The plant of Example 9.1 has the state equations given (9-1). Find the gain matrix K K

The plant of Example 9.1 has the state equations given (9-1). Find the gain matrix $K$ required to realize the closed-loop characteristic equation with zeros which have a damping ratio $\zeta$ of 0.46 and a time constant $\tau$ of $0.5s$. Use pole-assignment design.

Example 9.1

Equation 9-1

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As a second example, we consider the servo system utilized in several examples of pole- assignment design in Chapter 9. The system is shown in Fig. 11-2, and has the state model (see Example 9.1) with 0.0952 0.905 [10]x(k) where x(t) is the shaft angle and x(t) is the shaft angular velocity. We choose the cost function to be x(k + 1) y(k) = = 1 0 Q k=0 2x(K) J = x(k)Qx(k) + Ru(k) = 0.00484 0.0952 x(k) + 1 [8] R = 1 u(k) Thus velocity is ignored in the cost function. We are attempting to minimize the magnitude of the position x(k) without regard to the velocity x(k) required. The weight of the control, R, in the cost function is chosen arbitrarily. N is chosen equal to 2 so that the solution can be calcu- lated by hand. Thus, from (11-33),