109 lines
5.7 KiB
Markdown
109 lines
5.7 KiB
Markdown
All info from [[doyleFeedbackControlTheory2009]]
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# Chapter 1 - Introduction
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![[Pasted image 20241012132644.png]]
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Notable signals:
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- r - reference or command input
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- e - tracking error
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- u - control signal, controller output
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- d - plant distrurbance
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- y - plant output
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- n - sensor
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# Chapter 2 - Norms
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>[!note] Signal Norms
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>1-Norm:
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> $$||u||_1 = \int_{-\infty}^{\infty} |u(t)|dt$$
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> 2-Norm:
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> $$||u||_2 = \left(\int_{-\infty}^{\infty} u(t)^2 dt \right)^{1/2}$$
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> $\infty$-Norm
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> $$||u||_\infty = \sup_t |u(t)|$$
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> Power Signals (Not really a norm):
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> $$pow(u) = \left( \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^T u(t)^2 dt \right)^{1/2}$$
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> If the limit exists, u is called a *power signal*
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> > [!caution] Some Implications
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> >1. $||u||_2 < \infty \rightarrow pow(u) = 0$
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> >2. u is a power signal and $||u||_\infty < \infty \rightarrow pow(u) \leq ||u||_\infty$
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> >3. There's a third one in the book about the one norm. I'm ignoring it.
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>[!nnote] System Norms
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>$\hat G$ means the transfer function $G$ in the frequency domain.
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>2-Norm:
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>$$||\hat G||_2 = \left(\frac{1}{2\pi} \int_{-\infty}^\infty |\hat G(j\omega)|^2d\omega \right) ^{1/2} $$
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>$\infty$-norm
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>$$||\hat G||_\infty = \sup_{\omega} |\hat G(j\omega)|$$
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>>[!hint] Parseval's Theorem
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>> If $\hat G$ is stable, then
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>> $$ ||\hat G||_2 = \left(\frac{1}{2\pi} \int_{-\infty}^\infty |\hat G(j\omega)|^2d\omega \right) ^{1/2} = \left( \int_{\infty}^\infty |G(t)|^2 dt \right)^{1/2}$$
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![[Pasted image 20241012135404.png]]
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# Chapter 3 - Basic Feedback Loop
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![[Pasted image 20241014145054.png]]
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P, C, and F are system transfer functions. For a system to be **well-posed**, they cannot all be strictly proper. P is almost always strictly proper while the others aren't.
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Something interesting - If $\frac{1}{1+PCF}$ is proper, then this suggests that the system output goes to zero when $j\omega\rightarrow \infty$. This isn't true in reality, because real systems will behave in a not linear way at high frequencies.
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>[!tip] Nine System Transfer Functions
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>$$\left(\matrix{x_1 \\ x_2 \\ x_3}\right) = \frac{1}{1+PCF} \left[\matrix{1 & -PF & -F \\ C & 1 & -CF \\ PC & P & 1}\right] \left(\matrix{r \\ d \\ n}\right) $$
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>**Notable Properties**:
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>- All 9 transfer functions are strictly proper if 1+PCF is not strictly proper.
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>- If all 9 transfer functions are stable, then the system is **internally stable**
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>[!note] Internal Stability
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>A couple of theorems:
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>>[!hint] Theorem 1
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>>The feedback system is internally stable iff there are no closed-loop poles in $Res \geq 0$.
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>
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>>[!hint] Theorem 2
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>>The feedback system is internally stable iff the following two conditions hold:
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>>a) The transfer function 1+PCF has no zeros in $Res \geq 0$.
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>>b) There is no pole-zero cancellation in $Res \geq 0$ when the product PCF is formed.
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>
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>>[!warning] Nyquist Criterion
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>>The feedback system is internally stable iff the Nyquist plot does not pass through the point -1 and encircles it exactly n times counterclockwise.
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>
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The sensitivity function is defined as:
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$$ S = \frac{1}{1+L}$$ where L is the loop gain. The sensitivity function **is the transfer function from the reference input r to the tracking error e**.
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The number of zeros at the origin of S has a lot to do with asymptotic tracking. For perfect step tracking, this means one zero at the origin. For a ramp, this means two zeros at the origin.
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There is another function to understand. The complementary sensitivity function is defined as:
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$$ T = 1-S = \frac{L}{1+L}$$
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This function **is the transfer function from the reference input r to the output y**.
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Generally speaking:
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1. Performance specs that involve e result in weights on S
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2. Performance specs that involve u result in weights on S
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$$ \left[\matrix{e \\u}\right] = -\left[\matrix{PS & S \\ T & CS}\right] \left[\matrix{d \\ n}\right] $$
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# Chapter 4 - Uncertainty and Robustness
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>[!important] Multiplicative Disk Perturbation
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>$$\tilde{P} = (1+\Delta W_2)P$$
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> - P is a nominal plant transfer function
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> - $\Delta$ is a variable stable transfer function s.t. $||\Delta||_\infty <1$
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> - P and $\tilde P$ have the same unstable poles.
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> If $||\Delta||_\infty <1$:
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> $$ \left| \frac{\tilde P (j\omega)}{P(j\omega)} - 1 \right| \leq | W_2(j\omega) | \text{ , } \forall \omega$$
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$|W_2(j\omega)|$ is the uncertainty profile. This inequality describes a disk in teh complex plane: at each frequency the point P~/P lies in the disk with center 1, radius |W_2|.
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W_2 is basically a transfer function that will always be greater in magnitude than that P~/P -1
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>[!note] Robustness
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>A controller $C$ is robust to set of plants $\mathcal{P}$ with respect to a characteristic if this characteristic holds for every plant in $\mathcal{P}$.
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>> [!important] Robust Stability
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>> A system is robustly stable if it is internally stable for every plant in the set $\mathcal{P}$:
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>> $$|| \Delta W_2 T ||_\infty < 1$$
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>> ![[Pasted image 20241015172652.png]]
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>> Nominal performance is achieved simultaneously when
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>> $$ || \text{max} (|W_1S|, |W_2 T|)||_\infty < 1$$>
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>
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>>[!important] Robust Performance
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>>$$ |||W_1 S | + |W_2 T| ||_\infty < 1 $$
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>>![[Pasted image 20241015172708.png]]
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Something really helpful to think about came to mind as a result of watching a Steve Brunton video[^1]. Think about the way that loop gain works:
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$$ y = \frac{L}{1+L} r $$
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If at a certain frequency $\omega$, L approaches -1, big problems happen. What this means is that the denominator in the above equation gets really small, which means the gain from r to y actually gets really big. If it IS -1, immediate undefined blow up.
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This is where robustness comes from. The distance between L and -1 for all frequencies is what robustness is. Less distance, less room for plant perturbation that could make you unstable. More distance, safer response.
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[^1]: [[stevebruntonControlBootcampSensitivity2017]] |