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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.
A digital controller is usually cascaded with the plant in a feedback system. The rest of the system can either be digital or analog.
Typically, a digital controller requires:
The programs can take numerous forms and perform many functions
Although a controller may be stable when implemented as an analog controller, it could be unstable when implemented as a digital controller due to a large sampling interval. During sampling the aliasing modifies the cutoff parameters. Thus the sample rate characterizes the transient response and stability of the compensated system, and must update the values at the controller input often enough so as to not cause instability.
When substituting the frequency into the z operator, regular stability criteria still apply to discrete control systems. Nyquist criteria apply to z-domain transfer functions as well as being general for complex valued functions. Bode stability criteria apply similarly. Jury criterion determines the discrete system stability about its characteristic polynomial.
The digital controller can also be designed in the s-domain (continuous). The Tustin transformation can transform the continuous compensator to the respective digital compensator. The digital compensator will achieve an output that approaches the output of its respective analog controller as the sampling interval is decreased.
Tustin is the Padé(1,1) approximation of the exponential function :
And its inverse
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:
The digital controller can also be designed in the z-domain (discrete). The Pulse Transfer Function (PTF) represents the digital viewpoint of the continuous process when interfaced with appropriate ADC and DAC, and for a specified sample time is obtained as: [7]
Where denotes z-Transform for the chosen sample time . There are many ways to directly design a digital controller to achieve a given specification. [7] 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 and preserve the (scaled) zeros of the PTF numerator is to use the design equation: [8]
Where the scalar term ensures the controller 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 to the z-Transform of process output is then given by: [8]
Since process time delay manifests as leading co-efficient(s) of zero in the process PTF numerator , the synthesis method above inherently yields a predictive controller if any such delay is present in the continuous plant. [8]
The bilinear transform is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.
Control theory deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any delay, overshoot, or steady-state error and ensuring a level of control stability; often with the aim to achieve a degree of optimality.
Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant in which case they can be analyzed exactly using LTI system theory revealing their transfer functions in the frequency domain and their impulse responses in the time domain. Real-time implementations of such linear signal processing filters in the time domain are inevitably causal, an additional constraint on their transfer functions. An analog electronic circuit consisting only of linear components will necessarily fall in this category, as will comparable mechanical systems or digital signal processing systems containing only linear elements. Since linear time-invariant filters can be completely characterized by their response to sinusoids of different frequencies, they are sometimes known as frequency filters.
In engineering, a transfer function of a system, sub-system, or component is a mathematical function which theoretically models the system's output for each possible input. They are widely used in electronics and control systems. In some simple cases, this function is a two-dimensional graph of an independent scalar input versus the dependent scalar output, called a transfer curve or characteristic curve. Transfer functions for components are used to design and analyze systems assembled from components, particularly using the block diagram technique, in electronics and control theory.
In signal processing, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, the analog filter, which is typically an electronic circuit operating on continuous-time analog signals.
A phase-locked loop or phase lock loop (PLL) is a control system that generates an output signal whose phase is related to the phase of an input signal. There are several different types; the simplest is an electronic circuit consisting of a variable frequency oscillator and a phase detector in a feedback loop. The oscillator's frequency and phase are controlled proportionally by an applied voltage, hence the term voltage-controlled oscillator (VCO). The oscillator generates a periodic signal of a specific frequency, and the phase detector compares the phase of that signal with the phase of the input periodic signal, to adjust the oscillator to keep the phases matched.
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation.
Infinite impulse response (IIR) is a property applying to many linear time-invariant systems that are distinguished by having an impulse response which does not become exactly zero past a certain point, but continues indefinitely. This is in contrast to a finite impulse response (FIR) system in which the impulse response does become exactly zero at times for some finite , thus being of finite duration. Common examples of linear time-invariant systems are most electronic and digital filters. Systems with this property are known as IIR systems or IIR filters.
In system analysis, among other fields of study, a linear time-invariant system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined below. These properties apply to many important physical systems, in which case the response y(t) of the system to an arbitrary input x(t) can be found directly using convolution: y(t) = x(t) ∗ h(t) where h(t) is called the system's impulse response and ∗ represents convolution. What's more, there are systematic methods for solving any such system, whereas systems not meeting both properties are generally more difficult to solve analytically. A good example of an LTI system is any electrical circuit consisting of resistors, capacitors, inductors and linear amplifiers.
