Chaotic particle swarm optimization for optimal design of PID controllers in industrial systems

Abstract

Proportional integral derivative (PID) controllers are widely used in industrial control processes since they are simple and easy to implement and act as an effective measure to manipulate the dynamic properties of industry systems. Carrying out the optimal design of the PID controllers is an indispensable constituent of the premises of highly precious control of these systems. In order to solve the problem of designing the parameters of the PID controllers more effectively, we bring forward a chaotic particle swarm optimization (CPSO) approach which we call CP IDSO. In this approach, we introduce the combination of chaotic logistic dynamics, hierarchical inertia weight, enhancement learning strategy, mutation mechanism and a proportional integral derivative (PID) controller. The chaotic logistic map is used in the substitution of the two random parameters affecting the convergence behavior. The hierarchical inertia weight coefficients are determined in accordance with the present fitness values of the local best positions so as to adaptively expand the particles’ search space. The PID controller and enhancement learning strategy are simultaneously incorporated into standard PSO (SPSO) to efficiently enhance the particles’ local and global search exploration and exploitation abilities. For performance validation of CP IDSO, CP IDSO, together with other algorithms like chaotic catfish PSO (CCPSO), genetic algorithm (GA) and PSO, is exploited to design the parameters of a PID controller in a Kalman filter based cybernetic system. The simulation results illustrate that CP IDSO exhibits better performance than other algorithms and yields the best result in the parameter optimization design of the system.

Keywords

Chaotic logistic dynamics optimal design Kalman filter particle swarm optimization proportional integral derivative controller

1 Introduction

A PID controller is a generic control loop feedback mechanism widely used in industrial control systems. Based on the PID control rules to tune the three parameters, the controller can provide control action designed for specific process requirements. The response of the controller can be described in terms of the responsiveness of the controller to an error, the degree to which the controller overshoots the setpoint, and the degree of system oscillation. Accordingly, it has historically been considered to be the best controller. However, the use of the PID algorithm for control does not guarantee the optimal control of the system or system stability. A good many researchers and practitioners around the world attempt to pursue the better solutions to tune the parameters for the optimal system control. Over the years, several meta-heuristic methods such as manual tuning, Ziegler-Nichols and so on, have been proposed for the tuning of PID controllers. In general, it is often hard for these methods to determine optimal or near optimal PID parameters in many industrial systems. Accordingly, many artificial intelligence techniques and random search methods like neural network, fuzzy system, neural-fuzzy logic, GA and PSO have been employed to improve the controller performances for a wide range of industrial systems while retaining their basic characteristics. They have been widely applied to proper tuning of PID controller parameters [1 , 23–25].

It is worth noting that the PSO technique, which is first introduced by Kennedy and Eberhart [7, 9], is a stochastic population-based one of the modern heuristic algorithms. It was developed through simulation of a simplified social system and has been found to implement easily and to achieve high efficiency in tuning the parameters of PID controllers in industrial systems [2 , 25]. Therefore, it has drawn much attention from researchers and practitioners throughout the world. Bouallègue et al. proposed a new PID-type fuzzy logic controller tuning strategy using a PSO approach [2]. In [4], Chang and Shih presented an improved particle swarm optimization to search for the optimal PID controller gains for a class of nonlinear systems. In [24], Zhao et al. applied two lbests multi-objective PSO to designing multi-objective robust PID controllers for two MIMO systems, namely, distillation column plant and longitudinal control system of the super maneuverable F18/HARV fighter aircraft. Zamani et al. employed the PSO algorithm to carry out the optimal design of a fractional order PID controller in an automatic voltage regulator [25].

Over the past few years, great progresses with regard to the PSO technique have been made. Scholars have applied chaos to the PSO performance improvement by utilizing chaotic ergodic orbits to search the optima instead of using random orbits [3 , 20]. In [3], Cai et al. presented a CPSO method based on the Tent equation to solve economic dispatch problems with generator constraints. Compared with the traditional PSO method, the CPSO method has good convergence property accompanied by the lower generation costs and can result in great economic effect. In [5], Chuang et al. proposed CCPSO. Statistical analysis of the experimental results indicate that the performance of CCPSO is better than the performance of PSO, CPSO, catfish PSO. Liu et al. proposed a hybrid particle swarm optimization algorithm by incorporating logistic chaos and adaptive inertia weight factor into PSO, which reasonably combines the population-based evolutionary PSO search ability with chaotic search behavior [15]. In [20], Wang and Liu proposed a logistic CPSO approach to generate the optimal or near-optimal assembly sequences of products. The proposed method is validated with an illustrative example and the results are compared with those obtained by using the traditional PSO algorithm under the same assembly process constraints.

