Question

Let G(s) = 3.3/(s²). You want the system to display:

1. Ka = 11.9 or better
2. Overshoot at a unit step between 5.8% and 25.2%
3. Undamped natural frequency between 2.7 rad/s and 6.0 rad/s

Use matlab and the s-plane design method with discretization at the end to determine a discrete controller Gc(z), to be implemented in a PLC, that makes the system perform closer to the specified above.

Answer 1

To design a discrete controller using the s-plane design method, we need to find the poles of the desired closed-loop system based on the given specifications for gain, overshoot, and undamped natural frequency. The controller can be discretized using a suitable method, such as the bilinear transform.

Step-by-step explanation:

To design a discrete controller Gc(z) using the s-plane design method, we need to find the poles of the desired closed-loop system. In this case, we are given specifications for the gain, overshoot, and undamped natural frequency.

To achieve a gain of 11.9 or better, we can use a proportional controller. The transfer function of the controller is given by Kp = Ka / G(1), where Ka = 11.9 and G(1) is the value of the transfer function G(s) at s = 1.

To satisfy the overshoot specifications, we need to choose the damping ratio of the closed-loop system. The damping ratio can be calculated using the formula ζ = -ln(overshoot/100) / √(π^2 + ln(overshoot/100)^2). Once we have the damping ratio, we can find the natural frequency using the formula ωn = undamped natural frequency / √(1 - damping ratio^2). Finally, we can find the poles of the closed-loop system by using the formula s = -ζωn ± jωn√(1 - ζ^2). We can then discretize the controller using a suitable method, such as the bilinear transform.