SLOVENSKÁ TECHNICKÁ UNIVERZITA V BRATISLAVE FAKULTA ELEKTROTECHNIKY A INFORMATIKY NÁVRH PID REGULÁTORA PRE OBJEKT UDAQ.

Transcription

1 SLOVENSÁ TECHNICÁ UNIVERZITA V BRATISLAVE FAULTA ELETROTECHNIY A INFORMATIY NÁVRH PID REGULÁTORA PRE OBJET UDAQ Bakalárska práca Evidenčné číslo: FEI /011 Chu Duc Tung Son

2 NÁVRH PID REGULÁTORA PRE OBJET UDAQ Bakalárska práca Evidenčné číslo: FEI Študijný program: Priemyselná informatika Študijný odbor: Automatizácia, 9..9 Aplikovaná informatika Školiace pracovisko: ÚRPI FEI STU Školiteľ: prof. Ing. Vojtech Veselý, DrSc. onzultant: prof. Ing. Vojtech Veselý, DrSc. Bratislava 010/1011 Chu Duc Tung Son

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4 ANOTÁCIA BAALÁRSEJ PRÁCE Slovenská technická univerzita v Bratislave FAULTA ELETROTECHNIY A INFORMATIY Študijný program: Priemyselná informatika Autor: Chu Duc Tung Son (56070) Bakalárska práca: Návrh PID regulátora pre objekt UDAQ Vedúci bakalárskej práce: prof. Ing. Vojtech Veselý, DrSc. Mesiac, rok odovzdania: Máj, 011 ľúčové slová: PID regulátor, fázová bezpečnosť, amplitúdová bezpečnosť, maximálne preregulovanie, Bode diagram, Hallov fázový kruhový diagram. PID regulátor, ktorý sa skladá z proporcionálnej, integračnej a derivačnej zložky, je široko používaný v spätnoväzbových regulačných obvodoch priemyselných procesov. Pre návrh PID regulátora, parametre PID sú nastavené akoukoľvek konvenčnou metódou, aby sa zabezpečil správny referenčný signál, na uzavretý obvod sa získa filtráciou vhodne žiadaný krok signálu. V tomto článku, parametre PID regulátora pre objekt UDAQ sú návrhnuté pomocou LAFFCH a inžinierskou metódou. Prvým krokom v procese návrhu je identifikácia modelu pre obiekt UDAQ. Toto je dosiahnuté použitím krokov vstupu do skutočného procesu (udaq8/lt) a zapisovaním dát do Matlabu. Tieto dáta sú využívané na identifikáciu vhodného modelu pre proces. Parametre PID regulátora sú potom získané špecifický. Maximálne preregulovanie systému by malo byť približne 10%. Výsledky budú overené na reálnom objekte.

5 BACHELOR THESIS ABSTRACT Slovak University of Technology in Bratislava FACULTY OF ELECTRICAL ENGINEERIG AND INFORMATION TECHNOLOGY Study Programme: Industrial Informatics Author: Chu Duc Tung Son (56070) Bachelor Thesis: PID controller design for object UDAQ Supervisor: prof. Ing. Vojtech Veselý, DrSc Year, month: 011, May eywords: PID controller, phase margin, gain margin, maximum overshoot, Bode diagram, Hall diagram. The PID controller, which consists of proportional, integral and derivative elements, is widely used in feedback control of industrial processes. For controller tuning, the PID parameters are tuned by any conventional method in order to assure a good reference signal to the closed loop system is obtained by filtering appropriately the set-point step signal. In this paper, the parameters of PID controller for object UDAQ are tuned in the frequencydomain and using Ziegler-Nichols method. The first step in the tuning process is to identify a model for the object. This is achieved by applying steps input to the real process (udaq8/ LT) and logging the data to Matlab. This data is used to identify an appropriate model for the process. The PID controller parameters are then obtained with specification is maximum overshoot of the system should be approximately 10%. The results will be verified in real object.

