Maximum Allowable Delay Bound Estimation in Networked Control of Bounded Nonlinear Systems

Networked Control Systems (NCSs) has been recognized as an area where theory is behind the development of technology. The defining feature of NCSs can be considered as the information is exchanged through a network among control system components. So the network induced time delay is inevitable in NCSs. The time delay may degrade the performance of control systems and even destabilize the systems if they are designed without considering the effects of the time delays properly. Once the structure of a NCS is confirmed, it is essential to identify what the maximum time delay is allowed for maintaining the system stability which, in turn, is also associated with the process of controller design. This paper proposes a new method for estimating the maximum allowable time delay in networked control systems with norm bounded nonlinearity. The relation between the maximum nonlinearity and the maximum allowable delay is studied using the proposed method, and it is found that increasing the maximum nonlinearity bound reduces the maximum allowable delay. Furthermore, increasing the time delay leads to shrink the domain of attraction. The results of the maximum allowable delay bound and the maximum nonlinearity are compared with some of the published results.


Introduction
The advances in commun ication and network technology, and the availability of high-speed computers have resulted in an increasing interest in Netwo rked Control Systems (NCSs). This type of control systems can be defined as a control system where the control loop is closed through a real-t ime co mmunication netwo rk [1]. The term "Networked Control Systems" first appeared in Gregory C. Walsh's article in 1998 [2]. A typical organization of an NCS is shown in Figure 1. In Networked Control Systems, the reference input, p lant output and control input are exchange d through a real-time co mmunication network. The main advantages of NCSs are modularity, simp lified wiring, low cost, reduced weight, decentralization of control, integrated diagnosis, simp le installat ion, quick and easy for maintenan ce [3], flexible expandability (easy to add/remove sensors, actuators or controllers with lo w cost). NCSs are able to easily fuse global information to make intelligent decisions over large physical spaces.
As the control loop is closed through a commun ication network, the time delay and data dropout are unavoidable. This may degrade the performance of NCSs or even destabilize the system. In general, the control systems with time delays can be classified into time delay independent where the stability is not affected by the time delay and time delay dependent where the t ime delay affects the stability [4]. Time delay, no doubt, increases complexity in analysis and design of NCSs. Conventional control theories built on a nu mber o f standing assumptions; including synchronized control and non delayed sensing and actuation must be re-evaluated before they can be applied for NCSs [5].
In recent years, there are many results for the stability of linear network control systems [6]- [9]. However, there is not much wo rk reported in nonlinear networked control system analysis. Lyapunov functional, Lyapunov-Krasovski functional and Lyapunov-Razu mikhin functional based methods are most widely used to study the stability of linear Sensor Sensor n . . .

Plant
Controller Control of Bounded Nonlinear Systems and nonlinear networked control systems where the problem is usually formulated as Linear Matrix Inequalities (LMIs). In most of the published work in the literature, the aim is to find the maximu m allowab le delay bound for a given nonlinearity bound and to increase the conservativene ss of the maximu m allo wable delay bound results but on the expense of increasing the complexity. In [10] Razu mikh in and Lyapunov Theorem are used to derive a sufficient stability condition for the stability of a class of a nonlinear networked control system. The system under study is the linearized system with a bounded nonlinear function. A discrete-time approach for stabilizing a class of a nonlinear system is presented in [11] where the quadratic Lyapunov functional is used to derive a discrete linear controller for the affine nonlinear plant. In [12] a mu lti-input-mu lti-output continuous system is studied where the effects of the network induced time delay is modeled as an error state-vector which is regarded as a vanishing perturbation. The use of switching Lyapunov functional to derive stability conditions for a networked control system with bounded nonlinear uncertainty has been studied in [13].
In [14][15] the sampled-data approach is used for a networked control system with nonlinearity and the stability criteria is formu lated as LMIs. Their method can be used to calculate the maximu m nonlinearity bound for a given time delay and controller, but their results are conservative. The maximu m nonlinear bound is calculated by solving a constrained optimization p roblem. The Lyapunov -Krasovski functional is used in [16] to derive LMI to study the stability and for designing a stabilizing controller for a networked control system with time-delay, drop-outs and bounded time-varying nonlinearity. The fu zzy-logic approach has been addressed in many papers [17]. The Authors in [17] modeled a class of nonlinear networked control system using Takag i-Sugeno model. They use the approximate model of the discrete nonlinear system to represent the actual system model. In [18] the authors provided new results for stability analysis and stabilization of linear systems with norm bounded nonlinear perturbation.
Although the results of the maximu m nonlinearity bound are less conservative, the method is limited to free delay systems.
Most of the previously developed approaches require excessive load of co mputations, and also for higher-order systems; the load of computations will increase dramat ically. In practice, engineers may find it difficult to apply those available methods in control system design because of the complexity of the methods and the lack of a guideline in lin king between the design parameters and the system performance. Furthermo re, the design procedures highly depend on the post-design simulation to determine the design parameters. So there is a demand for a simp le design appr oach with cl ear g uidance f or pr actical applications. The time delay in real-time networks depends strongly on the network protocol and by scheduling the network the time delay can be made smaller and bounded. In this paper, a new simple method is proposed for estimat ing the Maximu m Allowable Delay Bound (MADB) in NCS with bounded nonlinearity. The method depends on using the fin ite difference appro ximation of the delay term and the problem is fo rmulated as LM I, which can be easily solved. Moreover, a simple analytical formu la relating the MADB with the maximu m nonlinearity bound is proposed.
The paper starts from the description of the proposed method for estimat ing the ma ximu m time delay for NCS with norm bounded nonlinearity. A few examples are illustrated, and the results are co mpared with those proposed in the previously published literature.

