I came across a non-linear MIMO system with coupled inputs which is to be modelled in state space form. The kinematics are as follows:
$$ \ddot{x} = F_Ecos(\psi)sin(\theta) + F_Esin(\psi)cos(\theta) +F_scos(\theta) \\ \ddot{y} = F_Ecos(\psi)cos(\theta) - F_Esin(\psi)sin(\theta) - F_ssin(\theta)\\ \ddot{\theta} = F_Esin(\psi)(l_1+l_n) - l_2F_s\\ $$
The standard state space form below assumes a linear time invariant system and the state vector q decoupled from the input vector u. $$ \dot{q} = Aq + Bu\\ y = Cq + Du $$
Assume that \$cos(\theta)=1, sin(\theta) = \theta\$, \$cos(\psi)=1, sin(\psi) = \psi\$ for small angles. In the kinematics above, inputs \$u = [F_E, \psi, F_s]\$. This gives rise to the following:
$$ \ddot{x} = F_E\theta + F_E\psi +F_s \\ \ddot{y} = F_E - F_E\psi\theta - F_s\theta\\ \ddot{\theta} = F_E\psi(l_1+l_n) - l_2F_s\\ $$
If the state \$q = [x, \dot{x}, y, \dot{y}, \theta, \dot{\theta} ]\$, this means that \$\dot{q}\$ is evaluated in terms of multiple inputs and states multiplied together (coupled). Example with \$\ddot{x}\$:
$$ \dot{q_2} = u_1q_5 +u_1u_2 + u_3 $$
However, state space standard form cannot represent this since the inputs are multiplied by constants in \$\dot{q}\$. My questions are;
- Can we still use state space for this system?
- If we still want matrices A, B, C, D, is there a decoupling method/transformation which is used to modify the kinematic equations?
- If we had to design 3 PIDs to control the 3 inputs, should we assume independence between the inputs and thus "decouple" the system by this assumption? I looked into Relative Gain Arrays for this part.
I searched books, tutorials and notes tirelessly, but they all seem to consider a simple example having inputs decoupled from the states and with a constant. An LQR is to be designed on a system similar to the above, but the state space form is needed for this. Any hints are appreciated.