(control:closed-loop:requirements-performance)=
# Closed-loop control: requirements and performance

| Requirement/Performance       | ...                                                    | ... |
| :---------------------------- | :----------------------------------------------------- | :-- |
| Stability                     | Nyquist criterion on the open-loop TF $L(s)$           |     |
| Robustness                    | Stability margins                                      |     |
| Reference tracking            | Type of the system. Integrators in open-loop TF $L(s)$ |     |
| Measurement noise suppression |                                                        |     |
| Input load                    |                                                        |     |
| Input noise suppression       |                                                        |     |
| Transient performance         |                                                        |     |


Let the transfer function of the system be

$$\mathbf{y} = \mathbf{G} \mathbf{u} + \mathbf{G}_d \mathbf{d} + \mathbf{G}_n \mathbf{n} \ ,$$

linking the control input $\mathbf{u}$, and the exogenous inputs ( process disturbances $\mathbf{d}$, and measurement noise $\mathbf{n}$) to the output $\mathbf{y}$. Let control disturbance $\Delta \mathbf{u}$ on the ideal control $\mathbf{u}_c$ be additive, $\mathbf{u} = \mathbf{u}_c + \Delta \mathbf{u}$. Let $\mathbf{F}$ be a feed-forward system from reference $\mathbf{r}$ to the reference output $\mathbf{y}_{\text{ref}}$ that is compared with the output of the system $\mathbf{y}$ to define the error $\mathbf{e} = \mathbf{y}_{\text{ref}} - \mathbf{y}$ feeding the regulator $\mathbf{R}$.

:::{figure} ./../../media/control/closed-loop-tfs.png
:alt: Transfer function of the closed-loop system
:align: center
:name: fig-closed-loop-tfs

Block diagram of the closed-loop system

:::

Closed-loop performance usually deals with the relation between the output $\mathbf{y}$, the tracking error $\mathbf{e}$, and the ideal control input $\mathbf{u}_c$ (the output of he regulator; that's what we need to care about for controller response, performance, saturation,...) w.r.t. the reference signal $\mathbf{r}$ and the exogenous inputs $\Delta \mathbf{u}$, $\mathbf{d}$, $\mathbf{r}$, in terms of:
* tracking of the reference 
* control performance
* disturbance suppression/noise rejection (on both the output and the input control)

For a MIMO

$$
\begin{bmatrix} \mathbf{y} \\ \mathbf{e} \\ \mathbf{u}_c \end{bmatrix} =
\begin{bmatrix} \left[ \mathbf{I} + \mathbf{G} \mathbf{R} \right]^{-1} & \cdot & \cdot \\ \cdot & \left[ \mathbf{I} + \mathbf{G} \mathbf{R} \right]^{-1} & \cdot \\ \cdot & \cdot &\left[ \mathbf{I} + \mathbf{R} \mathbf{G} \right]^{-1} \end{bmatrix}
\begin{bmatrix}
  \mathbf{G} \mathbf{R} \mathbf{F} & \mathbf{G} & \mathbf{G}_d & \mathbf{G}_n \\
                        \mathbf{F} & -            \mathbf{G} & -            \mathbf{G}_d & -            \mathbf{G}_n \\
             \mathbf{R} \mathbf{F} & - \mathbf{R} \mathbf{G} & - \mathbf{R} \mathbf{G}_d & - \mathbf{R} \mathbf{G}_n \\
\end{bmatrix}
\begin{bmatrix} \mathbf{r} \\ \Delta \mathbf{u} \\ \mathbf{d} \\ \mathbf{n} \end{bmatrix} \ .
$$

For a SISO

$$
\begin{bmatrix} y \\ e \\ u_c \end{bmatrix} =
\frac{1}{1+GR} \begin{bmatrix}
 GRF & G & G_d & G_n \\
   F & -   G & -   G_d & -   G_n \\
 R F & - R G & - R G_d & - R G_n 
 \end{bmatrix}
\begin{bmatrix} r \\ \Delta u \\ d \\ n \end{bmatrix} \ .
$$

**Two-degrees of freedom problem.** Typical design procedure:
* Design $R$ to provide load/noise performance
* Design $F$ to provide tracking performance

