The Nichols plot is a plot used in signal processing and control design, named after American engineer Nathaniel B. Nichols. [1] [2] [3] It plots the phase response versus the response magnitude of a transfer function for any given frequency, and as such is useful in characterizing a system's frequency response.
Given a transfer function,
with the closed-loop transfer function defined as,
the Nichols plots displays versus . Loci of constant and are overlaid to allow the designer to obtain the closed loop transfer function directly from the open loop transfer function. Thus, the frequency is the parameter along the curve. This plot may be compared to the Bode plot in which the two inter-related graphs - versus and versus ) - are plotted.
In feedback control design, the plot is useful for assessing the stability and robustness of a linear system. This application of the Nichols plot is central to the quantitative feedback theory (QFT) of Horowitz and Sidi, which is a well known method for robust control system design.
In most cases, refers to the phase of the system's response. Although similar to a Nyquist plot, a Nichols plot is plotted in a Cartesian coordinate system while a Nyquist plot is plotted in a Polar coordinate system.
Control theory is a field of control engineering and applied mathematics that 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.
In engineering, a transfer function of a system, sub-system, or component is a mathematical function that models the system's output for each possible input. It is widely used in electronic engineering tools like circuit simulators and control systems. In simple cases, this function can be represented as a two-dimensional graph of an independent scalar input versus the dependent scalar output. 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.
A phase-locked loop or phase lock loop (PLL) is a control system that generates an output signal whose phase is fixed relative to the phase of an input signal. Keeping the input and output phase in lockstep also implies keeping the input and output frequencies the same, thus a phase-locked loop can also track an input frequency. And by incorporating a frequency divider, a PLL can generate a stable frequency that is a multiple of the input frequency.
In physics, resonance refers to a wide class of phenomena that arise as a result of matching temporal or spatial periods of oscillatory objects. For an oscillatory dynamical system driven by a time-varying external force, resonance occurs when the frequency of the external force coincides with the natural frequency of the system. Resonance can occur in various systems, such as mechanical, electrical, or acoustic systems, and it is desirable in certain applications, such as musical instruments or radio receivers. Resonance can also be undesirable, leading to excessive vibrations or even structural failure in some cases.
In electrical engineering and control theory, a Bode plot is a graph of the frequency response of a system. It is usually a combination of a Bode magnitude plot, expressing the magnitude of the frequency response, and a Bode phase plot, expressing the phase shift.
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of systems, such as audio and control systems, where they simplify mathematical analysis by converting governing differential equations into algebraic equations. In an audio system, it may be used to minimize audible distortion by designing components so that the overall response is as flat (uniform) as possible across the system's bandwidth. In control systems, such as a vehicle's cruise control, it may be used to assess system stability, often through the use of Bode plots. Systems with a specific frequency response can be designed using analog and digital filters.
In control theory and stability theory, root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system. This is a technique used as a stability criterion in the field of classical control theory developed by Walter R. Evans which can determine stability of the system. The root locus plots the poles of the closed loop transfer function in the complex s-plane as a function of a gain parameter.
The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the outputs of a general system when its inputs change from zero to one in a very short time. The concept can be extended to the abstract mathematical notion of a dynamical system using an evolution parameter.
In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable.
The Butterworth filter is a type of signal processing filter designed to have a frequency response that is as flat as possible in the passband. It is also referred to as a maximally flat magnitude filter. It was first described in 1930 by the British engineer and physicist Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers".
In electronics engineering, frequency compensation is a technique used in amplifiers, and especially in amplifiers employing negative feedback. It usually has two primary goals: To avoid the unintentional creation of positive feedback, which will cause the amplifier to oscillate, and to control overshoot and ringing in the amplifier's step response. It is also used extensively to improve the bandwidth of single pole systems.
In signal processing, linear phase is a property of a filter where the phase response of the filter is a linear function of frequency. The result is that all frequency components of the input signal are shifted in time by the same constant amount, which is referred to as the group delay. Consequently, there is no phase distortion due to the time delay of frequencies relative to one another.
In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer Felix Strecker at Siemens in 1930 and the Swedish-American electrical engineer Harry Nyquist at Bell Telephone Laboratories in 1932, is a graphical technique for determining the stability of a dynamical system.
In control theory, quantitative feedback theory (QFT), developed by Isaac Horowitz, is a frequency domain technique utilising the Nichols chart (NC) in order to achieve a desired robust design over a specified region of plant uncertainty. Desired time-domain responses are translated into frequency domain tolerances, which lead to bounds on the loop transmission function. The design process is highly transparent, allowing a designer to see what trade-offs are necessary to achieve a desired performance level.
Bode's sensitivity integral, discovered by Hendrik Wade Bode, is a formula that quantifies some of the limitations in feedback control of linear parameter invariant systems. Let L be the loop transfer function and S be the sensitivity function.
Iso-damping is a desirable system property referring to a state where the open-loop phase Bode plot is flat—i.e., the phase derivative with respect to the frequency is zero, at a given frequency called the "tangent frequency", . At the "tangent frequency" the Nyquist curve of the open-loop system tangentially touches the sensitivity circle and the phase Bode is locally flat which implies that the system will be more robust to gain variations. For systems that exhibit iso-damping property, the overshoots of the closed-loop step responses will remain almost constant for different values of the controller gain. This will ensure that the closed-loop system is robust to gain variations.
In control engineering, the sensitivity of a control system measures how variations in the plant parameters affects the closed-loop transfer function. Since the controller parameters are typically matched to the process characteristics and the process may change, it is important that the controller parameters are chosen in such a way that the closed loop system is not sensitive to variations in process dynamics. Moreover, the sensitivity function is also important to analyse how disturbances affects the system.
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.
Negative imaginary (NI) systems theory was introduced by Lanzon and Petersen in. A generalization of the theory was presented in In the single-input single-output (SISO) case, such systems are defined by considering the properties of the imaginary part of the frequency response G(jω) and require the system to have no poles in the right half plane and > 0 for all ω in. This means that a system is Negative imaginary if it is both stable and a nyquist plot will have a phase lag between [-π 0] for all ω > 0.
Hall circles are a graphical tool in control theory used to obtain values of a closed-loop transfer function from the Nyquist plot of the associated open-loop transfer function. Hall circles have been introduced in control theory by Albert C. Hall in his thesis.
