1. Introduction

Recently, multi-mobile robot systems are widely used for military, surveillance, and transportation ^(1,2). When multiple robots move toward a target while maintaining a certain distance or angle, this is called formation ⁽³⁾. In the formation control techniques, the leader-following method has been adopted by many researchers ^(4,5,6). In this method, the leader tracks a predefined path and the follower maintains a desired geometric configuration with the leader. When multiple robots tasks, the inception of distributed robotics is important issues so that communication system has been extensively studied.

The communication network has a number of benefits simple installation and maintenance, and high reliability, increased flexibility and safety. Therefore, many of researchers are focused on this topic ^(7,8).

In networked control system, the input control is delayed according to network-induced delays. The network-induced delays usually consist of two kinds of delays: the communication delays between the controller and the following mobile robots and the communication delays between the controller, the actuator and sampler. The delay may cause instability and performance degradation so that the design of control scheme should be considered with aspects to performances of whole systems ⁽⁹⁾.

In this paper, we propose a sampled-data model predictive control for leader-following multi-mobile robots in network system. To derive the condition, the LPV model ⁽¹⁰⁾ is considered in continuous time which reduces the difference between the dynamics of the nominal closed-loop system and the actual evolution of the state. It is explicitly assumed that the LPV model is updated only at the sampling instants and that the control signal is kept constant between two consecutive sampled by means of a zero order holder, while the plant and the parameters evolve continuously in time. In the case of periodic and aperiodic sampling time, the robustness should be guaranteed so that a quadratic Lyapunov function is considered with new looped- functionals. To deal with the single integral term in the derivative of the Lyapunov function, a generalized free- weighting-matrix (GFWM) ⁽¹²⁾ gives a less conservatism. Finally, we demonstrate the effectiveness of the proposed approach via numerical simulation.

The main contributions of this paper are summarized as follows:

(1) In the modelling aspects, we attempt to consider the modelling of multi-mobile robots in continuous time which is more accurate than the discrete time. Moreover, The MPC technique is not only adequate for the Leader-Follower model represented by error dynamics, but also consider the input saturation constraint.

(2) In the sampled-data LPV systems, based on constructing new looped-functionals and using a GFWM integral inequality, the proposed sampled data MPC design method for LPV systems can get a larger sampling interval upper bound than the existing one ⁽¹³⁾.

Notations: Throughout this paper, ${\mathrm{ℝ}}^n$ denotes the n dimensional Euclidean space, and ${\mathrm{ℝ}}^{m\times n}$ is the set of all n×mreal matrices, For symmetric matrices A and B, the notation A $>$B(respectively, A $\ge$B) means that the matrix A-B is positive definite (respectively, non- negative). diag{...} denotes the block diagonal matrix. * denotes the symmetric part. I denotes identity matrix with appropriate dimensions. Sym(X) denotes X+X^T.

2. Problem formulation

Consider a multi robot system composed of a leader mobile robot and i=1,...n followers. The mobile robots in two dimensions are shown in Fig. 1.

The dynamics of each follower can be represented as

which $v_i\left(\frac{m}{sec}\right)$ is linear velocity, and $w_i\left(\frac{rad}{sec}\right)$ is angular velocity. The leader labeled as i=r has the same dynamics of the followers. To set up the problem, error coordinates between global and local coordination is considered by using the dynamics (1),

By differentiating (2) and substituting (1) into the result, the error dynamics is obtained as

Further on linearizing (3) around operating point $({\overline{x}}_{e,r,i}=0,{\overline{y}}_{e,r,i}=0,{\overline{\theta }}_{e,r,i}=0,{\overline{v}}_i=\hat{v},{\overline{w}}_i=\hat{w})$ results in the following linear model

where $X_{r,i}=[x_{e,r,i}-{\overline{x}}_{e,r,i},y_{e,r,i}-{\overline{y}}_{e,r,i},{\theta }_{e,r,i}-{\overline{\theta }}_{e,r,i}]$, and $U_{r,i}=[v_i-\hat{v},w_i-\hat{w}]$. The system matrices A and B are

To consider the system’s less uncertainty, the range of $\hat{w}\left(t\right)$ has $\hat{w}\in \left[w_r\right(t)-\tilde{w}{w}_r(t)+\tilde{w}]$, then the all solutions of ⁽⁴⁾ can be solved between A₁ and A₂

$A_1\left(t\right)=\left[\begin{array}{ccc}0& w_r\left(t\right)& -\tilde{w}0\\ -\left(w_r\left(t\right)-\tilde{w}\right)& 0& v_r\left(t\right)\\ 0& 0& 0\end{array}\right], A_2\left(t\right)=\left[\begin{array}{ccc}0& w_r\left(t\right)+\tilde{w}& 0\\ -\left(w_r\left(t\right)+\tilde{w}\right)& 0& v_r\left(t\right)\\ 0& 0& 0\end{array}\right]$

which A₁ is the minimum of the $\hat{w}$, and A₂ is the maximum of the $\hat{w}$.

