Question 10.25: Full-State Feedback Control of a Direct Current Motor–Driven...
Full-State Feedback Control of a Direct Current Motor–Driven Cart
Consider the full-state feedback control system discussed in Example 10.24, in which the state-space representation of the plant is
\dot{\mathbf{x}} = \begin{bmatrix} 0 & 1 \\ 0 & -16.883 \end{bmatrix}\mathbf{x} + \begin{bmatrix} 0 \\ 3.778 \end{bmatrix}u
y = \begin{bmatrix} 1 & 0 \end{bmatrix}\mathbf{x}
and the mathematical model of the controller is
u = −\mathbf{Kx}
with \mathbf{K} = \begin{bmatrix} 48.24 & 0.18 \end{bmatrix}. Build a Simulink block diagram of the feedback control system and find the closed-loop response if the cart is initially 1 \text{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,
\dot{x} = ax + bu
y = cx
then we can build a block diagram as shown in Figure 10.86, in which the statespace model is represented by using one Integrator block and three Gain blocks (\mathbf{A, B}, and \mathbf{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 each Gain block, define the corresponding matrix, and choose Matrix (K*u) 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 non-zero initial position, double-click the Integrator block and type [1; 0] for the Initial conditions parameter. Figure 10.87 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 state-variables must be available for a full-state feedback control system because the control signal is u = −\mathbf{Kx}. To simulate this, double-click the State-Space block and define 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 10.86. 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 10.88, which is the resulting displacement response y(t) due to the non-zero initial condition of 1 \text{m}. It is interesting to note that the curve in Figure 10.88 is a mirror image of the unitstep response curve in Figure 10.37 about the x-axis. The reason is left to the reader to find out.
