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Optimizing the Controller Design to Guide the Motion of a Maglev Train The challenge: To design a robust feedback controller and optimize the control parameters to guide the motion of a Maglev train along the guideway \302\251 Maplesoft, a division of Waterloo Maple Inc., 2008 Introduction Problem Definition 1. Model Development 1.1 Linear Model of the Magnet 1.2 Model for the Linearized Magnet Force 1.3 Model of Magnet Dynamics 2. Control System Design 2.1 PI Controller 2.2 Control Specifications and Measurements 2.3 Control System Transfer Function 2.4 Controller Gains & Closed Loop System 2.5 The Final Design 3. Vehicle Model 3.1 Equations of Motion 3.2 Response Plots Results 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Magnetically Levitated (Maglev) trains differ from conventional trains in that they are levitated, guided and propelled along a guideway by a changing magnetic field rather than by steam, diesel or electric engine. The absence of direct contact between the train and the rail allows the Maglev to reach record ground transportation speeds, which are on par to that of commercial airplanes. JSFH 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tEQN<JGlW_uO`lKx<TZIQ\\UjNhjTIoEAwB@OrlPl=sLdnt=Q^DycXpTUoqPueLq[dUMXj=<n;dW^pTg<R_lm_lStPvSXoQQUYqs_QOj<Rg\\P=eOT<jHHk]yMFAmgXhXwwoXeZqoEvjUic?pk>^`oveQn`^`bWagVgjIW]pVfqnp]wr?p`nWZA^j\\GeApb=vapnfmvj?^jEWaDhZW?ooofAppbWapYhR>`:^hDydIppdWr?gkZgZ^VfbQgAqpfWuhxda>s>^bZPZnXnc?moagqqhV_h=_a^`efqe:`Z^Gjg@yCxgavplwjKPZVGZsOmfNpIhv?_hiv`owiHvsPw`pwfqVpnWb`Vo@GefNkCFipxgbw`sWaEF`Qx`twnqNgnvptwpqnq[gpuwrq`iugbapsqhiiptswuqpiQwplvwqxi=H\\vwyq@juq`x>xmAjY>yn`^fNji>qdWfUVjy>yg@n=Xk<?a?xdrhjY?a@xfrpji?aAxhrtYkC\\IFhKv]kS\\iFhMvakc\\IGhOveks\\iGhQvikC]IHhSvFQDJ;B:q;:6:\"\{\}LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlUyMqRiotJS5CT1VORFNfSEVJR0hURzYjJCIlcUlGKi0lKUNISUxEUkVORzYi Figure 1: (A) Cross sectional view of cabin & (B) Expanded view of magnet support structure An Electromagnetic Suspension (EMS) Maglev train uses the attractive forces of magnets (& electro-magnets), positioned below a guideway, to levitate a train and guide it along the guideway (fig 1). Three phase AC propulsion coils mounted along the steel rails of the guideway provide a moving magnetic field that interacts with the periodic magnetic field created by the lifting magnets to propel the train forward. There are 24 lifting magnet pairs (48 magnets all together) mounted along the length of the train. The lifting magnets, which are canted at an 37\302\260 angle towards the steel rails, consists of high temperature superconducting (HTS) magnets and conventional (non-superconducting) coils (fig 2). A periodic magnetic field is generate by aligning the lifting magnet pairs so that their north and south poles alternate. There are many advantages and disadvantages associated with Maglev trains in comparison to conventional steel wheel on steel rail trains . The table (table 1) below highlights some of these. 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Figure 2: Expanded view of lifting magnets Advantages Disadvantages Record breaking ground transportation speeds reaching 500 km/h (300 mph) High infrastructure costs Less expensive to operate and maintain than traditional high-speed trains or even planes, and as a consequence lower cost per passenger per mile Not compatible with conventional wheeled tracks making it limited to only those places where maglev lines run. Silent at low speeds Increased noise pollution at high speeds Significant energy savings because the vehicle does not have to carry around the weight of its propulsion system No associated maintenance cost because there are no parts that are in contact with ground Improved ride comfort Table 1: Advantages and Disadvantages of Maglev trains over conventional trains

EMS Maglev trains require robust controllers to guide the train along the guideway. The controller prevents the train from colliding into the guideway by maintaining a constant air gap between the train and guideway regardless of inconsistencies in the guideway, change in direction or angle of the track, and environmental forces such as wind. Control of the vehicle is achieved by a combination of passive (i.e. through the orientation of the magnets along the vehicle) and active (i.e. with an active control system) techniques. Each magnetic coil has its own independent control system. This independence allows the pitching (forward and aft rotations) to be controlled by the magnets at the front and rear of the vehicle acting independently to maintain the air gap at 2 inches. Because the magnets are canted at 37\302\260, the lateral and yaw motions (motions that turn the vehicle left or right relative to the direction of travel) are also controlled automatically when the air gaps are maintained at the nominal 2 inches. For example, if a wind gust were to shift the vehicle to the right, the gap on the left side would become smaller and the one on the right side would become bigger. Each of the magnets would then correct for the motion by returning the magnets back to equilibrium. Lift (up and down motion) is controlled when the left magnet and right magnet move in unison. For example, when the rail rises it causes the gap on both the right and left side of the vehicle to get smaller. The magnet control systems respond to this change by adjusting the current flowing through the magnets to return the gap to the desired 2 inch setting. The control for roll (rotations to the left and right along the direction of travel) is done passively by offsetting each of the magnets pairs by 1 inch. This offset acts to provide a magnetic spring that naturally restores the vehicle so it is perpendicular to the guideway whenever a disturbance occurs. This worksheet will examine the design of a robust feedback controller to guide the motion of a EMS Maglev train along the guideway.

