Homework 1. System dynamics can be expressed in nonlinear state-space form as

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Reginald Gibbs
5 years ago
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1 Hoework arane s eqation and soe iortant systes This Hoework irst ives an eale o indin the dynaics o a certain canonical syste, then asks yo to do the sae or two other canonical systes odelin Physical Systes Syste dynaics can be eressed in nonlinear state-sace or as, y h, n with t R the internal state, t R the control int, and y t R the easred ott To obtain dynaical eqations o otion or systes we can se Hailton's eqations o otion or arane's eqation o otion d F, dt q q with q t the eneralized osition vector, q t the eneralized velocity vector, and Ft the eneralized orce vector The aranian is = K-U, the kinetic enery ins the otential enery et s illstrate the se o arane s eqations by obtainin the dynaics o an iortant canonical syste INVERTED PENDUU The inverted endl on a cart is reresentative o a class o systes that incldes stabilization o a rocket drin lanch, etc The osition o the cart is, the anle o the rod is the orce int to the cart is, the cart ass is, the ass o the bob is, and the lenth o the rod is The coordinates o the bob are, z z Inverted Pendl

2 We want to se arane's eqation The kinetic enery o the cart is K The kinetic enery o the bob is K z where, z so that, z Thereore, the total kinetic enery is K K K The otential enery is de to the bob and is U z The aranian is K U The eneralized coordinates are selected as q q q T so that arane's eqations are d dt d dt Sbstittin or and erorin the artial dierentiation yields arane Eqation For Deine atrices and vectors to write this in the arane eqation or as This is a echanical syste in tyical aranian or, with the inertia atri ltilyin the acceleration vector The ter is a centrietal ter and is a ravity ter State-Sace For

3 To write these eqations in state-sace or, irst invert the inertia atri and siliy to obtain Now, the state ay be deined as T T and the int as = Then the nonlinear state eqation ay be written as, Given this nonlinear state eqation, it is very easy to silate the inverted endl behavior on a diital coter We now want to linearize this and obtain the linear state eqation The noinal oint is =, where the rod is riht One cold ind Jacobians, bt it is easier to se the aroiations, valid near the oriin,, In addition, all sqared state coonents are very sall and so set eqal to zero This yields the linear state eqation B A The ott eqation deends on the easreents taken, which deends on the sensors available Assin easreents o cart osition and rod anle, the ott eqation is, h y The cart osition ay be easred by lacin an otical encoder on one o the wheels, and the rod anle by lacin an encoder at the rod ivot oint It is diiclt to easre the velocities,, bt this iht be achieved by lacin tachoeters on a wheel and at the rod ivot oint Then, the ott eqation will chane

4 Given the linear state-sace eqations, a controller can be desined to kee the rod riht Thoh the controller is desined the linear state eqations, the erorance o the controller shold be silated in a closed-loo syste the ll nonlinear dynaics, Proble - BA BAANCER For Proble yo will ind the dynaics o the ball balancer The inverted endl can be viewed as a two-derees-o-reedo robot ar with a risatic e etensible joint ollowed by a revolte e rotational joint It has only one actator-- on the risatic link The ball balancin on a ivoted bea can be viewed as a robot ar with a revolte link ollowed by a risatic link, also havin only one actator-- on the revolte link This is in soe sense a dal syste to the inverted endl The ball balancer is reresentative o a lare class o systes in indstrial and ilitary alications The osition o the ball is, the anle o the bea is the torqe int to the bea is, the inertia o the bea is J, and the ass o the ball is J Ball Balancer a Find the dynaics o the ball balancer by indin the kinetic and otential eneries and arane's eqation b Write in arane eqation or Proble - GANTRY CRANE The antry crane is a load ssended by a wire roe ro a ovin trolley The horizontal osition o the load is, the anle o the wire is the orce int to the trolley is, the ass o the trolley is, and the ass o the load is, and the lenth o the wire roe is Asse that the wire roe is sti so that it does not le or bend

5 Gantry Crane a Find the dynaics o the antry crane by indin the kinetic and otential eneries and arane's eqation b Write the dynaics in the arane eqation or 5

6 Hoework Project obile Robot Control & Potential Fields Potential Field Use ATAB to ake a -D lot o the otential ields described below Yo will need to se lot coands and aybe the esh nction The work area is a sqare ro, to, in the,y lane The oal is at, There are obstacles at 5,6 and 7,6 Use a relsive otential o Ki / r i or each obstacle, with ri the vector to the i-th obstacle For the taret se an attractive otential o Kr, T T with rt the vector to the taret Adjst the ains to et a decent lot Plot the s o the three otential ields in -D Potential Field Naviation For the sae scenario as in Proble, a obile robot starts at, The ront wheel steered obile robot has dynaics V y V V with,y the osition, the headin anle, V the wheel seed, the wheel base, and the steerin anle Set = a Cote orces de to each obstacle and oal Cote total orce on the vehicle at oint,y b Desin a eedback control syste or orce-ield control Sketch yor control syste c Use ATAB to silate the nonlinear dynaics assin a constant velocity V and a steerable ront wheel The wheel shold be steered so that the vehicle always oes downhill in the orce ield lot Plot the resltin trajectory in the,y lane 6

EE 4443/ Control System Design Project LECTURE 2

EE 4443/ Control System Design Project LECTURE 2 Coyiht F ewis 998 All ihts eseved EE 444/59 - Contol Syste De Poject ECTURE Udated:Tesday, ne 5, 4 SOE REPRESENTATIVE DYNAICA SYSTES We discss odelin o dynaical systes Seveal inteestin systes ae discssed