6 Acknowledgements I hereby state that I have written this thesis to the best of my knowledge with the help of supervisor and referenced literatures. I would like to thank Professor Vojtech Veselý for his regular consultations and useful advices. Bratislava Chu Duc Tung Son

7 CONTENT 1 Introduction Introduction Plant... 1 Basic Concepts of Feedback Control Feedback Control System Step Response of a System PID Controller The Algorithm Proportional Action Integral Action Derivative Action Identification Real Process for Regulating Temperature and Light Intensity in Object UDAQ Design and Implementation Identification Design of PID controller parameters Frequency-Domain Design of Control Systems Properties of PI controller and PD controller PI Controller PD Controller Design PID controller for a system Design PID Controller for ARMAX Model for Regulating Temperature Design PID Controller for ARX Model for Regulating Light Intensity The PID Controller Design Using Modification of Ziegler - Nichols method The First Ziegler - Nichols method The Second Ziegler Nichols method Modification of Ziegler - Nichols method Design PID Controllers for ARMAX Model for Regulating Temperature and ARX Model for Regulating Light intensity Design PID Controller for ARMAX Model for Regulating Temperature... 5

8 4..4. Design PID Controller for ARX Model for Regulating Light Intensity 8 5 Controller Implementation in Real Object UDAQ Conclusion References.34 Appendix

9 1 Introduction 1.1 Introduction The PID controller is the most form of feedback in use today. According to an estimate nearly 90% of the controllers used in industries are PID controllers. The family of PID controllers is rightly known as the building blocks of control theory owing to their simplicity and ease of implementation [1]. Designing and tuning a PID controller appears to be conceptually intuitive, but can to be hard in practice, if multiple (and often conflicting) objectives are to be achieved. A conventional PID controller with fixed parameters may usually derive poor control performance when it comes to system complexities. Since the gain and the time constants of the system change with the operating conditions the conventional PID controllers result in sub-optimal corrective actions and, hence, require frequent tuning adjustments. This stimulates the development of tools that can assist engineers to achieve the best overall PID control for the entire operating envelope of a given process [, 3]. Over the past 60 years, several methods for determining PID controller parameters have been developed. The basic idea in using controllers is to shape and mould the Bode diagram of the open-loop system in such a way that the given specifications are met. When Bode diagrams are used in design, the specifications have to be transformed into phase and gain margins and usually the steady-state error is also given (here also so-called error coefficients are used). In this presentation we concentrate on Bode diagram design exclusively. Other approaches that could be used are Nyquist diagrams, Nichols diagrams and root locus. Essentially the same principle applies: the given representations are shaped so that the given requirements are met. Lately also robustness issues are taken into account in a more careful manner. In 194 Ziegler and Nichols, both employees of Taylor Instruments, described simple mathematical procedures, the first and second methods respectively, for tuning PID controllers. These procedures are now accepted as standard in control systems practice. Both techniques make a priori assumptions on the system model, but do not require that these models be specifically known. Ziegler-Nichols formulae for specifying the controllers are based on plant step responses [4]. 1. Plant Thermo-optical plant udaq8/lt (Fig. 1) is multivariable system with three manipulated inputs and seven measurable outputs (Fig. ). System has three manipulated inputs: bulb voltage (0-5V) which represents heater and light source, fan voltage (0-5V) which can be used for temperature decreasing and voltage of led diode (0-5V) which represents another source of 1

10 light. Except of these 3 input voltages there exist still two parameters inputs for adjusting the sampling period and the time constant of the build in derivative filter. On the output is possible to measure seven variables: temperature insight the system (direct or filtrated), outsight temperature, light intensity (direct or filtrated), fan velocity and fan current [5]. Fig.1 Thermal plantudaq8/lt Fig. Basic electric diagram of thermo-optical plant udaq8/lt The advantage is that the whole system can communicate with a computer via USB interface. No special card is required. Since USB port can be usually found on all today s computers including notebooks, the introduced equipment can be easily wide-spread. The communication with the computer runs over the string exchange. The data transfer rate is 50kbit/s. The plant is shipped with the driver for Matlab/Simulink environment and therefore controls algorithms can be easily set via this programming tool. The next benefit consists in the fact that thanks to the plant construction the Matlab simulation scheme doesn t need to be compiled. It enables that the control algorithm can also be written using a common Matlab

11 Fcn block that can be found inside of Simulink. The plant enables to use a sampling period ms whereby considering the dynamics of the presented system 1 second should be sufficient for its quasi-continuous control Even though the user can measure 7 variables, we usually do not use all of them. They control the temperature inside the plastic cylinder that is influenced by the bulb heating and the ventilator cooling. In spite of the fact that both these variables can be used for control in the same time together, the plant is very often controlled only by the bulb voltage whereby the ventilator is considered as a disturbance factor. Another possibility is to control the light intensity that can be influenced by the voltage on the light diode. As it is possible to see the introduced plant offers a big variability of experiments that can be accomplished. In addition, except of control we have to solve tasks that are connected with plant identification, input-output data manipulation and communication with outer computer environment [6]. 3

