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WIP. Progress: (ME 561 @ Week4; ME 564 @ 0%)
This note combines content from ME 564 Linear Systems and ME 561 Discrete Digital Control
In this note, $f\in\mathbb{F}^\mathbb{G}$ stands for a function with domain in $\mathbb{G}$ and co-domain in $\mathbb{F}$, i.e. $f:\mathbb{F}\rightarrow\mathbb{G}$, $H(x)$ generally stands for Heaviside function (step function)

# Transforms

## Laplace Transform

• Definition: $F(s)=\mathcal{L}\{f(t)\}(s)=\int^\infty_0 f(t)e^{-st}\mathrm{d}t$

Note that the transform is not well defined for all functions in $\mathbb{C}^\mathbb{R}$. And the transform is only valid for $s$ in a region of convergence, which is usually separated by 0.

• Laplace Transform is a linear map from $(\mathbb{C}^\mathbb{R}, \mathbb{C})$ to $(\mathbb{C}^\mathbb{C}, \mathbb{C})$ and it’s one-to-one.
• Derivative: $f’(t) \xleftrightarrow{\mathcal{L}} sF(s)-f(0^-)$
• Integration: $\int^t_0 f(\tau)d\tau \xleftrightarrow{\mathcal{L}} \frac{1}{s}F(s)$
• Delay: $f(t-a)H(t-a) \xleftrightarrow{\mathcal{L}} e^{-as}F(s)$
• Convolution: $\int^t_0 f(\tau)g(t-\tau)\mathrm{d}\tau \xleftrightarrow{\mathcal{L}} F(s)G(s)$
• Stationary Value: $\lim\limits_{t\to 0} f(t) = \lim\limits_{s\to \infty} sF(s), \lim\limits_{t\to \infty} f(t) = \lim\limits_{s\to 0} sF(s)$

### Inverse Laplace Transform

Laplace transform is one-to-one, so we can apply inverse transform on functions in s-space

There’s several ways to calculate Laplace transform, the first one is directly evaluating integration while the latter two are converting the function into certain formats that are convenient for table lookup:

1. (Mellin’s) Inverse formula: $f(t)=\mathcal{L}^{-1}\{F(s)\}(t)=\frac{1}{2\pi j}\lim\limits_{T\to\infty} \int ^{\gamma+iT}_{\gamma-iT} e^{st}F(s)\mathrm{d}s$ where the integration is done along the vertical line $Re(s)=\gamma$ in the convex s-plane such that $\gamma$ is greater than the real part of all poles of $F(s)$.
2. Power Series: $F(s) = \sum^\infty_{n=0} \frac{n!a_n}{s^{n+1}}\xleftrightarrow{\mathcal{L}} f(t) = \sum ^\infty_{n=0} a_n t^n$
3. Partial Fractions: $F(s)=\frac{k_1}{s+a}+\frac{k_2}{s+b}+\ldots \xleftrightarrow{\mathcal{L}} f(t)=k_1 e^{-at} + k_2 e^{-bt} + \ldots$
• To calculate partial fractions, one can use Polynomial Division or following lemma:
• Suppose $F(s)=\frac{N(s)}{D(s)}=\frac{N(s)}{\prod^n_{i=1} (s-p_i)^{r_i}}$ where $\mathrm{deg}(N(s)) < \mathrm{deg(D(s))}$ and each $p_i$ is a distinct root of $D(s)$ (i.e. pole) with multiplicity $r_i$, then $F(s)=\sum^n_{i=1}\sum^{r_i}_ {j=1} \frac{k_{ij}}{(s-p_i)j}$ where $k_{ij}=\frac{1}{(r_i-j)!}\left.\frac{\mathrm{d}^{r_i-j}}{\mathrm{d}s^{r_i-j}}(s-p_i)^{r_i}F(s)\right\vert_{s=p_i}$

## Z-Transfrom

• Definition: $F(z)=\mathcal{Z}\{f(k)_ {k\in\mathbb{N}}\}(z)=\sum^\infty_{k=0} f(k)z^{-k}$

Notice that $f$ is defined on natural numbers. In time domain, it’s usually corresponding to $f(kT)$. Z-transform is also only valid for $z$ in certain region (usually separated by 1)

• Laplace Transform is a linear map from $(\mathbb{C}^\mathbb{N}, \mathbb{C})$ to $(\mathbb{C}^\mathbb{C}, \mathbb{C})$ and it’s one-to-one.
• Accumulation: $\sum^n_{k=-\infty} f(k) \xleftrightarrow{\mathcal{Z}} \frac{1}{1-z^{-1}}F(z)$
• Delay: $f(k-m) \xleftrightarrow{\mathcal{Z}} z^{-m}F(z)$
• Convolution: $\sum^k_{n=0}f_1(n)f_2(k-n) \xleftrightarrow{\mathcal{Z}} F_1(z)F_2(z)$
• Stationary Value: $\lim\limits_{t\to 0} f(t) = \lim\limits_{z\to \infty} F(z), \lim\limits_{t\to \infty} f(t) = \lim\limits_{z\to 1} (z-1)F(z)$

### Inverse Z-Transform

1. Inverse formula: $f(k)=\mathcal{Z}^{-1}\{F(z)\}(k)=\frac{1}{2\pi j}\oint _\Gamma z^{k-1}F(z)\mathrm{d}z$ where the integration is done along any closed path $\Gamma$ that encloses all finite poles of $z^{k-1}X(z)$ in the z-plane.
• According to residual theorem, we can write it as $f(k)=\sum_{p_i}Res(z^{k-1}f(z), pi)$ where $p_i$ are poles of $z^{k-1}f(k)$ and residual $Res(g(z),p)=\frac{1}{(m-1)!}\left.\frac{\mathrm{d}^{m-1}}{\mathrm{d}z^{m-1}}\left((z-p)^mg(z)\right)\right\vert_{z=p}$ with $m$ being the multiplicity of the pole $p$ in $g$.
2. Power Series: same as inverse laplace.
3. Partial Fractions: same as inverse laplace.

