# LINEAR CONTROL SYSTEMS PDF

Extensions. Steady-state error. Summary and plan. EE3CL4 C Introduction to Linear Control Systems. Section 3: Fundamentals of Feedback. Tim Davidson. PDF | Control theory in control systems engineering deals with the control of continuously operating dynamical systems in engineered. In this chapter, we describe a general process for designing a control system. A con- tem is the foundation provided by linear system theory, which assumes a.

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ing material for the course 'Control Theory for Linear Systems', given quadratic regulator problem and the H2 control problem, as well as the. Note: For time-invariant control systems – in the controllability definition – the initial time t0 can be set equal to zero. The Kalman rank condition. For linear. Superstable Linear Control Systems. II. Design. 1. B. T. Polyak and P. S. Shcherbakov. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences.

This generated a flurry of interest in the topic, during which Maxwell's classmate, Edward John Routh , abstracted Maxwell's results for the general class of linear systems.

## Description

The Wright brothers made their first successful test flights on December 17, 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. 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.

Open-loop and closed-loop feedback control[ edit ] A block diagram of a negative feedback control system using a feedback loop to control the process variable by comparing it with a desired value, and applying the difference as an error signal to generate a control output to reduce or eliminate the error.

Example of a single industrial control loop; showing continuously modulated control of process flow. Fundamentally, there are two types of control loops: open loop control and closed loop feedback control.

In open loop control, the control action from the controller is independent of the "process output" or "controlled process variable" - PV. 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.

In closed loop control, the control action from the controller is dependent on feedback from the process in the form of the value of the process variable PV. In the case of the boiler analogy, a closed loop would include a thermostat to compare the building temperature PV with the temperature set on the thermostat the set point - SP.

## Control theory

This generates a controller output to maintain the building at the desired temperature by switching the boiler on and off. A closed loop controller, therefore, has a feedback loop which ensures the controller exerts a control action to manipulate the process variable to be the same as the "Reference input" or "set point".

For this reason, closed loop controllers are also called feedback controllers.

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The controller is the cruise control, the plant is the car, and the system is the car and the cruise control. The system output is the car's speed, and the control itself is the engine's throttle position which determines how much power the engine delivers. A primitive way to implement cruise control is simply to lock the throttle position when the driver engages cruise control. However, if the cruise control is engaged on a stretch of flat road, then the car will travel slower going uphill and faster when going downhill.

Logic controllers are used to sequence mechanical operations in many applications. Examples include elevators, washing machines and other systems with interrelated operations.

## Linear Control Systems

An automatic sequential control system may trigger a series of mechanical actuators in the correct sequence to perform a task. For example, various electric and pneumatic transducers may fold and glue a cardboard box, fill it with product and then seal it in an automatic packaging machine.

PLC software can be written in many different ways — ladder diagrams, SFC sequential function charts or statement lists. On—off control uses a feedback controller that switches abruptly between two states. A simple bi-metallic domestic thermostat can be described as an on-off controller.

When the temperature in the room PV goes below the user setting SP , the heater is switched on. Another example is a pressure switch on an air compressor. When the pressure PV drops below the setpoint SP the compressor is powered. Refrigerators and vacuum pumps contain similar mechanisms. Simple on—off control systems like these can be cheap and effective. Linear control systems use negative feedback to produce a control signal to maintain the controlled PV at the desired SP. There are several types of linear control systems with different capabilities.

Proportional control is a type of linear feedback control system in which a correction is applied to the controlled variable which is proportional to the difference between the desired value SP and the measured value PV. Two classic mechanical examples are the toilet bowl float proportioning valve and the fly-ball governor. The proportional control system is more complex than an on—off control system, but simpler than a proportional-integral-derivative PID control system used, for instance, in an automobile cruise control.

On—off control will work for systems that do not require high accuracy or responsiveness, but is not effective for rapid and timely corrections and responses. Proportional control overcomes this by modulating the manipulated variable MV , such as a control valve , at a gain level which avoids instability, but applies correction as fast as practicable by applying the optimum quantity of proportional correction. A drawback of proportional control is that it cannot eliminate the residual SP—PV error, as it requires an error to generate a proportional output.

A PI controller can be used to overcome this. The PI controller uses a proportional term P to remove the gross error, and an integral term I to eliminate the residual offset error by integrating the error over time.

In some systems there are practical limits to the range of the MV. For example, a heater has a limit to how much heat it can produce and a valve can open only so far. Adjustments to the gain simultaneously alter the range of error values over which the MV is between these limits. The width of this range, in units of the error variable and therefore of the PV, is called the proportional band PB. When controlling the temperature of an industrial furnace , it is usually better to control the opening of the fuel valve in proportion to the current needs of the furnace.

