Term Project Gain Calculations

Authors: Damond Li, Christopher Or

Date Created: 12/4/2021

Date Modified: 12/6/2021

Found here are the steps used, based on lecture, to determine the gains for the Full State Feedback (FSFB) controller used for the ball balancing project.

Establish gain variables

syms s K1 K2 K3 K4;

A matrix from HW 0x03

A = [0          0           1       0
     0          0           0       1 
    -84.6      2.6      0       0
     1518.6   78.4     0       0];

B matrix from HW 0x03

B = [0
     0
     35.1365
    -630.6554];

Identity matrix

I = [1 0 0 0
1 0 0
0 1 0
0 0 1];

K matrix of gain values

K = [K1
     K2
     K3
     K4]';

Closed-loop system

A_cl = A - (B * K);

Characteristic polynomial matching

Z = s*I - A_cl;

detZ = det(Z);

C = coeffs(detZ, s);

C = C'

C =

Gain calculation using chosen parameters

w_n = 8.2;          % Natural frequency
zeta = 0.8;         % Damping ratio
lamda3 = -16;       % Pole
lamda4 = -17;       % Pole

Closed-loop polynomial

P = expand((s - lamda3) * (s - lamda4) * (s^2 + 2 * zeta * w_n * s + w_n^2))

P =

a_coeff = coeffs(P, s)';

T1 = C(1) == a_coeff(1);

T2 = C(2) == a_coeff(2);

T3 = C(3) == a_coeff(3);

T4 = C(4) == a_coeff(4);

[M, f] = equationsToMatrix([T1; T2; T3; T4], [K1; K2; K3; K4]);

Solve for values for gain matrix

K_matrix = vpa(inv(M) * f, 5)

K_matrix =