Modeling, Analysis and Speed Control Design Methods of a DC Motor

Transcription

1 Design Methods of a DC Motor Dr. Jaal A. Mohaed* Received on: 3/1/009 Accepted on:5 /1/011 Abstract Modern anufacturing systes are autoated achines that perfor the required tasks. The electric otors are perhaps the ost widely used energy converters in the odern achine-tools and robots. These otors require autoatic control of their ain paraeters (position, speed, acceleration, currents). With the help of an exaple, a DC otor syste, the use of MATLAB/Siulink for coprehensive study of odeling, analysis and speed control design ethods has been deonstrated. Keywords: DC otor, open loop, closed loop, syste, speed control. نمذجة وتحليل وطرق تصميم السيطرة على سرعة محرك التيار المستمر الخالصة: أنظمة التصنيع الحديثة ھي مكائن آلي ة تنج ز المھم ات المطلوب ة. المحرك ات الكھربائي ة ربم ا تمثل محوالت الطاقة األكثر استخداما في معدات المكائن الحديثة واإلنسان اآللي. تتطلب ھذه المحركات سيطرة آلية لمعامالتھا األساس ية (ك الموقع والس رعة والتعجي ل والتي ارات). في البحث الحالي تم استعراض دراسة شاملة لنمذجة وتحليل وطرق تصميم السيطرة على السرعة لمنظومة المحرك ذو التيار المستمر باستخدام برنامج ال.MATLAB/Siulink 1-Introduction: The echatronic systes, robots and low to ediu power achine-tools often use DC otors to drive their work loads. These otors are coonly used to provide rotary (or linear) otion to a variety of electroechanical devices and servo systes. There are several well known ethods to control DC otors such as: Proportional-Integral PI, Proportional-Integral-Derivative PID or bipositional [1]. Despite a lot of researches and the huge nuber of different solutions proposed, ost industrial control systes are based on conventional PID controllers. The purpose of developing a control syste is to enable stable and reliable control. Once the control syste has been specified and the type of control has been decided, then the design and analysis are done. There are three ajor objectives of syste analysis and design: producing the desired transient response, reducing steady-state error, and achieving stability []. A siple technique for designing a DC otor speed controller with continuous tie based on syste theory concepts was presented in [1]. In [3], Rusu, et al successfully designed and ipleented an open loop control syste to control the speed of a DC otor. In [4], Ayasun * Electroechanical Engineering Departent,University of Technology/Baghdad 141

2 and Nwankpa described the MATLAB/Siulink realization of the DC otor speed control ethods (field resistance, arature voltage and resistance) and feedback control syste for DC otor drives. In [5], Othan successfully designed a speed controller for closed loop operation of the BLDC otor. In [6], Allaoua, et al presented a new design ethod to deterine optial PID controller paraeters using the Particle Swar Optiization ethod. The current paper intends presenting coprehensive study for all the techniques of odeling, analyzing, and designing a DC otor speed controller based on syste theory concepts. The speed control design ethods used are studied under MATLAB and Siulink in the following sections. - MATLAB Control Design Method for Speed Modeling: i- Syste Equations: To perfor siulations of a syste, an appropriate odel needs to be established. This paper was presented the syste contains a DC otor based on the otor specifications needs to be obtained. This is achieved by developing the open loop transfer function of a DC otor with the use of the syste equations of the otor as given by Rashid [7]. The design ethod uses the concepts of the syste theory, such as signals and systes, transfer functions, direct and inverse Laplace transfors. This requires building the appropriate Laplace odel for each coponent of the whole control syste. In order to build the DC otor s transfer function, its siplified atheatical odel has been used. This odel consists of differential equations for the electrical part, echanical part and the interconnection between the. The electric circuit of the arature and the free body diagra of the rotor are shown in the Fig. 1. All the values for the physical paraeters are listed in Table A in the Appendix. The otor torque, T, is related to the arature current, i, by a constant factor K t. The back ef, e, is related to the rotational speed,θ & by the following equations [7]: T = Kt i.. (1) e = K & e θ.. () Assuing that, K t (torque constant) = K e (electrootive force constant) = K (otor constant). Fro Fig. 1 and Table A, the following equations can be written based on Newton s law cobined with Kirchhoff s law: J & θ + b & θ = K i.(3) L di dt T L + R θ & i = V K.. (4) a- Transfer Function Using Laplace Transfors, the above equations can be expressed in ters of s-doain. s ( J s + b) & θ ( s) = K I( s) T L ( s)..(5) ( L s + ) I( s) = V ( s) K s & θ ( s) R.(6) By eliinating I(s), the following open-loop transfer function can be obtained, where the rotational speed θ & is the output and the voltage V is the input. When the otor is used as a coponent in a syste, it is desired to describe it by the appropriate transfer function between the otor voltage and its speed. For this purpose assuing (load torque) T L =0 and (friction torque) T f =0, since neither affects the transfer function. Therefore 14