This helps avoid thermal shocks and applies heat more effectively. At low gains, only a small corrective action is applied when errors are detected.

The system may be safe and stable, but may be sluggish in response to changing conditions. Errors will remain uncorrected for relatively long periods of time and the system is overdamped.

## Related titles

If the proportional gain is increased, such systems become more responsive and errors are dealt with more quickly. There is an optimal value for the gain setting when the overall system is said to be critically damped.

Increases in loop gain beyond this point lead to oscillations in the PV and such a system is underdamped. Eventually as the PV falls back into the PB, heat is applied again, but now the heater and the furnace walls are too cool and the temperature falls too low before its fall is arrested, so that the oscillations continue. The temperature oscillations that an underdamped furnace control system produces are unacceptable for many reasons, including the waste of fuel and time each oscillation cycle may take many minutes , as well as the likelihood of seriously overheating both the furnace and its contents.

Suppose that the gain of the control system is reduced drastically and it is restarted. By carefully increasing the gain i. Doing this is known as 'tuning' the control system.

A well-tuned proportional furnace temperature control system will usually be more effective than on-off control, but will still respond more slowly than the furnace could under skillful manual control.

Apart from sluggish performance to avoid oscillations, another problem with proportional-only control is that power application is always in direct proportion to the error. To resolve these two problems, many feedback control schemes include mathematical extensions to improve performance.

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The most common extensions lead to proportional-integral-derivative control, or PID control. The derivative part is concerned with the rate-of-change of the error with time: If the measured variable approaches the setpoint rapidly, then the actuator is backed off early to allow it to coast to the required level; conversely if the measured value begins to move rapidly away from the setpoint, extra effort is applied—in proportion to that rapidity—to try to maintain it.

Derivative action makes a control system behave much more intelligently. On control systems like the tuning of the temperature of a furnace, or perhaps the motion-control of a heavy item like a gun or camera on a moving vehicle, the derivative action of a well-tuned PID controller can allow it to reach and maintain a setpoint better than most skilled human operators could.

If derivative action is over-applied, it can lead to oscillations too.

An example would be a PV that increased rapidly towards SP, then halted early and seemed to "shy away" from the setpoint before rising towards it again. The integral term magnifies the effect of long-term steady-state errors, applying ever-increasing effort until they reduce to zero. In the example of the furnace above working at various temperatures, if the heat being applied does not bring the furnace up to setpoint, for whatever reason, integral action increasingly moves the proportional band relative to the setpoint until the PV error is reduced to zero and the setpoint is achieved.

This option can be very helpful in stabilizing small boilers 3 MBTUH , especially during the summer, during light loads. It is possible to filter the PV or error signal. Doing so can reduce the response of the system to undesirable frequencies, to help reduce instability or oscillations. Some feedback systems will oscillate at just one frequency. By filtering out that frequency, more "stiff" feedback can be applied, making the system more responsive without shaking itself apart.

Feedback systems can be combined.

In cascade control , one control loop applies control algorithms to a measured variable against a setpoint, but then provides a varying setpoint to another control loop rather than affecting process variables directly.

If a system has several different measured variables to be controlled, separate control systems will be present for each of them.

Control engineering in many applications produces control systems that are more complex than PID control. Examples of such fields include fly-by-wire aircraft control systems, chemical plants, and oil refineries.

Model predictive control systems are designed using specialized computer-aided-design software and empirical mathematical models of the system to be controlled. Hybrid systems of PID and logic control are widely used.

The output from a linear controller may be interlocked by logic for instance. Fuzzy logic is an attempt to apply the easy design of logic controllers to the control of complex continuously varying systems.

Basically, a measurement in a fuzzy logic system can be partly true, that is if yes is 1 and no is 0, a fuzzy measurement can be between 0 and 1. The rules of the system are written in natural language and translated into fuzzy logic.

## Ee1256 – Control Systems

For example, the design for a furnace would start with: If the temperature is too low, increase the fuel to the furnace.Simple on—off control systems like these can be cheap and effective. Its name comes from the information path in the system: process inputs e. Linear control systems. This helps avoid thermal shocks and applies heat more effectively. The controller C then takes the error e difference between the reference and the output to change the inputs u to the system under control P.

Eventually as the PV falls back into the PB, heat is applied again, but now the heater and the furnace walls are too cool and the temperature falls too low before its fall is arrested, so that the oscillations continue. This reduces to Boolean arithmetic if values are restricted to 0 and 1, instead of allowed to range in the unit interval [0,1].

The most common extensions lead to proportional-integral-derivative control, or PID control.

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