New PI Controller Design and Applications

New PI Controller Design and Applications

Holiarimanga Mariette Randriamanjakony

Doctoral School Renewable Energies and Environment EDT-ENRE, University of Antsiranana

Diego Suarez, Madagascar

Jean Nirinarison Razafinjaka

Department of Electricity ESP- University of Antsiranana

Diego Suarez, Madagascar

Andrianantenaina Tsiory Patrick

Department of Electricity ESP- University of Antsiranana

Diego Suarez, Madagascar

Abstract This paper proposes a new PI controller design using the product form. The method is called General Method. It offers more possibilities to design the controller. One detailed application is on the DC Motor with Permanent Magnet. On this subject, some new considerations are taken into account. Simulation results show that all propositions are realizable and leads on good performances.

KeywordsPI controller, design, optimization, DC motor.

1. INTRODUCTION

   It is well-known that the method of Ziegler and Nichols [1] constitutes the bread and butter of PID tuning. It can be said that PID controller is widely used in all areas where control is needed and applied. Currently, several design methods are given for this controller to improve or to bring

   Where g is	the	gain o	f the controller, Ti the integral
   constant time,	Kp	and	Ki	represent respectively the

   proportional and integral constants and Tn is the proportioning integral time.

   The last expression in (1) is the product form. Having one of these expressions allows to pass to the others. It may be noted that the integral constant times Ti in the first and the third expression have not the same value.

   1. Case of a First Order System

      The flat and symmetrical criteria (FC, SC) [10] are commonly used in electric machine drives. Their design rest especially with the little constant time of the system. But these criteria are at fault with a first order system. The transfer function (TF) is:

      optimization of the system performance, especially for the PI controller [2], [3], [4], [5]. A Variable Gain PI (VGPI) using sum form is introduced in [6] and [7]. It uses an entire degree n for the gains Kp and Ki. In [8], a non-entire degree is proposed and applied on AC-DC converter with power factor correction. In [9], the combination with Fuzzy Logic and VGPI is given.

      G( p) K

      1 pT

      Where K is the static gain, T the constant time.

      • Using the sum form

        The second expression of (1) gives the TF:

        (2)

        In this paper, new PI controller design is presented and especially for linear system. An application for first order system with time delay is also given. First, theory about the

        GR ( p) Kp

        Ki . (3)

        p

        General Method (GM) is showed. Some examples are chosen to support the reasoning. Finally, a speed control of a DC

        Then, the transfer function in open loop (FTOL) is:

        motor with Permanent Magnet is taken. Some proposals are

        G ( p) K

        Ki . K

        (4)

        advanced to reduce the inrush current.

        o p

        p 1 pT

        i

2. NEW PI CONTROLLER DESIGN

   Three forms of the transfer functions of PI controller are often used, one product form and two sum forms as shown in (1

   The goal is to find the gains Kp and Ki. There are two possibilities:

   a. Compensating the constant time T and imposing a constant time Tf in closed loop. It gives,

   G ( p) g 1 1

   T

   R

   K

   pTi

   p K.T

   (5)

   f

   G ( p) K

   • Ki

     (1)

     R p

     p

     Kp

     Ki

     1 pT T

     R

     G ( p) n

     pTi

     1 b. The constant time T is not compensated. The damping 0.9 factor and the non-deadened natural pulsation n are 0.8 imposed. Then, 0.7 T 0.6 Ki n K (6) 0.5 K 2nT 1 0.4 p 0.3 K f positive constants are wanted, (7) must be respected. 0.2 0.1
     							setpoint b = 1 a = 1 b = 0.25 a = 1 b = 2 a = 1

     I

     T 1

     2n

     y(t)

     (7)

     0

     0 1 2 3 4 5 6 7 8 9 10

     t [s]

     It will see as follow that the GM generalizes these two possibilities.

     The adopted form of the PI controller is the third expression of (1):

     Fig. 1. Step response with different values of the constant b.

