An important consideration for the I term is its ability to quickly overwhelm the PID equation if the current measurement is far from the desired set points. This effect is known as windup and we will actually be placing a limit on the I term value to constraint its contribution in the implementation. As the error begins to approach and oscillate around zero after a disturbance, the integral term will help drive the system towards zero and improve accuracy. With regard to gain, too little or too much can result in a lack of accuracy and dramatic instability respectively. Finally, we have the D or derivative term which answers the question what is about to happen.

## Applied Control Theory for Embedded Systems

This value represents a prediction of what will be needed to counteract the disturbance based on the prior sample. As was the case with the previous PI terms, this term uses a constant gain that is multiplied by the rate of change of the error sample. As before, we will adjust the gain to get an optimal disturbance response from the system. The derivative has a large effect on the response time of the system, which also makes it quite sensitive to noise in the measured feedback signal.

For this term, too much gain will result in high frequency oscillations. For some control system applications, it is entirely reasonable and in fact necessary to use a gain of zero, which simply means a derivative term is not used since the nature of such control systems are not conducive to rapid changes. Remember the fourth beta term? This is the bias and it can be used to nudge the system or compensate for things like static loads such as gravity. Aside from nudging the control loop, we can also use this to inject the disturbance.

## Dynamical systems theory

This could be intentional as part of the application, but it is also quite useful for tuning as we will see later. Now that we waded a bit through the PID equation, it is time to see a close loop system example. In this application, we are using a voice coil actuator to approximate human motion in the context of striking a drum. The system uses an industrial voice coil actuator with very high acceleration and peak force. The linear motion is being translated to a rotary motion in order to move the striking implement about an axis.

This translation offers a magnification of the peak force while simultaneously providing a means to monitor its position with an optical quadrature encoder. The last ingredient is the use of a pulse width modulated signal and an H-bridge that allows us to control the magnitude and direction of the current through the voice coil actuator as directed by the PID controller. Starting at the top of the closed loop and moving clockwise, the first component is the quadrature encoder, which provides the rotation, speed, and direction information to the quadrature decoder.

The decoder translates the speed and direction information to the absolute location of the optical disc that is fixed to the axis of rotation. The PID controller takes the location at set point values and generates the direction and pulse width modulation signals for the H-bridge driver. Given the input signals from the controller, the driver adjusts the polarity and duty cycle of the current flowing through the voice coil actuator. The voice coil actuator in turn moves proportionally to the magnitude and direction of the current from the H-bridge driver.

Finally, the two degrees of freedom coupler translates the linear motion to rotary motion at the axis of rotation which closes the control loop. In this application, a striking implement is attached to the axis of rotation and therefore it can be precisely positioned as directed by the external set point. Additionally, the bias value can be used to induce a temporary disturbance or sustained offset of any magnitude and direction up to the maximum limit. A voice coil actuator is a continuous analog device that uses magnetic coupling with no backlash given the absence of gears.

Since all of the variables in the force equation are constant by design, the force is directly proportional to the current magnitude and direction. You might have also guessed that this is the same technology that you will find in a loud speaker. But instead of moving a paper cone to modulate air pressure and produce sound, this voice coil moves a shaft through a linear bearing.

Although the coil will stop when it bottoms out, there is nothing inherently in place to keep the coil from becoming airborne in the other direction. As a result, some type of mechanical stop must be in place to prevent extreme positions, unless your intent is to produce an expensive one time use projectile. Translating the linear actuator motion of the voice coil to a rotary motion provides an opportunity to magnify the peak force by increasing the torque.

Two degrees of freedom are used to couple the voice coil through the rotating axis that is held in a fixed bearing. The primary shaft as a rotary assembly can be use to mount a standard optical quadrature encoder as well as a striking implement. The quadrature encoder is a standard method to measure the speed and direction of motion, which in this case is a rotary configuration using an optical disc and encoder.

The encoder generates two signals whose phase relationship and the frequency indicates speed and direction. There is also an optional index signal that is active on each full rotation of the disk. The encoder signals are connected to a decoder that provides an absolute position count that will be used as our location variable, y t. The encoder operates from a single viable power supply but can be notoriously sensitive to both mechanical and electrical noise. Filtered power, good shielding, stable mounting, and precision alignment are all essential ingredients for getting good signals.

A pulse with modulation signal is used to set the average current through the coil and the H-bridge circuit provides a means to change the direction of the current.

