Sampled data systems
In most cases, we assume that the feedback control system behaves in a continuous-time fashion. Apart from that, there are also many other applications where a digital computer is included as an integral part of the control system.
These applications are used in aircraft autopilots, mass-transit vehicles, oil refineries, and many more. One big advantage of using a digital computer is that it increases the flexibility of the control program and improves decision-making.
The introduction of the digital computer to calculate the control action for a continuous-time system also needs the knowledge of sampling and quantization.
Sampling is necessary as digital computers can only manipulate discrete-time signals. Therefore, samples of the physical signals are taken such as position, or velocity, and are then used in the computer to calculate the appropriate control.
Coming to quantization, it occurs because digital computers operate with finite arithmetic. It means that the computer takes in numbers, stores them, performs the needed calculations, and returns them with finite accuracy. It introduces round-off errors into the calculations performed by the computer.
Sampled data systems are feedback control systems that use digital computers. They are also known as “hybrid systems”. Due to its hybrid nature, sampling in a purely continuous-time system or a purely discrete-time system becomes complex. It requires the combined use of both continuous-time and discrete-time analysis methods.
To understand better, let us consider a feedback control system in which the digital computer performs the controlling action. The analog-to-digital converter at the front end of the system acts on the continuous-time error signal and converts it into a stream of numbers for processing in the computer. The control calculated by the computer is a second stream of numbers, which is converted by the digital-to-analog converter back into a continuous-time signal applied to the plant.
- A/D converter: It is represented using an impulse sampler.
- Digital controller: The computer program responsible for the control is viewed as a difference equation, whose input-output effect is represented by the z-transform.
- D/A converter: The most common D/A converter is the zero-order hold. It simply holds the amplitude of an incoming sample constant for the entire sampling period, until the next sample arrives. The impulse response of the zero-order hold ranges from 0-1, 1 when 0<t<T, and 0 otherwise.
- Plant: The plant operates on the continuous-time control delivered by the zero-order hold to produce the overall system output. The plant is represented by the transfer function.
Closed-loop transfer function
Unlike how each functional block in a system has a transfer function of its own, the sampler does not have a transfer function, therefore determining the closed-loop transfer function, in this case, becomes a problem.
To find it, we need to relate the sampled version of the input to the sampled version of the output. By doing so we can analyze the behavior of the plant output at instants of sampling, however, we cannot gain any information regarding the output that varies between those instants.
To summarize the process, it will be as follows:
- Firstly, write the cause and effect equations using the Laplace transforms to obtain the closed-loop transfer function.
- Then convert the closed-loop transfer function to a discrete-time transfer function.
- Then use the z-plane analysis tool and find the system’s stability and performance.
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