Hello!
I have gone through the literature of optimal control. It has been stated in the literature that "mostly the relation between control law and performance indices will be non-linear. However, there are problems with non-linear relations of control law and performance indices. Therefore, for simplicity and some other reasons, quadratic relation between these two are the obvious choice."
I am not much clear about the nonlinear relations between control law and performance indices. Also, what are the reasons for choosing quadratic relation between them.
Thank you.
Hi Ankur,
The quadratic cost function (in terms of states) has been popular because of several of its useful properties:
In fact, so strong has been the popularity of the quadratic optimal control function, that there are complete control engineering areas (such as the LQR and LQG) that have been evolved essentially "around" it !
-Sanjay
Dear Ankur,
The quadratic term used in the optimal control is relative to the criteria. Why is the quadratic criterion: this fonctionnelle minimize on several variables (in the case of minimum unconstrained) minimum conditions can be derived from the Taylor expansion. We can then establish the minimum conditions using the Hessien was Jx = 0 v dx Jxx> 0 v dx.
Where a matrix is defined positive when it checks : xT.A.x > 0 v x
In the case of minimum constrained : We have j=( X1)2+(X2)2+(X3)2 under constrained g(x)= x1+x2+x3-1=0
It comes down to a constrained minimization problem to a minimization problem without constraint by introducing the Hamiltonian H (x, m, t)
H(x, m, t) = J(x) + mT.g(p)(x)
Seek the minimum of J (x) in the p constraints g (p) (x) = 0 is equivalent to search the minimum of H without constraint. With variations in x and independent of m; indeed :
Hx (x *) = Jxx *) + mT.gpx (x *) = 0 n relationship that provide x * (m).
Hm = g (p) (x) = 0 provides p relationships that give m
We must rigorously check Hxx> 0.
Best regards
as noted above, including a nonlinear term facilitates an interior solution; the alternative (in a linear model) is either a bang-bang or a singular solution.
Dear Ankur,
Sanjay's answer is correct. We use quadratic cost function mostly for ease of solution. In real applications, this cost function may not be desirable. But, in the small vicinity around the desired operating point, linear quadratic solution is the optimal solution. In large regions of state space, general nonlinearity is essential. Recently, we have introduced matching designs for feedback linearizable nonlinear systems to achieve optimality near the desired operating point and inverse optimality in entire state space. This gives you the best of both world, linear and nonlinear.
Data panezakrekok2001ba
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