Full-State Feedback Control Design Consider the DC motor–driven cart discussed in Example 10.12, in which a PD controller was designed to achieve the requirements: overshoot Mp

Question

Full-State Feedback Control Design

Consider the DC motor-driven cart discussed in Example 10.12, in which a PD controller was designed to achieve the requirements: overshoot [latex]M_{\mathrm{p}}<10 \%[/latex] and rise time [latex]t_{\mathrm{r}}<0.15 \mathrm{~s}[/latex].

a. Find a full-state feedback controller such that the closed-loop system achieves the same requirements.

b. Use MATLAB to find the control gain matrix [latex]\mathbf{K}[/latex].

Accepted Answer

a. The transfer function of the cart is given by

[latex]G(s)=\frac{Y(s)}{U(s)}=\frac{3.778}{s^{2}+16.883 s},[/latex]

where the output [latex]y[/latex] is the position of the cart and the input [latex]u[/latex] is the voltage applied to the DC motor. Using the position and the velocity as the state variables, that is, [latex]x_{1}=y[/latex] and [latex]x_{2}=[/latex] [latex]\dot{y}[/latex], we find the state-space model as

[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]

For a second-order system, the gain matrix [latex]\mathbf{K}[/latex] is [latex]1 imes 2[/latex] and

[latex] \mathbf{K}=\left[\begin{array}{ll} k_{1} & k_{2} \end{array} ight] . [/latex]

Applying Equation 10.71

[latex]\left|s\mathbf{I}-(\mathbf{A}-\mathbf{B}\mathbf{K}) ight|=0,[/latex] (10.71)

yields the theoretical characteristic polynomial

[latex] \begin{aligned} |s \mathbf{I}-\mathbf{A}+\mathbf{B K}| & =\left|\left[\begin{array}{cc} s & 0 \ 0 & s \end{array} ight]-\left[\begin{array}{cc} 0 & 1 \ 0 & -16.883 \end{array} ight]+\left[\begin{array}{c} 0 \ 3.778 \end{array} ight]\left[\begin{array}{ll} k_{1} & k_{2} \end{array} ight] ight| \ & =\left|\begin{array}{cc} s & -1 \ 3.778 k_{1} & s+16.883+3.778 k_{2} \end{array} ight| \ & =s^{2}+\left(16.883+3.778 k_{2} ight) s+3.778 k_{1} . \end{aligned} [/latex]

The time-domain specifications, [latex]M_{\mathrm{p}}<10 \%[/latex] and [latex]t_{\mathrm{r}}<0.15 \mathrm{~s}[/latex], indicate that [latex]\zeta>0.59[/latex] and [latex]\omega_{\mathrm{n}}>[/latex] [latex]12.33 \mathrm{rad} / \mathrm{s}[/latex]. Choose the same values of [latex]\omega_{\mathrm{n}}[/latex] and [latex]\zeta[/latex] as in Example 10.12, that is, [latex]\omega_{\mathrm{n}}=13.5 \mathrm{rad} / \mathrm{s}[/latex] and [latex]\zeta=0.65[/latex]. The desired closed-loop poles are then located at [latex]p_{1,2}=-\zeta \omega_{n} \pm j \omega_{n} \sqrt{1-\zeta^{2}}=[/latex] [latex]-8.775 \pm 10.26 \mathrm{j}[/latex]. Applying Equation 10.72

[latex](s-s_{1})\;(s-s_{2})...(s-s_{n})=0,[/latex] (10.72)

gives the desired characteristic polynomial

[latex]\left(s-p_{1} ight)\left(s-p_{2} ight)=s^{2}+2 \zeta \omega_{\mathrm{n}} s+\omega_{\mathrm{n}}^{2}=s^{2}+17.55 s+182.25[/latex]

Equating the two characteristic polynomials, we have

[latex] \begin{aligned} & 16.883+3.778 k_{2}=17.55 \ & 3.778 k_{1}=182.25 \end{aligned} [/latex]

which gives [latex]\mathbf{K}=\left[\begin{array}{ll}48.24 & 0.18\end{array} ight][/latex]. So the full-state feedback controller is

[latex]u=-\mathbf{K x}=-48.24 x_{1}-0.18 x_{2}.[/latex]

b. The following is the MATLAB session used to compute a full-state feedback control gain matrix [latex]\mathbf{K}[/latex] using pole placement design:

Question 10.22: Full-State Feedback Control Design Consider the DC motor–dri...

Related Answered Questions

Verified Answer:

Verified Answer:

Controllability and Observability Determine the controllability and observability for the second-order systems given in Example 10.20. ...

Verified Answer:

Verified Answer:

Stability Analysis in State Space a. Compute the poles of the system described by x = [0 1 0 -16.883]x + [0 3.778] u, y = [1 0]x. b. Verify the results by converting the state-space representation to a transfer function and then identifying the poles of the transfer function. ...

Verified Answer:

Proportional Control Design Using a Bode Plot Design a proportional controller for the cart system in Example 10.12 using the Bode plot technique. ...

Verified Answer:

Stability Analysis Using a Bode Plot Consider the feedback control system in Figure 10.63, where KG(s) = K 100/s(s+5)(s+20) ,and K is assumed to be 1. Plot the Bode magnitude and phase curves using MATLAB. Give comments on the stability of the closed-loop system. ...

Verified Answer:

Bode Plot Sketching Plot the Bode magnitude and phase for the system with the transfer function KG(s) = K s+2/s(s²+8s+400) , where K = 1. ...

Verified Answer:

Proportional Control Design Using Root Locus Design a proportional controller for the cart system in Example 10.12 using the root locus technique. ...

Verified Answer:

Analysis Using Root Locus Refer to the root locus obtained in Example 10.13. Comment on the stability and performance of the closed-loop system when K varies from 0 to ∞. ...

Verified Answer: