I designed a state feedback control with integral action for output tracking applied to a LPV system with 4 scheduling parameters using LMI in MATLAB. The LMI was synthesized upon Lyapunov function.

The system dynamics are given by :

dx(t)/dt =A(ρ)x(t)+B(ρ)u(t)+E(ρ)d

y(t) = Cx(t)

x = [x1 x2]

the LMI condition is expressed as follows :

P(θ) ≥ εI

[A_cl(θ) + A_cl(θ)' + 2αP(θ), P(θ)E(ρ);

E(ρ)'P(θ), -γI ] ≤ 0

where A_cl(θ) = A_aug(ρ)*P(θ) + B_aug(ρ)*Y(θ)

A_aug(ρ) = [A(ρ) zeros(1,2); -C 0]

B_aug(ρ) = [B(ρ) 0]

P(θ) and Y(θ) are both affine in θ (i.e., P(θ) = P0 + ∑θᵢ*Pᵢ)

For many α I tried to solve the LMI but it fails. Any suggestions to overcome this problem? Could you direct me to any other approaches to design the controller?