Normally We all know the state space model of the the form
der(x) = F*x(t)+G*u(t)
y = H*x(t)+J*u(t).
However I came across a state space model which has the following form
der(x) = F*x(t)+G*u(t) + c
y = H*x(t)+J*u(t).
Vector c is a constant matrix with same dimensions as Vector G. Actually this constant matrix was introduced in my Plant as due to the model of a gravity component which is always present and not associated with any of the input. We CANNOT assume c is a disturbance as we precisely know c.
One way i tackled it using the following form
der(x) = A*x(t)+[G c]*[u(t) 1]'
y = C*x(t)+D*u(t).
This made my system look like a two input system with second input as constant 1 and changed my G matrix from G to [G c].
However this technique is a serious problem in my full state feedback control and reference tracking as it does not behave normal. I haven't come across this type of SS models. i have searched a lot and just identified them as Affine Non Autonomous Systems Page 11. However I am unable to find some good resource on the web which deals with these types of systems. Every paper I have read is so much mathematically rigorous that it is beyond my understanding.
is there some control theory established for such systems which deals with control-ability, observe-ability, full state feedback and reference tracking? Has someone come across these types of systems?
I am using Matlab for my simulations and using lsim command.
In the system x(t) is 6 x 1, F is 6 x 6, c is 6 x 1, G is 6 x 1 and J = 0. u(t) is 1x1.
Thanks
The standard approach is to augment a new state vector to the system with a zero derivative. So let $x_1 = x$ and $x_2$ be your augmented state, then, $\xi = \left[\begin{array}{c} x_1 \\ x_2 \end{array}\right]$. Your new plant matrix will be $\begin{pmatrix} F & I \\ 0 & 0\end{pmatrix}$.
Your new control matrix will be $\begin{pmatrix} G \\ 0\end{pmatrix}$ and your new observability matrix will be $\begin{pmatrix} H & 0\end{pmatrix}$.
Alternatively, define $\hat{c}$ as the solution to $G\hat{c} = c$. Then you have $\dot{x} = Fx+G\hat{u}$, where $\hat{u} = u+\hat{c}$. Solve this control problem, then back out $u$ from $\hat{u}$.