A Comparison between Frequency Domain and Time

A Comparison between Frequency Domain and Time Domain Controller Synthesis Position Control of a DC Motor Michel Owayjan

Roy Abi Zeid Daou

Xavier Moreau

Dept. of Mechatronics Engineering American University of Science & Technology Beirut, Lebanon [email protected]

Dept. of Biomedical Technologies Lebanese German University Jounieh, Lebanon [email protected]

IMS - Laboratoire de l'Intégration, du Matériau au Système University of Bordeaux Bordeaux, France [email protected]

Abstract—Different control analysis and design methods exist using either time domain or frequency domain. This paper explores the robustness of four control techniques, two using the frequency domain (PID and CRONE – French acronym of Commande Robuste d’Ordre Non Entier) and two using the time domain (State-feedback and SMC – Sliding Mode Control). All four techniques are applied on a position control for a DC motor, which is a linear system. The robustness is introduced through uncertainty in the system’s gain. The responses and performance of the different techniques are then presented and compared. Keywords—PID; CRONE; State-Feedback; SMC; Robustness; Position Control of a DC Motor; Frequency Domain; Time Domain

I. INTRODUCTION Control Engineering is concerned with the analysis and design of controllers that force physical systems to behave according to desired specifications. The first step in control engineering is to model a physical system, also called process or plant, using a mathematical representation. This representation is usually a set of differential equations that can be either linear or nonlinear. In both cases, the differential equations can be of integer or non-integer (sometimes called fractional) order. On the other hand, the control is either linear or nonlinear.

aerospace and automotive control, industrial process control, and others [6]. There are two types of system nonlinearities; some are natural or inherent, and others are artificial or intentional, also called added [4, 7]. Natural nonlinearities are inherent in the plant model [7], such as operational amplifiers saturation, motor dead zone, backlash in gears, etc. [1]. This type of nonlinearities produces undesirable effects; thus control systems have to properly compensate them [4]. These nonlinearities are also classified as smooth (also called soft or continuous) or non-smooth (also called hard or discontinuous) nonlinearities depending on their mathematical properties [4, 6]. The latter causes parasitic effects, such as dry friction, or actuator saturation; and when it is significant, it requires specific treatment imposing constraints in the design [3, 6]. On the other hand, nonlinearities can be added by the designer in the controller for different reasons, such as improvement of existing control systems when dealing with large changes in soft nonlinearities, compensating for hard nonlinearities, dealing with model uncertainties, or even design simplicity [4]. Nonlinear control systems deal with the analysis and design of control systems containing either natural nonlinearities, added nonlinearities, or both [4, 6- 8].

A linear system possesses two properties: homogeneity and superposition [1-2]. A system possesses the homogeneity property if a scaled input by a certain factor produces an output scaled by the same factor. A system has the superposition property when it has the homogeneity and the additive properties [3]. When the time-shifting property is added, the systems are also called Linear Time-Invariant (LTI) systems. Linear control design techniques in the frequency domain are well-known and powerful techniques to achieve the control design problem and meet the desired specifications of LTI systems [4].

Two approaches are used in the analysis and design of controllers, frequency domain and time domain. Frequency domain techniques include the classical PID (ProportionalIntegral-Derivative) control, lag-lead compensation, CRONE control, etc. All of the frequency domain techniques are based on the frequency analysis and design tools such as the rootlocus, Bode diagrams, Nyquist plot, Nichols chart, etc. The system’s performance is evaluated according to its response for different frequencies. On the other hand, the time domain techniques such as the state-feedback control, the Sliding Mode Control (SMC), etc. are based on the analysis of the system and synthesis of controllers in the time domain based on the step response and how the system’s states are progressing with time [9].

However, most physical systems are nonlinear and time varying [3-5]. Nonlinear systems exist in various engineering and natural fields such as mechanical and biological systems,

