Compensation in Control System | Lag lead Compensation

Compensation in control system design shapes loop gain and phase so the closed-loop response can meet defined tracking, disturbance, noise, robustness and actuator constraints. A compensator must be designed with the plant and the complete control system, because a pole-zero network has no guaranteed effect in isolation.

Purpose of Compensation

1. Shape the loop transfer function to meet measurable performance and robustness targets.
2. Move closed-loop poles or frequency-response margins where the plant permits stabilisation.
3. Adjust transient response, including rise time, settling time and overshoot.
4. Increase low-frequency loop gain to reduce selected steady-state tracking or disturbance errors, while checking the resulting stability margins.
5. Add poles and zeros that change loop magnitude and phase across frequency. The final design must also respect noise amplification, model uncertainty and actuator limits.

Methods of Compensation

1. A compensator placed in the forward path between the error detector and plant is called series compensation.

Series Compensator

1. A dynamic element placed in a feedback path is called feedback compensation.

Feedback Compensator

1. A design may combine series and feedback elements. The term load compensation is not a universal name for every such combination and should be defined by the block diagram.

Load Compensator. A compensator can be implemented in analogue electronics or digital control software. Mechanical and hydraulic implementations also exist, and one design may combine technologies. Passive RC networks provide simple lead and lag transfer functions, while active circuits and digital controllers can realise the same pole-zero forms with gain.

Phase Lead Compensation

A first-order lead network has a left-half-plane zero at a lower break frequency than its left-half-plane pole. For compensation in a control system, a lead compensation network adds positive phase between those break frequencies and changes loop magnitude.
For the common form C(s) = K(s + z)/(s + p), lead action requires 0 < z < p. The zero is closer to the origin than the pole, but neither must be at the origin.
The circuit below is a passive phase lead compensation network.

Phase Lead Compensation Network
From above circuit we get,

Equating above expression of I we get,

Now let us determine the transfer function for the given network and the transfer function can be determined by finding the ratio of the output voltage to the input voltage.
So taking Laplace transform of both side of above equations,


On substituting the α = (R₁ +R₂)/ R₂ and T = {(R₁R₂) /(R₁ +R₂)} in the above equation.
Where, T and α are respectively the time constant and attenuation constant, we have

The above network can be visualized as an amplifier with a gain of 1/α. Let us draw the pole zero plot for the above transfer function.

Pole Zero Plot of Lead Compensating Network

In the plotted normalised form, the zero at -1/T is closer to the origin than the pole at -1/(αT), with 0 < α < 1. The network therefore contributes positive phase between its break frequencies.
Substitute s = jω into the transfer function to obtain its phase:
The frequency of maximum phase lead follows by differentiating the phase with respect to frequency and setting the derivative to zero:

Here, θ_m is the maximum phase lead angle. The magnitude at that frequency depends on the transfer-function normalisation, so use the same α convention as the displayed equations when applying the result for θ_m.

Effect of Phase Lead Compensation

1. The velocity constant K_v changes only if the compensator changes the relevant low-frequency loop gain.
2. The added positive phase can increase phase margin when the lead region is placed around gain crossover.
3. Gain crossover and bandwidth often move higher, subject to the plant and compensator gain.
4. The closed-loop response may become faster, but high-frequency noise and control effort can increase.

Advantages of Phase Lead Compensation

Typical reasons to use phase lead compensation include:

1. It can add phase near crossover and permit a higher crossover frequency.
2. It can improve damping and transient response when the design increases suitable stability margins.

Disadvantages of Phase Lead Compensation

Important trade-offs of phase lead compensation include:

1. Steady state error is not automatically improved, and higher high-frequency gain can amplify sensor noise and demand more actuator effort.

Phase Lag Compensation

A first-order lag network has a left-half-plane pole at a lower break frequency than its left-half-plane zero. In compensation in control system design, lag compensation is commonly used to increase low-frequency gain relative to high-frequency gain.
For C(s) = K(s + z)/(s + p), lag action requires 0 < p < z. The pole is closer to the origin than the zero. The network contributes negative phase between its break frequencies, so designers usually place that region below crossover when preserving phase margin matters.
The following circuit implements phase lag compensation.

Phase Lag Compensating Network
We will have the output at the series combination of the resistor R₂ and the capacitor C.
From the above circuit diagram, we get

Now let us determine the transfer function for the given network and the transfer function can be determined by finding the ratio of the output voltage to the input voltage.
Taking Laplace transform of above two equation we get,

On substituting the in the above equation (Where, T and β are respectively the time constant and DC gain), we have

The above network provides a high frequency gain of 1 / β. Let us draw the pole zero plot for the above transfer function.

Pole Zero Plot of Lag Network
In the plotted form, the pole lies closer to the origin than the zero. This ordering produces negative phase between the pole and zero break frequencies.
Substitute s = jω into the transfer function to obtain the phase:
The frequency of maximum phase lag follows by differentiating the phase with respect to frequency and setting the derivative to zero:

Here, θ_m denotes the maximum phase lag angle. Choose β from the required low-frequency gain ratio and the allowed phase effect. There is no general requirement that β exceed 10.

Effect of Phase Lag Compensation

1. Gain crossover frequency may move lower when the lag network attenuates the loop near crossover.
2. Bandwidth often decreases.
3. Phase margin can decrease because the network adds negative phase, although crossover relocation can alter the final margin.
4. A lower bandwidth usually gives a slower response with longer rise and settling times.

Advantages of Phase Lag Compensation

Typical reasons to use phase lag compensation include:

1. It raises low-frequency gain relative to high-frequency gain.
2. It can reduce selected steady-state errors and improve low-frequency disturbance rejection.

Disadvantages of Phase Lag Compensation

Important trade-offs of phase lag compensation include:

1. It can reduce bandwidth, slow the response and add phase lag that must be included in the stability-margin check.

Phase Lag Lead Compensation

A single lead or lag network may not meet both low-frequency accuracy and crossover-region phase requirements. A lag-lead design combines a lag section for low-frequency gain shaping with a lead section for phase and crossover shaping. The complete compensated loop must still be checked for stability, robustness, noise and actuator limits.
The following circuit is a phase lag- lead compensation network.

Lag Lead Compensating Network
Now let us determine transfer function for the given network and the transfer function can be determined by finding the ratio of the output voltage to the input voltage.

On substituting the αT₁ = R₁C₁, R₂C₂ = βT₂, R₁R₂C₁C₂ = αβT₁T₂ and T₁T₂ = R₁R₂C₁C₂ in the above equation (where T₁, T₂ and α, β are respectively the time constants and attenuation constants). We have

Let us draw the pole zero plot for the above transfer function.

Pole Zero Plot Lag Lead Network
The lag section places its pole below its zero, while the lead section places its zero below its pole. Their combined lag-lead compensation phase can be negative, positive or near zero at a given frequency. The design depends on all four break frequencies and the gain.

Advantages of Phase Lag Lead Compensation

Typical reasons to combine lag and lead compensation include:

1. The lead section can add phase near crossover while the lag section shapes low-frequency gain.
2. The combined network can meet accuracy and transient-response targets that one first-order section cannot meet alone.