The following differential equations represent a simplified model of an overhead crane:3 where mC is the mass of the trolley, mL is the mass of the hook/load, l is the rope length, g is the gravity acceleration, u is the force applied to the trolley, x1 is the position of the trolley, and x3 is the rope angle. Consider the position of the load y = x1 1 + sinx3 as the output. (a) Determine a linearized statespace model of the system about the equilibrium point x = 0 with state variables x1, x3, the first derivative of x1, and the first derivative of x3. (b) Determine a second statespace model when the sum of the trolley position and of the rope angle is substituted for the rope angle as a third state variable.
Consider the continuous-time model of the overhead crane proposed in Problem 7.10 with mc = 1000 kg, ml = 1500 kg, and l = 8 m. Design a discrete full-order observer state feedback control to provide motion of the load without sway.
Problem 7.10
The following differential equations represent a simplified model of an overhead crane:3
where mC is the mass of the trolley, mL is the mass of the hook/load, l is the rope length, g is the gravity acceleration, u is the force applied to the trolley, x1 is the position of the trolley, and x3 is the rope angle. Consider the position of the load y = x1 1 + sinx3 as the output.
(a) Determine a linearized statespace model of the system about the equilibrium point x = 0 with state variables x1, x3, the first derivative of x1, and the first derivative of x3.
(b) Determine a second statespace model when the sum of the trolley position and of the rope angle is substituted for the rope angle as a third state variable.
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