Application of PID Retuning Method for Laboratory ... - IEEE Xplore

Application of PID Retuning Method for Laboratory Feedback Control System Incorporating FO Dynamics ďubomír Dorþák

Ivo Petrᚹ

Institute of Control and Informatization of Production Processes, Faculty BERG, Technical University of Koššice, B. NČmcovej 3, 042 00 Koššice, Slovakia e-mail: [email protected]

Institute of Control and Informatization of Production Processes, Faculty BERG, Technical University of Koššice, B. NČmcovej 3, 042 00 Koššice, Slovakia e-mail: [email protected]

Emmanuel Gonzalez

Juraj Valsa

Center for Automation Research, College of Computer Studies, De La Salle University Manila, 2401 Taft Ave., Malate Manila 1004, Philippines Jardine Schindler Elevator Corporation, 8/F Pacific Star Bldg., Sen. Gil Puyat Ave. cor. Makati Ave., Makati City, 1209 Philippines, [email protected]

Faculty of Electrical Engineering and Computer Science, Brno University of Technology Kolejní 2906/4, 612 00 Brno, Czech Republic e-mail: [email protected]

Ján Terpák Institute of Control and Informatization of Production Processes, Faculty BERG, Technical University of Koššice, B. NČmcovej 3, 042 00 Koššice, Slovakia e-mail: [email protected] Abstract——Real objects in general are fractional-order (FO) systems, although in some types of systems the order is very close to integer order (IO). Since major advances have been made in the theory and practice of the identification of FO controlled objects and in the design of FO controllers, it is possible to consider also the real order of the dynamical systems and consider more quality criterion while designing the FO controllers with more degrees of freedom compared to their IO counterparts. In this paper, we present an application of the retuning method to design and apply new FO controller for the existing laboratory feedback control system with no modifications in the internal architecture of the original feedback control system. Along with the mathematical description, presented are also simulation results. Keywords——fractional calculus, fractional-order dynamic systems, fractional-order feedback control system, fractional-order controller, modeling, electronic realization, retuning method

I.

INTRODUCTION

Investigations in many different areas confirmed that the real objects are generally FO, however, for many of them the fractionality is very low. In those cases the description by FO differential equations (FODE) is more convenient and FO models are more adequate for the description of dynamical systems with distributed parameters [2] and also with concentrated parameters systems [3]. Because of the higher complexity and the absence of adequate mathematical tools,

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Monika ŽŽecová Institute of Control and Informatization of Production Processes, Faculty BERG, Technical University of Koššice, B. NČmcovej 3, 042 00 Koššice, Slovakia e-mail: [email protected]

FO dynamical systems were only treated marginally in the theory and practice of control systems, regardless of the reality and negative consequences caused by neglecting the real order of the system [1, 2, 3, etc]. The real objects were considered and identified as integer-order (IO) systems and the controllers were also of the IO type. Proportional-Integral-Derivative (PID) controllers have been the heart of control system engineering practice for decades because of its simplicity and ability to satisfactory control different types of systems in different fields of science and engineering in general. But the design of IO controllers for IO models of the FO controlled objects does not make it possible to consider more quality criteria [11-13] of control performance and in reality only during their implementation for the real controlled object they do not satisfy the quality criterion [3]. Since major advances [4-10] have been made in the theory and practice of the identification of FO controlled objects and in the design of FO controllers, it is possible to consider also the real order of the dynamical systems and consider more quality criteria while designing FO controllers with more degrees of freedom compared to their IO counterparts [10-13]. If, for existing control system, we want to additionally make provision for the FO of the controlled object PFO(s) and to design and apply new FO controller CR(s) instead of IO controller C(s) with no modifications in the internal architecture of the original feedback control system, we can do it through the method of retuning PI/PID controllers [14-15]