On the other hand, the modeling of physical systems and the synthesis of controllers are done based on differential equations that can either be of integer order or non-integer also

called fractional order [10]. The classical control theory based on integer order differential equations has been generalized into non-integer order, and the implementation of fractional order controllers has been recently on the rise [11]. The next section in the paper introduces two frequency domain methods, the classical PID controller and the CRONE controller. Then, two time domain methods are described, the state-feedback control, and the SMC. The next section presents the DC Motor position control model, followed by a section describing the synthesis of the four different controllers and the results of the simulations are presented for the four controllers with a nominal case, minimum, and maximum case to introduce robustness. The interpretation of the results is then done at the end of the paper along some conclusions and future works. II. FREQUENCY DOMAIN METHODS A. PID Control The PID controller has been the most widely used controller in the industry [12-13]. The PID controller used in feedback control is composed of three elements added together: a P (proportional) element indicating the “present” error in the system, an I (Integrator) element indicating the “past” error, and a D (Derivative) element indicating the “future” error as shown in Fig. 1. The transfer function of a PID controller is calculated as follow: (1) The transfer function of the PID controller can be rewritten as (2) where the zeros of the PD and PI compensators are clearly specified, and the gains, KD, KP, and KI, are calculated accordingly. (2) Using the Root-Locus technique, the synthesis of the PID control begins with the evaluation of the performance of the uncompensated system. Then, the design of a PD controller to meet the desired specifications is carried out by adding a zero to the system, and the system’s performance is evaluated. If the latter does not meet the specifications, a redesign is required. The last step is to design a PI controller in order to eliminate the steady-state error, and then calculate the PID controller’s gain [1]. B. CRONE The CRONE controller exists in three generations. When the phase and the margin are constant, the first generation is used. However, when the phase is only constant around the unit gain frequency, the second generation is used. Meanwhile, when both the phase and the margin are changing, the third generation is to be used. The system’s order for the first two generations is non-integer varying between 0 and 1 whereas, for the third generation, the complex order is found [14].

Fig. 1. PID Controller

In order to move from the rational or fractional representation (3) of the rational form transfer function (4), Oustaloup method is used. Interested authors can refer to the following reference for more details [15]. Fractional form: C ( s ) =

K0 s

⎛1 + s /ωa ⎜⎜ ⎝ 1 + s / ωb

s ⎛ ⎜1+ ' ωi K n Rational form: I N ( s ) = 0 ∏ ⎜ s i =1 ⎜ 1 + s ⎜ ωi ⎝ N

⎞ ⎟⎟ ⎠

n

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

(3)

(4)

where N represents the number of cells (usually each decade needs one cell which is composed of a pole and a zero with a minimum of four cells to reconstitute the fractional system), K0 is the gain responsible to set the unit gain frequency ωu, ωi and ω’i being respectively the poles and the zeros of the rational transfer function. Concerning the synthesis of the CRONE controller, the same method is used for the three generations. The a posteriori method is deployed where the transfer function of the controller is deduced after defining the behavior of the open loop system. Hence, the transfer function of the open loop is the following: n

n

⎛ 1+ s / ω b ⎞ b ⎛ 1+ s / ω h ⎞ − nh ⎟ ⎜ ⎟ (1 + s / ω h ) + s / ω 1 s / ω b ⎠ ⎝ b ⎠ ⎝

β (s) = β0 ⎜

(5)

This system can make sure that the following aspects are met: -

low frequencies in order to have good accuracy in the steady state; middle frequencies, especially around the frequency ωu, to get the stability degree robustness; high frequencies to have good input plant sensitivity.

Hence, ωb and ωh represent the low and high transitional frequencies, n is the fractional order varying between 1 and 2, nb and nh are the asymptotic order behaviours for low and high

frequencies and β0 is a constant that assure a unit gain at the frequency ωu. III. TIME DOMAIN METHODS A. State Feedback The state feedback technique with integral control is a time domain method for the synthesis of controllers offering many advantages over frequency domain techniques, whether using the root locus or frequency response. Among these advantages are the specification of all closed-loop poles and not only the second-order dominant poles, the application to systems with nonlinearities and to Multiple-Input, Multiple-Output (MIMO) systems, and the availability of a wide range of computational tools. On the other hand, state feedback with integral control cannot specify the location of closed-loop zero locations that can affect the transient response, does not provide the graphical insight into the design problem that frequency domain methods yield, and may prove to be sensitive to parameter variations [1]. In order to design the state feedback with integral control, the system should be represented as a state-space model in phase variable form similar to: (6) First, the controllability matrix for the system is calculated from (7). If its rank is the same as the order of the system, then the system is controllable. 2

1

(7)

The next step in the design is to feedback each state variable to the control, u, through a gain, ki, unlike a typical feedback system where the output, y, is fed back to the summing junction. This would produce n gains (where n is the system order) that could be adjusted to yield the desired closed-loop poles. The state equations for the closed-loop system become: (8) The characteristic equation for the closed-loop system represented in (8) is calculated, as well as the equivalent characteristic equation computed from the desired closed-loop poles. The coefficients in both equations are then equated to solve for the gains, ki. The steady-state error of the system is then calculated from: ∞

1

Fig. 2. State feedback with integral control block diagram

The integrator introduces an additional state variable, xN, and the error is the derivative of this variable. The new augmented state-space model is shown in (10).