3 & θ ( s) V ( s) T ( s) = 0 L = ( J s + b)( L s + R ) + K [1,8,9] (7) b. State-Space Representation: In the state-space for, Eqs. 3 and 4 can be expressed by choosing & θ and i as the state variable and V as an input. The output is chosen to beθ &. d dt b & θ J = i K L K J R L K & θ i L V & & θ θ = [ 1 0].(8) i ii- Design Requireents: Since the ost basic requireents of a otor are that it should rotate at the desired speed, the steady-state error e ss of the otor speed should be less than 1%. The other perforance requireent is that the otor ust accelerate to its steady-state speed & θ ss as soon as it turns on. In this case, it is desirable to have a settling tie T s in less than sec. Since speed faster than the reference ay daage the equipent, a peak overshoot Mp of less than 5% is wanted. The definition of these paraeters can be deonstrated as in [10]. iii- MATLAB Representation and Open-loop Response: The transfer function in Eq. 7 can be represented into MATLAB by defining the nuerator (nu) and denoinator (den) atrices as follows: nu=k and den=(js+b)(l s+r )+K,respectivel y, and the corresponding plot is shown as in Fig.. The otor is assued to be exposed at steady-state speed to step change in speed with 0.1 p.u. (10% of its steady-state speed). Fro the plot, it can be seen that when a step input voltage is applied to the syste at steady-state speed, the otor can only achieve a axiu speed of 0.11 p.u.; ten ties saller than our desired speed (1 p.u.). The otor paraeters were deterined on the basis of its step response about a steady-state operating point, naely input voltage V= 6 volts and speed θ & = 600 r/s. Also, it akes the otor takes 3sec to reach its steady-state speed; this does not satisfy the sec settling tie criterion. The syste can also be represented using the state-space Eqs. 8 to get the sae output plot obtained in Fig.. 3- Continuous PID Control Design Method: PID controllers are coonly used to regulate the tie-doain behavior of any different types of dynaic plants [8]. Considering the unity feedback syste architecture as shown in Fig. 3, where it can be assued that the plant is a DC otor whose speed ust be controlled and the Controller provides the excitation for the plant; designed to control the overall syste behavior. Different characteristics of the otor response (steady-state error, peak overshoot, rise tie, etc.) are controlled by selection of the three gains that odify the PID controller dynaics. First, taking a look at how the PID controller works in the closed-loop syste shown in Fig. 3. The variable e represents the tracking error; the difference between the desired input value R and the actual output Y. This error signal will be sent to the PID controller, and the controller coputes both the derivative and the integral of this error signal. 143