A lead–lag compensator is a component in a control system that improves an undesirable frequency response in a feedback and control system. It is a fundamental building block in classical control theory.
In applied mathematics, the starred transform, or star transform, is a discrete-time variation of the Laplace transform, so-named because of the asterisk or "star" in the customary notation of the sampled signals. The transform is an operator of a continuous-time function , which is transformed to a function in the following manner:
A switched capacitor (SC) is an electronic circuit element implementing a filter. It works by moving charges into and out of capacitors when switches are opened and closed. Usually, non-overlapping signals are used to control the switches, so that not all switches are closed simultaneously. Filters implemented with these elements are termed "switched-capacitor filters", and depend only on the ratios between capacitances. This makes them much more suitable for use within integrated circuits, where accurately specified resistors and capacitors are not economical to construct.
In discrete-time control theory, the dead-beat control problem consists of finding what input signal must be applied to a system in order to bring the output to the steady state in the smallest number of time steps.
The zero-order hold (ZOH) is a mathematical model of the practical signal reconstruction done by a conventional digital-to-analog converter (DAC). That is, it describes the effect of converting a discrete-time signal to a continuous-time signal by holding each sample value for one sample interval. It has several applications in electrical communication.
First-order hold (FOH) is a mathematical model of the practical reconstruction of sampled signals that could be done by a conventional digital-to-analog converter (DAC) and an analog circuit called an integrator. For FOH, the signal is reconstructed as a piecewise linear approximation to the original signal that was sampled. A mathematical model such as FOH is necessary because, in the sampling and reconstruction theorem, a sequence of Dirac impulses, xs(t), representing the discrete samples, x(nT), is low-pass filtered to recover the original signal that was sampled, x(t). However, outputting a sequence of Dirac impulses is impractical. Devices can be implemented, using a conventional DAC and some linear analog circuitry, to reconstruct the piecewise linear output for either predictive or delayed FOH.
The Smith predictor is a type of predictive controller designed to control systems with a significant feedback time delay. The idea can be illustrated as follows.
Pulse-density modulation, or PDM, is a form of modulation used to represent an analog signal with a binary signal. In a PDM signal, specific amplitude values are not encoded into codewords of pulses of different weight as they would be in pulse-code modulation (PCM); rather, the relative density of the pulses corresponds to the analog signal's amplitude. The output of a 1-bit DAC is the same as the PDM encoding of the signal. Pulse-width modulation (PWM) is a special case of PDM where the switching frequency is fixed and all the pulses corresponding to one sample are contiguous in the digital signal. For a 50% voltage with a resolution of 8-bits, a PWM waveform will turn on for 128 clock cycles and then off for the remaining 128 cycles. With PDM and the same clock rate the signal would alternate between on and off every other cycle. The average is 50% for both waveforms, but the PDM signal switches more often. For 100% or 0% level, they are the same.
Two dimensional filters have seen substantial development effort due to their importance and high applicability across several domains. In the 2-D case the situation is quite different from the 1-D case, because the multi-dimensional polynomials cannot in general be factored. This means that an arbitrary transfer function cannot generally be manipulated into a form required by a particular implementation. The input-output relationship of a 2-D IIR filter obeys a constant-coefficient linear partial difference equation from which the value of an output sample can be computed using the input samples and previously computed output samples. Because the values of the output samples are fed back, the 2-D filter, like its 1-D counterpart, can be unstable.
Sonar systems are generally used underwater for range finding and detection. Active sonar emits an acoustic signal, or pulse of sound, into the water. The sound bounces off the target object and returns an “echo” to the sonar transducer. Unlike active sonar, passive sonar does not emit its own signal, which is an advantage for military vessels. But passive sonar cannot measure the range of an object unless it is used in conjunction with other passive listening devices. Multiple passive sonar devices must be used for triangulation of a sound source. No matter whether active sonar or passive sonar, the information included in the reflected signal can not be used without technical signal processing. To extract the useful information from the mixed signal, some steps are taken to transfer the raw acoustic data.
Classical control theory is a branch of control theory that deals with the behavior of dynamical systems with inputs, and how their behavior is modified by feedback, using the Laplace transform as a basic tool to model such systems.
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