Automatic control of a dynamic system.

Transcription

1 EXAMENSARBETE INOM MECHANICAL ENGINEERING, GRUNDNIVÅ, 15 HP STOCKHOLM, SVERIGE 2016 Automatic control of a dynamic system. Positioning of a spherical object on a flat surface. JOHAN BLOMQVIST NILAS OSTERMAN KTH SCHOOL OF INDUSTRIAL ENGINEERING AND MANAGEMENT

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3 Automatic control of a dynamic system. Positioning of a spherical object on a flat surface. JOHAN BLOMQVIST NILAS OSTERMAN Bacherlor s Thesis in Mechatronics Supervisor: Martin Edin Grimheden Examiner: Martin Edin Grimheden Approved: TRITA MMKB 2016:27 MDAB088

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5 Abstract Controlling the value of a variable and compensating for external influences is a fundamental problem in a wide range of applications. This thesis investigates the issues of such control problems; it presents theory on control design and system modeling as well as the development of a demonstrator in the form of a ball-balancing platform to apply these concepts on. This project s purpose is to design a dynamic system and a state space controller that performs as well as possible with respect to response time and precision. The purpose of the project is achieved by analyzing the dynamic problem and from it create a theoretical model. This is then used to design a state space controller in order to continuously regulate the position of a ball on the platform. The final step is to build a demonstrator which will be used to verify that the designed controller fulfills the criteria that was assigned at the beginning of the project. The controller was tested by performing a step in the set-point with 15 millimeters. This made it possible to analyze the step response in order to determine the rise time, overshoot and the static error of the system. The result of the tests was a rise time of 0.3 seconds, with a overshoot of 4%, which fulfills the speed demands of the system. The static error had a peak of 3 millimeters offset from the set-point. The main part of the error was caused by a hardware issue in the form of the ball not moving even though the platform is tilting. This due to irregularities on the ball s surface as well as the resistive touch screens cushioning effect. The static error was therefore deemed as acceptable. iii

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7 Sammanfattning Bollbalancerande plattform Att kontrollera värdet av en variabel och kompensera för extern påverkan är ett grundläggande problem i en stor mängd tillämpningar. Detta examensarbete undersöker problemen hos den sortens reglerproblem; den presenterar teori för regulatordesign och systemmodellering tillsammans med dokumentation över utvecklingen av en bollbalancerande plattform på vilken koncepten tillämpas på. Syftet med detta projekt är att designa ett dynamiskt system och en tillhörande regulator för att kontrollera systemet med, baserad på en tillståndsmodell som presterar så bra som möjligt gällande snabbhet och precision. Syftet med projektet uppnås genom att analysera the dynamiska problemet och från detta ta fram en teoretisk modell. Den används för att skapa en regulator baserad på tillståndsvariabler för att kontinuerligt kunna kontrollera bollens position på plattformen. Sista steget är att bygga en demonstrator för att verifiera att den färdigställda regulatorn uppnår de kriterier som sattes under projektets början. Regulatorn testades genom att ändra referenspunkten med 15 millimeter. Detta gjorde det möjligt att analysera stegsvaret för systemet för att ta fram värden för stegtid, översläng och statiskt fel. Testen visade att stegtiden blev 0.3 sekunder med en 4% översläng, vilket uppfyllde de hastighetskrav som var satta. Det statiska felet hade ett maxvärde av 3 millimeter från referensvärdet. Den dominerade faktorn av felet bestod av att bollen inte rullar om vinkelutslagen från servomotorn är för små. Detta beror på att bollen inte har en helt slät yta och för att den resistiva tryckplattan som användes har en mjuk yta. Felet ansågs vara acceptabelt. v

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9 Preface We would like to thank Lars Svensson for the supervision and the feedback throughout the project. We would also like to show our gratitude to the entire KTH mechatronics department for all the equipment that made it possible to finalize the demonstrator. Many thanks to the student assistans that have helped us during these weeks. Last but not least, we want to thank the other students that we have been working side by side with, for all the moral support, help and feedback. Johan Blomqvist, Nilas Osterman Royal Institute of Technology, Stockholm May 2016 vii

