in the literature, for solving an optimal control problem, most of the times we consider the system (ode equation) as a constraint and solve the optimal control problem, here is the question: if I have the analytical form of the solution of my system (ode), is it possible to directly replace the analytic form of the solution into the cost function and solve an optimization problem to find the optimal control? (I approximate the solution of the ode by a Neural network, so I have an analytic form of the solution)
intuitively it seems that it is possible to do it. is there any article that consider such method?
Dear Ali,
You want to proceed as someone already has the solution to the end point and you go back in time to check your optimal trajectory, if that is so! I think you can do that and normally there is no problem.
Also i suggest you to see links and attached files in suject.
Analytical solutions to LQG homing problems in one dimension ...
www.tandfonline.com/doi/full/10.1080/21642583.2013.878886
Control and Optimization with PDE Constraints
https://books.google.dz/books?isbn=3034806310 -
Hi, in your problem the control variable u apeears in the right-hand-side of the ode. Thus, how to you determine an analytical solution of the ode ? Is you control parametrized ?
Hi Ali ,
In optimal control problems, if you consider a linear system, ODE becomes
Algebraic Riccati equation (ARE) equation subject to a cost function. You can directly use the analytical solution of the equation, for example you use Matlab to find ARE solution.
In optimal control problems for nonlinear systems, the ARE equation become an HJB equation which does not have analytical solutions. I introduce you adaptive dynamic programming (ADP) technique to approximate the solutions online.
Hi Ali,
I agree with prof. Epenoy. In an optimal control problem, there is a control function in the dynamical system of equations too, which we should decide on that...
dear Dr. Epony and Dr. Mehrpouya
If my system is x_dot=f(x), for linear system I have the analytical solution of phi(x0,t), and if f is nonlinear I can use the neural network to find the analytical solution for x(t). It is what Lagaris proposed in his paper:
"Lagaris, Isaac E., Aristidis Likas, and Dimitrios I. Fotiadis. "Artificial neural networks for solving ordinary and partial differential equations." IEEE Transactions on Neural Networks9.5 (1998): 987-1000."
now if my system is a control system x_dot=f(x,u) and if I could approximate the solution of the system in the analytical form of phi(x0,t,u) using neural network, is it possible to directly replace phi(x0,t,u) in the cost function (instead of x) and solve an unconstrained optimization to find the optimal control u(t)?
Sure....in this case, solving the optimal control problem reduce to solve an optimization problem in which, the parameters of the control function are decision variables.
I think that if you know the solution of de differential equation of the system, you can use it into the Hamilton equations and into the partial derivative of the Hamiltonian function with respect to the input, and hopefully you can arrive to the optimal control law if it exists. This procedure is used already for single non-linear systems (see the examples of the book of Jacob Engwerda, LQ Dynamic Optimization and Differential Games). Alternatively, as stated by Mohamed-Mourad Lafifi you can consider a discrete version of your optimal control problem and apply the optimal principle of Bellman or dynamic programing to solve in backward direction your problem. I think that this second approach is more easy.
dear Dr. Epony and Dr. Mehrpouya
you are right, in the right-hand-side of the ode, the control function appears, and I can not find the analytical solution for the ode unless I parameterize the control function (assume it as a function of state (x)). One possible form of the control is u=- kx which is the best form for linear systems. But what is the most general form that I can consider for u as a function of x? for example is the form
u(x)=a+b*x+c*x^2+....
a clever choice for a general nonlinear ode? this way I can find the optimal control by fining the optimal values of parameters a,b,c,..
what is the best form of parameterized control function for a general nonlinear ode?
Dear Ali,
Control parameterization techniques are a wide variety of methods where each of them have benefits and limitations. For instance, some are superior in obtaining a more accurate solution and some of them have less memory utilization. So, there is no conception called the best technique for solving an optimal control problem using control parameterization methods.
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