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;

  1. Can we still use state space for this system?
  2. If we still want matrices A, B, C, D, is there a decoupling method/transformation which is used to modify the kinematic equations?
  3. 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.

  • \$\begingroup\$ This looks like homework with zero attempt to solve, please provide solutions to the questions. Analyzing a nonlinear system is not trivial. You need to linearize it or pick an appropriate nonlinear controller. \$\endgroup\$ – Voltage Spike Jun 5 '17 at 21:27
  • \$\begingroup\$ The term in \$\psi \theta\$ renders the system nonlinear so, whilst it can be expressed in state space form, it can't be solved. I don't think the Jacobian can help in this case. \$\endgroup\$ – Chu Jun 6 '17 at 7:26
  • \$\begingroup\$ Thank you for your comments. I did attempt to solve it by making the strong assumption of independence between inputs whilst treating the others as 0, similar to the disturbance tests. The PIDs can be designed for each output in terms of different inputs and then summed. It's "ok" as a benchmark, but not an actual controller because we know that the assumption is in fact false. \$\endgroup\$ – Arex Jun 6 '17 at 9:37
  • \$\begingroup\$ You only linearized the equations in the states but not in the inputs. You need to completely linearize the system of ODEs with respect to a given operating point. \$\endgroup\$ – MrYouMath Jun 6 '17 at 15:22

You misunderstood state-space representation. While the standard form you've posted indeed targets linear systems but this doesn't mean the representation can't handle nonlinear system. Probably one of powerful tools of this representation is its capability for handling nonlinear system. Let's see how the system is represented in the state space (i.e. no matrix form).

Let's define \$q_1=x, q_2=\dot{x},q_3=y, q_4=\dot{y},q_5=\theta, q_6=\dot{\theta}\$, therefore, we obtain

$$ \begin{align} \dot{q}_1 &= q_2\\ \dot{q}_2 &= F_E\cos(\psi)\sin(q_5) + F_E\sin(\psi)\cos(q_5) +F_s\cos(q_5)\\ \dot{q}_3 &= q_4\\ \dot{q}_4 &= F_E\cos(\psi)\cos(q_5) - F_E\sin(\psi)\sin(q_5) - F_s\sin(q_5) \\ \dot{q}_5 &= q_6\\ \dot{q}_6 &= F_E\sin(\psi)(l_1+l_n) - l_2F_s\\ \end{align} $$ Now the system can be represented generally as

$$ \dot{q} = f(q, u), \quad q \in \mathbb{R}^3, u \in \mathbb{R}^3 $$ which can be solved numerically. I can solve the system if you provide me with the actual parameters of the system.


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