12 Basic Concepts of Feedback Control.1 Feedback Control System A controller makes the output y behave in a desired way by manipulating the input u. Control systems can be classified into two types: open-loop systems and closed-loop (feedback) systems. The controller of an open-loop system (Fig. 3) determines how to manipulate the input only using the information of the desired output, which is called reference, r. So this controller relies on system calibration. [7] Fig.3 Open-loop control system The controller of a feedback system, however, not only takes advantage of the desired output but also of the measurement of the output (Fig. 4). A feedback controller calculates the control effort, u, from the error e : r y. Fig.4 Closed-loop control system Feedback control is widely used because it: 1. Increases the accuracy of reference following.. Increases the stability of the system, or even stabilizes an originally unstable system. 3. Reduces the sensitivity of the system performance to system parameter variations. 4. Increases the ability of the system to reject external disturbances and noises. In summary, feedback control can change the dynamics of closed-loop control system. [7]. Step Response of a System A (unit) step signal is defined as: 4

13 0, t < 0, u ( t) (.1) 1, t 0. Fig.5 shows a typical step response: Fig.5 Typical step response of a control system There are several interesting characteristics of a system that can be derived from its step response, namely steady-state error e ss, maximum over-shoot, and setting time t s, among others: Steady-state error The steady-state value of the step response y(t) is defined as: y ss lim y( t) (.) t For a servo control system, we always want to the output, y(t), to follow a desired reference signal, r(t). Thus we can define the error as e( t) r( t) y( t) (.3) And consequently, the steady-state error is given by e ss lime( t) (.4) t Maximum overshoot: let y max denote the maximum value of y(t). The maximum overshoot of the step response y(t) is defined as Maximum overshoot y y max ss (.5) 5

14 The maximum overshoot is often represented as a percentage of the steady-state value, i.e. Maximum overshoot Percent maximum overshoot.100% y ss (.6) The maximum overshoot is often used to measure the relative stability of a system. A system with a large overshoot is usually undesirable. [7] Setting time: the setting time t s is defined as the time required for the response curve to reach and stay within a range of certain percentage (usually 5% or %) of the final value..3 PID Controller.3.1 The Algorithm Among all kinds of feedback controllers, PID (Proportional-Integral- Derivative) controllers are the most widely used, due to their simplicity and effectiveness. The control output (effort) of a PID controller is given by [8] u ( t) k e( t) + e( ) P t dτ + I 0 de dt ( t) τ D (.7) Where P, I and D are the proportional, integral, and derivative gains, respectively, and e(t) is the error signal already defined in expression (.3)..3. Proportional Action The proportional action causes an instantaneous response on the control effort due to error change. The larger the proportional gain, P, the larger the control effort caused by a given error value. This usually improves the response speed of the system, i.e. smaller rise time, and reduces the steady-state error. However, a large proportional gain also increases the overshoot. In fact, making the gain too large might even cause instability in the system. This means that the step response grows unbounded or oscillates sustainedly..3.3 Integral Action Even though the proportional action itself cannot eliminate the steady-state error, the integral action can achieve this goal with the proper selection of integral gain,. It does so by accumulating the error over time and increasing the control I 6

15 output in the direction that reduces the error, whenever error exists. However, the integral action can be destabilizing; it needs time to accumulate the error and might act at the wrong time, causing oscillation..3.4 Derivative Action The derivative action responds to the rate of change of the error. It can be viewed as additional damping to the system. It reduces overshoots and slows down the initial response of the system. The derivative action reduces the effect of the integral action on overshoots and oscillations. Even though it is a stabilizing action, the derivative action has no effect on steady-state error. 7

16 3 Identification Real Process for Regulating Temperature and Light Intensity in Object UDAQ 3.1 Design and Implementation The goal of the identification is research mathematical of a real plant. We created an identification scheme in Matlab environment for the purpose of identification real process. The Fig.6 shows a simulative scheme for obtaining the needed data for identification. We set the mean value bulb in identifying. We realize step bulb voltage by using blocks step1,, and 3 in the vicinity of the mean value. Sine Wave Sine Wave 1 Sine Wave Switch Sine Wave 3 Add Sine Wave 4 Ramp 0 Step 3 Constant Step control teplota Step 1 Temperature Step 5 Saturation Mux Temp intenzita 3.5 Bulb Fan Led Saturation 1 Saturation Ts/0 Optical -Thermal Plant I/O Interface Mux Light int. Light intensity prud _ventilatora Ts_deriv Clock cas.9 D_filt Fan rpm Fig.6 Simulative scheme for the purpose of identification real process We implement measurement to identify by simulative scheme Fan current a) Let s a constant and series of blocks Step to the input of process. b) Let s series of blocks Sine Wave to the input of process. We set value of amplitude and frequency in blocks Sine Wave c) We set a constant control for working point into constant. d) We set final value and step time in blocks Step. 8