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11 Contents Abstract Sammanfattning Preface Contents Nomenclature iii v vii ix xi 1 Introduction Background Purpose Scope Method Theory Physical model Control of a MIMO system Regulator design Controllability and observability Pole placement Sampling frequency Demonstrator Problem Formulation Software Timer interrupts Electronics System description Resistive touch input Hardware The mechanics ix

12 4 Results and conclusions Results Discussion Input issues State approximation issues System performance Possible error sources Conclusions Future work Future work Bibliography 25 Appendices A Calculations 27 A.1 Physical model A.2 State space controller A.3 MATLAB x

13 Nomenclature Symbols Symbols F x,y [N] M z [N m] d [m] L [m] r [m] g [m/s 2 ] m [kg] I G [kg m 2 ] x p [m] a x,y [m/s 2 ] v [m/s] θ [ ] α [ ] x Description Forces that are applied on the ball, separated into x and y directions. Torque around the z-axis. The distance of the drop arm from the motor. The distance from the center of the platform to where drop arm is fixed. The radius of the ball. The acceleration due to gravity. The mass of the ball. The ball s moment of inertia. The position of the ball. The acceleration of the ball, separated into x and y axis. Velocity. The angle of the servo motor. The angle of the platform. The state vector. Abbreviations Abbreviation CAD SISO MIMO Description Computer Aided Design Single Input Single Output Multiple Input Multiple Output xi

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15 Chapter 1 Introduction This chapter serves to give insight to the background and purpose of the project. It presents the scope and delimitations set, as well as the methods used to produce the results. 1.1 Background The subject of automatic control plays an important role in a wide range of applications in both industrial processes as well as products. In many of these applications it is desired to maintain one or more variables at some predefined value and counteract for external influences. A few examples of such problems could include controlling the position of a lens system in a camera to compensate for unwanted shaking motion in the camera body, maintaining constant velocity in automobiles while travelling on surfaces with varying inclination, or the control of temperature in a room with fluctuating external temperature. While the details of the aforementioned applications may differ, they all share the fact that the core of the problem lies in controlling a variable of interest in an autonomous fashion, with little human interaction. This fundamental problem in control theory can be realized in a multitude of ways. In this project the process variable (PV for short) is the measured position of a ball on a platform. The setpoint, the position which is to be maintained, is at the origin at the center of the platform. By varying the angular output of a servo motor, and by extension the inclination of the platform, the acceleration vector of the ball can be controlled. This makes it possible to control the position of the ball utilizing principles of feedback control to counteract for external influences in such a way that the proposed specifications for speed and precision (rise time T s and overshoot M) are satisfied. 1

16 CHAPTER 1. INTRODUCTION 1.2 Purpose This report investigates the application of feedback control on a dynamic system, it s abilities and limitations and how theory can be realized onto a physical system using servo motors and resistive touch technology as output and input respectively. The research questions in this project is to gather understanding on what the limiting factors consists of when controlling the position of the ball on the platform, and investigating ways to optimize performance of the system with regard to speed and precision. The goal of this project is to gather informations and find answers to the following research questions: What factors inherent to the system are limiting the control performance? How can the influence of these factors be compensated for in order to increase system performance? What accuracy can be achieved for the system? 1.3 Scope The demonstrator described in this report is to be able to control the position of a spherical object on a flat surface, and counteract for external forces of reasonable magnitude applied to the object. The system control is achieved by making a model of the systems physics, and from this designing a state feedback controller which is implemented on an Arduino Uno, described further in [Arduino, 2016]. The demonstrator s mechanic- and electronic system is designed and manufactured as a part of the project. The system is limited to two degrees of freedom by having the demonstrators platform suspended on two perpendicular axes in order to simplify construction and controller synthesis. This project follows the scope of a Bachelor thesis at KTH which is equal to ten weeks of work. 1.4 Method A demonstrator of the mechanical system is to be constructed and manufactured and will be controlled by an Arduino Uno, two electrical servo motors for controlling the inclination of the platform, and a resistive touch screen used to sense the position of the ball. By analyzing the dynamic system and deriving equations to describe the relationship between the ball s acceleration and the platform s angle, a state feedback controller 2