**Simplified model.** If 
* there's no feed-forward, i.e. $\mathbf{F} = \mathbf{I}$ or $F = 1$ for a SISO,
* the measurement noise is additive on an ideal output, the TF is $\mathbf{y} = \mathbf{G} \mathbf{u} + \mathbf{G}_d \mathbf{d} + \mathbf{n}$, i.e.$\mathbf{G}_n = \mathbf{I}$ or $G_n(s) = 1$ for a SISO,

for a MIMO

$$
\begin{bmatrix} \mathbf{y} \\ \mathbf{e} \\ \mathbf{u}_c \end{bmatrix} =
\begin{bmatrix} \left[ \mathbf{I} + \mathbf{G} \mathbf{R} \right]^{-1} & \cdot & \cdot \\ \cdot & \left[ \mathbf{I} + \mathbf{G} \mathbf{R} \right]^{-1} & \cdot \\ \cdot & \cdot & \left[ \mathbf{I} + \mathbf{R} \mathbf{G} \right]^{-1} \end{bmatrix}
\begin{bmatrix}
  \mathbf{G} \mathbf{R} & \mathbf{G} & \mathbf{G}_d & \mathbf{I} \\
             \mathbf{I} & -            \mathbf{G} & -            \mathbf{G}_d & - \mathbf{I} \\
             \mathbf{R} & - \mathbf{R} \mathbf{G} & - \mathbf{R} \mathbf{G}_d & - \mathbf{R} \\
\end{bmatrix}
\begin{bmatrix} \mathbf{r} \\ \Delta \mathbf{u} \\ \mathbf{d} \\ \mathbf{n} \end{bmatrix} \ ,
$$

and for a SISO

$$
\begin{bmatrix} y \\ e \\ u_c \end{bmatrix} =
\frac{1}{1+GR} \begin{bmatrix}
 G R & G & G_d & 1 \\
 1 & -   G & -   G_d & - 1 \\
 R & - R G & - R G_d & - R 
\end{bmatrix}
\begin{bmatrix} r \\ \Delta u \\ d \\ n \end{bmatrix} \ .
$$

For a SISO, beside process noise $\mathbf{d}$ that may have its own dynamics, the performance of the closed loop system is defined by 4 transfer functions only, **"the gang of the 4"**,
* sensitivity function ($\mathbf{y}(\mathbf{n})$, $\mathbf{e}(\mathbf{r})$, $\mathbf{e}(\mathbf{n})$)

  $$S = \frac{1}{1 + GR} \ ,$$

* complementary sensitivity ($\mathbf{y}(\mathbf{r})$, $\mathbf{u}_c(\Delta \mathbf{u})$)

  $$T = \frac{GR}{1 + GR} \ ,$$

* load sensitivity ($\mathbf{y}(\Delta \mathbf{u})$, $\mathbf{e}(\Delta \mathbf{u})$)

  $$PS = \frac{G}{1 + GR} \ ,$$

* noise sensitivity ($\mathbf{u}_c(\mathbf{r})$, $\mathbf{u}_c(\mathbf{n})$)

