LMIs in Control/pages/Unsat Inp Stabilization

LMI For Stabilization Condition for Systems With Unsymmetrical Saturated Control

The LMI in this page gives the feasibility conditions which, if satisfied, imply that the correstponding system can be stabilized.

The System

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+BSat(u(t)),\\x(0)&=x_{0},\\Sat(u)_{i}=,\end{aligned}}}

where ${\displaystyle x\in \|R^{n}}$  is the state, ${\displaystyle u\in \|R^{m}}$  is the control input.

For the system given as above, its symmetrical saturated control form can be derived by following the procedure in the original article. The new system will have the form:

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+{\tilde {B}}sat(z(t))+Ew\end{aligned}}}

where ${\displaystyle w_{i}=u_{i}-{\frac {\alpha _{i}-\beta _{i}}{2}},z_{i}=w_{i}{\frac {2}{\alpha _{i}+\beta _{i}}}}$

The Data

The system matrices ${\displaystyle (A,{\tilde {B}},E)}$ , the saturation bounds ${\displaystyle (\alpha _{i},\beta _{i})}$  of the control inputs. Positive scalars ${\displaystyle \rho ,\eta }$ .

The LMI: The Stabilization Feasibility Condition

{\displaystyle {\begin{aligned}&{\text{Find}}\;X,Y,Z:\\&{\text{subj. to: }}X>0,\\&\quad {\begin{bmatrix}AX+{\tilde {B}}(D_{s}Y+{\hat {D}}_{s}^{-}Z)\end{bmatrix}}+{\begin{bmatrix}AX+{\tilde {B}}(D_{s}Y+{\hat {D}}_{s}^{-}Z)\end{bmatrix}}^{\top }+{\frac {\eta }{\rho }}X+{\frac {1}{\eta }}EE^{\top }<0,\\{\begin{bmatrix}\mu &Z_{i}\\*X\end{bmatrix}}>0,i=1,...,{\bar {m}}\end{aligned}}}

Here ${\displaystyle D_{s}}$  is a diagonal matrix with a component either 0 or 1, and ${\displaystyle D_{s}+D_{s}^{-}={\frac {\Lambda +\mathrm {T} }{2}}}$  and ${\displaystyle {\hat {D}}_{s}^{-}=e_{f_{m}}\times D_{s}^{-}}$

Conclusion:

The feasibility of the given LMI implies that the system is stabilizable with control gains ${\displaystyle K=YX^{-1},H=ZX^{-1}}$ .

Implementation

A link to CodeOcean or other online implementation of the LMI