need to build the optimal controller, i.e. one that maximizes: J = <msubsup> &#x222B

glycleWogry

glycleWogry

Answered question

2022-06-22

need to build the optimal controller, i.e. one that maximizes:
J = 0 t f f ( u ) d t
For the following time-dependent system:
x ˙ = g ( x , u , t )
where x ( t ) R is the state, u ( t ) R the input, t R the time and l ( t ) R the time-varying lower bound for the state. Using the Pontryagin's maximum principle, I have defined the Hamiltonian as: H = ψ g ( x , u , t ) + f ( u ) and used the necessary condition ψ ˙ = H x . As you can see I have completely ignored the time-dependent inequality, so the solution I get is correct, but of course it doesn't enforce the lower bound l on the state value.
How can I rewrite the hamiltonian so that also the inequality is considered?

Answer & Explanation

Paxton James

Paxton James

Beginner2022-06-23Added 25 answers

I think we need to add λ ( t ) ( x ( t ) l ( t ) ) to the Hamiltonian, where λ ( t ) is a time-dependent Lagrange multiplier. Then we need to find a triplet of functions x , u , λ, such that the HJB-equations are satisfied and λ ( t ) 0 and λ ( t ) = 0 if x ( t ) > l ( t ).

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