Obviously, the CPSO algorithms have recently received much interest for achieving high efficiency and searching global optimal solution in nonlinearly multi-dimensional problem spaces. As a result, we herewith attempt to pursue an effective CPSO algorithm to tune the parameters of PID controllers in industrial systems. We put forward a novel CPSO algorithm which we call the chaotic logistic dynamics, hierarchical inertia weight, enhancement learning strategy, mutation mechanism and a PID controller hybridized chaotic PSO approach (CP IDSO). In order to verify the effectiveness of CP IDSO, we apply it to tuning the parameters of a PID controller in a Kalman filter based cybernetic system together with other algorithms like CCPSO, GA and PSO. Simulation results prove that CP IDSO is superior to CCPSO, GA and PSO in tuning the parameters of a PID controller in a Kalman filter based cybernetic system and that it is a more efficient approach to optimal design in engineering and sciences.

The rest of the paper is organized as follows. Section 2 depicts the derivation of CP IDSO and its enhancement learning strategy, mutation mechanism and the whole procedure. Section 3 presents the experimental study of applying CP IDSO to the optimal design of a PID controller in a Kalman filter based cybernetic system. Section 4 gives the conclusions and future work.

2 Representation of CP IDSO

In this part,we discuss the stability of CP IDSO, design a PID controller, depict the enhancement learning strategy and mutation mechanism, and give a full description of the procedure of CP IDSO in turn.

2.1 Analyzing the stability of CP IDSO and designing a PID controller

SPSO is a kind of typically stochastic standard algorithm to search for the best solution by simulating the movement of the flocking of birds or fish. It works by initializing a flock of birds or fish randomly over the searching space, where each bird or fish is called a “particle”. These particles fly with certain velocities and find the global best position after some generations. At each generation, they are dependent on their own momentum and the influence of their own local and global best positions x_lbest and x_gbest to adjust their own next velocity v and position x to move in turn. SPSO is clearly depicted as follows:

$\begin{matrix} v (t + 1) & = & ω_{pso} \cdot v (t) + c_{1} \cdot {rand}_{1} \cdot (x_{lbest} - x (t)) \\ + c_{2} \cdot {rand}_{2} \cdot (x_{gbest} - x (t)) \end{matrix}$ (1) $x (t + 1) = v (t + 1) + x (t),$ (2) where ω_pso, c₁ and c₂ denote the inertia weight coefficient, cognitive coefficient and social coefficient, respectively, and rand₁, rand₂ are both random values between 0 and 1. Besides, v is clamped to a given range [-v_max, +v_max].

Adjusting the Equations (1) and (2) to the Equations (3) and (4).

$\begin{matrix} v (t + 1) - v (t) \\ = (ω_{pso} - 1) \cdot v (t) \\ - (c_{1} \cdot {rand}_{1} + c_{2} \cdot {rand}_{2}) \cdot x (t) \\ + (c_{1} \cdot {rand}_{1} \cdot x_{lbest} + c_{2} \cdot {rand}_{2} \cdot x_{gbest}) \end{matrix}$ (3)

$\begin{matrix} x (t + 1) - x (t) \\ = v (t + 1) \\ = ω_{pso} \cdot v (t) \\ - (c_{1} \cdot {rand}_{1} + c_{2} \cdot {rand}_{2}) \cdot x (t) \\ + (c_{1} \cdot {rand}_{1} \cdot x_{lbest} + c_{2} \cdot {rand}_{2} \cdot x_{gbest}) \end{matrix}$ (4)

Supposing φ₁ = c₁ · rand₁, φ₂ = c₂ · rand₂, φ = c₁ · rand₁ + c₂ · rand₂ and $θ = \frac{φ_{2}}{φ}$ , the Equations (3) and (4) can be transformed into the following differential evolutionary SPSO equations (5) and (6) since $v (t + 1) - v (t) \approx \frac{d v}{dt}$ and $x (t + 1) - x (t) \approx \frac{dx}{dt}$ . $\begin{matrix} \frac{d v}{dt} & \approx & (ω_{pso} - 1) \cdot v (t) - φ \cdot x (t) \\ + (φ_{1} \cdot x_{lbest} + φ_{2} \cdot x_{gbest}) \end{matrix}$ (5) $\begin{matrix} \frac{dx}{dt} & \approx & ω_{pso} \cdot v (t) - φ \cdot x (t) \\ + (φ_{1} \cdot x_{lbest} + φ_{2} \cdot x_{gbest}) \end{matrix}$ (6)

Provided the initial outsets V (0) ≈0 and X (0) ≈0, the following formulae are obtained after the Laplace transformation of differential evolutionary SPSO equations. $V (s) \approx \frac{s}{s + 1} \cdot X (s)$ (7)