17 1.4. METHOD can be designed to control the system according to the specifications set upon it. The synthesized controller was then tested using simulations in MATLAB to ensure that satisfactory performance and stability was achieved before implementation on the demonstrator. The first prototype of the demonstrator was designed with only one degree of freedom to ensure quick testing of the control algorithm since the system was designed as separate but identical controllers for each axis this allowed for easier tuning and optimization in the one dimensional environment before implementation on the final demonstrator. To evaluate the performance of the finalized controller, the system was monitored in MATLAB to gather information on the system reacting to a step in the set point and to disturbances imposed. The experiment was carried out by allowing the ball to come to rest at the center of the platform, after which a defined step of change in the set point was made in the controllers software. The ball was again allowed to come to rest at this new set point, while the position of the ball was plotted over time in MATLAB. These plots was then evaluated to assess the speed and accuracy achieved by the controller in terms of the step response characteristics. These characteristics is typically used to specify and assess the performance of a control system as described in [Torkel Glad, 2006]. In order to be able to determine if the controller performs well enough, specifications for the step response characteristics mentioned above are set for the controller: Rise time T r less or equal to 0.5 seconds. A overshoot M less than 5%. The static error e 0 should be equal to zero. The goal is that the controller will be able to fulfill these demands. 3

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19 Chapter 2 Theory This chapter presents the theoretical foundation for the project divided in two sections: Physical system model and controller design. 2.1 Physical model To be able to theoretically describe the dynamic system, Newtons laws are applied. F = m a (2.1) By making a schematic figure of the system represented in figure 2.1 it is possible to define the different forces that affect the various parts of the system. In order to see the direction of each force a coordinate system is inserted in the figure. This makes it possible to separate the problem into a system of equations: F x = mẍ G, F y = mÿ G, M z = I G θ. In order to solve this system two assumptions are made. Since the ball is rolling on the surface, it is assumed that the ball is not moving upwards relative to the platform, making the term F y equal to zero. While this is a simplification of the true system it s assumed to work well when the angular acceleration is reasonably small. Secondly, is that it is assumed that the ball is not sliding on the surface which is assumed to be true since the friction between the surface of the platform and the ball is sufficiently large. By solving the system equations the ball s acceleration on the x-axis is given as a function of the servo angle given in equation A.5. 5

20 CHAPTER 2. THEORY Figure 2.1. defined. Schematic of the dynamic system with force vectors and variables 2.2 Control of a MIMO system There are several ways to deal with the control of a MIMO system which is the task in this project. The position of the ball is given in two dimensions (x,y) and the two servo motors requires one input signal each that determine the tilt of the platform on each of its two axes. With two inputs and two outputs that can, according to the theory presented in section 2.1, be separated into two independent physical systems, we take on the simple approach of coupling each of the inputs to each of the outputs resulting in two SISO controllers, one for each axis. 2.3 Regulator design A state space controller is a feedback-controller that only use the time domain which works well since it is to be used in order to control a dynamic system. The open loop system is described by transforming the theoretical model into a equation of 6