4 Therefore, the PID controller is defined by the relationship between the controller input e and the controller output u that is applied to the otor arature: de u = K e + K e dt K (9) P I + dt, where K P = Proportional gain, K I = Integral gain, and K D =Derivative gain. The signal u will be sent to the plant, and the new output Y will be obtained and sent back to the sensor again to find the new error signal e. The controller takes e and coputes its derivative and it s integral again. This process goes on and on. By adjusting the weighting constants K P, K I, and K D, the PID controller can be set to give the desired perforance. Effects of each of these paraeters on a closed-loop syste are suarized as in [8]. Taking the Laplace transfor of Eq. 9 gives the following transfer function: U ( s) K( s) = = K E( s) K + s D I P + K D s + KP s + KI = [8] s.(10) This transfer function clearly illustrates the proportional, integral, and derivative gains that ake up the PID copensation. With adding 0.1 p.u. step input to the steady-state speed and the design criteria previously entioned, the PID controller can be designed and added into the syste, recalling the transfer function of a PID controller in Eq. 10. i. Design Steps of PID Controller: To designing a PID controller for a given syste, the following steps ust be followed to obtain a desired response. K D s 1. Obtaining an open-loop response and deterining what needs to be iproved.. Adding a K P to iprove the rise tie. 3. Adding a K D to iprove the overshoot (increase daping). 4. Adding a K I to eliinate the steady-state error. 5. Adjusting each of K P, K I, and K D until obtaining a desired response. The MATLAB is used to obtain a closed-loop transfer function directly fro the open-loop transfer function. Using K P with a gain of 100, the step response can be shown as in Fig. 4. Fro the plot in Fig. 4, it can be seen that, both the steady-state error and the overshoot are too large. Adding an integral ter will eliinate the steady-state error and a derivative ter will reduce the overshoot. Trying a PID controller with sall K I and K D, the resulting response will be plotted as in Fig. 5. ii. Tuning the Gains: By tuning the gains of the PID controller and producing the optiu response using trial and error ethod (it provides siplest way to achieve a good copensator [11]), the siulation start with the best initial gains. The settling tie shown in Fig. 5 is too long. So it is necessary to increase K I to reduce the settling tie. Changing K I to 00 akes the response looks as in Fig. 6. The response in the Fig. 6 sees uch faster than before, but the large K I has worsened the transient response (big overshoot). Therefore, K D ust increase to reduce the overshoot. Changing K D to 10 can get the plot seen in Fig. 7. So, it ust be known that using (after several trial and error runs) a PID 144

5 controller with K P =100, K I =00, and K D =10, all of the design requireents will be satisfied, providing the desired response. 4- Root Locus Control Design Method: For designing a controller using the root locus ethod, it should follow the following steps: i. Open-loop Root Locus: The ain idea of root locus design is to find the closed-loop response fro the open-loop root locus plot. Then by adding zeros and/or poles to the original plant, the closed-loop response can be odified. So, first, it is necessary viewing in the root locus for the plant. Three arguents ust be founded are the daping ratio (zeta) ter (ξ > 0.8 corresponds to an overshoot Mp < 5%), the natural frequency (ω n ) ter (= 0 corresponds to no rise tie criterion), and the siga ter (4.6/ =.3 sec). Using these criteria results a root locus plot shown in Fig. 8. The following three equations are used in continuous syste designs: ξ ω n 4.6 / T s (Siga ter) (11) ω n 1.8 / T r.(1) ξ [((lnmp /π) /(1+(lnMp /π) )] 0.5.(13), where ξ: Daping ratio, ω n : Natural frequency, T s : Settling tie, T r : Rise tie and Mp: Peak overshoot [11]. ii. Finding the Gain: Recalling that, the settling tie and the overshoot ust be as sall as possible. Large daping corresponds to points on the root locus near the real axis. A fast response corresponds to points on the root locus far to the left of the iaginary axis. Matlab can find the gain corresponding to a point on the root locus. The gain can be found and the step response plotted using this gain all at once. When a point is selected on the root locus plot in Fig. 8, half-way between the real axis and the daping requireent, say at i, the MATLAB should return the output siilar to the following: Selected point= i, gain; K= , poles= ±.0400i, Mp %= and the ξ = Noticing that, the values returned in MATLAB results ay not be exactly the sae, but should at least have the sae order of agnitude. With the above results shown in Fig. 9, it can be seen that, there is no overshoot and the settling tie is about 1sec, so the overshoot and settling tie requireents are satisfied. The only proble can be seen fro this plot is that; the steadystate error is about 50%. If the gain is increased to reduce the steady-state error, the overshoot becoes too large. So it is necessary adding a Lag Controller (in transfer function for) to reduce the steady-state error. iii. Adding a Lag Controller: The plot in Fig. 8 shows a very siple root locus. The daping and settling tie criteria were et with the proportional controller. The steady-state error is the only criterion not et with the proportional controller. A Lag Copensator [11] can reduce the steady-state error. By doing this, it ight however increase the settling tie. By using the following lag controller first: (s + 1) / (s ).(14), the root locus will be shown as in Fig. 10, which looks very siilar to the original one. iv. Closed-loop Response: By closing the loop and seeing the closed-loop step response, the 145