$\begin{matrix} X (s) & \approx & \frac{φ_{1} \cdot (s + 1)}{s^{2} + s \cdot (1 - ω_{pso} + φ) + φ} \cdot X_{lbest} (s) \\ + \frac{φ_{2} \cdot (s + 1)}{s^{2} + s \cdot (1 - ω_{pso} + φ) + φ} \cdot X_{gbest} (s) \\ = & \frac{φ_{1} \cdot (s + 1)}{s \cdot (s + 1 - ω_{pso})} \cdot (X_{lbest} (s) - X (s)) \\ + \frac{φ_{2} \cdot (s + 1)}{s \cdot (s + 1 - ω_{pso})} \cdot (X_{gbest} (s) - X (s)) \\ = & \frac{φ \cdot (s + 1)}{s \cdot (s + 1 - ω_{pso})} \cdot (((1 - θ) \cdot X_{lbest} (s) \\ + θ \cdot X_{gbest} (s)) - X (s)) \end{matrix}$ (8)

Supposing $G (s) = \frac{φ \cdot (s + 1)}{s \cdot (s + 1 - ω_{pso})}$ , the closed-loop transfer function for the input X (s) and the combination output of X_lbest (s) and X_gbest (s) is the following formula (9). $\frac{X (s)}{((1 - θ) \cdot X_{lbest} (s) + θ \cdot X_{gbest} (s))} = \frac{G (s)}{1 + G (s)}$ (9)

Provided X_lbest (s) = X_gbest (s), the Equation (9) is changed into the Equation (10). $\frac{X (s)}{X_{gbest} (s)} = \frac{G (s)}{1 + G (s)}$ (10)

Thus, the evolutionary relationship between the position X (s) and its global best position X_gbest (s) in a closed-loop scheme is displayed in Fig. 1. It clearly illustrates their evolutionary relationship which denotes a second order transfer function. In order to advance the evolutionary relationship, we add one PID controller between X_gbest (s) and X (s) as in Fig. 1, where the PID controller is expressed by the Equation (11) [12] and appears in the dashed framework. $G_{PID} (s) = k_{p} + \frac{k_{i}}{s} + k_{d} \cdot s$ (11)

Accordingly, we obtain the following formula (12). $\frac{X (s)}{X_{gbest} (s)} = \frac{G_{PID} (s) \cdot G (s)}{1 + G_{PID} (s) \cdot G (s)}$ (12)

The corresponding eigenvalue function is expressed below by the Equation (13) $1 + G_{PID} (s) \cdot G (s) = 0,$ (13) namely, $\begin{matrix} (1 + k_{d} \cdot φ) \cdot s^{3} + (1 + k_{p} \cdot φ + k_{d} \cdot φ) \cdot s^{2} \\ + (- ω_{pso} + φ \cdot k_{p} + φ \cdot k_{i}) \cdot s + k_{i} \cdot φ = 0 . \end{matrix}$

According to Routh-Hurwitz’s stability criterion, the inequalities are obtained below. $- ω_{pso} + k_{p} \cdot φ + k_{d} \cdot φ > 0$ (14)

$\begin{matrix} (k_{p}^{2} + k_{i} \cdot k_{p} + k_{d} \cdot k_{p}) \cdot φ^{2} \\ + (k_{p} - ω_{pso} \cdot (k_{p} + k_{d})) \cdot φ - ω_{pso} > 0 \end{matrix}$ (15)

On the other hand, the updated X (s) is as follows.

$\begin{matrix} X (s) & = & \frac{φ_{1} \cdot (s + 1) \cdot G_{PID} (s)}{s \cdot (s + 1 - ω_{pso})} \cdot (X_{lbest} (s) - X (s)) \\ + \frac{φ_{2} \cdot (s + 1) \cdot G_{PID} (s)}{s \cdot (s + 1 - ω_{pso})} \cdot (X_{gbest} (s) - X (s)) \\ = & \frac{φ \cdot (s + 1) \cdot G_{PID} (s)}{s \cdot (s + 1 - ω_{pso})} \cdot (((1 - θ) \cdot X_{lbest} (s) \\ + θ \cdot X_{gbest} (s)) - X (s)) \end{matrix}$ (16)

After being combined with the Equation (16), the Equation (7) is presented below.

$\begin{matrix} V (s) & = & \frac{φ \cdot G_{PID} (s)}{s \cdot (s + 1 - ω_{pso})} \cdot (((1 - θ) \cdot X_{lbest} (s) \\ + θ \cdot X_{gbest} (s)) - X (s)) \end{matrix}$ (17)

