Question

Problem 8: A simplified model of a glider is where y is the flight path angle in radians, v is the airspeed in m/sec, n -L/mg is the load factor, L is the lift in Newtons, m is the mass in kg, and k 61.6594 and k 4.8747x103 are constants for the glider. (a) Given that y -0.15 rad, and the airspeed is 50.8691 m/sec, find the necessary load factor to maintain equilibriunm (b) Let the state vector be [7 v]T, let the input be n, and let the output of interest be v. Derive the linearized system about the equilibrium point obtained from above (c) From the linearized system dynamics, derive the transfer function from input n to output v, and the transfer function from input n to output y.(Hint: Use tf and ss in Matlab to check your answers.) (d) Plot the unity gain negative feedback root locus of the linearized system using Matlab. Give the locations of the open loop poles, and the range of gains for which the closed-loop system is stable.

Answer 1

Well, you may apply some calculus in here. First of all, you know $\dot{\gamma}$ and $\dot{v}$ are corresponding to derivates. So there are many ways you could solve this. The first thing comes to my head is creating a relation between both equation to create an equilibrium point. Here to do it, also you have some different way to do it. You could calculate the integral to one of this, you choose which one after that you may replace one equation into the other, so you could find a good relation, after that, just replace the data you got, and simple clear up the you need.

It is more calculus in here, but don't worry. It is not to complicate as it looks. Just make some calculus tools like derivates and integrals and that's set. I suggest you to minize the equation so maybe you could see it easier, there are lot of terms repeated. And for the 3 and 4, well, the 3 they are asking you to make an transpose matrix, after you get the values, just make a matrix as they ask you, so later you have the matrix, apply all the transpose properties which is actually invert the Aij to Aji. Also you could read about this and 4 you don't specified what simulator are you using to write the code of this. Hope enjoy.