin{aligned} P_{i}-\mu _{i}P_{j} \le 0, \end{aligned}$$

(12.52)

$$\begin{aligned} \varOmega _{i13}= & {} \left[ \begin{array}{cc} A_{i}^{\top }X_{i}^{\top }+C_{i}^{\top }B_{Fi}^{\top } &{} A_{i}^{\top }Z_{i}^{\top }+C_{i}^{\top }B_{Fi}^{\top }\\ A_{Fi}^{\top } &{} A_{Fi}^{\top } \end{array}\right] , \varOmega _{i14}=\left[ \begin{array}{c} G_{i}^{\top }-C_{i}^{\top }D_{Fi}^{\top }\\ -C_{Fi}^{\top } \end{array}\right] , \\ \varOmega _{i23}= & {} \left[ \begin{array}{cc} E_{i}^{\top }X_{i}^{\top }+D_{i}^{\top }B_{Fi}^{\top }&E_{i}^{\top }Z_{i}^{\top }+D_{i}^{\top }B_{Fi}^{\top } \end{array}\right] , \varOmega _{i24}=L_{i}^{\top }-D_{i}^{\top }D_{Fi}^{\top }, \end{aligned}$$

and

$$\begin{aligned} P_{i}= & {} \left[ \begin{array}{cc} P_{1i} &{} P_{2i} \\ *&{} P_{3i} \end{array}\right] \ge 0, R_{i}=\left[ \begin{array}{cc} X_{i} &{} Y_{i} \\ Z_{i} &{} Y_{i} \end{array}\right] . \end{aligned}$$

Then, for any MDADT switching signal satisfying (12.20), the system (12.46)–(12.47) is FTB and has a prescribed $\mathscr {H}_{\infty }$ performance with respect to $(0,c_{2},R,d,\gamma _{s},N,\sigma )$, where $\gamma _{s}$ is given in (12.21). Moreover, for $i \in \mathscr {M}$, the filter gains are given by

$$\begin{aligned} \left[ \begin{array}{cc} A_{fi} &{} B_{fi} \\ C_{fi} &{} D_{fi} \end{array}\right] = \left[ \begin{array}{cc} Y_{i}^{-1} &{} 0 \\ 0 &{} I \end{array}\right] \left[ \begin{array}{cc} A_{Fi} &{} B_{Fi} \\ C_{Fi} &{} D_{Fi} \end{array}\right] \end{aligned}$$

(12.53)

Proof

Omitted.

Example 12.2

Consider the discrete-time switched system (12.41)–(12.43) with two subsystems. The system parameters are same as the ones in Example 8.2.

Set $\alpha _{1}=\alpha _{2}=1.005$, $\mu _{1}=1.011$ $\mu _{2}=1.009$, and $\tau _{m1}=\tau _{m2}=2$. Moreover, consider $T=6$, $c_{2}=20$, $d=0.27$, $N=80$ and $R=I$. By using Theorem 12.2, one can obtain feasible solutions with $\tau _{a\min }^{*}=9.3542$, and $\gamma _{s}=0.7824$. The minimum value of $\gamma _{s}$ obtained by Corollary 12.2 is 0.9382, which is larger than the one obtained by Theorem 12.2.

The filter gains obtained by Theorem 12.2 are given as follows:

$$\begin{aligned} A_{f1}(0)= & {} \left[ \begin{array}{cc} 0.9792 &{} -0.1140 \\ -0.2678 &{} 0.0282 \end{array}\right] , B_{f1}(0)=\left[ \begin{array}{c} 0.4658 \\ 1.6687 \end{array}\right] , \\ C_{f1}(0)= & {} \left[ \begin{array}{cc} 0.0158&0.0024 \end{array}\right] , D_{f1}(0)=1.2585; \\ A_{f1}(1)= & {} \left[ \begin{array}{cc} 0.9464 &{} 0.0758 \\ -0.3609 &{} 0.2851 \end{array}\right] , B_{f1}(1)=\left[ \begin{array}{c} 0.1412 \\ 0.8650 \end{array}\right] , \\ C_{f1}(1)= & {} \left[ \begin{array}{cc} -0.0154&0.0984 \end{array}\right] , D_{f1}(1)=1.0091; \\ A_{f1}(2)= & {} \left[ \begin{array}{cc} 0.9021 &{} 0.1328 \\ -0.4000 &{} 0.3083 \end{array}\right] , B_{f1}(2)=\left[ \begin{array}{c} -0.0455 \\ 0.7343 \end{array}\right] , \\ C_{f1}(2)= & {} \left[ \begin{array}{cc} -0.0226&0.1163 \end{array}\right] , D_{f1}(2)=0.9685; \\ A_{f2}(0)= & {} \left[ \begin{array}{cc} 0.4289 &{} 0.0973 \\ 0.6584 &{} 0.2152 \end{array}\right] , B_{f2}(0)=\left[ \begin{array}{c} -1.2685 \\ 1.2487 \end{array}\right] , \\ C_{f2}(0)= & {} \left[ \begin{array}{cc} 0.3162&0.0910 \end{array}\right] , D_{f2}(0)=1.6819; \\ A_{f2}(1)= & {} \left[ \begin{array}{cc} 0.8590 &{} -0.5170 \\ 0.7571 &{} -0.2309 \end{array}\right] , B_{f2}(1)=\left[ \begin{array}{c} 0.5333 \\ 2.2639 \end{array}\right] , \\ C_{f2}(1)= & {} \left[ \begin{array}{cc} 0.7280&-0.3417 \end{array}\right] , D_{f2}(1)=3.1883; \\ A_{f2}(2)= & {} \left[ \begin{array}{cc} 0.6779 &{} -0.2808 \\ 0.9117 &{} -0.3028 \end{array}\right] , B_{f2}(2)=\left[ \begin{array}{c} -0.1749 \\ 2.6388 \end{array}\right] , \\ C_{f2}(2)= & {} \left[ \begin{array}{cc} 0.6113&-0.2160 \end{array}\right] , D_{f2}(2)=2.8030. \end{aligned}$$

Fig. 12.6

A designed switching signal $\sigma (k)$ with $\tau _{a\min }\ge 10$ and $T=6$

Fig. 12.7

The origin output and estimation output responses

Fig. 12.8

The filtering error responses

Fig. 12.9

The history $x^{\top }(k)Rx(k)$

Fig. 12.10

The actual disturbance attenuation performance $\gamma _{l}(k)$

Consider the exogenous disturbance input is $\omega (k)=0.08\sin (0.5k)$. The designed switching signal satisfying the above constraints is given in Fig. 12.6. Under zero initial condition, the origin output and estimation output responses are displayed in Fig. 12.7. The filtering error responses are displayed in Fig. 12.8. The history of $x^{\top }(k)Rx(k)$ is displayed in Fig. 12.9, which is far less than the given bound. The evolution of $\gamma _{l}(k)$ is shown in Fig. 12.10. The values of $\gamma _{l}(k)$ in the concerned interval is no greater than 0.3552, which is less than the prescribed $\mathscr {H}_{\infty }$ performance index. The above simulation results illustrate the effectiveness of designed filters.

4 Conclusion

In this chapter, finite-time performance for discrete-time switched linear systems is investigate with MDPDT switching. Firstly, Sufficient conditions are addressed, which can guarantee that the switched system is FTB with prescribed $\mathscr {H}_{\infty }$ performance. Based on the finite-time $\mathscr {H}_{\infty }$ performance criteria, control and filtering schemes are developed, respectively. Finally, numerical examples are provided to illustrate the effectiveness of the developed methods.