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Question

1. Your boss says that your long awaited pay-raise depends on your success in "stabilizing" the Ax + bu where /-10 1 0 0 1 0| , b | 1 |, using a linear control-law of following system: the form u k1x1 +R2x2 +kx, k constant. Using the "tools" presented in this course, show (by appropriate calculations) that this system can or cannot be stabilized by state- feedback control. Identify the "dynamics" and associated eigenvalues of any "part of the system that is "totally uncontrollable." Explain each step in your procedure so the boss will understand Consider the nth-order system x Ax + Bu where B-constant, u-r-vector and the constant matrix A has the special property A2-I.Compute, analytically, the corresponding state-transition matrix ?(t, to) for this system. Determine the conditions for complete controllability of this system, in terms of A and B "ranks and dimensions Design an idealized linear control-law u order system y +wyun constant>0,"mimic" the ideal-model 1s-order system Nm + y,n-0, where constant > 0. Assume the specification: y(t)-, y,n(t) with a 99% settling-time of T units of time. Show the solution derivation using "subspace-stabilization techniques. Express the closed-loop x -state "gains" k in terms of (wn. T.T). Sketch the typical behavior of the state-trajectories xi (t), x2(t) [Xy-5% 3. -kM + kzx2 that will make the time-invariant, , ] in the ??, X2 state-space Show all intermediate steps For the case A-S ?] satisfies the steady-state Riccati equation PA+ATP-PbR-1bP+Q scalar > 0, and0[9:91 9192 analytical form, in terms of .41,2) and verify that P- PT>0 ; b = 11, compute analytically the 2 x 2 matrix P-PT > 0 that 0 where Rr 192 9292 0: 92 0. Write out each element of py of P, in

Explanation / Answer

1. Here we have used matlab for easier calculation. first we enter the coeffient matrix and used the command to check for controlability.Here we have compared the length. We know that when the rank(A) is not equal to the rank (Pc) the system is not controlable. we have generated the eigen values of A also.

2.A system can be controlled by using a feedback only when itis absolute controlable. here we use simple feedback u= kx. and tried to calculate the values of k . but since the system is not controllable it is not posisble to define K.

3. State transition matrix is also calculated using matlab . Asystem is completlety controllable when rank(a) is equal to rank (Pc). When a system is said to be fully controllable then its output can be fully controlled by varying its input vectors.

a=[1 1 0;0 1 0;0 0 1]
b=[1;1;1]

pc=ctrb(a,b)
n= rank(pc)
if n== length(a)
disp('contollable')
else
disp('not controllable')
end
e=eig(a)

s=size(a)
if s(1)==rank(pc)
p1=[0 0 1]*inv(pc)
p=[p1;p1*a;p1*a*a]
ja=poly(a)
a1=ja(2)
a2=ja(3)
a3=ja(4)
jj=poly(j)

b1=jj(2)
b2=jj(3)
b3=jj(4)
kc=[b3-a3;b2-a2;b1-a1]'
k=kc*p
k1=k(1)
k2=k(2)
k3=k(3)
end
[V,D] = eig(a)
statematrix = V*diag(exp(diag(D)))/V
  
  
ANSWER
  
a =

1 1 0
0 1 0
0 0 1

b =

1
1
1

pc =

1 2 3
1 1 1
1 1 1

n =

2

not controllable

e =

1
1
1

s =

3 3

V =

1.0000 -1.0000 0
0 0.0000 0
0 0 1.0000

D =

1 0 0
0 1 0
0 0 1

statematrix =

2.7183 0 0
0 2.7183 0
0 0 2.7183

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