17 3. Identification When the measurements are finished, the data measured are saved in workspace (teplota, intenzita, prud_vzduchu, cas) and consequently, it is allowed to display data and to identify real process. We can create transfer function by running program IdentUDAG_intenzita.m. Realize selection of identification method (by ARX or ARMAX), and system order (-4 order) by click on button in screen. In this experiment, a system model for regulating temperature is identified using data collected as can be seen in Fig.7 Fig.7 Input and output signals Identification results real process for regulating temperature using ARMAX method and second-order is: ( s G P 1 s) (3.1) s s

18 And comparison of time response of process output and ARMAX model output is in Fig process output: temperature model output: ARMAX 40. y [rad/s] t [s] Fig.8 Comparison of time response of process output and ARMAX model output A system model for regulating light intensity is identified using data collected as can be seen in Fig.9 Fig.9 Input and output signals 10

19 Identification results real process for regulating light intensity using ARX method and second-order is: (.849s G P s) (3.) s s + 11 And comparison of time response of process output and ARX model output is in Fig process output: light intensity model output: ARX 7 y [rad/s] t [s] Fig.10 Comparison of time response of process output and ARX model output 11

20 4 Design of PID controller parameters 4.1 Frequency-Domain Design of Control Systems The design of feedback control systems in industry using frequencyresponse methods is more popular than any other. This is primarily because the frequency response method provides good designs in the face of uncertainty in the plant model and can easily use experimental information for design purposes. Classical control focuses on the frequency domain properties of control systems and has developed design methods based on simple but powerful graphical tools such as the Nyquist plot, Bode plots, and Nichols Chart. These techniques are well known and are popular with practicing engineers. The PID controller combines the advantages of the derivative and integral controls. We can make use of the controller equation established as follows: I I Gc ( s) P + Ds + ( 1 + D1s) P + (4.1) s s The relationships between P, D, and I, and D1, P,and I are given in expressions (4.) through (4.4) + (4.) P D D1 I P (4.3) D1 P I I (4.4) 1

21 4.1.1 Properties of PI controller and PD controller PI Controller Fig.11 Bode diagram of PI controller From Bode diagram of PI controller, we have: P I ω c π tan ϕ I (4.5) PI [ DB] 0log P 1+ I P 1 ωc (4.6) 13

22 PD Controller Fig.1 Bode diagram of PD controller From Bode diagram of PI controller, we have: D, c ( ϕ ) ω ' tan (4.7) D, [ ] ( ) DB 0log 1+ D c PD ω (4.8) 4.1. Design PID controller for a system We design PID controller whose PI component and PD component will add phases ϕ I, ϕ D to original system. The frequency domain design procedure PID compensation to realize a given maximum overshoot X max is as follows: G P of the uncompensated system is made with the open-loop gain set according to the steady-state performance requirement. Step 1: The Bode plot of open-loop transfer function ( s) Step : By using expressions (4.9), (4.10) and (4.11) we determine a phase margin ϕ0 of the system with PID controller. 14

23 ( ω ) 100 [%] ( ω ) 184 [%] X max 0.9ϕ 0 0 X max 30 (4.9) X.51ϕ 30 X max 70 (4.10) max 1 0 In which: ϕ ( ω 0 ) is all phase of system. ( ) ϕ0 180 ϕ ω 0 (4.11) Step 3: By using expression (4.1) we determine a new gain-crossover frequencyω 0. For a specified frequencyω 0, the new phase margin ϕ corresponding to this frequency is found on the Bode-diagram. π t π ω 4 0 (4.1) s t s In which: t s is setting time. Step 4: For a specified phase margin ϕc ϕ0 + ϕcontroller. If ϕ c ϕ then D1 0. Else we have: ϕ D ϕc ϕ (4.13) From expression (4.7): D 1 ( ) ϕ D tan (4.14) ω 0 Step 5: Draw Bode-diagram of G p ( s)( + Ds) ( jω)( jω) 1. The gain G m of G p 1+ D. corresponding to frequency ω0 is found on the Bode-diagram. The magnitude plot of the compensated transfer function must pass through the 0 db at this gain-crossover frequency ω 0. Thus, the PI controller must provide an attenuation of G m at ω0 in order to bring the magnitude curve up to 0 db at this frequency and we have ϕ ϕ I controller Step 6: According to step 5 and expressions (4.5) and (4.6) we have: P I ω 0 π tan ϕ I (4.15) I I 1 0log P Gm 1+ P (4.16) ω0 From expressions (4.15) and (4.16) we can determine values of P and 15

