F(t)=kP​θ(t)+kI​∫0t​θ(t′)dt′+kD​θ(t) F(t)=kp​θ(t)+kt​∫0t​θ(ti)dt∗+kb​θ(t) where kp​,kl,​ and k3​ are the gain factors fo

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task 1 please help
F(t)=kP​θ(t)+kI​∫0t​θ(t′)dt′+kD​θ(t)
F(t)=kp​θ(t)+kt​∫0t​θ(ti)dt∗+kb​θ(t) where kp​,kl,​ and k3​ are the gain factors for the proportional, integral and derivative control components, respectively. Please note that these gain factors differ from one another in their respective dimensions. Figure 1. Schematic of an inverted pendulum attached to a cart. Consider the simplified system of Fig.?, with a cart of negligible mass. Derive the differental equation governing &. for this purpose, fust represent the force as F(t). (That is, don't assume anything yet about a control scheme.) HiNT uf: The sum of the moments about the mass center is equal to the moment of inertio about the mass center of the pendulum times the angular ticeicration. HINT H2: There is an eosy formsio for the moment of inertio of o point mass about a given axis. HINT u3: The center of mass of the pendulum is probebly where all of the mass of the prendulum is