Hybrid APSO–spiral dynamic algorithms with application to tuning of filtered PPI controller in a wirelessHART environment

Abstract

The accelerated particle swarm optimisation (APSO) is an improved variant of the PSO algorithm that guarantees convergence through the use of only global best to update both velocity and position of particles. However, like its predecessor, the APSO is also prone to being trapped in local minima. Therefore, this paper proposes two hybrid algorithms synergizing the social ability of the APSO and the exploitative ability of both spiral dynamic algorithm (SDA) and Adaptive SDA (ASDA). The exploration phase of the proposed algorithms APSO-SDA and APSO-ASDA, will be achieved through the APSO algorithm. The exploration phase solutions of the APSO are then fed to the SDA and ASDA to achieve the exploitation phase. The proposed algorithms have been evaluated with benchmark function and have also been used to tune a filtered predictive proportional-integral (FPPI) controller for WirelessHART networked control systems (WHNCS). The results obtained from Friedman’s rank test show that the proposed APSO-SDA and APSO-ASDA outperformed their constituent algorithms. Time domain analysis of the FPPI controller also show that the APSO-SDA and APSO-ASDA outperformed the APSO, SDA and ASDA in terms of settling times and overshoot.

Keywords

Accelerated PSO Hybrid optimisation algorithm predictive PI controller spiral dynamic algorithm WirelessHART

1. Introduction

Metaheuristic algorithms combine both simple local search and higher-level search strategies to seek best solutions for optimisation problems. In such algorithms, an iteratively generated process directs gradient free subordinate heuristics by intelligently combining several explorations and exploitation concepts within the search space [1]. These algorithms are often inspired by natural phenomena such as birds flocking [2], natural selection [3], bacterial foraging [4], path seeking behaviour of ants [5], natural spiral phenomena [6, 7], whale optimisation [8, 9], etc. Key advantage of these algorithms is that high quality solutions can be obtained within short time without the need for gradient of the optimisation problem. This indicates that the differentiability of the search space is not required by such algorithms. Furthermore, metaheuristics can be used as an efficient tools for solving broad range of intractable problems.

The APSO is a simplified form of the PSO, which is guaranteed to have the same or even better optimisation ability. In the APSO, the velocity update equation uses only global best as against using both global and local best in the standard PSO algorithm. This speeds up the convergence of the optimisation process. In this simplified version, the position update can be written without the velocity term in order to further hasten the convergence. However, despite the advantages of convergence speed, the APSO like its parent algorithm, is still prone to being trapped in local minima. Recently, researchers have either applied or attempted to improve on the original APSO algorithm. For example in [10], the algorithm has been applied to optimise parameters of wire electrical discharge machining (WEDM) to achieve minimum surface roughness, maximum Material Removal Rate (MRR) and minimum kerf width. Authors in [11] have applied the algorithm to achieve maximum power point (MPPT) in a partially shaded PV system. A dynamic spectrum sensing algorithm based on APSO is proposed in [12]. In that work, it was found that the proposed method compared to the existing ones, gives more efficient results. Wang et al. [13] proposed an improvement to the APSO algorithm by hybridising it with differential evolution (DE) algorithm. In the paper, instead of using random walk, the mutation operator of DE is used to tune the update of velocity and positions of each particle. In another work by Guedria [14], an improvement to the APSO algorithm through use of individual particle memories to update positions of particles in the swarm has been presented. Results comparison showed that the improved algorithms can perform even better than the compared algorithms.

The spiral dynamic algorithm (SDA) was proposed first for two-dimensional problems by Tamura and Yasuda in [6]. The authors later extended the algorithm for n-dimensional problems in [7]. The algorithm mimics natural spiral phenomena like galaxy, tornadoes and hurricane. The SDA compared to other known metaheuristic algorithms, is simple hence it has short computational time and faster convergence speed. A unique feature of the SDA is that in the initial stage of the algorithm, diversification strategy is employed to coarsely explore wider region in the search space. In the later stage of the algorithm, intensification strategy is employed to intensively and finely search around region of good solution. The strategy of diversification and intensification is a pointer to an effective meta-heuristic optimisation [1, 7]. Most of the improvements to the algorithm are geared towards adaptation and hybridization. In [15] several adaptation schemes for the spiral radius ranging from fuzzy, linear, quadratic and exponential functions have been proposed. These schemes are all based on the fitness values to change dynamically the spiral radius. In a related development, Nasir et al. [16] proposed adaptation of both the spiral radius and spiral angle of rotation using linear function. Majority of the hybridization effort made for the algorithm are with bacterial foraging algorithm (BFA). For instance authors of [1] proposed hybridization of bacteria-chemotaxis with SDA. Here, the exploration is achieved through the BFA while the exploitation phase is handled by the SDA. In other works also by same authors [17, 18], the elimination and dispersal ability of the BFA has been employed to improve the search ability of the SDA. Furthermore, a proposal to update the bacterial swim in the BFA algorithm in a spirally based on SDA has been proposed in [19]. A key drawback of these hybridizations is that they are all based on the BFA which is complex and requires high computational time. It should be noted that to our knowledge, there is no reported case on synergy between SDA and APSO algorithm.