  $$CS = \frac{R}{1 + GR} \ ,$$


```{dropdown} Details

**Output.**

$$\begin{aligned}
  \mathbf{y}
  & = \mathbf{G} \mathbf{u} + \mathbf{G}_d \mathbf{d} + \mathbf{G}_n \mathbf{n} = \\
  & = \mathbf{G} \left( \mathbf{u}_c + \Delta \mathbf{u} \right) + \mathbf{G}_d \mathbf{d} + \mathbf{G}_n \mathbf{n} = \\
  & = \mathbf{G} \mathbf{R} \mathbf{e} + \mathbf{G} \Delta \mathbf{u} + \mathbf{G}_d \mathbf{d} + \mathbf{G}_n \mathbf{n} = \\
  & = \mathbf{G} \mathbf{R} \left( \mathbf{y}_{\text{ref}} - \mathbf{y} \right) + \mathbf{G} \Delta \mathbf{u} + \mathbf{G}_d \mathbf{d} + \mathbf{G}_n \mathbf{n} = \\
  & = \mathbf{G} \mathbf{R} \mathbf{F} \mathbf{r} - \mathbf{G} \mathbf{R} \mathbf{y} + \mathbf{G} \Delta \mathbf{u} + \mathbf{G}_d \mathbf{d} + \mathbf{G}_n \mathbf{n} \ ,
\end{aligned}$$

and thus

$$\begin{aligned}
\mathbf{y}
  & = \left[ \mathbf{I} + \mathbf{G}\mathbf{R} \right]^{-1} \mathbf{G} \mathbf{R} \mathbf{F} \,        \mathbf{r} + \\
  & + \left[ \mathbf{I} + \mathbf{G}\mathbf{R} \right]^{-1} \mathbf{G}                       \, \Delta \mathbf{u} + \\
  & + \left[ \mathbf{I} + \mathbf{G}\mathbf{R} \right]^{-1} \mathbf{G}_d                     \,        \mathbf{d} + \\
  & + \left[ \mathbf{I} + \mathbf{G}\mathbf{R} \right]^{-1} \mathbf{G}_n                     \,        \mathbf{n} \ .
\end{aligned}$$

**Error.**

$$\begin{aligned}
  \mathbf{e}
  & = \mathbf{F} \mathbf{r} - \mathbf{y} = \\
  & = \mathbf{F} \mathbf{r} - \left(  \mathbf{G} \mathbf{u} + \mathbf{G}_d \mathbf{d} + \mathbf{G}_n \mathbf{n} \right) = \\
  & = \mathbf{F} \mathbf{r} - \mathbf{G} \mathbf{u}_c - \mathbf{G} \Delta \mathbf{u} - \mathbf{G}_d \mathbf{d} - \mathbf{G}_n \mathbf{n} = \\
  & = \mathbf{F} \mathbf{r} - \mathbf{G} \mathbf{R} \mathbf{e} - \mathbf{G} \Delta \mathbf{u} - \mathbf{G}_d \mathbf{d} - \mathbf{G}_n \mathbf{n} \ ,
\end{aligned}$$

and thus

$$\begin{aligned}
  \mathbf{e}
  & = \left[ \mathbf{I} + \mathbf{G} \mathbf{R} \right]^{-1} \mathbf{F}   \, \mathbf{r} + \\
  & - \left[ \mathbf{I} + \mathbf{G} \mathbf{R} \right]^{-1} \mathbf{G}   \, \Delta  \mathbf{u} + \\
  & - \left[ \mathbf{I} + \mathbf{G} \mathbf{R} \right]^{-1} \mathbf{G}_d \, \mathbf{d} + \\
  & - \left[ \mathbf{I} + \mathbf{G} \mathbf{R} \right]^{-1} \mathbf{G}_r \, \mathbf{r} \ .
\end{aligned}$$


**Input.**

$$\begin{aligned}
  \mathbf{u}_c 
  & = \mathbf{R} \mathbf{F} \mathbf{r} - \mathbf{R} \mathbf{y} = \\
  & = \mathbf{R} \mathbf{F} \mathbf{r} - \mathbf{R} \left( \mathbf{G} \mathbf{u} + \mathbf{G}_d \mathbf{d} + \mathbf{G}_n \mathbf{n} \right) = \\
  & = \mathbf{R} \mathbf{F} \mathbf{r} - \mathbf{R} \mathbf{G} \mathbf{u}_c - \mathbf{R} \mathbf{G} \Delta \mathbf{u} - \mathbf{R} \left( \mathbf{G}_d \mathbf{d} + \mathbf{G}_n \mathbf{n} \right) \ ,
\end{aligned}$$

and thus

$$\begin{aligned}
  \mathbf{u}_c
  & = \left[ \mathbf{I} + \mathbf{R} \mathbf{G} \right]^{-1} \mathbf{R} \mathbf{F}   \, \mathbf{r} + \\
  & - \left[ \mathbf{I} + \mathbf{R} \mathbf{G} \right]^{-1} \mathbf{R} \mathbf{G}   \, \Delta  \mathbf{u} + \\
  & - \left[ \mathbf{I} + \mathbf{R} \mathbf{G} \right]^{-1} \mathbf{R} \mathbf{G}_d \, \mathbf{d} + \\
  & - \left[ \mathbf{I} + \mathbf{R} \mathbf{G} \right]^{-1} \mathbf{R} \mathbf{G}_r \, \mathbf{r} \ .
\end{aligned}$$


```

**Algebraic constraint on sensitivity and complementary sensitivity.** It's not possible to make $S(s)$ and $T(s)$ small at the same time, as

$$S(s) + T(s) = 1$$

Usually,

* $|S(j \omega)|$ small at low frequency $\omega$ to get small error in the band of the reference and filtering low-frequency measurement noise on the inp 
* $|T(j \omega)|$ small at high frequency $\omega$ to filter high-frequency noise from $\Delta u$ to $u$ (usually reference signal $r$ has no high-frequency content, so there should be little issues in filtering out the effect of reference signal to the output).