### Modified Z-Transfrom

• Definition: $F(z,m)=\mathcal{Z}_m(f,m)=\mathcal{Z}(\left\{f(kT-(1-m)T)\right\} _{k\in\mathbb{N}^+})(z)$
• We denote corresponding continuous form $\mathcal{L}(f(t-(1-m)T)\delta_ T(t))$ as $F^*(s,m)$
• Residual Theorem: $\mathcal{Z}_m(f,m)=z^{-1}\sum _{p_i} Res(\frac{F(s)e^{mTs}}{1-z^{-1}e^{Ts}}, p_i)$
• ModZ Transform is usually used when there’s delay in the system, use this transform to shift the signal with proper $m$ value.

## Starred Transform

• Definition: $F^* (s)=\sum^\infty_{n=0}f(n*T)e^{-nTs}$

Starred Transform is defined only continuous s-domain, but it only aggregates on discrete s values defined periodically by sampling time T, like Z-Transform

• Sometimes we also see * as an operator to sample a continuous signal. It converts a continuous signal to discrete delta functions. (See the “Sampler” section below)
• Calculation from Laplace Transform
• $F^*(s)=\sum_{p_i\in\{poles\;of\;F(\lambda)\}} Res\left(F(\lambda)\frac{1}{1-e^{-T(s-\lambda)}}, p_i\right)$
• $F^*(s)=\frac{1}{T}\sum^\infty_{n=-\infty}F(s+jn\omega_s)+\frac{e(0)}{2}$ where $\omega_s=\frac{2\pi}{T}$
• Properties:
• $F^*(s)$ is periodic in s plane with period $j\omega_s=\frac{2\pi j}{T}$
• If $F(s)$ has a pole at $s=s_0$, then $F^*(s)$ must have poles at $s=s_0+jn\omega_s$ for $m\in\mathbb{Z}$
• $A(s)=B(s)F^* (s) \Rightarrow A^* (s)=B^* (s)F^* (s)$, while usually $A(s)=B(s)F(s) \nRightarrow A^* (s)=B^* (s)F^* (s)$

## Fourier Transform

Fourier transform is basically to substitute $s=j\omega$ into Laplace transform. Additional properties are not discussed here.

• One important theorem (Shannon Nyquist Sampling Theorem): Suppose $e:\mathbb{R}_+\rightarrow\mathbb{R}$ has a Fourier Transform with no frequency components greater than $f_0$, then $e$ is uniquely determined by the signal $e_s$ generated by ideally sampling $e$ with period $\frac{1}/{2}f_0$.

## Conversions

• Starred Transform and Z-Transform: $F(z)=\left.F^*(s)\right|_{e^{sT}=z}$

# Continuous System

## Continuous State Space Representation

### Transfer function

• Definition: transfer function is the mapping of the input to the laplace transform of the output when $x(0)=0$
• For LTI case, $\frac{Y(s)}{U(s)} = C(sI-A)^{-1}B+D$

This can be derived by take laplace transform of both sides of state equations

# Discrete System

## Continuous State Space Representation

### Discretization

$\dot{x}=Ax+Bu \Rightarrow x_{k+1}=e^{AT}x_k+A^{-1}(e^{AT}-I)Bu_k$

### Transfer function

• Transfer function between the sampled input and output at the sampling instants is usually called Pulse Transfer Function.
• For LTI case, $H(z)=C(zI-A)^{-1}B+D$

# Discretization

## Illustrating Example

The following image shows a minial example of sampling and hold.

## Sampling (A/D)

• Ideal sampler (a.k.a impulse modulator) converts a continuous signal $e: \mathbb{R}_+ \rightarrow \mathbb{R}$ to a discrete one $\hat{e}: \mathbb{N}\rightarrow \mathbb{R}$, such that $$\hat{e}=e(t)\delta(t-kT)=e(t)\delta_T(t); \forall k\in \mathbb{N}$$
• Ideal sampler is actually applying starred transform.

## Reconstruction/Hold (D/A)

• Zero order hold (ZOH): $ZOH(\{e(k)\}_{k\in\mathbb{N}})(t) = e(k)\;for\;kT\leq t\leq (k+1)T$
• Alternative form: $ZOH(\{e(k)\})=\sum^\infty_{k=0}e(k)(H(t-kT)-H(t-(k+1)T))$
• Its Laplace Transform: $G_{ZOH}(s)=\frac{1-e^{-Ts}}{s}$
• First order hold (FOH): (delayed version) $$FOH(\{e(k)\}_ {k\in\mathbb{N}})(t)=\sum_ {k\in\mathbb{N}}\left[e(kT)+\frac{t-kT}{T}(e(kT)-e((k-1)T)) \right]\left[H(t-kT)-H(t-(k+1)T) \right]$$

# Controllers

## MPC

Treat me some coffee XD