In spite of the limitations of rigidity, high installation and maintenance cost associated with wired industrial monitoring and control applications, earlier wireless sensor networks (WSNs) such as ZigBee, Wi-Fi and Bluetooth are not suitable candidates to overcome these limitations. This is because, they are also lacking in reliability, security, device interoperability and scalability which are a requirement for application in the industry [20 –22]. Consequently, recent development in the emergence of wireless industrial standards such as WirelessHART, WIA-PA and ISA100 Wireless has rekindled effort towards minimising these problems [23, 24]. The three standards are guaranteed to overcome the limitations of both the wires and earlier WSNs. The first to get approval of the International Electrotechnical Commission (IEC) among these standards is the WirelessHART, which is based on the traditional wired highway addressable remote transducer (HART) protocol that has millions of devices installed globally [25]. However, like other wired and wireless networked systems, the WirelessHART suffers from network stochastic delay that is capable of degrading control performance [26 –28]. To overcome this, researchers have proposed several methods including our recently proposed predictive PI (PPI) controller [29 –34]. A drawback of the PPI is that it is affected by noise. Furthermore, if in the presence of noise, or if higher order plants are used, the response becomes oscillatory [34]. Hence, the need for filtering and also tuning of both the filter and the controller parameters.

The first objective of this paper is to propose synergy between the APSO algorithm and both the SDA and adaptive SDA (ASDA). The second aim of the paper is to demonstrate the effectiveness of this proposal through tuning of a filtered PPI (FPPI) controller in WirelessHART environment. Assessment of the performance of proposed approaches will be based on comparison with predecessor algorithms i.e. APSO, SDA and ASDA.

The reminder of this paper is organised as follows: In Section 2, brief about the WirelessHART network as well as preliminary experiments are discussed. Section 3 presents the development of the proposed algorithms while Section 4 gives validation with benchmark functions. In Section 5, the developed algorithms are applied to optimise FPPI controller. Lastly, conclusion is presented in Section 6.

2. WirelessHART networked control system

This section will first discuss the structure of the WirelessHART networked control system (WHNCS) and the PPI controller. The second part of the section will give the procedure for preliminary experiments to determine the network delay.

2.1. WirelessHART control loop

Fig. 1 :figure1 shows the representation of a single loop WHNCS. From the figure, the delay associated with the network can be written as $τ_{N} = τ_{ca} + τ_{sc}$ (1) where τ_ca and τ_sc are the controller-to-actuator and sensor-to-controller delays respectively. With the assumption of commutativity between various terms in the control loop, dead-time of the process (L_p) can be added to τ_N to obtain the loop delay L as follows

Fig. 1

Block diagram representation of a single loop WHNCS.

L = τ_{N} + L_{p}

(2)

If G_c (s) is a PPI controller, the control action is given as $u (t) = K_{c} (1 + \frac{1}{{pT}_{i}}) e (t) - \frac{1}{{pT}_{i}} (u (t) - u (t - L))$ (3) where p is a differential operator $\frac{d}{dt}$ . It can be seen that the first term of (3) is a PI controller while the second term is the prediction. Equation (3) can be written in the s domain as $U (s) = K_{c} E (s) + \frac{1}{1 + {sT}_{i}} e^{- sL} U (s)$ (4) where K_c and T_i are the PPI controller gain and time constants respectively. These parameters are related to the plant gain K and time constant T as follows: $K_{c} = \frac{α}{K}$ (5) $T_{i} = β T$ (6) For the purpose of (3) and (4), α and β are chosen to be both unity [35, 36]. However, these assumptions only work well for first order system under constant delay. With higher order systems, the controller response becomes oscillatory. Another area where (4) cannot be used without modification is when the delay is varying as in the case of WirelessHART network. To curtail the oscillation caused by either variable delay or use of higher order plant or both, the control action given in (4) will be filtered using first order filter [37] as given in (7) Consequently, the proposed APSO-SDA and APSO-ASDA will be used to select controller parameters K_c, T_i and filter time constant T_f. $U (s) = (K_{c} E (s) + \frac{1}{1 + {sT}_{i}} e^{- sL} U (s)) F (s)$ (7)F (s) in (7) is a filter transfer function given as $F (s) = \frac{1}{1 + {sT}_{f}}, T_{f} > 0$ (8) where T_f is the filter time constant.

