An optimization-based methodology for the multiobjective control of a large class of nonlinear systems is performed.
Consider a nonlinear, continuous-time system
![{\displaystyle {\dot {y}}=A(x)+B_{u}(x)u,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed35e3ba9890f448b43d700a2eda5b53b68d8eb0)
![{\displaystyle y=C_{y}(x)+D_{yu}(x)u,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d180ff3845f90421686cd0734c308da61327805)
where
is the state vector,
is the input and
is the output.
is the state vector,
is the input and
is the output.
are multivariable functions of x.
(that is, 0 is an equilibrium point of the unforced system associated with the system).
and
have no singularities at the origin.
=
+ ![{\displaystyle {\begin{bmatrix}B_{p}\\D_{yp}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7157125e9e348b5f0e3857ea6d0529a9895170c)
![{\displaystyle \bigtriangleup (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4796f8933b395398bf046f4364fc1664d633bbaa)
![{\displaystyle {\begin{bmatrix}C_{q}&D_{qu}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cf1ddd0238a92d8e8583230fd8b3702ae3ba65f)
For a given scalar \sigma > 0, we associate with the Linear differential inclusion,
![{\displaystyle {\dot {x}}=Ax+B_{u}u+B_{p}p,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b13d0c5e58854229d18ebc54826c8b2a28c063f8)
![{\displaystyle q=C_{q}x+D_{qu}u+D_{qp}p,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/684a726de131a5362ac299593ef923b228dcd177)
![{\displaystyle y=C_{y}x+D_{yu}u+D_{yp}p,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ad41c75c9e0d52f6fd602dac2d2396b510cf3d)
![{\displaystyle p=\bigtriangleup (t)q,\|\bigtriangleup (t)\|\leq \sigma ^{-}1,\bigtriangleup (t)\in D(r),t\geq 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bbe3b224aad2d039c54cdb3ed9830bde75a6dc6)
The LMI: Control of Rational Systems using Linear-fractional Representations and Linear Matrix Inequalities
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For given \sigma > 0, the LDI system is quadratically stable if there exists P, S, and G such that the LMIs
![{\displaystyle P>0,S>0,G=-G^{T},S,G\in B(r),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e16c2ad46a8e4cb58d9d716146852bd82ce972d1)
hold. Then, for every
such that
![{\displaystyle det(I-D_{pq}\bigtriangleup )^{-1}\neq 0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c67721a9468d741dbc29980b2759f23bd349b6)
![{\displaystyle {\begin{bmatrix}y_{max}^{2}I&C_{y}\\C_{y}^{T}&P\end{bmatrix}}\geq 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85a3bc2bc557ea495a52b3b5b483f1670440f84f)
< 0
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.