The force from the magnet is a function of the gap distance LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and the current in the coils LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjSWNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn. The force can be developed from first principles using Maxwell's equations, or by using the inductance of the magnets, but because of the iron in the core, the actual force is a complex function of the geometry of the magnet, the amount of iron in the magnet and the rail, and the inherent hysteresis of the iron. The nonlinear effects of the magnet can be approximated to first order so that the magnet force is proportional to the square of the current flowing in the magnet coils and inversely proportional to the square of the gap between the rail and the magnet. Thus: PkkiZkc2ImYqNiRJInpHRiRJI0ljR0YkRiQ2JEkpb3BlcmF0b3JHRiRJJmFycm93R0YkRiQqKEkiS0dGJCIiIjkkISIjOSUiIiNGJEYkRiQ=

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print(); # input placeholder

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print(); # input placeholder

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Displacement: Pkklb2RlMkc2Ii8tSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiekdGJDYjSSJ0R0YkRi0tSSJ2R0YkRiw=

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print(); # input placeholder

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Suppose the weight of each car in a two car Maglev train is 135,000 pounds. Using the same units, the acceleration of gravity is defined as 32.2 ft/s/s. For now let's assume the proportionality constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is unity. This then gives the parameters for the model as: print(); # input placeholder JSFH QyQ+SSdwYXJhbXNHNiI3JS9JIktHRiUiIiIvSSJnR0YlJCIkQSQhIiIvSSJtR0YlLCQtSSIvRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGLCInK104Ri4=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

The first step in obtaining an accurate model of the vehicle is to model the nonlinear magnet dynamics. Therefore, we will create, using the DynamicSystems package, the nonlinear differential equation model, using the coupled differential equations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElb2RlMUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElb2RlMkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= defined above, with the parameters defined above. Pkklb2RlMUc2Ii1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRGI0kncGFyYW1zR0Yk

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInZHNiI2I0kidEdGKkYsIiIiJCIkQSQhIiJGLSwmKiYtSSJ6R0YqRishIiMtSSNJY0dGKkYrIiIjJCIrJj0mPSZRIyEjOC1JI0ZkR0YqRitGOQ==

Pkklb2RlMkc2Ii1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRGI0kncGFyYW1zR0Yk

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ6RzYiNiNJInRHRilGKy1JInZHRilGKg==

QyQ+SSdzeXNfZGVHNiItJkkvRHluYW1pY1N5c3RlbXNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0ktRGlmZkVxdWF0aW9uR0YlNiU3JEklb2RlMUdGJUklb2RlMkdGJS9JLmlucHV0dmFyaWFibGVHRiU3JC1JI0ljR0YlNiNJInRHRiUtSSNGZEdGJUY3L0kvb3V0cHV0dmFyaWFibGVHRiU3JC1JInpHRiVGNy1JInZHRiVGNyEiIg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%; To see what the object sys_de looks like we use the PrintSystem command in the DynamicSystems package. LSZJL0R5bmFtaWNTeXN0ZW1zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJLFByaW50U3lzdGVtR0YoNiNJJ3N5c19kZUdGKA==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

We now have a differential equation system object with the inputs LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EjSWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi9GMi8lMGZvbnRfc3R5bGVfbmFtZUdRL09yZGVyZWR+TGlzdH4xRicvRjlRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EjRmRGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi9GMi8lMGZvbnRfc3R5bGVfbmFtZUdRL09yZGVyZWR+TGlzdH4xRicvRjlRJ25vcm1hbEYn, and outputs LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EiekYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGL0YyLyUwZm9udF9zdHlsZV9uYW1lR1EvT3JkZXJlZH5MaXN0fjFGJy9GOVEnbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EidkYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGL0YyLyUwZm9udF9zdHlsZV9uYW1lR1EvT3JkZXJlZH5MaXN0fjFGJy9GOVEnbm9ybWFsRic=. The proportionality constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in equation is derived from the fact that the magnetic force is given by the cross product of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= field with the current producing the magnetic field. Since the current produces both the lifting field and the field it interacts with (through the closed loop around the C shaped magnet and the rail), the lifting force is proportional to the square of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= field over the area A of the magnet's pole face. The proportionality constant for this force is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYkUSIyRidGMi1JJW1zdWJHRiQ2JS1JI21pR0YkNiVRJyYjOTU2O0YnLyUnaXRhbGljR1EmZmFsc2VGJ0YyLUYjNiQtRi82JFEiMEYnRjJGMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjIvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlEvJSliZXZlbGxlZEdGQy1GPjYjUSFGJ0Yy. Thus the force from the magnet is: QyQvSSJGRzYiKihJIkJHRiUiIiNJIkFHRiUiIiItSSIvRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJCZJI211R0YlNiMiIiFGKEYqISIi

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); TheLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YrRi8=field is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJyZzZG90O0YnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLUYsNiVRIkhGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0Yy where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy is the magnetic permeability of the iron in the magnet and rail (due to the hysteresis of the iron, this term is not constant) and the magnetic flux is given by the current in the coil multiplied by the number of turns in the coil. Thus, the constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is given by: QyQvSSJLRzYiKipJI211R0YlIiIjSSJOR0YlRihJIkFHRiUiIiItSSIvRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJCZGJzYjIiIhRihGKyEiIg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print();

Based on a simple understanding of the physics, it should be clear that when there is no current in the coils the vehicle will drop under the influence of gravity causing the undercarriage to hit the guideway. Similarly, when the current in the magnet is very large, the magnet will be attracted to the rail and hit the rail (in fact the assumed force function is infinite when the gap z is zero). Therefore, as we analyze the nonlinear differential equation we should expect that the dynamics are unstable. To investigate the dynamics, the force term LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInpGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0YvRjJGQUZBRkE= must be linearized. The basic superconducting magnet was designed to have a nominal value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJyZzZG90O0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMLUYsNiVRIklGJ0YvRjJGOQ== equal to 50,000 ampere turns. The control coils, which regulate the current through the superconducting magnets, were designed to allow a maximum of 15,000 ampere turns. In addition, the nominal air gap for the system is 2 inches. These values establish the numerical value for force and therefore for the term LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The linearization of the nonlinear term in , from the Taylor series expansion around the nominal 2 inch gap and 50,000 ampere turns current in the coils, is obtained from the partials QyQvSSNLekc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJmR0YlNiRJInpHRiVJI0ljR0YlRi0hIiI=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder and QyQvSSNLSUc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJmR0YlNiRJInpHRiVJI0ljR0YlRi4hIiI=

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print(); # input placeholder JSFH LUklZGlmZkclKnByb3RlY3RlZEc2JC1JImZHNiI2JEkiekdGKEkjSWNHRihGKg==

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= print(); # input placeholder

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= print(); # input placeholder