The network-induced input delay is considered, so the control input is defined

where K_r,i is the control gain matrix for t∈[t_k,r,i t_k+1,r,i). Without loss of generality, it is assumed that the sampled time interval is bounded by

where h_r,i(t)=t_k+1,r,i-t_k,r,i and h_M,r,i is the maximum sampled delay. Using sampled signals, the LPV systems of mobile robot (4) is reformulated as delayed LPV model,

To maintain the constant distance between the leader and followers,

where l_x,i,l_y,i,l_θ,i is the safety distance between the leader and th robot in each coordinate.

Lemma 1.⁽¹¹⁾ For any constant matrices of appropriate dimensions Θ₁,Θ₂,Ψ and a scalar τ(t)∈[0,τ_M], the following two conditions are equivalent:

Lemma 2.⁽¹²⁾ Consider X is a differentiable in [a,b]∈ $\mathrm{ℝ}$_n . For matrices R $>$0 and any matrices L and H, the following inequality holds:

where ε₀ is any vector and ε₁=X(b)-X(a),

${ϵ}_2=X\left(b\right)+X\left(a\right)-\frac{2}{b-a}{\int }_a^{b}X\left(s\right)ds.$

Remark 1. From the proof of Lemma 5⁽¹²⁾, the generalized free-matrix inequality can be modified as

${\int }_a^b{ϵ}_0^T\left(X^T{R}^{-1}X\right){ϵ}_{0}ds=(b-a){ϵ}_0^T\left(X^T{R}^{-1}X\right){ϵ}_0$

and

${\int }_a^b\lambda \left(s\right){ϵ}_0^T\left(H^T{R}^{-1}H\right)\lambda \left(s\right){ϵ}_{0}ds=\frac{(b-a)}{3}{ϵ}_0^T\left(H^T{R}^{-1}H\right){ϵ}_0$

respectively.

3. Main Results

The main purpose of this paper is to design a sampled- data MPC. the essence of a MPC scheme is to optimize predictions of process behavior over a sequence of future control inputs. Therefore, the objective function to be minimized can be stated as a quadratic function of the states and control inputs:

where Q,R are weighting matrices. For the given performance index, if the following condition is satisfied

then the upper bound of the performance index can be derived instead of directly minimizing performance index.

Before deriving conditions, following notations are defined.

d_r,i(t)=t-t_k,r,i,

e₁=[I 0 0 0], e₂=[0 I 0 0], ··· , e₄=[0 0 0 I],

F₁=[I -I 0 0], F₂=[I I 0 -2I],

$\overline{f}$_s=[A_s M BY -M 0], s=1,2,

f_s=[A_s BK -I 0],

${\zeta }_{r,i}\left(t_{k,r,i}\right)=\left[X_{r,i}^T\right(t\left)X_{r,i}^T\right(t_{k,r,i}\left)\dot{X_{r,i}^T}\right(t)\frac{1}{t-t_{k,r,i}}{\int }_{t_{k,r,i}}^t{X}_{r,i}^T\left(s\right)ds{]}^T.$

Theorem 1. For a given with maximum sampling interval h_M, the continuous system (1) is asymptotically stabilizable if there exist matrices $\dot{P}>0$, $S=\left[\begin{array}{cc}{S}_1& S_2\\ *& S_3\end{array}\right]>0$, M, $\overline{L}$, $\overline{H}$, U_M,r,i and the control input at time instant t_k,r,i guarantees the performance index (13) with γ_r,i.

where

$U_{M,r,i}=\left[\begin{array}{cc}{u}_{M,r,i}^{11}& 0\\ *& u_{M,r,i}^{22}\end{array}\right],\begin{array}{c}{u}_{M,r,i}^{11}\le {\left(v_i-\hat{v}\right)}_{\max }^2\\ u_{M,r,i}^{22}\le {\left(w_i-\hat{w}\right)}_{\max }^2\end{array}$,

${\Xi }_1^s=\left[\begin{array}{cccc}{\Xi }_{11}^s& {\Xi }_{12}^{}& h_{M,r,i}·\overline{L}& h_{M,r,i}·\overline{H}\\ *& {\Xi }_{22}^{}& 0& 0\\ *& *& -h_{M,r,i}·{\overline{S}}_3& 0\\ *& *& *& -h_{M,r,i}·{\overline{S}}_3\end{array}\right]$,

${\Xi }_2^s=\left[\begin{array}{cc}{\Xi }_{11}^{2s}& {\Xi }_{12}^{}\\ *& {\Xi }_{22}^{}\end{array}\right]$,

${\Xi }_{12}^{}=diag[M,Y]$,

${\Xi }_{22}^{}=diag[-{\gamma }_{r,i}{Q}^{-1},-{\gamma }_{r,i}{R}^{-1}]$,

with

$$\begin{gathered} {\Xi }_1=e_1^T{P}_0{e}_3+(e_1^T{P}_0{e}_3{)}^T+(e_1-e_2{)}^T{\overline{P}}_1{e}_3+\left(\right(e_1-e_2{)}^T{\overline{P}}_1{e}_3{)}^T \\ +Sym({\overline{Z}}_1{F}_1+{\overline{Z}}_2{F}_2)-e_2^T{\overline{S}}_2(e_1-e_2)-\left(e_2^T{\overline{S}}_2\right(e_1-e_2){)}^T \\ +(e_1+\alpha e_4{)}^T{\overline{f}}_i+\left(\right(e_1+\alpha e_4{)}^T{\overline{f}}_i{)}^T \end{gathered}$$,