Mathematical Analysis
A nonlinear system is given by: where α > 0 is the nonlinearity bounding parameter and H is a constant matrix. For any given H; The constraint (2) can be interpreted as [18]; Stabilizing the system with a linear controller which is given by; (4) A typical networked control system model is shown in Figure 2. The t ime delay may be constant, variable or even random. In NCSs, the time delay is composed of the time delay fro m sensors to controllers, time delay in the controller and controllers to actuators time delay, which is given by: where τ sc is the time delay between the sensor and the controller τ c is the time delay in the controller, τ ca the time delay fro m the controller to the actuator. For a general formulat ion the packet dropouts can be incorporated in (5): dh ca c sc where d is the number of dropouts and h the sampling period. And by (6) the data dropouts can be considered as a special case of the time delay [9].
It is supposed that the follo wing hypotheses hold. Hypothesis 1 (H.1 ): • Sensors are clock driven.
• The controllers and the actuators are event driven.
• The data are transmitted as a single packet.
• The old packets are discarded.
• All the states are available for measurements and hence for transmission.
• The t ime delay τ is small enough for the fin ite difference approximat ion to be hold. Before we proceed to the analysis we will use the following Lemma: Lemma 1:(Schur Co mp lement): [4] For a given symmetric matrix where Ω 11 , Ω 12 and Ω 22 are block matrices, and Ω 11 is a square matrix. The following three conditions are equal in value: 0 If H.1 holds, then the time delay term can be approximated using the finite difference approximat ion by Tyler Series expansion. The exp ression for can be obtained by Taylor Expansion as: Where x (n) (t) is the n th order derivative. The second order approximation of the delay term is given by; (9) Fro m (9) it can be seen that R 3 (x,τ) depends on the time delay, τ, and the higher-order derivatives of x(t ) which can be neglected if the t ime delay and the norm of R 3 (x,τ) are small. For small t ime delay and slowly time varying nonlinear perturbation the second derivative can be approximated as: Substituting (9) and (10) into (7); Equation (11) can be written as: According to (2) with the time delay the quadratic inequality can be written as; which can be interpreted as; The constraint can be written as; Choosing the quadratic Lyapunov functional candidate and taking its derivative; This can be written as; Following the approach in [18] by co mbining (15) and (17) we get; The objective for the next step is to find the range of τ that will ensure [20]. Taking the derivative of (19) along with the system trajectory (12), If there exists P = P T > 0 and Q = Q T > 0, satisfying: Substituting (21) into (20) we get: For Using (23) and (24) Fro m (25) it can be found that if , the system will be robustly stable with degree α . We can see fro m Coro llary 1 that the MADB decreases with increasing α . Setting 0 ≈ α and neglecting the second order term then Corollary 1 reduces to Corollary 1 in [22] as follows; which means the MADB on the boundary is approximately ze ro.
In NCS with nonlinearity it is important to find or estimate the domain of attraction. The do main of attraction is defined as the reg ion where the limit o f every t rajectory of the nonlinear system orig inating in R A is the equilibriu m point. R A is shown in Figure 3. It is assumed that the origin is asymptotically stable.
In [21] the do main of attraction of the equilibriu m point (the origin) is defined as: is the init ial state at t = 0. It is difficult to find the do main of attraction but we can estimate a region Ω c , that is Ω c ⊂ R A , using Lyapunov's method. The estimate o f the domain of attraction Ω c in [21] is defined as: where c is a positive constant. Since; For ) (x V to be positive c can be chosen as: Fro m (28) we can draw some conclusions on the relation between the time delay and the domain of attraction because of the dependence of λ min (P(τ)) on the time delay through (21). A few references [13] [14] reported that increasing the time delay decreases the nonlinearity bound. In the following section, a number of examp les are picked-up fro m literature for co mparison and discussion.

