Robust Finite-Time Output Feedback $H_\infty$ Control for Stochastic Jump Systems with Incomplete Transition Rates

Abstract

This article aims to investigate the problem of robust finite-time output feedback $H_\infty$ control for stochastic jump systems with incomplete transition rates. Firstly, for the nominal stochastic jump systems, the sufficient conditions for the finite-time boundedness and finite-time output feedback stabilization are developed, respectively. Then, a robust finite-time $H_\infty$ output feedback controller is designed by means of linear matrix inequalities. A key point of this work is to relax the special requirement of completely known transition rates to more general form that mixes two cases of completely known and completely unknown transition rates. Finally, a numerical example is given to demonstrate the applicability of the main results.

Appendix

1.1 Proof of Lemma 2

Proof

For the stochastic jump system (1) ($u(t)=0, v(t)=0$, and $F(t, r_t)=0$), choose a Lyapunov function candidate as (30). Then, by Definition 4, we obtain

$$\begin{aligned} \mathcal {L}{V(x(t),i)}&= x^T(t)\bigg [A_i^TP_i+P_iA_i+\sum _{j=1}^N{\pi _{ij}}P_j\bigg ]x(t) \nonumber \\&+\,x^T(t)G_i^TP_iG_ix(t). \end{aligned}$$

(61)

If the transition rates are not accessible completely, the following equation hold for arbitrary symmetric matrices $Q_i$ due to $\sum _{j=1}^N{\pi _{ij}}Q_i=0$

$$\begin{aligned} \mathcal {L}{V(x(t),i)}&= x^T(t)\bigg [A_i^TP_i+P_iA_i+{\sum _{j=1}^N}{\pi _{ij}}P_j-\sum _{j=1}^N{\pi _{ij}}Q_i\bigg ]x(t)\nonumber \\&+\,x^T(t)G_i^TP_iG_ix(t) \nonumber \\&= x^T(t)\bigg [A_i^TP_i+P_iA_i+\sum _{j\in L_k^i}{\pi _{ij}}(P_j-Q_i)+\sum _{j\in L_{uk}^i}{\pi _{ij}}(P_j-Q_i)\nonumber \\&+\,G_i^TP_iG_i\bigg ]x(t). \end{aligned}$$

(62)

If $i\in L_k^i$(the elements of the diagonal are known), by inequalities (14) and (15), the following inequality holds

$$\begin{aligned} \displaystyle {} \mathcal {L}{V(x(t),i)}\le \alpha V(x(t),i). \end{aligned}$$

(63)

If $i\in L_{uk}^i$(the elements of the diagonal are unknown), according to inequalities (14–16), the inequality (63) holds. Multiplying (63) by $e^{-\alpha t}$ yields

$$\begin{aligned} \mathcal {L}({e^{-\alpha t}}{V(x(t),i)})\le 0. \end{aligned}$$

(64)

According to Dynkin’s formula for (64), we get

$$\begin{aligned} \displaystyle {}{e^{-\alpha t}}{V(x(t),i)}-{V(x_0,t_0)}\le 0, \end{aligned}$$

(65)

which shows

$$\begin{aligned} V(x(t),i)\le e^{\alpha t}V(x_0,t_0). \end{aligned}$$

(66)

This together with $\tilde{P}_i={H_i^{-1/2}}P_i{H_i^{-1/2}}$ gives rise to

$$\begin{aligned} \begin{array}{ll} V(x(t),i)\le e^{\alpha t} {c_1} \lambda _{\max }(\tilde{P}_i). \end{array} \end{aligned}$$

(67)

Consider

$$\begin{aligned} V(x(t),i)=x^T(t){P_i}x(t) \ge \lambda _{\min }(\tilde{P}_i) x^T(t){H_i}x(t). \end{aligned}$$

(68)

For $\forall t\in [0, T]$, we obtain

$$\begin{aligned} \mathbb {E}\{x^T(t){H_i}x(t)\}\le {c_1}e^{\alpha t}\frac{{\lambda _{\max }(\tilde{P}_i) }}{\lambda _{\min }(\tilde{P}_i)}<c_2. \end{aligned}$$

(69)