     The more the constant b decreases, the more the response is fast.

     R

     G ( p) 1 pTn

     (8)

     a = 0

     pTi

     By the GM, Tn and Ti are given as follow:

     Tn a.T

     Ti

     b.K.T

     With a 0 and b > 0

     (9)

     The controller is not truly a PI one but has an effect I.

     G ( p) 1

     R pbT

     The TFOL is then,

     1 1

     (13)

     The TFOL is then,

     Go ( p)

     pbT

     .

     1 pT

     (14)

     G ( p) 1 paT . K 1 paT . 1 (10)

     The characteristic equation in closed loop is a second order

     o pbKT

     1 pT pbT

     1 pT

     polynomial:

     According the value of a, three possibilities are presented.

     . d ( p) p² p 1 1 0

     (15)

     a = 1

     It means that the constant time T is compensated. The

     c T bT ²

     The canonic form, dc*(p) of the characteristic equation is,

     TFOL Go(p) and the transfer function in closed loop (TFCL)

     d*( p) p² 2

     p 2 0

     (16)

     H (p) are both first order.

     G ( p) 1

     (11)

     c n n

     With the damping factor and n the non-deadened

     And,

     o pbT

     natural pulsation, (17).

     By comparing (15) and (17),

     H ( p) 1 . (12)

     1 pbT

     2 1

     n T

     (17)

     It is clear here that the constant b determines the speed of

     2 1

     the step response as shown in Fig. 1. In this example,

     n

     bT 2

     K = 2 and T = 1 [s].

     Relation (17) allows to obtain and n. For instance,

     2 b 2 . (18)

     2

     a > 0

     In this case, it is assumed that a 1. The TFOL is,

     G ( p)

     1 paT

     . (19)

     o pbT 1 pT

     This TF has one zero zo and two poles p1 and p2,

     1

     2

     a b

     z

     o aT

     (20)

     n bT

     (23)

     p 0;

     p 1

     2 1

     1 2 T

     n

     bT 2

     Fig.2 and Fig.3 show the root locus map for different values of a with K = 2 and T = 1 [s].

     By the second expression of (23), it can be noticed that the step response speed depends of the value of the constant b. Fig. 4 shows step responses with different values of a.

     RootLocus

     0.2

     1.4

     1.2

     0.15

     0.1 1

     Imaginary Axis

     0.05

     0

     y(t)

     0.8

     0.6

     setpoint

     b = 1 a = 1

     b = 1 a = 0.25

     b = 1 a = 1.5

     -0.05

     0.4

     -0.1

     0.2

     -0.15

     -0.2

     -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5

     Real Axis

     0

     0 1 2 3 4 5 6 7 8 9 10

     t [s]

     Fig. 4. Step responses according to a.

     Fig. 2. Root locus with a > 1.

     It is highlighted that the case in Fig.3 will generate an oscillatory deadened response and the case in Fig.2, an aperiodic response one.

     By (19), the TFCL is,

     H ( p) 1 paT p²bT ² pT (a b) 1

     (21)

     The more a decreases, the more the overshoot increases. It may be noted, that changing the value of the constant b gives more possibilities.

     Fig.5 presents the effect of varying b with a fixed. When b decreases, the response is fast and the overshoot increases. When b > 1, the response becomes aperiodic. It can be said that there are several possibilities for the combinations.

     						setpoint
     						b = 1 a = 0.5 b = 0.25 a = 0.5
     						b = 2 a = 0.5

     1.4

     Root Locus

     4

     3

     2

     .

     Imaginary Axis

     1

     0

     -1

     -2

     1.2

     1

     y(t)

     0.8

     0.6

     0.4

     0.2

     -3

     -4

     -14 -12 -10 -8 -6 -4 -2 0 2

     Real Axis

     Fig. 3. Root locus with 0 < a < 1

     0

     0 1 2 3 4 5 6 7 8 9 10

     t [s]

     Fig.5. Step responses according to b with a fixed.