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JSFH When the nominal values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjSWNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= are substituted into the expansion, the linearization becomes (in this and the subsequent analysis, the variations in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjSWNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM5MTY7ekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEpJiM5MTY7SWNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn, are written as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjSWNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn respectively): QyQvLUkiZkc2IjYkLUkiekdGJjYjSSJ0R0YmLUkjSWNHRiZGKiwmKiZJI0t6R0YmIiIiRihGMUYxKiZJI0tJR0YmRjFGLEYxRjEhIiI= The nominal value for the vehicle mass, and the nominal value of K gives Kz as 281,000 pounds/foot and the nominal value of KI as 3372000 pounds/kiloampere-turn. This then provides the linear dynamics as: QyQ+SSVvZGUzRzYiLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ2R0YlNiNJInRHRiVGLiwoKihJI0t6R0YlIiIiLUkiekdGJUYtRjJJIm1HRiUhIiJGMiooSSNLSUdGJUYyLUkjSWNHRiVGLUYyRjVGNkYyKiYtSSNGZEdGJUYtRjJGNUY2RjJGNg==

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print(); # input placeholder JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= QyQ+SSdwYXJhbXNHNiI3Jy9JIktHRiUiIiIvSSJtR0YlIiYrMCYvSSJnR0YlJCIkQSQhIiIvSSNLekdGJSInKzVHL0kjS0lHRiUiKCs/UCRGMQ==

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print(); Now we evaluate the equation at the operating point derived above: Pkklb2RlM0c2Ii1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRGI0kncGFyYW1zR0Yk

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print(); # input placeholder

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JSFH

The analyses in the previous section allow us to begin the design of the control system. The first step in this is an investigation of the underlying dynamics in order to understand how best to control the vehicle. In this section we will use the equations defined above to develop a controller which will effectively control the behavior of our system to external perturbations, such as changes in guideway and disturbance forces (such as wind)), according to a set of well defined control criteria. To start, we define the differential equation system object that we will be working with using LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElb2RlMkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElb2RlM0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= above: QyQ+SSdzeXNfZGVHNiItJkkvRHluYW1pY1N5c3RlbXNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0ktRGlmZkVxdWF0aW9uR0YlNiU3JEklb2RlMkdGJUklb2RlM0dGJS9JLmlucHV0dmFyaWFibGVHRiU3JC1JI0ljR0YlNiNJInRHRiUtSSNGZEdGJUY3L0kvb3V0cHV0dmFyaWFibGVHRiU3JC1JInpHRiVGNy1JInZHRiVGNyEiIg==

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%; LSZJL0R5bmFtaWNTeXN0ZW1zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJLFByaW50U3lzdGVtR0YoNiNJJ3N5c19kZUdGKA==

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print(); # input placeholder

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For the design of the control system, the transfer function poles and zeros for the linear differential equation need to be derived. We start by using DynamicSystems to get the ZeroPoleGain version of sys_de: Pkkkc3lzRzYiLSZJL0R5bmFtaWNTeXN0ZW1zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJLVplcm9Qb2xlR2FpbkdGJDYjSSdzeXNfZGVHRiQ=

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print(); # input placeholder

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Zero-Pole-Gain Analysis Zero-Pole Plot Zeros 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 Figure 3: Zero-Pole Plot JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkc3lzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOi1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ6RidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5

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= print(); # input placeholder

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Poles LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkc3lzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOi1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJwRidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5

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= print(); # input placeholder

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Gains LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkc3lzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOi1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJrRidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

= print(); # input placeholder

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

From the zero-pole plot we see immediately that there are two poles: one in the right half plane and one in the left. As we deduced from first principles, the system is unstable. Just for completeness, if only the the transfer function for the system were needed, it would be obtained from: Pkkkc3lzRzYiLSZJL0R5bmFtaWNTeXN0ZW1zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJMVRyYW5zZmVyRnVuY3Rpb25HRiQ2I0knc3lzX2RlR0Yk

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print(); # input placeholder

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Pkkkb3V0RzYiNyNfSSRzeXNHRiRJI3RmR0Yk

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyMtSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JJCVpZEc2IiIpZ10pRyc=

Now we can design the control system.

Superconductor magnets do not like rapidly varying currents. We can use the superconductor as an integrator (this comes from physics). The magnet has essentially zero resistance, so the differential equation for the magnet is: QyQvKiYmSSJMRzYiNiNJI3NjR0YnIiIiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JJUljc2NHRic2I0kidEdGJ0YyRiotSSJ2R0YnRjEhIiI=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder Thus, for any voltage LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== applied to the superconductor, the current in the magnet is the integral of the voltage. Making the control signal the voltage, the superconductor will integrate this control and the gain on this signal will be the integral control gain, which we denote by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS2lGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn. Similarly, the control coil has the model: QyQvLCYqJiZJIkxHNiI2I0kiY0dGKCIiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSNJY0dGKDYjSSJ0R0YoRjNGK0YrKiYmSSJSR0YoRilGK0YwRitGKy1JInZHRihGMiEiIg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder and of course in steady state the current in the control coil is simply 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 multiplied by the proportional control gain (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS3BGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn).

KiYtSSJ2RzYiNiNJInRHRiUiIiImSSJSR0YlNiNGJCEiIg==

This analysis shows that the control system is a proportional plus integral controller with gains LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS3BGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS2lGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn, but we still have not addressed how to stabilize the lift magnets. Let's investigate the measurements that will be required to stabilize the magnets.

When a Maglev "flies" above the guideway (and below the rails in this design), it is moving along in inertial coordinates that are only slightly constrained by the proximity of the guideway. This means that a slight change in the rail location relative to the vehicle - one that is caused by an irregularity in the rail for instance - can be ignored. Thus, measurements of the gap between the magnet and the rail are not sufficient to tell the control system what to do. In most attractive Maglev designs, therefore, a measurement of the gap is supplemented with a measurement of the acceleration of the magnet in the direction of the rail (remember in this design the magnets are canted at an angle so the force is not applied purely along the vertical (z) axis). The control system is therefore required to null accelerations. Acceleration alone can not be the only feedback. The guideway (and the rail along with it) must follow the terrain and the desired route (turning left and right). This means that the gap between the magnets and the rail must be measured too. Furthermore, the availability of acceleration as a measurement would allow derivative control (i.e. the control could be full state feedback PID control) simply by integrating the acceleration. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The control criteria is therefore: At a fast scale (response times between 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSJzRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lMGZvbnRfc3R5bGVfbmFtZUdRLEJ1bGxldH5JdGVtRidGLw==), zero the acceleration. At a slow scale (times greater than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSJzRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lMGZvbnRfc3R5bGVfbmFtZUdRLEJ1bGxldH5JdGVtRidGLw==), follow the gap and maintain it at 2 inches. The break between these two actions is approximately where the superconductor and the control coils are active (i.e. up to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSNIekYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRicvJTBmb250X3N0eWxlX25hbWVHUSxCdWxsZXR+SXRlbUYnRi8= use the superconductor currents change to modify the "lift" and thereby keep the gap constant, and at faster scales (up to about 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) the control coils try to zero the vertical accelerations).