21 2.3. REGULATOR DESIGN matrices [ẋ ] [ ] [ ] a11 a = 12 b1 x + u (2.2) ẍ a 21 a 22 b 2 ] y = [c 1 c 2 x (2.3) which more simplified can be described as x = A x + Bu y = C x. The state space model gives the input signal as (2.4) u = L x + l 0 r (2.5) that will be used to control the system. The term L x is a scalar product and r is the reference signal and l 0 is a scalar that makes sure that the static gain of the closed loop system is unity. This results with the equation of matrices becoming x = (A BL) x + B r (2.6) y = C x (2.7) which is the closed loop system. It is now possible to place the poles so that the controller fulfills the performance demands. This is done by choosing the right values of the scalar product since the poles of the system is determined by the eigenvalues of (A BL) Controllability and observability In order to have full freedom in the placement of the poles it is necessary to check the controllability of the system. This is done by determining the controllability matrix S [Torkel Glad, 2006], which is given by S = [ ] B AB A 2 B A n 1 B (2.8) where n is the number of states. If the determinant of the matrix S is separated from zero, the placement of the poles are not restricted and it is possible to place the poles so that the controller fulfills the demands. Another important factor is if all the states can be measured. This is important since the controller needs to have all the information in order to successfully control the system. A more formal way of describing this is that it is necessary that given the systems output signal, it should be possible to fully describe the behaviour of the system. This is called the observability of the system. To check this a observability matrix O is defined such as 7

22 CHAPTER 2. THEORY O = C CA CA 2. CA n 1 (2.9) Where n is the number of states. If the determinant of the matrix O is non-zero, the system is observable. If not, some approximation of the states are needed in order for the system to function properly as described in [Torkel Glad, 2006] Pole placement To find initial values for the L vector which determines the placement of the poles of the system, the controller has been modeled as continuous time for which the following theory applies. To be able to choose the correct pole placements a rule of thumb is that the poles shall be placed entirely on the left half of the complex coordinate system and with an angular difference of maximum 46 degrees (between the real and complex axis) see figure 2.2. This prevents the system from becoming unstable, and decreases the overshoot as mentioned in [Torkel Glad, 2006]. To make the system as fast as Figure 2.2. Pole placement. 8

23 2.4. SAMPLING FREQUENCY possible the poles should be placed as far away from the real axis as possible. But since this creates overshoot it is important to find a good balance so that the system has a good response time but also minimal overshoot. 2.4 Sampling frequency Formally, sampling is the reduction of an continuous signal into a discrete signal as shown in 2.3. Since the system to be controlled in this project is in continuous time, while the implemented controller works in discrete time, a sampling of the output signal from the system will have to be implemented. In order to get a good approximation of the behaviour of the dynamic system, the sampling frequency must be chosen as high as possible to reduce the lag in the controlling process. When Figure 2.3. Figure illustrating the sampling of a continuous signal S(t) with sampling interval T. controlling a continuous time system with a discrete controller, the system will have a time lag equal to the sampling time interval. This implies that the system will not react to fluctuations in the input signal between these discrete time steps. Since a signal often can not be measured as often as is needed, there will be a approximation of the signal. This is done by measuring the current signal with a given time interval T. As illustrated in 2.3, the signal is separated into discrete points with a fixed time step in between. To be able to still get a good system response from the signal input, choosing the frequency of these measurements are important. Optimal would be to have the period time as low as possible, this is called oversampling. However this causes some problems, first of all the requirements of the processor is a big factor since the smaller the period the bigger the clock frequency of the processor must be. And still the errors are just reduced but can not be eliminated entirely. So choosing the correct sampling frequency is of significant importance in order to get a good approximation of the signal, which is discussed more in [Torkel Glad, 2006]. 9

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25 Chapter 3 Demonstrator 3.1 Problem Formulation The following has been taken into consideration when constructing the demonstrator: The centerline of the platform s axis must be level with the platform s surface in order for the mechanics to cohere to the modeled system. The two axes on which the platform is suspended on must be centered with respect to each other. The touch screen must be fixed on the platform. The servo motors must be attached to the platform on a line perpendicular to, and passing through the center of the axis of rotation. The platform must be level when at resting position. The demonstrator must be easy to assemble and manufacture with high precision. 3.2 Software As stated earlier, a state space controller is used to control the position of the ball. The states used in the controller is the ball s position and velocity. The Arduino gets the position from the touch screen and interpolates the velocity of the ball. The controller then calculates the input to the servo motor by multiplying the states with the scalar vector and comparing the term with the given set point (see appendix A for calculations). In figure 3.1 a flowchart of this process is seen. The input to the motor can become too high if the ball s positional difference from the set point is too large. This could cause the servo to be set at an angle which is restricted by the mechanics of the demonstrator which creates limitations on the magnitude of the 11