${\Xi }_2=e_4^T{\overline{P}}_1{e}_1+(e_4^T{\overline{P}}_1{e}_1{)}^T-e_2^T{\overline{S}}_1{e}_2-e_4^T\overline{Z}{e}_2-(e_4^T\overline{Z}{e}_2{)}^T$,

${\Xi }_3={\left[\begin{array}{c}{e}_2\\ e_3\end{array}\right]}^T\overline{S}\left[\begin{array}{c}{e}_2\\ e_3\end{array}\right]+e_1^T\overline{Z}{e}_2+{\left(e_1^T\overline{Z}{e}_2\right)}^T$.

In addition, the state feedback gains are given as K_r,i=YM^-1.

Proof. Choosing the following Lyapunov function for t∈[t_{k, r, i}, t_{k+1,r, i}] yields

where

$V_{1,r,i}\left(t\right)=X_{r,i}^T\left(t\right)M^{-T}{P}_0{M}^{-1}{X}_{r,i}\left(t\right)$,

$V_{2,r,i}\left(t\right)={\left[\begin{array}{c}{X}_{r,i}\left(t\right)-X_{r,i}\left(t_{k,r,i}\right)\\ {\int }_{t_{k,r,i}}^t{X}_{r,i}\left(s\right)ds\end{array}\right]}^T{P}_1\left[\begin{array}{c}{X}_{r,i}\left(t\right)-X_{r,i}\left(t_{k,r,i}\right)\\ {\int }_{t_{k,r,i}}^t{X}_{k,r,i}ds\end{array}\right]$

$V_{3,r,i}\left(t\right)=\left(t_{k+1,r,i}-t\right)\left({\int }_{t_{k,r,i}}^t{\left[\begin{array}{c}{X}_{r,i}\left(t_{k,r,i}\right){]}^T\\ {\dot{X}}_{r,i}\left(t\right)\end{array}\right]}^{T}S\left[\begin{array}{c}{X}_{r,i}\left(t_{k,r,i}\right)\\ {\dot{X}}_{r,i}\left(t\right)\end{array}\right]ds\right)+2{\int }_{t_{k,r,i}}^t{X}_{r,i}^T\left(s\right)dsZ{X}_{r,i}\left(t_{k,r,i}\right)$

Differentiate the Lyapunov function

From Lemma 2, the following holds

where L, H are auxiliary variables. Taking into account system dynamics (8),

Summing up from (22) to (26) leads to

where

Pre-and post-multiplying with a matrix ${\gamma }_{r,i}^{\frac{1}{2}}\times \{diagM,M,M,M\}$, the followings are satisfied with Lemma 1.

where $\overline{S}=MSM,\overline{L}=MLM,\overline{H}=MHM,$ and K_r,i=YM^-1. Using Schur complement, the equations in (18) and (19) are equivalent to those of (11) and (12). For every sampling instance, V_2,r,i and V_3,r,i vanish. Then, the upper bound of Lyapunov function is expressed in terms of V_1,r,i.

where γ_r,i denotes the bound of optimal performance index. The input saturation is considered similar to the method in [14]⁽¹⁴⁾. This ends the proof.

4. Numerical Examples

The dynamical equation (8) is considered as

where

$A_1\left(t\right)=\left[\begin{array}{ccc}0& w_r\left(t\right)-0.03& 0\\ -\left(w_r\left(t\right)-0.03\right)& 0& v_r\\ 0& 0& 0\end{array}\right]$,

$A_2\left(t\right)=\left[\begin{array}{ccc}0& w\left(t\right)+0.03& 0\\ -\left(w_r\left(t\right)+0.03\right)& 0& v_r\left(t\right)\\ 0& 0& 0\end{array}\right]$,

$B=\left[\begin{array}{cc}-1& 0\\ 0& 0\\ 0& -1\end{array}\right]$.

The model parameters are calculated with a sampling time 0.8s. The sampling time h_r,1 is less than 0.8s. Along the reference trajectory (v_r=0.2, w_r=0.2), the input is constrained to $-0.3\le (v_1-\hat{v})\le 0.3$ and $-0.3\le (w_1-\hat{w})\le 0.3(\tilde{w}=0.03)$. The corresponding controller gain matrix is

Fig. 2 and Fig. 3 show the simulation results which are obtained with the above controller gain, taking Q=I, R=I, α=0.1.

5. Conclusions

The sampled-data MPC method for multi-mobile robot systems have been investigated by considering polytopic LPV model. Based on the quadratic Lyapunov function approach, sufficient conditions for the sampled-data MPC controller are derived by constructing new looped-functionals. The proposed method guarantees a performance and stability in much longer sampling delay than the existing paper. The effectiveness of the presented method has been verified by illustration numerical simulation.