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As depicted in the schematic diagram, enabling the pair of power MOSFET Q1 and Q4 while simultaneously disabling Q2 and Q3 will allow the current to flow in one direction through the motor, or in this case a voice coil. Conversely, by disabling Q1 and Q4 and enabling Q2 and Q3, the current is able to flow in the opposite direction. For this application, the PWM frequency is 20 kilohertz and the duty cycle resolution is 8 bits or zero to decibel. In this simplified bare metal example, the function main continually reads the location, updates the controller by calling the PID function, adjust the H-bridge driver by setting the H-bridge direction and PWM duty cycle and sleeps for a sample period before repeating the same sequence again indefinitely.

In this trivial example, we will take for granted that the PID gains, K-sub-P, K-sub-I and K-sub-D are set to pre-tuned default constants and that the desired position is being specified and updated by a separate thread of execution.

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The readLocation, pwm and usleep functions are rather self-explanatory. As we have seen in the previous slide, the PID implementation is a single function that is called on a periodic basis that is equal to the sample rate of the location. This value is passed into the function as the LOC variable, which is the current location of the system. This function includes a reset feature that allows the calling code to clear the previous location sample and the integral value on demand.

As we can see, the error is calculated by simply subtracting the current location from the desired position, which is the equivalent of e t equals R t minus y t. Recall that the goal of the PID controller is to eliminate the error and restore the system to the set point following a disturbance. Note that both the previous and integral variables are declared static in order to maintain their values between calls. Continuing with the PID function, we can see the integration of the error value along with windup or protection. The derivative is calculated by subtracting the current location from the previous location sample which represents the rate of change.

## Applied Control Theory for Embedded Systems Image at Mighty Ape NZ

All of the terms including the bias are assigned to the sum or u t. Note that each of the terms are multiplied by their respective gain constants that were passed in to the PID function, which gives us the ability to adjust the gains at runtime. Finally, the direction is determined based on the sign of the sum, which will be used to control the direction of current flow through the H-bridge.

In the final fragment of the pid function, the sum is clamp to limit the PWM duty cycle value and the sign is removed to establish the magnitude of the current through the coil.

Last but not the least, we save the current location since it will be needed to compute the derivative on the next call. Chapter 10 , Software Implications, presents the special issues that are raised when one uses a digital computer to implement controllers. It covers the various data types that can be used to represent signals; it covers the problems associated with quantization and overflow; it discusses resources issues such as memory usage and processor speed; and it gives a nice set of controller implementation examples in the three most popular data types.

Chapter 11 , Afterword lists the software tools and gives an annotated bibliography for the book. Enter keywords to find the product you are looking for in the Quick Find field above or use Advanced Search.

Applied Control Theory for Embedded Systems. This product was added to our catalog on Thursday 17 January, Customers who purchased this product also bought:. Embedded Software: The Works. Embedded Multitasking. Shopping Cart. What's New?

Tell someone you know about this product. Write a review on this product! List of all our Products. In it, the learner's mind reaches a state of disequilibrium where old patterns have broken down. This is the phase transition of cognitive development. Self-organization the spontaneous creation of coherent forms sets in as activity levels link to each other. Newly formed macroscopic and microscopic structures support each other, speeding up the process.

These links form the structure of a new state of order in the mind through a process called scalloping the repeated building up and collapsing of complex performance. This new, novel state is progressive, discrete, idiosyncratic and unpredictable. Dynamic systems theory has recently been used to explain a long-unanswered problem in child development referred to as the A-not-B error.

The application of Dynamic Systems Theory to study second language acquisition is attributed to Diane Larsen-Freeman who published an article in in which she claimed that second language acquisition should be viewed as a developmental process which includes language attrition as well as language acquisition. From Wikipedia, the free encyclopedia. Main article: Dynamical system definition. Main article: Nonlinear system.

Main article: Dynamic approach to second language development. List of dynamical system topics Baker's map Biological applications of bifurcation theory Dynamical system definition Embodied Embedded Cognition Fibonacci numbers Fractals Gingerbreadman map Halo orbit List of types of systems theory Oscillation Postcognitivism Recurrent neural network Combinatorics and dynamical systems Synergetics Systemography. Retrieved Busemeyer , "Dynamic Systems". To Appear in: Encyclopedia of cognitive science , Macmillan. Retrieved 8 May Journal of Educational and Developmental Psychology. PLoS One.

Preservation of fertility. Interdisciplinary Sciences: Computational Life Sciences. Accessed Child Development. Trends in Cognitive Sciences. Applied Linguistics. Areas of mathematics. Category theory Information theory Mathematical logic Philosophy of mathematics Set theory. Abstract Elementary Linear Multilinear. Calculus Real analysis Complex analysis Differential equations Functional analysis.