2.2. Preliminary experimentation

To ascertain the level of WirelessHART networked induced delays τ_ca and τ_sc, Dust Networks-Smart Mesh WirelessHART network starter kit developed by Linear Technology is used. In the experiment, upstream and downstream network delays t_u and t_d for communication between WirelessHART mote and gateway and verse versa were measured. The measurement is conducted in the same manner to the methods reported by [29, 38] and [39]. The delays are recovered by the gateway by specifying the MAC address of the communicating mote employing the getLatency MACaddress command [40]. Each delay is measured using timestamps on communication messages. Thus, the delay is computed as the difference between the received timestamps generated at the gateway and the sent timestamps embedded on the arrival message. Network devices experience clock drift during operation despite each having internal clock. This effect is handled by network-wide time synchronization prescribed by the WirelessHART standard which ensures all devices have the same reference time as the gateway within less than 1ms accuracy. To avoid battery drain of the motes, 4s packet refresh rate is used throughout the experiment. The set-up to measure the network delays is shown in Fig. 2.

Fig. 2

WirelessHART network delay measurement experimental set-up

3. Development of hybrid APSO-SDA and APSO-ASDA

This section will present the hybridization of APSO algorithm with SDA and ASDA. First, brief highlight on the constituent algorithms namely APSO, SDA and ASDA will be given. Then, the development of hybrid APSO-SDA and APSO-ASDA will be presented.

3.1. APSO Algorithm

The standard PSO algorithm is based on the swarm behaviour of particles and is governed by two update equations (9) and (10) that represents the velocity and position of each particle in the swarm [41]. At the initial stage of the algorithm, particles are randomly distributed in the n-dimensional search space. Then the velocity and position are updated with the corresponding values of the previous iterations. $v_{i} (t + 1) = v_{i} (t) + c_{1} φ_{1} (X_{i} (t) - x_{i} (t)) + c_{2} φ_{2} (X_{g} (t) - x_{i} (t))$ (9) $x_{i} (t + 1) = x_{i} (t) + v_{i} (t + 1)$ (10) where,

c₁ and c₂ are constants representing learning rates,

φ₁ and φ₂ are uniformly distributed random numbers,

v_i (t) and x_i (t) are velocity and position of the i^th particle and

X_i (t) and X_g (t) are local and global best positions of the particle.

As seen in (9) and (10), the algorithm uses current values of both local and global positions to update the velocity. This was done to increase the diversity of solution. However, Yang [3, 42] suggested that since it is possible to employ some randomness to simulate the diversity, then local best is not needed in the velocity update. This is to avoid the particles being trapped in a local minimum. This improved version of the PSO i.e. the Accelerated PSO (APSO), uses only the global best for velocity update, hence accelerating the convergence of the algorithm. The velocity and position update equations of the APSO are given as follows $v_{i} (t + 1) = v_{i} (t) + ρ ε_{n} + σ (X_{g} (t) - x_{i} (t))$ (11) $x_{i} (t + 1) = x_{i} (t) + v_{i} (t + 1)$ (12) To further step-up the convergence, (12) can be written in a single step [43] as follows: $x_{i} (t + 1) = (1 - σ) x_{i} (t) + σ X_{g} (t) + ρ ε_{n}$ (13) where σ is a constant chosen between 0.1 and 0.7 for most applications, ε_n ∈ N (0, 1) is a random number and ρ is a randomness parameter.

The randomness parameter ρ can be selected as constant or using decay function such as $ρ = ρ_{0} e^{- γ t}, (0 < γ < 1)$ (14) or $ρ = ρ_{0} γ^{t}, (0 < γ < 1)$ (15) where ρ₀ ≈ 0.5 - 1 is the initial randomness parameter, γ is a control variable and t is the number of iterations such that t ∈ [0, t_max]. In this work (14) is used for APSO and in both hybrid algorithms.

3.2. SDA

Since the SDA mimics natural phenomena such as hurricane, tornadoes and galaxy that are spiral in nature, search points in the algorithm are made to move from outermost layer towards a centre spirally. At the initial stage of the algorithm, search points are moved using large step-size to achieve diversification. This occurs at the outermost layer of the spiral. As the search points approach the centre of the spiral, the step-size is dynamically reduced to achieve intensification at the innermost layer of the spiral. This diversification and intensification is depicted in Case 1 of Fig. 3. With this strategy, the algorithm will potentially reach global optimum on the search space.

Fig. 3

Effect of varying spiral angle θ and radius r in SDA based on parameters of Table 1

Movement of a search point i is modelled using n-dimensional spiral model as follows $x_{i} (k + 1) = S_{n} (r, θ) x_{i} (k) - (S_{n} (r, θ) - I_{n}) x^{*}$ (16) where x^* is the spiral centre, k is the iteration number, r is the rotation radius, θ is the spiral angle, I_n is n × n identity matrix and S_n (r, θ) is a stable matrix given as $S_{n} (r, θ) = {rR}^{n} (θ)$ (17) where Rⁿ (θ) is n × n rotational matrix.

