Design of Robust Fractional-Order PID Controller for DC Motor Using the Adjustable Performance Weights in the Weighted-Mixed Sensitivity Problem

Received Feb 2, 2018 Revised Apr 15, 2018 Accepted Apr 30, 2018 This paper deals with the robust series and parallel fractional-order PID synthesis controllers with the automatic selection of the adjustable performance weights, which are given in the weighted-mixed sensitivity problem. The significant contribution of the paper is to achieve the good trade-off between nominal performances and robust stability for DC motor regardless its nonlinear dynamic behavior, the unstructured model uncertainties and the effect of the sensor noises on the feedback control system. The main goal is formulated as the weighted-mixed sensitivity problem with unknown adjustable performance weight. This problem is then solved using an adequate optimization algorithm and its optimal solution leads to determine simultaneously the robust fractional PID controller, which is proposed by the series and the parallel fractional structures, As well as, the obtained optimal solution determines the corresponding adjustable performance weight. The proposed control technique is applied on DC motor where its dynamic behavior is modeled by unstructured multiplicative model uncertainty. The obtained performances are compared in frequencyand time-domains with those given by both integer controllers such classical PID and H∞ controllers. Keyword:


INTRODUCTION
Recently, one of the most desired aspects in DC control motor is to achieve a good trade-off between RS and NP of the feedback control system [1], [2]. Due to inaccurate modeling, component aging of mechanical part of DC motor, sensor noises, exterior conditions, and others, all proposed DC motor models unavoidably incorporate uncertainties and external disturbances.
In control engineering, the controller synthesis using the integer PID controller-structures is still widely recognized as one of the simplest yet most effective control strategies in industry [3][4][5]. However, the obtained H ∞ performances analysis does not guarantee both RS and NP, and optimal trade-off between them. Hence this trade-off should be enhanced when DC motor is subjected to parametric uncertainties and measurement noises.
To avoid this problem, Matrix Inequality LMI based H ∞ control techniques or Algebraic Riccati equations AREs are usually preferred over other methods [6]- [7], due to its computational simplicity and efficiency. The controller parameters are designed from solving the weighted-mixed sensitivity problem where all the above-mentioned effects are presented using some weights in the weighted-mixed sensitivity So that the Riemann-Liouville definition is given as shown in (2).

Γ
is the Euler's gamma function that given by ( ) Noticing that, the implementation FO-PID controller needs to approximate its fractional part of powers γ by the usual integer transfer functions with a similar behavior. The method is based on approximating s γ in a specified frequency range [ , ] h b ω ω ω = and of integer order N by a rational transfer function obtained in the following manner (4) [19]: From (4) the zeros, poles and gain are respectively defined as shown in (5-7).
In some fractional controller-structure, due to the commutative property of the fractional operator s α and order 1 α ≥ , it can be approximated by

ROBUST PARALLEL/SERIES FO-PID DESIGN CONTROLLER
Some feedback control systems implement a FO-PID controller function on serial form, while others use the parallel form. The aim of this paper is to observe differences between them for the DC motor, and to see the performances of each one in time and frequency domains.

Robust Parallel FO-PID Controller
The Robust parallel FO-PID controller called also PFO-PID is the general case of the classical parallel integer one. In time domain, the differential equation is defined by (8)  ( ) According to (9), the controller parameters are given by the design vector , , , , where its derivative part is usually replaced by the term d K s 1 s µ τ + in order to attenuate the noise amplification effect. We get (10): So, there are six parameters to be tune, which are given by the design vector , , , , ,

Robust Series FO-PID Controller
The Robust series FO-PID controller called also SFO-PID becomes the general form the classical series integer PID controller. The differential equation is therefore given in time domain by (11) [22], [23]: where the weighted-mixed sensitivity problem is solved using the same prevois vector.