24 Step 7: the Bode plot of the compensated system is investigated to see if the performance specifications are met Design PID Controller for ARMAX Model for Regulating Temperature Controller parameters design on the basis of transfer function obtained from identification. Consider the following second-order process: s s) s s G ( P 1 The specification of the system is the follow 1. The maximum overshoot of the system should be approximately 10 With the maximum overshoot is 10% we can choose the phase margin of the system is approximately is By using expression (4.1) with the setting time ω 0.4rad / 0 s t s 1s, we choose We can choose ϕ controller 30 the specified phase margin ϕ ϕ + ϕ (4.17) c 0 controller The Bode diagram of G P1 ( jω) is shown in Fig.13. From the Fig.13, at frequency ω 0.4rad / 0 s the phase is ϕ ϕ ϕ ϕ.09 ω tan(0.09 ) (4.18) G 1 D c ( s) G ( s) G ( s) 0 D1 0 D1 ( s )( s) P 1 D1 (4.19) s s Fig.13 shows the Bode diagram of ( jω). The PI controller must reduce the magnitude of G 1 ( jω) to 0 db at ω 0.4rad / 0 s. From the magnitude plot of G 1 ( jω), the value of G 1 ( jω) at ω 0.4rad / 0 s is db. Thus, the PI controller must provide an attenuation of db at ω 0.4rad / 0 s in order to bring the magnitude curve up to 0 db at this frequency and we have ϕ ϕ 30. G 1 I controller 0log I I P + P (4.0) ω0 ω0 16

25 P I P ( ) ω 0 tan (4.1) I From expressions (4.0) and (4.1) we have 5. P 168 and 1. I 193. By using expressions (4.) through (4.4), the parameters of the PID controller are determined to be 5. 17, , and The transfer function of the PID controller is thus P D C 1 ( s) s (4.) s G T The open-loop transfer function of the PID-compensated system is: s G( s) s (4.3) s s s The Bode diagram of the compensated G ( jω) is shown in Fig.13. The phase margin of the PID-compensated system is actually I 61. The unit-step response of the compensated system is shown in Fig.14. The maximum overshoot in this case is 13.5%. 17

26 Fig.13 Bode diagram of ( s) s G P 1 with PD and PID s s compensations 1. Step Response Amplitude Time (sec) Fig.14 Unit-step response of the compensated system Design PID Controller for ARX Model for Regulating Light Intensity Consider the following second-order process: G P.849s s s + 11 The specification of the system is the follow: 1. The maximum overshoot of the system should be approximately 10 With the maximum overshoot is 10%, we can choose the phase margin of the system is approximately is By using expression (4.1) with the setting time ω 0 10rad / s t s 0. 5s, we choose We can choose ϕ controller 31 the specified phase margin ϕ ϕ + ϕ c 0 controller 18

27 The Bode diagram of G P ( jω) is shown in Fig.15. From the Fig.15, at frequency ω 0 10 rad / s the phase is ϕ 9.38 Thus, we have: ϕ ϕ 0 c D1 The PI controller must reduce the magnitude of G P ( jω) to 0 db at ω s. From the magnitude plot of G ( jω P ), the value of ( jω) 10rad / 0 G P at ω 10rad / 0 s is 14.4 db. Thus, the PI controller must provide an attenuation of db at ω 10rad / 0 s in order to bring the magnitude curve up to 0 db at this frequency and we have ϕ I ϕ 31 controller 0log I I P + P (4.4) ω0 ω0 P I P ( ) ω 0 tan (4.5) From expressions (4.4) and (4.5) the parameters of the PI controller are determined to be: 0. P 161 and I The transfer function of the PID controller is thus I C 1 ( s) (4.6) s G L The open-loop transfer function of the PI-compensated system is: G.849s (4.7) s s + 11 s ( s) The Bode diagram of the compensated G ( jω) is shown in Fig.15. The phase margin of the PI-compensated system is actually 61. The unit-step response of the compensated system is shown in Fig.16. The maximum overshoot in this case is 4.46%. 19

