A stochastic multi-parameters divergence method for online auto-tuning of fractional order PID controllers

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Journal of the Franklin Institute 351 (2014) 2411–2429 www.elsevier.com/locate/jfranklin

A stochastic multi-parameters divergence method for online auto-tuning of fractional order PID controllers Celaleddin Yeroğlun, Abdullah Ateş Computer Engineering Department, Engineering Faculty, İnönü University, Malatya, Turkey Received 9 July 2013; received in revised form 18 November 2013; accepted 10 December 2013 Available online 19 December 2013

Abstract This paper presents a stochastic multi-parameters divergence method for online parameter optimization of fractional-order proportional–integral–derivative (PID) controllers. The method is used for auto-tuning without the need for exact mathematical plant model and it is applicable to diverse plant transfer functions. The proposed controller tuning algorithm is capable of adaptively responding to parameter fluctuations and model uncertainties in real systems. Adaptation skill enhances controller performance for real-time applications. Simulations and experimental observations are carried on a prototype helicopter model to confirm the performance improvements obtained by the online auto-tuning of fractional-order PID structure in laboratory conditions. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction There has been renewed interest in fractional-order systems due to their practical outcomes in engineering and applied sciences [1]. The first discussion on a fractional-order derivative can be traced back to the 1690s, when L’Hospital wrote to Leibniz speculating about whether the order of a derivative could become a non-integer [2]. Although fractional order calculus attracted the attention of Euler and Lagrange, systematical labors were started at the beginning and middle of the 19th century by Liouville (1832), Holmagren (1864), Rieman (1953) [3,4]. In a practical sense, some approximate definitions for the calculation of fractional-order integrations and derivations were given by Grünwald–Letnikov, Rieman–Liouville and Caputo [5–7]. In the last n

Corresponding author. E-mail addresses: [email protected], [email protected] (C. Yeroğlu), [email protected] (A. Ateş). 0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.12.006

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several decades, with a better theoretical understanding of fractional calculus and subsequent developments in the computing technologies, fractional calculus has begun to be widely utilized in various science and engineering areas [8]. There is also a growing trend toward the use of fractional order systems in control [9,10] and robotic applications [11]. Some studies present relation between fractional calculus and performance or flexibility of a new control system approaches [12]. Satisfactory results, obtained for fractional-order systems, have encouraged researchers and engineers to continue working in this direction. Recent research efforts have been focused on developing analysis and design methods for the use of fractional order controllers to extend classical control methods [10]. Some of these studies aim to develop a tuning method that is simple, practical, robust, and one which provides a satisfactory performance in real-world applications [13,14]. Most of the design efforts intend to achieve optimum system performance. Optimization methods greatly helps to provide effective controller design maintaining controller performance for optimum control system applications in real-world conditions. Due to importance of optimum control structures, several studies, related with different optimization algorithms, have been proposed to solve different control problems [15]. For example, neurofuzzy control was used for underwater vehicle-manipulator systems in [16]. Methodology of the single parameter tuning to obtain PI controller coefficients by using Dipohantine equations for practical application was discussed in [17]. Synthesis of control vector optimization and genetic algorithms for mixed-integer dynamic optimization was presented in [18]. In recent studies, promising research efforts use optimization algorithms for different controller design methods [19]. In many tuning methods, the tuning process requires a plant transfer function [20,21]. Unfortunately, an exact mathematical modeling of a physical system is not always possible due to the unpredictable environmental factors that affect the system outputs. That is why, in realworld control applications, adaptive controllers that are capable of online auto-tuning of controller coefficients, are necessary to deal with the negative impacts of temporal parameter fluctuations on control performance. Indeed, the adaptation skill is one of main concerns in controller design technology for satisfactory performance and robust control in real applications. This paper presents a Stochastic Multi-parameters Divergence-based Optimization (SMDO) method for the online auto-tuning of fractional order PIDs (PI λ Dμ ), which lead to the development of Adaptive-PI λ Dμ (A-PI λ Dμ ) controller structures. This optimization basically follows the descent of an object function by consecutive sets and trials of parameters. The performance of the A-PI λ Dμ controller using SMDO was demonstrated in simulations and experimental work conducted on a Twin-Rotor Multi-Input/Multi-Output (MIMO) System (TRMS) experimental setup. The TRMS setup provides a nonlinear and complex helicopter flight control test platform to simulate several flight modes – such as hovering and taking off – in laboratory conditions. In fact, flight control problems [14,22] involve serious complications, due to the complex mechanisms of aircrafts, nonlinear and inaccessible aerodynamics, and changes in flight conditions depending on altitude, payload, and weather conditions. That is why, adaptive control skill based on online auto-tuning of controller parameters is required to maintain control performance under the variable conditions found in real-time flight applications. The novelty of the paper lies in introducing SMDO method for adaptive fractional order controller design for real time flight application in laboratory condition. The rest of the paper is organized as follows; Section 2 presents a brief introduction to the proposed SMDO method. In Section 3, simulation results are presented demonstrating the control performance of PI λ Dμ controllers using the SMDO method for a second-order plant

