Control Systems - Root Locus Notes

Steady State Error Due to Step Disturbance

• Last week recap:-

  • Poles of transfer function indicate stability; specifically, poles in the left-half plane (LHP) denote a stable system, while poles in the right-half plane (RHP) indicate instability. Poles on the imaginary axis can lead to marginal stability.

  • Proportional gain (K) can alter stability; increasing K might stabilize or destabilize a system, depending on the system's open-loop transfer function.

  • Routh's table helps determine the range of K for stability by analyzing the coefficients of the characteristic equation. This method provides conditions for all roots to have negative real parts.

  • Steady-state error analysis is relevant only for stable systems because unstable systems do not reach a steady state.

  • Unity feedback configuration is commonly used; non-unity feedback can be converted into a unity feedback system for easier analysis and design.

• Disturbance:-

  • Uncontrolled input that can occur at any time, affecting system performance by adding unwanted signals.

  • Requires a robust controller that can minimize the impact of disturbances and maintain desired performance levels.

• System Analysis:-

  • Error (E) is the difference between input (R) and output (C):$<br>E = R - C$

  • Output (C) is affected by both disturbance (D) and error (E):$<br>C = DG<em>2 + EG</em>1G*2$

• Derivation of Equations:-

  • Substituting $E = R - C$ into the equation for C:$<br>C = DG<em>2 + (R - C)G</em>1G*2$

  • Rearranging to solve for C:$<br>C(1 + G<em>1G</em>2) = DG<em>2 + RG</em>1G*2$

  • Transfer function: $<br>C = R \frac{G<em>1G</em>2}{1 + G<em>1G</em>2} + D \frac{G<em>2}{1 + G</em>1G*2}$

    • The above equation decomposes the output (C) into two parts: one due to the reference input (R) and one due to the disturbance (D). The principle of superposition allows for such decomposition.

  • Transfer function for input:$\frac{G<em>1G</em>2}{1 + G<em>1G</em>2}$

  • Transfer function for disturbance: $\frac{G<em>2}{1 + G</em>1G*2}$

• Error Calculation:-

  • Error equation: $E = R - C$

  • Substituting the expression for C:$<br>E = R - (RT + DT_D)$

  • Where $T = \frac{G<em>1G</em>2}{1 + G<em>1G</em>2}$ (transfer function for input) and $T<em>D = \frac{G</em>2}{1 + G<em>1G</em>2}$ (transfer function for disturbance).

  • Error due to disturbance: $-DT_D$

• Step Disturbance:-

  • $D(s) = \frac{1}{s}$

  • Error due to step disturbance: $<br>E<em>D(s) = - \frac{G</em>2(s)}{1 + G<em>1(s)G</em>2(s)} \cdot \frac{1}{s}$

• Final Value Theorem:-

  • Used to evaluate steady-state error:$<br>e<em>{ss} = \lim</em>{s \to 0} sE(s)$

  • Steady-state error due to disturbance:$<br>e<em>{ss,D} = \lim</em>{s \to 0} s E<em>D(s) = \lim{s \to 0} - \frac{G2(s)}{1 + G1(s)G2(s)} = \lim</em>{s \to 0} \frac{-1}{\frac{1}{G<em>2(s)} + G</em>1(s)}$

• Example Interpretation:-

  • If the error is negative, a positive disturbance causes the output to increase above the input. This means the system's natural response to the disturbance is to overshoot the desired value.

Non-Unity Feedback Systems

• Conversion to Unity Feedback:-

  • Non-unity feedback systems can be converted to unity feedback systems to apply standard error analysis techniques.

• Long Way (Direct Approach):-

  • Compute the closed-loop transfer function: $<br>\frac{G}{1 + GH}$

  • Analyze stability using the Routh table to ensure the system is stable before proceeding with error analysis.

  • Apply the Final Value Theorem to find the error:

    • For a step input: $<br>\lim_{s \to 0} s \cdot E(s)$, where $E(s) = \frac{R(s)}{1 + G(s)H(s)}$

• Example:-

  • Given system with $G$ and $H$.

  • Compute overall transfer function.

  • Construct Routh table to ensure stability.

  • Apply Final Value Theorem to error expression: $<br>\lim_{s \to 0} s \cdot \frac{1}{1 + G(s)H(s)} \cdot \frac{1}{s}$

• Equivalent Unity Feedback System:-

  • Add and subtract a feedback loop to create an equivalent system.

  • Original error: $R - CH$

  • Modified error: $R - CH - C + C = R - CH$

• Transformation Steps:-

  • Incorporate the -1 feedback loop.

