Obsidian/4 Qualifying Exam/3 Notes/Feedback Control Theory.md

All info from doyleFeedbackControlTheory2009

Chapter 1 - Introduction

! Notable signals:

• r - reference or command input
• e - tracking error
• u - control signal, controller output
• d - plant distrurbance
• y - plant output
• n - sensor

Chapter 2 - Norms

[!note] Signal Norms 1-Norm:

||u||_1 = \int_{-\infty}^{\infty} |u(t)|dt

2-Norm:

||u||_2 = \left(\int_{-\infty}^{\infty} u(t)^2 dt \right)^{1/2}

$\infty$-Norm

||u||_\infty = \sup_t |u(t)|

Power Signals (Not really a norm):

pow(u) = \left( \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^T u(t)^2 dt \right)^{1/2}

If the limit exists, u is called a power signal

[!caution] Some Implications

1. ||u||_2 < \infty \rightarrow pow(u) = 0
2. u is a power signal and ||u||_\infty < \infty \rightarrow pow(u) \leq ||u||_\infty
3. There's a third one in the book about the one norm. I'm ignoring it.

[!nnote] System Norms \hat G means the transfer function G in the frequency domain. 2-Norm:

||\hat G||_2 = \left(\frac{1}{2\pi} \int_{-\infty}^\infty |\hat G(j\omega)|^2d\omega \right) ^{1/2}

$\infty$-norm

||\hat G||_\infty = \sup_{\omega} |\hat G(j\omega)|

[!hint] Parseval's Theorem If \hat G is stable, then

||\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}

!

Chapter 3 - Basic Feedback Loop

! 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.

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.

[!tip] Nine System Transfer Functions

\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)

Notable Properties:

• All 9 transfer functions are strictly proper if 1+PCF is not strictly proper.
• If all 9 transfer functions are stable, then the system is internally stable

[!note] Internal Stability A couple of theorems:

[!hint] Theorem 1 The feedback system is internally stable iff there are no closed-loop poles in Res \geq 0.

[!hint] Theorem 2 The feedback system is internally stable iff the following two conditions hold: a) The transfer function 1+PCF has no zeros in Res \geq 0. b) There is no pole-zero cancellation in Res \geq 0 when the product PCF is formed.

[!warning] Nyquist Criterion The feedback system is internally stable iff the Nyquist plot does not pass through the point -1 and encircles it exactly n times counterclockwise.

The sensitivity function is defined as: 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. 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.

There is another function to understand. The complementary sensitivity function is defined as:

T = 1-S = \frac{L}{1+L}

This function is the transfer function from the reference input r to the output y.

Generally speaking:

1. Performance specs that involve e result in weights on S
2. Performance specs that involve u result in weights on S

\left[\matrix{e \\u}\right] = -\left[\matrix{PS & S \\ T & CS}\right] \left[\matrix{d \\ n}\right]

Chapter 4 - Uncertainty and Robustness

[!important] Multiplicative Disk Perturbation

\tilde{P} = (1+\Delta W_2)P

• P is a nominal plant transfer function
• \Delta is a variable stable transfer function s.t. ||\Delta||_\infty <1
• P and \tilde P have the same unstable poles. If ||\Delta||_\infty <1:

\left| \frac{\tilde P (j\omega)}{P(j\omega)} - 1 \right| \leq | W_2(j\omega) | \text{ , } \forall \omega

|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|.

W_2 is basically a transfer function that will always be greater in magnitude than that P~/P -1

[!note] Robustness 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}.

[!important] Robust Stability A system is robustly stable if it is internally stable for every plant in the set \mathcal{P}:

|| \Delta W_2 T ||_\infty < 1

! Nominal performance is achieved simultaneously when $$ || \text{max} (|W_1S|, |W_2 T|)||_\infty < 1$$>

[!important] Robust Performance

|||W_1 S | + |W_2 T| ||_\infty < 1

!

Something really helpful to think about came to mind as a result of watching a Steve Brunton video¹. Think about the way that loop gain works:

y = \frac{L}{1+L} r

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.

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 ↩︎