Controlo Em Espaço de Estados. First Test

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1 Mestrado Itegrado em Egeharia Electrotécica e de Computadores Cotrolo Em Espaço de Estados 13/1 First Test April, 1, h. room C1 Duratio hours Its ot allowed either cosultatio of ay kid or programmable calculators Quotação: P1 a)1 b) c) d)1 P a) 3 b) 1 c) P3 a) 1 b) 1 c) 1 d) 1 P-. P1. Figure P1-1 represets a side cut view of a water delivery caal i which there is a gate ( comporta ) that is moved by a motor. u, comado do motor ível da água, y Superfície da água Motor Comporta Água Água Fudo do caal Figure P1-1. Problem 1. The trasfer fuctio that relates the sigal u (gate commad)with the gate positio, v, is 1 G ( 1 s). s s 3 The trasfer fuctio that relates the gate positio, v, with the water level y measured by a sesor located upstream the gate is s G ( s). 3 s 3s s Aswer the followig questios: 1

2 a) Write a state realizatio of G 1. b) Write a state realizatio of G. c) Write a state realizatio of the global system (give by the series of G 1.ad G ), ad such that the state of the global system is the cocateatio of the states of G 1.ad G that you have defied i a) ad b). d) Draw a block diagram of G, usig oly itegrators, gais ad algebraic sums, all scalar. P. I relatio to the liear state model x = Ax, cosider the matrices umbered from 1 to : A 1 = [ 1. ] A 1.5 = [ 1.5 ] A 5/3 /3 3 = [ /3 5/3 ] A = [ 1 ] Cosider also the phase portraits show i figure P-1 ad idetified with the letters A, B, C ad D. A B x x C D x x

3 a) State which matrix matches which plot. Justify.. b) I relatio to A compute a expressio that yields the state as a explicit fuctio of time, kowig that the iitial coditio is x() = [ 5 ]. c) What really happeed at the Waterloo battle. The Waterloo battle took place i 1 Jue 115 (close to the village with the same ame, i Belgium). I this battle the allied army, commaded by the Duke of Welligto (most soldiers i this army were british, but 17 differet laguages were spoke!) beated the frech army commaded by Napoleo. The victory oly felt to the side of Welligto after the arrival of geeral Bucher (a prussia that exclaimed whe he visited Lodo: What a beatifull city to be saked ) ad his troops, that joied the allied forces. Matrix A above represets the state model for the evolutio of the umber of frech ad allied troops durig the battle. I this model, represets the umber of frech soldiers (divided by, meaig that = 1 meas frech soldiers) ad x the umber of allied soldiers (also divided by ). Iicially, the frech had 3 soldiers ad the allies 5. Geeral Bucher brough soldiers that joied the allies. Show that, if Bucher did ot arrived, Napoleo would wi the Waterloo battle (meai that, whe time passed, the troops would reach a situatio i which > ad x = ), but, with the reiforcemet of Bucher soldiers, is Welligto that wis (meaig that, whe time passes, oe reaches a situatio i which = ad x > ). I order to simplify thigs, assume that Bucher ad his soldiers arrived at the begiig of the battle. Use a sketch of the phase portrait to justify your aswer. 3

4 P3. A heat exchager allows to trasfer heat from a fluid (heat source) to aother that we wat to heat. It cosists of two separate circuits. I oe of the circuits circulates the fluid to heat ad i the other circulates the fluid that is the heat source. By adjustig the flow of the hot fluid oe ca vary the quatity of eergy trasferred to the fluid to heat, by uit of time, thereby chagig its temperature. Figure P3-1 shows a schematic view of a heat exchager. u Comado da válvula vapor Fluido a aquecer Válvula y Temperatura à saída sesor de temp. Fluido aquecido Fig. P3-1 Schematic view of a heat exchager. I this problem, we wat to desig a liear state feedback cotroller to regulate the temperature of the outlet fluid, by actig o a steam valve. For that purpose, we kow the followig state model that relates icremets aroud a workig poit i the steam admissio valve commad u with the icremets y of the fluid temperature at the outlet: x( x( u(.1.1 y 1 x( Aswer the followig questios: a) Desig a state feedback cotroller that places the closed-loop poles at.5 j.7 b) Tell if the system is or is ot cotrollable. Justify your aswer. c) Desig a state observer with the poles of the estimatio error dyamics at.15 j. d) Tell if the system is or is ot observable. Justify your aswer.

5 P. Cosider the system described by the liear state equatio where dx Ax bu, with iitial coditio x (), x R (colum vector) is the state, u R (scalar), is the iput, t R deotes time, ad variables A R ad b R are matrices of parameters. Usig the chage of At x( e z(, where z R is a ew state variable, get a expressio for the solutio of the state equatio as a fuctio of t, of the iitial coditio, the iput ad of the matrices that defie the system. Aids: d e d At At Ae, M ( N( MN MN Glossary Comporta Gate Água Water Vapor Steam Nível da água Water level Comado do motor Motor commad Fluído a aquecer Fluid do heat Fluído aquecido Heated fluid Válvula Valve Temperatura à saída Output temperature 5