Page - (000076) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

70 9 StateFeedback 9.1 RegulationExample In theprior chapteron regulation, I analyzed theprocess inEq.6.8as P(s)= 1 s2+αs+β, (9.1) withα=0.1andβ =1.This processhas a resonancepeaknearω=1.The state- spacemodel for thisprocess is x˙1 = x2 x˙2 =−βx1−αx2+u (9.2) y= x1, inwhich the dynamics are equivalent to a second-order differential equation, x¨+ αx˙+βx =u,with y= x. For a state-space regulation problem, the design seeks to keep the states close to their equilibriumvalues.We can use equilibriumvalues of zerowithout loss of generality.Whenthestatesareperturbedawayfromtheirequilibrium,weadjust the input control signal,u, todrive the statesback to their equilibrium. Thecost functioncombines thedistancefromequilibriumwithregardtothestate vector,x, and the energy required for the control signal,u. Distances and energies are squared deviations from zero, whichwe canwrite in a general way in vector notationas J = ∫ T 0 ( u′Ru+x′Qx)dt, (9.3) inwhichR andQ arematrices that give the costweightings for components of the statevector,x= x1,x2,... , andcomponentsof the inputvector,u=u1,u2,... . In theexamplehere, thereisonlyoneinput.However,state-spacemodelseasilyextend tohandlemultiple inputs. FortheregulationprobleminFig.9.1, thegoalistofindthefeedbackgainsforthe states given in thematrixK thatminimize the cost function. The full specification of theproblemrequires thestateequationmatricesforuse inEq.2.6,whichwehave fromEq.9.2as A= ( 0 1 −β −α ) B= ( 0 1 ) C= (1 0), (9.4) andthecostmatrices,RandQ. Inthiscase,wehaveasingleinput,sothecostmatrix for inputs,R, canbeset toone,yieldingan input cost term,u2. For the state costs,wecould ignore the secondstate, x2, leavingonly x1 = y, so that the state costwould beproportional to the squaredoutput, y2 = e2.Here, y is