A client wishes you to design a control system for a crane system, modelled as s

Question

A client wishes you to design a control system for a crane system, modelled as shown in the Figure below:

The trolley is moved by an input F(t) in order to control x(t) and (t). The block diagram of the crane is shown in the figure below:

Client Goals: Zero steady state error for a step input, a percent overshoot of less than 1% along with a settling time of less than 2 seconds.

Use Matlab to solve the following:

(a) Discuss the stability of the crane system if a proportional controller were used

(b)Discuss the order of the crane system if a proportional controller were used

Explanation / Answer

The problem is that beyond a critical gain, discrete-time first-order systems with proportional feedback will oscillate, and at even high gains, will become unstable. Continuous-time first-order systems with proportional feedback never oscillate, and in principle, allow infinite gain. In practice, no system is identically first-order, but many continuous-time systems are close enough to allow very high gains.

In the second Crane problem and in the second lab, we examined second-order systems. Such systems have complex natural frequencies, resulting in oscillatory behavior. Determining the impact of feedback gains in second- and higher-order systems is more complicated, yet such systems are unavoidable. They arise, for example, whenever changes in position are the result of force actuation, like when we raise the copter-levitated arm by increasing the copter thrust, or reposition a a crane by controlling its acceleration and deceleration. Feedback control in second- and higher-order systems is a big leap from proportional feedback in first-order systems. In the first-order case, we used one proportional gain to nudge one natural frequency along the real line, but in the higher order systems (such as the copter-arm and the accelerartion controlled Crane), we used proportional and delta (derivative) feedback, together with proportional and delta (derivative) gains, to position the feedback system's natural frequencies at desirable locations in the two-dimensional complex plane.

But, we do not "gain" enough, even with two gains. In this third lab, we will add sum (integral) feedback to the discrete-time feedback controller for the propeller arm, and in the next few weeks, we will move beyond controllers based on weighted combinations of proportional, integral, and derivative (PID) feedback.

In this prelab, we will start by introducing some software tools for creating complicated system models by composing subsystems described by transfer functions. This transform-based approach will obviate the need to deploy your error-free algebra skills, and should allow you to focus on the issues of controller design. In what follows, we will use matlab examples (along with python examples), as the matlab control system toolbox is mature, extremely comprehensive, and widely used by control system designers. We are continuing to develop python tools for control system design, with functionality better suited to learning about control, and will distribute them as they become available. But, as you saw last week it is a work in progress.

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