If the random parameters rand₁ and rand₂ in the Equation (1) of SPSO are chaotic, they can ensure the optimal ergodicity throughout the search space. Furthermore, there are no fixed points, periodic orbits, or quasi-periodic orbits in the behaviors of the chaotic systems. Therefore, they are necessarily substituted by the two sequences Cr^(t) and (1 - Cr^(t)) generated via the following logistic map Equation (18) ${Cr}^{(t + 1)} = μ \cdot {Cr}^{(t)} \cdot (1 - {Cr}^{(t)}), i = 0, 1, 2, \dots, n .$ (18) where Cr⁽⁰⁾ is generated randomly for each independent run, but it is not equal to {0, 0.25, 0.5, 0.75, 1}, and μ is equal to 4. Obviously, Cr^(t) is distributed in the interval (0, 1.0). So the driving parameter μ of the logistic map controls the behavior of Cr^t as the iteration number t goes to infinity. So φ and θ are changed into the following formulae (19) and (20). $φ = c_{1} \cdot {Cr}^{(t)} + c_{2} \cdot (1 - {Cr}^{(t)})$ (19) $θ = \frac{c_{2} \cdot (1 - {Cr}^{(t)})}{c_{1} \cdot {Cr}^{(t)} + c_{2} \cdot (1 - {Cr}^{(t)})}$ (20)

Thus, the Equation (17) is turned into the following time-varying function formula (21) after the inverse Laplace Transformation. Consequently, our proposed CP IDSO is comprised of the Equations (21) and (2).

$\begin{matrix} v (t + 1) \\ = ω_{pso} \cdot v (t) + φ \cdot ((1 - θ) \cdot (k_{p} \cdot (x_{lbest} - x (t)) \\ + k_{i} \cdot \int_{0}^{t} (x_{lbest} - x (t)) \cdot dt + k_{d} \cdot \frac{d (x_{lbest} - x (t))}{dt}) \\ + θ \cdot (k_{p} \cdot (x_{gbest} - x (t)) + k_{i} \cdot \int_{0}^{t} (x_{gbest} - x (t)) \cdot dt \\ + k_{d} \cdot \frac{d (x_{gbest} - x (t))}{dt})) \end{matrix}$ (21)

Being different from the Equation (1) in SPSO, the Equation (21) not only includes the proportional terms of (x_lbest - x (t)) and (x_gbest - x (t)), but also encompasses their integral terms and derivative terms. These terms enable CP IDSO to achieve a proper response, eliminate the steady-state errors, and improve particles’ evolutionary dynamics simultaneously so that CP IDSO enhances the diversity of the swarm and converges fast to the global best position. Moreover, The chaotic logistic map is used in the substitution of the two random parameters rand₁ and rand₂.

Based on the above inequalities (14) and (15) and our professional experiences, we design the following three coefficients of the PID controller, where t is the present generation, and MaxT is the maximum generation. $k_{p} = e^{(ω_{pso} - 1) \cdot \frac{t}{MaxT}}$ (22) $k_{i} = \frac{e^{(ω_{pso} - 1) \cdot \frac{t}{MaxT}}}{1 + e^{(ω_{pso} - 1) \cdot \frac{t}{MaxT}}}$ (23) $k_{d} = [e^{(ω_{pso} - 1) \cdot \frac{t}{MaxT}}]^{2}$ (24)

Concerning the inertia weight coefficient, we adopt the following hierarchical formula (25) [15]

$\begin{matrix} w_{pso} \\ = {\begin{matrix} {ω_{pso}}_{\min} + \frac{({ω_{pso}}_{\max} - {ω_{pso}}_{\min}) (f - f_{\min})}{f_{avg} - f_{\min}}, & f \leq f_{avg} \\ {ω_{pso}}_{\max}, & f > f_{avg} \end{matrix}, \end{matrix}$ (25) where ω_pso_max and ω_pso_min represent the maximum and minimum of ω_pso, f is the current objective value of the particle, f_avg and f_min are the average and minimum objective values of all particles, respectively. In addition, the cognitive coefficient is supposed to decrease linearly from 2 to 0 while the social coefficient is supposed to increase linearly from 0 to 2.

2.2 Enhancement learning of CP IDSO

In order to promote the evolutionary process of CP IDSO, we adopt an enhancement learning strategy (ELS) to help particles perform comprehensive learning from their own local and neighboring best positions, other local best positions, and their global best position. It is evident that particles are easily trapped into local optima after some iterations. Therefore, for a specific local best particle, we first randomly select some other local best particles from the population whose total number is equal to their dimensional number. Then, we take turns to choose one different dimension from the selected local best particles, whose position is $x_{lbest}^{j} (k)$ and velocity vector is $v_{lbest}^{j} (k)$ . Next, we use $x_{lbest}^{j} (k)$ to replace the same dimension (j) of the specific local best particle as a temporary specific local best particle (x_lbest′). Successively, we compare the fitness value of a specific local best particle with the fitness(x_lbest′). If the fitness value of a specific local best particle > the fitness(x_lbest′), the position of the specific local best particle in the dimension j is moved to $x_{lbest}^{j} (k)$ . The same method is also exploited to learn from their neighboring best positions.

