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 x = [0 1 0 -16.883]x + [0 3.778 ]u, y = [1 0]x, and the mathematical model of the controller is u = −Kx with

Question

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

[latex] \begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{cc} 0 & 1 \ 0 & -16.883 \end{array} ight] \mathbf{x}+\left[\begin{array}{c} 0 \ 3.778 \end{array} ight] u, \ & y=\left[\begin{array}{ll} 1 & 0 \end{array} ight] \mathbf{x}, \end{aligned} [/latex]

and the mathematical model of the controller is

[latex]u=-\mathbf{K x}[/latex]

with [latex]\mathbf{K}=\left[\begin{array}{ll}48.24 & 0.18\end{array} ight][/latex]. Build a Simulink block diagram of the feedback control system and find the closed-loop response if the cart is initially [latex]1 \mathrm{~m}[/latex] away from the equilibrium position.

Accepted Answer

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,

[latex] \begin{aligned} & \dot{x}=a x+b u, \ & y=c x \end{aligned} [/latex]

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 [latex]\mathbf{K}[/latex], are in matrix form. This should be specified explicitly in Simulink. Double-click on each Gain block, define the corresponding matrix, and choose Matrix [latex]\left(K^{*} u ight)[/latex] for the Multiplication parameter. Recall that we have chosen [latex]x_{1}=y[/latex] and [latex]x_{2}=\dot{y}[/latex]. 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 [latex]\left[\begin{array}{ll}1 & 0\end{array} ight][/latex] 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 [latex]u=-K \mathbf{x}[/latex]. To simulate this, double-click on the state-Space block and define [latex]\mathrm{C}[/latex] as an identity matrix and [latex]D[/latex] as a zero matrix with compatible dimensions (eye [latex](2)[/latex] for [latex]C[/latex] and zeros [latex](2,1)[/latex] for [latex]D[/latex] in this example). The parameter of Initial conditions has the same value as defined in Figure 10.82. To obtain the output [latex]y[/latex], 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 [latex]-\mathbf{K}[/latex], not [latex]\mathbf{K}[/latex].

Running both simulations yields the same curve as shown in Figure 10.84, which is the resulting displacement response [latex]y(t)[/latex] due to the nonzero initial condition of [latex]1 \mathrm{~m}[/latex]. 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 [latex]x[/latex]-axis. The reason is left to the reader to find out.

Question 10.23: Full-State Feedback Control of a DC Motor–Driven Cart Consid...

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