Controller Design and Analysis of Multi-agent Linear Systems Based on $H_{\infty }$ Index

Abstract

This study deals with the multi-agent linear system, which is a more realistic and accurate discrete model with disturbance terms. Based on linear matrix inequality technology, Lyapunov stability theory and $H_\infty$ index, we give the convergence result. Results of simulation show that the proposed method is effective.

1 Introduction

Linear matrix inequality (LMI) is an effective mathematical tool, and is widely used in the design of control systems with constrained control inputs [1,2,3]. G. Grimm et al. designed a dynamic compensator based on LMI for the general stability model system, which guaranteed the stability of the system. The system output is guaranteed to have $L_2$ gain to external interference [4]. H. S. Hu et al. studied the gain characteristics and stable region of $L_2$ based on LMI for general systems [5]. H. N. Wu et al. used LMI to realize the fuzzy control under control input constraints for the coupled model of ordinary differential and partial differential. S. D. Majumder entailed a finite-element design of a 3d-autopilot missile synthesis of multiple objective controls by solving the inequalities encountered while using a linear matrix [6]. In paper [7], the regression problem is considered. The key point of the paper is finding a matrix approximation of the Koopman operator from data. They formulated the regression problem as a convex optimization problem subject to LMI constraints.

An LMI approach to $H_{-}$ index and mixed $H_{-}/H_{\infty }$ fault detection observer design is proposed in [8]. The finite-horizon $H_\infty$ containment control issue for a general discrete time-varying linear multiagent systems with multileaders is investigated in [9]. A sufficient condition is established to ensure the desired $H_\infty$ containment performance. Then the desired controller and observer parameters are obtained by solving two coupled backward recursive Riccati difference equations [9]. Containment control problems for high-order linear time-invariant multi-agent systems with fixed communication time-delays are investigated in [10]. Based on the linear matrix inequality method and Lyapunov-Krasovskii functional method, the feedback gains, sufficient conditions on the communication digraph and the allowed upper bound of the delays are given [10]. The lecture [11] investigates finite-time containment control problem for second-order multi-agent systems with norm-bounded non-linear perturbation. The discontinuous control protocol is designed and the appropriate value range of control parameter is obtained by applying finite-time stability analysis [11]. The $H_{\infty }$ control problem for Lur’e singular systems with time delays is considered in [12]. By using Lyapunov stability theory, sufficient conditions for the system to be exponentially stable and satisfy the performance index of $H_{\infty }$ are obtained based on the LMI method.

The main contributions and primary distinctions of this paper with other works can be given as follows.

• 1. A more accurate and realistic discrete time model is proposed which is relevant for many practical sampled data systems;

• 2. Sufficient conditions of the convergence result is established based on LMI and the $H_\infty$ index.

• 3. Results of simulation show that the proposed method is effective.

The organization of this paper is as follows: In Sect. 2, we give the systems definition about the multi-agent linear systems. In Sect. 3, we present the convergence result and give the LMI. In Sect. 4, a simulation example is presented.

2 Systems Definition

We consider the multi-agent linear systems

$$\begin{aligned} \dot{x}_i=A_ix_i(t)+B_i(u_i+d_i),\ i=1,\ldots ,N, \end{aligned}$$

(1)

where $x_i=[x^1_i\ x^2_i]^{T}$, $u_i$ are the state and control input of the i th agent, $d_{i}$ is a disturbance term.

$$\begin{aligned} u_i=K_ix_i,\ i=1,\ldots ,N, \end{aligned}$$

(2)

where $K_i=[k^1_i\ k^2_i]$. Our aim is to design LMI to solve the $K_i$ and to achieve that $x_i\rightarrow 0$.

3 The Convergence Result and LMI

Design the Lyapunov function as following:

$$\begin{aligned} V_i=x^T_iP_ix_i,\ i=1,\ldots ,N, \end{aligned}$$

(3)

where $P_i>0$, $P_i=P^T_i$.