6 following Matlab results will be obtained: selected point = i, gain; K= , poles= ±4.4445i, and When propted to select a point, pick one that is near the daping requireent (diagonal dot line in Fig. 10). The resultant plot will be shown as in Fig. 11. The gain should be about 0. It can be seen fro the figure that the response is not quite satisfactory. It ay also note that even though the gain was selected to correlate with a position close to the daping criterion, the overshoot is not even close to 5%. This is due to the effect of the lag controller; its pole is uch slower. What this eans is that it can go beyond the dotted lines that represent the liit, and getting the higher gains without worrying about the overshoot. Keep trying until getting a satisfactory response. It should look siilar to the following (using a gain of around 50): Selected point= i, gain; K = , poles = ± i and Fig. 1 shows the corresponding response with gain =50. It can be seen fro the figure that the steady-state error is saller than 1%, and the settling tie and overshoot requireents have been et. As result, it concluded that, the design process for root locus is very uch a trial and error process. That is why it is nice to plot the root locus, picking the gain, and plotting the response all in one step. If it has not been able to get a satisfactory response by choosing the gains, it can have tried a different lag controller, or even added a Lead Controller. 5- Frequency Control Design Method: i. Original Bode Plot: The ain idea of frequency-based design is to use the Bode plot of the open-loop transfer function to estiate the closed-loop response. Adding a controller to the syste changes the open-loop Bode plot, therefore changing the closed-loop response. First, the Bode plot for the original open-loop transfer function is drawn as shown in Fig. 13. ii. Adding Proportional Gain: Fro the bode plot in Fig. 13, it can be seen that the phase argin Ф [11] can be greater than about ( =60 0 ) if ω is less than 10 r/s. Adding gain to the syste so the bandwidth frequency [11] is 10 r/s, which will give Ф of about The gain at 10 r/s can be found by reading off the Bode plot in Fig. 13 (it looks to be slightly ore than -40dB, or 0.01 in agnitude) and the exact agnitude: ag = To have an open-loop gain of 1 (0dB) at 10 r/s, it ay be needed to add a control gain of 1/ ( 37dB) to get the Bode plot shown as in the Fig. 14. Fro this plot: (ag=0.0139, phase = , and ω=10 r/s). iii. Closed-loop Response: Fro the plot in Fig. 14, it can be seen that the phase argin is now quite large and the corresponding closed-loop response looks like as in Fig. 15. The figure shows that the settling tie is fast enough, but the overshoot and the steady-state error are too high. iv. Adding a Lag Controller: Reducing the gain to 50, and trying the lag controller in Eq. 14, which should reduce the steady-state error by a factor of 1/0.01=100 (but could increase the settling tie) gives the bode plot shown in Fig. 16 with 146

