Digital control

Digital control is a branch of control theory that uses digital computers to act as system controllers. Depending on the requirements, a digital control system can take the form of a microcontroller to an ASIC to a standard desktop computer. Since a digital computer is a discrete system, the Laplace transform is replaced with the Z-transform. Since a digital computer has finite precision (See quantization), extra care is needed to ensure the error in coefficients, analog-to-digital conversion, digital-to-analog conversion, etc. are not producing undesired or unplanned effects. Since the creation of the first digital computer in the early 1940s the price of digital computers has dropped considerably, which has made them key pieces to control systems because they are easy to configure and reconfigure through software, can scale to the limits of the memory or storage space without extra cost, parameters of the program can change with time (See adaptive control) and digital computers are much less prone to environmental conditions than capacitors, inductors, etc.

Digital control is a branch of control theory that uses digital computers to act as system controllers. Depending on the requirements, a digital control system can take the form of a microcontroller to an ASIC to a standard desktop computer. Since a digital computer is a discrete system, the Laplace transform is replaced with the Z-transform. Since a digital computer has finite precision (See quantization), extra care is needed to ensure the error in coefficients, analog-to-digital conversion, digital-to-analog conversion, etc. are not producing undesired or unplanned effects. Since the creation of the first digital computer in the early 1940s the price of digital computers has dropped considerably, which has made them key pieces to control systems because they are easy to configure and reconfigure through software, can scale to the limits of the memory or storage space without extra cost, parameters of the program can change with time (See adaptive control) and digital computers are much less prone to environmental conditions than capacitors, inductors, etc. s = \frac{2(z-1)}{T(z+1)} Tustin is the Padé (1,1) approximation of the exponential function \begin{align} z &= e^{sT} \end{align} : \begin{align} z &= e^{sT} \\ &= \frac{e^{sT/2}}{e^{-sT/2}} \\ &\approx \frac{1 + s T / 2}{1 - s T / 2} \end{align} And its inverse \begin{align} s &= \frac{1}{T} \ln(z) \\ &= \frac{2}{T} \left[\frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} \right)^3 + \frac{1}{5} \left( \frac{z-1}{z+1} \right)^5 + \frac{1}{7} \left( \frac{z-1}{z+1} \right)^7 + \cdots \right] \\ &\approx \frac{2}{T} \frac{z - 1}{z + 1} \\ &= \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}} \end{align} Digital control theory is the technique to design strategies in discrete time, (and/or) quantized amplitude (and/or) in (binary) coded form to be implemented in computer systems (microcontrollers, microprocessors) that will control the analog (continuous in time and amplitude) dynamics of analog systems. From this consideration many errors from classical digital control were identified and solved and new methods were proposed: Marcelo Tredinnick and Marcelo Souza and their new type of analog-digital mapping Yutaka Yamamoto and his "lifting function space model" Alexander Sesekin and his studies about impulsive systems. M.U. Akhmetov and his studies about impulsive and pulse control G(z) =\frac{B(z)}{A(z)} = \frac{(z-1)}{z}Z\biggl(\frac{G(s)}{s}\Biggr) Where Z() denotes z-Transform for the chosen sample time T . There are many ways to directly design a digital controller D(z) to achieve a given specification. For a type-0 system under unity negative feedback control, Michael Short and colleagues have shown that a relatively simple but effective method to synthesize a controller for a given (monic) closed-loop denominator polynomial P(z) and preserve the (scaled) zeros of the PTF numerator B(z) is to use the design equation: D(z) =\frac{k_p A(z)}{P(z) - k_p B(z)} Where the scalar term k_p = P(1)/B(1) ensures the controller D(z) exhibits integral action, and a steady-state gain of unity is achieved in the closed-loop. The resulting closed-loop discrete transfer function from the z-Transform of reference input R(z) to the z-Transform of process output Y(z) is then given by: \frac{Y(z)}{R(z)} =\frac{k_p B(z)}{P(z)} Since process time delay manifests as leading co-efficient(s) of zero in the process PTF numerator B(z) , the synthesis method above inherently yields a predictive controller if any such delay is present in the continuous plant. ==References== FRANKLIN, G.F.; POWELL, J.D., Emami-Naeini, A., Digital Control of Dynamical Systems, 3rd Ed (1998). Ellis-Kagle Press, Half Moon Bay, CA KATZ, P. Digital control using microprocessors. Englewood Cliffs: Prentice-Hall, 293p. 1981. OGATA, K. Discrete-time control systems. Englewood Cliffs: Prentice-Hall,984p. 1987. PHILLIPS, C.L.; NAGLE, H. T. Digital control system analysis and design. Englewood Cliffs, New Jersey: Prentice Hall International. 1995. M. Sami Fadali, Antonio Visioli, (2009) "Digital Control Engineering", Academic Press, . JURY, E.I. Sampled-data control systems. New-York: John Wiley. 1958. de:Digitaler Regler

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