2.3 Mutation of CP IDSO

It has been observed that the normal SPSO is easily stagnated in local optimum because of the lack of diversity of the population. Thus, particles remain in a local optimum for unpredictable generations. In order to increase search diversity and avoid getting trapped in local optima, many leaping-out mechanisms are proposed [13 , 22]. However, the performance can be affected in many factors and is hard to predict after introducing the leaping-out algorithms. In order to improve the diversity of the particles and prevent them from being premature, we adopt an effective measure to easily implement. Compared with other mutation mechanisms, our mutation is produced by randomly selecting the local best particle and carrying out the acrossover operation with a random number of a Gaussian distribution. Its advantages are the twofold operational randomness properties. These random operations may result in both the position improvement of the global best particle and the diversity promotion of the swarm.

In CP IDSO, we first randomly select the local best particle (x_lbest (k)) out of the population. If it is not the global best particle (x_gbest), we randomly choose one dimension (j) from the selected particle, whose position is $x_{lbest}^{j} (k)$ and velocity vector is $v_{lbest}^{j} (k)$ . Thereafter, we use $x_{lbest}^{j} (k)$ to replace the same dimension (j) of the global best particle as a temporary global best particle (x_gbest′). Otherwise, we use the following formula $x_{lbest}^{j} (k) + (x_{\max}^{j} - x_{\min}^{j}) \cdot Gaussian (μ, σ^{2})$ (26) to do it, where the search range $[x_{\min}^{j}, x_{\max}^{j}]$ is the same as the lower and upper bounds of the problem, and the Gaussian(μ, σ²) is a random number of a Gaussian distribution with a zero mean μ and a standard deviation σ. Similar to some time-varying neural network training schemes, it is suggested that σ be linearly decreased with the generation number, which is given by $σ = 1 - \frac{t}{MaxT},$ (27) where t is the present generation and MaxT is the maximum generation. Next, we compare the fitness(x_gbest) with the fitness(x_gbest′). If the fitness(x_gbest)> the fitness(x_gbest′), the position of the global best particle in the dimension j is moved to $x_{{gbest}^{'}}^{j}$ and its updated fitness(x_gbest) value is equal to the fitness(x_gbest′).

2.4 Procedure of CP IDSO

Consequently, based on the aforementioned contexts, our proposed CP IDSO can be depicted below in detail.

Step 1: Initialize parameters including the number PN of particles, dimensional size D of each particle, maximum generation number MaxT, initial chaotic logistic values Cr⁽⁰⁾, initial position x and velocity v of each particle, inertia weight coefficient w_pso, and cognitive coefficient c₁, social coefficient c₂. Calculate the initial fitness of each particle, and set the initial local best position x_lbest and global best position x_gbest.

Step 2: If the specific local optimal value x_lbest (k) does not evolve for some certain iterations, improve the specific local best position by the above-mentioned ELS. Thereafter, according to the equations (22)–(24), calculate the three parameters k_p, k_i and k_d of the PID controller. Then in terms of the equations (21) and (2), calculate the next velocity v (t) and position x (t) of each particle. Next, calculate the fitness of each particle, set the local best position x_lbest and the global best position x_gbest. Thereafter, update the global best position x_gbest with the temporary global best mutation position x_gbest′ if the fitness (x_gbest)> the fitness(x_gbest′).

Step 3: Observe if the global best fitness(x_gbest) meets the given stopping threshold or not, or observe if the maximum generation number MaxT reaches or not. If not, go back to Step 2.

Step 4: Otherwise, the operation can be terminated. Finally, output the global best position x_gbest, and its corresponding global best fitness as well as convergent generation number.

The pseudo-code for CP IDSO is presented below in Algorithm 1.

1: /*initialize the swarm.*/

2: fori = 1 → PNdo

3: create particle p_i with dimension D, velocity v_i and position x_i from 1 to PN.

4: set x_lbest (i) = x_i

5: calculate fitness (x_i).

6: end for

7: set x_gbest = best(x_lbest (i))

8: calculate inertia coefficient w_pso, cognitive coefficient c₁ and social coefficient c₂.

9: set maximum generation number MaxT and chaotic variable Cr⁰.

10: / *update velocity and position with an evolutionary PID style strategy.*/

11: fort = 1 → MaxTdo

12: calculate PID controller parameters: k_p, k_i and k_d.

13: fori = 1 → PNdo

14: / *improve local best position at a given generation.*/

15: ifrepeat _ num (i) > =5 then

16: set tmp _ x_lbest (i) = x_lbest (i)

17: randomly create a D dimensional array ar between 1 and PN.

18: forii = 1 → Ddo

19: set $tmp_x_{lbest}^{ii} (i) = x_{lbest}^{ii} (ar (ii))$

20: iffitness (tmp _ x_lbest (i)) > fitness (x_lbest (i)) then

21: set $tmp_x_{lbest}^{ii} (i) = x_{lbest}^{ii} (i)$

22: end if

23: end for

24: set x_lbest (i) = tmp _ x_lbest (i)

25: end if

26: calculate velocity v_i and position x_i, according to the equations (21) and (2).

