LMI Stability Condition for NCS with Packet

Delay and Event-triggered Control

M. Sami Fadali

EBME Department, University of Nevada, Reno, NV 89557, U.S.A.

Keywords: Networked Control, Stability, Packet Delay, Linear Matrix Inequality (LMI).

Abstract: This paper presents a controller design for networked control systems (NCS) with packet delay and event-

triggered control. The total network delay is assumed to be an integer multiple of a fixed sampling period so

systems where the controller receives information

from the plant and delivers control commands to the

actuator through a communication network (Antaklis,

et al., 2007; Li, et al., 2015). The shared network

connection between different components of the

control loop yields a flexible architecture and reduced

network by using event-triggered control (Yang,

With event-triggered control or with packet delay,

the interval between updates of the control signal

varies. This variation results in a system that switches

current control update. Switching requires careful

design to ensure that the switched system remains

stable and perform satisfactorily.

arbitrary packet delays. Although the result was

intended for the design of gain scheduled control

systems, a special case of the result allows us to

exploit it for the design of NCS. The NCS model is

corresponding to different packet delays. The

switching regime associated with the packet delays.

In addition, if the system is designed for switching at

multiples of the sampling period, it is robust with

respect to random variations in the sampling period.

Thus, the sampling period need not be known exactly.

(Montestruque and Antaklis, 2004) and some

properties of switched systems. Section 3 presents our

controller design methodology, which is the main

result of this paper. Section 4 presents simulation

Consider the linear plant

. We adopt

the NCS model of (Montestruque and Antaklis, 2004)

and investigate the stability and controller design for

The plant model is not exactly known, and the

nominal model of the system with the matrices of the

same order as their true counterparts in (1) and (2) is

Subtracting the nominal from the actual dynamics

gives the error dynamics

sufficiently faster sampling than the Nyquist

.

is a variable but bounded integer,

.

two consecutive arriving packets does not

Discretizing the system with different sampling

between the models

(19)

Theorem 1. 0(Zhai et al., 2002) If all state matrices

exponentially stable under arbitrary switching.

The result clearly applies in the case of Schur

switched system (19) is globally exponentially stable

under arbitrary switching between sampling periods

The stability condition is valid if the sampling period

varies randomly because the state matrices remain

properties of the matrix exponential.

under arbitrary switching is for all subsystem

matrices to be Schur stable. Otherwise, switching to

an unstable subsystem and subsequently remaining

2017) provides stability conditions for a system that

switches between different linear models.

Theorem 2. (Pandey and Oliveira, 2017)

Consider a time-varying discrete-time linear system

of the form

The system is stable with the control

We form the matrices of the augmented system

event triggered control or packet delay. The model

solver. The solutions can be improved by imposing

constraints on the norms of the matrices. Solving the

LMIs gives the gain matrices

The design obtained by solving the LMIs results in a

Using 100 Monte Carlo simulations for the system

under different conditions, we compare the

simulated (a) with switching between sampling

1 and 2. Figure 3 shows the random switching

Because the LMI for the delay that results in doubling

the sampling period gives a faster response, contrary

to intuition, the response is faster for the lower

well for both probabilities, as do others not included

in the paper.

Figure 2: Plot of x

variation Δh around h,Δh~U[−0.1h,0.1h].