Multivariate fitting of linear equation

I want to do multivariable fitting for the following linear equation

P = K * T

Where:

P is matix of known and controlled variable

T is matix of known, measured variable

K is the matrix of unknown coefficients

P is 4x1 matrix

P = [P1; P2; P3; P4]

K is 4X6 matrix

K = [K1_1 K1_2 K1_3 K1_4 K1_5 K1_6;

K2_1 K2_2 K2_3 K2_4 K2_5 K2_6;

K3_1 K3_2 K3_3 K3_4 K3_5 K3_6;

K4_1 K4_2 K4_3 K4_4 K4_5 K4_6]

T is 6x1 matrix

T = [T1; T2; T3; T4; T5; T6]

I have readings at multiple values of P (aprox 68) and similarly (aprox 68) reading for T at each P value.

How can I do this?

Answer 1

[Bruno Luong]

Your system is underdetermined system, there is only 4 equations for 6 unknowns.

There exists not only one solution but infinite solutions (difference belong to a vectorial space of dimension at least 2).

To pick one solution you can do

T_A = lsqminnorm(K, P)

or

T_B = K\P

Any vector of the form

N = null(K)
T_A + N * x % or T_B + N * x

where x is an arbitrary vector of size (size(N,2),1) will be solution of your problem

[Muhammad Hassan]

I know the values of P and the resulting values of T (from measurment)(linear relation). I want to find the value for K that best fits these values of P and T.

[Bruno Luong]

T=rand(5,1);
P=rand(4,1)

P = 4×1

0.3629 0.5981 0.2752 0.8331

K=T'.*(P./(T'*T));
K*T

ans = 4×1

0.3629 0.5981 0.2752 0.8331