     The characteristic equation is given by (22).

     a b 1

     1. First Order System with Time Delay

        The TF is here,

        e_ pTo

        dc ( p) p² p.

        0

        bT bT ²

        (22)

        G( p) K

        1 pT

        (24)

        In comparison with (16),

        Where, To is the time delay, K the gain and T the constant time.

        If To << T, the TF can be approximated as,

        G( p) K

        1 pTo 1 pT

        (25)

        Fig. 6 and Fig. 7 show the simulation results obtained by

        T aT

        (19) and (20) using the flat criteria (FC) [10] and GM.

        GM:

        n 1

        (28)

        1.4

        1.2

        Ti bKTp

        Fig. 8 shows the simulation results obtained by the two methods.

        1 1.5

        y(t)

        0.8

        0.6

        1

        y(t)

        0.4

        0.2

        0

        ref

        Meplat : a= 1 b = 2

        GM : a = 1.1 b = 2.25

        0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

        t [s]

        0.5

        ref

        FC : Tn = 4Tp Ti =8KTp*Tp/T1

        Fig. 6. Simulation results with PI controllers using (24)

        GM : a = 1.2 b = 2

        GM : a = 1.25 b = 3

        By Fig.6 and Fig.7, it is highlighted that GM gives more possibilities and can improve performances obtained by the

        0

        0 1 2 3 4 5 6

        t [s]

        Fig. 8. Step responses by SC and GM

        FC.

        1.4

        1.2

        The characteristics of the responses are resumed in Table

        1.

        TABLE I CHARACTERISTICS

        				ref Meplat : a= 1 b = 2
        			GM : a = 1.1 b = 2.25 GM : a = 1.25 b = 1.8

        	Step response characteristics
        	PI – SC	PI- GM a= 1,2 b = 2	PI- GM a = 1,25 b = 3
        D1	43,3 %	21,2 %	11,87 %
        tp	0,59 [s]	0,55 [s]	0,74 [s]
        Tr (± 5%)	1,53 [s]	1,38 [s]	1,98 [s]

        1

        y(t)

        0.8

        0.6

        0.4

        0.2

        0

        0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

        t [s]

        Fig. 7. Step responses with PI controllers using (25)

        Putting aside the use of Smith predictor, [11], [12] propose a PI controller design using the pole dominant method (POLDOM) in the domain frequency and taking into account To.

     2. System with Integral Behavior

     For one application, a system defined by the TF given in

     (26) is chosen.

     D1: overshoot tp : peak time Tr settling time for ±5%

     It is here highlighted that the SC leads to a high overshoot. For the GM, the more b increases, the more D1% decreases but the response becomes slower. In anytime, the GM presents better performances and permits more possibilities.

3. APPLICATION ON DC MOTOR SPEED CONTROL

   The DC Motor is with Permanent Magnet one. The speed control of the motor needs two loops: an inner loop constituted by the current loop and the principal loop which is the speed loop. The PI control design needs the modeling of the system. The motor provided with an inertial load and a viscous friction is taken into account.

   Such system is rather current and presents also a teaching interest and industrial applications. Fig. 9 shows the system.

   G( p) K

   pT1 1 pTp

   (26)

   For this kind of system [10] recommends that the FC is not applicable. The PI controllers obtained by symmetric criteria (SC) and GM design are applied.

   Tn 4Tp

   SC:

   T 2

   (27)

   i

   T 8K p

   T1

   Fig. 9. The DC-Motor with its load

   With Ra, La the resistance and inductance armatures, Tm the motor torque, f the constant friction, J the inertia and Ui the back electromotive force.