  • Compute the transfer function of the inner loop: $H - 1$

  • The equivalent forward loop transfer function becomes: $<br>G_{eq} = \frac{G}{1 + (H-1)G}$

• Advantages:-

  • Allows use of standard unity feedback formulas for error analysis.

  • Simplifies transfer function calculation.

• Procedure:-

  • Compute the equivalent transfer function $<br>G_{eq}$.

  • Determine the appropriate error constant (e.g., position constant).

  • Apply the Final Value Theorem. For a step input:$<br>e<em>{ss} = \frac{1}{1 + K</em>p}$

    where$<br>K<em>p = \lim</em>{s \to 0} G_{eq}(s)$

• General Formula From the Same Situation: $<br>G_{eq} = \frac{G}{1 + GH - G}$

Root Locus

• Definition:-

  • Graphical representation of how the poles of a closed-loop system move as the gain (K) varies. It plots the trajectories of the closed-loop poles in the complex s-plane as K changes from 0 to infinity.

  • Illustrates the effect of changing the gain on system stability and performance; specifically, it shows how stability margins (such as gain and phase margins) change with K.

• Purpose:-

  • Understand system behavior by visualizing how the closed-loop poles shift with variations in the gain K, which directly impacts system response characteristics.

  • Design controllers to achieve desired performance, such as settling time, overshoot, and steady-state error, by selecting appropriate gain values that place the closed-loop poles in desired regions of the s-plane.

• Complex Number Representation:-

  • Complex number: $s = a + jb$ where a is the real part, b is the imaginary part, and $j = \sqrt{-1}$.

  • Magnitude: $<br>|s| = \sqrt{a^2 + b^2}$

  • Angle: $<br>\angle s = \arctan(\frac{b}{a})$

• Phase Calculation:-

  • For a ratio of complex numbers, the phase is the difference between the phases of the numerator and denominator. This is crucial in determining the angle of the open-loop transfer function, which is a key factor in root locus analysis.

• Polar Form:-

  • Alternative representation: $s = r e^{j\theta}$

    where$<br>a = r \cos(\theta)$

    $<br>b = r \sin(\theta)$

• Second-Order Systems:-

  • Poles are complex conjugates and are characterized by their damping ratio ($\zeta$) and natural frequency ($\omega_n$).

  • Expressed in terms of damping ratio ($\zeta$) and natural frequency ($\omega<em>n$):$s = -\zeta \omega</em>n \pm j \omega_n \sqrt{1 - \zeta^2}$

  • Constant $\omega_n$ traces a circle in the s-plane, indicating that the magnitude of the pole remains constant.

  • Constant $\zeta$ traces a line from the origin, with the angle determining the damping ratio; higher angles correspond to lower damping ratios.

  • The angle $\phi$ is related to the damping ratio: $<br>\sin(\phi) = \zeta$

• Implications:-

  • Pole locations determine transient response characteristics such as settling time, peak time, and overshoot. Poles closer to the imaginary axis result in slower responses and higher overshoot, while poles further to the left result in faster, more damped responses.

  • Desired specifications can be translated into pole locations; for instance, specific overshoot requirements can be mapped to a desired damping ratio, which in turn dictates the angle of the pole location in the s-plane.

  • Gain (K) can be modified to place poles at desired locations along the root locus, allowing engineers to tune the system's performance to meet specific requirements.

• Example System:-

  • Unity feedback system with plant $G$ and controller $K$.

  • Closed-loop transfer function: $<br>\frac{KG}{1 + KG}$

  • Root locus illustrates the movement of the closed-loop poles as K varies, providing insights into how the system's stability and response change with different gain values.

• Rules for Sketching Root Locus:-

  • Provides a basis for sketching, must consult additional reliable sources. Root locus plots start at the open-loop poles and end at the open-loop zeros, following specific rules related to angles of departure, arrival, and asymptotes.

• Definition:-

  • A graphical method of tracing the path of the poles as system gain changes in a feedback system. It visually represents how the closed-loop poles move in the complex plane as the gain K is adjusted.

• Feedback System:-

  • Transfer function: $<br>\frac{KG}{1 + KGH}$

• Poles:-

  • Must satisfy the characteristic equation: $<br>1 + KGH = 0$

  • Magnitude condition: $<br>|GH| = \frac{1}{K}$

  • Angle condition: $<br>\angle GH = (2k + 1)180^\circ$, where k is an integer. This condition ensures that the sum of the angles from all open-loop poles and zeros to a point on the root locus is an odd multiple of 180 degrees