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

@@ -46,3 +46,32 @@ Something interesting - If $\frac{1}{1+PCF}$ is proper, then this suggests that
 >**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 - Plant Uncertainty