Feedback Characteristics

In this section, we describe how to determine the stability, transient response, and steady- state error for feedback systems. For the remainder of this section, we use_R(_P), andθ(_P) to denote, respectively, the radius and angle (in radians) of the pole_P in polar-complex form, and we use_Pm to denote the closed-loop pole that is farthest from the origin.

Stability. Recall from Section 2.4.1 that a system is stable if every bounded reference input value causes the system’s steady-state error to be bounded. The test for stability is based on the location of the closed-loop poles in the complex plane: a system isstable if all closed-loop poles are within the unit circle in the complex plane, i.e., _Pm <1. A system is unstable if

any closed-loop pole is outside the unit circle, i.e., _Pm>1. A system is marginally stable, in

which case the output neither converges nor diverges, if at least one pole is on the unit circle and no pole is outside of the unit circle, i.e., _Pm = 1. Thus, in the system from Figure 2.6,

ifa1 = 2 and a2 = 1, then system is stable if b ∈(0,2/3), the system is marginally stable if b= 0 or b= 2/3, and the system is unstable if b >2/3 or b <0.

Transient response. Two of the most important types of feedback systems arefirst- and

second-order systems. A feedback system is considered to be afirst-order system if it has one closed-loop pole and a system is considered to be asecond-order system if has two closed-loop poles. These two types of systems are important because both have a set of simple formulas for determining their transient response. (Higher-order systems often have a transient response that is too complex to determine without approximation.)

The transient response of a system is usually evaluated by examining the behavior of the output when the system incurs astep input, i.e., the reference input value suddenly increases to a given value. Since feedback systems use previous results to predict future results, a step reference input value represents the worst-case scenario—a sudden change from one value to

another. For a first-order system, the transient response is characterized by its settling time

(i.e., the time it takes for the output to attain and stay within 2% of its steady-state value) and whether the output “overshoots” its final value. For example, such a scenario is depicted by the curve labeled under-damped in Figure 2.5. For a second-order system, the transient response is characterized by its settling time and whether it is under-damped, over-damped, orcritically-damped (all three are depicted in Figure 2.5.) The settling time (where time is measured in terms of samples) of the system is given by the standard formula

−4

ln(_R(_Pm))

.

(This formula has a ceiling because time is discrete.)

For first-order systems, the output overshoots its final value if _Pm < 0. For first-order

systems, it is typically undesirable for the output to overshoot its final value. (This is not the case for second-order systems since for second-order systems overshooting may be the only way to achieve the specified settling time. For most first-order systems, it is possible to achieve a desired settling time without overshooting the output.)

If a second-order system is over-damped, then the output will never overshoot the reference input value for a step input. If a second-order system is under-damped, then the output will overshoot the reference input value for a step input. For under-damped systems, thepercent overshoot is an additional characteristic of transient response. If a second-order system is critically-damped, then the settling time is as small as possible without causing the output to overshoot the reference input value. Whether a system is under-, over-, or critically- damped depends on the location of the closed loop poles. If both poles are unique, real, and positive, then the system is over-damped. If both poles have the same radius, are real, and are positive, then the system is critically-damped. Otherwise, the system is under-damped. For under-damped systems, the percent overshoot is given by

whereζ is a value called thedamping ratio and is given by

ζ = q −ln(R(Pm))

θ(_Pm)2+ln2(_R(_Pm))

For example, in the system from Figure 2.6, ifa1 = 2 anda2 = 1, then the system is under- damped for any value of b _∈ (0,2/3). Alternatively, if b = 1, a1 ≈ 0.102, and a2 ≈0.3035, then the system is critically-damped and has a settling time of 6 time units. Finally, ifb= 1, a1 = 1.4008, and a2 = 1.60238, then the system is under-damped, the settling time is 5 time units, and the percent overshoot is approximately 10.3%.

Steady-state error. Finally, we turn our attention to steady-state error. The steady-state error of a system is measured based on the system’s response to a step and/or a ramp input. The ramp input simulates a reference input value that constantly increases by a rate of_T per time unit. The steady-state error for a system is determined by using thefinal value theorem, which states that ifE(z) is thez-transform of a system’s error, then the steady state error is given by

lim

z→1 z₋1

z E(z). (2.6)

Since E(z) can be defined as

E(z) =R(z)₋M(z), (2.7)

whereR(z) is the reference input value andM(z) is the output, andM(z) can be defined as

M(z) =E(z)G(z),

whereG(z) is the open-loop transfer function, we get

E(z) = R(z)

1 +G(z). (2.8)

Since the z-transform of the step input is given by R(z) = _z−z₁, from (2.6) and (2.8), we

can derive the steady-state error of a system in response to a step input as

lim

z→1 1

1 +G(z). (2.9)

Since thez-transform of the ramp input is given byR(z) = _(zz₋T₁₎2, from (2.6) and (2.8), we

can derive the steady-state error of a system in response to a ramp input as

T

limz→1(z−1)G(z)

. (2.10)

The value to which the (z₋1) term is raised in the denominator of the open-loop transfer function, G(z), is called the system type, and it is used to quickly determine if the steady- state error is zero, some non-zero constant, or _∞. Specifically, if G(z) has a system type of zero (i.e., it has no (z₋1) terms in its denominator), then the system has a steady-state error of some constant value for the step input and _∞ for any ramp input. If G(z) has a system type of one (i.e., it has one (z₋1) term in its denominator), then the system has a steady-state error of 0 for the step input and a constant for any ramp input. Finally, ifG(z) has a system type of two (i.e., it has two (z₋1) terms in its denominator), then the system has a steady-state error of 0 for any step or ramp input.

For example, in the system depicted in Figure 2.6, since there is one (z₋1) term in the denominator of G(z), the system type is one. Thus, it has a steady-state error of zero for a step input and a constant steady-state error for any ramp input. Specifically, if b = 0.5, a1 = 2 and a2 = 1, then the steady-state error for the ramp response is 2T. Alternatively, if b = 1, a1 ≈ 0.102, and a2 ≈ 0.3035, then the steady-state error for the ramp input is approximately 3.295_T. Finally, ifb= 1,a1 = 1.4008, anda2 = 1.60238, then the steady-state error is approximately 0.624_T.