From the discussions above, the transfer function for the controller is a PID control with acceleration feedback. Assume that the acceleration gain is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS2FGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn, the derivative gain in the PID controller is the velocity gain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS3ZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn, and of course the integral gain is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS2lGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn. Instead of placing a gain on the position, we assume that the entire feedback control is multiplied by a loop gain of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS2ZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn (instead of a position gain it actually is in the forward loop and is a gain for the entire control system). Thus the feedback controller is: PkkjSGNHNiJmKjYjSSJzR0YkRiQ2JEkpb3BlcmF0b3JHRiRJJmFycm93R0YkRiQqJi1JI0tmR0YkNiMsJiomSSNLYUdGJCIiIjkkIiIjRjIqJkkjS3ZHRiRGMkYzRjJGMkYyLCYqJkkjS2lHRiRGMkYzISIiRjJGMkYyRjJGJEYkRiQ=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Zio2I0kic0c2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYtSSNLZkdGJTYjLCYqJkkjS2FHRiUiIiI5JCIiI0YwKiZJI0t2R0YlRjBGMUYwRjBGMCwmKiZJI0tpR0YlRjBGMSEiIkYwRjBGMEYwRiVGJUYl

The control system will also have the time constants associated with the control coils and the sensors (the gap sensor and the accelerometer that will be used to get the velocity). For now we will neglect these. The complete linear dynamics block diagram of the control system is therefore as shown in Figure 4 below. 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Figure 4: Block Diagram of the Linear Controller for a Maglev Magnet The control consists of an integral controller and feedback of the magnet acceleration, velocity and position with respect to the rail.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Knowing the control system structure allows us to analyze the block diagram and derive the closed loop transfer function. The first step is to do some simple block diagram manipulations to obtain the block diagram shown in Figure 5. This immediately gives the closed loop transfer function for the displacement LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== with respect to the desired gap as the third order system: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= PkkkR2NsRzYiZio2I0kic0dGJEYkNiRJKW9wZXJhdG9yR0YkSSZhcnJvd0dGJEYkKi5JI0tmR0YkIiIiSSRLSWNHRiRGLSwmOSRGLUkjS2lHRiRGLUYtRjAhIiIsJiokRjAiIiNGLSomSSNLekdGJEYtSSJtR0YkRjJGMkYyLCZGLUYtKi4sKComSSNLYUdGJEYtRjBGNUYtKiZJI0t2R0YkRi1GMEYtRi1GLUYtRi1GLEYtRi5GLUYvRi1GMEYyRjNGMkYtRjJGJEYkRiQ=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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Before we begin determining the gains in the control system, we need to discuss controlling an unstable system and how the instability affects the specification for the overall closed loop system. 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Figure 5: Simplified Block Diagram In the design of unstable linear systems, the gain needs to be large enough to bring the unstable pole into the left half plane. One very useful trick that has been used in many unstable systems (including aircraft that are longitudinally unstable) is to simplify the transfer function by selecting the gain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS2lGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn of the PID control so that the zero introduced by the integral compensation cancels the left half plane pole of the unstable dynamics. The motivation for this is twofold. First, this cancellation will result in closed loop dynamics that are second order, and secondly the bandwidth of the closed loop servo can be set to any desired value (independent of the magnet dynamics). As a second order system, the gains for the measurements and the proportional feedback can be selected to make the magnet response optimal in the LQ sense (i.e. we can select the closed loop bandwidth to be LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSJ+RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEjSHpGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGLw== as required in the specifications, and we can set the damping to the optimal value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1JJm1zcXJ0R0YkNiMtRiM2KS1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRLFsxMjgsMCwxMjhdRicvJSxwbGFjZWhvbGRlckdGQi9GOFEnaXRhbGljRictRiM2J0YzRjpGPUZARkgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlEvJSliZXZlbGxlZEdRJmZhbHNlRidGOkY9Rjc=) . The cancellation of the zero with the left half plane pole of the unstable dynamics requires that 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder The following simplifications are done to get the closed loop transfer function into the required form: LUknZmFjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMtSShhbGdzdWJzR0YnNiQvSSNLaUdGJy1JJXNxcnRHRiQ2IyomSSNLekdGJyIiIkkibUdGJyEiIi1JJEdjbEdGJzYjSSJzR0Yn

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

KiwsJkkic0c2IiIiIiokKiZJI0t6R0YlRiZJIm1HRiUhIiIjRiYiIiNGJkYmSSNLZkdGJUYmSSRLSWNHRiVGJkYqRiYsMiosRi5GJkYvRiZGKkYmSSNLYUdGJUYmRiQiIiRGJiouRi5GJkYvRiZGKkYmRjJGJkYkRi1GKEYsRiYqLEYuRiZGL0YmRipGJkkjS3ZHRiVGJkYkRi1GJiouRi5GJkYvRiZGKkYmRjZGJkYkRiZGKEYsRiYqKkYuRiZGL0YmRipGJkYkRiZGJioqRi5GJkYvRiZGKkYmRihGLEYmKiZGJEYzRipGJkYmKiZGJEYmRilGJkYrRis=