The convergence effect of x can be effectively adjusted by the design of $P_i$. It is also beneficial to the solution of LMI. Then

$$\begin{aligned} \dot{V}_i= & {} \dot{x}_i^{T}P_ix_i+x_iP_i\dot{x}_i\\= & {} (A_ix_i+B_iu_i+B_id_i)^TP_ix_i+x^T_iP_i(A_ix_i+B_iu_i+B_id_i)\\= & {} (A_ix_i+B_iu_i+B_id_i)^TP_ix_i+x^T_iP_i(A_ix_i+B_ik_ix_i+B_id_i)\\= & {} x^T_i(A_i+B_iK_i)^T P_ix_i+x^T_iP_i(A_i+B_iK_i)x_i+(B_id_i)^TP_ix_i+x^T_iP_i(B_id_i)\\= & {} x^T_i(Q_i^1)^Tx_i+x^T_iQ_i^1x_i+(B_i^TP_ix_i+x_i^TP_iB_i)d_i, \end{aligned}$$

where $Q_i^1=P_i(A_i+B_iK_i)$, $Q_i=Q_i^1+(Q_i^1)^T$.

Let $\eta _i=[x^T\ d_i]^T$, then

$$\begin{aligned} \eta _i=\left[ \begin{array}{ccc} x_i\\ d_i \end{array}\right] = \left[ \begin{array}{ccc} x^1_i\\ x^2_i\\ d_i \end{array}\right] ,\ \ \eta _i^T=[x^1_i\ x^2_i\ d_i]. \end{aligned}$$

(4)

Hence, we have

$$\begin{aligned} \dot{V}_i= & {} x^T_i Q_ix_i+\eta ^T_i\left[ \begin{array}{ccc} 0 &{} P_iB_i\\ B^T_iP_i&{} 0 \end{array}\right] \eta _i = \eta ^T_i\left[ \begin{array}{ccc} Q_i &{} P_iB_i\\ B^T_iP_i&{} 0 \end{array}\right] \eta _i, \end{aligned}$$

(5)

where

$$\begin{aligned} x^T_i Q_i x_i=[x^T_i\ d_i]\left[ \begin{array}{ccc} Q_i &{} 0 \\ 0 &{} 0 \end{array}\right] \left[ \begin{array}{ccc} x_i\\ d_i \end{array}\right] =\eta _i^T\left[ \begin{array}{ccc} Q_i &{} 0\\ 0&{} 0 \end{array}\right] \eta _i, \end{aligned}$$

(6)

$$\begin{aligned} (B^T_iP_ix+x^T_iP_iB_i)d_i= & {} [d_i B^T_iP_i\ \ x^TP_iB_i]\left[ \begin{array}{ccc} x_i\\ d_i \end{array}\right] \\= & {} [x^T_i\ d_i]\left[ \begin{array}{ccc} 0&{} P_i B_i\\ B^T_iP_i&{} 0 \end{array}\right] \left[ \begin{array}{ccc} x_i\\ d_i \end{array}\right] \\= & {} \eta ^T_i\left[ \begin{array}{ccc} 0 &{} P_i B_i\\ B^T_iP_i&{} 0 \end{array}\right] \eta _i, \end{aligned}$$

then assume that the output is $Z_i=C_ix_i$ and $H_\infty$ index takes

$$\begin{aligned} \int _0^t Z^T_iZ_idt<\int ^t_0 \xi ^2_id^2_i(t)dt+V(0), \end{aligned}$$

(7)

where $\xi _i>0$ and $C_i=\left[ \begin{array}{ccc} 1 &{} 0\\ 0&{} 1 \end{array}\right]$.

