Why do you write for shooting method XN and lambda0 on left hand side of the equation and turn it upside down on the other hand? And why the matrix M has this N at the exponent?
The two-point BVP displayed at 0:56 is a prescription for computing x_1 and lambda_{N-1} if x_0 and lambda_N are known. Typically x_0 is known but lambda_N is generally unknown. But ignore this fact at first (and work just symbolically). Now that we have "computed" x_1 and lambda_N-1, we can repeat the previous computation (multiplication by a matrix) and compute x_2 and lambda_N-2. And so on, untill we get x_N and lambda_0. The matrix that is labelled as M at about 1:24 gives us a relation (a linear one) between x_N and lambda_0 on on side and x_0 and lambda N on the other. Out of the four guys, two are given here: x_0 and x_N, and two are unknown: lambda_N and lambda_0. The known guys are not both on the same side but this is not a major problem. We can still solve for the two unknowns (and the procedure is shown in the video). In fact, computing the lambda_N is enough because once we get it, we can solve for full trajectories of x and lambda.
aa4cc Thank you so much for the explanation, it was clear for me! My last question would be: I've been following nicely the introduction to MPC and optimization videos, but since it started this part of the series about the "2 point BVP" and "indirect approach" for optimal control, I come into confusion of what is this method for... I mean, is it an extension for MPC control or is it purely optimization math to compute the optimal inputs "u"? Hope you can clarify me the applications of this indirect approach and this BVP.
@@jesusmanuellevinsonrondon5107 These are just alternative approaches. Different ways to formulate and solve a problem of optimal control. It is similar to what happens if you want to minimize a function of a scalar argument: either you can view it DIRECTly as minimization (and come up with some numerical techniques) or you can reformulate it as a problem of finding a root of a nonlinear equation (remember, we set the derivative - provided it exists - to zero) and by solving this reformulated problem you are solving the original problem INDIRECTly.
Thank you so much for your efforts and for sharing this ... god bless you!!
Why do you write for shooting method XN and lambda0 on left hand side of the equation and turn it upside down on the other hand? And why the matrix M has this N at the exponent?
The two-point BVP displayed at 0:56 is a prescription for computing x_1 and lambda_{N-1} if x_0 and lambda_N are known. Typically x_0 is known but lambda_N is generally unknown. But ignore this fact at first (and work just symbolically). Now that we have "computed" x_1 and lambda_N-1, we can repeat the previous computation (multiplication by a matrix) and compute x_2 and lambda_N-2. And so on, untill we get x_N and lambda_0. The matrix that is labelled as M at about 1:24 gives us a relation (a linear one) between x_N and lambda_0 on on side and x_0 and lambda N on the other. Out of the four guys, two are given here: x_0 and x_N, and two are unknown: lambda_N and lambda_0. The known guys are not both on the same side but this is not a major problem. We can still solve for the two unknowns (and the procedure is shown in the video). In fact, computing the lambda_N is enough because once we get it, we can solve for full trajectories of x and lambda.
aa4cc Thank you so much for the explanation, it was clear for me! My last question would be: I've been following nicely the introduction to MPC and optimization videos, but since it started this part of the series about the "2 point BVP" and "indirect approach" for optimal control, I come into confusion of what is this method for... I mean, is it an extension for MPC control or is it purely optimization math to compute the optimal inputs "u"? Hope you can clarify me the applications of this indirect approach and this BVP.
@@jesusmanuellevinsonrondon5107 These are just alternative approaches. Different ways to formulate and solve a problem of optimal control.
It is similar to what happens if you want to minimize a function of a scalar argument: either you can view it DIRECTly as minimization (and come up with some numerical techniques) or you can reformulate it as a problem of finding a root of a nonlinear equation (remember, we set the derivative - provided it exists - to zero) and by solving this reformulated problem you are solving the original problem INDIRECTly.
Thank you both for your comments, it really cleared up this part for me.
I think that there s sth wrong here ... if you replace k in that expression with either 0 or N-1, it won't give you that result