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Digital control design

Digital control loop structure

A common continuous closed-loop has the following structure:

where all the elements are analog.

If we consider now a digital controller, the main closed-loop is:

Now the loop contains both analog and digital signals. The implementation of the controller is done by means of a digital computer. The interface between analog and digital signals is done by the AD/DA converters:

A/D converter: sampling of the analog signals to convert it into a digital signal.

D/A converter: generation of an analog signal from a digital signal. Observe that apart from generating the analog values at he sample times, the D/A also provide values between sample times (so that a reconstruction of the signal is done). In general zero-order hold (zoh) is used for signal reconstruction. This element is based on mantaining the value of the last sample time until next sample time update. The transfer function of a zero order hold is:

An ilustration of the sampling and hold process is presented in the figure:

Analysis of a digital control loop

For the analysis of a digital control loop the following diagram can be considered:

If we focus the analysis on the output values at the sample-time intants, a sampling element should be included in the output:

Or, equivalently:

where 

Implementation issues

Programming algorithm

The controller will be implmentes by means of a computer. The controller will be an executable program running in real-time. The algorithm is:

  1. Read the input variable y(kT) using a sensor/transmiter and the A/D converter.
  2. Compute the error e(kT)
  3. Obtain the control law u(kT) and send the resulting value to the actuator by means of the D/A converter.
  4. Update variables and execute storage actions if necessary.
  5. Wait until t=(k+1)t.
  6. k=k+1
  7. go to 1
The computation of the control output in step 3 is done by antitransformation of U(z) variable. The procedure for this is:
so that,
Then,
 
From this expression the difference equation is easily obtained, and u(kT) can be isolated to define the control law. 

Example: Obtaining the control law u(kT) for a simple transfer function

Assume that the controller of a closed-loop system is

Obtain the control law u(kT).

 

 

Digital control design methodologies

The video shows the different design approaches when designing a digital control system.

 

Discretization Methods

Different methods of discretization are available to calculate the approximate discrete equivalent model for a continuous model. Then, the objective is to obtain the discrete transfer function or difference equation or discrete state variable representation of a continuous system. Some of the most used methods are: 

  • Euler approximation: it is based on taking the differential equation and substituting the derivatives by difference approximations. Por example: dy(t)/dt ≈ (y(k)-y(k-1))/T.
  • Bilinear transformation: it is based on taking G(s) and substituting s by    

                                         

  • Other approximations: impulse invariant approximation, step invariant approximation, ramp step invariant approximation.