Question 5.4.4: Simulink Model of a Two-Mass System The state-variable model...
Simulink Model of a Two-Mass System
The state-variable model of the two-mass system discussed in Example 5.1.3 is
\dot{z} = Az + B f(t)where
A = \begin{bmatrix} 0 & 1 & 0 & 0 \\ -1 & -\frac{12}{5} & \frac{4}{5} & \frac{8}{5} \\ 0 & 0 & 0 & 1 \\ \frac{4}{3} & \frac{8}{3} & -\frac{4}{3} & – \frac{8}{3} \end{bmatrix} B = \begin{bmatrix} 0\\ 0 \\ 0\\ \frac{1}{3} \end{bmatrix}and
z = \begin{bmatrix} z_{1}\\ z_{2} \\ z_{3}\\ z_{4} \end{bmatrix} = \begin{bmatrix} x_{1}\\ \dot{x}_{1} \\ x_{2}\\ \dot{x}_{2} \end{bmatrix}Develop a Simulink model to plot the unit-step response of the variables x_{1} and x_{2} with the initial conditions x_{1}(0) = 0.2, \dot{x}_{1}(0) = 0, x_{2}(0) = 0.5, and \dot{x}_{2}(0) = 0.
First select appropriate values for the matrices in the output equation y = Cz + D f (t). Since we want to plot x_{1} and x_{2}, which are z_{1} and z_{3}, we choose C and D as follows.
C = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} D = \begin{bmatrix} 0 \\ 0 \end{bmatrix}To create this simulation, obtain a new model window. Then do the following to create the model shown in Figure 5.4.7.
1. Select and place in the new window the Step block from the sources category. Doubleclick on it to obtain the Block Parameters window, and set the Step time to 0, the Initial and Final values to 0 and 1, and the Sample time to 0. Click OK.
2. Select and place the State-Space block. Double-click on it, and enter [0, 1, 0, 0; -1, -12/5, 4/5, 8/5; 0, 0, 0, 1; 4/3, 8/3, -4/3, -8/3] for A, [0; 0; 0; 1/3] for B, [1, 0, 0, 0; 0, 0, 1, 0] for C, and [0; 0] for D. Then enter [0.2; 0; 0.5; 0] for the initial conditions. Click OK. Note that the dimension of the matrix B tells Simulink that there is one input. The dimensions of the matrices C and D tell Simulink that there are two outputs.
3. Select and place the Scope block.
4. Once the blocks have been placed, connect the input port on each block to the outport port on the preceding block as shown in the figure.
5. Experiment with different values of the Stop time until the Scope shows that the steadystate response has been reached. For this application, a Stop time of 25 is satisfactory. The plots of both x_{1} and x_{2} will appear in the Scope.