3.3. ASDA

In the standard SDA, r and θ are the key parameters that control the movement of search points towards centre of the spiral. This is illustrated using six different scenarios in Fig. 3. From the figure, Case 1-3 shows the effect of varying the angle θ while Case 4-6 shows the effect of varying r. The different values of r and θ used for this illustration are given in Table 1. The figure clearly depicts how choice of these parameters can influence the shape of the spiral, hence how a search point will move. Thus, poor selection of the parameters will potentially make the algorithm skip global optimal point.

Table 1
SDA Illustration parameters, x₀ = (25, 25)

Fixed r = 0.95 Fixed θ = π/6

Case 1 θ = π/6 Case 4 r = 0.95

Case 2 θ = π/4 Case 5 r = 0.90

Case 3 θ = π/2 Case 6 r = 0.85

Fixed r = 0.95	Fixed θ = π/6
Case 1	θ = π/6	Case 4	r = 0.95
Case 2	θ = π/4	Case 5	r = 0.90
Case 3	θ = π/2	Case 6	r = 0.85

In an attempt to adapt r and θ to the change of fitness of the associated search point, Nasir et al. [16] proposed the following formulations for adaptive radius (r_a) and angle (θ_a): $r_{a} = \frac{(r_{l} - r_{u})}{1 + (\frac{c_{r}}{| f (x_{i} (k)) - minf (x (k)) |})} + r_{u}$ (18) $θ_{a} = \frac{(θ_{l} - θ_{u})}{1 + (\frac{c_{a}}{| f (x_{i} (k)) - minf (x (k)) |})} + θ_{u}$ (19) where,

c_r and c_a are positive constant representing fitness deviation rate of change

f (x_i (k)) is fitness of a given search point

minf (x (k)) is the current iteration’s global best fitness

r_l and r_u are the lower and upper values of the spiral radius chosen between 0 and 1

θ_l and θ_u are the minimum and maximum values of the spiral trajectory angle chosen between 0 and 2π

With these adaptations, when a high fitness point in the search space is detected, this leads to large radius and angle that will yield longer step size. This will move the point to a better location. On the other hand, if low fitness point is detected, smaller radius and angle that will eventually yield shorter step size are obtained. Consequently, this point continue around to search for optimal solutions. Thus, apart from the updates of r and θ with (18) and (19), all other steps in the ASDA are similar to those of the SDA.

3.4. Hybrid APSO-SDA and APSO-ASDA

A common advantage of the APSO, SDA and the ASDA is that they are all simple, easy to implement with fewer parameters and have faster convergence rate. On one hand, the APSO algorithm has greater diversity and good exploration ability due to the social behaviour of the particles involved. However, a key problem of the algorithm is that it gets trapped in local minima. Hence, the APSO has poor exploitation ability. On the other hand, both the SDA, and ASDA have good exploitation ability but poor exploration capability. Therefore, each of the proposed new hybrid algorithm will leverage on the strength of its constituent algorithms.

In the exploration phase of the new proposed APSO-SDA and APSO-ASDA, the APSO will be used to search for optimal local solutions. Afterwards, in the exploitation phase, the best solution of the APSO will be used as the initial search points in the SDA and ASDA to exploit around this good region for global solutions. In other words, the SDA and ASDA will be used to fine-tune the solutions of the APSO. The implementation flowchart of the proposed algorithms is given in Fig. 4 while the description of various parameters of the algorithms are given in Table 2. It is worth noting that the only difference between the implementation of APSO-SDA and APSO-ASDA is the update of r and θ with (18) and (19) in the latter. This stage has been indicated using dashed outline in Fig. 4.

Fig. 4

Flowchart for implementation of APSO-SDA and APSO-ASDA.

Table 2

APSO-SDA and APSO-ASDA Parameters

Parameter	Descripton
N	Number of birds
birdStep	Number of bird steps
D	Dimension of the problem
J	Fitness funtion
X	Bird position
v	Bird velocity
m	Number of spiral search points
r	Convergence rate or spiral radius
R ⁿ	Rotation matrix of dimension n × n
x ^*	Center
x	Rotation point
k _max	Max iteration number
I _n	Identity matrix
θ	Rotation angle
S _n	Stable matrix
r _a	Adaptive radius
θ _a	Adaptive angle

4. Validation with benchmark functions

This section will attempt to validate the two proposed algorithms with some selected benchmark functions as reported in [44, 45]. The validation will be done in comparison to APSO, SDA and ASDA. The validation functions considered in include Six-hump camel, Shubert, Booth and Easom functions. The formulations for these functions are shown in Table 4. The functions have been selected to reflect various shapes such as valley, many local minima, plate, and steep ridges.