7 (K=50 34dB) with good phase argin ( 70 0 ). Closing the loop gives the step response shown in Fig. 17, which eets the design requireents. It can see that adding a lag controller can reduce the steady-state error and at the sae tie, the overshoot can be reduced by reducing the gain. 6- A State-Space Control Design Method: For the ain proble, the dynaic equations in state-space for are given as in the Eqs. 8. i. Designing the Full-state Feedback Controller: The scheatic for a full-state feedback syste is shown as in the Fig. 18. The characteristic polynoial for this closed-loop syste is the deterinant of (si-(a-bk)) where s is the Laplace variable. Since the atrices A and BK are both x atrices, there should be poles for the syste. By designing a full-state feedback controller, these two poles can ove anywhere wanted the. The first try is to place the at: -5±i ;( poles= -ξω n ± jω n (1-ξ ) 0.5 ) [11] corresponds to ξ = 0.98 which gives 0.1% overshoot and a siga=ξω n =5 which leads to 1sec settling tie by using Eqs. (11-13). Once coing up with the wanted poles, MATLAB will find the controller atrix, K. After adding the K atrix into the syste in Fig. 18, the state-space equations becoe: X & = ( A BK ) X + Bu Y = CX [11].(15), and the plot of the closed-loop response can be shown as in Fig. 19. ii. Adding a Reference Input: Fro the plot in Fig. 19, it can be seen that the steady-state error is too large. Coputing the input can be done in one step by adding a constant gain Nbar after the reference R. The Matlab can find the scale factor N which will eliinate the steady-state error to a step reference for a continuous-tie, single-input syste with full-state feedback using the scheatic shown in Fig. 0. The corresponding step respond plot is shown in the Fig. 1 and the MATLAB results are: Nbar= and poles = ±1.0000i. Now, the steady-state error is uch less than 1% and all the other design criteria have been et as well. 7- Discrete PID Control Design Method: A digital DC otor odel can be obtained fro conversion of the analog odel. The controller for this exaple will be designed by a PID ethod. To discretise the controller, the integral and derivative ters of the controller have to be approxiated. The integral becoes a su and the derivative becoes a difference. The continuous tie signal e is sapled at a saple period T [8]. i. Continuous to Discrete Conversion: The first step in designing a discrete control syste is to convert the continuous transfer function in Eq. 7 to a discrete one [1] as follows: & θ ( z) 0.009z = V ( z) z z (16) The nuz of Eq. 16 shown above, has one extra zero in the front, so, ust rid of it before closing the loop. 147

8 By generating the vector of discrete output signals and connecting the, the closed-loop response without any control will be shown as in Fig.. Fro the design requireent, the sapling tie T = 0.1*τ = 0.1*0.6T s = 0.1sec, (which is 1/10 the tie constant τ of the syste with a settling tie T s of sec) [8]. ii. PID Controller: For apping the continuous-tie transfer function for a PID controller in Eq. 10 fro the s-plane to z-plane, the bilinear transforation [10] will be used. According to the PID design ethod for the DC otor in Sec. 4, K P =100, K I =00 and K D =10 are satisfied the design requireent. All of these gains will be used in this exaple. Finally, the close-loop stair step response will be plotted as shown in Fig. 3. It can be seen fro this figure that, the closed-loop response of the syste is unstable. Therefore, there ust be soething wrong with copensated syste. So, it should take a look at root locus of the copensated syste shown in Fig. 4. Fro this root-locus plot, it can be seen that the PID controller has a pole at -1 in the z-plane, outside the unit circle and the syste will be unstable. The pole location can be changed by changing the copensator design. It is chosen to cancel the zero at This will ake the syste stable for at least soe gains. Furtherore, an appropriate gain can be chosen fro the root locus plot to satisfy the design requireents. The new z-plane will have a pole at instead of -1, which alost cancels the zero of uncopensated syste as shown on the root-locus plot in Fig. 5. The MATALB results are: ξ =1 and Mp=0. Then MATALB will return the appropriate gain and the corresponding copensated poles, and plot the closed-loop copensated response as shown in Fig. 6, where: selected point= i, K=68.387, poles=-0.665, ±0.194i, and The plot shows that the settling tie is less than sec and the overshoot is around 3%. In addition, the steady-state error is zero. Therefore this response satisfies all of the design requireents. 8- Siulink Design Method for Speed Modeling: A basic feedback control syste shown in Fig. 3 represents a very coon block diagra for. For the proposed syste, the Plant will be a DC otor. i. Building the Model: The otor syste shown in Fig. 1 will be odeled by suing the torques acting on the rotor inertia and integrating the acceleration to give the velocity. Also, Kirchoff's laws will be applied to the arature circuit. First, the integrals of the rotational acceleration and of the rate of change of arature current will be odeled: d θ dθ di =, and i dt dt = (17) dt Next, both Newton's law and Kirchoff's law will be started to odel. These laws applied to the otor syste to give the following equations: d θ dθ J = T b dt dt d θ 1 dθ = Kt i b.(18) dt J dt L di = dt R i + V e 148