28 Fig.15 Bode diagram of ( s) G P 1.849s with PI compensations s s Step Response 1. 1 Amplitude Time (sec) Fig.16 Unit-step response of the compensated system 0

29 4. The PID Controller Design Using Modification of Ziegler - Nichols method 4..1 The First Ziegler - Nichols method One tuning method is based on process information in the form of the openloop step response obtained from a bump test. This method can be viewed as a traditional method based on modelling and control where a very simple process model is used. The step response is characterized by only two parameters, L the delay time and T the time constant, as shown in Fig.17 Fig.17 Characterization of a step response in the first Ziegler-Nichols method Using the parameters L and T, we can set the values of, T i and T d according to the formula shown in the tab.1 [9] Type of controller P PI PID T i T d T 0 L T L T L L L 0.5L Tab.1 Ziegler-Nichols recipe first method Theses parameters will typically give us a response with an overshoot about 5% and goog setting time. We may then start fine-tuning the controller using the basic rules that relate each parameter to the response characteristics. 1

30 4.. The Second Ziegler Nichols method The second method targets plants that can be rendered unstable under proportional control. The technique is designed to result in a closed loop system with 5% overshoot. This is rarely achieved as Ziegler and Nichols determined the adjustments based on a specific plant model. The steps for tuning a PID controller via second method are as follows: Using only proportional feedback control: 1. Reduce the integrator and derivative gains to 0.. Increase k p from 0 to some critical value k p kcr at which sustained oscillations occur. If it does not occur then another method has to be applied. 3. Note the value k cr and the corresponding period of sustained oscillation, T cr The controller gains are now specified in the tab. [9] Type of controller k T i T d P 0.5k cr 0 PI 0.45k cr 1 0 T cr 1. PID 0.6k cr.5tcr Tcr Tab. Ziegler-Nichols recipe second method 4..3 Modification of Ziegler - Nichols method A PID controller with the transfer function: G c 1 1 (4.8) ( s) k + + T s d Ti s Is employed to control the process. Hence, G c 1 1 (4.9) ( jω) k + + T jω d Ti jω

31 With PID controller it is possible to move the critical point C [ 1, j0] Nyquist curve to the point A [ x, jy] as is indicated in Fig.18 on the Fig.18 Compensation critical point C using controller PID In which: A is the intersection of circle ψ and straight line ζ As mentioned above for the PID controller, the essence of the method is based in the shift frequency characteristics of the open-loop control system with critical point C in A x, jy. On the border of stability is valid [10]: ω to point [ ] cr 1 G( jω cr ) kcr 1 G( jωcr ) (4.30) kcr Or with PID controller 1 cr G c ( jω ) x jy cr (4.31) G c k T T ω 1 cr d (4.3) jω T T ω d i cr ( jω ) k + + jkω T k + j k cr Substitute into the characteristic polynomial: cr i i cr 3

32 k k cr 1+ TdTiω cr 1 j x T iωcr jy (4.33) We get: k kcr x (4.34) k k cr TdTiω cr 1 y T ω i cr (4.35) From expression (4.34) we can determine k We can assume that: Ti T d (4.36) 16 Then we get a quadratic equation in the form: 16. y z z 16 0 (4.37) x π In which: z Tiω cr, ωcr (4.38) Tcr From expressions (4.36), (4.37) and (4.38) we can determine T i and T d Design PID Controllers for ARMAX Model for Regulating Temperature and ARX Model for Regulating Light intensity Design a PID controller to make the maximum overshoot of the system to be approximately 0% or less. Solution: We start design the PID controller by applying modification of Ziegler-Nichols. With the maximum overshoot is 0%, we can choose the phase margin of the system is approximately is 50 P M rad. P M 1 1 arcsin M [11] (4.39) T M T By using expression (4.39) with phase margin determine M T With M 1. 19, we get Hall circular diagram: T P M rad, we can 4

33 U + V (4.40) Thus, we have to move the critical point C [ 1, j0] the point A [ 0.97, j0.456] as is indicated in Fig.19 on the Nyquist curve to Fig.19 Compensation critical point C using controller PID From expressions (4.34) (4.36), (4.37) and (4.38), we can set the values of k, T i and T d according to the formula shown in the tab.3 Type of controller PID k T i 0.97k cr.0tcr T d T cr Tab Design PID Controller for ARMAX Model for Regulating Temperature The model for regulating temperature is: G P 1 ( s) s s s The open-loop transfer function with P controller is: 5