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transfer function. Section 4 is devoted to the experimental demonstration of A-PI λ Dμ controllers, using SMDO in a TRMS experimental setup. Section 5 includes conclusions. 2. Stochastic multi-parameters divergence method for the auto-tuning of controllers 2.1. Motivation Fractional-order controllers are commonly considered as a generalization of PID controller. Fractional-order PIDs are widely known as PI λ Dμ controllers, where λ and m are the fractional orders of integration and differentiation respectively [20]. It has been reported that PI λ Dμ controllers demonstrate a better control performance in comparison with classical PID controllers, because of the involvement of extra real parameters (λ, m) in the control process [10,20]. Because (λ, m) parameters enlarged output span of classical PID controller, which obtains a better-fitting control signal for enhanced plant control performance. Due to the widespread use of classical PID controllers in industry, several methods have been proposed for achieving best performance of the system and practical tuning of classical PID controllers. Nowadays, the tuning problems of PI λ Dμ controllers is a promising research area. The PI λ Dμ controller transfer function was widely expressed as [20] CðsÞ ¼ k p þ

ki þ k d sμ sλ

ð1Þ

The fractional-order controller parameters, namely proportional gain – k p , integration gain – ki , derivative gain – kd , fractional order of integration – λ, and fractional order of derivative – μ, define a five-component parameter space. Consider the basic block diagram of an adaptive control system in Fig. 1. Here, GðsÞ represents the transfer function of the plant being controlled, CðsÞ represents the transfer function of the controller, and the optimizer block performs optimization of the parameters for the enhancement of control performance. The motivation for the present paper comes from the understanding of importance of online auto-tuning of PI λ Dμ controller in real-time applications. Auto-tuning of controllers defines a parameter-optimization problem on the basis of a performance evaluation function which should converge towards minima or maxima of the function. In our literature survey, we encountered several optimization methods. A holistic multi-objective optimization method, using a multiobjective evolutionary algorithm, has been proposed for controller tuning [14]. A self-tuning controller incorporating a gain and phase margin formula has been implemented and tested [23]. A self-tuning internal model PID controller with gain and phase margin specifications has been Optimizer Parameters

r e

G(s)

C(s) u

o

Fig. 1. Basic block diagram of adaptive unity feedback control system.

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presented for the real-time monitoring of gain and phase margins of PID control systems [24]. An improved evolutionary Non-dominated Sorting Genetic Algorithm (NSGA-II), augmented with a chaotic Henon map is used for the multi-objective optimization based design procedure in [25]. Optimal tuning of fractional order PID controllers successfully combined with stochastically varying network delays and packet dropouts in networked control system applications in [26,27]. There are many other potential optimization techniques for the tuning of parameters, depending on performance precedence. This study proposes the SMDO optimization algorithm that is simple to implement for multi-parameter optimization, yet effective for fast convergence to the minima, easy adaptation to real time application and stochastic in nature, providing continuous search capability.

2.2. Problem formulation Let us denote PI λ Dμ controller coefficients at time instance t ¼ nh by the vector X n ¼ ½knp kni k nd λn μn  in a parameter space X. Here, parameter h represents the sampling period of discrete state sampling, and n ¼ f1; 2; 3; :::g is the discrete time increment. In order to adapt a PI λ Dμ controller to the system, X n controller parameter vector has to be persistently changed to a X m vector in time that satisfies the following adaptation condition, EðX m Þ  EðX n Þo0

ð2Þ

where Eð:Þ is a positive real-valued error function that is used to evaluate the performance of PI λ Dμ controllers in the parameter optimization. Let p ¼ m  n40 refer to the adaptation period of the PI λ Dμ controller, where n and m represent the current and a future sampling respectively. During the adaptation period, the PI λ Dμ controller transforms to a better controller function, in terms of minimization of the error function. Adaptation degree of the controller – namely the adaptation rate – can be evaluated by the rate of ΔEm;n =p. The ΔE m;n indicates the convenience of a parameter vector divergence in terms of the error function and it can be defined as ΔE m;n ¼ EðX m Þ  EðX n Þ. Let ΔX m;n denote the degree of deviation in X vector from sampling n to sampling m. The remainder of this section is given to provide some basic definitions and remarks upon this process. Definition 1. If a deviation ΔX m;n satisfies Eq. (2), this divergence will be considered a convenient divergence in the parameter space at time instance nh. Otherwise, ΔX m;n is not a convenient divergence in the parameter space of the A-PI λ Dμ controller. Lemma 1. Given any positive real valued error function EðXÞ : R-Rþ , a vector divergence ΔX m;n in the parameter space of a controller leads to a deviation in descent of EðX n Þ, only if it satisfies Eq. (2) for any discrete sampling n and m, where m4n40. Proof. A parameter deviation resulting in a descent of the positive real valued EðX n Þ in time can be expressed as, ð∂E=∂tÞo0. Considering the limit definition of the derivatives, one can reorganize it as limΔt-0

ΔE o0 Δt

ð3Þ

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For any discrete time sampling of n and m for m4n40, a Δt time interval can be always found such that Δtomh  nh. In this case, one can easily write the following inequality, ΔE ΔE olimΔt-0 o0 ð4Þ ðm  nÞh Δt then, for a state sampling at n and m for m4n40, if ΔE ¼ EðX m Þ EðX n Þ is considered, the condition of EðX m Þ  EðX n Þo0 is valid. Remark 1. For a vector divergence ΔX m;n in the parameter space of a controller, the controller is said to be adapting in accordance with the error function E, only if ΔE m;n =p is negative and real. 2.3. SMDO method The condition given by Lemma 1 assures the evolution of X n vectors in the parameter space towards the descent of the error function. This persistent evolution of X n vectors along the descent of the error function potentially can continue to a minima of the error function. The SMDO presents a stochastic parameter divergence in the parameter space X for the persistent minimization of the error function. The aim of the controller adaptation process is a transformation from the initial X 0 parameter vector to the optimal vector X opt , which can accommodate to the minima of the error function in a predefined search range (X s A X) of parameter space and this can be expressed as EðX opt Þ ¼ minX n A X s fEðX n Þg

