Thread: Maximising the response of a system

1. Maximising the response of a system

Hi there! I hope this isn't to vague, I'm rubbish at describing problems...

I have a nonlinear problem that I'm modelling in Matlab, and I want to maximise some of the states of the equation, while minimising others in response to a user input. To do this, I intend to linearise the equation about its equilibrium condition and write it in state space form, i.e.,

$\displaystyle \dot{x}(t)=A(t)x(t)+B(t)u(t)$
$\displaystyle y(t)=C(t)x(t)+D(t)u(t)$
$\displaystyle z(t)=E(t)x(t)+F(t)u(t)$

with my observed states y and z maximised and minimised respectively subject to a user input u. This means that I need to find an optimum user input u, and optimum controller matrix B(t) (with location and size constraints) for the system.

Does anyone know how to even begin solving this problem? I'm looking everywhere, but I don't know the key words to use to start looking for books. I've found loads of books titles 'optimal linear control' and such like, but nothing covering what I'm requiring.

2. just a sec...
u need optimum user input of "step function" or it's just coincidence, that your input is just any u(t) (as any input variable) ?

3. I'm sorry, I'm not quite sure I understand you comment. I've also decided my original post isn't very helpful, so I wanted to clear something up.

I have a system, and I want to find the optimum location and size of controller and user input such that certain states of the system are maximised, and others minimised. Therefore, my first equation looks like this:

$\displaystyle \dot{x}(t)=A(t)x(t)+B(t,S,L)u(t)$

where S is the size of the controller, L is the location, and u(t) is the user input. A is a nxn matrix where n is the number of states I have, and B is a nxm matrix, where m is the user of user inputs I have on the system. Of course, I have a limitation on where the controller is located, putting constraints on L, and I have certain limitations on how big a user input I can use. The size of the controller is not a problem, as part of the optimisation should ensure it is not abnormally large (though some constraints might be needed here too).

So, some of the entries of the x vector I want to maximise and some I want to minimise, but as you can see, I have 3 variables which I'm able to change. I'm not only required to find the optimum user input u, but also the optimum location and size, which will of course change my B matrix. I know that there are a number of standard methods to maximise the performance given that the B matrix is constant, or changes with time, but I'm not sure how to include the changes in the B matrix into the optimisation strategy. That said, I've not read a huge about optimisation strategies, so there might be a very simple way of dealing with this, hence why I've posted here.