Assume we have a control system in state space as: xdot1=u1, xdot2=u2, where x is state and u is control. I am interested to know the possibility of complex valued control signal to control such a system or other different systems. Do not mind the system itself. Only the philosophical possibility of such complex valued (imaginary) control signal. For example with regard to the system xdot1=u1, and xdot2=u2 such possibility for candidate control-laws u1=-i*x2 , and, u2=i*x1, where i is the imaginary unit (or complex variable unit). Only discuss about the possibility of such control laws for some certain systems, irrespective of the control system which I exemplified. But you can also put the basis of your analysis over the exemplified system. Thank You.
Hi Saeb AmirAhmadi Chomachar,
If you have a system such that i·x" = u, where i is the imaginary unit, then it is theoretically possible to stabilize the system with a complex-valued control law:
u = − 2i·x' − i·x.
I wonder what the physical manifestation of the complex-valued control signal would look like... 🤔
Hi dear Yew-Chung. Thank you. It was a good example you mentioned, however, I think what you mentioned is also possible for totally real system with no imaginary coefficient. With regard to the system I considered in my question: xdot1=u1, and, xdot2=u2, if, u1=-i*x2, and, u2=i*x1, then we have xdot1=u1=-i*x2, and, xdot2=u2=i*x1. But from these identities, we may conclude xdot1=-i*x2, and, xdot2=i*x1. Therefore, if x1" is the second derivative of x1, then:
x1"=-i*(xdot2)=-i*(i*x1)=-i^2*x1=x1. Hence, x1"=x1, which means x1 could be probably made stable as we find: x1(t)=c1 e^t +c2 e^-t.
Furthermore, it seems x1 is a real-valued signal, and due to symmetry of the state space model, we may conclude x2, as well is real-valued. But, notice that the corresponding control laws are imaginary. Thus the real system xdot1=u1, xdot2=u2, which is completely a real-valued system, might be stabilizable by control laws u1=-i*x2, and, u2=i*x1, which are totally imaginary.
Have you ever seen a text book on differential equations involving imaginary unit?. If yes, please let me know. Regards.
@Yew-Chung Chak. Similar to you, I cannot perceive and imagine, what is the physical manifestation of a complex-valued or imaginary control signal. However, I think it is worthy of being pondered more.
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