ð5Þ

For illustration proposes, let us assume X~ ¼ ½x1 x2  to be two component vectors in the parameter n space X~ . Fig. 2(a) shows a representation of the evolution of a two-dimensional X~ to the X~ opt vector nþ1;n using consecutive ΔX~ divergences in the parameter space X~ . Fig. 2(b) illustrates the divergence n n region (DR) for a X~ vector using a dashed circle. The divergence region of a X~ vector specifies the nþ1;n upper limit for the magnitude of ΔX~ divergence vectors in the parameter space. A shrink in the size of divergence regions mostly results in a slower approximating to the X~ opt vector during the adaptation process. Conversely, enlarging the divergence region may result in a faster adaptation, however it may also decrease the accuracy of the optimization because of the highly volatile

x2

~ Xn

~ n,n+1 ΔX

x2 2

~ X n+1 3

1 4

Xopt

DR ~ Xn x1

x1

Fig. 2. (a) An illustration of two component X~ ¼ ½x1 x2  vectors evolution to X~ opt . (b) A forward divergence of x1 and x2, indicated by labels 1 and 2, and a backward divergence of x1 and x2, indicated by labels 3 and 4, respectively.

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E

E

DR

DR

X

Xn X

X

n

n+1

n+1

X Xopt

n+2

X

X

n+2

X X opt

n+3

X

Fig. 3. (a) Convergence to X opt by using a fixed step ΔX nþ1;n . (b) Convergence to X opt by using a variable step ΔX nþ1;n . n nþ1;n convergence of X~ to the X~ opt vector. A variable step ΔX~ in the divergence region can be used to improve the accuracy of the convergence to X~ opt . Fig. 3(a and b) illustrate the convergence potentials of a fixed step divergence and a variable step divergence for the same convergence regions. nþ1;n n A fixed step ΔX~ can be stuck in the vicinity of X~ opt when the distance between X~ opt and X~ is nþ1;n less than the constant step length. Whereas, a variable step ΔX~ can provide a finer approximation of X~ opt . The SMDO method applies random step divergences to generate variable steps in n approximating the X~ opt . The random steps always have a possibility of X~ ¼ X~ opt . In fact, in the case n of convex error functions, the probability of X~ ¼ X~ opt goes to 1 for n-1, so, limn-1 P n ðX~ ¼ X~ opt Þ ¼ 1. The SMDO method is based on stochastic bidirectional componential set and trial (ST) operations. It searches for a convenient divergence vector satisfying the condition given by Eq. (2). For these purposes, a component of X vector is tested initially for a random forward step. If it does not result in a convenient divergence (ΔE nþ1;n Z 0), then the component is tested for a random backward step. If the backward step test also fails to yield a convenient divergence of X n , the component keeps its original value, which implies a non-divergent stochastic bidirectional componential ST operation in the optimization. Fig. 2(b) illustrates possible forward and backward divergence vectors of the two-component space. Fig. 4(a) illustrates the trajectory of a twocomponent X~ vector on the contour plot of a convex error function, while converging the global minima by the proposed SMDO method. Fig. 4(b) shows decreases in the error function throughout the convergence toward the global minima shown in Fig. 4(a). SMDO method reaches the vicinity of the global minima with an adaptation rate (ΔE 101;1 =100) of  0.07 up to the 100th iteration. SMDO does not include any sophisticated computation block (sigmoid, tanh,…etc.) and processes (reproduction, mutation, recombination) in any part of the algorithm, which may reduce computation complexity and makes the method simple. Fig. 4 illustrates that the SMDO reaches to minima of a convex error function about 100 iterations (ST).

2.4. Adaptive PI λ Dμ controller with SMDO method Adaptive control has been preferred in many systems with high complexities and uncertainties due to its flexibility. Adaptive control enables to design a controller which can achieve the prespecified control objectives for a given class of systems with uncertain dynamics. Many valuable

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E

7 6 5

x2

4

6

~ X 150

5 4 3

2

2

1 0

7

Global Minima

3

0

~ X1

2

4

6

2417

1

7 6

Adaptation interval

5

E 4 3 2

Adapted

1 0

x1

50

100

150

n

n Fig. 4. (a) A simulation result demonstrating the trajectory of X~ ¼ ½xn1 xn2  vector to convergence toward the global 2 minima of a convex error function ðEðx1 ; x2 Þ ¼ ððx1  5Þ þ ðx2  5Þ2 Þ0:5 Þ by SMDO method. (b) Error function during iterative approximation to the global minima.

studies have been proposed recently to introduce fractional order operators and systems in the schemes of the adaptive control theory. Charef et al. proposed a new scheme of fractional order adaptive controller via high-gain output feedback for a class of linear, time-invariant, minimum phase and single input-single output systems of relative degree one [28]. Petráš introduced a possible adaptation algorithm for the fractional order controllers [29]. Das et al. have been studied an adaptive gain and order scheduling of optimal fractional order PID controllers with radial basis function neural-network [30]. In the SMDO method, the number of divergence vectors to be tested consecutively for a full search of all components in a k-parameter system is 2k. During the adaptation process of the PI λ Dμ controllers, five components were subsequently tested, and only the convenient divergences are allowed to modify X n vectors in the five-dimensional parameter space of PI λ Dμ controllers. Let a divergence vector for A-PI λ Dμ controller be written as ΔX n ¼ ϕnv ½Δknp Δkni Δknd Δλn Δμn 

