Imagine we have an ODE system
x_dot=[f1(x,u), f2(x,u), f3(x,u),....fn(x,u)]
where f1,..,fn are nonlinear functions of control input u and states x,
x is member of R^n and u is member of R^m
under which conditions we can change the dynamic of the system to arbitrary dynamic x_dot=[f1_des(x) f2_des(x),...,fn_des(x)]
where f1_des(x) f2_des(x),...,fn_des(x) are arbitrary pre-defined nonlinear functions.
is input to state controllability enough to do that?
Example
consider the 2-D system:
x1_dot=f1(x1,x2)+g1(x1,x2)*u1
x2_dot=f2(x1,x2)+g2(x1,x2)*u2
by choosing
u1=(f1_des(x1,x2)-f1(x1,x2))/g1(x1,x2)
u2=(f2_des(x1,x2)-f2(x1,x2))/g2(x1,x2)
we can change the dynamic of the system to
x1_dot=f1_des(x1,x2)
x2_dot=f2_des(x1,x2)
Hi Ali Namadchian,
Naturally, for control-affine nonlinear systems as in your 2-D example, as long as the condition gi(x1, x2) ≠ 0 is satisfied, then you can transform the nonlinear system. For other cases, the conditions are problem-dependent. If you are interested, you are encouraged to read about the application of Lie bracket to the analysis of nonlinear systems.
Dear Dr. Yew-Chung Chak
gi(x1, x2) ≠ 0 is necessary or sufficient condition for our control-affine nonlinear system?
I think it could not be sufficient condition, consider the previous example for just one control input
x1_dot=f1(x1,x2)+g1(x1,x2)*u1
x2_dot=f2(x1,x2)+g2(x1,x2)*u1
Here gi(x1, x2) ≠ 0, but is it possible to find u1(x1,x2) such that alters the dynamics of the system to
x1_dot=f1_des(x1,x2)
x2_dot=f2_des(x1,x2)
?
Dear Ali Namadchian,
The answer for your question is dependent on classes of the nonlinear systems and control targets you are considering.
For a class of nonlinear systems in strict feed-back form and (robust, adaptive) optimal control, you can use back-stepping techniques to transform the system into equivalent affine nonlinear systems, and then design the optimal controller.
The particular condition is based on the particular problem, for example, there exit a stable controller, the lipschitz condition, smooth nonlinear functions, bounded functions,...
Best,
Hi Ali,
Your question is very general and, as a general rule, I don't like to deal with the general solution for general problem.
Your example instead is very specific. The control signals are separate (affine system) and therefore easy to use and you also assume, even without mentioning, full-state availability. Adding to it controllability, then yes, you can do about everything with your system.
dear Itzhak Barkana
you are correct
these assumptions are necessary
-control affine system
-availability of states
- controllability
beside these assumptions, do you think if the dimension of control is less than the dimension of states, is it possible to do such transformation?
Yes, it is. Controllability, or more precise input-to-state controllability, means that you can force all states behave as you desire, even for a single input.
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