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Spring mass

Plant Models

Continuous Model

The transfer function is
\begin{equation}
P(s)=\frac{1}{s^2+1}
\end{equation}
The open loop system is visibly only marginally stable. A minimal realization is
\begin{equation}
\dot{x_p} =
\begin{pmatrix} 0& 1 \\ -1 &0 \end{pmatrix}x_p
+ \begin{pmatrix} 0 \\ 1 \end{pmatrix} u
\end{equation}
\begin{equation}
y=\begin{pmatrix} 1 & 0 \end{pmatrix}x_p
\end{equation}

Discretized Model

Euler Discretization at period $.01$:
\begin{equation}
x_{p,k+1} =
\begin{pmatrix} 1& .01 \\ -.01 & 1 \end{pmatrix} x_{p,k}
+ \begin{pmatrix} 5\cdot10^{-5} \\ 0.01 \end{pmatrix} u_k = A_px_{p,k}+B_pu_k
\end{equation}\begin{equation}
y_k=\begin{pmatrix} 1 & 0 \end{pmatrix}x_{p,k} = C_px_{p,k}
\end{equation}

lead-lag controller Model

Continuous Model

We choose a lead-lag controller of the following transfer function:
\begin{equation}
T(s)=1280\frac{(s+1)(s+5)}{(s+50)(s+0.1)}
\end{equation}
A state space realization of it is:
\begin{equation}
\dot{x_c} =
\begin{pmatrix} -50.1 & -5 \\ 1 & 0 \end{pmatrix} x_c
+ \begin{pmatrix} 100 \\ 0 \end{pmatrix} y
\end{equation}
\begin{equation}
u= \begin{pmatrix} 564.48 & 0 \end{pmatrix} x_c+ 1280 y
\end{equation}

Discretized Model

First order Euler discretization of the state-space at period $0.01$:
\begin{equation}
x_{c,k+1} =
\begin{pmatrix} 0.499 & -0.05 \\ 0.01 & 1 \end{pmatrix} x_{c,k}
+ \begin{pmatrix} 1 \\ 0 \end{pmatrix} y_k =A_cx_{c,k}+B_cy_k
\end{equation}
\begin{equation}
u_k= \begin{pmatrix} 564.48 & 0 \end{pmatrix} x_{c,k}+ 1280 y_k = C_cx_{c,k}+D_cy_k
\end{equation}

Stability and Margins

The full model includes a saturation on the distance to the desired rate as follows:

Bounded Input/ Bounded State stability of the controller in open loop

Assuming $x_{c,0}=0$, and looking for a matrix $P=P^\mathrm{T}>0$ such that
\begin{equation}
\forall y \in \mathbb{R}, x_c \in \mathbb{R}^2: ||y||\leq 1\ \&\ x_c^\mathrm{T}Px_c\leq 1 \implies (A_cx_c+B_cy)^\mathrm{T}P(A_cx_c+B_cy)\leq 1
\end{equation}
amounts, by S-procedure, to finding a $0\leq\mu\leq 1$ and a $P=P^\mathrm{T}>0$ such that :
\begin{equation}
\begin{pmatrix} A_c^\mathrm{T}PA_c - \mu P & PB_c \\ B_c^\mathrm{T}P & B_c^\mathrm{T}PB_c- (1- \mu ) \end{pmatrix} \leq 0
\end{equation}
which is valid for $P=10^{-3}\begin{pmatrix}0.6742 & 0.0428 \\ 0.0428 & 2.4651 \end{pmatrix}$ and $\mu=0.9991$.

Closed loop stability of the state-plant interconnection:

Now considering the interconnection of the discrete plant and the discrete controller, with the desired value of $y$ as input, we again set out to establish bounded input/ bounded state stability for the whole, i.e. with $x_k = \begin{pmatrix} x_{c,k} \\ x_{p,k} \end{pmatrix}$:

\begin{equation}
x_{k+1} =
\begin{pmatrix} A_c & 0_{2x2} \\ B_pC_c & A_p \end{pmatrix} x_k
+ \begin{pmatrix} B_c \\ B_pD_c \end{pmatrix} \mathbf{SAT}(Cx_k - y_{d,k}) =A x_k+B \mathbf{SAT}(Cx_k - y_{d,k})
\end{equation}
\begin{equation}
y_k= \begin{pmatrix} 0_{1x2} & C_p \end{pmatrix} x_k = C x_k
\end{equation}

Consider matrix $P=\begin{pmatrix}3.33128 &16.21541 & -0.30161 &0.26079 \\
16.21541 & 86.55698 & -0.89735 & 1.27046 \\
-0.30161 & -0.89735 & 0.11996 & -0.02276\\
0.26079 & 1.27046 & -0.02276 & 0.02048 \end{pmatrix}$. We show that under the assumption $|y_{d,k}|\leq 0.5$, the ellipsoid defined by $\mathcal{E}_P=\{x\in\mathbb{R}^4: x^\mathrm{T}Px\leq 1 \}$ is an invariant set for this system.

Files

All files available on the C-SAL subversion

Updated by Romain Jobredeaux over 8 years ago ยท 77 revisions