ð6Þ

A forward divergence of components can be written as X nþ1 ¼ X n þ ϕnv ½Δk np Δk ni Δknd Δλn Δμn 

ð7Þ

A forward test for a convenient divergence in the parameter space of the PI λ Dμ controller can be arithmetically expressed as follows: EðX n þ ΔX n Þ EðX n Þo0

ð8Þ

where, ϕv is the divergence rate vector. Similarly, a backward divergence of components can be expressed as X nþ1 ¼ X n  ϕnv ½Δk np Δkni Δknd Δλn Δμn 

ð9Þ

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A backward test for a convenient divergence in the parameter space of the PI λ Dμ controller can be arithmetically expressed as follows: EðX n  ΔX n Þ  EðX n Þo0

ð10Þ

In tuning the PI λ Dμ controller using the proposed SMDO method, φv vector subsequently takes the values of {½ η 0 0 0 0T , ½ 0 η 0 0 0T , ½ 0 0 η 0 0T , ½ 0 0 0 η 0T , ½ 0 0 0 0 ηT } for v ¼ f1; 2; 3; 4; 5g. The value of η parameter in each step is obtained from a random number generator with a uniform distribution in the range of ½0; 1. Sub-index v indicates the components of X vector which is tested in a bidirectional componential ST operation. For example, ϕ1 denotes the vector ½ η 0 0 0 0T , which is used in the divergence of the first component knp in PI λ Dμ controller tuning. The vector of ½Δknp Δkni Δknd Δλn Δμn  specifies the divergence-length of the parameter, which directly affects the adaptation rate. Fig. 5 illustrates the proposed algorithm for auto-tuning API λ Dμ controllers by employing the SMDO method. Parameter Initialization

E( X n) > Emin

Adapted

No

Yes Adaptation

v = v +1 v=0 v >5

Yes

No Compose

ΔX n X n+1 = X n + Δ X n

Forward test:

E(X n +Δ X n) −E(X n) < 0

X n+1 = X n −Δ X n

Yes

No Backward test:

Yes

E(X n −Δ X n) − E(X n) < 0

No

Fig. 5. An algorithm for online auto-tuning of A-PI λ Dμ controllers, employing a bidirectional componential set and trial method. The index v is the number of parameters of the optimization function (v¼ 5 was used in A-PI λ Dμ tuning).

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Forth-order continued approximation of the fractional derivative and integration is used to implement PI λ Dμ and A-PI λ Dμ controllers in the experimental study of the proposed algorithm in Fig. 5. Forth-order continued approximation of a fractional differentiation operator can be calculated using the continued fraction expansion (CFE) method, The CFE being one of the most important approximations for fractional order systems can be expressed in the following form [31]: ð1 þ xÞα ¼

1 αx ð1 þ αÞx ð1  αÞx ð2 þ αÞx ð2  αÞx 1 1 þ 2 þ 3 þ 2 þ 5 þ :::

ð11Þ

In this formulation x ¼ s 1 used for the computation of sα [32]. Several methods for rational approximations of fractional order systems such as, Carlson0 s method, Matsuda0 s method, Oustaloup0 s method, the Grünwald–Letnikov approximation, Maclaurin series based approximations, time response based approximations etc., are proposed in the literature [3,7,9]. The CFE exhibits faster response and converges in a wider range of complex plane according to the power series expansion based methods, [7]. In our previous study [32] we presented that the forth order approximation, which is obtained with CFE method, exhibits almost same frequency response with the exact value obtained by frequency response of fractional order system by substituting s ¼ jω in the Laplace transform. The paper [32] also proves that the forth order CFE approximation of the system provides almost same stability region for the parameter space of the PI controller with the real one. Consequently, the forth order CFE approximation of the fractional order derivative and integration will exhibit true approach in SMDO algorithm. In the proposed algorithm (see Fig. 5), the SMDO method performs online auto-tuning during the adaptation interval, unless the error function is lower than a predefined error limit. This means that the algorithm has two operational modes, namely, the adaptation mode and the adapted mode. The following short-time average error function with a backward time window length of L (Batch error) is used in the SMDO optimization of an A-PI λ Dμ controller: Z 1 t 2 e ðtÞ dt ð12Þ E¼ L tL Eq. (12) provides a real valued error function for the SMDO method in order to reduce the closed-loop error signal magnitude in Fig. 1. The SMDO method tests all components, one by one, for both the forward and backward directions. The method observes the impacts of random divergence of each component for both directions on the error function (Eq. (12)). Thus, it decides upon the best divergence, resulting in a decrease of the error function. The SMDO method presents the advantage of being independent of plant transfer functions, however it requires a large enough trial period (L) in ST operations for a reliable evolution of the vector divergences ΔX n in control applications. However, use of unnecessarily large trial periods has the side effect of slowing down the adaptation rate of the controller in real applications. A satisfactory adaptation performance is possible when the trial period of the divergence is slightly larger than the periodicity of the reference signal. Therefore, in this case, the result of the error function defined by Eq. (12) becomes consistent in each trial. In order to have good initial values of X 0 , one can also perform SMDO on a simulation model of the controller, in an off-line manner. This may avoid undesired A-PI λ Dμ controller responses