WEIGHTED-MIXED SENSITIVITY FORMULATION PROBLEM
Let consider the feedback control system shown in Figure 1 where ( ) is the normalized uncertainty that is assumed to satisfy (13): is the maximal singular value of at the frequency point and min max In robust control theory the trade-off between RS and NP depends heavily by satisfying two following conditions, which are:

RS Condition
The robust FO-PID controller should guarantee the RS that means the closed-loop system must remain stable in presence of all possible uncertainties. In order to secure the suitable RS, the complementary sensitivity transfer function ( , ) T s x has been used. Based upon the small gain theorem the RS condition for an uncertain system subject to the unstructured multiplicative uncertainty is defined by (14) [24], [20]: where [ ] where its parameters are chosen similar to that given in [21].

NP Condition
During the design procedure, relatively fast responses, small overshoots and robustness against the model uncertainties can be assumed as suitable performances. Consequently, acquiring the NP is a crucial factor that should be fulfilled by optimization. To ensure this goal, the sensitivity transfer function ( , ) S s x , has been used.
Noticing that, the sufficient small singular values ( , ) S s x in specific frequency ranges can satisfy precise performance characteristics. Moreover, all these characteristics can be obtained by selecting the performance weight ( ) S W s , which is used to shape the sensitivity function as follow (17) [24], [20]: denotes the sensitivity function, which defines the transfer function from both inputs ( ) y d s and ( ) r s to the output ( ) e s . In this paper the performance weight S W will be assumed as the adjustable transfer function that defined in (18). Its parameters are jointly optimized to those of the desired controller using some rules given that decribed later [25]. We get (18)

SIMULATION RESULTS AND DISCUSSIONS 5.1. DC Motor Model
The DC machines are characterized by their simplicities and flexibilities. By means of various combinations of the shunt, series and the separately excited field windings, they can be designed to display a wide variety of volt-ampere or speed-torque characteristics for both dynamic and steady-state operation. The systems of DC machines have been frequently used in many applications requiring a wide range of motor speeds and a precise output motor control. The diagram of typical DC motor is shown by Figure 2.
The nominal values of DC motor are summarized by Table1. We get    W j x σ ω * at some frequencies except, in frequency range [0.07 , 1.33] ω ∈ radians/seconds. This can be explained in the time domain by higher sensitivity to sensor noises. Noticing that the better NP margin is given when the maximum singular values plot of the sensitivity function are small as much as possible at low-frequency range. So that, the robust SFO-PID controller ensures the better margin then the H ∞ controller. Furthermore, in low frequency range when 0.005 ω ≤ radians/seconds, the curve of max [ ( , )] S j x σ ω * is below 80 dB − , which means that the load disturbances are attenuated more than 10000 times at plant output.    According to figure 9, it is easy to see that the singular values plot, which is given by the PFO-PID controller is reduced at frequencies beyond the system bandwidth in order to secure robustness at high frequency range. Furthermore, for frequencies above 3200 ω = radians/seconds, this plot is below -60 dB in which the sensor noises are suppressed more than 1000 time at the plant output. Consequently, the PFO-PID controller provides the better RS margin compared with the one given by the SFO-PID controller. To confirm the above results in time domain, the set-point reference that assumed a unit-step is used. Therefore, figure 10 shows the obtained tracking dynamic of the closed-loop system given by the H ∞, integer PID, robust PFO-PID and robust SFO-PID controllers. So that, the better tracking properties are ensured by the robust PFO-PID controller, which are characterized by the fast settling time with the reasonable overtaking.

CONCLUSION
In this paper, comparisons between two fractional controller-structures have been presented for DC speed control motor. The design controllers have been achieved using the series and parallel fractional order PID configurations. Each controller has been designed with the automatic selection of corresponding performance weight. The parameters of both controller and weight have been determined from solving the weighted-mixed sensitivity problem by the fminimax function of the Matlab software. The obtained simulation results have been compared with those given by two conventional controllers. The obtained simulation results show the notable improvement that the proposed control strategy. However, it is also clear that further improvements in weighted-mixed sensitivity formulation step will require introducing the fractional weights to enhance the controller performances for the wide variation of the model parameters.