Linear Systems And Controls

[Hebe Zuelke]

Linearcontrol are control systems and control theory based on negative feedback for producing a control signal to maintain the controlled process variable (PV) at the desired setpoint (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.

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. 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. Adjusting gain to achieve critically damped behavior is known as tuning the control system.

In the underdamped case, the furnace heats quickly. Once the setpoint is reached, stored heat within the heater sub-system and in the walls of the furnace will keep the measured temperature rising beyond what is required. After rising above the setpoint, the temperature falls back and eventually heat is applied again. Any delay in reheating the heater sub-system allows the furnace temperature to fall further below the setpoint and the cycle repeats. The temperature oscillations that an underdamped furnace control system produces are undesirable.

In a critically damped system, as the temperature approaches the setpoint, the heat input begins to be reduced, the rate of heating of the furnace has time to slow and the system avoids overshoot. Overshoot is also avoided in an overdamped system but an overdamped system is unnecessarily slow to initially reach a setpoint response to external changes to the system, e.g. opening the furnace door.

Pure proportional controllers must operate with residual error in the system. Though PI controllers eliminate this error they can still be sluggish or produce oscillations. The PID controller addresses these final shortcomings by introducing a derivative (D) action to retain stability while responsiveness is improved.

On control systems involving 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. If a derivative action is over-applied, it can, however, lead to oscillations.

The integral term magnifies the effect of long-term steady-state errors, applying an ever-increasing effort until the error is removed. 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.

Some controllers include the option to limit the "ramp up % per minute". This option can be very helpful in stabilizing small boilers (3 MBTUH), especially during the summer, during light loads. A utility boiler "unit may be required to change load at a rate of as much as 5% per minute (IEA Coal Online - 2, 2007)".[1][failed verification]

It is possible to filter the PV or error signal. Doing so can help reduce instability or oscillations by reducing the response of the system to undesirable frequencies. Many systems have a resonant frequency. By filtering out that frequency, stronger overall feedback can be applied before oscillation occurs, 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 field applications 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.

2. ( Not Applicable ) An ability to apply engineering design to produce solutions that meet specified needs with consideration of public health, safety, and welfare, as well as global, cultural, social, environmental, and economic factors

4. ( Not Applicable ) An ability to recognize ethical and professional responsibilities in engineering situations and make informed judgments, which must consider the impact of engineering solutions in global, economic, environmental, and societal contexts

5. ( Not Applicable ) An ability to function effectively on a team whose members together provide leadership, create a collaborative and inclusive environment, establish goals, plan tasks, and meet objectives

Outcome 1 (Students will demonstrate expertise in a subfield of study chosen from the fields of electrical engineering or computer engineering):

1.Solve linear, time-invariant differential equations.

2.Model physical systems by the state-space approach.

3.Analyze reachability, controllability and observability of linear systems.

Outcome 2 (Students will demonstrate the ability to identify and formulate advanced problems and apply knowledge of mathematics and science to solve those problems):

1.Design feedback controllers for closed-loop stability and eigenvalue assignment.

2.Design Luenberger observers for output feedback.

Advanced methods of analysis and synthesis of linear systems, continuous and discrete-time systems, analytical approach to linear control theory. Prerequisite: Graduate standing or consent of instructor.

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The state equation is a first-order linear differential equation, or (more precisely) a system of linear differential equations. Because this is a first-order equation, we can use results from Ordinary Differential Equations to find a general solution to the equation in terms of the state-variable x. Once the state equation has been solved for x, that solution can be plugged into the output equation. The resulting equation will show the direct relationship between the system input and the system output, without the need to account explicitly for the internal state of the system. The sections in this chapter will discuss the solutions to the state-space equations, starting with the easiest case (Time-invariant, no input), and ending with the most difficult case (Time-variant systems).

We can see that this equation is a first-order differential equation, except that the variables are vectors, and the coefficients are matrices. However, because of the rules of matrix calculus, these distinctions don't matter. We can ignore the input term (for now), and rewrite this equation in the following form:

We call the matrix exponential eAt the state-transition matrix, and calculating it, while difficult at times, is crucial to analyzing and manipulating systems. We will talk more about calculating the matrix exponential below.

If, however, our input is non-zero (as is generally the case with any interesting system), our solution is a little bit more complicated. Notice that now that we have our input term in the equation, we will no longer be able to separate the variables and integrate both sides easily.

This is the general Time-Invariant solution to the state space equations, with non-zero input. These equations are important results, and students who are interested in a further study of control systems would do well to memorize these equations.

The state transition matrix, eAt, is an important part of the general state-space solutions for the time-invariant cases listed above. Calculating this matrix exponential function is one of the very first things that should be done when analyzing a new system, and the results of that calculation will tell important information about the system in question.

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