https://github.com/leopits/linear-mpc-implementation
About This is the MATLAB code for a brief tutorial for Model Predictive Control (MPC) for water tank system with constrained states and inputs.
https://github.com/leopits/linear-mpc-implementation
matlab model-predictive-control mpc-control
Last synced: 4 months ago
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About This is the MATLAB code for a brief tutorial for Model Predictive Control (MPC) for water tank system with constrained states and inputs.
- Host: GitHub
- URL: https://github.com/leopits/linear-mpc-implementation
- Owner: LeoPits
- Created: 2023-10-19T16:21:45.000Z (about 2 years ago)
- Default Branch: main
- Last Pushed: 2023-11-03T16:18:06.000Z (about 2 years ago)
- Last Synced: 2025-02-10T09:17:47.103Z (11 months ago)
- Topics: matlab, model-predictive-control, mpc-control
- Language: MATLAB
- Homepage:
- Size: 437 KB
- Stars: 1
- Watchers: 1
- Forks: 0
- Open Issues: 1
-
Metadata Files:
- Readme: README.md
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README
# MPC-water-tank-implemention
This tutorial shows an implementation of a linear quadratic model predictive controller (MPC) to control a quadruple-tank model, simulated in Matlab/Simulink.
Consider that the plant to control is modeled as a linear time-invariant system given by
$$
\begin{align}
x^+&=Ax+Bu \\
y&=Cx+Du
\end{align}
$$
where $x\in R^n$ his the current state of the system, $u \in R^m $
is the current input, $y \in R^p$ is the controlled output and $x^+$
the successor state.
This linear model is derived from the linearization of the model at an operation equilibrium
point given by the triplet $(x_0, u_0, y_0)$. Therefore, denoting
as $(x(j), u(j), y (j))$ the state, inputs and outputs of the
plant, we have that $x(j) = x(j) − x_0$, $u(j) = u(j) − u_0$
and $y(j) = y (j) − y_0$.
The objective is to implement an MPC control law $U(j) =
\kappa(x(j), R)$ such that the controlled system is asymptotically
stable and the controlled variables $y (j)$ converge to the
reference $R$ if this is reachable.
For a given state of the prediction model $x = x−x_0$ and a
reference $r = R −y_0$, the MPC control law is derived from
the online solution of the following optimization control
problem:
$$
\begin{align}
V_{\text{N}}(x,y_{sp};\mathbf{u})&= \sum_{j=0}^{N-1}\parallel x(j)-x_r\parallel_{Q}^{2} + \parallel u(j)-u_r\parallel_{R}^{2} + \parallel x(N)-x_{r}\parallel_{P}^{2} \\
s.t.& \quad x(j+1)=Ax(j)+Bu(j),\\
& \quad x(0)=x, \\
& \quad u_{min} \leq u \leq u_{max}, \\
& \quad (x_r,u_r)=M_r, \\
& \quad x(N) =0
\end{align}
$$
where $M$ is a suitable matrix that maps the steady state and input given by the reference. Once a solution $u^{∗}(x, r)$
is obtained, the control law is calculated by the receding horizon technique as follows $\kappa_N (x, r) = u^{*} (0; x, r)$. The
control law implemented in the real plant will be $u(k) =\kappa_N (x(j) − x_O, R − y_0) + u_0$.
## Lifted System Dynamics
the notation of a lifted system dynamics
$$
\mathbf{x}=G_x x(j) + G_u \mathbf{u}
$$
is used, where the whole state sequence can be determined with the aid of the input sequence $\mathbf{u}$ for a given initial state $x(j)$. The state sequence and the input sequence are
$$
\mathbf{x}=\left[\begin{array}{c}
x_{0}\\
x_{1}\\
\vdots\\
x_{N}
\end{array}\right],u=\left[\begin{array}{c}
u_{0}\\
u_{1}\\
\vdots\\
u_{N-1}
\end{array}\right]
$$
The lifted system matrix and the lifted input matrix are
$$
G_{x}=\left[\begin{array}{c}
I\\
A\\
A^{2}\\
\vdots\\
A^{N}
\end{array}\right],G_{u}=\left[\begin{array}{cccc}
0 & 0 & \cdots & 0\\
B & 0 & \cdots & 0\\
AB & B & \cdots & 0\\
\vdots & \vdots & \vdots & \vdots\\
A^{N-1}B & A^{N-2}B & \cdots & B
\end{array}\right]
$$
## Quadratic Program
The implementation with the MATLAB built-in functions quadprog is shown here.
$$
\begin{align}
\underset{\mathbf{u}}{\min}V_{N}(x,x_{r};\mathbf{u}) &= (\mathbf{x}-\mathbf{x_r})^TQ(\mathbf{x}-\mathbf{x_r})+ (\mathbf{u}-\mathbf{u}_r)^TR(\mathbf{u}-\mathbf{u}_r)\\
s.t.& \quad \mathbf{x}=G_xx(j)+G_u\mathbf{u},\\
&\quad \widetilde{G}\mathbf{u} \leq \widetilde{g}
\end{align}
$$
$$
\begin{align}
\underset{\mathbf{u}}{\min}V_{N}(x,x_{r};\mathbf{u})& = \mathbf{u}^T(G_u Q G_u + R )\mathbf{u} + 2( x(j)^{T} G_{x}^{T} Q G_u)\mathbf{u}-2( x_{ R}^{T} Q G_u - \mathbf{u_{r}}^{T} R )\mathbf{u}\\
s.t.& \quad \mathbf{x}=G_xx(j)+G_u\mathbf{u},\\
&\quad \widetilde{G}\mathbf{u} \leq \widetilde{g}
\end{align}
$$
$$
\begin{align}
\underset{\mathbf{u}}{\min}V_{N}(x,x_{r};\mathbf{u}) &= \mathbf{u}^T H \mathbf{u} + 2(F_1-F_2)\mathbf{u}\\
s.t.& \quad \mathbf{x}=G_xx(j)+G_u\mathbf{u},\\
&\quad \widetilde{G}\mathbf{u} \leq \widetilde{g}
\end{align}
$$
## Results