based on closed-loop model and using only one additional controller. On the other hand with this method we can also solve the problem of improving the robustness characteristics, such as guaranteeing gain cross-over frequency and phase margin specifications, and robustness to system gain variations, of an existing unity-feedback closed-loop system incorporating a stable IO plant PIO(s) and an IO controller C(s) of PID family type, i.e. P, PI, PD, and PID, by adding a new additional FO controller CR(s). This method includes the use of an existing reference and output signals as well as the parameters of the original PID controller to come up with a new additional FO controller satisfying a given set of performance characteristics. The idea is to determine the type of controller CR(s) best suited for its application and tuning of its parameters using any preferred means such as the well-known Ziegler-Nichols method, etc. Particularly in [14], the original controllers used on the synthesis method of CR(s) are based on the classical IO PI and classical PID approaches. In [15] was extended the work [14] to incorporate FO dynamics using FO controller in the original closed-loop system with IO plant by using a retuning heuristic. New FO controllers are obtained from the heuristic [15] such as PIȜ and PIȜDP controllers, where Ȝ; P  (0; 2) are the order of the integrator and differentiator, respectively. The parameters of a new controller CR(s) are functions of the existing classical PI/PID controller constants in C(s) and the new controller incorporated in the system does not modify the original system’’s internal architecture. Aside from this, the values of the new constants in CR(s) would also depend on the measured or calculated closed-loop system model that will dictate the type of FO controller to be used whether it is an FO PIȜ or PIȜDį controller. Methods presented in [10-13, 16-18] can be used in such cases. In this paper, we extend the work [15] and present an application of the retuning method for the existing laboratory feedback control system with FO plant and IO controller using additional FO controller CR(s) with no modifications in the internal architecture of the original feedback control system. Also presented, along with the mathematical description, are simulation results (in Matlab and in Micro-Cap 9 software). II.

DEFINITION OF THE FO CONTROL SYSTEM

We consider a simple unity feedback control system shown in Fig.1 where P(s) denotes the transfer function of the controlled system (plant) which is either IO type PIO(s) or more generally FO type PFO(s) and C(s) is the transfer function of the controller, also either IO type CIO(s) or FO type CFO(s).

Closed-Loop System R(s)

+

E(s)

D(s) U(s)

C(s)

Y(s) P(s)

-

Figure 1. Simple unity feedback control system

Y(s) denotes the output of the controlled system and U(s) its input. R(s) is desired value of the output of the system and E(s) is the error or deviation between R(s) and Y(s). We could consider also disturbances D(s) at the input and output of the plant. A. Definition of the Model of the Plant and Conteoller The basic mathematical models of the FO plant (1), (2) and FO controllers (3), (4) are FO differential equations in the time domain (1), (3) and FO Laplace transfer functions in s domain (2), (4). Frequently, as a model of the plant, is the three-term FODE or corresponding Laplace transfer function: a2 y (D ) (t )  a1 y ( E ) (t )  a0 y(t ) u(t )   PFO ( s)

1   a2 sD  a1s E  a0

where D, E are generally real numbers, a2, a1, a0 are arbitrary constants, y(t) denotes the output of the controlled system and u(t) its input. Similarly the FO PIODP controller can be described by the FO integro-differential equation (FOIDE) or the following FO Laplace transfer function [7]: O







O

P

 

P

 



 

where K is a proportional constant, Ti is an integration constant, Td is a derivation constant, O is an integral order and P is a derivation order. B. Considered Parameters of the Plant and of the Conteoller In work [3] were published results of earlier commonly used process of identification of the FO plant as an IO system. To the considered FO plant with the parameters a2 = 0.8, a1 = 0.5, a0 = 1, D= 2.2, E = 0.9 were obtained the following parameters values a2 = 0.7414, a1 = 0.2313, a0 = 1, D= 2, E = 1 of the corresponding IO model. For the obtained IO model of the plant, as an approximation to the FO system, was designed IO PD controller so that a unit step at the input of the closed regulation system in Fig.1 would induce at the output an oscillatory unit step response with stability measure equal to 2 and damping measure equal to 0.4. The obtained IO PD controller parameters have the following values K = 20.5, Td = 2.7343, P=1. C. Definition of the Laboratory Plant and used Conteoller Solution of many practical problems, such as the study of FO dynamical systems, FO controllers (FOC), can be simplified with the help of electrical or electronic realization of the plant, or so called analog electronic models utilizing so called FO element (FOE) or constant phase element (CPE)

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based on various principles [20-28]. With this purpose was created an electronic realization of the real working FO control system with the above described mathematical models of the plant and controller [24]. The electronic realization is in work [24] based on the equivalence of the Laplace transfer functions of active electronic circuits shown in Fig. 2 and Fig. 3 to the Laplace transfer functions of the mathematical models of the plant (1), (2) and controller (3), (4) - both IO or FO. FOE1 , D1sD1

Fig.4), and with another decrease of Td (Td < 1.1) it would be unstable, whereas the system, case B, would be stable. It can also be well seen on the changes of the phase margin in Fig. 4.

