< Day Day Up > |
1.4 Plant Characteristics
In the context of control systems, a plant is a system to be controlled. From the controller's point of view, the plant has one or more outputs and one or more inputs. Sensors measure the plant outputs and actuators drive the plant inputs. The behavior of the plant itself can range from trivially simple to extremely complex. At the beginning of a control system design project, it is helpful to identify a number of plant characteristics relevant to the design process.
1.4.1 Linear and Nonlinear Systems
| Important Point | A linear plant model is required for some of the control system design techniques covered in the following chapters. In simple terms, a linear system produces an output that is proportional to its input. Small changes in the input signal result in small changes in the output. Large changes in the input cause large changes in the output. A truly linear system must respond proportionally to any input signal, no matter how large. Note that this proportionality could also be negative: A positive input might produce a proportional negative output. |
1.4.2 Definition of a Linear System
Advanced Concept
Consider a plant with one input and one output. Suppose you run the system for a period of time while recording the input and output signals. Call the input signal u1(t) and the output signal y1(t). Perform this experiment again with a different input signal. Name the input and output signals from this run u2(t) and y2(t), respectively. Now perform a third run of the experiment with the input signal u3(t) = u1(t) + u2(t).
The plant is linear if the output signal y3(t) is equal to the sum y1(t) +y2(t) for any arbitrarily selected input signals u1(t) and u2(t).
Real-world systems are never precisely linear. Various factors always exist that introduce nonlinearities into the response of a system. For example, some nonlinearities in the automotive cruise control discussed earlier are:
The force of air drag on the vehicle is proportional to the square of its speed through the air.
Friction (a nonlinear effect) exists within the drive train and between the tires and the road.
The speed of the vehicle is limited to a range between minimum and maximum values.
However, the linear idealization is extremely useful as a tool for system analysis and control system design. Several of the design methods in the following chapters require a linear plant model. This immediately raises a question: If you do not have a linear model of your plant, how do you obtain one?
The approach usually taught in engineering courses is to develop a set of mathematical equations based on the laws of physics as they apply to the operation of the plant. These equations are often nonlinear, in which case it is necessary to perform additional steps to linearize them. This procedure requires intimate knowledge of plant behavior, as well as a strong mathematical background.
In this book, I don't assume this type of background. My focus is on simpler methods of acquiring a linear plant model. For instance, if you need a linear plant model but don't want to develop one, you can always let someone else do it for you. Linear plant models are sometimes available from system data sheets or by request from experts familiar with the mathematics of a particular type of plant. Another approach is to perform a literature search to locate linear models of plants similar to the one of interest.
System identification is an alternative if none of the above approaches are suitable. System identification is a technique for performing semiautomated linear plant model development. This approach uses recorded plant input signal u(t) and output signal y(t) data to develop a linear system model that best fits the input and output data. I discuss system identification further in Chapter 3.
Simulation is another technique for developing a linear plant model. You can develop a nonlinear simulation of your plant with a tool such as Simulink and derive a linear plant model on the basis of the simulation. I will apply this approach in some of the examples presented in later chapters.
Perhaps you just don't want to expend the effort required to develop a linear plant model. With no plant model, an iterative procedure must be used to determine a suitable controller structure and parameter values. In Chapter 2, I discuss procedures for applying and tuning PID controllers. PID controller tuning is carried out with the results of experiments performed on the system consisting of plant plus controller.
1.4.3 Time Delays
Sometimes a linear model accurately represents the behavior of a plant, but a time delay exists between an actuator input and the start of the plant response to the input. This does not refer to sluggishness in the plant's response. A time delay exists only when there is absolutely no response for some time interval following a change to the plant input.
For example, a time delay occurs when controlling the temperature of a shower. Changes in the hot or cold water valve positions do not have immediate results. There is a delay while water with the adjusted temperature flows up to the shower head and then down onto the person taking the shower. Only then does feedback exist to indicate whether further temperature adjustments are needed.