Since $Z^T_i Z_i-\xi ^2_id^2_i=x^T_iC^T_iC_ix_i-\xi ^2_id^2_i$, and

$$\begin{aligned} \eta ^T_i\left[ \begin{array}{ccc} C^t_iC_i &{} 0\\ 0&{} -\xi ^2_i \end{array}\right] \eta _i= & {} [x^T_i\ \ d_i]\left[ \begin{array}{ccc} C^t_iC_i &{} 0\\ 0&{} -\xi ^2_i \end{array}\right] \left[ \begin{array}{ccc} x_i\\ d_i \end{array}\right] \\= & {} [x^T_i C^T_iC_i-\xi ^2_i d_i]\left[ \begin{array}{ccc} x_i\\ d_i \end{array}\right] \\= & {} x^T_i C^T_iC_ix_i-\xi ^2_i d_i^2, \end{aligned}$$

then

$$\begin{aligned} Z^T_i Z_i -\xi ^2_id^2_i=\eta ^T_i\left[ \begin{array}{ccc} C^T_iC_i&{} 0\\ 0 &{}-\xi ^2_i \end{array}\right] \eta _i. \end{aligned}$$

(8)

Moreover, we find that

$$\begin{aligned} \dot{V}_i+Z^T_iZ_i-\xi ^2_id^2_i=\eta _i^T\left[ \begin{array}{ccc} C^T_iC_i+Q_i&{} P_iB_i\\ 0 &{}-\xi ^2_i \end{array}\right] \eta _i. \end{aligned}$$

(9)

Supposed that

$$\begin{aligned} \Theta =\left[ \begin{array}{ccc} C^T_iC_i+Q_i&{} P_iB_i\\ 0 &{}-\xi ^2_i \end{array}\right] <0, \end{aligned}$$

(10)

then

$$\begin{aligned} \dot{V}_i+Z^T_iZ_i-\xi ^2_id^2_i\le 0. \end{aligned}$$

(11)

Integral (11) from 0 to t, we get

$$\begin{aligned} V_i(t)-V_i(0)+\int ^t_0Z^T_iZ_idt\le \int ^t_0\xi ^2_id^2_idt. \end{aligned}$$

(12)

Assume that $d_i$ is a decreasing disturbance signal, and we choose that $\int ^\infty _0d^2_idt\le (\xi _i)_{\max }^{-2}\alpha ^{max}_{i}$, and the definitions of $\alpha ^{max}_{i}$ and $P_i^{\min }$ are similar to those of $\nu _{max}$ and $P_{min}$ in [13]. Due to $0\le \int ^t_0Z^T_iZ_idt$. It is clear that

$$\begin{aligned} V_i(t)-V_i(0)\le (\xi _i)^2_{\max }\int ^t_0d^2_idt\le \alpha ^{max}_{i},\end{aligned}$$

(13)

$$\begin{aligned} V_i(t)\le \alpha ^{max}_{i}+V_i(0). \end{aligned}$$

(14)

Thus, we have

$$\begin{aligned} P_i^{\min }\Vert x_i\Vert ^2\le x^T_iP_i x_i\le \alpha ^{max}_{i}+V_i(0), \end{aligned}$$

(15)

and the convergence result is given by

$$\begin{aligned} \Vert x_i\Vert ^2 \le \frac{\alpha ^{max}_{i}+V_i(0)}{P_i^{\min }}. \end{aligned}$$

(16)

It is not hard to get following theorem.

Theorem 1

Assume that $H_\infty$ index satisfying

$$\begin{aligned} \int _0^t Z^T_iZ_idt<\int ^t_0 \xi ^2_id^2_i(t)dt+V(0), \end{aligned}$$

(17)

and the output $Z_i=C_{i}x_{i}$, where $\xi _i>0$ and $C_i=\left[ \begin{array}{ccc} 1 &{} 0\\ 0&{} 1 \end{array}\right]$, then

$$\begin{aligned} \Vert x_i\Vert ^2 \le \frac{\alpha ^{max}_{i}+V_i(0)}{P_i^{\min }}. \end{aligned}$$

(18)

From (10), we know that

$$\begin{aligned} \left[ \begin{array}{ccc} P_i A_i+P_i B_i K_i+A^T_iP_i +K^TB^T_iP_i&{} P_iB_i\\ 0 &{}-\xi ^2_i \end{array}\right] <0, \end{aligned}$$

