I'm trying to reproduce the numerical solution given in the paper by Curtis and Beard, Successive collocation: An approximation to optimal nonlinear control, Proceedings of the American Control Conference 2001. But without success. Probably I must have misunderstood a few concept here. I have try solving using Pontryagin's Minimum Principle and Riccati equation.
The problem is to minimize the cost J(x_0)=integral_0^10 (x'x + u'u) dt
subject to the dynamic dot(x) = [0 , 1; -1, 2] x + [0 , 1]' u
and initial condition x_0=[-12 , 20]'.
x=[x_1 , x_2]' is the state variable, u the control and prime denote matrix transpose.
The answer that I obtain is always 2346.5 but the numerical optimal value given in the paper is J*=2221. Can someone please confirm that the given J* is correct.
I consider the final state x(10) to be unspecified, i.e. the co-state lambda(10)=0. Is this correct?
Dear Amran Hussin. If I solve this example using Boundary Condition Iteration method, I get the same cost as you, so I guess there's a typo in the original paper.
Thanks Radoslav. Initially, I really though I have miss something here because the system is not stable. The eigenvalues are 1,1. You see I'm trying to work on a new numerical method similar to that paper. I want to make sure first that the method works for linear system before moving to non-linear system. The technique that you use is new to me. Can you please lead me to some literatures on Boundary Condition Iteration.
Dear Amran Hussin. Method of Boundary Condition Iteration is a method for resolution of Two-point Boundary Value Problem one obtains after explicit use of Pontryagin's principle.
It works as: Initially, the guess lambda(t0) is used to integrate the set of ODEs for state and adjoint variables forward in time. Values of lambda(tf ) obtained by integration are then compared with optimality condition from Pontryagin's principle. If these are not in accord, a new guess of initial conditions for adjoint variables is made and the whole procedure is repeated until they are equal within specified tolerance. This update may be done e.g. using evaluated sensitivity of the λ(t) trajectory to this initial condition.
The method looks simple enough some sort like the shooting method for BVP. Do the method required the transition matrix to move forward in time? This is interesting because I'm also working on how to effectively compute the state-transition matrix. See my other question thread.
Yes, the transition matrix has to be propagated through time. The method does simply a simulation of the set of differential equations for state and adjoint variables. If you aim for higher-order systems, then I am not sure if this is the right method to use. It might not converge that well.
Dear Amran,
Did you consider using direct numerical methods to solve the optimal control(OC)? Though there are advantages of using indirect numerical methods for OC, to use direct collocation method is straightforward. Also, I did not find much researches of combining these two approaches.
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