Closed-loop performance objectives

Figure 4-5: Unity-feedback control system

In addition to the internal stability, a closed-loop system should provide several other performance requirements such as robustness, asymptotic tracking, disturbance attenuation and noise rejection, as stated before. To investigate the performance objectives of a closed loop control system, one should study the relationship between a set of reference exogenous signals and their corresponding steady-state error. For simplicity assume that ܨ = 1 (i.e. the unity-feedback loop) as shown in Figure 4-5. Here ݎ is the system reference

input,ݕ is the system output and ݁ is the control error defined as the difference between the ideal response,ݎ, and the measured response, ݕ_௠:

݁ = ݎ− ݕ௠

݁= ݎ− ݕ − ݊ _(4-19)

The output from the controller is

ݑ = ܭ(ݎ− ݕ − ݊) (4-20)

and the output from the plant is

ݕ = ܩ(ݑ + ݀) (4-21)

Substitution of(4-20)into(4-21)yields

ݕ = ܩܭ(ݎ− ݕ − ݊) + ܩ݀ (4-22)

Here,ܩ݀is the effect of the (actuator) disturbance on the output.

Hence, the closed-loop response can be written in terms of three exogenous inputsݎ,݀ ,and ݊ as

ݕ =_{(1 + ܩܭ)}1 [ܩܭ ݎ+ ܩ ݀ − ܩܭ ݊] (4-23)

Closed-loop performance could be investigated by focusing on the response of the system to the three exogenous inputs ݎ,݀, and ݊ (Assadian F., 2011). For example, the closed-loop transfer function from the actuator disturbance (ܩ݀) to the plant outputݕ is called sensitivity function ܵ.

ܵ =_{1 + ܩܭ =}1 _{1 + ܮ}1 (4-24)

where ܮ denotes the (open) loop transfer function, ܮ = ܩܭ. The closed-loop transfer function from reference input ݎ to the plant output ݕ is called

complementary sensitivity functionܶ:

From definition of(4-24)and(4-25), one can conclude that:

ܵ+ ܶ = 1 (4-26)

One way to quantify how sensitive ܶ is to variation in ܩ is to take the limiting ratio of a relative perturbation inܶ (i.e., ∆ܶ/ܶ) to a relative perturbation in ܩ (i.e., ∆ܩ/ܩ). Considering of ܩ as a variable and ܶ as a function of it, we get

lim ∆ீ→଴ ∆ܶ/ܶ ∆ܩ/ܩ = ݀ܶ ݀ܩ ܩ ܶ = 1 1 + ܩܭ = 1 1 + ܮ = ܵ (4-27)

In this way, ܵ is also the sensitivity of the closed-loop transfer function ܶ which defines howܶ changes as a result of a change in ܩ.

Employing the above definitions forܵ and ܶ, the plant output (Eq.(4-23)) can be written in terms ofܵ and ܶ as:

ݕ = ܶ ݎ+ ܵܩ݀ − ܶ ݊ (4-28)

The first term in(4-28)is the closed-loop function between control reference and plant output (so called, tracking performance), while the second term is the effect of the disturbance (so called, disturbance attenuation performance) and the third term is the effect of the measurement noise (so called, noise rejection performance) on the output respectively.

Similarly, the control error݁ can be written as:

݁= ݎ− ݕ− ݊ = ܵݎ− ܵܩ݀ − ܵ݊ (4-29)

and the corresponding controller output (actuator input) signal ݑ in terms of ܵ andܶ is:

ݑ = ܭܵݎ− ܭܵܩ݀ − ܭܵ݊ (4-30)

The closed-loop transfer function from reference input ݎ to the actuator input ݑ is called Youla parameterܻ:

which is the measure of actuator effort (Assadian F. , 2011). It is concluded from Eqs. (4-27), (4-28), (4-29), and (4-30) that all control performance problems can be summarised in terms of ܵ, ܶ, ܻ or some combination of them. The main control design issue is a trade-off between making ܵ small and making ܶ small: Ideally we want ܵ small to obtain the benefits of feedback (good robustness as well as small control error for command and disturbances), and we want ܶ equal to one for good command following at low frequencies and small to avoid sensitivity to noise which is one of the disadvantages of feedback at high frequencies. Moreover, from a practical point of view, we are also interested in keeping ܻ as small as possible. As shown in Eq. (4-26), these requirements cannot be met simultaneously, asܵ and ܶ are related to each other by ܵ+ ܶ = 1. Fortunately, the conflicting design objective mentioned above are generally in different frequency ranges and the objectives can be fulfilled by using a large

loop gain |ܮ| at low frequencies below crossover, and a small gain at high frequencies above crossover.