27: iffitness (x_i) < fitness (x_lbest (i)) then

28: set x_lbest (i) = x_i

29: else

30: set repeat _ num (i) = repeat _ num (i) +1

31: end if

32: iffitness (x_lbest (i)) < fitness (x_gbest) then

33: set x_gbest = x_lbest (i)

34: set fitness (x_gbest) = fitness (x_lbest (i)

35: /*mutation of global best position.*/

36: randomly select k between 1 and PN

37: set x_gbest′ = x_gbest

38: ifk ! = ithen

39: randomly select j between 1 and D, and crossover between $x_{{gbest}^{'}}^{j}$ and $x_{lbest}^{j} (k)$

40: else

41: calculate standard deviation σ, randomly select j between 1 and D, and crossover between $x_{{gbest}^{'}}^{j}$ and the equation (26)

42: end if

43: iffitness (x_gbest) < fitness (x_gbest′) then

44: set $x_{{gbest}^{'}}^{j} = x_{gbest}^{j}$

45: end if

46: set x_gbest = x_gbest′

47: end if

48: end for

49: /*operation termination.*/

50: if goal threshold or maximum generation number MaxT reaches then

51: break

52: end if

53: end for

54: output results.

3 Optimal design of PID controller in a Kalman filter based cybernetic system

In this part, we conduct a detailed experimental study to optimize the parameters of a PID controller in a Kalman filter based cybernetic system by using CP IDSO. The experiment includes the description of the Kalman filter based cybernetic system and experimental setup, parametric optimization design and experimental results as well as optimization design validation.

3.1 Description of the Kalman filter based cybernetic system and experimental setup

The PID cybernetic system based on a Kalman filter is shown in Fig. 2. In Fig. 2, the discrete linear system is modeled by the following eqations $\begin{matrix} x (k) & = & Ax (k - 1) + B (u (k - 1) + w (k - 1)) \\ y_{v} (k) & = & Cx (k) + D + v (k), \end{matrix}$ where A is the system matrix, B is the input matrix, C is the output matrix, D is the transitional matrix, w (k) is the process disturbance noise signal, and v (k) is the measured noise signal. The discrete Kalman filter is described below $\begin{matrix} M_{n} (k) & = & \frac{P (k) C^{T}}{CP (k) C^{T} + R} \\ P (k) & = & AP (k - 1) A^{T} + {BQB}^{T} \\ p (k) & = & (I_{n} - M_{n} (k) C) P (k) \\ x (k) & = & Ax (k - 1) + M_{n} (k) (y_{v} (k) - CAx (k - 1)) \\ y_{e} (k) & = & Cx (k) + D, \end{matrix}$ where Q is the covariance of w (k), R is the covariance of v (k), P (k) is the initial error covariance of the discrete Kalman filter. Accordingly, the error covariance of the discrete Kalman filter is expressed below. $errcov (k) = CP (k) C^{T}$

The controlled object is expressed by the following transfer function. $G_{p} (s) = \frac{133}{s^{2} + 25 s}$

As a result, the above-mentioned known matrixes are as follows. $A = [\begin{matrix} 0 & 1 \\ 0 & - 25 \end{matrix}] B = [\begin{matrix} 0 & 133 \end{matrix}]$ $C = [\begin{matrix} 1 & 0 \end{matrix}] D = [\begin{matrix} 0 \end{matrix}]$

Besides, the sampling time is 1ms. The process disturbance noise signal w (k) and the measured noise signal v (k) are both white noise signals with their amplitudes 0.002 while the input signal is a step signal. The covariance Q of w (k) is set at 1, and the covariance R of v (k) is set at 1. The whole simulation time is set within 1s.

In order to evaluate the performance of CP IDSO, we conduct the experiments to compare four state-of-the-art algorithms including our proposed CP IDSO, CCPSO, GA and PSO for the optimal design of the PID controller in the Kalman filter based cybernetic system. Their population size PN is set at 20, maximum generation number MaxT is set at 50, and the three dimensional (3-D) parameters k_p, k_i and k_d are retrieved around the ranges [0, 10], [0,1] and [0, 10], respectively. Their settings of other important parameters are summarized in Table 1. The fitness function for 3-D optimization design is determined below $J = \int_{0}^{'} (w_{1} | e (t) | + w_{2} u^{2} (t) + w_{3} | e (t) |) dt,$ where e (t) = y_d (t) - y_e (t), y_d is the ideal input signal, y_e is the output signal, w₁, w₂ and w₃ are weight coefficients, and w₃ ⪢ w₁. w₁, w₂ and w₃ are set at 0.999, 0.001 and 10, respectively.