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at the beginning of adaptation, particularly error-critical control applications. It was observed in our experimental tests that the online auto-tuning with SMDO can also further enhance controller performance when it is initialized with an off-line optimized X 0 using the SMDO method according to a simulation model of the plant. 2.5. A real-time instability prevention mechanism during adaptation process Online auto-tuning of controllers without pre-knowledge of plant model parameters bears the risk of system instability during the optimization of the controller parameters. Due to the lack of plant modeling, it is very difficult to specify a parameter range, where the system stays stable. In this case, it is very important to develop a real-time instability detection mechanism in order to avoid control system instability during online auto-tuning of PI λ Dμ controllers using the SMDO method. We used a simple error-threshold mechanism which is based on Lyapunov stability criterion. Hurwitz stability suggests that as poles of a system approach the complex axis or right half plane, system output diverges from the reference, and this causes a sharp rises in the error function. By using a predefined error threshold (δs ), the pole approximation to complex axis was detected in real-time by monitoring the condition of EðX n Þ4δs according to the Lyapunov stability criterion. When EðX n Þ4δs occurs, the SMDO method turns back to the best parameters,

1.2 1 0.8 0.6

Reference PID PIλDμ A-PIλDμ

0.4 0.2 0 0.6

0.65

0.7

0.75

Time [s]

1.4

1.2

1.2

1

1

0.8

0.8

0.6

Reference PID PIλDμ A-PIλDμ

0.4 0.2 0 -0.2

0.6

Reference PID PIλDμ A-PIλDμ

0.4 0.2 0

2.6

2.65

2.7

Time [s]

2.75

5

5.05

5.1

5.15

Time [s]

Fig. 6. (a) Step responses of the controller at 0.6 s. (b) Step responses of the controller at 2.6 s. (c) Step responses of the controller at 5 s.

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providing a minimum error value during the optimization, in the following manner: ( ) u n fX g EðX Þ4δ min 0ouon s Xn ¼ Xn EðX n Þr δs

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ð13Þ

The δs restricts the optimization into a predefined error function limit, and it should be configured according to error tolerances of real-time applications. This mechanism can effectively prevent unstable responses of control system during online auto-tuning unless exceedingly large divergence lengths are used in SMDO. 3. Simulation results for online auto-tuning of A-PIλ Dμ controllers In this section, responses of the proposed adaptive controller (A-PI λ Dμ ) are compared with a classic PID and a non-adaptive PI λ Dμ controller, which are used for controlling the following two-pole plant transfer function: K GðsÞ ¼ ð14Þ 2 a2 s þ a1 s þ a0 Parameters of the classic PID (kp ¼ 1, ki ¼ 2, kd ¼ 1) are calculated using known methods in the literature [33,34]. The proposed SMDO produces the parameters of non-adaptive PI λ Dμ (kp ¼ 1, k i ¼ 2, k d ¼ 1, λ ¼ 0:95, μ ¼ 0:3) and A-PI λ Dμ (kp ¼ 1, k i ¼ 1:5, kd ¼ 7:85, λ ¼ 0:95, μ ¼ 0:3) controllers to simulate the control performance for the plant GðsÞ in Eq. (14). SMDO method, which tunes fractional-order PIDs in this paper, was also successfully used in our previous paper demonstrating tuning of PID controller for a smooth rotor control application [35]. A reference signal in the form of a periodical square wave was used to test the performance of these controllers in two cases of GðsÞ: a constant coefficient GðsÞ and a dynamically perturbing coefficient GðsÞ. Fig. 6 illustrates the online auto-tuning of the A-PI λ Dμ controller in the case of the constant coefficient plant transfer function GðsÞ. The proposed A-PI λ Dμ controller and nonadaptive PI λ Dμ controller present similar step responses in Fig. 6(a) because they are performed with almost the same controller coefficients ðk np k ni knd λn μn Þ at the beginning. After a 2.6 s

Fig. 7. (a) Alteration of A-PI λ Dμ controller coefficients during 12-second simulation. (b) Alteration of average error from start to the end of simulation.

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interval, the step responses of the controllers are demonstrated in Fig. 6(b). The A-PI λ Dμ controller improves the maximum overshoot as a result of online auto-tuning, while preserving its settling time. There is a need for further tuning efforts to have a better rise and fall time for the adaptive controller0 s unit step response. Fig. 6(c) clearly reveals the improvement in maximum overshoot and rise time of the proposed controller at the 5th second. Fig. 7(a) shows the alteration of controller coefficients during these adaptation efforts. Evolution of average control error from the start of the simulation is demonstrated in Fig. 7(b). The results represented in Fig. 7 demonstrate the online auto-tuning potential of the SMDO method in control applications. In order to exhibit adaptation skill of the PI λ Dμ controller using the SMDO method, the parameters of the plant GðsÞ were perturbed in the range of a0 ¼ ½  1; 1, a1 ¼ ½1; 3 and K ¼ ½3; 5 respectively to show adaptation skill of the proposed optimization algorithm. Fig. 8 demonstrates adaptation capability of A  PI λ Dμ controller for all parameter perturbations of this plant. A-PI λ Dμ controller still preserves stability of the closed loop system for this perturbation interval of the plant parameters. The auto-tuning of the A-PI λ Dμ , classical PID, and non-adaptive PI λ Dμ controllers for the dynamically perturbed GðsÞ in Fig. 8(a), (b), and (c) shows that the perturbation in GðsÞ results in deteriorations in control of the classical PID and the non-adaptive PI λ Dμ controllers. However, the adaptive fractional controller was observed to enhance its response thanks to the auto-tuning

Fig. 8. Step responses of the controllers for a perturbing plant transfer function. (a) Step responses of controllers at 0.2 s. (b) Step responses of controllers at 3.8 s. (c) Step responses of controllers at 11.4 s.

