Question 10.23: Full-State Feedback Control of a DC Motor–Driven Cart Consid...
Full-State Feedback Control of a DC Motor-Driven Cart
Consider the full-state feedback control system discussed in Example 10.22, in which the statespace representation of the plant is
\begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{cc} 0 & 1 \\ 0 & -16.883 \end{array}\right] \mathbf{x}+\left[\begin{array}{c} 0 \\ 3.778 \end{array}\right] u, \\ & y=\left[\begin{array}{ll} 1 & 0 \end{array}\right] \mathbf{x}, \end{aligned}
and the mathematical model of the controller is
u=-\mathbf{K x}
with \mathbf{K}=\left[\begin{array}{ll}48.24 & 0.18\end{array}\right]. Build a Simulink block diagram of the feedback control system and find the closed-loop response if the cart is initially 1 \mathrm{~m} away from the equilibrium position.
There are two ways to construct a Simulink block diagram of a full-state feedback control system. If we treat the state-space model as its scalar counterpart,
\begin{aligned} & \dot{x}=a x+b u, \\ & y=c x \end{aligned}
then we can build a block diagram as shown in Figure 10.82, in which the state-space model is represented by using one Integrator block and three Gain blocks (A, B, and C). Note that all the gains, including the control gain \mathbf{K}, are in matrix form. This should be specified explicitly in Simulink. Double-click on each Gain block, define the corresponding matrix, and choose Matrix \left(K^{*} u\right) for the Multiplication parameter. Recall that we have chosen x_{1}=y and x_{2}=\dot{y}. Thus, the physical position variable is the same as the first state variable. To specify the nonzero initial position, double-click on the Integrator block and type \left[\begin{array}{ll}1 & 0\end{array}\right] for the Initial conditions parameter.
Figure 10.83 presents an alternative Simulink block diagram, in which the state-space model is built using the State-Space block instead of the Integrator block. Note that all statevariables must be available for a full-state feedback control system because the control signal is u=-K \mathbf{x}. To simulate this, double-click on the state-Space block and define \mathrm{C} as an identity matrix and D as a zero matrix with compatible dimensions (eye (2) for C and zeros (2,1) for D in this example). The parameter of Initial conditions has the same value as defined in Figure 10.82. To obtain the output y, a Gain block is included to define the real matrix C. It should be pointed out that the value of the Gain block corresponding to the full-state feedback controller is -\mathbf{K}, not \mathbf{K}.
Running both simulations yields the same curve as shown in Figure 10.84, which is the resulting displacement response y(t) due to the nonzero initial condition of 1 \mathrm{~m}. It is interesting to note that the curve in Figure 10.84 is a mirror image of the unit-step response curve in Figure 10.35 about the x-axis. The reason is left to the reader to find out.
