## 2 LS-SVM Time-Delay Prediction Model  
The time delay in the networked control system includes: the sensor for the controller time delay \(\tau_{sc}\), the controller to the actuator time delay \(\tau_{ca}\) and the controller calculated time delay \(\tau_{c}\), shown as in Figure 1.  
The sensor for the controller time delay and the controller to actuator time delay do not have the same variation rules, and the effect on the system performances is not the same either. Under normal circumstances at time \(k\), the sensor to the controller time delay \(\tau_{sc}(k)\) can be measured, and the controller to actuator time delay \(\tau_{ca}(k)\) and controller computing time \(\tau_{c}(k)\) are unknown, but just moments before \(k\) is known, so the historical time delay can be used to predict the current network time delay. In the networked control system discussed in this paper, the controller and actuator are event driven, and the sensor is clock-driven; therefore, the forward and feedback channel time delay of networked control system can be combined for analysis [13]. The system total delay can be expressed as \(\tau(k)=\tau_{ca}(k)+\tau_{sc}(k)+\tau_{c}(k)\). Because the time delay can be combined for analysis, the total system time delay \(\tau(k)\) can be predicted through an appropriate prediction model. Figure 2 gives 500 groups' time delay test data collected from an IEEE 802.11b-based networked control system. From the figure, it can be seen that time delay has a strong, nonlinear characteristic, and the mean value of the time delay sequence is within a range of variation over a longer time, but, in a certain interval during a shorter time. Figure 1: Networked control system with time delay. Figure 2: Typical random time delay of a network. The predictability and similarity of the time-delay sequence can be measured by the Hurst parameter; the time-delay sequence has self-similarity when \(H\in\left(0.5,1\right);\) the greater \(H\) is, the greater is the sequence similarity. In this paper, the \(R/S\) (rescaled range analysis) method is used to calculate the Hurst parameter of the time-delay sequence, and it can be described as:  
\[\left(R\,/\,S\right)_{n}=An^{H} \tag{1}\]  
where in \(n\) is the number of samples, \(R\) is the rescaling range, that is \(R=\max(X_{{}_{t,n}}-\min\,X_{{}_{t,n}}),t=1,2,\cdots,n\), \(S\) is the standard deviation, \(H\) is the Hurst parameter and \(A\) is a constant. Given the chart of the relationship between \(\lg(R\,/\,S)_{n}\) about \(\lg\,n\), the least squares method is used for calculating the slope, and then, the slope is Hurst parameter. The Hurst parameter of the time-delay sequence is commutated by the \(R/S\) method, as shown in Figure 3. Using a straight line linear fitting of these data points, the slope of the line is the Hurst parameter, \(H\) is 0.936, apparently satisfying \(0.5<H<1\), which also shows that the time-delay sequence has self-similarity. Therefore, the time-delay sequence can be predicted through the LS-SVM model. The standard SVM algorithm is complex; based on the standard SVM, LS-SVM sets the loss function of SVM as square errors, converts the inequality constraints into equality constraints and reduces the undetermined parameters. At the same time, it transforms the quadratic programming problem into linear equations, which can reduce the complexity of the solution [17]. LS-SVM uses the following function to estimate the random time delay of the networked control system [18]:  
\[y=w^{T}\Phi(x)+b \tag{2}\]  
Nonlinear function \(\Phi(\cdot)\) can transform the input space to a high dimensional feature space, so that the nonlinear fitting of input space can be seen as a linear fitting of the high dimensional feature space. Let \(\{x_{i},y_{i}\},i=1,2,\cdots,N\) be a training set, based on the risk minimization principle; the regression problem can be expressed as a constrained optimization problem:  
\[\min J(w,e)=\frac{1}{2}w^{T}w+\frac{1}{2}\gamma\sum_{i=1}^{N}e_{i}^{2},\gamma>0 \tag{3}\]  
Figure 3: Hurst parameter of the time delay sequence.