26 CHAPTER 3. DEMONSTRATOR Figure 3.1. Flow chart for software. output from the controller. To make sure that these limitations are not crossed, a maximum and minimum output angle is set in the software. Since the demonstrator does not have a sensor which can measure the speed of the ball an approximation must be used. This was done by using the built-in timer interrupts in the Arduino platform. 3.3 Timer interrupts The timers on the Arduino works by having a counter that is incremented on the tick of the Arduinos clock. There are different modes for the timers. The mode used in this project is called clear timer of compared match (CTC) which allows the counter to increment until it reaches a compare match value. When this happens, the counter resets and the interrupt sequence is executed. Using timer interrupts allows for taking measurements at precise intervals (i.e. constant sampling frequency). When the interruption is done, the programs main loop will continue from where it was stopped. It is possible to control the frequency of the interruptions by setting a prescaler which determines how fast the counter is incremented by dividing the clock frequency at 16 MHz by some constant. The compare match value can also be altered. By doing this, a chosen time difference dt between the position samples allows for an approximation of the ball s velocity to be calculated. During the interruption the difference between the ball s last known position x 0 and current position x 1 is 12

27 3.4. ELECTRONICS divided by the time difference according to v = x 0 x 1 dt (3.1) which gives an approximated velocity. 3.4 Electronics The system is built around an Arduino Uno which is a programmable open-source platform based on the 8-bit ATmega328P microcontroller from Atmel, with a 16 MHz clock frequency, 14 digital- and 6 analog pins System description The Arduino is wired to the different components as is described by the hardware block chart in figure 3.4. Two of the digital output pins is connected to two corners of the touch panel to enable reading of x- and y-coordinates by switching the voltage on the pins between V CC (high) and GND (low), this is explained more in depth in section below. The screens probe pin is wired to one of the Arduinos analog pins, which is used to read the position. The servo motor(s) is controlled via a digital output pin that controls the angle of the servo shaft, and the microcontroller also supply all the voltage for the system Resistive touch input For sensing the position of the object on the platform, a 5-wire resistive touch panel is used. These types of touch screens consists of a flexible top layer and a stiff bottom layer that has been sandwiched together with spacerdots in between. A resistive coating is applied uniformly over each of the surfaces. When pressure is applied to the screen, an electrical connection is made between the two layers as illustrated in figure 3.2. The bottom layer has one wire to each of its four corners; by applying 5V to two corners and grounding the opposite two, the voltage flows uniformly across the surface as illustrated in figure 3.3. The top layer is used to probe the voltage at the point of contact, thus giving an analog reading of the position on one of the platform s axes which is then converted to a digital value between by the Arduino s 10-bit A/D-converter. By altering which corners that are grounded, the other axes position can be read in the same manner and the position of the object on the panel can be determined. 3.5 Hardware This section presents the hardware that the demonstrator is made up of. A schematic representation is shown in figure 2.1 of the different components. 13

28 CHAPTER 3. DEMONSTRATOR Figure 3.2. The composition of the touch screens layers [Elotouch, 2014] Figure 3.3. Schematic of a five wire touch screen seen from above. [Elotouch, 2014] 3.6 The mechanics The mechanical construction is largely made up of CO 2 -laser cut pieces of PMMAplastic sheet material that are secured together with hot glue. The platform is suspended on two steel shafts that is fastened to a rigid frame. In 3.5 a CAD model of the one dimensional demonstrator is shown. One of the most critical aspects in construction of the demonstrator is to get these two shafts axially aligned with each 14

