Difference between revisions of "CDS 110b: Optimal Control"

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{{cds110b-wi08 lecture|prev=Two Degree of Freedom Control Design|next=Linear Quadratic Regulators}}
 
This lecture provides an overview of optimal control theory.  Beginning with a review of optimization, we introduce the notion of Lagrange multipliers and provide a summary of the Pontryagin's maximum principle. __NOTOC__
 
This lecture provides an overview of optimal control theory.  Beginning with a review of optimization, we introduce the notion of Lagrange multipliers and provide a summary of the Pontryagin's maximum principle. __NOTOC__
  

Latest revision as of 03:30, 2 March 2008

CDS 110b Schedule Project Course Text

This lecture provides an overview of optimal control theory. Beginning with a review of optimization, we introduce the notion of Lagrange multipliers and provide a summary of the Pontryagin's maximum principle.

References and Further Reading

Frequently Asked Questions

Q: Could you please explain what the psi function is or what it means if psi(x(T))=0 versus what it means if psi(x(T))=x(T)?

The psi function represents a general form of terminal constraint for the state variables. It gives a way of indicating which states have a terminal cost attached to them. For example, by defining psi_i(x(T))=x_i(T)-x_i,f for i=1,2,...n, we can impose terminal costs on all states (a fully constrained case) by letting p=n (n being the # of states). When we optimize over time and want x(T)=x_f, then x(T)-x_f=0, and so psi(x(T))=0. If we take x_f=0, then psi(x(T))=x(T).

Luis Soto, 22 Jan 08

Q: In Problem 2.4d, are the boundary conditions for the differentially-flat trajectory correct?

Please ignore the boundary conditions given in part 2.4d for the differentially-flat trajectory and instead use x(0)=1 for the initial condition and x(1)=0 for the condition at final time t=1. Moreover, use c=100 instead of c=1. Note: the x(t_f) of the optimal solution won't be exactly 0, but will be close enough for the intent of this problem.

Luis Soto, 21 Jan 08

Q: In problem 2.4(d) of the homework, to what positive value should the parameter b be set?

Use b = 1 for part d when solving for and comparing the two trajectories found symbolically in previous parts.

Julia Braman, 18 Jan 08

Q: In the example on Bang-Bang control discussed in the lecture, how is the control law for \(u\) obtained?

Pontryagin's Maximum Principle says that \(u\) has to be chosen to minimise the Hamiltonian \(H(x,u,\lambda)\) for given values of \(x\) and \(\lambda\). In the example, \(H = 1 + ({\lambda}^TA)x + ({\lambda}^TB)u\). At first glance, it seems that the more negative \(u\) is the more \(H\) will be minimised. And since the most negative value of \(u\) allowed is \(-1\), \(u=-1\). However, the co-efficient of \(u\) may be of either sign. Therefore, the sign of \(u\) has to be chosen such that the sign of the term \(({\lambda}^TB)u\) is negative. That's how we come up with \(u = -sign({\lambda}^TB)\).

Shaunak Sen, 12 Jan 06

Q: Notation question for you: In the Lecture notes from Wednesday, I'm assuming that \(T\) is the final time and \(T\) (superscript T) is a transpose operation. Am I correct in my assumption?

Yes, you are correct.

Jeremy Gillula, 07 Jan 05