Use the example from Notes pages 149-151, solve the followings:

1. Understand the linearised system, and describe in detail.
2. Find out whether linearised system is controllable and observable.
3. Find the Gain Matrix K by using the
   1. pole placement and
   2. Ackermann's formula.
4. Write the Matlab code to implement the above three mentioned steps.
5. Plot the open loop and closed loop responses in Matlab.

Write the above tasks (1) - (3) in latex or Word file with detailed derivation. Task (4) - (5) implement and submit MATLAB files. You can use the attached m file as a starting point.

M-File Code:

(M) Rx Consider a two dimensional pioblem where the pendulum is comstrained, to move in the vertical plane. For inis system, The contore input is frrce ' f ' That moves the cart horizontally and The outputs are angular position of the pardulum ' ' and the horizontal positun of the cart' x '. in (1) summing forces in the freebody diagram of cast in honzontal direction, Mx+bx+1x we cansum vertical direction forces useful. Now summig the forces in The free-body diggram of Pendulum in horizonial direction, we get exprestion for the ' reactive. force N N=mx+mlcosml2sin Sussosticute \& (2) in (1) Mx+bx+mx+mlcosml2sin=F(M+m)x1+bx+mlcosml2sin=f. (2+ml2)mgl=mlx(10)(M+m)x+bxml=u(11)(u=f). (u=f). pace: xx=00000I(M+m)+mml2(I+ml2b)0I(m+m)+nml2mlbI(M+m)+Mml2m2gl20I(M+m)+nmL2mgl+0I(m+m)+mml2I+ml20I(M+m)+mml2mly=[10000100]xx+[00]u M=.5; m=0.2; b=0.1; I =0.006; g=9.8; I=0.3; P=I(M+m)+MmI2; sdenominator for the A and B matrices A=00001(I+ml2)b/p0(mlb)/p0(m2gl2)/p0mgl(M+m)/p0;0;1;0]; B=[0; (I+ml2)/p; 0 ; m1/p]; c=[1000; 0010]; D=[0; 0] ; states ={ 'x' 'x_dot' 'phi' 'phi_dot' }; inputs ={ 'u' }; outputs ={ 'x'; 'phi' }; sys_ss = ss ( A,B,C,D, 'statename', states, 'inputname', inputs, 'outputname', out eig(A) rank(ctrb(A,B))