Hi! I have a system that loosely looks like this-
$\dot{x}_{1}=f_{1}(x_{1},x_{2},x_{3},x_{4})$,
$\dot{x}_{2}=f_{2}(x_{1},x_{2},x_{3},x_{4})$,
$\dot{x}_{3}=x_{4}$,
$\dot{x}_{4}=f_{4}(x_{1},x_{2},x_{3},x_{4})+bu_{1}$,
$y=x_{3}$,
I am designing a direct fuzzy adaptive controller to control the state x3. I wish to know what should be inputs to the fuzzy system that will approximate the ideal controller? Is it going to be all the states i.e $x_{1},x_{2},x_{3},x_{4}$ or $e,\dot{e}$? And finally what kind of adaptive law will ensure that the error is driven to zero. In my real system, I have got 9 states, and the state that I am interested in controlling has a relative degree of two. So, Shall I take all the states as input to my adaptive fuzzy controller? If I choose that, then will it not be exhorbitantly computationally expensive- considering the fact that there are 3 MFs per input-- resulting in $3^{9}$ rules? Kindly provide your inputs. Thanks in advance.
Hi Amardeep,
Your questions can be improved by rephrasing them to provide better clarity for the readers.
There must be some design flaws in your direct adaptive fuzzy control approach. To understand the working of adaptive control, there are less complicated methods that can be tried before resorting to fuzzy logic. For example, you can develop the approximators based on the physical knowledge of the patterns or structures of the underlying nonlinearities, f1(x1, x2, x3, x4) and f2(x1, x2, x3, x4).
The inputs to the approximators should be based on measurable signals, {x1, x2, x3, x4}. To stabilize the adaptive control system, you have to design the control law and the parameter update law such that an error-based radially unbounded Lyapunov function V(e) is chosen to satisfy V' < − κ1V + κ2, where κ1 and κ2 are positive constants. This ensures that the error, e is bounded.
Hi Yew!
I have tried various approaches- i.e gradient descent based adaptation(labiod et al.), lyapunov based adaptation(wang et al), however unfortunately for some reason I am not able to track the desired trajectory.
Hi Yew! Actually I have an airship as my system, It has 6 DoF resulting in 12 differential equations(V,alpha,beta,p,q,r,phi,theta,psi,x,y,z). I wish to control the attitude- theta and psi. I have choosen 9 states(V,alpha,beta,p,q,r,phi,theta,psi) as inputs to my fuzzy system that approximates an ideal controller. However even after trying out various adaptation laws, I am still not able to track desired theta and psi(theta_Desired=.8 and psi_desired=.9). Here are the error profiles of theta and psi. As we can see, instead of driving the error to zero, it oscillates. PFA
Question #1:
One possibility is due to poor approximation. Have you really compared the output of the approximated function with the output of the actual function? If the approximation error is large or deviating from zero, there could be various reasons.
Question #2:
Another possibility is that the performance of the baseline control (nonlinear state feedback) is poor, slow, or sluggish. Have you really compared the results of the adaptive control with the results of the baseline control with known functions f1 & f2?
Question #3:
Have you really checked if the residual is sufficiently compensated by robustifying term in the adaptive control? Please check the V' signal, where V' is the time derivative of Lyapunov function.