Embedded Systems November 2000 Vol13_12

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SPECTRA e FIGURE 1 Control Disturbance Command + ~·6~ Feedback square wave and examine the out- put, it will always exhibit a fixed delay. We can quantify the delay in degrees by noticing at what point in the pe riodic cycle it begins and con- tinues as the t·eference signal does. Just because a delay is fixed in time- does not mean that phase lag is fixed as well. For example, if our fixed delay is half a second and we input a square wave with a pe riod of two ec- onds, the phase lag is 90 degrees. But, if we input a square wave with a o ne second pe ri od instead , th e phase lag becomes 180 degrees. This is were th e trouble a rises. Zero phase lag is impossible in a system such as the one we are describ- ing, because each of the system's ele- ments conu·ibute. The motor itself contributes an L/ R time constant; cur- rents and velocity do not change instantaneously with the application of power. The feedback system creates delay. Fil ter , drivers, and processes such as integration also cause delay. A system's stabili ty and bandwidths depend on the choices you make in each of these areas. Step response As with other signal processing activi- ti es, control theory is based in the fre- quency domain and reflected in the time domain . But in control th eory, perhaps more than signal processing, time domain responses are important and often used to judge the perfor- mance of a system. Here, we' ll use the step response to describe a system in practical or industrial applications becau e it's easy to measure with an oscilloscope. Typically, th e stability of a system measured by its step response. Step response is a sys tem's response to any instantaneous change at the input. The change might be a com- mand to a motor to move to a new location and stop: a square wave. The characte ri stic we are most con- cerned with is ove rshoot. Overshoot is th e ra ti o of th e peak of the re ponse to th e command. The amo unt of acceptable ove rshoot varies with the application. System stabi lity is dependent on delays and phase. In order for a con- trol system to work, the feedback must be negative at the summing node you see in Figure l. If it is not, the ystem will become forward-driven, the error will be amplified by the gains applied, and the system wi ll go out of control. Anyone who has ever reversed the encoder wiring on a servo knows what this is like. Building up enough phase lag throughout the system can produce a similar result. When this is the case, the feedback will appear to go posi- tive at a certain frequency-phase lags are frequency depende nt-and 22 NOVEMBER 2000 Embedded Systems Programming Response Machine destabi lize the system. (Remember, all it takes for the feedback to appear positive at the summing node is a phase-lag of 180 degrees.) Such a sys- tem can be very sensitive to noise, since these are usually high frequency elements, and may cause the applica- tion to perform poorly even at low frequencies. As I describe the compo- nents of the PID algorithm, I will indicate how they affect the phase in the system. Proportional control Proportional control, or P control, is th e simplest control technique. I t takes the sum of the negative feedback and the command as its inpu t. The error is the diffe rence between the commond and the actual position. The control applies a gain and the result is sourced to the drive: U[t] = K P * e[t ] P control is easy to work with , but it makes no compensation fo r distur- bances such as temperature and fric- tion, among others. These long-term errors are sometimes referred to as DC errors. The phase lag of a proportion- al gain is zero degrees. To tune a system such as this, apply the square wave step command and raise the gain until it becomes unsta-

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