0

(10)

0 Since the order of the system is increased, a pole is usually added to the characteristic equation (at a location more than five times the real part of the dominant poles) and the characteristic equations, one derived from (10), and the other from the desired specifications, are compared and the gains, ki and ke, are then calculated. B. Sliding Mode Control (SMC) In the 1950s and 1960s, researchers such as Emelyanov in the former Soviet Union initiated Variable Structure Control (VSC) which includes a number of control laws and a decision rule [16-17]. This introduces a discontinuity in the control action making the control nonlinear [18]. The theory was then developed by Utkin and Young [19]. Sliding Mode Control is a type of VSC where the discontinuous feedback switches the structure of the system according to the evolution of its state vector resulting in a sliding surface/manifold [20-21]. SMC’s main advantage is that the system becomes totally insensitive to some uncertainties such as parameters variations, disturbances rejections, and bounded nonlinearities [19, 21-22]. The main limitation of SMC is due to the nature of the discontinuous control signal, which is switching between two values as depicted in Fig. 3. This switching causes high frequency oscillations around the sliding surface/manifold, called chattering, and its causes can be explored in [20].

(9)

The next step is to introduce an integrator to the system in order to increase the system type and therefore reduce the steady-state error to zero. This is done by feeding back the output and adding it to the input to form the error which is integrated and multiplied by a gain, ke, to form the input to the controller (Fig. 2).

0 1

Fig. 3. Design of a SMC System

There exist two versions of the SMC, a first-order SMC C is based on the and a higher-order SMC. The first-order SMC idea of keeping the sliding variable (an auxiliary output variable) zero. This represents the error from tthe specifications and depends on the first time derivative of thee sliding variable. On the other hand, higher-order SMC are characterized by discontinuous control acting on the higgher order time derivatives of the sliding variable in orderr to remove the chattering problem [23]. we start with the For the synthesis of a first-order SMC, w state-space model of the system in phase variaable form (6). The design of a first-order SMC consists of two stages. The first stage is to choose a sliding surface or mannifold where the system exhibits defined by the specifications. The second stage is the controller design where a control actiion is defined to steer the state trajectories onto the sliding surfface, after a finite transient [24-25].

Fig. 4. DC Motor skematic diagram TABLE I.

Parameter N1 N2 JL Ja DL Da Kt Kb Ra

For the sliding surface design, first the slidding surface (11) is introduced as a function of the state variablees. :

0

DC MOTO OR PARAMETERS

(11)

Valu ue 25 250 1 kg--m2 0.123 3 kg-m2 0.1 N-m-s/rad N 0.005 588 kg-m2 0.5 N-m/A N 0.5 V-s/rad V 8Ω

For a second-order system, (11) becomes (12). ,

(12)

Since we have a second-order system, onnly the first time derivative is needed (first-order SMC), and tthe coefficients S are selected to meet the desired specifications oof the system. The next stage is to derive the control law w that drives the system trajectories into the sliding surface inn finite time. The control law, u can be written as, where U is a sufficiently large positive constant: , ,

0 0

(13)

0.00688

(16)

To complete the transfer functtion of the motor, (14) is multiplied by the gear ratio to arriive at the transfer function relating load displacement to armatu ure voltage: .

0.1

.

(17)

In order to study the robustn ness of the four different techniques, an uncertainty in thee gain is introduced. The nominal gain value used to syn nthesis the four different controllers is 0.2803. Two extreme cases c are studied, the first is a variation of the gain to its third, i.ee., 0.0934, and the second is a variation of the gain by a factor off three, i.e., 0.8409. V. CONTROLLERS SYNTHESIS

IV. SYSTEM PRESENTATION In order to compare the frequency domain and time domain techniques, a DC motor with gears and a looad is used as a position control system, where the input to the motor is the armature voltage, ea(t), and the output is the angular displacement of the load, θL(t) (Fig. 4). The pparameters of the DC motor and its load are shown in Table I. The DC motor is an electromechanical syystem that can be represented by the following transfer functtion relating the armature displacement to the armature voltagee [1]: ⁄

. .

(14)

where the equivalent inertia, Jm, and equuivalent viscous damping, Dm, are calculated as: 0.0223

.

.

