Chance constraint

[chengzll]

Hello, 

I would like to add chance constraints to the MPC problem(ie. the examples of Model predictive control-Basics, the disturbance d{k} is random variables ), and  the constraints with random variables are established with a certain probability,

 but I have not found useful relevant information in the forum. Is there any related tutorials or examples for reference?

Thanks in advanced.

Best regards, 

chengzhe

[Johan Löfberg]

Not in the public release yet, so no documentation

[chengzll]

Dear Prof. Johan

  I want to use analytical methods to solve model predictive control problems with chance constraints, assume that the uncertainty variable d follows normal distribution, the optimization problem has the following form:

    min J=βΣ(e(k+h|k))^2+(1-β)Σ(u(k+h|k))  subject to 

1.     x(k+h)=A*x(k+h-1)+B*u(k+h-1)+E*d(k+h-1)
2.     Pr[ |y(k+h|k)-yref(k+h|k)|<=e(k+h|k) ]>=α
3.     Smin<=x(k+h|k)<=Smax
4.     Umin<=u(k+h|k)<=Umax

    the chance constraint 2 Pr[-e(k)<=C*(A*x(k)+B*u(k)+E*d(k))-yref(k+1)<=e(k)]>=α, the uncertain variable d can be separated, the constraint can be transformed as follows:

    (CE)^-1(-e-CA*x(k)-CB*u(k)+yref(k+1))<=φ^-1(1-α)，(CE)^-1(e-CA*x(k)-CB*u(k)+yref(k+1))>=φ^-1(α), φ^-1 is Inverse function of cumulative distribution function，

  I am not sure the above derivation is feasible, and I don't know how to combine the analytic method with the model predictive control state space expression with the uncertain variable d in 1. 

  I hope you can give some advice. Thanks in advanced.

[Johan Löfberg]

That does not look correct. with the double-sided representation of abs, you will have to deal with a joint chance constraint as you want to say that the probability that both constraints are satisfied hold with a certain probability, which is much harder than two single chance constraints dealt with individually

Having said that, you ask if the derivation is correct. The answer is that you don't have any derivation, you only made an unsubstantiated claim

[chengzll]

 If I use e1 and e2 to represent the positive and negative deviations of the output trajectory and the reference trajectory , can it replace the abs function above?

   (CE)^-1(-e1(k)-CA*x(k)-CB*u(k)+yref(k+1))<=φ^-1(1-α)，(CE)^-1(e2(k)-CA*x(k)-CB*u(k)+yref(k+1))>=φ^-1(α), e1(k)>=0, e2(k)>=0.

[Johan Löfberg]

modelling absolute value is the trivial part. The problem is that once you've done that, your single chance constraint has evolved into a joint chance constraint, and there is no closed-form expression for that as far as I know.