Motor position control theory

We consider Shannon sampling theorem for continuous-to-discrete transformation.

Our sampling rate is well designed as 1000 Hz. We consider control diagram for angle position as below. The output of this system is the actual rotation angle  , which is _a disturbed by human noise,  . _h The control block diagram is shown in Fig. 5-8. By combining human normal gait data as control command,  , with left summing point, _c we can get the angle error,  , as a controller input. _e

Fig. 5-8 Mode 1 control block diagram.

We use PID controller in this work. This controller has good stability and high reliability. Its structure is easy to implement and understood. A PID controller includes three data feedback processing system, which is proportional, integral and derivative control. Proportional control can enlarge or shrink error to track input signal, which can enlarge system over shoot. Integral control can gradually stack error, which can eliminate steady state error. However, integral control is too slow to handle instant system variation. In contrast, derivative control can estimate error variation in advance.

However, derivative control can be easily disturbed by high frequency noise. This disadvantage should be compensated by integral control.

The angle position control model would be easier to have two extra poles than common control system. To stabilize these two characteristics, we use two zeros from

PID to handle this problem. The PID controller structure yields the following two After deriving controller structure, we can derive plant structure. For rotation system, the mass is the moment of inertia. We estimate the system effective moment of inertia as

  parameters. Some information is from MAXON datasheet [42].

Table 5-1 Known parameters.

Although the BLDC is a three-phase motor, which means it is composed of three

Fig. 5-9 Schematic of system plant.

The transfer function of angle position over voltage for one of phases for BLDC

Analysis of the electric, including the back electromotive voltage , leads to

(5)

The Laplace transform of equation (6) is

3 2

In control theory, standard mass damper spring system (MCK system) has second order differential equation. Its transfer function standard form is

2 frequency (_n) and damping ratio (). These two properties play an important role in system response. Although variable of equation (8) has third order term, these properties still can be used because of our system has constants for common MCK system problem.

In our case, these two properties can be written as:

b t

Sorting equation (8) with two characteristic properties, the transfer function for the motor is readily found to be

2 amplifier circuit to control exoskeleton power unit directly, we should consider transfer function in discrete domain for digital controller such as computer or micro controller.

To discretize the transfer function for angle position over voltage, we must understand zero order hold, which is a common way in digital-to-analog conversion. Zero order hold is a mathematical model of the practical signal reconstruction. That is, it describes the effect of converting a discrete-time signal to a continuous-time signal by holding each sample value for one sample interval. To get this mathematical model, we constructed two-step functions. One is normal step function, and the other is step function with one sample interval delay. After subtracting these two functions, we get a constant within one sample interval, which means discrete signal in pre sample interval is held to present sample interval. As a result,

1 e ^sT 1 e ^sT

s s s

 

   (13)

To obtain discrete transfer function, which is so called z domain transfer function, we should consider new system dynamics (NSD), which is the multiplication of zero order hold (equation (13)) and continuous transfer function (equation (12)).

2

There is a useful transformation matching table between time domain, s domain, which is continuous frequency domain, and z domain, which is digital domain [44]. To use this

table for quickly getting transformation equation representation on z domain, we must do partial fraction expansion with continuous transfer function as

2 2

The first and second terms can be directly transformed from s domain into z domain by applying transformation table. For the last term, it is difficult to transform because our parameter on the damping ratio is over one, which implies an over-damping system.

There is only under-damping system for table, and its component has 1² term. If we directly substitute s domain last term into transform pair, it would show imaginary term in our transfer function, which is difficult to programmable of our transfer function for control root locus design. Fortunately, this useful table also can match time domain transform to z domain transformation pair. Therefore, we can do inverse transform with last term of partial fraction expansion on s domain transfer function.



 



Transferring the above time domain equation into z domain gives

 

denominator form of denominator has three poles. We define these three poles as

After reduction of fractions to a common denominator, we can get

   

Then we can sort each z polynomial term on numerator. After sorting with simple linear combination, we can get the discrete transfer function on rotated angle over voltage,

We have already known all coefficients for above equation by parametric Table 5-1 as previous described. Substituting these known parameters into equation (24) yields

2