4.(Axes Synchronization)Each axis of a two axes system is described by2Xi(s)=0.02s+1U1(s),i=1,2Verify that the state equation for this system is品图=[50-01]+[8°180J]The control objective is to obtain a state feedback controller of the formu(t)=-Kx(t),K∈R2x2which minimizes the cost functionalJ=∫{x()+x()+y(x1(0)-x2()2+p(u()+u()}dtwhere y and p are weights and the third term y(x(t)-x2(t))penalizes thesynchronization error between the two axes;i.e.the larger the value of y,the more thecontrol action will try to make the synchronization error (x,(t)-x2(t))small.(a)Set p=1.Determine (analytically)the optimal feedback gain K and the resultingclosed loop poles of A.=A-BK for y =1.(Hint:first conver the objectivefunction to a standard formulation of infinite-horizon linear quadratic optimal controlproblem;find matrices Q and R (see lecture notes #7);then solve the ARE.)(b)Still set p=l.Use the MATLAB function“lqr’to determine the optimal feedbackgain K and the resulting closed loop poles of Ac A-BK for y =0 and y=100.Eigenvalues of a matrix can be obtained using the MATLAB function“eig”.