34 G 01 ( s) k k s ( s ) s ( jω ) ( ω ) jω k( 1.105ω ) ( ω ) + ( 5.437ω ) s jω G k ( jω) ( jω ) k ω jω ( jω ) [( ω ) jω] ( ω ) + ( 5.437ω ) kω( ω.05) j U + jv ( ω ) + ( 5.437ω ) + 01 V ω.05 0 ω cr 3.88 Tcr 1.6 U ( ωcr ) + ( 5.437ω cr ) ( 1.105ω ) 1 k cr cr With tab.3, we determine the parameters of the PID controller as follows: k T T i d cr 0.97k cr 4.0T 6.51 cr 0.51T The transfer function of the PID controller is thus 1 G C T ( s) s (4.41) 6.51s The open-loop transfer function with the PID controller is: G s s s P1 C T (4.4) 3 s s s ( s) G ( s) G ( s) 6

35 Fig.0 Nyquist plot of the open-loop transfer function G ( jω) G ( jω) P1 C T The unit-step response of the closed-loop system is shown in Fig.1. The maximum overshoot in this case is 4.33%. Fig.1 Unit-step response of the closed-loop system 7

36 4..4. Design PID Controller for ARX Model for Regulating Light Intensity The model for regulating light intensity is: G P.849s s s + 11 The open-loop transfer function with P controller is: G 0 ( s) k k s (.849s ) s + 11 (.849 jω ) k ( ω + 11) jω k( ω ) ( ω + 11) + ( 15.91ω ) + s jω G 0 ( jω) (.849 jω ) k ω jω + 11 (.849 jω )( ω jω) ( ω + 11) + ( 15.91ω ) kω(.849ω 13693) j U + jv ( ω + 11) + ( 15.91ω ) V 0.849ω ω cr Tcr U ( ωcr + 11) + ( 15.91ω cr ) ( ω ) 1 k cr With tab.3, we determine the parameters of the PID controller as follows: k T T i d cr 0.97k 1.65 cr 4.0T T 0.03 cr The transfer function of the PID controller is thus: 1 G C L ( s) s (4.43) 0.366s The open-loop transfer function with the PID controller is: G s s s P C L (4.44) 3 s s + 11s ( s) G ( s) G ( s) 8

37 Fig. Nyquist plot of the open-loop transfer function G ( jω) G ( jω) P C L The unit-step response of the closed-loop system is shown in Fig.3. The maximum overshoot in this case is 13.88%. Fig.3 Unit-step response of the closed-loop system 9

38 5 Controller Implementation in Real Object UDAQ In the previous section, two types of controller have been designed via simulation. However, it was not enough to ensure that all the design controllers are exactly capable to control object UDAG until it was implemented practically. The simulink block diagram of the system with PID controller is represented in Fig.4 ref_intenzita control teplota 10 w PID Temperature set point of light PID Controller 3 Bulb 0 Saturation Mux Temperat intenzita Fan Saturation 1 Led Saturation Optical-Thermal Plant I/O Interface Mux Light intensity1 Ts/0 Light int. prud _vzduchu Ts_deriv.9 D_filt Fan rpm Fan current Clock cas Fig.4 Simulink block diagram with PID controller Fig.5, Fig.6, Fig.7 and Fig.8 show steps response of the experimental system employing the controllers which are tuned in the frequency-domain and using modification of Ziegler-Nichols method. 30

39 Fig.5 Temperature process response from experiment with PID C s G T ( s) s Fig.6 Temperature process response from experiment with PID 1 G C T ( s) s 6.51s 31

40 Fig.7 Light intensity process response from experiment with PID C 1 s G L ( s) Fig.8 Light intensity process response from experiment with PID 1 G C L ( s) s 0.366s From Fig.5 and Fig.6, the processes output show high overshoot with setting time is around 1000 seconds corresponding to step input reference. From Fig.8, the process output is unstable with high amplitude. By comparing the figures, we can see that the outputs from real experiments are not similar with the output attained from simulations which produce very small steady-state error and fast response time. It indicates that the object UDAQ was not identified exactly 100%. 3

