Control of uncertainly multi-variable system withfractional PID

Salah ChenikherDept of Electrical EngineeringTebessa University, Algeriaphone: +213 7 95 66 67 13

s.chenikher@mail.univ-tebessa.dz

Samir AbdelmalekDept of Electrical Engineering

Tebessa University, Algeriaphone: +213 6 61 36 86 22

s.abdelmalek@mail.univ-tebessa.dz

Moussa SedraouiDept of Electrical Engineering

Guelma University, Algeriaphone: +213 5 55 53 60 10

msedraoui@Gmail.com

Abstract— This paper proposes a novel method to designfractional order controller with robust stability and disturbanceattenuation. The Fractional PID problem is formulated as anH∞ problem with a controller structure constraint and the con-troller parameters are optimized to achieve both user-specifiedrobust stability and performance. This problem is solved byOptimization with constraints to minimize a cost function subjectto H∞-norm. Which permit to design a robust fractional orderPID technique with robust performances. Better performancesusing fractional order PID controller can be obtained and isvalidated through multi-variable process. A comparison to theclassical integer order PID controllers for controlling systemswith multiple delays is achieved.

I. INTRODUCTION

The PID controller is by far the most dominating formof feedback in use today. Due to its functional simplicityand performance robustness, Specifications, stability, design,applications and performance of the PID controller have beenwidely treated [1] and [2]. On the other hand, in recentyears it is remarkable the increasing number of studies relatedto the application of fractional controllers in science andengineering areas [12], [13], [21], [20] and [22]. In whatconcerns automatic control theory the FC concepts wereadapted to frequency based methods. The frequency responseand the transient response of the non-integer integral and itsapplication to control systems was introduced by Manabe [3]and more recently in [4]. Oustaloup studied the fractionalorder algorithms for the control of dynamic systems anddemonstrated the superior performance of the CRONE Controlmethod over the PID controller [5] and [6]. More recently,Podlubny [7] proposed a generalization of the PID controller,namely the PIλDµ controller [23], involving an integrator oforder λ and a differentiator of order µ. A frequency domainapproach by using fractional PID controllers is also studied in[8]. In [9] the extension of differentiation and integration orderfrom integer to non integer numbers provides a more flexibletuning strategy and therefore an easier achieving of controlrequirements with respect to classical controllers. In [10] anoptimal fractional order PID controller based on specified gainmargin and phase margin with a minimum ISE criterion hasbeen designed by using a differential evolution algorithm. Anexperimental investigation has been presented in [11], wherea fractional PID control has been applied for active reduction

of vertical tail buffeting. A fractional order control strategyhas also been successfully applied in the control of a powerelectronic buck converter [12] and [13], more concretely afractional sliding mode control.Another approach is the use of a new control strategy tocontrol first-order systems with delay [14] based on a DβIα

controller with PIα fractional order integral and derivativeparts. Besides, it is being developed another method for plantswith long dead-time based on the use of a PIλDµ controllerwith a fractional integral part of order see [15].In this paper, a new robust multi-variable fractional order PIDcontroller is presented. Its particularity consists in the noninteger of the derivatives and integrals acting. A systematicmethod for tuning this above controller for MIMO plant pro-posed. The PID tuning problem is formulated as an H∞ prob-lem with a controller structure constraint and the controllerparameters are optimized to achieve both user-specified robuststability and nominal requirement performance. This problemis solved by a non linear optimization approach. The fractionalorder multi-variable PID is examined by simulation, on HVACpilot and given results are compared with those obtained by aninteger order multi-variable PID. The remainder of this paperis organized as follows. Section II describe the Fractional-Order Controller. Section III is devoted to the formulation ofrobust fractional order PID problem. The design examples andresults are presented in section IV. This work is concluded insection V.

II. FRACTIONAL-ORDER CONTROLLER

Consider the control diagram as shown in figure 1. Them ×m plant is modeled as a transfer function matrix G(s).The most common form of a fractional order PID controlleris the PIλDµ controller [22], involving an integrator of orderλ and a differentiator of order µ, where λ and µ can be anyreal numbers. The transfer function of such a controller hasthe following form, which can be arranged in a matrix formas:

K(s) =

K11(s) ... K1m(s)... ... ....

