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| marvin:ecp2 [2009/01/29 10:15] – rieper | marvin:ecp2 [2009/01/29 10:34] – rieper |
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| ====Our Angle of Approach Towards Control Theory==== | ====Our Angle of Approach Towards Control Theory==== |
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| Basically there are three approaches to our control problem, which we will describe briefly in an increasing order of complexity. The first method requires no more than an intuitive understanding of the tuning parameters in a given control loop as we consider the control plant as a black box. We simply implement a control loop by following the directions of the control loop as indicated in the following section “Introducing the PID Controller”. In this case we make no attempts to precalculate the stability issues e.g. using bode plots to observe phase margin as a function of our bandwidth. Instead we use a trial and error principle based on our intuitive understanding of the tuning parameters. The problem with this approach is that lack of knowledge about the dynamics of the control plant (the robot) if we do not succeed. We have no theoretical foundation.\\ | Basically there are three approaches to our control problem, which we will describe briefly in an increasing order of complexity. The first method requires no more than an intuitive understanding of the tuning parameters in a given control loop as we consider the control plant as a black box. We simply implement a control loop by following the directions of the control loop as indicated in the following section [[#Introducing the PID Controller|Introducing the PID Controller]]. In this case we make no attempts to precalculate the stability issues e.g. using bode plots to observe phase margin as a function of our bandwidth. Instead we use a trial and error principle based on our intuitive understanding of the tuning parameters. The problem with this approach is that lack of knowledge about the dynamics of the control plant (the robot) if we do not succeed. We have no theoretical foundation.\\ |
| The second approach is again to consider the control plant as a black box, but here we wish to estimate the characteristic transfer function of the plant. Recall that if we give an impulse to a filter we are given the filter characteristic by means of the impulse response. The analogy to control theory is to apply a step function and observe the transient response, which is illustrated on the figures in the following section “Introducing the PID Controller”. Depending on the order of the system we may need to apply a ramp or a parabola, but the principle is the same. By observing the rise time and settling time we may be able to estimate the transfer function and thereby to create a more solid foundation as to why the system is unstable and how large bandwidth we can obtain. This method is more complex since we need to be able to measure the transient response, which often requires expensive and precise equipment.\\ | The second approach is again to consider the control plant as a black box, but here we wish to estimate the characteristic transfer function of the plant. Recall that if we give an impulse to a filter we are given the filter characteristic by means of the impulse response. The analogy to control theory is to apply a step function and observe the transient response, which is illustrated on the figures in the following section [[#Introducing the PID Controller|Introducing the PID Controller]]. Depending on the order of the system we may need to apply a ramp or a parabola, but the principle is the same. By observing the rise time and settling time we may be able to estimate the transfer function and thereby to create a more solid foundation as to why the system is unstable and how large bandwidth we can obtain. This method is more complex since we need to be able to measure the transient response, which often requires expensive and precise equipment.\\ |
| The third method is to derive a mathematical model of the dynamics of the control plant and this can be somewhat complex. In the case of a balancing robot it is helpful to look at the mathematical modelling of an inverted pendulum and there next to add the physical dimensions of the balancing robot. When working with modelling we end up with non linear equations, which is a problem as our controllers are linear. One common method is to use the state space representation and then to make a linearization around the steady state point or the equilibrium. If our mathematical model is precise and close to the true physical model we obtain a great theoretical foundation for creating a stable system. This is exactly what Yorihisa Yamamoto(([[http://www.mathworks.com/matlabcentral/fileexchange/19147| Yorihisa Yamamoto's]])) has done and the result is outstanding. In our case however, the modelling is too comprehensive and we have seen several examples that prove a stable balancing robot. We therefore rely on our intuitive understanding and use method one to implement our solution. We have decided to use the PID controller as we have already obtained some practical/intuitive understanding of the tuning parameters during LAB4(([[http://wiki.aasimon.org/?id=marvin:lab4|Lab 4]])). | The third method is to derive a mathematical model of the dynamics of the control plant and this can be somewhat complex. In the case of a balancing robot it is helpful to look at the mathematical modelling of an inverted pendulum and there next to add the physical dimensions of the balancing robot. When working with modelling we end up with non linear equations, which is a problem as our controllers are linear. One common method is to use the state space representation and then to make a linearization around the steady state point or the equilibrium. If our mathematical model is precise and close to the true physical model we obtain a great theoretical foundation for creating a stable system. This is exactly what Yorihisa Yamamoto(([[http://www.mathworks.com/matlabcentral/fileexchange/19147| Yorihisa Yamamoto's]])) has done and the result is outstanding. In our case however, the modelling is too comprehensive and we have seen several examples that prove a stable balancing robot. We therefore rely on our intuitive understanding and use method one to implement our solution. We have decided to use the PID controller as we have already obtained some practical/intuitive understanding of the tuning parameters during LAB4(([[http://wiki.aasimon.org/?id=marvin:lab4|Lab 4]])). |
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