41 6 Conclusion In this report, I have presented a frequency-domain design method and a modification of Ziegler-Nichols method for tuning PID controllers. This approach was applied to the control object UDAQ. It is not possible to say exactly which method is better. The easier way to find PID parameters is using modification of Ziegler-Nichols. At the beginning of my project I focused on identification process. First at all I have input and output signals from object UDAQ. The ARX model and ARMAX model structures are used for all identification. Two approximated second-order plant models are used in the present design, since many of the industrial plants can be modelled using a second-order model. The PID controllers are developed on these models. A frequency-domain design and a modification of Ziegler-Nichols method are implemented in the models to see how the models response is. The results were good as were shown in Fig.14, Fig.16, Fig.1 and Fig.3. The overall results provide the advantages of two methods in sense of fast response, less overshoot, smaller setting time. Finally, the theoretical results were applied experimentally to a real thermooptical plant in laboratory condition. The results were illustrated in Fig.5, Fig.6, Fig.7, and Fig.8. From the figures, we can see that the experimental results are not similar with the theoretical results. It indicates that the object UDAQ was not identified exactly 100%. The future work will design PID controller directly in real plant. 33

42 References [1] Sung, S.W., Lee, I.-B. and Lee, J., 1995.Modified Proportional-Integral- Derivative (PID) Controller and a New Tuning Method for the PID Controller, Ind. Eng. Chem. Res., 34, pp [] Salami, M. and Cain, G., An Adaptive PID Controller Based on Genetic Algorithm Processor, Genetic Algorithms in Engineering Systems: Innovations and Applications, 1-14 September, Conference Publication No. 414, IEE. [3] Asriel, U.L. and Narendra,.S. January Control of Non-linear Dynamical Systems using Neural Networks-Part II: Observation, Identification and Control, IEEE Transactions on Neural Networks, Vol. 7, No. 1. [4] Brian R Copeland, M The Design of PID Controllers using Ziegler Nichols Tuning. < > [5] Osuský Jakub, M Robust SISO Control Design for Thermo-Optical plant UDAQ8/LT. < > [6] atarína Žáková, - Matej ohút, J Matlab Based Remote Control of Thermo-Optical Plant. [online]. Bratislava < [7] JongEun Choi, 005. PID Control Gain Tuning Of a Drive Position Servo System. California < > [8] Aidan O Dwyer, 006. Handbook of PID And PI Controller Tuning Rulers. London :Imperial College Press, 006, 533. ISBN [9] Bian R Copeland, M.008. The Design of PID Controller using Ziegler Nichols Tuning < [10] Vojtech Veselý, 007. Inžinierske metódy nastavovania parametrov PID regulátorov [11] Vojtech Veselý, Ladislav Harsányi. Robust control for dynamic systems 34

43 Appendixes Appendix A m-files Metoda1.m function [P,I,D]metoda1(SYS,overshoot) if overshoot>0 & overshoot<30, phase180-(100+overshoot)/0.9; elseif overshoot>30 & overshoot<70, phase180-(184+overshoot)/1.51; end ficontrollerinput('enter the phase of controller ') ficphase + ficontroller; tsinput('enter the settting time ts '); wminpi/ts wmax4*pi/ts w0input('enter a frequency in interval [wmin wmax] w0 ') [MAG,PHASE] BODE(SYS,w0); fiphase+180 if fic<fi d10; fii90-(fi-phase); atand(fii)/w0; i1/(mag*sqrt(a^+1/w0^)) pi*a; else d1tand(fic-fi)/w0 Ftf([d1 1],[1]); FF*SYS; [MAG1,fi1] BODE(F,w0); atand(90-ficontroller)/w0; i1/(mag1*sqrt(a^+1/w0^)) pi*a end Pd1*i+p; Dd1*p; Ii; Metoda.m function [P,I,D]metoda(SYS,overshoot) if overshoot>0 & overshoot<30, phase180-(100+overshoot)/0.9; elseif overshoot>30 & overshoot<70, phase180-(184+overshoot)/1.51; end Mt1/(*sind(phase/)); rmt/(1-mt^); amt^/(1-mt^); 35

44 a1(a^-r^)/a; b1-sqrt(a1*a-a1^); kb1/a1; r1abs(a-r); x-sqrt(r1^/(k^+1)); yk*x; korenroots([1-16*y/x -16]); zkoren(1); [Gm,Pm,Wcg,Wcp] MARGIN(SYS); k-gm*x; Tiz/Wcg; TdTi/16; Pk; Ik/Ti; Dk*Td; 36