1.6

A

1.4

FOE3 , D3sD3

FOE2 , D2sD2

Step responses - Matlab

1.8

B

1.2

R2

R4 Uin

R3

OPA1

OPA3

OPA2

C

1

y(t)

R1

0.8

Uout

R5

0.6 0.4

A=-1

OPA4 0.2 0

0

1

2

Figure 2. Diagram of the electronic realization [24] of the plant (1), (2)

OPA5 FOE 4

D4s

OPA6

R10 OPA8

OPA7

D4

Figure 3. Diagram of the electronic realization [24] of the controller (3), (4)

40

-20 -40

A

0

B

-45 -90 -135

C 1

10 Frequency (rad/sec)

Magnitude (dB)

10

2

Bode Diagram

50

open-loop system 0

-50 0

Phase (deg)

It follows from the simulations (Fig. 4) that the dynamic properties of the closed regulation system with the FO model and an IO controller designed on the IO approximation of the FO plant, are considerably worse (step response A and Bode diagram A in Fig. 4) than with the IO model of the FO system and the same IO controller (step response B and Bode diagram B in Fig. 4). The expected overshoot, case B, is in case A greater by 18%. The system, case A, stabilizes later and is much more sensitive to changes in parameters. For example, at the change of Td (or also coefficient a2) to value Td =1.2 the system, case A, is just near the border of stability (course C in

0

-180 0 10

D. Analyzis of the Control System Properties We have analyzed two closed-loop control systems in time and in frequency domain. The IO controller designed on the IO model of the plant, as an approximation to the FO system, was applied first on the IO model of the controlled object and then on the FO model of the controlled object. The parameters values of the plants models and controllers used in the analysis are given in chapter II.B. The simulation results, step responses and Bode diagrams, are depictured in Fig. 4.

5

closed-loop system

20

-60 45

U

Phase (deg)

R8 C1

R11

R9

R7

R6

E

Magnitude (dB)

R12

4

Bode Diagram

40

C2

3

Time (sec)

-45

A B

-90 -135 -180 -225 0 10

C 1

10 Frequency (rad/sec)

10

2

Figure 4. Step responses and Bode plots of the closed- and open-loop system

2013 14th International Carpathian Control Conference (ICCC)

Hence disregarding the FO of the original controlled system, its approximation by a nearby IO system and an application of a controller designed for the approximating IO system to the original FO system is not adequate for many cases. Moreover, with the FO controllers, it is possible to consider also the real order of the dynamical systems and accordingly the retuning heuristic [15] to consider also more quality criteria of control performance while designing the FO controllers [11, 12, 13], including e.g. the gain crossover frequency, gain margin, phase margin, robustness to system gain variations, etc. III.



Original Closed-Loop System

D(s)

+

+

Y(s) C(s)

+

P(s)

-

\



 

R(s) +

E(s)

D(s) Y(s)

U(s)

CR(s)+1

C(s)

P(s)

-

Figure 6. Equivalent retuned control system

IV.

DETERMINATION OF THE ADDITIONAL CONTROLLER

So if we want indirectly replace the original (initial) IO or FO controller C(s) in original control system (Fig. 1) with the better (known or just designed) IO or FO controller C*(s) - (7), (8), etc. - without changing the original closed-loop system, and also without changing the structure and parameters of original controller C(s), we can do it through the architecture depictured in Fig. 5 and one additional PIȜ, PDį, or PIȜDį controller CR(s). We can replace IO (Ȝ, ȝ  Z) or FO (Ȝ, ȝ  R) controller C(s) with the FO (ij, ȥ  R) controller C*(s), of the following structures:

CR(s) +



Equivalent Controller C*(s)

RETUNING ARCHITECTURE OF THE CONTROL SYSTEM

The new system architecture including the original closedloop system, one additional controller CR(s) and utilizing the existing reference and output signals is depictured in Fig. 5.

M

where K*, Ti*, Td* > 0 and 0 < ij, ȥ < 2 are assumed without loss of generality. Since the objective is to have a FO PIȜDį controller, it is import that satisfy the conditions 0 < ij, ȥ < 2. From work [3] we have the new FO PD controller parameters K*=20.5, Td*=3.7343, ȥ*=1.15 for the later example.

We can improve existing original closed-loop system and consider more quality criteria without modifications in the internal architecture of the original system (Fig. 1) using the method of retuning PI/PID controllers [14-15] based on closedloop model and using only one additional IO (Ȗ, į  Z) or FO (Ȗ, į  R) controller CR(s).

R(s)



E(s) -

1.) initial PIȜ with the new PIij or PIijDȥ controller, 2.) initial PDP with the new PDȥ or PIijDȥ controller or 3.) initial PIȜDP with the new PIijDȥ controller.

