Control Strategies Shuvra Das University of Detroit Mercy

Topics  Feedback Control Systems  Bang-Bang Control  PID Control  Digital Control

Control System Components

Plant Plant is the system whose performance we wish to control. Such a system may have multiple inputs and outputs. The essence of control is to determine which control inputs affect the performance in which we are interested, and to produce those inputs which cause the plant to exhibit the desired performance, as observed in the outputs of the system.

Controller The controller is the heart of the control system. The control engineer determines some way to provide inputs to the plant, which will cause the plant to behave as desired. The derivation of this strategy, called the control law, is the primary focus of the control engineer. Implementation of control often takes place in electronic circuitry, which performs mathematical operations of integration, differentiation, multiplication, addition and subtraction.

Controller The same operations can be accomplished through computer code for a digital controller, which can be as simple as a single chip processor, or as complex as a high-end Pentium workstation. The error signal is the difference between the desired (reference input) and the measured outputs. A controller uses this error signal to determine the appropriate control action. The larger the error, the more control effort will be needed to achieve the desired performance.

Control System Components

Actuators Actuators convert the controller’s output signal (usually an electrical signal) into a signal that has physical relevance to the system we wish to control. Some examples of actuators include motors, hydraulic pistons, and relays.

Sensors Sensors are designed to convert some observed physical quantity to a signal that can be processed by the controller. For example, a potentiometer (variable resistance) can be used to sense the orientation of an antenna. A sonar sensor can be used to sense the distance of an autonomous material-handling robot from an obstacle in its path. To measure temperature in a furnace used for the heat treating of metals, a thermocouple can be used.

Advantages of Feedback  Providing the controller with information about the plant's actual behavior allows for automatic adjustments to maintain performance within acceptable limits.  Decreases sensitivity to variations in parameters and to external disturbances  Caution: feedback can produce instability feedback can introduce instability

Feedback Control Concepts: Open loop control No feedback Controller receives no information about whether or not it is providing the correct information Input Actuator Actuators Controlle r Plant Outpu t

Open-loop control Open-loop control of the position of a robot arm. The desired angular position is set using a dial or other input device.  denotes the actual arm position.

Open-loop control The control unit produces a voltage needed to drive the motor (actuator) for an amount of time dependent only on the input setting. There is no way to automatically correct for error in the arm position. If someone accidentally disturbed the robot such that its zero position was offset by a few degrees, the controlled system would exhibit the same offset error.

Closed-Loop/Feedback Control: In a closed-loop system there is feedback. The output is monitored by a sensor. The sensor measures the system output and converts this measurement into an electrical signal, which passes back to the controller. The feedback allows the controller to make any adjustments necessary to keep the output at or near its desired value. Closed-loop control is also known as feedback control.

Closed-loop control This time a potentiometer (a.k.a. “pot”) has been connected to the motor shaft. As the shaft turns, the pot resistance changes. The resistance is converted to a voltage and then fed back to the controller. Thus, we have a measurement of the actual arm position _ + Set point (30 0 ) ErrorMotor Pot Control unit Figure 4

Closed-Loop control To command the arm to an angular position of 30°, a set-point voltage corresponding to 30° is sent to the controller. This is compared to the measured arm position, and the controller drives the motor in a direction to reduce the error. Feedback decreases a system’s sensitivity to disturbances. Suppose the arm is bumped and it comes to rest at 35 instead of 30 degrees. With a closed-loop system, the effect of the disturbance is measured and is passed back to the controller, which compensates for the error.

Closed-loop control To maintain the level in the vessel, we have to monitor the level itself and adjust the inlet valve if the level deviates from the desired value. Such a feedback strategy is error driven in that the control effort is a function of the difference between the required level and the actual level.

Control Law The relationship between the error and the control effort is called the control law. Feedback control can provide regulation against unmeasured disturbances.

An Application of Feedback Control: Regulation A control system for maintaining the plant output constant at the desired value in the presence of external disturbances is called a regulator. Disturbances will cause the plant output to deviate and the regulator must apply control action or control effort to attempt to maintain the plant output at the reference value with minimum error. A good regulator will minimize the effects of disturbances on the plant output.

Regulation  D(s) 1 G(s) e  r   Compensation Disturbance, w Plant

Regulation As an example, consider a temperature control system for a furnace. The goal is to maintain a constant temperature equal to that specified as the reference input. Opening the furnace door creates a disturbance - a rush of cool air entering the chamber. The performance of the system is related to how quickly the system is able to compensate for this disturbance.

Stability Simply and imprecisely, a system is unstable if its output is out of control. Goal of control system design is to cause the controlled system to possess a desired output characteristic, an unstable system is undesirable. If the plant is inherently unstable, then the controller must stabilize the system. If the plant is naturally stable, then the controller is to enhance some characteristic of the system response without causing instability. Note that closing the feedback loop can affect the stability of a system.

Stability Picture an overhead crane with a wrecking ball attached to the end of its chain. If the wrecking ball is given an initial push and allowed to swing freely, it will eventually come to rest at its equilibrium point. Now consider the problem of balancing a broomstick in an upright (or inverted) position. This is an example of an unstable equilibrium position.

