Many control systems must operate in two dimensions, for example, the x- and the y-axes. A two-axis control system is shown in Figure P3.10, where a set of state variables is identified. The gain of each axis is K1 and K2, respectively.

(a) Obtain the state differential equation.

(b) Find the characteristic equation from the A matrix.

(c) Determine the state transition matrix for K1 = 1 and K2 = 2.

K 1/s Y(s) R(s) X(s) K2 1/s R(s) Y(s) X(s) R(s) K (a) R(s) K FIGURE P3.10 Two-axis system. (a) Signal-flow graph. (b) Block diagram model. (b) 911 Y(s) X(s) 115 Y(s) X(s)