This completes the proof. $\square$

1.2 Proof of Lemma 3

Proof

For the stochastic jump system (1) ($u = 0$ and $F(t, r_t)=0$), choose a Lyapunov function candidate as (30). Based on Definition 4, we have

$$\begin{aligned} \mathcal {L}{V(x(t),i)}&= x^T(t)\big [A_i^TP_i+P_iA_i+\sum _{j=1}^N{\pi _{ij}}P_j+G_i^TP_iG_i\big ]x(t)\nonumber \\&+\,v^T(t)E_{xi}^TP_ix(t)+x^T(t)P_i E_{xi}v(t). \end{aligned}$$

(70)

Since $\sum _{j=1}^N{\pi _{ij}}Q_i=0$ is always true for arbitrary symmetric matrices $Q_i$, (70) can be rewritten as

$$\begin{aligned} \mathcal {L}{V(x(t),i)}&= x^T(t)\left[ A_i^TP_i+P_iA_i+{\sum _{j=1}^N}{\pi _{ij}}P_j-\sum _{j=1}^N{\pi _{ij}}Q_i+G_i^TP_iG_i\right] x(t)\nonumber \\&+\,v^T(t)E_{xi}^TP_ix(t) \nonumber \\&+\,x^T(t)P_i E_{xi}v(t)\nonumber \\&= x^T(t)\Big [A_i^TP_i+P_iA_i+\sum _{j\in L_k^i}{\pi _{ij}}(P_j-Q_i)+\sum _{j\in L_{uk}^i}{\pi _{ij}}(P_j-Q_i)\nonumber \\&+\,G_i^TP_iG_i\Big ]x(t) \nonumber \\&+\,v^T(t)E_{xi}^TP_ix(t)+x^T(t)P_i E_{xi}v(t). \end{aligned}$$

(71)

Notice that $\pi _{ij}\ge 0$ for all $i\ne j$, and $\pi _{ii}=-\displaystyle {}{\sum _{j=1,i\ne j}^{N}}{\pi _{ij}}<0$ for all $i\in \mathcal {M}$, if $i\in L_k^i$ (the elements of the diagonal are known), by inequalities (18) and (19), the following inequality holds

$$\begin{aligned} \displaystyle {} \mathcal {L}{V(x(t),i)}\le \alpha V(x(t),i)+ \gamma ^2 v^T(t)v(t). \end{aligned}$$

(72)

If $i\in L_{uk}^i$(the elements of the diagonal are unknown), according to inequalities (18–20), the inequality (72) holds. Multiplying (72) by $e^{-\alpha t}$ yields

$$\begin{aligned} \mathcal {L}({e^{-\alpha t}}{V(x(t),i)})\le \gamma ^2 e^{-\alpha t}v^T(t)v(t). \end{aligned}$$

(73)

Using Dynkin’s formula to (73), we obtain

$$\begin{aligned} \displaystyle {}{e^{-\alpha t}}{V(x(t),i)}-{V(x_0,t_0)}\le \gamma ^2 \int _{0}^{t}e^{-\alpha s}v^T(t)v(t) ds, \end{aligned}$$

(74)

which in turn shows

$$\begin{aligned} V\big (x(t),i\big )&\le e^{\alpha t}V(x_0,t_0)+\gamma ^2 e^{\alpha t} \int _{0}^{t}e^{-\alpha s}v^T(s)v(s) ds \nonumber \\&\le e^{\alpha t}\left[ V(x_0,t_0)+\gamma ^2 d \frac{1-e^{-\alpha t}}{\alpha }\right] . \end{aligned}$$

(75)

This together with $\tilde{P}_i={H_i^{-1/2}}P_i{H_i^{-1/2}}$ gives rise to

$$\begin{aligned} \begin{array}{ll} V(x(t),i)\le e^{\alpha t} \big [{c_1} \lambda _{\max }(\tilde{P}_i)+\gamma ^2d \frac{(1-e^{-\alpha t})}{\alpha }\big ]. \end{array} \end{aligned}$$

(76)

Taking into account the fact that

$$\begin{aligned} V(x(t),i)=x^T(t){P_i}x(t) \ge \lambda _{\min }(\tilde{P}_i) x^T(t){H_i}x(t), \end{aligned}$$

(77)

$\forall t\in [0, T]$, we have

$$\begin{aligned} \mathbb {E}\{x^T(t){H_i}x(t)\}\le \frac{e^{\alpha t}\big [{c_1}{\lambda _{\max }(\tilde{P}_i)+\gamma ^2d \frac{(1-e^{-\alpha t})}{\alpha }}\big ]}{\lambda _{\min }(\tilde{P}_i)}<c_2. \end{aligned}$$

(78)

This completes the proof. $\square$