Stability Even if we place the broomstick at this position initially, once we let go, the broomstick will not be able to maintain the desired position without some kind of control effort. If you were trying to balance the broomstick on your palm, you would need to make continuous adjustments to keep it from falling. A special kind of apparatus called an inverted pendulum allows us to design and test control laws that allow for the balancing of the pendulum without human intervention.

Robustness When we design control systems, we often do not have a perfect mathematical model of the plant. Some of the parameters of the system may be uncertain. Robustness is the property that the dynamic response (including stability) is satisfactory not only for a nominal plant transfer function used for the design but also for the entire class of mathematical models that express the uncertainty of the designer about the dynamic environment in which the real controller is expected to operate.

Transient Response The overall response of any dynamic system will consist of two distinct parts. The transient response is the part of the total response that normally diminishes as time proceeds without any subsequent sudden changes in inputs or disturbances to the system. The steady-state response is what is left when the transient has indeed died away to zero. Note that for stable systems, the transient response always approaches zero as time approaches infinity.

Response Steady-state response Figure 7 Transient

Response Steady-state response Overall response Figure 8 Transient response

Steady State Accuracy Steady-state accuracy refers to how well the output, y, tracks the reference input, r, once the initial transients of the system shown in Figure 9 die out. The difference between r and y is called the system error, e. As time goes on (in the absence of further disturbances or abrupt changes in the reference input - steady-state), this error should at best, decrease to zero, but should at least approach some acceptable finite constant value.

Steady State Accuracy D(s) 1 G(s)  e  r Compensation y

Frequency Response When a sinusoidal signal is the input to a linear dynamic system, the output will be a sinusoid of the same frequency, but most likely of differing magnitude and phase (Fig. 10). Furthermore, the magnitude and phase of the output may vary with the frequency of the input signal. Collecting this frequency response data and plotting them give the control engineer insight as to the behavior of the system, and approaches to designing appropriate compensation.

Acos(  t)Bcos(  t+  ) Linear System

BANG-BANG CONTROL There are many different strategies for designing feedback control for a dynamic system. Depending on the knowledge (or lack thereof) of a model for the system we want to control, we may choose different strategies. Bang-bang control refers to the operation of actuators in a control system. Also known as "on- off" control, this strategy is used when a smooth system response is not critical.

BANG-BANG CONTROL Consider, for example, the cooling system in your automobile. A temperature sensor (thermostat) monitors the temperature of the engine coolant. When the temperature exceeds the desired operating point, the corresponding signal from the thermostat initiates the opening of a valve to allow coolant to re-circulate, drawing cool liquid from the reservoir to keep the engine temperature at the desired level. The valve is either on or off, thus the name "on-off" or "bang-bang" control.

BANG-BANG CONTROL In the absence of a precise mathematical model for the plant, bang-bang control is sometimes utilized to provide gross control of movement. A third mode for each of the motors will be "stop". (I suppose this makes the problem one of bang-bang-bang control…)

PID CONTROL PID stands for Proportional-Integral-Derivative and a PID controller can be useful in providing a level of control more precise than the gross control provided by a bang-bang strategy. Electronic components called operational amplifiers (or op-amps) can be configured to perform the following operations on an electrical signal: integration, differentiation, and multiplication by a proportional gain factor.

PID CONTROL Thus op-amps can be used to implement a differential equation electrically. This allows a control engineer to modify the basic response of a dynamic system. The output of the electronic controller is a control signal, which drives the actuators in such a way as to provide the desired plant output. A PID controller consists of a proportional gain element, an integrator and a differentiator, all working together

PID CONTROL  Proportional control: Increasing proportional gain usually improves steady-state accuracy at the cost of less stable transient response dynamics.  Integral control: Improves steady-state accuracy, at the cost of undesirable transient response characteristics (slow settling time, less stable)

PID CONTROL  Derivative control: Controller output proportional to rate of change of actuating error signal. Must always be used with proportional control. PD allows for controller with high sensitivity, but is susceptible to noise. Improves stability, because it kicks in before actuating error gets too large.  PID: combines effects of all three. Designer determines what values to set for the three gain parameters - not as easy as it sounds. Control theory gives the designer some strategies for picking these values.

PID vs. Bang Bang –MATLAB Demo…. PID Controller –Compare PID to bang-bang:  Discuss the ND cart steering. Jerky, as opposed to PID smooth.

Digital Control In many applications, computers of some kind are used to control devices and systems. In fact a network of op-amps configured to implement differential equations is called an analog computer, which came before the digital computers with which we are all familiar. Today's digital computers operate on discrete pieces of information, rather than on a continuous electrical signal. Such computers are used to implement difference equations, the discrete-time counterpart to differential equations. In digital computers, the control design is implemented using algorithms and code, rather than electronic hardware.

Digital Controls-Advantages  Less complex hardware (assuming the computer is there already)  Flexibility (change software, not hardware)  Parameters don't drift with temperature, age  Accuracy - digital signals have more noise immunity (1's and 0's - e.g., CD audio vs. vinyl)  Reliable - no wires, parts to go bad

Project In the project, you will be using a STAMP2 single-board computer as the controller. The sensor will feed back to the computer information about the distance from the wall, and will use that data to decide how to adjust the direction (or position in the case of the sensor) of each of the servomotors. Part of your job is to decide on a strategy for the decisions made by the computer and then generate the code to accomplish this. (Talk about the need for D/A and A/D when using a computer.)