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efforts of SMDO algorithm. These figures confirm the adaptation skills of the PI λ Dμ controller with SMDO in response to changing controller operating conditions. In order to present the advantage of SMDO we compared the results with an adaptive optimization algorithm given in the literature. Consider the following unstable time delay plant in [26]: P1 ðsÞ ¼

e  0:2s s 1

ð15Þ

PID controller parameters were computed for P1 ðsÞ using Differential Evolutionary algorithm as kp ¼ 2:725339, k i ¼ 1:624656, kd ¼ 0:026691 that are given in Table 2 in [26]. Fractional order PIDparameters were also computed using Differential Evolutionary algorithm for the same plant as k p ¼ 2:425323, ki ¼ 1:365907, k d ¼ 0:181526, λ ¼ 0:989938, μ ¼ 0:828449 in Table 3 in [26]. Proposed SMDO method in this manuscript produces PI λ Dμ parameters for the plant P1 ðsÞ as kp ¼ 3:973686, ki ¼ 2:144711, kd ¼ 0:134777, λ ¼ 0:932582 and μ ¼ 1:251988. Fig. 9 demonstrates that the SMDO method improves the control performance of the plant P1 ðsÞ. 4. Experimental studies 4.1. Experimental setup Fig. 10(a) depicts a TRMS setup installed in a laboratory. The TRMS is composed of two propellers, which are perpendicular to each other. The propellers are joined by a shaft pivoted on its base, and this shaft can rotate freely on both the horizontal and vertical planes. The motion of the shaft can be controlled by the input voltages that adjust the rotational speed of these two propellers, which are driven by DC electric motors. The main rotor has a control on the vertical angle of the shaft, and the tail rotor controls the horizontal angle of the shaft. A pendulum counterweight is attached to the propeller shaft. This counterweight adjusts the angular momentum of the main rotor motion on the vertical plane, and it is used for changing the operating conditions of the TRMS. The TRMS setup mimics the flight controls of a helicopter in

Fig. 9. Step response of , Fractional order PID obtained with Differential Evolutionary Algorithm for P1 ðsÞ and step re PID sponse of obtained by SMDO for P1 ðsÞ.

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Tail Rotor Levels of Pendulum Counterweight

Shaft ½

1

0

SMDO

kp ki λ kd μ

Main Rotor

r e

kp +

u ki + kd s μ λ s

TRMS

o

PI λ D μ

Fig. 10. (a) Picture of the TRMS experimental setup. (b) Block diagram of TRMS control setup using SMDO.

a laboratory setting. Controlling a TRMS involved some difficulties, due to the mutual interaction between the two axes, and the nonlinearity in motion of mechanisms [36,37]. Fig. 10(b) shows a block diagram of the experimental system developed for A  PI λ Dμ tuning using the SMDO method. The vertical and horizontal angle control were tested in our experiment. The control objective was to make the TRMS move quickly and accurately to the desired attitudes and directions, by controlling the vertical and horizontal angles [14,36]. The Matlab/Simulink modules are used to obtain SMDO setup for TRMS. The sampling frequency is selected as 1 kHz for simulating and real time experiment. Forth order rational approximations of the fractional differentiation operator is used to simulate real time A-PI λ Dμ [31]. 4.2. Experimental results In the literature, studies generally compare the developed methods with classical one on TRMS experimental setup. For example, robust dead band control results are compared with traditional PID for TRMS experimental setup in [38]. Advantages of the controller design method based on a real-value-type genetic algorithm over conventional PID was shown via TRMS experiment in [39]. In order to show the advantages of proposed fractional order PI controller over integer order modified Ziegler–Nichols PID controller, experiments were performed on small or micro unmanned aerial vehicle under extreme conditions in [40]. In this manuscript, the proposed SMDO method is experimentally tested on our TRMS system in the laboratory. SMDO algorithm in Fig. 5 is implemented for this system, using Matlab, realtime windows and real-time workshop in a computer. The control and the data acquisition are implemented using the equipped “Advantech PCI 1711” interface cards on a computer. The experiments are carried out in room conditions. The input signal is chosen to enable the system to reach its steady state. Perturbation of the controller parameters in the SMDO optimization are restricted within the range of 0okp r 20, 0oki r 20, 0okd r 20, 0:5 r λr 1:5, 0:5r μ r 1:5 for this experimental study. In order to demonstrate control performance improvements when using the SMDO method in TRMS experiments, the proposed adaptive controller A-PI λ Dμ are compared with a classic PID and a non-adaptive PI λ Dμ controller in TRMS system. In a real time application, random initial values of the online optimization parameters may cause unexpected response and may damage the real time running machinery. However, initial parameters of the online running SMDO optimization in the experimental study are obtained via simulation model of the plant. The PID controller in the experiment was configured to PID coefficients recommended by Feedback Instruments Ltd. [37]. Coefficients for the non-adaptive PI λ Dμ controller were obtained via off-line optimization with a model of the TRMS in a MATLAB Simulink [37]. In order to demonstrate control performance