Note that the possibility of having a common factor in the numerator and denominator is not obvious, but it is possible to select the values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS2FGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS2ZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS2lGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn to make the denominator of this transfer function equal to 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 , where the values of the desired closed loop denominator are: QyQvJkklWmV0YUc2IjYjSSNDTEdGJiwkLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjIiIjIyIiIkYwISIi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= QyQvJkkmb21lZ2FHNiI2I0kjQ0xHRiYsJEkjUGlHJSpwcm90ZWN0ZWRHIiM/ISIi JSFH To simplify the algebra, let's assume that the zeros introduced by the acceleration gain and velocity gain are both at the same location denoted by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLDYlUSJ6RicvRjZRJXRydWVGJy9GOVEnaXRhbGljRidGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjg=. This assumption makes the acceleration and velocity gains as follows: QyQvSSNLYUc2IiokJkkmb21lZ2FHRiU2I0kiekdGJSEiIyEiIg== and QyQvKiZJIktHNiIiIiJJInZHRiZGJyokJkkmb21lZ2FHRiY2I0kiekdGJiEiIkYu JSFH While it might seem like a large set of simultaneous nonlinear equations need to be solved to completely specify the closed loop dynamics, we have systematically been eliminating the non-linearities in the solution as we constrained the gains. Therefore, the values for the gains are completely specified by the location of the zeros, which because of our assumption is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUYvNiVRInpGJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIuRidGNS8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlhGNQ== Define the open loop undamped natural frequency as 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 Then QyQvJkkmb21lZ2FHNiI2I0kiekdGJiomLSZGJTYjSSNDTEdGJjYjLCYiIiJGMC1JJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2IywmRjBGMCooJkYlNiNJI09MR0YmRjBGKyEiIiwmJkklWmV0YUdGM0YsIiIjKiZGOUYwRitGPEYwRjBGMEYwRjBGPUY8Rjw=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder The values for the feedback gains LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS2FGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS3ZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn and the integral gain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS2lGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn are completely determined from the above, and the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS2ZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn is: QyQvJkkiS0c2IjYjSSJmR0YmKigmSSZvbWVnYUdGJjYjSSNDTEdGJiIiIkkkS0ljR0YmISIiLCZGLkYuLUkiXkclKnByb3RlY3RlZEc2JComRipGLiZGKzYjSSJ6R0YmRjAiIiNGLkYwRjA=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print();

We can now verify that the design is correct by examining the system response to disturbance forces and guideway changes and errors. Toward that end, let's find the numerical values for the gains based on the equations we have derived above. Substituting

SS1TdWJzdGl0dXRpbmdHNiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJnpldGE7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtRi82JVEjQ0xGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82I1EhRidGPQ==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLDYlUSNDTEYnL0Y2USV0cnVlRicvRjlRJ2l0YWxpY0YnRjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y4, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEnJiM5Njk7RicvJSdmYW1pbHlHUSpTYW5zU2VyaWZGJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYlUSNPTEYnL0Y5USV0cnVlRicvRj9RJ2l0YWxpY0YnRj4vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ from the specification completely determines the gains that, along with the other parameters, can be used to create the system model that we want to examine. Thus the parameters are: QyQ+SSdwYXJhbXNHNiI3KS9JIm1HRiUkIiZEPiUhIiIvSSNLekdGJSIoKz9QJC9JJEtJY0dGJSImJSl6Ii9JI0thR0YlJCImbWUiISIpL0kjS3ZHRiUkIiRfIyEiJS9JI0tpR0YlJCInKmYkR0Y7L0kjS2ZHRiUkIiZLWSNGK0Yr

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); The state space model for the complete linear system can be written from the block diagram in Figure 3, in the following manner: We start at the right side of the diagram, moving from integrator to integrator for each of the states. The first integrator (for the velocity) gives: QyQ+SSVvZGUxRzYiLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ4R0YlNiNJInRHRiVGLi1JInZHRiVGLSEiIg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder The next integration (of the acceleration) is a little complex because there is an algebraic loop involved. Thus the differential equation for the acceleration has a term on the right that involves the derivative of the velocity (i.e. the acceleration) that must be removed by bringing the term to the right and dividing through by the resulting coefficient (which is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIitGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkctSSZtZnJhY0dGJDYoLUYjNigtSSNtaUdGJDYlUSNLZkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GMzYuUScmc2RvdDtGJ0ZTRlZGNkY5RjtGPUY/RkFGQy9GRlEmMC4wZW1GJy9GSUZmbi1GUDYlUSNLYUYnRlNGVi1GMzYtRlpGL0Y2RjlGO0Y9Rj9GQUZDRmVuRmduLUZQNiVRJEtJY0YnRlNGVkYvLUYjNiQtRlA2JVEibUYnRlNGVkYvLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zqby8lKWJldmVsbGVkR0Y4Ri8=), so: Pkklb2RlMkc2Ii8tSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkidkdGJDYjSSJ0R0YkRi0sLCoqLCZJI0t6R0YkIiIiKiZJI0tmR0YkRjJJJEtJY0dGJEYyISIiRjJJIm1HRiRGNiwmRjJGMioqRjRGMkkjS2FHRiRGMkY1RjJGN0Y2RjJGNi1JInhHRiRGLEYyRjIqLkY0RjJJI0t2R0YkRjJGNUYyRjdGNkY4RjZGKkYyRjYqLEY0RjJGNUYyRjdGNkY4RjYtSSN4M0dGJEYsRjJGMiosRjRGMkY1RjJGN0Y2RjhGNi1JImdHRiRGLEYyRjIqJi1JI0ZkR0YkRixGMi1GNzYjRjhGNkYy

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ2RzYiNiNJInRHRilGKywsKiosJkkjS3pHRikiIiIqJkkjS2ZHRilGMEkkS0ljR0YpRjAhIiJGMEkibUdGKUY0LCZGMEYwKipGMkYwSSNLYUdGKUYwRjNGMEY1RjRGMEY0LUkieEdGKUYqRjBGMCouRjJGMEkjS3ZHRilGMEYzRjBGNUY0RjZGNEYnRjBGNCosRjJGMEYzRjBGNUY0RjZGNC1JI3gzR0YpRipGMEYwKixGMkYwRjNGMEY1RjRGNkY0LUkiZ0dGKUYqRjBGMComLUkjRmRHRilGKkYwLUY1NiNGNkY0RjA=

The last equation comes from the integrator in the integral compensation, and this also is complex because the right hand side also has the acceleration in it. We let Maple do the work of solving for this and eliminating it as follows: QyQ+SSVvZGUzRzYiLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSN4M0dGJTYjSSJ0R0YlRi4sKiooSSNLaUdGJSIiIkkjS2FHRiVGMi1GKDYkLUkidkdGJUYtRi5GMiEiIiooSSNLdkdGJUYyRjFGMkY2RjJGOComRjFGMi1JInhHRiVGLUYyRjgqJkYxRjItSSJnR0YlRi1GMkYyRjg=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder We now substitute all of the parameter values into these equations: Pkklb2RlMUc2Ii1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRGI0kncGFyYW1zR0Yk

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ4RzYiNiNJInRHRilGKy1JInZHRilGKg==

Pkklb2RlMkc2Ii1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRGI0kncGFyYW1zR0Yk