Many industrial processes exhibit time delays. Control system design methods that rely on linear plant models can't directly work with time delays, but it is possible to extend a linear plant model to simulate the effects of a time delay. The resulting model is also linear and captures the approximate effects of the time delay. Linear control system design methods are applicable to the extended plant model. I discuss time delays further in Chapter 3.
1.4.4 Continuous-Time and Discrete-Time Systems
A continuous-time system has outputs with values defined at all points in time. The outputs of a discrete-time system are only updated or used at discrete points in time. Real-world plants are usually best represented as continuous-time systems. In other words, these systems have measurable parameters such as speed, temperature, weight, etc. defined at all points in time.
The discrete-time systems of interest in this book are embedded processors and their associated input/output (I/O) devices. An embedded computing system measures its inputs and produces its outputs at discrete points in time. The embedded software typically runs at a fixed sampling rate, which results in input and output device updates at equally spaced points in time.
Two basic approaches are available for developing control algorithms that run as discrete-time systems. The first is to perform the design entirely in the discrete-time domain. For design methods that require a linear plant model, this method requires conversion of the continuous-time plant model to a discrete-time equivalent. One drawback of this approach is that it is necessary to specify the sampling rate of the discrete-time controller at the very beginning of the design process. If the sampling rate changes, all the steps in the control algorithm development process must be repeated to compensate for the change.
An alternative procedure is to perform the control system design in the continuous-time domain followed by a final step to convert the control algorithm to a discrete-time representation. With the use of this method, changes to the sampling rate only require repetition of the final step. Another benefit of this approach is that the continuous-time control algorithm can be implemented directly in analog hardware if that turns out to be the best solution for a particular design. A final benefit of this approach is that the methods of control system design tend to be more intuitive in the continuous-time domain than in the discrete-time domain.
For these reasons, the design techniques covered in this book will be based in the continuous-time domain. In Chapter 8, I discuss the conversion of a continuous-time control algorithm to an implementation in a discrete-time embedded processor with the C/C++ programming languages.
1.4.5 Number of Inputs and Outputs
The simplest feedback control system, a single-input-single-output (SISO) system, has one input and one output. In a SISO system, a sensor measures one signal and the controller produces one signal to drive an actuator. All of the design procedures in this book are applicable to SISO systems.
Control systems with more than one input or output are called multiple-input-multiple-output (MIMO) systems. Because of the added complexity, fewer MIMO system design procedures are available. Only the pole placement and optimal control design techniques (covered in Chapters 5 and 6) are directly suitable for MIMO systems. In Chapter 7, I cover issues specific to MIMO control system design.
In many cases, MIMO systems can be decomposed into a number of approximately equivalent SISO systems. For example, flying an aircraft requires simultaneous operation of several independent control surfaces, including the rudder, ailerons, and elevator. This is clearly a MIMO system, but focusing on a particular aspect of behavior can result in a SISO system for control system design purposes. For instance, assume the aircraft is flying straight and level and must maintain a desired altitude. A SISO system for altitude control uses the measured altitude as its input and the commanded elevator position as its output. In this situation, the sensed parameter and the controlled parameter are directly related and have little or no interaction with other aspects of system control.
The critical factor that determines whether a MIMO system is suitable for decomposition into a number of SISO systems is the degree of coupling between inputs and outputs. If changes to a particular plant input result in significant changes to only one of its outputs, it is probably reasonable to represent the behavior of that I/O signal pair as a SISO system. When the application of this technique is appropriate, all of the SISO control system design approaches become available for use with the system.
However, when too much coupling exists from a plant input to multiple outputs, there is no alternative but to perform a control system design with the use of a MIMO method. Even in systems with weak cross-coupling, the use of a MIMO design procedure will generally produce a superior design compared to the multiple SISO designs developed assuming no cross-coupling between I/O signal pairs.
No comments:
Post a Comment