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improvements when using the SMDO method in TRMS experiments, the proposed adaptive controller, a classic PID, and a non-adaptive PI λ Dμ controller were tested with the TRMS system. Controller transfer functions used for the classic PID and the non-adaptive fractional PI λ Dμ controllers respectively, are given by 8 CPID ðsÞ ¼ 5 þ þ 10s s CPI λ Dμ ðsÞ ¼ 9:6052 þ

ð16Þ

7:7863 þ 11:8976s1:0788 s1:0053

ð17Þ

For this real time optimization experiment, the optimizer produces non-adaptive fractional PI λ Dμ controllers parameters close to classical PID controller except the proportional constant. Here the optimizer obtains better controller performance at these digits. In fact, we experienced during this real time experimental study that very small change in fractional orders of derivatives and integrals may result in significant change in system behavior. The proposed A-PI λ Dμ controller used the coefficients of the non-adaptive PI λ Dμ controller in Eq. (17) as initial values and performed an online optimization to improve the performance of the experimental controller in real-time. Backward time window length (L) was used as 40 s in all real-time optimizations. A reference signal in the form of a step function with an amplitude of 0.5 was used to control the main rotor and tail rotor positions. In order to perturb TRMS system operating conditions, the location of the pendulum counterweight was altered between the bottom and the top positions of the pendulum, which considerably changed the angular momentum of the main rotor during takeoff and hovering. Fig. 11 compares the responses of the main rotor and the tail rotor respectively for the proposed adaptive controller, the classic PID, and the non-adaptive PI λ Dμ controller. It was clearly observed that the adaptive controller exhibits a superior control performance compared to the other controllers in tests of the main and tail rotors. Especially, the A-PI λ Dμ controller exhibited a lower overshoot with a shorter settling time when tracing the reference signal in real-time TRMS applications. Fig. 12(a) and (b) shows the alteration of controller coefficients and the error function values recorded during the adaptation process. Peaks in the short-time error function seen in Fig. 12(b) indicate inconvenient parameter divergences. The SMDO method chooses only convenient parameter divergences, 0.7 0.7

0.6

0.6

0.5

0.5

0.4

0.4

Reference PID PIλDμ A-PIλDμ

0.3 0.2

Reference PID PIλDμ A-PIλDμ

0.3 0.2 0.1

0.1 5

10

15 Time [s]

20

25

0

0

5

10

15

20 25 Time [s]

30

35

40

Fig. 11. Responses of the main rotor (a) and the tail rotor (b) controlled by the , the classic PID and non-adaptive PIλDμ controllers.

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kp

10 9 8 10 8 6

ki

14 kd 12 10 1.5 1 λ 0.5 1.5 1 0.5

μ

0.048 0

5

10

15

20

25

30

35

40

0.046 0.044

0

5

10

15

20

25

30

35

40

0.042 0.04

0

5

10

15

20

25

30

35

40

0.038 0.036 0.034

0

5

10

15

0

5

10

15

20

25

30

35

40

20

25

30

35

40

Adapted

Adaptation State 5

10

15

20

25

30

35

Iteration

Iteration

Fig. 12. (a) Alteration of A-PI λ Dμ controller coefficients during the main rotor control experiment. (b) Alteration of short-time average error for adaptation state and adapted state.

0.7

0.7

0.6

0.6 0.5

0.5

0.4

0.4

0.3

Reference PID PIλDμ A-PIλDμ

0.3 0.2

0.1

0.1 0

Reference PID PIλDμ A-PIλDμ

0.2

0 5

10 Time [s]

15

20

5

10

15

20

Time [s]

Fig. 13. Controller responses when pendulum counterweight positioned in the middle of the pendulum bar (a) and at the top of pendulum bar (b).

reducing the short-time error function (in Eq. (12)) in each ST. In this experiment, SMDO reached the adapted state at about 28 iterations. Fig. 13 demonstrates experimental results obtained after changing the position of pendulum counterweight. The location of counterweight was changed to the middle and to the top of pendulum in the main rotor control experiments in Fig. 13(a) and (b). These changes affected the vertical angular momentum considerably. It was observed that the adaptive controller performed auto-tuning and improved its step responses. A-PI λ Dμ , optimized with SMDO, in the experiments with altered pendulum counterweights, are listed in Table 1. Table 2 shows the average error values of closed loop system in Fig. 10, when positioning of the pendulum counterweight was varied. The average errors for each test were calculated by the formula jEj ¼ 1=N∑Nn ¼ 1 jeðnÞj, where the parameter N represents the total number of samples captured from the experimental measurement of step response. (N ¼ 20000 was used in the experiments).