Table 3
Benchmark functions

Function Formula Search Space Global minimum

Six-hump camel $f (x) = (4 - 2.1 x_{1}^{2} + \frac{x_{1}^{4}}{3}) x_{1}^{2} + x_{1} x_{2} + (4 x_{2}^{2} - 4) x_{2}^{2}$ -2 ≤ x₂ ≤ 2 -3 ≤ x₁ ≤ 3, -1.0316

Shubert $f (x) = \prod_{i = 1}^{2} \sum_{j = 1}^{5} jcos ((j + 1) x_{i} + j)$ -5.12 ≤ x_i ≤ 5.12 -186.7309

Booth f (x) = (x₁ + 2x₂ - 7) ² + (2x₁ + x₂ - 5) ² -10 ≤ x_i ≤ 10 0

Easom f (x) = - cos(x₁) cos(x₂) exp(- (x₁ - π) ² - (x₂ - π) ²) -100 ≤ x_i ≤ 100 -1

Function	Formula	Search Space	Global minimum
Six-hump camel	$f (x) = (4 - 2.1 x_{1}^{2} + \frac{x_{1}^{4}}{3}) x_{1}^{2} + x_{1} x_{2} + (4 x_{2}^{2} - 4) x_{2}^{2}$ -2 ≤ x₂ ≤ 2	-3 ≤ x₁ ≤ 3,	-1.0316
Shubert	$f (x) = \prod_{i = 1}^{2} \sum_{j = 1}^{5} jcos ((j + 1) x_{i} + j)$	-5.12 ≤ x_i ≤ 5.12	-186.7309
Booth	f (x) = (x₁ + 2x₂ - 7) ² + (2x₁ + x₂ - 5) ²	-10 ≤ x_i ≤ 10	0
Easom	f (x) = - cos(x₁) cos(x₂) exp(- (x₁ - π) ² - (x₂ - π) ²)	-100 ≤ x_i ≤ 100	-1

Table 4

Statistical results for 100 simulation runs of benchmark functions

NO.	Problem	Dim.		APSO	SDA	ASDA	APSO-SDA	APSO-ASDA
1	Six-hump camel	3	Mean	-1.0257	-0.9034	-0.8794	-1.0270	-1.0273
			Best	-1.0316	-1.0036	-0.9994	-1.0316	-1.0316
			Worst	-0.9720	-0.4896	-0.4675	-1.0029	-0.9911
			SD	0.0091	0.0913	0.1051	0.0059	0.0080
		10	Mean	-1.0246	-0.8998	-0.8823	-1.0273	-1.0266
			Best	-1.0316	-1.0057	-0.9941	-1.0316	-1.0316
			Worst	-0.9151	-0.5595	-0.5020	-0.9780	-0.9885
			SD	0.0175	0.0769	0.1001	0.0074	0.0068
		30	Mean	-1.0254	-0.9016	-0.8775	-1.0259	-1.0282
			Best	-1.0316	-1.0182	-0.9971	-1.0316	-1.0316
			Worst	-0.8708	-0.5949	-0.4380	-0.9310	-0.9786
			SD	0.0169	0.0836	0.1056	0.0122	0.0065
2	Shubert	3	Mean	-183.0735	-139.2888	-135.9428	-183.5993	-183.6704
			Best	-186.7308	-186.3891	-186.5624	-186.7309	-186.7308
			Worst	-123.0031	-55.5230	-47.7837	-123.5655	-123.5309
			SD	11.0900	34.6026	33.5879	11.0035	9.9167
		10	Mean	-182.7526	-133.5925	-135.8872	-182.8331	-183.6709
			Best	-186.7309	-184.8262	-186.1619	-186.7309	-186.7309
			Worst	-106.3559	-45.0788	-51.9855	-121.0531	-110.8807
			SD	12.9030	32.8562	35.0346	12.6377	11.0533
		30	Mean	-182.4027	-140.4786	-135.3453	-182.7113	-183.4608
			Best	-186.7308	-186.7276	-186.4072	-186.7308	-186.7309
			Worst	-121.7422	-45.8797	-56.6269	-116.9337	-121.9951
			SD	12.9474	33.7231	34.6460	13.5574	11.3906
3	Booth	3	Mean	0.5990	0.1558	0.0944	0.0172	0.0119
			Best	0.2631	0.0026	6.2242×10^-4	4.6759×10^-7	7.6309×10^-5
			Worst	1.0596	0.6808	0.7325	0.3243	0.0628
			SD	0.1724	0.1408	0.1236	0.0379	0.0158
		10	Mean	0.5657	0.1358	0.0856	0.0174	0.0177
			Best	0.1218	0.0024	0.0022	6.8371×10^-5	8.7069×10^-6
			Worst	0.9860	0.5940	0.5323	0.1885	0.2181
			SD	0.1607	0.1201	0.0965	0.0280	0.0293
		30	Mean	0.5359	0.1355	0.0865	0.0177	0.0198
			Best	0.1427	0.0099	7.6172×10^-4	1.1823×10^-4	8.1695×10^-6
			Worst	0.9787	0.5301	0.6143	0.3088	0.2263
			SD	0.1630	0.1213	0.1029	0.0366	0.0343
4	Easom	3	Mean	-0.0263	-2.1869×10^-5	-2.1032×10^-5	-0.0175	-0.9998
			Best	-0.3081	-2.8918×10^-5	-2.8902×10^-5	-0.2644	-1.0000
			Worst	-8.0105×10^-5	-1.3694×10^-5	-1.1837×10^-5	-8.0392×10^-5	-0.9986
			Std.Dev	0.0759	3.6644×10^-6	4.1116×10^-6	0.0579	2.0853×10^-4
		10	Mean	-0.0322	-2.2047×10^-5	-2.1546×10^-5	-0.0292	-0.9998
			Best	-0.2457	-2.9625×10^-5	-2.9528×10^-5	-0.2493	-1.0000
			Worst	-8.0512×10^-5	-1.3124×10^-5	-1.0547×10^-5	-7.9791×10^-5	-0.9987
			Std.Dev	0.0770	3.3992×10^-6	3.9198×10^-6	0.0746	2.4365×10^-4
		30	Mean	-0.0423	-2.1773×10^-5	-2.1251×10^-5	-0.0265	-0.9998
			Best	-0.2800	-2.8970×10^-5	-2.9277×10^-5	-0.2467	-1.0000
			Worst	-8.0399×10^-5	-1.1655×10^-5	-1.0158×10^-5	-8.0536×10^-5	-0.9983
			Std.Dev	0.0916	3.8563×10^-6	3.9806×10^-6	0.0723	2.5336×10^-4
Friedman mean rank test of the algorithms based on average mean values				3.3	4.1	4.4	2	1.3