   Gcm

   ( p) Kcm . (34)

   1 pTcm

   The equations governing the system are as follows:

   Several cases will be taken into account by the consideration of this MOC. Fig. 11 shows the functional

   u (t) R .i (t) L

   a

   di (t)

   u (t)

   diagram.

   a a a a dt i

   Tem (t) KT ia (t)

   u (t) K .(t)

   i V

   (29) +

   _

   PI

   Controller

   Current

   Loop Mechanical set

   Speed Sensor

   With KV, the speed constant and KT the constant torque It is here assumed that KV and KT are the same,

   KV KT

   The motion equation is:

   (30)

   Fig. 11. Functional diagram

   As already said, the current loop will be studied first then the speed loop will be analyzed.

   T (t) T (t) J d (t )

   f (t) . (31)

   em r dt

   Where, Tr is a resistive torque.

   1. Current Loop Analysis

      The current loop diagram is given by Fig. 12.

      Using the Laplaces transformation, these equations are given by (32) and (33):

      Ua ( p) Ui ( p) Ra pLa Ia ( p)

      Ic +

      _

      PI

      Controller

      Control Unit + Conveter

      Electrical Ia System

      And,

      Tem ( p) KT Ia ( p)

      Ui ( p) KV .( p)

      Tem ( p) Tr ( p) Jp f ( p)

      (32)

      (33)

      Current Sensor

      Fig. 11. Current loop diagram

      For the speed control, a cascade scheme is adopted which contains two loops: the current loop and the speed one. First, the inner loop, which is the current loop is studied then the

      set,

      Equation (34) gives the TF of the MOC. For the electrical

      speed loop. The technical diagram is showed in Fig. 10.

      Where 1: DC-Motor, 2: buck DC converter, 3: control unit, 4: Current PI controller, 5: current sensor, 6: current controller, 7: speed sensor, 8: setpoint circuit, 9: Permanent

      G ( p) 1

      a a

      i R pL

      The current sensor is assumed to be ideal. Its TF is,

      (35)

      Magnet

      Gcs ( p) 1. (36)

      Relation (35) can be written as,

      With,

      Gi ( p)

      Ka

      1 pTa

      . (37)

      1

      Ka R

      T

      a

      a

      La Ra

      (38)

      Fig. 10. Technical diagram of the complete system

      The Motor Organ Command (MOC) is constituted by the control unit, the buck DC converter. It is defined by the TF,

      Relation (8) is for the PI controller. The TFOL of the electrical set with the MOC is,

      G ( p) K1 . (39)

      ci 1 pT

      1 pT

      cm a

      Reponse indicielle du courant

      						ic ia: a =0.2 b= 2 ia: a =1 b= 2
      						ia: a =0.5 b= 1.25

      3.5

      With, K1 Kcm.Ka .

      Thus, the TFOL with the PI controller is given by (40).

      3

      2.5

      Goi

      ( p) 1 pTn .

      pT

      1 pT

      K1

      1 pT

      Ia [A])

      (40) 2

      i cm a

      Several cases will be considered according the MOC TF.

      • The little constant time Tcm is taken into account

        FC and GM are applied.

        1.5

        1

        0.5

        0

        FC:

        GM:

        Here, Tcm << Ta.

        Tn Ta

        2.K T

        Ti 1 cm

        Tn a.Ta

        b.K T

        Ti 1 cm

        (41)

        (42)

        0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

        t[s]

        Fig. 13. Current curves according (a, b).

        The responses are largely slower than the system with the little constant time Tcm. For the speed loop, the couple (a=0,5; b = 1,25) will be chosen as an application.

        • Compensating the little time constant Tcm.

          Fig. 12 shows the simulation results of these cases.

          R?ponse indicielle du courant

          							Ic FC: a = 1 b= 2 GM: a = 1.85 b= 3
          						GM: a = 2 b= 2.3

          3.5

          3

          In [10], it is formally recommended not to compensate the little constant-time by the reason that noises in high frequencies are badly known. However, an equivalent constant-time is proposed.

          2.5

          Ia [A])

          2

          1.5

          1

          np

          Tp Tpj

          j 1

          Where Tpj is a small constant-time. Then, it can be posed,

          Tn Tp

          (45)

          0.5

          0

          0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

          bK T

          Ti 1 a

          . (46)

          t[s]

          Fig. 12. Current curves with FC and GM

          -3

          x 10

          In this case, Tp = Tcm.