Figure 5. System architecture of the retuned original control system

The new additional controller CR(s) measures the input reference signal R(s) and the output signal Y(s) in determining the error signal E(s). This error is then processed by the new controller and is fed as part of the new reference signal to the original closed-loop system. The new controller CR(s) does not modify the internal architecture of the original feedback control system and only the reference signal to the original closed-loop system is manipulated. Using simple block diagram algebra, incorporating CR(s) into the system would result in an equivalent unity-feedback system depicted in Fig. 6. Given the original specifications of C(s), the objective is to determine the structure and appropriate parameters of CR(s) to be able to improve existing original closed-loop system. The new equivalent feedback control system controller C*(s) in Fig. 6 is then a function of the original classical PID controller and new controller CR(s). The models of the equivalent controllers are also represented as 



\



 

The structure and parameters of the original controller C(s) are known and fixed. We must derive the adequate mathematical formulation of the additional controller CR(s) in the structure architecture in Fig. 5 and to determine the parameters values of the controller CR(s) from the parameters of the new, known or designed, controller parameters C*(s) and original parameters of C(s). The base equations for derivation are eqs. (4), (8), (9) 

C * ( s)

CR (s)  1 C (s),

C R (s)

C * ( s) 1.  C ( s)

 

If we want for the most complex case 3 replace the original PIȜDP controller C(s) with the new PIijDȥ controller C*(s) the formula for the additional IO or FO controller CR(s) is 

C R ( s)

Ti* s O M  Td* s O \  Td s O  P  ( K *  K ) s O  Ti    , ( Ks O  Ti  Td s O  P )

where K, Ti, Td, K*, Ti*, Td* > 0 and 0 < Ȝ, ȝ < 2, 0 < ij, ȥ < 2 are assumed without loss of generality. Another particular

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types of replacing and the adequate formulas for the additional CR(s) controller together with the proofs we can find in [15]. V.

NUMERICAL EXAMPLE

Consider the original closed-loop system (Fig. 1) with the FO plant (2) and initial IO PD controller. Electronic realization [24] of such closed-loop system with diagrams depicted in Fig. 2, 3 represents our laboratory object with the parameters values of the plant and controller given in chapter II.B. The step response and Bode plots of such system are depicted in Fig. 4. In consequence of rather bad dynamic properties of the closed regulation system (chapter II.D) we want to improve them in such a way as if we replace the initial IO controller C(s) with better FO controller C*(s). In work [3] were published the parameters of one of the simplest FO controller with the following parameters values K* = 20.5, Td* = 3.7343, ȥ*=1.15. In case of replacement the original IO PD controller with the new FO PDȥ controller the equation (10) has form 

CR ( s)

Td* s\  Td s  ( K *  K )    ( K  Td s)

From the comparison of the in Matlab and in Micro-Cap 9 software simulated step responses and Bode plots (Fig. 7, 8) of the laboratory control system with controller CR(s) we can see rather good qualitative and quantitative agreement. The dynamical properties of the FO controller are better comparing to original IO controller.

 

where K, Td, K*, Td* and ȥ are known parameters of C(s) and C*(s) controllers introduced in the previous parts. Step response - Matlab

1.6 1.4 1.2

A

FO controller, FO plant

B

IO controller, FO plant

closed-loop system

y(t)

1 0.8 0.6 0.4 0.2 0

0

1

2

3

4

5

Time (sec)

Magnitude (dB)

10

Figure 8. Step responses and Bode plots of the closed-loop system ASI mer.

Bode Diagram A

0

B

-10

VI.

-20 -30

Phase (deg)

-40 0 -45 -90 -135 0 10

1

10 Frequency (rad/sec)

10

2

Figure 7. Step responses and Bode plots of the initial and retuned system

42

CONCLUSIONS

In this paper we have describe, extend and verified the retuning method on the existing laboratory feedback control system with IO or FO plant and initial IO or FO controller using new additional FO controller CR(s) with no modifications in the internal architecture of the original feedback control system. The simulation results confirmed that the method is suitable for such purposes in cases if we do not have possibility to directly replace the initial controller and we decided to do it through this additional controller in retuned original control system. One of the reasons of such procedure is the ability to make the system robust having constant overshoots or to consider another quality criteria of control performance. Measurements on laboratory object are under preparation.

2013 14th International Carpathian Control Conference (ICCC)

ACKNOWLEDGMENT This work was partially supported by grants VEGA 1/0746/11, 1/0729/12, 1/0497/11, 1/2578/12, and APVV-048211 from the Slovak Grant Agency, the Slovak Research and Development Agency, and by the Center for Automation Research under the College of Computer Studies in De La Salle University Manila. REFERENCES [1]

[2]

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[8]

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