Linear Control System Analysis And Design Conventional And Modern

[Rode Neagle]

TheNyquist stability criterion has been used extensively in science and engineering to assess the stability of physical systems that can be represented by sets of linear equations. The mathematical foundations of the criterion can be found in many advanced mathematics or linear control theory texts such as Wylie and Barrett (1982), D'Azzo and Houpis (1975) or Willems (1970). The criterion has found popularity partly because of its method of application, which is through a relatively straight-forward geometrical construction. Consider the simple dynamic feedback system represented in the frequency-domain by the following block diagram, Figure 1. This system response dy can be related to the external forcing function δyext by the frequency-domain equation,

The quotient Ф(s) can usually be written so that no singularities appear in the denominator. The system is stable, for a bounded forcing function, if the real parts of the zeros of the denominator have negative real parts. This is because if transformed back into time-domain it is these which form the indices of the exponentials in the solution. The stability of a linear system is thus determined by examining the signs of the real parts of the roots of the characteristic equation. For the simple feedback system described here the characteristic equation is,

The Nyquist criterion considers the variation of the characteristic equation with variation of s over a closed contour C which is formed by a semicircle in the right-hand half of the s-plane and the imaginary axis. Provided that the characteristic equation is analytic everywhere on the closed contour then the net variation of the argument of Ec(s) can be related to the number of poles P and zeros Z of Ec(s) inside C by the following equation,

In geometric terms the number of clockwise encirclements of the origin of the mapping of Ec(s), for a single traverse of s in the s-plane around the contour C, is equal to the difference between the number of zeros and the number of poles of Ec(s) inside C. If the semicircle, or Nyquist "D-contour", is then enlarged to an infinite size, so as to take up the entire right-hand half of the s-plane, the method can be used to check if the characteristic equation has any roots in the right-hand half of the plane. As mentioned earlier it is normally the case that Ф(s) is rearranged so as to remove any poles from Ec(s). It is assumed that this has been done and hence P = 0. If any roots lie in the right-hand half of the plane then they have positive real parts and the system is, therefore, unstable.

For Laplace transforms of most physical systems, the application of the criterion is simplified by the fact that the mapping of s into Ec(s) along the semicircular portion of C collapses onto the origin as the radius of the semicircle tends to an infinite size (see Laplace Transformations.) Furthermore, the mappings along the positive and negative branches of the imaginary axis of s into Ec(s) are symmetric about the real axis. For Laplace transformed systems the application of the Nyquist criterion amounts simply to plotting the locus produced by the mapping of the characteristic equation in its plane (the w-plane), for variation of s along the positive branch of the imaginary axis in the s-plane. In algebraic terms the locus in the w-plane is formed by evaluating,

An example of the application of the criterion is to the stability analysis of boiling channels [Lahey and Podowski (1989)]. A sample Nyquist diagram which shows two loci calculated for a boiling channel is given below. The locus that encircles the origin shows an unstable system and the locus which does not, shows a stable system. (See also Instability, Two-phase.)

Linear control of systems governed by an equation such as xn+1=Axn+Bun (where x and u belong to finite dimensional vector spaces) is widely known [4]. Control design is more complex when dealing with nonlinear systems (for instance [1] or [3]). A well-known method consists in linearizing the system at positions that occur the most often, then solving this serie of linear control problems, and finally patching such local controls together [8].

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.

To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or set point (SP). The difference between actual and desired value of the process variable, called the error signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are controllability and observability. Control theory is used in control system engineering to design automation that have revolutionized manufacturing, aircraft, communications and other industries, and created new fields such as robotics.

Extensive use is usually made of a diagrammatic style known as the block diagram. In it the transfer function, also known as the system function or network function, is a mathematical model of the relation between the input and output based on the differential equations describing the system.

Control theory dates from the 19th century, when the theoretical basis for the operation of governors was first described by James Clerk Maxwell.[1] Control theory was further advanced by Edward Routh in 1874, Charles Sturm and in 1895, Adolf Hurwitz, who all contributed to the establishment of control stability criteria; and from 1922 onwards, the development of PID control theory by Nicolas Minorsky.[2]Although a major application of mathematical control theory is in control systems engineering, which deals with the design of process control systems for industry, other applications range far beyond this. As the general theory of feedback systems, control theory is useful wherever feedback occurs - thus control theory also has applications in life sciences, computer engineering, sociology and operations research.[3]

A notable application of dynamic control was in the area of crewed flight. The Wright brothers made their first successful test flights on December 17, 1903, and were distinguished by their ability to control their flights for substantial periods (more so than the ability to produce lift from an airfoil, which was known). Continuous, reliable control of the airplane was necessary for flights lasting longer than a few seconds.

By World War II, control theory was becoming an important area of research. Irmgard Flgge-Lotz developed the theory of discontinuous automatic control systems, and applied the bang-bang principle to the development of automatic flight control equipment for aircraft.[9][10] Other areas of application for discontinuous controls included fire-control systems, guidance systems and electronics.

Sometimes, mechanical methods are used to improve the stability of systems. For example, ship stabilizers are fins mounted beneath the waterline and emerging laterally. In contemporary vessels, they may be gyroscopically controlled active fins, which have the capacity to change their angle of attack to counteract roll caused by wind or waves acting on the ship.

The Space Race also depended on accurate spacecraft control, and control theory has also seen an increasing use in fields such as economics and artificial intelligence. Here, one might say that the goal is to find an internal model that obeys the good regulator theorem. So, for example, in economics, the more accurately a (stock or commodities) trading model represents the actions of the market, the more easily it can control that market (and extract "useful work" (profits) from it). In AI, an example might be a chatbot modelling the discourse state of humans: the more accurately it can model the human state (e.g. on a telephone voice-support hotline), the better it can manipulate the human (e.g. into performing the corrective actions to resolve the problem that caused the phone call to the help-line). These last two examples take the narrow historical interpretation of control theory as a set of differential equations modeling and regulating kinetic motion, and broaden it into a vast generalization of a regulator interacting with a plant.

In open-loop control, the control action from the controller is independent of the "process output" (or "controlled process variable"). A good example of this is a central heating boiler controlled only by a timer, so that heat is applied for a constant time, regardless of the temperature of the building. The control action is the switching on/off of the boiler, but the controlled variable should be the building temperature, but is not because this is open-loop control of the boiler, which does not give closed-loop control of the temperature.

In closed loop control, the control action from the controller is dependent on the process output. In the case of the boiler analogy this would include a thermostat to monitor the building temperature, and thereby feed back a signal to ensure the controller maintains the building at the temperature set on the thermostat. A closed loop controller therefore has a feedback loop which ensures the controller exerts a control action to give a process output the same as the "reference input" or "set point". For this reason, closed loop controllers are also called feedback controllers.[11]

The definition of a closed loop control system according to the British Standards Institution is "a control system possessing monitoring feedback, the deviation signal formed as a result of this feedback being used to control the action of a final control element in such a way as to tend to reduce the deviation to zero."[12]

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