3.2 Parametric optimization design and experimental results

Figure 3 presents the median convergence and optimal design characteristics of diverse involved algorithms for 3-D optimal design problem above. The results of the proposed CP IDSO are depicted by bold solid lines in Fig. 3. Note that the results of Y axis in Fig. 3(a) are logarithmic. Table 2 shows the median results of diverse selected algorithms for 3-D optimal design problem. The best results among the four selected algorithms are shown in bold in Table 2. In order to determine whether the results obtained by CP IDSO are statistically different from the results generated by other algorithms, the nonparametric Wilcoxon rank sum tests are conducted between the CP IDSO’s result and the best result achieved by other algorithms for each test function. The h_t-tests presented in the last column of Table 2 is the result of t-tests. An h_t-tests of 1 indicates that the performances of the two variants are statistically different with 95% certainty, whereas an h_t-tests of 0 implies that the performances are not statistically different.

From the results in Table 2 and the graphs in Fig. 3, we clearly notice that CP IDSO achieves the best result while GA does worst. CCPSO yields a better result than PSO. CP IDSO shows its comparatively better mean best fitness value than CCPSO and PSO in Table 2 though it is not obvious in Fig. 3(a). Furthermore, for CP IDSO, CCPSO and PSO, PD control is more suitable to tune the performance of the Kalman filter based cybernetic system instead of either PID control or PI control.

3.3 Optimization design validation

To verify the optimal results of the four algorithms, the median parametric results about k_p, k_i and k_d are exploited to tune the performance of the Kalman filter based cybernetic system. The concrete results of the verification experiments are presented in Fig. 4. Note that the results of Y axes in Fig. 4 are logarithmic. The position tracking error is defined as the square sum of difference between the step input signal and the output signal. Figure 4 presents the position tracking responses and the position tracking errors by CCPSO, GA, PSO and CP IDSO in accordance with step input signal while the Kalman filter is considered or not. Figure 4(a) and (b) present the position tracking responses and the position tracking errors by CCPSO, GA, PSO and CP IDSO when the Kalman filter is not considered. Figure 4(c) and (d) present the the position tracking responses and the position tracking errors by CCPSO, GA, PSO and CP IDSO when the Kalman filter is considered. Table 3 presents the response indicators of diverse involved algorithms for 3-D optimal design problem in accordance with step input signal while the Kalman filter is considered. In the meantime, the time unit is second.

From the graphs in Fig. 4(a), (b), (c) and (d), one may observe that the tuning performances by CCPSO, GA, PSO and CP IDSO when the Kalman filter is considered are much better than those when no Kalman filter is considered. Moreover, the position tracking error by CP IDSO is smallest whilst the one by GA is biggest. The result by CCPSO is comparably better than that by PSO. From the results in Table 3, We find that GA gets the smallest overshoot and the shortest delay time while the self regulating times of all the algorithms are same. The fact indicates that though selecting chaotic random parameters consumes run time and causes the response fluctuation, it has no effect on the whole self regulating time. On the other hand, the reason why CP IDSO yields better results than other involved algorithms for the optimal design of the PID controller is that the combination of the time varying PID controller, chaotic random parameters and mutation mechanism has effectively improved the evolutionary dynamics of particles and enhanced the particles’ local and global search exploration and exploitation abilities. Despite of these, all the involved algorithms can be utilized to tune the performance of the Kalman filter based cybernetic system.

4 Conclusions and future work

In order to solve the problem of designing the parameters of the PID controllers in industrial cybernetic systems more effectively, we present a novel CPSO variant which we called CP IDSO, where we attempt to introduce the combination of chaotic logistic dynamics, hierarchical inertia weight, enhancement learning strategy, mutation mechanism and a PID controller into SPSO. Successively, CP IDSO, together with CCPSO, GA and PSO, is used in the optimal design of the PID controller in the Kalman filter based cybernetic system. The experimental results illustrate that our proposed CP IDSO outperforms CCPSO, GA and PSO for the parameter optimization of the PID controller in the Kalman filter based cybernetic system and is regarded as a more effective tool for optimization computation and search problem solving in engineering and sciences since it enhances the diversity of the swarm and has good convergence efficiency.

Future work will further the enhancement learning ability of CP IDSO and the performances of PID controllers. Moreover, we will apply the proposed CP IDSO to other practical engineering and science applications.

Footnotes

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities in China. The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.

References

Ayala

H.V.H.

and Coelho

L.D.S.

, Tuning of PID controller based on amultiobjective genetic algorithm applied to a robotic manipulator, Expert Systems with Applications39(10) (2012), 8968–8974.

Bouallègue

, Haggège

, Ayadi

and Benrejeb

, PID type fuzzy logic controller tuning based on particle swarm optimization, Engineering Applications of Artificial Intelligence25(3) (2012), 484–493.

Cai

, Ma

, Li

and Peng

, Chaotic particle swarm optimization for economic dispatch considering the generator constraints, Energy Conversion and Management48(2) (2007), 645–653.

