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Daniel Alazard

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Optimal control

Simplified case : Llnear system, quadratic performance index, fixed horizon and final state

Problem :

Let us consider the linear system :

From a given initial state , the objective is to bring back the state to 0 within a given time horizon () while minimizing the quadratic performance index :

where and are given weighting matrices with and .

Solution using Pontryagin’s minimum principle :

where is the costate vector.

• the optimal control minimizes  :

• Costate dynamics :

• State-costate dynamics : (1), (2) and (3) leads to :

with

is the Hamiltonian matrix associated to such a control problem. (4) can be intregrated taken into account boundary conditions on the state-costate augmented vector  :

• initial conditions on  : (5),
• terminal conditions on  : (6).

The set of equations (4), (5) and (6) is also called a two point boundary-value problem.

• Integration of the two point boundary-value problem :

where , are the 4 submatrices partionning (WARNING !! : ).

Then one can easily derive the initial value of the costate :

where depends only on the problem data : , , , , and not on .

• Optimal control initial value : from equation (2) :

• Closed-loop optimal control at any time  : at time , assuming that the current state is known (using a measurement system), the objective is still to bring back the final state to () but the time horizon is now . The calculus of the current optimal control is the same problem than the previous one, just changing by and by . Thus :

with :

the time-varying state feedback to be implemented in closed-loop according to the following Figure :

Remark : is not defined since and is not invertible.

• Optimal state trajectories : The integration of equation (4) between 0 and () leads to (first row) :

where :

is called the transition matrix.

• Optimal performance index :

For any and a current state one can define the cost-to-go function (or value-function) as :

and the optimal cost-to-go function as :

From equation (4) : one can derive that :

Thus (after simplification) :

Thus :

From this last equation, on can find again the definition of the costate used to solve the Hamilton–Jacobi–Bellman equation ; i.e. : the gradient of the optimal cost-to-go function w.r.t.  :

The optimal performance index is : .

Exercises

• Exo #1 : show that is the solution of the matrix Riccati differential equation :

also written as :

• Exo #2 : considering now that , compute the time-variant state feedback gain and the time-variant feedforward gain of the optimal closed-loop control law to be implemented according to the following Figure.

Groupe(s) de recherche

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