29 3.6. THE MECHANICS Figure 3.4. Block diagram for hardware setup. other with it s center leveled with the touch pads surface to ensure that the mechanics of the system coheres with the model. This is achieved by taking advantage of the laser cutters precision, and by designing the pieces of the demonstrator in such a way that all the critical dimensions are determined by the geometry of the parts fit together. The servo motor is fixed to the frame underneath the platform. It is connected to the platform by a plexiglas arm with a 1-DoF joint in each end to allow the rotational motion of the servos shaft to control the tilt of the platform. Figure 3.5. Isometric view of the CAD model of the one dimensional demonstrator. 15

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31 Chapter 4 Results and conclusions This chapter describes the achieved results and performance of the demonstrator, as well as a discussion around what conclusions can be drawn relating to the scientific questions posed for the project. 4.1 Results Tests on the demonstrator were carried out to assess the performance of the system and determine whether they live up to the proposed specification. To test the system performance, two identical steps of a magnitude of 15 mm in the set point was introduced in software, and by monitoring the position of the ball over time using MATLAB the rise time T r, overshoot M and static error e 0 was measured. The response of the system is shown in figure 4.1. Figure 4.1. Measured step response of the system. 17

32 CHAPTER 4. RESULTS AND CONCLUSIONS The acquired results is presented in 4.1. Table 4.1. Measured results T r [s] M [%] e 0 [mm] By calculating the eigenvalues of the observability and controllability matrices, the result shows that both their eigenvalues are separated from zero, which leads to the system always being both observable as well as controllable. 4.2 Discussion This section sheds light on some of the issues faced during the project, and discusses possible causes and solutions, as well as a general discussion on the performance of the system Input issues. The only sensor the demonstrator contains is a 5-wire touch screen. It is used to track the position of the ball in the x- and y-axis. One of the big flaws is that the positional measurements is not independent of each other. When the ball is held at a fixed position on one of the axes, the measured value is affected by movement along the other axis. This obviously causes problems in the controller since the measured system behaviour along one of the axes can differ greatly from reality depending on the ball s velocity along the other. In order to fix this, another approximation was made in order to keep the x- coordinate fix while only moving in the y direction. By measuring the difference of the x-coordinate when moving the ball in the y-axis, it became clear that the error was approximately linear. By measuring both coordinates and adjusting the position using a linear model of the error based from measurements, the measured error was reduced. However, to be completely eliminated a deeper understanding of the error must be found or a better touch screen used State approximation issues. One of the main issues faced in this project was dealing with input noise during implementation of the controller. Despite the use of a resistive touch screen which measures the position relatively accurately, some small fluctuations in position while the ball is at rest causes large spikes in the approximated velocity. 18

33 4.2. DISCUSSION Since the prototype does not have any sensors to measure the velocity of the ball directly, all the information regarding the state of the system is dependent on the same data. While this has it s advantages in only dealing with noise from one sensor, it also has a backside in that noise which is on an acceptable level for the position is amplified when approximating the velocity state as the time step is small (due to high sampling frequency). There are of course other ways of solving this. One approach would be to implement some other type of sensor which only functions to measure the velocity of the ball and would be independent of the positional sensor. Most such solutions however, would be expensive to implement in two dimensions, and does not guarantee a better approximation without further filtration. To handle the problem in software is both more economical and time efficient. In this case the solution was relatively simple since the fluctuations in position was constant when the ball was at rest resulting in a spike of constant amplitude in velocity. A simple filter which restricts changes in velocity when the difference in position is smaller than a defined value eliminates most of the noise when the system is close to being at rest which is when the approximation issues has the largest noticeable effect System performance The performance of the controller is very satisfying. As the results show, both the rise time as well as the overshoot is low meaning that the controller is both fast as well as stable. This meets the demands that the controller needed to fulfill. It is possible to make the system even faster, but this at the cost of having more overshoot which leads to a decreased settling time and was therefore deemed as unacceptable since the system needs to be as fast as possible while still being stable. To be able to decrease the overshoot, the speed of the system would have to be reduced. This would however influence on the usage of the controller since it is needs to be able to handle outer disturbances as fast as possible. It also became clear that by decreasing the overshoot, the static error became larger as a result, which is worse since the control error is the main problem for the controller. As described earlier some problems occurred with the ball not moving even though the platform is tilting which causes a small error (e 0 ). The error has a peak at three millimeters which seems highly acceptable since the error is only caused by the hardware and physical factor. This is shown in the plot of the step response. When the ball is laying still on a position, the angle of the motor is still separated from zero. This means that the controller is trying the reposition the ball but is unable since the ball does not move at such small angles. 19

