Phase Lead/Lag Controllers

Interestingly, a Phase Lead Controller is much like a PD Controller in that it acts to speed up response and increasing stability by shifting the Root Locus to the left. Furthermore, a Phase Lag Controller is much like a PI Controller in that it attempts to eliminate steady-state error by shifting the Root Locus to the right.

For myself, I like to think of Phase Lead/Lag Controllers as hybrid Controllers. In the case of a Phase Lead Controller, we start with the following general equation but set p greater than z:

𝐶_{𝐿𝑒𝑎𝑑/𝐿𝑎𝑔}(𝑠) =_{𝑠 + 𝑝}𝑠 + 𝑧

If we arbitrarily set z equal to 1 and p equal to 10 and compare it to a PD Controller with unity gains we get the following Bode Diagram:

Figure 4.0 – Bode Diagram of Phase Lead Controller vs PD Controller

Phase Lead Controller PD Controller

64 When comparing the Bode Diagrams in Figure 4.0 you’ll notice that both Magnitude plots start to ramp at 20dB per decade at 1rad/s or the location of the zero. However, because of the pole in the denominator of the Phase Lead Controller, its Magnitude plot levels off at 10rad/s where the Magnitude plot for the PD Controller continues to ramp indefinitely. You’ll also notice one of the drawbacks of the Phase Lead Controller in the reduction in magnitude at low frequencies. Furthermore, the Phase Lead Controller produces a hump in its Phase plot centered in between its zero and pole locations, starting at 0 degrees and ending at 0 degrees where the Phase plot for the PD Controller starts at 0 degrees and ramps up to 90 degrees where it levels off. The length, height and centre point of this hump is the focus of Phase Lead Controller design in the effort to increase Phase Margin.

Conversely, a Phase Lag Controller produces a valley centered about its zero and pole locations (Figure 4.1). This is the consequence of having z greater than p. This valley is actually the main drawback of Phase Lag Controller design. Instead, the focus of Phase Lag Controller design is to balance the side- effects of the phase lag distortion by appropriately designing the size and location of this valley in order to boost the gain of the system at low frequencies, thereby improving steady-state tracking performance and disturbance rejection. As shown in Figure 4.1, both Controllers boost low frequency gain.

Figure 4.1 – Bode Diagram of Phase Lag Controller vs. PI Controller PI Controller

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4.1.2 Introduction to Gain and Phase Margins

The concept of Gain and Phase Margins are best explained with the use of an example. Consider the following closed-loop system:

Figure 4.2 – Block Diagram of Example Closed-Loop System

Using Matlab’s margin(sys) function for an open-loop system where: 𝑠𝑦𝑠 = 𝐼(𝑠) ∙ 𝐶(𝑠) ∙ 𝑃(𝑠) And for our example:

𝑠𝑦𝑠 = 1.5 ∙ 1 𝑠 + 1∙

2 𝑠2_{+ 2𝑠 + 1}

We get the Bode Diagram shown in Figure 4.3. Matlab tells us that the Gain Margin is 8.52dB at the Phase Crossover Frequency 𝜔𝑐𝑝 of 1.73rad/s and I’ve added an arrow to indicate this. The Gain Margin

is calculated using the following equation:

𝐺𝑎𝑖𝑛 𝑀𝑎𝑟𝑔𝑖𝑛 = 0𝑑𝐵 − 𝐺𝑎𝑖𝑛

Where Gain is the distance between the curve and 0dB at the Phase Crossover Frequency 𝜔𝑐𝑝. Notice

that if the Gain Margin is below 0dB at the Phase Crossover Frequency the Gain is negative resulting in a positive Gain Margin and if it’s above 0dB the Gain is positive, resulting in a Gain Margin that’s negative. If the Phase plot never dips below -180°, the Gain Margin is considered to be infinite.

Matlab has also calculated our Phase Margin to be 41.7° at the Gain Crossover Frequency 𝜔𝑐𝑔

of 1.04 rad/s and I’ve added an arrow to indicate this. The Phase Margin is calculated using the following equation:

𝑃ℎ𝑎𝑠𝑒 𝑀𝑎𝑟𝑔𝑖𝑛 = 180° + 𝑃ℎ𝑎𝑠𝑒

Where Phase is the value of the curve at the Gain Crossover Frequency 𝜔𝑐𝑔. Notice that if the Phase

Margin is above -180° the Phase Margin is positive and if the Phase Margin is below -180° the Phase Margin is negative. If the Magnitude plot never dips below 0dB, the Phase Margin is considered to be infinite.

.

66 Figure 4.3 – Bode Diagram of Example Closed-Loop System

When designing a Controller, the conservative rule of thumb for stability margins are: 𝐺𝑎𝑖𝑛 𝑀𝑎𝑟𝑔𝑖𝑛 ≥ 10𝑑𝐵 and

𝑃ℎ𝑎𝑠𝑒 𝑀𝑎𝑟𝑔𝑖𝑛 ≥ 60° However, some texts promote typical stability margins as low as:

𝐺𝑎𝑖𝑛 𝑀𝑎𝑟𝑔𝑖𝑛 ≥ 6𝑑𝐵 and 𝑃ℎ𝑎𝑠𝑒 𝑀𝑎𝑟𝑔𝑖𝑛 ≥ 30°

The idea behind the Gain and Phase Margins are that they provide a factor of safety. If you were walking along the a cliff, would you feel comfortable with walking right along the edge, or would say 10 feet from the edge make you feel more comfortable? With experience you’ll know when you can get away with less conservative stability margins but for now it would probably be wise to stick with the conservative approach.

Gain Margin

Phase Margin 𝜔_𝑐𝑔

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4.1.3 Phase Lead Controller Design

Following Figure 4.4, if your Proportional Controller produces too much overshoot but the steady state error is acceptable, apply Phase Lead Control.

Figure 4.4 – Flowchart for Phase Lead/Lag Controller Design

Following the procedure laid out by Prof. Dan Davison in his SYDE 352 Course Notes [13], our objective is to satisfy the specifications using:

𝐶(𝑠) = 𝐾 ∙ 𝐶_{𝐿𝑒𝑎𝑑}(𝑠) = 𝐾 1 √𝑎 1 + 𝑎𝑇𝑠 1 + 𝑇𝑠 Where: K = Controller gain

a = ratio of pole position to zero positon (a > 1) T = pole time constant (T > 0)

(4.0)

Apply Phase Lead Control Too much Overshoot?

Too much Steady State Error?

Apply Phase Lag Control

Yes

No

Apply Phase Lead and Lag Control Yes

68 Which results in the following Bode Diagram:

Figure 4.5 – Bode Diagram of General Phase Lead Controller

Step 1: Sketch the Bode plot of the open-loop system. As shown in Figure 2.0, our open-loop system