          The FTOL with the PI controller is,

          By combination of (a, b), GM offers more possibilities. Generally, the overshoot increases with a.

          G ( p) 1

          (47)

          • The MOC is assumed to be ideal

          oi pbT

          1 pT

          In this cas, Tcm is not taken into account. Then, the MOC TF is as,

          a a

          Simulation results are given in Fig. 14.

          Gcm ( p) Kcm

          The TFOL becomes,

          (43)

          					Ic FC: a = 1 b= 2 GM: a = 1 b= 3 GM: a = 1 b= 2.3

          3.5

          3

          R?ponse indicielle du courant

          Gci

          ( p)

          K1

          1 pTa

          (44)

          2.5

          Ia [A])

          2

          K1 is always given by (39).

          SC or FC are at fault. The PI controller can be only given by GM. The reasoning is exactly in section II-A.

          For the PI controller, Tn and Ti are obtained by (9). The speed responses is dictated by the constant time

          Ta. Fig. 13 shows the simulations results with different combinations of (a, b).

          1.5

          1

          0.5

          0

          0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

          t[s]

          Fig. 14. Current curves with different values of b; (a =1)

          With the same value of a, the speed response is slower when b increases.

   2. Speed loop analysis

   The mechanical set is defined by the TF as below,

   	PI FC	PI GM a = 1,25 b= 2,25	PI GM a= 0,98 b= 2,25
   D1 overshoot	8,15 %	12,28 %	3,37 %
   tp Peak-time	0,34 [ms]	0,3 [ms]	0,36 [ms]
   tr (±5%)	0,4 [ms]	0,4 [ms]	0,26 [ms]

   TABLE II. CHARACTERISC RESPONSES

   With,

   G ( p) 1

   m Jp f

   Km

   1 pTm

   (48)

   With a fixed value of b, the overshoot decreases when the constant a decreases by GM. Current curves in closed loop

   1

   f

   Km

   (49)

   are showed in Fig. 16

   			Iref FC FC: a =1 b =2
   			GM : a =0.98 b =2.25 GM : a =1.25 b =2.25

   5000

   T J

   4000

   m f

   3000

   Fig. 11 shows the general functional diagrm. The speed

   loop is analyzed according to the three cases:

   • Case 1: Tcm is taken into account

     Using (38), and (41), the TFLO with a = 1 is,

     I [A] ]

     2000

     1000

     G ( p) 1

     0

     (50)

     oi pbT

     1 pT

     cm cm

     -1000

     0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

     Here (41) is a particular case with b = 2. Then, the TFCL is,

     t [s]

     Fig. 16. Currents in speed closed loop

     -3

     x 10

     H ( p) 1

     (51)

     It is here highlighted that the inrush currents are too high.

     i pbT 2

     • pbT 1

       Even the duration is very short, the values are not

       cm cm

       Because Tcm has a very small value (Tcm = 33,3 [s]), it be assumed that,

       realistic. It is due by taking into account the little constant-time Tcm which needs a fast control. Inserting a filter at the setpoint constitutes a solution but in this case inrush current is always high. Two methods to reduce this

       i

       H ( p) 1

       (52)

       inrush current are now proposed.

       pbTcm 1

       Fig. 15 gives the speed curves obtained with FC and GM.

       			Wref FC: a =1 b =2 GM : a =0.98 b =2.25 GM : a =1.25 b =2.25

       180

       160

       140

       120

       W [rd/s ]

       100

       80

       60

       40

       • Case 2: The MOC is assumed as ideal

     Relation (43) gives the FT of the MOC. It is a simple gain and this method is already a usual one. For the current loop, FC and SC are in fault but the GM can be applied. Fig. 17 and Fig. 18 give respectively speed curves and currents in closed loop.