34 CHAPTER 4. RESULTS AND CONCLUSIONS Possible error sources One of the limiting factors for the predictability of the system model is the need to linearize the sine term from the servo angle. Since dealing with nonlinear terms in control theory is beyond the scope of this project, the term had to be linearized with a Taylor expansion. The problem is that this approximation only holds true for small angles no greater than 4,7 degrees around zero. When the output angle from the servo is greater than that, the theoretical model will start to disassociate from the physical problem. Since the project is based on balancing a ball around a equilibrium position, this approximation works well when the disturbance is small enough since the angular output will be sufficiently small for the model to handle. For greater disturbances however, the predictability of the system will decrease and the performance of the controller might suffer. Another factor that causes problems in the performance is the static friction between the ball and the surface. When the ball is close to the set point, the angular output of the servos does not produce the predicted reaction from the system and the ball stays put resulting in a static error. The surface of the touch screen is soft by nature, causing the ball to dent the surface slightly which counteracts the rolling motion. When the ball is close to the set point and the angle is small, this causes a problem since the system behaviour does not follow the one predicted by the model. Using a lighter ball might improve on this problem, or by tweaking the first value in the L-vector in the controller to allow the position state to influence the output more. The latter however has been observed to increase the overshoot of the system, this might motivate the implementation of a integral part in the controller. Further, a more rigorous theoretical model might improve the controllers performance with respect to this as well. 20

35 4.3. CONCLUSIONS 4.3 Conclusions Here follows a discussion on what conclusions can be drawn relating to the research questions posed in section 1.2. What factors inherent to the system are limiting the control performance? The single biggest factor limiting control performance was found to be the bounds on the control signal θ, the angle of the servos output. The bounds are set in the software to account for the restrictions imposed by the mechanical construction of the demonstrator. The bounds affect the systems performance when subjected to disturbances or steps of a high enough magnitude. Since the model of the controller has no knowledge of any bounds on the angle, it will try to control the system with a control signal of arbitrarily large magnitude. This results in the control signal reaching it s bounded value, and the angle of the servo no longer follows the values calculated by the controller. This in turn affects the speed of the system, since the angle of the platform is directly related to the acceleration vector of the ball, and when a large disturbance is imposed on the system it will not have the means to respond as well as the controller predicts. How can the influence of these factors be compensated for in order to increase system performance? The bounds on the control signal is as stated mostly a question of mechanical design. The servos rotational freedom can be increased to ±90 if the servo is mounted in a position to allow this. Other than this, the point at which the servo is connected to the platform has a major influence on how the bounds on the control signal limits performance. If the distance L in figure 2.1 is decreased, a change in the control signal affects the inclination of the plane more than if L is large. This can be seen in equation A.4. While this does not directly influence the bounds on θ it allows for a smaller control signal to affect the system more, which increases the magnitude of disturbances which can be handled without saturating the control signal. This shows that the mechanical construction of the demonstrator will affect the behaviour of the system, and care must be taken so that the systems performance specifications can be met. What accuracy can be achieved for the system? As is seen in table 4.1 the specification on control error for a step input was not met. The possible reasons for this has been discussed in the section and will not be analyzed in further detail. The result however is not unexpected since some simplifications was made in the modeling of the system, and deviations from the theoretical model are somewhat inevitable. 21

