Problem 1 (20 pts) Consider a first order unstable plant with a pole at p: K...

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Problem 1 (20 pts) Consider a first order unstable plant with a pole at p: K G(s) S-P Let's say that it is controlled by a 1st order dynamic controller: C(s) O-S Where the pole for C(s) is in the left-hand complex plane. The controller and plant are combined in the usual way in a unity negative feedback loop. Physically, it makes sense that the controller must be able to act more quickly than the plant in order to stabilize it. Prove this claim mathematically. Hint: recall the relationship between a dynamic pole's location and the speed of its response. Problem 1 (20 pts) Consider a first order unstable plant with a pole at p: K G(s) S-P Let's say that it is controlled by a 1st order dynamic controller: C(s) O-S Where the pole for C(s) is in the left-hand complex plane. The controller and plant are combined in the usual way in a unity negative feedback loop. Physically, it makes sense that the controller must be able to act more quickly than the plant in order to stabilize it. Prove this claim mathematically. Hint: recall the relationship between a dynamic pole's location and the speed of its response.