Km1(s) ... Kmm(s)

(1)

with: Kij(s) =(KPij

+KIijs−λ +KDij

sµ), (2)

978-1-4673-0784-0/12/$31.00 ©2012 IEEE 1079

Where (λ, µ � 0), KP is the proportional constant, KI is theintegration constant and KD is the differentiation constant.Clearly, selecting λij = 1and µij = 1, a classical PIDcontroller can be recovered. Using λij = 1, µij = 0 ,and λij = 0, µij = 1, respectively, corresponds to theconventional PI and PD controllers. All these classical typesof PID controllers are special cases of the PIλDµ controllergiven by [4]. It can be expected that the PIλDµ controller mayenhance the systems control performance. One of the mostimportant advantages of the PIλDµ controller is the possiblebetter control of fractional order dynamical systems. Anotheradvantage lies in the fact that the PIλDµ controllers are lesssensitive to changes of parameters of a controlled system [15].Among many different definitions, two commonly used forthe general fractional integral-differential operation are theGrnwaldLetnikov (GL) definition and the RiemannLiouville(RL) definition [20]: The GL definition is given in [20]:

αDρt = limk→ 0h

−λ[(t−a)/h]∑

j

(−1)je(t− jh)

(ρ

)(3)

Where [(t − a)/h] means the integer part, while the RLdefinition is [21]:

αDρt e(t) =

1

Γ(n− ρ)

dn

dtn

∫ φ=0

φ=t

e(φ)

(t− φ)ρ−n+1 dφ (4)

For {n− 1 ≺ ρ ≺ n} and Γ(n− ρ) where is Eulers gammafunction. For convenience, Laplace domain notion is com-monly used to describe the fractional integro-differentialoperation. The Laplace transform of the RL fractionalderivative/integral under zero initial conditions for orderρ {0 ≺ ρ ≺ 1} is given by [20]:

L(αD

+ρt f(t)

)= s+F (s). (5)

III. FORMULATION OF ROBUST FRACTIONAL ORDER PIDPROBLEM

The most common form of a fractional order PID controlleris the PIλDµ controller [21], involving an integrator of orderλ and a differentiator of order µ, where λ and µ can be anyreal numbers. The transfer function of such a controller hasthe following form [21]:

K(s) = Kp +Kis−λ +Kds

µ, (6)

Consider the Multi-input Multi-output (MIMO) feedbackcontrol system of Figure 1, in which G(s) is the nominalplant, ∆(s) is represented a multiplicative uncertainty . It isassumed that system ∆(s) is stable with its maximum singularvalue bounded by:

σmax [∆(s)] ≺ |w1(s)| ,∀w ∈ R. (7)

The user-defined performance is specified as the limit offrequency-weight H∞-norm:

‖W1(s)S(s)‖ ≺ 1, (8)

Fig. 1. A feedback control system with multiplicative uncertainty.

where S(s) is the sensitivity function, which denotes thetransfer function from set point r to control error e, or fromdisturbance -d to e and n noise. w1 is a frequency dependentweighting function, which penalizes the control error e. Fromthe Small Gain Theorem [16], the closed-loop system will berobustly stable under the uncertainties bounded by equation 7if the following condition is satisfied:

‖W2(s)T (s)‖ ≺ 1, (9)

where: W1(s) = w1(s)Im×m, (10)

and: T (s) = [I +G(s)C(s)]−1G(s)C(s), (11)

is the complementary sensitivity function. Define:

Tcl =

(γW1(s)S(s)W2(s)T (s)

), λ � 0, (12)

The desirable robust PID controller K(s) should maxi-mize γ for best performance while satisfies ‖Tcl‖∞ ≺ 1forrobustness. To represent different controller specifications ina unified framework, the model G(s) is augmented into atwo-port generalized plant P (s), which includes G(s) andweighting functions W1(s) and W2(s) :

P (s) =

[P11(s) P12(s)P21(s) P22(s)

](13)

With : P11(s) = [W1; 0];P12(s) = [−γW1(s)G(s);W2(s)G(s)];P21(s) = I; P22(s) = −G(s).

Transfer function matrix Tcl is now represented by the lowerLinear Fractional Transformation (LFT) [6]:

Tcl = Fl(P (s),K(s))

= P11(s) + P12(s)K(s) [I − P22(s)K(s)]−1P21(s), (14)

Assume ks is the set of stabilizing controllers with Frac-tional PID structure. The problem of designing a FractionalPID controller K(s) optimized for controller performancewhile maintaining robust stability and performance can beformulated as an H∞ optimization problem with constraintson the controller structure:

minK(s)∈ks ‖Tcl‖∞ =minK(s)∈ks ‖Fl(P (s),K(s))‖∞

= min

[γW1(s)T (s)W2(s)S(s)

](15)

1080

In this work we solve the optimization problem with con-straints. The optimization problem is presented as:

MinimizeF (x) = min(f1(x), ..., fm(x)] (16)

Subject to: gi(x) ≤ 0;hj(x) = 0;xl ≤ x ≤ xuwhere, i=1,...,m;j=1,...,p; F (x) is the objective function to

be optimized, hj is the set of equality constraints, gi is the setof inequality constraints, x is the vector of dependent variable,

