LMIs in Control/Click here to continue/Applications of Non-Linear Systems/Control of Rational Systems using Linear-fractional Representations and Linear Matrix Inequalities*

Introducation

An optimization-based methodology for the multiobjective control of a large class of nonlinear systems is performed.

The System

Consider a nonlinear, continuous-time system
${\dot {y}}=A(x)+B_{u}(x)u,$
$y=C_{y}(x)+D_{yu}(x)u,$
where $x\in \mathbb {R} ^{n}$ is the state vector, $u\in \mathbb {R} ^{n_{u}}$ is the input and $y\in \mathbb {R} ^{n_{y}}$ is the output.

The Data

$x\in \mathbb {R} ^{n}$ is the state vector, $u\in \mathbb {R} ^{n_{u}}$ is the input and $y\in \mathbb {R} ^{n_{y}}$ is the output.
$A,B_{u},C_{y},D_{yu}$ are multivariable functions of x.
$A(0)=0$ (that is, 0 is an equilibrium point of the unforced system associated with the system).
$C_{y}(0)=0$ and $B_{u},D_{yu}$ have no singularities at the origin.

Proof

• ${\begin{bmatrix}A(x)&B_{u}(x)\\C_{y}(x)&D_{yu}(x)\end{bmatrix}}$ = ${\begin{bmatrix}A&B_{u}\\C_{y}&D_{yu}\end{bmatrix}}$ + ${\begin{bmatrix}B_{p}\\D_{yp}\end{bmatrix}}$$\bigtriangleup (x)$${\begin{bmatrix}C_{q}&D_{qu}\end{bmatrix}}$
  

For a given scalar \sigma > 0, we associate with the Linear differential inclusion,
${\dot {x}}=Ax+B_{u}u+B_{p}p,$
$q=C_{q}x+D_{qu}u+D_{qp}p,$
$y=C_{y}x+D_{yu}u+D_{yp}p,$
$p=\bigtriangleup (t)q,\|\bigtriangleup (t)\|\leq \sigma ^{-}1,\bigtriangleup (t)\in D(r),t\geq 0.$

The LMI: Control of Rational Systems using Linear-fractional Representations and Linear Matrix Inequalities

For given \sigma > 0, the LDI system is quadratically stable if there exists P, S, and G such that the LMIs
$P>0,S>0,G=-G^{T},S,G\in B(r),$

${\begin{bmatrix}A^{T}P+PA+C_{q}^{T}SC_{q}&PB_{p}+C{q}^{T}G+C_{q}^{T}SD_{q}p\\B_{p}^{T}P-GC_{q}+D_{qp}^{T}SC_{q}&D_{qp}^{T}SD_{qp}G-\sigma ^{2}S+D_{qp}^{T}G-GD_{q}p\end{bmatrix}}<0$

hold. Then, for every $\bigtriangleup \in D(r)$ such that $\|\bigtriangleup (t)\|\leq \sigma ^{-}1,$ $det(I-D_{pq}\bigtriangleup )^{-1}\neq 0,$

${\begin{bmatrix}y_{max}^{2}I&C_{y}\\C_{y}^{T}&P\end{bmatrix}}\geq 0.$

${\begin{bmatrix}AQ+QA^{T}+B_{p}TB_{p}^{T}+B_{u}Y+T^{T}B_{u}^{T}&QC_{q}^{T}+Y^{T}D_{qu}^{T}+B_{p}TD_{qp}^{T}+B_{p}H\\C_{q}Q+D_{qu}Y+D_{qp}TB_{p}^{T}-HB_{p}^{T}&D_{qp}TD_{qp}^{T}-\sigma ^{2}T+D_{qp}H-HD_{qp}^{T}\end{bmatrix}}$ < 0

Conclusion

The above LMIs provide a unified setting, as well as an efficient computational procedure, for answering (possibly conservatively) several control problems pertaining to a quite generic class of nonlinear systems. This method makes an explicit and systematic connection (via LFRs and LMIs) between robust control methods and nonlinear systems.