Limit on the change rate of the variable in fmincon function

Hi,

In fmincon function, how can I put lower and upper bounds on the change rate of a signal, i.e., ?

Thanks

[Matt J]

What are the unknown variables in this problem and how is related to them?

Answer 1

[Matt J]

You would build linear inequality matrices like the following.

E=diff(eye(numel(u)));

A=[E;-E];  b=[ub(:);-lb(:)];  %linear inequality matrices

[Mohammad]

That doesn't seem to work. Assume u has 3 elements. Then, the code generates the following inequality,

[Matt J]

Yes, and the inequalities you wrote are precisely

lb(t) <= u(t+1)-u(t) <=ub(t)  

for 1<=t<=2. I don't see what you think is incorrect.

[Walter Roberson]

Is u the function being minimized, which would have to be a scalar for each point? Or is u the inputs to that function, which could be a vector? Is the question about the rate of change of the signal a question about the values of all of the elements of the Jacobian, that all of the partial derivatives are within the same lb ub pair? Is the question about the difference between adjacent locations that fmincon is permitted to examine?

Because what this proposed formulation seems to accomplish is to limit the difference between adjacent input variables, like saying in f = x^2 - y^2 as an objective function that lb <= x-y <= ub, which is not what I would understand as the rate change of a signal.

[Matt J]

I think u(t), t=1,2,...N have to be the design variables. If that were not the case, there would have been no reason for Mohammad to have interpreted my A,b data as imposing constraints on u, as he did here,

[Walter Roberson]

But he also said those do not work. Which is correct if the goal is to limit the elements of the Jacobian. If you are trying to limit the Jacobian you can only do that through linear constraints for functions that are at most quadratic.

[Mohammad]

u(t) is the design variable. How about using lb < diff(u) < ub in a nonlinear constraint as an argument of the fmincon function? Does that bound the slew rate change of u(t) such that lb < u(t+1) - u(t) < ub?

[Matt J]

You should not tell fmincon that a constraint is nonlinear when it is, in fact, linear. That prevents fmincon from recognizing and taking advantage of the linearity of the constraint.

[Catalytic]

You should not tell fmincon that a constraint is nonlinear when it is, in fact, linear.

The one exception to this that I can think of is when it is more computationally manageable to represent the operation A*x in operator form, similar to what pcg allows you to do. But here, I don't see any reason to do that. Note that Matt J's A,b matrices can also be constructed in sparse form if memory limits are a problem.

E=diff(speye(numel(u)));

A=[E;-E];  b=[ub(:);-lb(:)];  %linear inequality matrices

[Mohammad]

That's right. But the reason I defined it as a nonlinear one is that it's not possible to defined a time-dependent constraint using A, and b. Basically, diff(u) is not something multiplied by the variable u. Would be any way to get arund that?

[Matt J]

Basically, diff(u) is not something multiplied by the variable u.

Yes, it is. E*u is the same as diff(u). That is why my original proposal is valid.

>> u=rand(5,1);
>> E=diff(speye(5));
>> E*u

ans =

   -0.3930
    0.2690
    0.0212
    0.4964

>> diff(u)

ans =

   -0.3930
    0.2690
    0.0212
    0.4964

[Mohammad]

Ok. Thanks. So now I try both methods to see the differences. As you mentioned, I would expect the linear representation results in more accurate solution.

[Matt J]

You're welcome, but please Accept-click the answer that works for you.

[Mohammad]

I figured out a problem. I actually don't have u prior to run the fmincon. Hence, I cannot define E beforehand!

[Walter Roberson]

Could you confirm that what you want is that for design variables u1, u2, u3, ... that lb <= u2-u1 <= ub, and lb <= u3-u2 <= ub, and lb <= u4-u3 <= ub, and so on ? If so then that can be implemented through the A b matrix... but it is not what I would call "rate of change of the signal".

[Matt J]

I figured out a problem. I actually don't have u prior to run the fmincon. Hence, I cannot define E beforehand!

You don't need u beforehand. You just need to know the number of unknowns, N

E=diff(speye(N));

[Mohammad]

@Walter and Matt: u is a vector in each sampling time t. I want to bound the slew rate of this vector ovet time, i.e., I want u(t+1) - u(t) to lie between two bounds. I don't want to put bounds on the difference of the entries of u.

Hence, I see that none of the diff(u) nor E*u don't refer to what I want which is the slew rate of the vector u ( u(t+1) - u(t) ).

Is there a way, probably defining non-linear constraint to get that?

[Walter Roberson]

Is it correct that you have a function of a single variable?

fmincon evaluates the function with different values of the input variables. fmincon cannot know ahead of time what response there will be to the different inputs, so it cannot guarantee that on two adjacent calls that the function response will change only within certain bounds no matter what the step size is. But if you want to constrain the difference between values of the design variables that are to be attempted, then you can configure https://www.mathworks.com/help/optim/ug/fmincon.html#busog7r-options DiffMaxChange DiffMinChange FiniteDifferenceStepSize

I still think that what you really want is limits on the derivative(s) of the function at function at the location to be evaluated. If I am correct then except for functions that are at most quadratic, you would need to implement this through nonlinear constraints that know how to evaluate the derivative of the function.

[Matt J]

Is there a way, probably defining non-linear constraint to get that?

This is still a linear constraint!

I will assume that your total matrix of unknown variables is MxN and is of the form

U=[u(1, u(2),...,u(N)] 

where each u(t) is an Mx1 vector. I will assume, similarly that you have Mx(N-1) matrices LB and UB of lower and upper bounds, respectively. Then the appropriate linear inequality matrices are

E=kron(diff(speye(N)), speye(M));

A=[E;-E]
b=[UB(:);-LB(:)];

[Mohammad]

@Walter and Matt: The constraint can be like . This way there is no need for the fmincon function to know the signal ahead of time. How can I implement that?

[Mohammad]

@Matt: I don't have a set of those signals like an MxN matrix. At each sampling time I have an Mx1 vector u(t) and I want to put a limit on u(t) - u(t-1).

[Matt J]

It sounds like you're saying that at time u(t), the values of u(t-1) are already known. Then the bounds can be rewritten

lbnew <= u(t) <= ubnew

where lbnew and ubnew are the known bound vectors

ubnew=ub+u(t-1);
lbnew=lb+u(t-1);

They are known since u(t-1) is known. So, you now have plain ordinary bounds on u(t) and there is nothing special about the way you set up the problem.

But I need to emphasize that, if you have an objective f(t,u(t)) at each time t, the approach you are pursuing will not give you the minimum of if that was your goal in all this. There is no way to do that by solving an independent problem at each t.

Answer 2

[Walter Roberson]

See

https://www.mathworks.com/matlabcentral/answers/489165-how-can-i-bound-rate-of-changes-dx-dt-in-fmincon-optimization#answer_399837