Robust Unconstrained Model Predictive Control with State Feedback
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Model Predictive Control is an open-loop control design procedure where at each sampling time k, plant measurements are obtained and a model of the process is used to predict future outputs of the system. Using these predictions,
control moves
are computed by minimizing a nominal cost
over a prediction horizon
. The objective is to minimize the nominal cost function.
We consider the nominal cost function as:
![{\displaystyle \min _{u(k+i),i=0,1,...,m-1}J_{p}(k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2865be5e9c7c441a8d14894469b28df63fc231c4)
where,
![{\displaystyle J_{p}(k)=\Sigma _{i=0}^{p}[x(k+i|k)^{T}Q_{1}x(k+i|k)+u(k+i|k)^{T}Ru(k+i|k)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5f377179da665b5be08cf16db968e5c488863b)
and ![{\displaystyle R>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57914127d03a5cea02c60a32cfbb22f34904f00d)
and
are positive definite weighting matrices.
In this case, we take
. This is also called infinite horizon MPC.
Here, we consider system uncertainties that are modeled as polytopic uncertainties or structured uncertainties.
The set
is the polytope
Where,
denotes the convex hull.
The operator
is a block-diagonal:
![{\displaystyle \Delta ={\begin{bmatrix}\Delta _{1}&&&\\&\Delta _{2}&&&\\&&\ddots &\\&&&\Delta _{r}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ab9619385c5cc13c36e23f564a0471b12c5e818)
Each
can be a repeated scalar block or a full block.
Consider a linear time-varying(LTV) system:
![{\displaystyle x(k+1)=A(k)x(k)+B(k)u(k),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ad925e63e65a2c81989ee7de01a7708ffae4f8)
![{\displaystyle y(k)=CX(k),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f12f401e883110df203e5e441075d6af0a6ace65)
![{\displaystyle {\begin{bmatrix}A(k)&B(k)\end{bmatrix}}\in \Delta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/80349c1c32c2e8364dc6fd580301d53ae0dd304d)
Here,
is the control input,
is the state of the plant and
is the plant output and
is uncertainty set that is either polytopic system or structured uncertainty.
We modify the minimization of the nominal cost function to a minimization of the worst-case objective function.
The modified objective function minimizes the robust performance objective as follows:
![{\displaystyle \min _{u(k+i),i=0,1,...,m-1}\max _{[A(k+i)B(k+i]\in \Delta ,i\geq 0}J_{\infty }(k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f98907a2194f660e35f4b1ea273a81b85e8b4c54)
where,
![{\displaystyle J_{\infty }(k)=\Sigma _{i=0}^{\infty }[x(k+i|k)^{T}Q_{1}x(k+i|k)+u(k+i|k)^{T}Ru(k+i|k)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5fd370d24cd45d5e4168ad0acbb954906fbc78c)
The LMI:Robust Unconstrained Model Predictive Control with State Feedback for polytopic uncertainty
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subject to
![{\displaystyle {\begin{bmatrix}1&x(k|k)^{T}\\x(k|k)&Q\end{bmatrix}}\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd637fec97115a968feaa54ab513b4ca993e07b9)
and
![{\displaystyle {\begin{bmatrix}Q&QA_{j}^{T}+Y^{T}B_{j}^{T}&QQ_{1}^{1/2}&Y^{T}R^{1/2}\\A_{j}Q+B_{j}Y&Q&0&0\\Q_{q}^{1/2}&0&\gamma I&0\\R^{1/2}Y&0&0&\gamma I\end{bmatrix}}\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e46a4c7f9832c13329e8eaea2c1a5ca4793ab95)
The LMI:Robust Unconstrained Model Predictive Control with State Feedback for structured uncertainty
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subject to
![{\displaystyle {\begin{bmatrix}1&x(k|k)^{T}\\x(k|k)&Q\end{bmatrix}}\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd637fec97115a968feaa54ab513b4ca993e07b9)
![{\displaystyle {\begin{bmatrix}Q&Y^{T}R^{1/2}&QQ_{1}^{1/2}&QC_{q}^{T}+Y^{T}D_{qu}^{T}&QA^{T}+Y^{T}B^{T}\\R^{1/2}Y&\gamma I&0&0&0\\Q_{1}^{1/2}Q&0&\gamma I&0&0\\C_{q}Q+D_{qu}Y&0&0&\Lambda &0\\AQ+BY&0&0&0&Q-B_{p}\Lambda B_{p}^{T}\end{bmatrix}}\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62514034a28c082b3c5cd2ba3faf41a14ff21763)
where
![{\displaystyle \Lambda ={\begin{bmatrix}\lambda _{1}I_{n1}&&&\\&\lambda _{2}I_{n2}&&&\\&&\ddots &\\&&&\lambda _{r}I_{nr}\end{bmatrix}}>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f072b2b59409298a40000c2144a06c8bf370f54)
The state feedback matrix F in the control law
that minimizes the upper bound
on the robust performance objective function at sampling time
is given by :
![{\displaystyle F=YQ^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd458601abcff4bf1c4e0b63de238e47a8abd75)
where
and
are obtained from the solution of the above LMI.