AUTOMATIC CONTROL SYSTEMS II HOMEWORK #3

Submitted by: Barkan ÇİLEK

Submission Date: 21/01/2011

Submitted to:Zeynep ÇAKIR

TASK 1

We have the same plant as in homework 2 but this time we will analyze the system

response using an LQR Controller and see if it satisfies Level 1 Handling Qualities.

Our plant is:

With the cost function which we want to minimize and using Ricatti Equation

MATLAB will calculate the optimum trajectory with the inputted weighting matrices(Q,R)

and find the feedback gains associated with it.I want to control the slower modes and make

them faster such that they satisfy Level 1 Handling Qualities.Open loop eigenvalues are:

>> eig(A)

ans =

0.0667 + 0.2240i

0.0667 - 0.2240i

-0.3705

-0.3563

I want to intervene to the slower modes which are actually phugoid mode given by the

unstable eigenvalues.In this mode we have large amplitude variation of pitch angle,altitude

and air velocity.Pitch attitude and air velocity refers to W which is the vertical velocity in

longitudunal axis while pitch angle is represented by θ.Construction my weight matrix F,i will

assign them larger values to supress the diverging ossilations.For G matrix,for simplicty i will

assign idendity matrix.The weighting matrices are:

Q=[100 0 0 0;0 1 0 0;0 0 1 0;0 0 0 100] Q = 100 0 0 0 0 1 0 0 0 0 1 0 0 0 0 100

R=1 R = 1

My LQR controller gains,S matrix and closed loop eigenvalues are found as:

[K,S,e]=lqr(A,B,Q,R) K = -9.9805 -0.0255 4.2514 58.7906 S = 1.0e+003 * 0.0182 -0.0000 0.0529 -0.0521 -0.0000 0.0015 0.0004 -0.0005 0.0529 0.0004 0.3523 0.0304 -0.0521 -0.0005 0.0304 2.0281 e = -10.7715 -0.2466 + 2.2193i -0.2466 - 2.2193i -0.3249

Closed loop eigenvalues are:

damp(e) Eigenvalue Damping Freq. (rad/s) -1.08e+001 1.00e+000 1.08e+001 -2.47e-001 + 2.22e+000i 1.10e-001 2.23e+000 -2.47e-001 - 2.22e+000i 1.10e-001 2.23e+000 -3.25e-001 1.00e+000 3.25e-001

Observe that I moved my unstable conjugate eigenvalues to the LHS of s-plane and

their damping values(seen in red) do satisfy Level 1 qualities which were defined to be

Longitudinal Phugoid: ξ > 0.04

My short period mode which is (-1.08e+001) also satisfies the short period

requirement which were defined to be:

Short Period Specifications: ξ > 0.04 , wn > 0.28

TASK 2

In this task,we will analyze the response of our both open loop and closed loop

systems and simulate the response using SIMULINK.The logic behind this is simply this

equation:

x(t t) x(t) [Ax(t) Bu(t)]t

a) Using state space block,i will input my matrices and see the response for 4 different

states.To input the desired input in specified time boundary i will use signal builder.I will give

δ = 0.0174 rad=1 degree

By using different sensor matrices(C) and arrange them as a row vector to get single

output for each state the following plots are obtained.

For u(longitudunal velocity) the following is obtained.

For w(z-axis component of the velocity) the following is obtained.

For q(pitch rate) the following is obtained.

For θ(pitch attitude) the following is obtained.

b) Now i will input the new closed loop matrix -Anew- using the basic feedback law :

Anew=A-B*K Anew = -10.2976 -0.0112 5.6775 28.3837 -0.8916 -0.3263 0.4788 3.7570 1.6654 0.0035 -0.9657 -9.8063 0 0 1.0000 0

Now inputting these matrix to my state space box,i will obtain the state responses as following.

To see the steady-state response,i will now simulate the system for 50 seconds.

For u,the following plot is obtained.

For w(z-axis component of the velocity) , plot below is obtained.

For q(pitch rate),the following plot is obtained.

For pitch attitude(θ) , the following is obtained.

CONCLUSION

In the second homework by specifying a characteristic equation we calculated feedback gains

using simple feedback law.We got 2 complex conjugate roots for the closed loop however in

this LQR controller,i tried many weighting matrices other than i showed here yet neither of

them gave two complex conjugate roots.Hence,i can conclude that our optimal closed-loop

system with the minimum so called cost function does not produce such 2 complex conjugate

roots(ofcourse 4 roots at total).Hence,my calculated gains and closed loop system in the

previous homework was indeed not the optimal one.