Stability Analysis Case Studies
In general, two approaches are applied to controller design for NCSs. The first design approach is to estimate the maximu m allo wable delay bound for the system and then the network is scheduled to limit the time delay to be less than the MADB. The second approach is to design the controller while taking the time delay and data dropouts into account. In this paper, the first approach has been adopted. In this section, a number of examp les are studied to demonstrate the approach proposed and compare it with the previously published cases. In particular, the results derived using the method proposed in this paper has been compared with the results using the LMI method given in [14] [15].
Example 1 The first examp le has been studied in [14][15] with the sampled-data approach, the system is given by The maximu m nonlinearity bound given in [14][15] is 0.0013. In [24] α max =0.1636 with 0.2509 s time delay. However Corollary 1 and Theorem 1 still g ive conservative results the method is very easy co mpared with the method in [14], [15] and [24]. It is clear that the results of Theorem 1 are less conservative than the results of Coro llary 1.The trajectory of the system is shown in Figure 4. Fo r co mparison the nonlinear function and the initial conditions for the simu lation are g iven by; In [14], [15] with 0. , the maximu m nonlinear bound is α max =0.1365, using Coro llary 1 α max =0.0256 while using Theorem 1 α ma x =0.2555. Here Theorem 1 gives less conservative results than the published ones. The MADB as a function of the nonlinearity is given in Figure 5. It can be easily seen that as the nonlinearity increases the MADB decreases. This example has been studied in [17] with the sampled-data approach, the system is given by  . Setting 0 ≈ α and using theorem 1 the MADB is 0.0601 s. The maximu m nonlinearity bound for the delay free system is 4.3. The MADB as a function of the nonlinearity is shown in Figure 6. The system response with 0.03 s and 3 nonlinearity is shown in Figure 7. Fro m Figure 5 and Figure 6 it is clear that increasing the nonlinearity bound decreases the MADB. The same relation has been noticed in [14] [15]. We have carried out many simu lation and we found that increasing the nonlinearity reduces the MADB. A lthough the method is still conservative, but it can be easily applied. Exa mple 3: The last example is the estimat ion of the region of attraction using the proposed method. A nonlinear system is given by: In [10] the system is stabilized with a linear controller as: ( ) The equilib riu m points of the system are: 0 = x and 2 = x , we will study the domain of the attraction at the orig in; for the system to be stable we must have; 2 The domain o f attraction is estimated through using (28) We can see that increasing the time delay decreases the domain of attraction and when the time delay approaches the MADB then the domain of attraction becomes very small.
The system response with different time delays is shown in Figure 8. Fro m Figure 8, the system is still stable even with 0.75 s, wh ich shows that the results of Theorem 1 are still conservative. Figure 9 and Figure 10 show the system response with 0.76 s time delay and 0.15 and 0.5 init ial condition respectively.  In Figure 9, the init ial condition is 0.15, and we can see that the system is stable while in Figure 10 the init ial condition is 0.5, and the system is unstable. Increasing the initial condition reduces the MADB, and this is the same conclusion obtained from Coro llary 1 also in [10] the authors show that increasing the time delay reduces the domain of attraction.

Conclusions
The main contribution of the paper is to have derived a new method for estimating the maximu m t ime delay in NCSs with norm bounded nonlinearity. The most attractive feature of the new method is that it is simp le in structure and easy for applications, which can be clearly interpreted to design engineers in industrial sectors. The results obtained in this method are co mpared with those obtained through the methods introduced in other literatures. The method has demonstrated its merits in using less computation time due to its simple structure and giving less conservative results while showing good agreement with other methods. The method is used to estimate the MADB for a given nonlinearity bound which can be used as a guiding tool for the network scheduling. We found that increasing the nonlinearity bound reduces the MADB also increasing the time delay reduces the domain of the attraction fo r the NCS with bounded nonlinearity. The system response x Time Delay (sec)