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ2RzYiNiNJInRHRilGKywsLUkieEdGKUYqJCErISpRTFpPISInRickISsoZXImWyoqISIpLUkjeDNHRilGKiQiK29lJXklUkYxLUkiZ0dGKUYqRjctSSNGZEdGKUYqJCIrKW82X1EjISM4

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Pkklb2RlM0c2Ii1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRGI0kncGFyYW1zR0Yk

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSN4M0c2IjYjSSJ0R0YpRissKi1GJDYkLUkidkdGKUYqRiskIStNPGUqXCUhIzdGLyQhKVtwWXIhIiktSSJ4R0YpRiokIScqZiRHISIlLUkiZ0dGKUYqJCInKmYkR0Y7

QyQ+SSdzeXNfZGVHNiItJkkvRHluYW1pY1N5c3RlbXNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0krU3RhdGVTcGFjZUdGJTYlNyVJJW9kZTFHRiVJJW9kZTJHRiVJJW9kZTNHRiUvSS5pbnB1dHZhcmlhYmxlR0YlNyQtSSNGZEdGJTYjSSJ0R0YlLUkiZ0dGJUY4L0kvb3V0cHV0dmFyaWFibGVHRiU3JC1JInhHRiVGOC1JInZHRiVGOCEiIg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%; JSFH LSZJL0R5bmFtaWNTeXN0ZW1zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJLFByaW50U3lzdGVtR0YoNiNJJ3N5c19kZUdGKA==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

We can verify the behavior of our controller by examining how it responds to disturbances and guideway changes and/or irregularities. If we want to see what happens when there is a change in the force applied to the vehicle (as for, example, a vertical gust), the Fd(t) input can be set to a step input, and if we want to see what happens when there is an downward slope, the guideway input LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== can be set to be a ramp. JSFH Control Response to Disturbance Forces and Guideway Changes Gap Error,Gap Velocityuse DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. end use; JSFH use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. SetProperty(CheckBox2, value, false); end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. #CheckBox3 SetProperty(CheckBox3, value, false); end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. if DocumentTools:-GetProperty(ComboBox0, 'value') = "Gap Error" then GapError := 'true': InputTitle := "Gap Error": lunits:= "inches" else GapError := 'false': InputTitle := "Gap Velocity": lunits:= "inches/sec" end if: if DocumentTools:-GetProperty(CheckBox2, 'value') = "true" then SlopeOn := 1: else SlopeOn := 0: end if: if DocumentTools:-GetProperty(CheckBox3, 'value') = "true" then WindOn := 1: else WindOn := 0: end if: Wind := WindOn*Step(10000, 1): #Slope := SlopeOn*Ramp(-1, 1, 0): Slope := SlopeOn*Ramp(-1.2, 1, 0): InputOutputPlot := Array(1 .. 1, 1 .. 2): if `and`(WindOn = 1, SlopeOn = 0) then InputOutputPlot[1, 1] := plot(Wind, color = blue, thickness = 2, gridlines = true, title = "Wind", view = [0 ..10, 0 .. 10020], labels = ["time [sec]", "Wind [lbs]"], labeldirections = [horizontal, vertical], font = [HELVETICA, 10], labelfont = [HELVETICA, 10]): if GapError = true then OutputType := 12.0*x: else OutputType := 12.0*v: end if: ylabel := cat(InputTitle," [", lunits ,"(" , convert(OutputType, string), ")]"): elif `and`(WindOn = 0, SlopeOn = 1) then InputOutputPlot[1, 1] := plot(Slope, color = blue, thickness = 2, gridlines = true, title = "Slope", view = [0 .. 10, -9 .. 0], labels = ["time [sec]", "Slope [inches]"], labeldirections = [horizontal, vertical], font = [HELVETICA, 10], labelfont = [HELVETICA, 10]): if GapError = true then #OutputType := (x-Ramp(-1, 1)): OutputType := (x-Ramp(-1.2, 1)): else #OutputType := (v-Step(-1.2, 1)): OutputType := (v-Step(-1.2, 1)): end if: ylabel := cat(InputTitle," [", lunits,"]"): end if: InputOutputPlot[1, 2] := DynamicSystems[ResponsePlot](sys_de, [Wind, Slope], output = OutputType, duration = 10, numpoints = 1000, gridlines = true, thickness = 2, title = InputTitle, color = red, labels = ["time [sec]", ylabel], labeldirections = [horizontal, vertical], font = [HELVETICA, 10], labelfont = [HELVETICA, 10]): DocumentTools:-Do(%Plot3 = InputOutputPlot[1, 2]); DocumentTools:-Do(%Plot2 = InputOutputPlot[1, 1]); end use; The response obtained below was calculated for a 10,000 pound aerodynamic gust of wind, and for a guideway slope that descends 1 ft over 10 seconds of travel. 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Figure 6B: System Response Plot A quick look at the control response plots shows that the controller is able to correct for air gap changes due to disturbance forces, such as a 10,000 pound aerodynamic gust of wind, in about 0.1 seconds. Moreover, the air gap due to such a strong wind is only 0.004 inches. In addition, we can see that the controller is able to maintain the air gap even when the guiderail path is perturbed. These results show that the controller meets the design criteria specified in Section 2.2.

By maintaining the air gap between the guideway and train we expect the control system to be able control the motion of the vehicle in terms of heave, sway, pitch and yaw. In this section we will use the magnet model and control system equations determined previously to derive the equations of motion for the entire Maglev system. We will then examine the Maglev train system response to disturbance forces and guideway changes.

Figure 7 shows the magnets as they are grouped along the sides of the vehicle. Twelve groups of four magnets are mounted along the vehicle (4 magnets are grouped as a single assembly and the force they apply is along the axis of the magnets through a compliant joint). Because of the cant angle the vertical forces (along the z-axis) are given by 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 and the horizontal forces along the y axis are given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtc3ViR0YkNiUtSSNtb0dGJDYtUSJ+RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRiM2JC1GLDYlUSJpRidGL0YyRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y4LUYsNiVRJHNpbkYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyZiZXRhO0YnRi9GMkYyRjJGMg==, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy is the magnet cant angle. 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Figure 7. Maglev train geometry Location of the 12 magnet modules, their produced forces f1 through f12, and the three lever arms for the moments (magnets are placed symmetrically so these 3 define all 6 lever arms). The moments and forces created by the magnets depend, of course, on the geometric location of the magnets. All of the magnets are canted at an angle of 35\302\260. Thus the attractive force of the magnet (which is what has been modeled so far) must be resolved into vertical and sideways (sway) forces. Once this has been done, the moments created by the magnets can be defined. We create a vector of forces for the magnets and then also create a vector of lever arms for each of the rotational degrees of freedom. When defining the equations we assume that all of the rotational angles are small so the gyroscopic forces and Coriolis forces can be neglected. 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print(); # input placeholder