For the purpose of this validation and performance analysis of the proposed algorithms, 3, 10 and 30 dimensions of the benchmark functions for 100 independent simulation runs are considered. The APSO parameters used for all algorithms are defined as N = 100, birdStep = 100, ρ_o = 0.5, σ = 0.7, γ = 0.7. Likewise, to ensure fairness, the SDA and ASDA parameters are defined such that the same number of iteration are obtained with the APSO. Thus, their parameters are defined as m = 100, k_max = 100, r = 0.95, θ = π/6, r_l = 0.1, r_u = 1, θ_l = 0.1, θ_u = 2π, c_r = 1 and c_a = 100. The selection of these parameters is done through trial and error and mainly guided by the works reported in [6 , 46]. It should also be noted that the initial particles positions and spiral search points are selected randomly.

The results of 100 simulation run conducted with all the compared algorithms for each of the four benchmark functions is presented in Table 4. The results are given in terms of best, worst and in terms of statistical mean and standard deviation (SD). The statistics is to show how consistent and accurate an algorithm is. As indicated in the table, the best of the mean, best, worst and standard deviation is highlighted in bold. For the Six-hump camel function, it can be observed that interms of the Best run, the APSO, APSO-SDA and APSO-ASDA algortihms have achieved the global optimum of -1.0316 for all the 3, 10 and 30 dimensions. For this same function, the APSO-ASDA has best mean for dimensions 3 and 30 while APSO-SDA achieved the best mean for dimension 10. In terms of SD and best of Worst, the same pattern is observed with APSO-ASDA having 2 out of 3 for dimensions 10 and 30 while APSO-SDA has 1 for dimension 3. Results of the Shubert function indicated that the APSO-ASDA algorithm has the best mean and SD for the three dimensions considered. The APSO-SDA has 2 out the 3 best mean for the Booth function while the APSO-ASDA has the one corresponding to 3 dimensions. On the other hand, APSO-ASDA takes 2 of the best SD of the Booth function while the APSO-SDA gets the one corresponding to dimensions 10. The best of mean, best and worst of the optimisation with Easom function have all been dominated by the APSO-ASDA. Although the SDA did not fare well in terms of the mean, best and worst, it won all the SD corresponding to the dimensions considered for the Easom function.

Analysis of the performance of each algorithm with the benchmark function is given in Table 4 and Fig. 5. The table shows frequencies of best results for each algorithm in terms of mean, best, worst and standard deviation while the figure shows pictorial depiction in terms of histogram. It can be seen from both the figure and table that the two proposed algorithms APSO-SDA and APSO-ASDA have the highest frequencies in terms of mean and best. The ranking of algorithms as conducted through Friedman rank test is also provided in Table 4. The algorithms best on the test are ranked in the following order: APSO-ASDA, APSO-SDA, APSO, SDA and ASDA. The best rank of 1.3 is achieved by the APSO-ASDA while the least ranked algorithm is ASDA with a value of 4.4.

Fig. 5

Chart summary for performance of various algorithms.

