Root Locus Techniques

Design via Root Locus

• Root locus techniques are discussed

Root Locus Techniques

• Root locus is graphically presenting closed-loop poles as a system parameter k is varied
• Root locus is the graph of all possible roots of the equation when K is the variable parameter
• The root locus provides information about the stability and transient response of feedback control systems
• Zeros of T(s) are zeros of G(s) and poles of H(s)
• Poles of T(s) depends on gain K
• The closed-loop control function (CLCF) is a function of K
• Root Locus graphically shows poles of T(s) as K varies

Evaluation of a Complex Function via Vectors

• Any complex number, σ + jω, described in Cartesian coordinates can be graphically represented by a vector
• M is the product of the zero lengths divided by the product of the pole lengths
• Θ is the sum of zero angles minus the sum of the pole angles

Defining the Root Locus

• Explains defining a root locus using a camera system as an example
• Gain less than 25, over-damped
• Gain = 25, critically damped
• Gain over 25, under-damped
• Stable system, as no pole on right-hand plane
• During underdamped, real parts are same; so settling time (which is related to real part) remains the same
• Damping frequency (imaginary part) increases with gain, resulting in reduction of peak time

Properties of Root Locus

• The closed-loop transfer function T(s) = KG(s) / 1+KG(s)H(s)
• So is a pole if 1 + KG(so)H(so) = 0 which means KG(so)H(so) = −1 = 1 angled at (2k + 1)180° where k = 0, ±1, ±2, ...
• |KG(s)H(s)| = 1 means that K = 1 / |G(s)||H(s)|
• The sum of zero angle minus the sum of pole angle equals 56.31°+71.57°-90°-108.43° = -70.55°
• Since it is not a multiple of 180º, -2 + j3 is not in the root locus (can't be a pole for some value of K)

Sketching the Root Locus

• Number of branches: Equals the number of closed-loop poles
• Symmetry: Symmetrical about the real axis (conjugate pairs of poles, real coefficients of the characteristic equation polynomial)
• Real axis segments: For K > 0, root locus exists to the left of an odd number of real axis poles and/or zeros (angle condition)
• Start and end points: The root locus begins at finite and infinite poles of G(s)H(s) and ends at finite and infinite zeros of G(s)H(s)
• Asymptotes: The root locus approaches straight lines as asymptotes as the locus approaches infinity, where the equation of the asymptotes is: σa = (∑ finite poles – ∑ finite zeros) / (≠ finite poles - ≠ finite zeros)
• θa= ((2k + 1)π) / (≠ finite poles - ≠ finite zeros) for k = 0, ±1, ±2, ...

Sketching the Root Locus: Example

• Explains how to sketch a root locus for a system shown in a figure by calculating asymptotes to find real axis intercept
• There are four poles and one finite zero, since Root locus begins at poles and ends at zeros with three zeros at infinity (at the ends of the asymptotes)

Refining the Sketch

• Real-Axis Breakaway and Break-in Points shows the breakaway point (leave the real axis) and break-in point (return to the real axis)
• Breakaway point: at maximum gain on the real axis between -2 and -1
• Break-in point: at minimum gain (increases when moving towards a zero) between +3 and +5

Finding Breakaway and Break-in Points

• One method to find break-in and break-away points is a variation on the transition method
• Breakaway and break-in points satisfy the relationship: ∑ (1 / σ +Zi) = ∑ (1 / σ +Pi)
• From the example: σ = −1.45 and σ = 3.82 (negative of zero and pole values of G(s)H(s))
• Method 2 utilizes the following equal zero: (11σ² – 26σ – 61) / (σ² – 8σ + 15)² = 0

Imaginary-Axis Crossing

• Stability needs system's poles in the left half-plane up to a particular value of gain K
• For a system: solve for the frequency and gain, K, for which the root locus crosses the imaginary axis to determine what range of K is the system stable for
• For K> 0, only s¹ row can be zero which gives -K²-65K+720 = 0 which simplifies to K = 9.65
• Forming the even polynomial by using the s² row gives (90 −K)s² + 21K = 80.35s² + 202.7 = 0 giving s = ±j1.59
• The root locus crosses the imaginary-axis at ±j1.59 at a gain of 9.65, therefore, the system is stable for 0 ≤ K < 9.65

Angles of Departure and Arrival

• Deals with complex poles
• Includes angle of departure given as the sum of zero and pole angles and is equal to (2k+1)180°

Improving System Response

• System response can be improved
• To speed up the response at A to that B, without affecting the percent overshoot, compensation by adding poles and zeros serves to move the root locus and put to the desired pole on the root locus for some value of gain
• Dynamic compensators can meet transient and steady-state error specifications simultaneously
• Transient response is better with the addition of differentiation, and steady-state error is better with the addition of integration into the forward path

Compensators

• Ideal compensators, pure integration PI for improving steady-state error or pure differentiation PD for improving transient response, use active amplifiers
• Compensators implemented with passive elements Lead and Lag are not ideal compensators (steady-state error is not driven to zero)
• If a reasonable design cannot be made by the proportional gain alone, then dynamic compensator is used as compensation
• Both methods change the open-loop poles and zeros, thereby creating a new root locus that goes through the desired closed-loop pole location (cascade or feedback)

Ideal Integral Compensation (PI)