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JSFH The row vectors 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and 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are each an array of distances (or +1's and -1's in the case of the forces) that multiply the forces to give the forces or moments (summed together at the center of gravity) for the 12 magnet modules. They are given by: Row Vectors 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and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiTEYnLyUnZmFtaWx5R1EmQXJpYWxGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNidRInpGJ0YyRjVGOEY7RjJGNS9GPFEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0YyRjVGQw==: QyQvJkkiTEc2IjYjSSZUaGV0YUdGJi0mSSdWZWN0b3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0knY29sdW1uR0YmNiMvSSQlaWRHRiYiKUszPGwhIiI= QyQvJkkiTEc2IjYjSSRwc2lHRiYtJkknVmVjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNJJ2NvbHVtbkdGJjYjL0kkJWlkR0YmIiknZmdzJyEiIg== QyQvJkkiTEc2IjYjSSJ5R0YmLSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjSSdjb2x1bW5HRiY2Iy9JJCVpZEdGJiIpM1gxbiEiIg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= QyQvJkkiTEc2IjYjSSJ6R0YmLSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjSSdjb2x1bW5HRiY2Iy9JJCVpZEdGJiIpISl6L28hIiI= The equations of motion defined above are combined with the magnet module equations to obtain the closed-loop system equations that describe the behavior of the Maglev train. We rename the state variable x3(t) to aux(t): PkksY29udG9ybE9ERXNHNiItSSVzdWJzRyUqcHJvdGVjdGVkRzYkLy1JI3gzR0YkNiNJInRHRiQtSSRhdXhHRiRGLDclSSVvZGUxR0YkSSVvZGUyR0YkSSVvZGUzR0Yk

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print(); # input placeholder

NyUvLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInhHNiI2I0kidEdGKkYsLUkidkdGKkYrLy1GJTYkRi1GLCwsRigkISshKlFMWk8hIidGLSQhKyhlciZbKiohIiktSSRhdXhHRipGKyQiK29lJXklUkY1LUkiZ0dGKkYrRjstSSNGZEdGKkYrJCIrKW82X1EjISM4Ly1GJTYkRjlGLCwqRjAkIStNPGUqXCUhIzdGLSQhKVtwWXJGOEYoJCEnKmYkRyEiJUY9JCInKmYkR0ZP

Next, the equations are duplicated 12 times for each of the magnet modules: QyQ+SS1saXN0VmFybmFtZXNHNiI3JkkieEdGJUkidkdGJUkkYXV4R0YlSSJnR0YlISIi

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder QyQ+SSlhbGxjb2lsc0c2IjcjLUkkc2VxRyUqcHJvdGVjdGVkRzYkLUkjb3BHRik2Iy1JJXN1YnNHRik2JC1JJG1hcEdGKTYkZio2I0kmZHVtbXlHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS85JChGO0kiaUdGJUYlRiVGJUktbGlzdFZhcm5hbWVzR0YlSSxjb250b3JsT0RFc0dGJS9GPTsiIiIiIzchIiI=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder Expression for the force generated per module (for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= modules) is defined as: PkkuY29pbEZvcmNlRXhwckc2Ii8tSSJGR0YkNiNJInRHRiQqKEkibUdGJCIiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ2R0YkRihGKUYsSSJuR0YkISIi

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkZHNiI2I0kidEdGJSooSSJtR0YlIiIiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInZHRiVGJkYnRipJIm5HRiUhIiI=

Parameter values are substituted: PkkqY29pbEZvcmNlRzYiLUklc3Vic0clKnByb3RlY3RlZEc2JTwjL0kibkdGJCIjN0kncGFyYW1zR0YkSS5jb2lsRm9yY2VFeHByR0Yk

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkZHNiI2I0kidEdGJSwkLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInZHRiVGJkYnJCIrKyt2JFwkISIo

Creating the force expressions for y and z axes: PkkrWUZvcmNlRXhwckc2Ii8tSSNGeUdGJDYjSSJ0R0YkKiYtSSJGR0YkRigiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0klYmV0YUdGJEYt

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0Z5RzYiNiNJInRHRiUqJi1JIkZHRiVGJiIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSSViZXRhR0YlRis=

PkkrWkZvcmNlRXhwckc2Ii8tSSNGekdGJDYjSSJ0R0YkKiYtSSJGR0YkRigiIiItSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0klYmV0YUdGJEYt

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JI0Z6RzYiNiNJInRHRiUqJi1JIkZHRiVGJiIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSSViZXRhR0YlRis=

Substituting parameter values and generating the force for each individual module: QyQ+SS1saXN0VmFybmFtZXNHNiI3JkkiRkdGJUkjRnlHRiVJI0Z6R0YlSSJ2R0YlISIi

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder QyQ+SSdZRm9yY2VHNiItSSRtYXBHJSpwcm90ZWN0ZWRHNiRJJWV2YWxHRig3Iy1JJHNlcUdGKDYkLUklc3Vic0dGKDYkLUYnNiRmKjYjSSZkdW1teUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLzkkKEY7SSJpR0YlRiVGJUYlSS1saXN0VmFybmFtZXNHRiUtRjA2JUkqY29pbEZvcmNlR0YlPCMvSSViZXRhR0YlLUkmZXZhbGZHRig2Iy1JKGNvbnZlcnRHRig2JiIjTi5JJnVuaXRzR0YlLkkkZGVnR0YlLkkkcmFkR0YlSStZRm9yY2VFeHByR0YlL0Y9OyIiIiIjNyEiIg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%; QyQ+SSdaRm9yY2VHNiItSSRtYXBHJSpwcm90ZWN0ZWRHNiRJJWV2YWxHRig3Iy1JJHNlcUdGKDYkLUklc3Vic0dGKDYkLUYnNiRmKjYjSSZkdW1teUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLzkkKEY7SSJpR0YlRiVGJUYlSS1saXN0VmFybmFtZXNHRiUtRjA2JUkqY29pbEZvcmNlR0YlPCMvSSViZXRhR0YlLUkmZXZhbGZHRig2Iy1JKGNvbnZlcnRHRig2JiIjTi5JJnVuaXRzR0YlLkkkZGVnR0YlLkkkcmFkR0YlSStaRm9yY2VFeHByR0YlL0Y9OyIiIiIjNyEiIg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%; Vehicle Dynamics Heave The vehicle dynamics in the z-direction is then given by summing up all of the Z forces: Pkklb2RlWkc2Ii1JJWV2YWxHJSpwcm90ZWN0ZWRHNiMtSSVzdWJzR0YnNiVJJ3BhcmFtc0dGJEknWkZvcmNlR0YkLyomSSJtR0YkIiIiLUklZGlmZkdGJzYkLUkiekdGJDYjSSJ0R0YkLUkiJEdGJzYkRjgiIiNGMS1JJGFkZEdGJzYkSSJ4R0YkL0ZANyMtSSRzZXFHRic2JC0oSSNGekdGJEkiaUdGJEY3L0ZJO0YxIiM3