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Table 1 A-PI λ Dμ -optimized controller coefficients with SMDO in the experiments for changing pendulum counterweight positioning. Level of pendulum counterweight (see Fig. 10(a))

kp

ki

kd

λ

μ

0 1/4 1/2 3/4 1

9.8965 8.8939 9.8939 8.8881 9.5436

7.1349 7.6710 8.3063 8.0648 8.3829

11.8178 11.2421 11.6235 11.8976 11.0123

0.8136 1.0053 0.9352 0.9139 0.8053

1.1356 1.0082 1.0135 1.0155 1.0138

Table 2 Average performance error comparison of PID, PIλDμ and A-PIλDμ controllers in separate test seasons for changing pendulum counterweight positioning. These values of errors are reached about 100 iterations. Level of pendulum counterweight (see Fig. 10(a))

PID

PIλDμ

A-PIλDμ

0 1/4 1/2 3/4 1

0.0261 0.0193 0.0196 0.0199 0.0195

0.0205 0.0179 0.0183 0.0231 0.0200

0.0202 0.0172 0.0153 0.0176 0.0176

As provided in Table 2, the A-PI λ Dμ controller, which optimized with proposed SMDO method, has presented a better error performance than other controllers tested. Although the classic PID controller was less sensitive to change in the location of the counterweights, in general its average error is not as good as that of A-PI λ Dμ . Some remarks on the results of the experimental study:

  

Non-adaptive PI λ Dμ controllers can exhibit better control performance compared to classic PID controllers, however A-PI λ Dμ controllers can improve the performance of non-adaptive PI λ Dμ controllers. A-PI λ Dμ controllers provide more robust control performance in TRMS tests against model perturbations and fluctuations in system parameters. TRMS test results show that online auto-tuning using the SMDO method is simple and practical, yet provides satisfactory performance improvements.

5. Conclusions The paper presents an online optimization of PI λ Dμ controller coefficients using the SMDO method. This method performed the online auto-tuning of fractional-order controller coefficients under varying conditions within a closed loop control system. The proposed method is independent of the exact mathematical plant model. This makes it suitable for use in real-world applications. The SMDO method chooses stochastic parameter divergences, resulting in a descent of error function. The SMDO was inspired from gradient descent methods and evolutionary algorithms. It can effectively perform a real-time auto-tuning of PI λ Dμ controllers

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with low computational complexity. The adaptation rate of PI λ Dμ controllers can be adjusted using divergence-lengths of the controller coefficients. Although a high divergence-length may prevent the SMDO algorithm from getting stuck at a local minima of the error function, it increases the risk of unstable driving of the plant in some ST operations. It might be noticed that a low divergence-length should be preferred, particularly in mission-critical control applications. Simulation results and experimental observations using the TRMS confirmed that adaptation skill is increased when using the SMDO method. These findings hold promise for the technological potential of real control applications. The proposed stochastic tuning method can deal with complicated controller design problems such as fractional order controller or non-linear control. This optimization method can be particularly useful for flight control applications where system model parameters vary depending on uncertain environmental conditions. In future work, PI λ Dμ controllers using SMDO method should be tested for a multi-constraint auto-tuning of PI λ Dμ controllers in order to obtain more desirable responses from PI λ Dμ controllers. Acknowledgment We are thankful to Inonu University Computer Engineering Department's Flight Control Laboratory for experimental setup and B. Baykant Alagoz for his contribution. References [1] R.E. Gutiérrez, J.M. Rosário, J.A.T. Machado, Fractional Order Calculus: Basic Concepts and Engineering Applications, Mathematical Problems in Engineering, Hindawi Publishing Corporation, 2010 (2010) 1–9. [2] G.W. Leibnitz, Letter from Hanover, Germany, September 30, 1695 to GA L’hospital, Leibnizen Mathematische Schriften Olms Verlag, First Published in 1849, Germany, Hildesheim, 1962. [3] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, USA, 1999. [4] B. Ross, Fractional Calculus and its Applications, Springer, Verlag-Berlin, New York, 1975. [5] J. Sabatier, O.P. Agrawal, J.A.T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, Netherland, 2007. [6] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Nort-Holland, 2006. [7] S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, Berlin, Heidelberg, New York, 2007. [8] Y. Luo, Y.Q. Chen, H.S. Ahn, Y.G. Pi, Fractional order periodic adaptive learning compensation for state-dependent periodic disturbance, IEEE Trans. Control Syst. Technol. 20 (2012) 465–472. [9] R. Caponetto, G. Dongola, L. Fortuna, I. Petras, Fractional Order Systems Modeling and Control Applications, World Scientific Series on Nonlinear Science, World Scientific Publishing Co. Pte. Ltd., Singapore, 2010. [10] H.S. Li, Y. Luo, Y.Q. Chen, A fractional order proportional and derivative (FOPD) motion controller: tuning rule and experiments, IEEE Trans. Control Syst. Technol. 18 (2010) 516–520. [11] M.G. Marcos, F.B.M. Duarte, J.A.T. Machado, Fractional dynamics in the trajectory control of redundant manipulators, Commun. Nonlinear Sci. Numer. Simulation 13 (2008) 1836–1844. [12] I. Pan, S. Das, Intelligent fractional order systems and control, Stud. Comput. Intell. 438 (2013). [13] K.H. Ang, G. Chong, Y. Li, PID control system analysis design and technology, IEEE Trans. Control Syst. Technol. 13 (2005) 559–576. [14] G.R. Meza, S.G. Nieto, J. Sanchis, F.X. Blasco, Controller tuning by means of multi-objective optimization algorithms: a global tuning framework, IEEE Trans. Control Syst. Technol. 21 (2013) 445–458. [15] S. Panda, B.K. Sahu, P.K. Mohanty, Design and performance analysis of PID controller for automatic voltage regulator system using simplified particle swarm optimization, J. Franklin Inst. 349 (2012) 2609–2625. [16] B. Xu, R.S. Pandian, N. Sakagami, F. Petry, Neuro-fuzzy control of underwater vehicle-manipulator systems, J. Franklin Inst. 349 (2012) 1125–1138.

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