• Steady-state error can be improved (without appreciably affecting the transient response) by placing an open-loop pole at the origin, because this increases the system type by one
• To solve the problem, a zero is added close to the pole at the origin, to where the angular contribution of the compensator zero and compensator pole cancel out

Example1

• An example is given for system with a damping ratio of 0.174
• The addition of the ideal integral compensator reduces the steady-state error to zero for a step input without appreciably affecting transient response
• The root locus for the uncompensated system is shown
• The dominant poles are 0.694 ± j3.926 for a gain, K, of 164.6 with Position constant Kp = 8.23 with steady-state error: e(∞) = 0.108

Example 1 Conted

• An example of adding an ideal integral compensator with a zero at - 0.1 and how it leads to a better root Locus
• Poles are gain arc approximately the same, and it promotes same transient response
• Steady State Error is ZERO where Damping Ratio is unchanged (with K = 158:2)

Implementation of an ideal integral compensator

• Deals with the value and adjustments of Zero to keep the error and the integral of the error fed forward to the plant G(s), making steady-state error zero.

Lag Compensation

• Works to reduce steady-state error
• Similar to the Ideal Integrator, however it has a pole not on the origin but close to the origin (fig c) due to the passive networks.
• It explains the compensation process before (using static error constants for a system) and after compensation techniques

Example2

• Compensating a system to improve the steady-state error by a factor of 10, if the system is operating with a damping ratio of 0.174
• The uncompensated system error was 0.108 with KP = 8.23
• A tenfold improvement results in e(∞) = 0.0108
• The improvement in K, from the uncompensated system to the compensated system is the required ratio of the compensator zero to the compensator pole

Example 2 Conted

• Includes a comparison of Lag-Compensated and the Uncompensated Systems on the ζ= 0.174 line
• Discusses the third and fourth closed-loop poles
• The second-order dominant poles are at −0.678 ±j3.836 with gain K=158.1

Example 2 Conted

• The transient response of both systems is approximately the same with reduced steady-state error

Ideal Derivative Compensation (PD)

• Objective: design a response that has a desirable percent overshoot and a shorter settling time than the uncompensated system as is achieved by Ideal derivative compensation (which uses active elements)
• In comparison Lead Compensation (uses passive elements) approximates compensation by adding a zero and a more distant pole to the forward-path transfer function allowing a less sensitive application
• One way to speed up the original system is to add a single zero to the forward path.
• The transient response of a system can be selected by choosing an appropriate closed-loop pole location on the s-plane

Ideal Derivative Compensation (PD)

• See how compensation affects a system when (a) uncompensated (b) compensated, zero at -2 (c) zero at -3 (d) zero at -4 (operating with a damping ratio of 0.4
• Observations and facts:
• In each case gain K is chosen such that percent overshoot is same
• Compensated poles have more negative real part (smaller settling time) and larger imaginary part (smaller peak time)
• Zero placed farther from the dominant poles, compensated dominant poles move closer to the origin

Example - 3

• Designing an ideal derivative compensator to yield a 16% overshoot with a threefold reduction in settling time is demonstrated
• For the uncompensated Root-Locus: (a long damping ratio line); (ζ = 0.504 with the Dominant second-order pair of poles to reach Settling time of Ts =3.320), with the gain of the dominant poles k = 43.35 and a Third pole at -7.59

Example - 3 Contd

• Process of compensating the system for the desired angle
• Design the location of the compenstor zero to achieve the design

Example - 3 Contd

• Root-Locus After Compensation: Includes the parameters along with the Improvement in the transient response

Example - 3

• Deals with using the Ideal derivative compensator's implementation
• Highlights how to use a Proportional-plus-derivative (PD) controller and the value of design after compensating the location of the zero.

Lead Compensation

• States the Basic Idea regarding the difference between 180° and the sum of the angles (which must be the angular contribution required of the compensator)
• A choice from many infinite possibilities affects the Static Error Constants, Required gain to reach the design point on the root locus or Difficulty in justification a second order approximation

Example 4

• The design for three lead compensators to reduce the system in Figure to reduce the settling time by a factor of 2 while maintaining 30% overshoot
• A characteristic of the uncompensated system is operating at 30% overshoot with the lead compensator allowing a smaller settling time with faster action

Example 4 Contd

• Discusses a Lead compensator Design regarding the placement of the zero on real axis, and the sum of angles
• Shows where to the find the location of the compensator pole and how it leads to improved compensation with approximation, so long as the values are valid for the case

Improving Steady-State Error and Transient Response

• Outlines how to to improve Steady-State Error and Transient Response
• The order is: First improve the transient response using PD or lead compensation techniques
• Followed by, improving the steady-state response, using PI or lag compensation techniques
• Two Alternatives: PID (using active elements); Lag-Lead Compensator or Lead-Lag Compensator (using passive elements)

PID Controller Design

• States the function of the functions of the the compensator and the design procedure.
• The Transfer Function of the compensator is given as, and the design follows steps to achieve the necessary function.
• These are: (1) Figure out the desired pole location to meet transient response specifications (2) Design the PD controller to meet transient response specs (3) Check validity by simulation

Lag-Lead Compensator Design

• A Cheaper Solution than PID where the the first design is lead compensator used to improve the transient response by designing the the lag compensator to increase meet the steady-state error requirements.