IV. RESULTS AND DISCUSSIONS

This section presents the determination of the parametersof the robust fractional order multi-variable PID controller forMIMO plant with important multiples delay. The proposedcontroller should be satisfying, simultaneously, the stabil-ity robustness and nominal requirement performance of thefeedback system. The room temperature MIMO plant wasdescribed by [18] , and we compare between results obtainedby the integer order multi-variable PID controller is thatproposed by [18].The transfer function of the plant is given asfollows:

G(s) =

(e−23.2s

−2680.4s−22.4e−63.3s

−7697.2s−70.6e−15s

−7059.8s−51.5e−14.0s

−1902.6s−19.8

)(17)

The Fractional PID controller is designed to achieve thefollowing robustness and performance specifications:

1) Robustness: the closed-loop system including the plantand the controller is robustly stable with the followingmultiplicative uncertainty ∆(s):

[∆max(jw)] ≺∣∣∣ (s+104)100(100s+1)

∣∣∣.2) Performance: this nominal performance is expressed as:∥∥∥∥100(100s+ 1)

s+ 104S(s)

∥∥∥∥∞≺ 1 (18)

According to above specifications, the weighting functions inthe problem of equation 15 are chosen as:W1 = diag

((100(100s+1)

s+104

); W2 = diag

((2500(10s+1)

105s+1

).

To solve the problem in equation 15, we can use the fmin-imax function, available in the toolbox Optimization/Matlab[19]. We initialized the optimization problem by x0 = 0.01,γ = 0.35 and bounds conditions as follow:• 0 ≤ λ ≤ 1; 0 ≤ µ ≤ 1; −∞ ≤ Kp ≤ +∞;• −∞ ≤ Ki ≤ +∞; −∞ ≤ Kd ≤ +∞.

The given optimal xopt solution permits to determine thetransfer function matrix K̃(s) as follow:

K̃(s) =

(K̃11(s) K̃12(s)

K̃21(s) K̃22(s)

)(19)

Such as:• K̃11(s) = −74.73− 0.67

s1.0 − 69.447s0.39;• K̃12(s) = −9.55− 1.59

s0.78 − 9.60s0.26;• K̃21(s) = −4.02 + 2.94

s0.53 ;• K̃22(s) = −55.05− −2.57s0.88 − 51.94s0.38.

We notice that, the above transfer function matrix of thecontroller carried the general structure:

K̃(s) =

(PIDµ PIλDµ

PIλ PIλDµ

)(20)

Knowing that, the transfer function matrix of the integer ordermulti-variable PID controller is that proposed by [18] with:

K0(s) =

(K11(s) K12(s)K21(s) K22(s)

)(21)

• K11(s) = −43.36− 0.42s + 293.21s;

• K12(s) = 16.4− 0.19s − 75.9s;

• K21(s) = 11.25 + 0.16s − 75.44s;

• K22(s)− 46.69− 0.58s + 212.73s.

In order to improve the above dynamics, the feedbackcontrol system by the proposed controller should be provide

¯σ(T (s)) and ¯σ(S(s)) smallest as possible for, respectively,the low and the high frequencies. These improvements arebe possible when we do a good choice for the weightingfunctions W1(s) and W2(s). The figure 2 compares betweenthe maximal singular values of the complementary sensitivityfunctions obtained by the above two controllers, and also, thecurve of the inverse of the weighting function. The figure 3compares between the maximal singular values of the directsensitivity functions obtained by the above two controllers,and also, the curve of the inverse of the weighting function.According to the figures 2 and 3, we can notice that, the bet-ter noise-measurements rejections and attenuate perturbationdynamics are those provided by the feedback control systemwith ˜K(s) controller. From the block diagram of figure 1, thefollowing inputs are being uses:• Reference inputs (to be tracked): represent from an ech-

elon vector with r = [r1 = 1; r2 = 0];• Perturbation inputs (to be rejected): represent from re-

tarded echelon with amplitude 0.35;• Noise measurement inputs (to be minimized): represent

from white noise with mean = 0 and variance = 0.25.The figure 4 present the obtained outputs signals of thefeedback control system with the two controllers: ˜K(s), K0(s).According to these above figures, we can notice that, the

˜K(s) controller represents the advantage of a butter rise time,best settling time and little time interval for attenuated thedisturbance inputs; it also produce controls little bit sensitiveto the noise-measurements inputs.

V. CONCLUSION

This paper proposed a methodology based in optimizationapproach for synthesis a robust multi-variable fractional orderPID controller; this controller is applied to MIMO plant withimportantly multiple delays. The feedback control system withproposed controller can be guarantied a better compromisebetween the stability robustness with a best requirement nom-inal performances. These specifications are be possible with agood choice of the weighting functions. The obtained resultsshow the efficiency of the proposed method in the time andin frequency domains which is not feasible with a standardmulti-variable PID.

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Fig. 2. Stability robustness.

Fig. 3. Requirement Nominal Performance

Fig. 4. Dynamics of tracking with disturbance and noise minimization

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