### Utilization of fractional order $$textrmPI^lambda textrmD^mu$$ controller as AQM algorithm

Table of Contents

This part provides the theoretical background for our analysis. Very first, we examine the product which we use in the simulation. The design is a close approximation of a solitary actual-lifestyle router we use in our testbed ecosystem. Both, the product and unit we use are centered on the one packets queue. The 2nd component of the part presents the strategy of calculation the response of (textrmPI^lambda textrmD^mu) controller utilised as AQM system.

### Simulation model of network node

The description of the AQM queue was presented in^{29} as an example of an IoT gateway. In general, we product the process incorporating the AQM system as the queue with the controller decides about the packet acceptance or rejection. Graphical illustration of the model we use is presented in Fig. 1.

In the offered design, there are two key parameters. The (lambda) is the price of packets arrivals—the average selection of packets incoming to the method per device of time. The (mu) is the level of packets service—the regular amount of packets leaving the device for each device of time. When the charge of acquiring packets ((lambda)) is greater than the price of sending packets out of the queue ((mu)), the queue in the gadget grows. On each and every occasion of the enqueuing packet, the AQM algorithm calculates the likelihood of dropping it dependent on the queue length. Primarily based on this likelihood, the packet is approved or rejected. In common alternatives, the dropping chance function is a regular mathematical purpose, for illustration, linear or exponential. We change the chance purpose with the reaction of (textrmPI^lambda textrmD^mu) controller. The controlled signal is the queue duration.

### (textrmPI^lambda textrmD^mu) controller as AQM mechanism

The function of (textrmPI^lambda textrmD^mu) controller is to continue to keep the worth of the managed signal on the offered stage. The non-integer order controllers have been established greater at this occupation in before performs^{30}. In buy to hold the sign at the preferred amount, the error price from the current price of the sign and the desired 1 is calculated as a distinction concerning the two. In get to use (textrmPI^lambda textrmD^mu) controller as AQM system as a managed signal we use the present length of the packets queue in the router. In the standard PID controller, the integral element is a easy sum of historical queue lengths (Eq. 5). Likewise, the spinoff portion is the change between the recent and previous queue lengths (Eq. 4). Equally the integral and spinoff elements are calculated in the similar way. The calculated worth is derivative for favourable purchase, although for the unfavorable order, it is an integral. Equations (6) and (7) present that in the scenario of fractional buy calculus, instead of a easy sum, we get the sum of the queue length multiplied by the coefficients similar to integral and spinoff order. Those people coefficients boost the general performance of the AQM mechanism by improving its adjustment to visitors ailments.

A description of how to use the response of the (textrmPI^lambda) controller is offered in^{31}. The chance of dropping the packets on arrival is specified by the controller reaction and is provided by the Eq. (1).

$$beginaligned p = max(, -( K_pe_ + K_isum _jin V_i nu _i(j)e_j +K_dsum _jin V_d nu _d(j)e_j ) ) finishaligned$$

(1)

the place:K_{p}—amplification of proportional portion, K_{i}—amplification of an integral portion, K_{d}—amplification of spinoff portion,e(_textrmj) = q(_textrmj)—setpoint—the price of j-th mistake, setpoint—the desired queue duration,q(_textrmj)—j-th queue size,V_{i}—set of integral part coefficients,V_{d}—set of spinoff section coefficients.

Things of the sets V_{i} and V_{d} are specified as:

$$start offaligned nu _i(j) = {still left start outarrayll 1&j= \ nu _i(j-1)(1 – (frac1+gamma j)) & j>0 stoparrayideal. stopaligned$$

(2)

where by (gamma) = (lambda) for an integration purchase or (gamma) = (mu) for a derivation buy.

The packet administration in the AQM routers is performed at the discrete factors at the time of packet arrival. Thus, we consider the queue product as a case of discrete systems. We use Grunwald-Letnikov’s definition of the discrete vary-integrals of fractional buy in our calculations.

For a given sequence f(_), f(_1),…, f(_textrmj),…, f(_textrmk):

$$commencealigned Delta ^gamma f_k = sum ^k_j= (-1)^j genfrac().0ptgamma j f_k-j closealigned$$

(3)

where (gamma in textbfR) is a non-integer, fractional buy, f(_textrmk) is a differentiated discrete perform and (genfrac().0ptgamma j) is generalized Newton image.

For (gamma = 1), we get the formula for the first-get derivative:

$$beginaligned Delta ^1 x_k = 1x_k – 1x_k-1 + 0x_k-2 + 0x_k-3… finishaligned$$

(4)

For (gamma = -1), we get the method for the first-order discrete integral:

$$start outaligned Delta ^-1 x_k = 1x_k + 1x_k-1 + 1x_k-2 + 1x_k-3… finishaligned$$

(5)

For a fractional-purchase by-product and integral buy, we get the weighted sum of all samples, for case in point:

$$commencealigned Delta ^-1.2 x_k = 1x_k + 1.2x_k-1 + 1.32x_k-2 + 1.408x_k-3… endaligned$$

(6)

$$start offaligned Delta ^-.8 x_k = 1x_k + .8x_k-1 + .72x_k-2 + .672x_k-3… conclusionaligned$$

(7)

The calculations presented above have to have a considerable variety of floating place operations, which are problematic to conduct on products with limited calculational abilities. To overcome the dilemma of floating stage functions, the solution proposed even further in the doc estimates the (textrmPI^lambda textrmD^mu) controller value employing only integer figures in the calculation method. The principal thought guiding the alternative calculation approach may perhaps be expressed as follows: applying the bit change, an integer amount can approximate a fractional variety with the provided resolution. In the proposed solution, all controller parameters are integer numbers with an assumed shift of 13 bits to the remaining. The change value of 13 bits enables to set the controller parameters with the resolution of 1/2(^13). For instance, use of benefit 1 suggests that we are working with a parameter of price 1/2(^13), and use of price 2(^13) is equal to 1.. Because in the calculation process, at most two parameters are multiplied, calculated (textrmPI^lambda textrmD^mu) controller worth is equivalent to the precise benefit, which would be attained making use of floating level functions, and the benefit of 2(^26). The drawback of the proposed method is the calculation mistake, specially for values near to the resolution benefit or lesser. Because of to this point, it is essential to identify the calculation mistake effect on the general controller performance. The calculation mistake also influences the duration of the error history. The for a longer time the mistake background, the lesser values are employed in the calculation approach, and the mistake grows, which cancels out the advantages of remembering more historical problems.