9 di 1 dθ = R i + V K.(19) dt L dt ii. Open-loop Response: To siulate the otor syste, first, an appropriate siulation tie ust be set. By selecting paraeters fro the Siulation enu and entering 3 in the Stop Tie field, the open-loop response can be viewed. The physical paraeters ust be set as J=0.01; b=0.1; K =0.01; R =1; and L =0.5. Running the siulation odel shown in the Fig. 7 will give the output on the Scope shown in the Fig. 8. iii. Extracting a Linear Model into MATLAB: A linear odel of the syste (in state space or transfer function for) can be extracted fro a Siulink odel into MATLAB. To verify the odel extraction, an open-loop step response of the extracted transfer function will be generating in MATLAB. The corresponding plot will be exactly siilar to that shown in Fig. which is equivalent to the Scope's output. iv. Ipleenting Lag Copensator Control: In the otor speed control root locus exaple in Sec. 5, a Lag Copensator was designed with the following transfer function: 50(s+1)/(s+0.01). To ipleent this in Siulink, the open-loop syste odel in Fig. 7 will be contained in a Subsyste block, and inserting a Lag Copensator into a closed-loop around the plant odel by feeding back the plant output as shown in Fig. 9. The output of the Su block will provide the error signal, which will be feed into a Lag Copensator. Finally, a step input is applying for viewing the output on the scope. v. Closed-loop Response: To siulate the proposed syste, first, an appropriate siulation tie ust be set, then selecting paraeters fro the Siulation enu and entering 3 in the Stop Tie field. 3sec is suitable for siulation, since it is ore than the desired settling tie. The physical paraeters ust be set. As a result, the Scope output will be shown as in Fig Conclusions: There are any otor control syste design ethods that ay be ore or less appropriate to a specific type of application. The designer engineer ust choose the best one for his work. Therefore, the current work akes coprehensive study of the analysis, odeling and speed control design techniques of a DC otor. An iportant advantage of the root-locus ethod is that the roots of the characteristic equation of the syste can be obtained directly, which results in a coplete and accurate solution of the transient and steady-state response of the controlled variable. The frequency-response approach yields enough inforation to indicate whether the syste needs to be adjusted or copensated and how the syste should be copensated. PID controller enables the otor to reach the speed soothly and within an acceptable period of tie. It found that using (after several trial and error runs) a PID controller with K P =100, K I =00, and K D =10, all of the design requireents will be satisfied, providing the desired response It has observed that both the transfer function and the PID control could have a large influence upon the response of the syste. 149