To study closed-loop performance over a range of frequencies, the frequency response of the loop transfer functions ܮ(݆߱), ܶ(݆߱) and ܵ(݆߱) can be employed. One of the advantages of the frequency domain analysis compared to the time domain analysis, is that it considers the system over a broader class of signals (sinusoids of any frequency). This makes it easier to characterise feedback properties, and in particular system behaviour below the crossover (bandwidth) region.

The traditional performance measures in frequency domain are the Gain Margin (ܩܯ ) and Phase Margin (ܲܯ ), which can be used as control design criteria (Ogata, 2010). Defining the phase crossover frequency ߱_ଵ଼଴, to be the frequency at which the phase angle of the open loop transfer function ܮ equals to −180° (where the Nyquist curve of ܮ(݆߱) crosses the negative real axis between -1 and 0, as shown in Figure 4-4,b ), the gain margin is defined as:

ܩܯ =_|ܮ(݆߱1

For a stable system, the ܩܯ indicates how much the gain |ܮ(݆߱)| can be increased before the closed-loop system becomes unstable (see Figure 4-4,b). Theܩܯ is thus a direct safeguard against steady-state gain uncertainty (error). The phase margin is defined as:

ܲܯ = ∠ܮ(݆߱௖) + 180° (4-33)

Definition2: the gain crossover frequency߱_௖ is the frequency at which |ܮ(݆߱)| first crosses 1 from above, that is:

|ܮ(݆߱௖)| = 1 (4-34)

The phase margin is the amount of additional phase lag (negative phase) which can be added to the loop at frequency ߱_௖ before the phase at this frequency becomes −180° which corresponds to closed-loop instability, as shown in Figure 4-4,a. Therefore, for a minimum phase system to be stable, the phase margin should be positive (see Figure 4-4,b). The phase margin is a direct safeguard against time delay uncertainty: the system becomes unstable if we add a time delay of:

ߠ௠ ௔௫ = ܲܯ /߱௖ (4-35)

where ߠ_{௠ ௔௫} is the maximum time delay in sec (if ߱_௖ is in rad/sec and ܲܯ is in rad).

From the above arguments, we see that gain and phase margins provide stability margins for gain and delay uncertainty. In short, the gain and phase margins are used to provide the appropriate trade-off between performance and stability. As a common rule of thumb, for achieving a satisfactory performance, the phase margin should be between 30° and 60°, and the gain margin should be greater than 2 (6 dB) (Ogata, 2010).

Interestingly, the gain and phase margins are closely related to the peak values of ܵ(݆߱) and ܶ(݆߱) and are therefore useful in terms of performance (Assadian F., 2011). Define maximum peaks sensitivity and complementary sensitivity functions as

ܯௌ= max_ఠ |ܵ(݆߱)|; ܯ் = max_ఠ |ܶ(݆߱)| (4-36)

the relationship between these maximum peak and the gain and phase margins are (Skogestad & Postlethwaite, 2007)

ܩܯ ≥_ܯܯௌ ௌ− 1 ; ܲܯ ≥ 2 sin ିଵ_൬ 1 2ܯௌ൰≥ 1 ܯௌ [rad] (4-37) ܩܯ ≥ 1 +_ܯ1 ் ; ܲܯ ≥ 2 ݏ݅݊ ିଵ_൬ 1 2ܯ்൰≥ 1 ܯ் [ݎܽ݀] (4-38)

For example, with ܯ_ௌ= 2 we are guaranteed ܩܯ ≥ 2 and ܲܯ ≥ 29.0° and similarly, with ܯ_் = 2 we have ܩܯ ≥ 1.5 and ܲܯ ≥ 29.0°. Therefore requiring ܯௌ< 2 implies the common rule of thumb ܩܯ ≥ 2 and ܲܯ ≥ 30°. Typically it is required that ܯ_ௌ is less than about 2 (6dB) and ܯ_் is less than about 1.25 (2dB). A large value of ܯ_ௌ and ܯ_் (larger than about 4) indicates poor performance as well as poor robustness.