Page 372 - The Mechatronics Handbook

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FIGURE 18.12 System stability.

on the thermostat, the output remains off. Once room temperature has increased to the setpoint plus
half the deadband, then the cooling system output goes to fully on. As the room cools, the output stays
fully on until the temperature reaches the setpoint minus half the deadband. At this point the cooling
system output goes fully off.

18.14 System Response

Sensors and actuators respond to inputs that change with time. Any system that changes with time is
considered a dynamic system. Understanding the response of dynamic systems to different types of inputs
is important in mechatronic system design. The most important concept in system response is stability.
The term stability has many different definitions and uses, but the most common definition is related to
equilibrium. A system in equilibrium will remain in the same state in the absence of external disturbances.
A stable system will return to an equilibrium state if a “small” disturbance moves the system away from
the initial state. An unstable system will not return to an equilibrium position, and frequently will move
“far” from the initial state.
Figure 18.12 illustrates three stability conditions with a simple ball and hill system. In each case an
equilibrium position is easily identified—either the top of the hill or the bottom of the valley. In the
unstable case, a small motion of the ball away from the equilibrium position will cause the ball to move
“far” away, as it rolls down the hill. In the stable case, a small movement of the ball away from the equi-
librium position will eventually result in the ball returning, perhaps after a few oscillations. In the third
case, the absence of friction causes the ball to oscillate continuously about the equilibrium position once
a small movement has occurred. This special case is often known as marginal stability, since the system
never quite returns to the equilibrium position.
Most sensors and actuators are inherently stable. However, the addition of active control systems can
cause a system of stable devices to exhibit overall unstable behavior. Careful analysis and testing is required
to ensure that a mechatronic system acts in a stable manner. The complex response of stable dynamic
systems is frequently approximated by much simpler systems. Understanding both first-order and second-
order system responses to either instantaneous (or step) changes in inputs or sinusoidal inputs will suffice
for most situations.

18.15 First-Order System Response

First-order systems contain two primary elements: an energy storing element and an element which
dissipates (or removes) energy. Typical first-order systems include resistor–capacitor filters and resistor–
inductor networks (e.g., a coil of a stepper motor). Thermocouples and thermistors also form first-order
systems, due to thermal capacitance and resistance. The differential equation describing the time response
of a generic first-order system is

dyt() 1
------------ ---- y t() = ft()
+
dt τ

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