(19)

then multiply $\left[ \begin{array}{ccc} P_i^{-1}&{} 0\\ 0 &{}I \end{array}\right]$ to the left and right side of (19), we have

$$\begin{aligned} \left[ \begin{array}{ccc} A_i P^{-1}_i+B_i K_i P^{-1}_{i}+P^{-1}_iA^T_i +P^{-1}_iK^TB^T_i&{} B_i\\ B^T_i &{}-\xi ^2_i \end{array}\right] <0. \end{aligned}$$

(20)

Let $F_i=K_iP^{-1}_i$, $N=P^{-1}$, it is easy obtain the first LMI,

$$\begin{aligned} \left[ \begin{array}{ccc} A_i N_i+B_iF_i+N_iA^T +F^{T}_{i}B_i^T&{} B_i\\ B^T_i &{}-\xi ^2_i \end{array}\right] <0. \end{aligned}$$

(21)

According to the definition of $P_i$, we give the second LMI,

$$\begin{aligned} N_i=N_i^T, N_i>0. \end{aligned}$$

(22)

Through the two LMI (21) and (22), we could get $K_i$.

4 Simulation

We consider the system

$$\begin{aligned} \ddot{\theta }_1=-15\dot{\theta }_1+113(u_1(t)+d(t)),\end{aligned}$$

(23)

$$\begin{aligned} \ddot{\theta }_2=-20\dot{\theta }_2+123(u_2(t)+d(t)),\end{aligned}$$

(24)

$$\begin{aligned} \ddot{\theta }_3=-25\dot{\theta }_3+133(u_3(t)+d(t)), \end{aligned}$$

(25)

according to (1), choose $d(t)=0.1e^{-6t}$, $\xi _1=3,\xi _2=5,\xi _3=7$ the initial values are $x_i(0)=[0,01\ \ 0]$, for $i=1,2,3$. And

$$\begin{aligned} x_1^1=\theta _1,\ x_1^2=\dot{\theta }_1, x_2^1=\theta _2,\ x_2^2=\dot{\theta }_2, x_3^1=\theta _3,\ x_3^2=\dot{\theta }_3,\end{aligned}$$

(26)

$$\begin{aligned} b_1=\frac{15}{113},\ J_1=\frac{1}{113}, b_2=\frac{20}{123},\ J_2=\frac{1}{123}, b_3=\frac{25}{133},\ J_3=\frac{1}{133}. \end{aligned}$$

(27)

$$\begin{aligned} A_i=\left[ \begin{array}{ccc} 0&{} 0\\ 0 &{}-\frac{b_i}{J_i} \end{array}\right] , B_i=\left[ \begin{array}{ccc} 0\\ -\frac{b_i}{J_i} \end{array}\right] ,\ i=1,2,3. \end{aligned}$$

(28)

By using Matlab to solve LMI (21) and (22), we find that

$$\begin{aligned} K_1=[-0.0116\ \ 0.1244], K_2=[-0.0107\ \ 0.1550], K_3=[-0.0099\ \ 0.1809]. \end{aligned}$$

(29)

Furthermore, we give the figures (Figs. 1, 2, 3, 4, 5 and 6) of state responds and control input for the system in the following.

Fig. 1.

State responds

Fig. 2.

Control input

Fig. 3.

State responds

Fig. 4.

Control input

Fig. 5.

State responds

Fig. 6.

Control input

5 Conclusion

The establishment of $H_\infty$ control theory has changed the tendency that modern control theory is too mathematical, so it is more convenient for engineers to learn and design. The main mathematical tools has used include modern algebra and operator theory. Whether the design index can be realized or not depends on the accuracy of the mathematical model. However, due to the objective reality, there are various uncertain factors which do not meet the ideal assumptions. Therefore, it is impossible to obtain an accurate mathematical model. This model uncertainty directly affects the application of control theory in practice. $H_\infty$ control is based on system design of uncertain imprecise model. In this paper, by using linear matrix inequality technology, Lyapunov stability theory and $H_\infty$ index, we obtain the convergence result for the multi-agent linear system with disturbance.