International Journal of Scientific Engineering Research Volume 4 Issue 8 August 2013 1290. ISSN 2229 5518, With the disturbance term N s set to zero the final input. output transfer function of the system after substitution is. Vsh s 5 126s 5 126,V1 s s 2 089s2 1 528s 0 439,2 2 Azimuth model. The Azimuth model obtained from the system prototype. shown in figure 2 below yields the set of first order ODE s. shown below,Figure 1 Elevation equivalent circuit, The Elevation model obtained from the system prototype. shown in figure 1 above yields the following set of first. order ODE s shown below, And in equivalent state space representation x Ax Bu we Figure 2 Azimuth equivalent circuit. 1 1 1 2 1 1, Where k1 k2 are constants grouping some parameters 2 2. defined by 2, This model takes for its input v1 and gives a response. which is equivalent to the shaft velocity Vsh After. transforming the first order ODE s the resulting transfer. function of the elevation system which links these two With equivalent state space representation given as. variables evaluates to,3 4 1 L1 2,2 k4 V1 s N s 0,1 1 sL1 R1 2 2. Where N s is a disturbance term which is due to the mass. hanging down while the panel load is at a maximum The azimuth model takes v2 as its input and the response is. incline to the vertical K3 r n1n2Mtot and k4 observed on the platform s angular velocity Wpl After. r n1n2k1L1 making necessary substitutions for the overall Azimuth. IJSER 2013,http www ijser org, International Journal of Scientific Engineering Research Volume 4 Issue 8 August 2013 1291. ISSN 2229 5518, transfer function the following form is obtained for the. system transfer function in open loop,3 1 Elevation dynamic parameters. Relating the output platform angular velocity to the input. supply voltage the transfer function for the azimuth. A resume of the identified elevation dynamic parameters is. dynamics is obtained but not shown here for lack of space. given below in Table 1, Table 1 Open loop Elevation identification parameters. It must be noted that this coarse form of the azimuth Parameters Ziegler Nichols Broida. transfer function made little or no simplifications to the 0 0 2 365. response time T 0 0 1 741, Azimuth model obtained from the system dynamic studies. The parameters specified in this equation are, K1 r2 j2 k2 Frr Jr2 k3 Frr Jpl with electric time constant. 2 L2 R2 Employing Ziegler Nichols method where t1 is time at 28. of max response and t2 is time at 40 of max response. As a first approximation before system identification all The open loop response gives a Time delay of about 0 1. the constants are set to unity for open loop test and seconds The time constant 63 of output response for the. analysis This generates a system transfer function which is system in open loop is 3 89sec. strictly proper having the degree of the transfer function. characteristic polynomial s denominator greater than A Matlab simulink model for the elevation dynamics was. degree of numerator used to obtain the open loop response shown in figure 3 the. closed loop response of the system is obtained from the. az signal block diagram given by figure 4 with the response. s6 4s5 9s4 16s3 19s2 12s 3, for the closed loop system shown in figure 7 below. 3 MODELING AND SYSTEM IDENTIFICATION, The system identification for the elevation dynamic follows. from the method of Ziegler Nichols and Broida 1 From. the response curve seen in figure 3 the system is,approximately a first order system Using Broida. parameters the system response time T and time constant. is calculated following the formulas given below,2 8 1 1 8 2. Open loop analysis was carried out on both elevation and. azimuth models obtained previously in the analog or Figure 3 elevation open loop response. continuous domain We then proceed to identify the system. parameters which have utility in synthesizing a suitable. controller for each subsystem taking into consideration the. system dynamics involved,IJSER 2013,http www ijser org. International Journal of Scientific Engineering Research Volume 4 Issue 8 August 2013 1292. ISSN 2229 5518,0 0 1 7626,response time T 0 0 1 1165. 4 MODELING AND CONTROLLER SYNTHESIS, Since the elevation response follows a first order system. response curve a PI controller is chosen with gains tuned. Figure 4 closed loop elevation diagram showing feedback following the Chiens Hrones Reswick formulas in tracking. From the open loop analysis the elevation controller. synthesis follows,Kp 0 35x T 0 257,3 2 Azimuth dynamic parameters Ti 1 2xT 2 089. from where the closed loop control response is shown. Similarly for the azimuth dynamics we obtain the open. loop response shown in figure 5 as obtained from, simulation experiments in open loop while the resume of. the identified azimuth parameters is given in Table 2. Figure 6 Closed loop elevation response,The azimuth controller synthesis follows. Kp 0 35x T 0 222,Ti 1 2xT 1 3398,4 1 Closed loop system. After controller synthesis the controller is incorporated. Figure 5 Open loop Azimuth response