10 It is found the design process for root locus is very uch a trial and error process. Future suggestion is to replace the conventional speed controller by adaptive self tuning of PID controller. 10- References: [1] M. S. RUSU, and L. Graa, The Design of a DC Motor Speed Controller, Fascicle of Manageent and Tech. Eng., Vol. VII (XVII), 008, pp [] J. J. Jordan, A DC Motor Drive for a Dyno-Microcontroller and Power Electronics, University of Queensland, Australia, BSc. thesis, Oct [3] J. Santana, J. L. Naredo, F. Sandoval, I. Grout, and O. J. Argueta, Siulation and Construction of a Speed Control for a DC Series Motor, Mechatronics, Vol. 1, issues 9-10, Nov.-Dec. 00, pp [4] S. Ayasun and C. O. Nwankpa, DC Motor Speed Control Methods Using MATLAB/Siulink and Their Integration into Undergraduate Electric Machinery Courses, IEEE Trans Educ, March, (007), [5] A. S. Othan, Proportional Integral and Derivative Control of Brushless DC Motor, European Journal of Scientific Research, Vol. 35 No. (009), pp [6] B. Allaoua, B. Gasbaoui and B. Mebarki, Setting Up PID DC Motor Speed Control Alteration Paraeters Using Particle Swar Optiization Strategy, Leonardo Electronic Journal of Practices and Technologies, Issue 14, Jan.-June 009, pp [7] M. H. Rashid, Power Electronics, Circuits Devices and Applications, nd Edition, Prentice- Hall International, Inc., New Jersey, [8] B. Shah, Field Oriented Control of Step Motors, MSc. Thesis, SVMIT- Bharuch, India, Dec [9] W. P. Aung,'' Analysis on Modeling and Siulink of DC Motor and its Driving Syste Used for Wheeled Mobile Robot'', World Acadey of Science, Engineering and Technology 3, 007, pp [10] B. Kuo, Autoatic Control Systes, Prentice-Hall, Englewood, Cliffs. NJ, [11] N. S. Nise, Control Systes Engineering (3 rd Edition), John Wiley and Sons Inc., New York, 000. [1] A. Klee, Developent of a Motor Speed Control Syste Using MATLAB and Siulink, Ipleented with a Digital Signal Processor, MSc. Thesis, Orlando, Florida,

11 + V - R L T e =K θ. + - θ bθ. J Fig. 1: The electric circuit of the arature and the free body diagra of the rotor for a DC otor Fig. : The Open-loop step response of the otor syste using T. F. or state-space; with initial speed=0.1 p.u. at steady state R + - e Controller u Plant Y Fig. 3: Feed back syste scheatic Fig. 4: Closed-loop, step response; K P = 100 Fig. 5: Step response using a PID controller with sall K I and K D Fig. 6: Step response using a PID controller with large K I 151

12 ξ = 0.8 Fig. 7: Step response using a PID controller: K P =100, K I =00, K D =10 Fig. 8: Root locus of the open-loop syste ξ = 0.8 Fig. 9: Open-loop syste, step Fig. 10: Root locus with - response with gain; K=10 a lag controller Fig. 11: Closed-loop step-response with a lag controller and gain=0 Fig. 1: Closed-loop, step-response with a lag controller and gain=50 15

13 Ф Ф Fig. 13: Bode plot of the original open Fig. 14: Bode plot of the plant - loop transfer function with proportional gain=7 Ф Fig. 15: Step response of the open Fig. 16: Bode plot of the plant - loop syste with gain=7 with a lag controller and gain=50 R - u. x = Ax + By y = Cx y K x Fig. 17: Step response with a lag controller and gain=50 Fig. 18: The scheatic for a full - state feedback syste R N + - u. x = Ax + By y = Cx y K x Fig. 19: Closed-loop step response with a K controller Fig. 0: Scheatic for a full-state feed -back syste with adding Nbar gain=10 153

14 Fig. 1: Closed-loop, step-response with adding Nbar =10 Fig. : Closed-loop, stair-step response with no control p = - 1 z = Fig. 3: Close-loop stair-step response Fig. 4: Root locus of the copensated syste with discrete PID controller p = Fig. 5: Root locus of the stable syste Fig. 6: Stair-step stable response with PID controller Fig. 7: Siulation of the open-loop otor syste odel Fig. 8: Open-loop step response; scope output 154

15 Fig. 9: Closed-loop syste using lag copensator control Fig. 30: Closed-loop step response; scope output Appendix Table A: Physical paraeters of the DC otor J Moent of Inertia of the Rotor (kg. ) 0.01 Daping ratio of the b Mechanical Syste 0.1 (Ns) K Motor Constant (N/A) 0.01 R Motor Electric Resistance (Ω) 1 L Motor Electric Inductance (H) 0.5 V Input Voltage (Volt) 6 θ & Rotating Speed (r/s) 600 Fig. A: Transient response (closed loop) of a DC otor 155