Lecture State Space Copyright Paul Oh DCP Block

Lecture: State Space © Copyright Paul Oh

DCP Block Diagram: Open-Loop Transfer Function [rad] OLTF: (1) © Copyright Paul Oh

Example DCP Open-Loop Transfer Function Given Nm/V m (1*) kgm 2 kg Laplace domain OL Transfer function Nms/rad © Copyright Paul Oh

State Space Realization Previously, showed that the OLTF for this particular DCP has: Alternatively (2) One could solve (2) as an ordinary differential equation (ODE). For higher order ODEs, this can be quite tedious. State Space Realization converts ODEs like (2) into a compact (matrix) form. Doing so simplifies the math – and solving the ODE First, define two state variables e. g. and © Copyright Paul Oh

Then, one can re-write (2) as (3) State space form given by matrices: (4) Next, can re-express (3) in the form of (4) as: (5) © Copyright Paul Oh

State space realization (5), transfer function (1) and differential equation (2) are the same representation of the real-world system. As such, they should all have the same characteristic equation i. e. same poles The characteristic equation from state space (5) is defined as: (6) Using the values from (1*) into (6) yields: (7*) Which is the same as the denominator given in (1*) No surprise! That’s why it’s call the characteristic equation © Copyright Paul Oh