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37 Chapter 5 Future work This chapter presents some ideas on future work to continue building on this project, and what has been left out that could be improved upon. 5.1 Future work Because of time restrictions, not much time has been spent on analysing the system using both axes for the platform, and measurements have been taken with the ball moving in one dimension only. In order to perfect performance of the system in two dimensions, further work is needed to make the implementation as good as possible. This project has laid down a foundation for this, and since the two axes are regulated separately, the implementation should be straight forward from here. Future work could include applying up to 6 degrees of freedom to the system. This would make it possible to balance the ball up, down, and diagonally. This is needed in order for the system to be applied in all situations and to be used for simulations of some real world systems. This would call for a restructure of the design and a more advanced controller. 23

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39 Bibliography [Arduino, 2016] Arduino (2016). Arduino uno. Available from: arduino.cc/en/main/arduinoboarduno [cited ]. [Elotouch, 2014] Elotouch (2014). Compare all resistive touch technologies (4-, 5-, 6-, 7- and 8-wire explained.). Available from: Technologies/compare_resist.asp [cited ]. [Torkel Glad, 2006] Torkel Glad, L. L. (2006). Reglerteknik - grundläggande teori. Studentlitteratur AB, 4th edition. 25

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41 Appendix A Calculations This chapter presents the calculations that were made in the design of the controller. A.1 Physical model By taking the system of equations in each dimension separated, it is possible to find forces in each direction resulting in F x = mẍ G, mg sin β + f = mẍ G, F y = mÿ G, N mg cos β = mÿ G, fr = I G θ. F z = I G θ. And since there will be no acceleration in y, the second row in the system of equations is redundant. Since mentioned earlier it is assumed that the ball will not slide on the surface as it rolls on the platform. This causes the acceleration in the x-direction to be described as ẍ G = r θ (A.1) It is then possible to describe θ using ẍ G, this is used for describing the friction force f, f = I Gẍ G which will be placed in the first row of the system. This will produce the result ẍ G = r 2 mr2 I G + mr 2 g sin β (A.2) (A.3) Since there is a non-linear term, some sort of linearization must be used for use in the controller. The sine term is therefore linearized using Taylor expansion, causing the sine term to be described as sin β = d L θ (A.4) 27

42 APPENDIX A. CALCULATIONS where d is the length of the arm of the servo motor, and L is the length from the center of the platform, to the position where the arm is fixed to the platform. During this linearization the angle of the platform β has also been described as the angle of the motor θ, since it is this angle that will be controlled. Hence the resulting in the function G is defined as G = dmgr 2 L(I G + mr 2 ) θ (A.5) which maps the input signal θ (angle of the motor) to the acceleration of the ball ẍ G. A.2 State space controller With G defined, it is possible to find the matrices that will be used in the state space controller. The controller will use two states in order to control the system. The chosen states is the position of the ball p x and the ball s velocity v x, hence the state vector x is defined as x = [ ] px v x (A.6) The next step will be to define the matrices A, B and C. The relation between input and output is known from the function given in equation A.5. Hence it is easy to find the correct matrices which describes the system. In the theory chapter the equation was described as x = A x + Bu (A.7) y = C x Where u is the input signal θ, and y is the output. The resulting state space model is defined as [ ] [ ] [ ] [ ] vx 0 1 px 0 = + θ (A.8) a x. 0 0 G v x [ ] [ ] p y = 0 1 x v x (A.9) When the state space model has been set up, the first step will be to check the controllability and observability. In order to do this, the two matrices S and O must first be defined. After which the determinant of these matrices must be separated from zero. S = [ ] 0 G, O = G 0 [ ] (A.10) 28

43 A.3. MATLAB A.3 MATLAB The poles were chosen by using MATLAB as a tool in order to check the step response of the system for different placements of the poles. When the step responses fulfilled the demands set on the controller, the given pole placements were used to calculate the scalar vector L. The state space uses the scalar vector L as well as a reference point in order to determine the input signal θ θ = Lx + l 0 r (A.11) L is now determined by taking the eigenvalues of A BL. l 0 it then determined so that the static gain of the closed loop system is equal to one. This will eliminate the errors. All of this was done using MATLAB as a tool, by taking the chosen poles and placing them using the acker method the L vector is determined. The closed loop system is then defined and the scalar l 0 is calculated using the dcgain method. 29

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