Chang

W.-D.

and Shih

S.-P.

, PID controller design of nonlinear systems using an improved particle swarm optimization approach, Communications in Nonlinear Science and Numerical Simulation15(11) (2010), 3632–3639.

Chuang

L.Y.

, Tsai

S.T.

and Yang

C.H.

, Chaotic catfish particle swarm optimization for solving global numerical optimization problems, Applied Mathematics and Computation217(16) (2011), 6900–6916.

Dounis

A.I.

, Kofinas

, Alafodimos

and Tseles

, Adaptive fuzzy gain scheduling PID controller for maximum power point tracking of photovoltaic system, Renewable Energy60 (2013), 202–214.

Eberhart

R.C.

and Kennedy

, A new optimizer using particle swarm theory, in: Proceedings of 6th International Symposium on Micromachine Human Science1996,pp. 39–43.

Fang

M.-C.

, Zhuo

Y.-Z.

and Lee

Z.-Y.

, The application of the self-tuning neural network PID controller on the ship roll reduction in random waves, Ocean Engineering37(7) (2010), 529–538.

Kennedy

and Eberhart

R.C.

, Particle swarm optimization, in: Proceedings of IEEE International Conference Neural Networks1995, pp. 1942–1948.

10.

Kim

S.-M.

and Han

W.-Y.

, Induction motor servo drive using robust PID-like neuro-fuzzy controller, Control Engineering Practice14(5) (2006), 481–487.

11.

T.D.

, Kang

H.-J.

, Suh

Y.-S.

and Ro

Y.-S.

, An online selfgain tuning method using neural networks for nonlinear PD computed torque controller of a 2-dof parallel manipulator, Neurocomputing116 (2013), 53–61.

12.

Levine

W.S.

, The control handbook, CRC Press/IEEE Press, Piscataway, New Jersey, 1996.

13.

Liang

J.J.

, Qin

A.K.

, Suganthan

P.N.

and Baskar

, Comprehensive learning particle swarm optimizer for global optimization of multimodal functions, IEEE Transactions on Evolutionary Computation10(3) (2006), 281–295.

14.

Lin

and Lian

R.-J.

, Hybrid fuzzy-logic and neural-network controller for MIMO systems, Mechatronics19(6) (2009), 972–986.

15.

Liu

, Wang

, Jin

Y.H.

, Tang

and Huang

D.-X.

, Improved particle swarm optimization combined with chaos, Chaos, Solitons and Fractals25(5) (2005), 1261–1271.

16.

Montes de Oca

M.A.

, Stutzle

, Birattari

and Dorigo

, Frankenstein’s PSO: A composite particle swarm optimization algorithm, IEEE Transactions on Evolutionary Computation13(5) (2009), 1120–1132.

17.

Panda

, Multi-objective PID controller tuning for a FACTSbased damping stabilizer using Non-dominated Sorting Genetic Algorithm-II, International Journal of Electrical Power and Energy Systems33(7) (2011), 1296–1308.

18.

Ratnaweera

, Halgamuge

S.K.

and Watson

H.C.

, Selforganizing hierarchical particle swarm optimizer with timevarying acceleration coefficients, IEEE Transactions on Evolutionary Computation8(3) (2004), 240–255.

19.

Truong

D.Q.

and Ahn

K.K.

, Force control for press machines using an online smart tuning fuzzy PID based on a robust extended Kalman filter, Expert Systems with Applications38(5) (2011), 5879–5894.

20.

Wang

and Liu

J.H.

, Chaotic particle swarm optimization for assembly sequence planning, Robotics and Compter-Integrated Manufacturing26(2) (2010), 212–222.

21.

Wang

Y.-X.

, Xiang

Q.-L.

and Zhao

Z.-D.

, Particle swarm optimizer with adaptive tabu and mutation: A unified framework for efficient mutation operators, ACM Transactions on Autonomous and Adaptive Systems5(1) (2010), 1556–1565.

22.

Zhan

Z.-H.

, Zhang

, Li

and Chung

H.S.H.

, Adaptive particle swarm optimization, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics39(6) (2009), 1362–1381.

23.

Zhang

B.T.

and Pi

Y.G.

, Enhanced robust fractional order proportional-plus-integral controller based on neural network for velocity control of permanent magnet synchronous motor, ISA Transactions52(4) (2013), 510–516.

24.

Zhao

S.-Z.

, Iruthayarajan

M.W.

, Baskar

and Suganthan

P.N.

, Multi-objective robust PID controller tuning using two lbests multi-objective particle swarm optimization, Information Sciences181(16) (2011), 3323–3335.

25.

Zamani

, Karimi-Ghartemani

, Sadati

and Parniani

, Design of a fractional order PID controller for an AVR using particle swarm optimization, Control Engineering Practice17(12) (2009), 1380–1387.