     					Wref GM: a =1 b =2 GM : a =1 b =1.5 GM : a =1.25 b =1.5

     180

     160

     140

     120

     W [rd/s ]

     100

     80

     20 60

     40

     0

     0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

     t [s]

     Fig. 15. Speed responses with FC and GM

     -3

     x 10 20

     0

     The speed responses are very fast. The characteristics are

     resumed in Table II.

     0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

     t [s]

     Fig. 17 Speed curves with GM

     					Iref FC GM: a =1 b =2 GM : a =1 b =1.5 GM : a =1.25 b =1.5

     40

     35

     30

     25

     I [A] ]

     20

     15

     10

     5

     0

     -5

     0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

     t [s]

     Fig. 18. Currents in closed loop with GM

     The inrush currents decrease considerably but the control becomes slower. The constant times in question are only the electrical and mechanical constant times Ta and Tm.

   • Case 3: Compensation of Tcm

   In this case, PI controller parameters are designed as follows in current loop,

4. CONCLUSION

In this paper, new PI controller design is proposed. It is called General Method (GM) which uses the product form. It is seen that GM offers more possibilities and can be applied on first order system. In another way, it can improve FC and SC. The application on DC Motor with Permanent Magnet shows that taking into account the little constant time of the MOC leads to a very high inrush current. Simulation shows that compensating this little constant time is one possibility to reduce this inrush current. But, it may be said that considering the MOC as ideal and defined as a simple gain is the better solution.

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bKT

Tii a

For the speed loop, since Ta << Tm, it may be assumed,

Tnn Tm

b K T

Tin 1 1 p

With, Tp = b.Ta.

(53)

(54)

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Fig. 19 and Fig.20 show speed responses and current curves in closed loop.

			Wref FC: a =1 b =2 GM : a =0.98 b =2.25 GM : a 1.25 b =2.25

180

160

140

120

W [rd/s ]

100

80

60

40

20

0

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

t [s]

Fig. 19. Speed responses with FC and GM

It can be noticed that the inrush currents leads to better performances by the combination of a and b.

35

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« New variable Gain PI Controller with Non-Entire Degree for AC-DC Converter with Power Factor Correction », 5th International of Renewables Energies (CIER-2017), Proceeding of Engineering and Technology, Copyright IPCO-2017, ISSN 2356-5608.

[9] Tsiory Patrick Andrianantenaina, Hervé Mangel, Jean Nirinarison Razafinjaka, Fuzzy-Variable Gain PIControl of WECS based on a Doubly Fed Induction Generator,The International Conference on Modelling and Applied Simulation, 16th Edition , September 18-20, 2017, Genoa, Italia.

[10] H. Bühler , « Conception des systèmes automatiques », Presses Polytechniques Romandes, Lausanne, 1988.

[11] Jean Nirinarison Razafinjaka, « Automatisation et synthèses des régulateurs standards et polynomiaux Application à la commande adaptative », thèse pour lobtention du diplôme de Doctorat 3è cycle, ESP Antsiranana et EPFL Lausanne, 1991.

[12] T. Hagglund and K. Aström, « Industrial Adaptive Controllers Based on frequency response Techniques », Automatica, Vol. 27, N°4, GB, pp

Iref

FC

599-609,1991.

30

FC: a =1 b =2

GM : a =0.98 b =2.25

25 GM : a =1.25 b =2.25

20

I [A] ]

15

10

DC-Motor parameters: Pn = 440 [W]; Nn= 1500 [tr/mn]; In = 3 [A];

Un = 175 [V]; Tem = 3 [N.m]; Ra = 5 [ La

J = 0,004 [kg.m²]; f = 0,0016 uSI; Ta = 4,86 [ms]; Tm = 2,5 [s];

Kcm = 055; Tcm = 33,3 s]; Kt = 0,986 [Nm/A];

5

0

-5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

t [s]

Fig. 20. Currents with FC and GM in closed loop