into the system with the resulting system diagram shown in. figure 6 below, Table 2 Open loop Azimuth identification parameters. Parameters Ziegler Nichols Broida,IJSER 2013,http www ijser org. International Journal of Scientific Engineering Research Volume 4 Issue 8 August 2013 1293. ISSN 2229 5518,With the constants defined as,K1 k4kpkdL1. K3 k4kpkiL1, Figure 7 Azimuth closed loop block schematic K5 R1 k2L1 k4kpkdL1. K6 k2R1 k3k4L1 k4kpL1,K7 k4kpkiL1, Stability of the system can be inferred from the poles of the. TF denominator Since all the eigenvalues will be real and. 4 2 Stability analysis strictly negative the system is eigenvalue stable as all its. characteristic poles will be found on the left half of the. Various methods exist to test the stability of dynamic complex plane. systems such as that describing the elevation model of the. sun tracker Some widely used methods are,Routh Hurwitz criterion. Eigenvalue analysis 4 4 Stability analysis using Exponential time growth. Lyapunov method and LaSalle s invariance, principle The stability analysis looks at the dynamics of the system in. Small or short time stability analysis exponential time If the error of the system decays to zero as. time goes to infinity then the system is said to be stable. This is seen from the plot given in figure 7 of the closed. loop dynamic controller response of the elevation system. 4 3 Stability determination using Eigenvalue,analysis 5 EXPERIMENTAL RESULTS AND ANALYSIS. To perform stability analysis on the elevation system the Experiments were carried out in simulation and the results. closed loop response will have to be made use of Recalling are shown here with utilized parameters The system is. controllable using a simple PID controller with gain tuning. the form of the transfer function in open loop for the. elevation subsystem given by equation 3 we can as shown in figure 6 with various tuned values for. controller gains From the open loop response of the. elevation system we see that divergence from the nominal. Incorporate the behavior of the PID controller recalling that step input occurs The voltage equivalent for this response. the new input from the PID controller is given by means a sharp rise in input voltage to the elevation motor. which is not supposed to be The controller addresses this. 1 situation by forcing the output to be within the desired. response of 12 volt, with U s substituting Vin in equation 3 and the error term. from feedback action E s being replaced by Vd s Vsh s. The azimuth open loop response shows a wide divergence. we will have a new closed loop transfer function given by. in the motor behavior This is seen through the strong. Vsh s s2K1 sK2 K3 oscillatory response observed Using a simple PID might. Vd s s K4 s2 K5 sK6 K7,IJSER 2013,http www ijser org. International Journal of Scientific Engineering Research Volume 4 Issue 8 August 2013 1294. ISSN 2229 5518, not be ideal for the control of this system and ongoing tests Modeling Simulation and Control of Dynamic systems. are in place to resolve the issue of the azimuth control Elsevier s Press. 6 DISCUSSION AND CONCLUSION,4 Bideaux Eric, Modeling Simulation and Control of Dynamic systems. The modeling analysis and control of the sun tracker. INSA de Lyons France, system are looked at from a control system stand point with. some results of the performed experiments given Gain. tuning is employed to adjust the system response to within 5 Ovie E Bond Graph approach to Dynamic system. the desired values Modeling NASRDA Abuja,March 2013. While standard methods 1 exist and are used to compute 6 Lamport L LaTeX A Documentation Preparation. the start gain parameters for the controller it is seen that System User s Guide and Reference Manual. working gain values could widely diverge from the Addison Wesley Pub Co 2nd edition August. calculated or nominal as was the case with this work This 1994. is being attributed to the unknown system dynamics. ignored as part of this analysis as the parts used were. sourced for locally in the market without a thorough. knowledge of the internal components especially as it. concerns frictional effects reduction gear ratios and other. unaccounted for parameters which could play a role in this. divergence, In summarizing the goal of building a system model and. carrying out analysis needed for controller synthesis was. achieved through this paper with the dissemination of such. knowledge to interested scientists and engineers, Further work will be to carry out detailed numerical. simulations to gain further insight into the system behavior. References,1 K Hamiti A Voda Besanqon and H Roux Buisson. POSITION CONTROL OF A PNEUMATIC,ACTUATOR UNDER THE INFLUENCE OF STICTION. UJFG LIME B P 53 38041 Grenoble Cedex France,Laboratoire d Automatique de Grenoble ENSIEG B P. 38402 Saint Martin d Heres France,2 Henry M Paynter. An Epistemic Prehistory of Bond Graphs,Elsevier s Press. 3 Karnoop Rosenberg Margolis,IJSER 2013,http www ijser org.

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