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LywkLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkiekc2IjYjSSJ0R0YsRi5GLiQiJkQ+JSEiIiw6LUYlNiQtSSN2MUdGLEYtRi4kIitiQyI+J0chIigtRiU2JC1JI3YyR0YsRi1GLkY3LUYlNiQtSSN2M0dGLEYtRi5GNy1GJTYkLUkjdjRHRixGLUYuRjctRiU2JC1JI3Y1R0YsRi1GLkY3LUYlNiQtSSN2NkdGLEYtRi5GNy1GJTYkLUkjdjdHRixGLUYuRjctRiU2JC1JI3Y4R0YsRi1GLkY3LUYlNiQtSSN2OUdGLEYtRi5GNy1GJTYkLUkkdjEwR0YsRi1GLkY3LUYlNiQtSSR2MTFHRixGLUYuRjctRiU2JC1JJHYxMkdGLEYtRi5GNw==

JSFH Sway The vehicle dynamics in the y-direction is then given by summing up all of the Y forces: JSFH Pkklb2RlWUc2Ii1JJWV2YWxHJSpwcm90ZWN0ZWRHNiMtSSVzdWJzR0YnNiVJJ3BhcmFtc0dGJEknWUZvcmNlR0YkLyomSSJtR0YkIiIiLUklZGlmZkdGJzYkLUkieUdGJDYjSSJ0R0YkLUkiJEdGJzYkRjgiIiNGMS1JJGFkZEdGJzYkSSJ4R0YkL0ZANyMtSSRzZXFHRic2JComKSEiIkkiaUdGJEYxLShJI0Z5R0YkRklGN0YxL0ZJO0YxIiM3

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print(); # input placeholder

LywkLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkieUc2IjYjSSJ0R0YsRi5GLiQiJkQ+JSEiIiw6LUYlNiQtSSN2MUdGLEYtRi4kISt1RSRSKyMhIigtRiU2JC1JI3YyR0YsRi1GLiQiK3VFJFIrI0Y5LUYlNiQtSSN2M0dGLEYtRi5GNy1GJTYkLUkjdjRHRixGLUYuRj4tRiU2JC1JI3Y1R0YsRi1GLkY3LUYlNiQtSSN2NkdGLEYtRi5GPi1GJTYkLUkjdjdHRixGLUYuRjctRiU2JC1JI3Y4R0YsRi1GLkY+LUYlNiQtSSN2OUdGLEYtRi5GNy1GJTYkLUkkdjEwR0YsRi1GLkY+LUYlNiQtSSR2MTFHRixGLUYuRjctRiU2JC1JJHYxMkdGLEYtRi5GPg==

Pitch The dynamic equation is a function of the level arm distance from the center of gravity for each of the modules. Collecting the level arm into a list: QyY+SSdMUGl0Y2hHNiI3KCQhJG4mISIiRickISVfNiEiI0YqJCEkdyZGLEYtRik+RiQ3JC1JI29wRyUqcHJvdGVjdGVkRzYjRiQtRjI2Iy1JJG1hcEdGMzYkZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQ5JEYpRiVGJUYlRiRGKQ== The equation of motion of the pitching motion, with augmented forces generated by the motion of the modules relative to the rail, is given by: 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print(); # input placeholder

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JSFH The closed-loop system equations defining the behavior of the entire Maglev train in terms of heave, sway, pitch and yaw is obtained by combining the twelve magnet module equations and the four equations of motion defined above: QyQ+SStzeXNPdmVyYWxsRzYiLSZJL0R5bmFtaWNTeXN0ZW1zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNJKk5ld1N5c3RlbUdGJTYlNyctSSNvcEdGKjYjSSlhbGxjb2lsc0dGJUklb2RlWUdGJUklb2RlWkdGJUkpb2RlVGhldGFHRiVJJ29kZVBzaUdGJS9JLmlucHV0dmFyaWFibGVHRiU3JC1JI0ZkR0YlNiNJInRHRiUtSSRzZXFHRio2JC0oSSJnR0YlSSJpR0YlRj0vRkU7IiIiIiM3L0kvb3V0cHV0dmFyaWFibGVHRiU3Ji1JInlHRiVGPS1JInpHRiVGPS1JJnRoZXRhR0YlRj0tSSRwc2lHRiVGPUZI

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%;

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JSFH

The heave response plots to a vertical gust of wind and a descending guideway slope of 0.1ft/sec are shown in Figures 8 and 9, respectively. From the plots we see that the control system developed for each magnet is able to work together to minimize the heave displacement. 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Figure 8A: Input Signal 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Figure 8B: Input Signal Response to Descending Guideway JSFH 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Figure 9A: Input Signal 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Figure 9B: Input Signal

This worksheet documented the design and development of a controller to guide a EMS Maglev train along a guideway. In section 1, we determined a mathematical model to describe the magnet dynamics of the lifting magnets. In section 2, we developed a PID controller with feedback acceleration to maintain an air gap between the train and the guideway of 2 inches and thus counteract the inherent unstable nature of the lifting magnets (ie. when there is no current in the coils the vehicle will drop under the influence of gravity causing the undercarriage to hit the guideway and when the current in the magnet is very large, the magnet will be attracted to the rail and hit the rail). We showed that the behavior of our controller adhered well to a set control criteria by investigating the system response to disturbance forces and guideway changes. In section 3, we used the magnet model and control system equations to derive the equations of motion for the entire Maglev system. These equations were then used to examine the Maglev train's response to disturbance forces and guideway changes. We were able to show that the control system defined in section 2 was able to control the motion of the train in terms of heave displacement.

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