5. Application to tuning of filtered PPI (FPPI) controller

This section presents the application of APSO, SDA, ASDA, APSO-SDA and APSO-ASDA to tuning of FPPI controller presented in Section 2 for application in a WirelessHART networked environment. Similar algorithm parameters chosen for the benchmark functions are used here except N, birdStep, m and k_max that are chosen as 50, 50, 20 and 50 respectively. These parameters are arrived at after exhaustive test based on trial and error guided by works reported in [46] and [7]. In a similar way to the validation with benchmark section, the initial positions of particles and spiral search points are selected randomly.

The controller parameters are tuned using the compared algorithms with the objective of minimising the integral time absolute error (ITAE) given in (20). The ITAE is chosen because of its advantages of producing better performance than other indices such as integral absolute error (IAE), integral square error (ISE), mean square error (MSE) etc. This is because the multiplication of the error term with time augments its effect at higher values. This helps in reducing settling time (t_s) and percentage overshoot (%OS) [47]. $J = \int_{τ = 0}^{t} τ | e (τ) | d τ$ (20)

The optimisation structure of the controller is shown in Fig. 6. As seen in the figure, the cost ITAE, which is a function of the error, is used in the optimisation algorithm to select controller parameters K_c, T_i and T_f.

Fig. 6

Block diagram of the optimisation structure of the FPPI controller for WirelessHART networked system.

5.1. Plant model selection and experimental network delay

In this work, the plant models given in Equations (21) and (22) are used for simulation with the FPPI controller optimised with APSO, SDA, ASDA, APSO-SDA and APSO-ASDA. These models represent first and second order plus dead time systems as reported in [48] and [49]. $P_{1} (s) = \frac{5}{1 + 2 s} e^{- 4 s}$ (21) $P_{2} (s) = \frac{1}{(1 + 1.5 s) (1 + 0.4 s)} e^{- 5 s}$ (22)

Fig. 7 shows the delay between the gateway and a single mote (node) in the WirelessHART network. Mote-to-gateway delay is the upstream delay t_u while the gateway-to-mote delay is the downstream delay t_d. Here, t_u is taken as τ_sc while t_d is taken as τ_ca see Fig. 1. The stochastic variation of upstream delay over time is as a result of communication between the mote and gateway. To reduce power consumption and preserve battery since usually battery powered, the motes will go into idle state after each communication cycle with the gateway or other motes is completed. In this state, the motes takes longer time to process signal due to lower processing capability. This contributes to the upstream delays variation. On the other hand, the gateway is always in active state since it is the host in the network, hence constant downstream delay.

Fig. 7

WirelessHART network delay profile.

The average values of the two delays t_d and t_u from the statistical information of Fig. 7 given in Table 5. The total loop delay L for each plant configuration is calculated based on (1) and (2) Thus the estimate for the L are given as 6.5959, 7.5959, 7.5959 and 12.5959s for first, second, third and fourth order plants respectively. These values are used to design the FPPI controller for each plant. Each of the four plants is simulated to a step signal of unit magnitude and a disturbance of 50% is injected at the input at 80s to test for robustness of the controller.

Table 5

Statistics of WirelessHART network delay

Delay (s)	Min	Max	Mean	Std.
t _u	1.0220	1.6600	1.3159	0.1173
t _d	1.2800	1.2800	1.2800	0.0000

5.2. First order plant

The tuned controller parameters using the five compared algorithms are given in Table 6. The closed loop response of the plant with the controller optimised with these algorithms is shown in Fig. 8. The performance analysis with respect to rise time t_r, settling times before and after disturbance t_{s
₁}, t_{s
₂} and percentage overshoot %OS is also given in Table 6. From the responses as well as the table, the controller with SDA settles faster before disturbance at 13.3204s. The APSO algorithm produced the fastest response with t_r=3.1181 and settled faster after disturbance at 99.0945s. However, this is at the cost of very high overshoot of more than 10%. On the other hand, the two hybrid algorithms produced very less overshoots of around 0% each despite slow rise and settling times. Here it is observed that for the FPPI controller, low overshoot values comes at the expense of longer settling and rise times. This can be observed clearly with ASDA and hybrid algorithms. In addition, the controller optimised with the hybrid algorithms give smoother signals compared to the other algorithms.

Fig. 8

Response of first order plant to tuned controller with various algorithms.

