EC-3.3 Systems of Equations, and the State Model

Work through these

• Why one second-order equation becomes two first-order ones

  Naming the unknown and its derivative as two separate quantities turns a second-order equation into a pair of first-order equations with no loss of information. The same trick reduces any order to first order, at the cost of more unknowns.

  NPTEL: Differential Equations for Engineers · Course

• The state form, and what each matrix in it does

  The state equation says how the internal quantities evolve and how the input pushes them, and the output equation says which combination of them is actually measured. Separating those two statements is what makes the form general.

• Eigenvalues decide the behaviour, and this is why they were worth learning

  Each eigenvalue of the system matrix contributes one exponential mode, so their real parts decide whether the system settles and their imaginary parts decide whether it oscillates. This is the moment the linear algebra earns its place.

  MIT OpenCourseWare 18.03: Differential Equations · Course

• The matrix exponential, and the solution it produces

  The scalar first-order solution generalises directly: the state at any time is the matrix exponential applied to the initial state, plus an integral carrying the input. Computing it by diagonalising is the case worth doing by hand.

• Write one circuit as a state model, choosing the states deliberately

  Capacitor voltages and inductor currents are the natural choice because they are the quantities that cannot change instantaneously. Choosing them makes the matrices fall out of the two circuit laws without algebra.

  NPTEL: Differential Equations for Engineers · Course

• Equilibrium and stability, read from the eigenvalues alone

  An equilibrium is a state the system would stay in, and it is stable when every eigenvalue has a negative real part. One eigenvalue on the wrong side is enough to make the whole system unstable, regardless of the others.