(15)

In order to compare the fou ur control techniques, the following specifications are used for the position control of the DC motor: • • •

Percentage Overshoot, %OS = 10%, Settling Time, Ts = 2 s, and Steady-state error, e(∞)) = 0.

A. PID Control on control system, the For the DC motor positio uncompensated system is studied to find out if the mple gain adjustment. The specifications can be met using sim open-loop transfer function is show wn in (17). A PD design is first conducted by adding a zero at a -5 in order to meet the transient response specifications (%OS and settling time). uce the steady-state error to Then, a PI design is added to redu zero. The gain is calculated to be 8.29 making the controller’s transfer function: 8.29

.

(18)

Therefore the step response of the closed-loop system with the three cases, minimum, nominal, and maximum gains is shown in Fig. 5. And the Bode plots of the systems are displayed in Fig. 6. Plant output y(t) 1.5

C. State-Feedback For the state-feedback controller synthesis with integral operation to remove the error, the gains of the feedback states and the integrator are calculated as K = [411.451 103.829] and Ke = 4085.27. The state space representation of the closed-loop system becomes (20) and the corresponding transfer function is shown in (21).

1

0

0 411.451 0.2803

Minimum Nominal Maximum

0.5

0

1

2

3

4

5 time (s)

6

7

8

9

10

0 0 1

(20) (21)

.

Fig. 5. Step responses for the PID Controller Phase (deg) Magnitude (dB)

1 0 104 4085.27 0 0 0.2803 0 0

The step response and the Bode plot of the closed-loop system with the three cases, minimum, nominal, and maximum gains is shown in Fig. 9 and Fig. 10 respectively.

Bode Diagram 50 0

Plant output y(t) 1.4

-50 0

1.2 1

-90

0.8

-180 -2 10

0.6

-1

0

10

1

10

2

10

10

0.2 0

Fig. 6. Bode Plots for the PID Controller

/

.

.

1

/ .

/12.02

(19)

The step response and the Bode plot of the closed-loop system with the three cases, minimum, nominal, and maximum gains is shown in Fig. 7 and Fig. 8 respectively.

Phase (deg) Magnitude (dB)

/ . / .

0

1

2

3

4

5 time (s)

6

7

8

9

10

Fig. 9. Step responses for the State-Feedback Controller

B. CRONE Referring to the above specifications, the open loop transfer function is the following: 8.292

Minimum Nominal Maximum

0.4

Frequency (rad/s)

Bode Diagram 200 0 -200 0 -180 -360 -2 10

-1

10

0

1

10

10

2

10

3

10

4

10

Frequency (rad/s)

Fig. 10. Bode Plots for the State-Feedback Controller

Plant output y(t) 1.4 1.2 1

D. Sliding Mode Control (SMC) The calculations of the sliding surface and the control law are based on [26]. The equations are given in (22) and (23).

Minimum Nominal Maximum

0.8 0.6 0.4 0.2 0

0

2

4

6

8

10 time (s)

12

14

16

18

20

(22)

Fig. 7. Step responses for the CRONE Controller

1.71

2

(23)

1.71

Fractional (blue) and Rational (red) controllers Bode diagrams Magnitude (dB)

80

The equations become (24) and (25) after substituting λ = 35, Ks = 1, and Kp = 65.

60 40 20 0 -3 10

-2

10

-1

10

0

10

1

10

2

10

829

Phase (deg)

45

(24) 98.29

(25)

The step response of the closed-loop system with the three cases, minimum, nominal, and maximum gains is shown in Fig. 11.

0 -45 -90 -3 10

35

3

10

-2

10

-1

10

0

10 Frequency (rad/s)

Fig. 8. Bode Plots for the CRONE Controller

1

10

2

10

3

10

[9]

Plant output y(t) 1.4 1.2

[10]

1 0.8 0.6 0.4

Minimum Nominal Maximum

0.2 0

0

1

2

3

4

5 time (s)

6

7

8

9

10

[11]

Fig. 11. Step responses for the SMC Controller

VI. CONCLUSIONS AND FUTURE WORK We have presented in this paper a comparison between four different controllers where two of them (PID and CRONE) are synthesized using frequency domain analysis whereas the two remaining (Sliding Mode Control and Feedback control) are synthesized using time domain analysis. The positioning control of a DC motor was then presented. The results showed that the controllers calculated the frequency domain constraints were more robust concerning the first overshoot. As a future work, a comparison between integer order and fractional order controllers can be proposed. Added to that, a nonlinear plant can be taken into consideration to study the power of each controller.

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