Table 6

Tuned controller parameters and performance analysis for P₁ (s)

Algorithm	Optimised controller parameters			Plant control performance
	K _c	T _i	T _f	J _min	t _r	t _{s ₁}	t _{s ₂}	%OS
APSO	0.2241	2.3768	0.1172	716.9085	3.1181	22.7385	99.0945	10.6525
SDA	0.1930	1.9412	0.1576	785.9947	5.2204	13.3204	103.1610	1.6891
ASDA	0.1416	1.3064	0.1945	837.7454	10.0192	21.9902	110.3579	0.4064
APSO-SDA	0.1890	2.2952	0.6218	766.6157	6.3924	21.3873	108.6295	0.0162
APSO-ASDA	0.1842	1.4132	1.2366	781.7953	7.6084	24.5817	111.1369	0.0100

5.3. Second order plant

In a similar fashion to the first order, the controller parameters tuned with the compared algorithms are given in Table 7. The closed loop responses of the plant with controller optimised with these algorithms is shown in Fig.9. The numerical analysis of the performance with respect to t_r, t_{s
₁}, t_{s
₂} and %OS is also given in Table 7. From the figure and the table, it can be seen that unlike in the case of first order, the controller tuned with hybrid algorithms performed better in terms of both settling times and overshoot. For example, best settling times t_{s
₁} and t_{s
₂} are obtained with the APSO-SDA algorithm, while the overshoot is second best to APSO-ASDA. On the other hand, The APSO-ASDA produced the best overshoot of 0.0024% while achieving second best settling time before disturbance. The APSO algorithm notwithstanding maintains its lead in terms of response speed with t_r=3.4562s but with overshoot of more than 10% just as in the first order plant. Considering the control signals, the two hybrid algorithms produced smooth control signal compared to the others. This is in congruent with the first order plant.

Fig. 9

Response of second order plant to tuned controller with various algorithms.

Table 7

Tuned controller parameters and performance analysis for second order plant

Algorithm	Optimised controller parameters				Plant control performance
	K _c	T _i	T _f	J _min	t _r	t _{s ₁}	t _{s ₂}	%OS
APSO	1.0410	1.9019	0.4197	877.4035	3.4562	20.1515	107.4818	10.4740
SDA	1.0858	3.0899	2.1745	1093.5270	5.1132	31.8760	119.8577	5.7149
ASDA	0.7607	1.5853	2.1729	1016.1714	18.1427	41.4342	125.2754	0.0445
APSO-SDA	0.9675	2.1839	1.4097	917.4698	4.8924	17.9435	106.8817	0.0042
APSO-ASDA	0.9821	2.1773	2.4817	1045.6743	7.3575	18.3783	108.1609	0.0024

5.4. Summary

Based on the performance of the FPPI controller optimised with APSO, SDA, ASDA, APSO-SDA and APSO-ASDA, the following observations are made:

The proposed hybrid algorithms i.e., APSO-SDA and APSO-ASDA, produced responses with good settling times both before and after disturbance. This is because, the optimal local solutions from the APSO has been used as initial search space in the SDA for finding the global best.

The APSO and SDA produced the fastest responses for both plants. However, this speed is at the cost of increased overshoot especially for the APSO.

The proposed APSO-ASDA algorithm produced response with the best overshoot for both plants.

For the two plants, it is noted that faster rise time leads to higher overshoot while smaller overshoots leads to slow response.

6. Conclusion

Two hybrid algorithms synergizing the APSO with SDA on one hand and APSO with ASDA on the other hand have been proposed. In the proposed algorithms, the best solutions of the APSO are used as initial search points of either the SDA or the ASDA to achieve APSO-SDA and APSO-ASDA respectively. Performance of the proposed algorithms in comparison to their constituents algorithms has been evaluated using several benchmark functions. Mean rank test conducted on the compared algorithms through Friedman’s rank test suggests that the algorithm with the best rank is APSO-ASDA. This is followed by APSO-SDA, then APSO. The last two in the ranking are the SDA and ASDA. This ranking result indicates that the proposed algorithms have improved on their constituent algorithms. The proposed hybrid algorithms have been used to tune parameters of FPPI controller designed for application in a wireless networked environment. Results have shown that the proposed algorithms produce controller with good time domain performance of faster settling times as well as less overshoot compared to constituent algorithms.

As a part of future study, the implementation and evaluation of proposed optimal control strategy on real-time process plant will be done.

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Hybrid APSO–spiral dynamic algorithms with application to tuning of filtered PPI controller in a wirelessHART environment

Abstract

Keywords

1. Introduction

2. WirelessHART networked control system

2.1. WirelessHART control loop

3.1. APSO Algorithm

Table 1 SDA Illustration parameters, x0 = (25, 25) Fixed r = 0.95 Fixed θ = π/6 Case 1 θ = π/6 Case 4 r = 0.95 Case 2 θ = π/4 Case 5 r = 0.90 Case 3 θ = π/2 Case 6 r = 0.85

6. Conclusion

References

Table 1
SDA Illustration parameters, x₀ = (25, 25)

Fixed r = 0.95 Fixed θ = π/6

Case 1 θ = π/6 Case 4 r = 0.95

Case 2 θ = π/4 Case 5 r = 0.90

Case 3 θ = π/2 Case 6 r = 0.85