Unit 2 - Notes

ECE305 10 min read

Unit 2: Modelling and Representations of Control Systems

Block Diagram Representation and Reduction Techniques

A block diagram is a pictorial representation of the functions performed by each component of a system and the flow of signals. It provides a visual understanding of the cause-and-effect relationships between the input and output.

Basic Components

Block: Represents the transfer function of a system element. The input-output relationship is Output(s) = G(s) * Input(s).
TEXT
```
    Input(s) ----->| G(s) |-----> Output(s)
    
```
Summing Point: Represents an algebraic summation of two or more signals. The output is the sum of the inputs with their associated signs.
TEXT
```
      X1(s) ---+
               v
      X2(s) --->(+)---> Y(s) = X1(s) + X2(s) - X3(s)
               ^
      X3(s) ---(-)
    
```
Take-off Point: Represents a point from which a signal is taken and applied to other blocks or summing points. It allows the same signal to be used in multiple places.
TEXT
```
                   .------> X(s) (to another path)
                   |
    X(s) ----------.------> X(s) (continues on main path)
    
```

Canonical Form of a Feedback Control System

This is the standard representation of a single-loop feedback system.

TEXT

       R(s)      E(s)               C(s)
Input ----- O ----->|   G(s)   |-----> Output
          ^ -       -----------
          |         (Forward Path)
          |                       
          |_________|   H(s)   |_______
                    -----------
                    (Feedback Path)

R(s): Reference Input (Setpoint)
C(s): Controlled Output
G(s): Forward Path Transfer Function (Plant)
H(s): Feedback Path Transfer Function (Sensor)
E(s): Error Signal, E(s) = R(s) - B(s)
B(s): Feedback Signal, B(s) = H(s) * C(s)

The Closed-Loop Transfer Function T(s) = C(s) / R(s) is derived as:
C(s) = G(s) * E(s)
E(s) = R(s) - H(s) * C(s)
C(s) = G(s) * [R(s) - H(s) * C(s)]
C(s) = G(s)R(s) - G(s)H(s)C(s)
C(s) [1 + G(s)H(s)] = G(s)R(s)

TEXT

T(s) = C(s) / R(s) = G(s) / (1 + G(s)H(s))  (For Negative Feedback)

For positive feedback, the denominator becomes (1 - G(s)H(s)).

Block Diagram Reduction Rules

The goal is to simplify a complex diagram into a single block representing the overall transfer function.

Rule	Original Diagram	Equivalent Diagram
1. Blocks in Cascade (Series)	`X(s) -> [G1] -> [G2] -> Y(s)`	`X(s) -> [G1*G2] -> Y(s)`
2. Blocks in Parallel	`+---->[G1]----+` `X(s) ->\| \|->(+) -> Y(s)` `+---->[G2]----+`	`X(s) -> [G1 + G2] -> Y(s)`
3. Eliminating a Feedback Loop	`R(s) -> O -> [G] -> C(s)` `^ - \|` `\|____[H]____\|`	`R(s) -> [G / (1 + GH)] -> C(s)`
*4. Moving a Summing Point after* a Block**	`X1 -> O -> [G] -> Y` `^ +` `X2 -\|`	`X1 -> [G] -> O -> Y` `^ +` `X2 -> [G] -\|`
*5. Moving a Summing Point before* a Block**	`X1 -> [G] -> O -> Y` `^ +` `X2 --\|`	`X1 -> O -> [G] -> Y` `^ +` `X2 -> [1/G] -\|`
*6. Moving a Take-off Point after* a Block**	`X -> . -> [G] -> Y` `\|` `V` `Z`	`X -> [G] -> . -> Y` `\|` `v` `[1/G] <- Z`
*7. Moving a Take-off Point before* a Block**	`X -> [G] -> . -> Y` `\|` `v` `Z`	`X -> . -> [G] -> Y` `\|` `v` `[G] -> Z`
8. Interchanging Summing Points	`X1 -> O -> O -> Y` `^ + ^ +` `X2 -\| X3 -\|`	`X1 -> O -> O -> Y` `^ + ^ +` `X3 -\| X2 -\|`

Signal Flow Graphs (SFG)

An SFG is a graphical representation of a set of linear algebraic equations. It consists of nodes and branches and is a convenient alternative to block diagrams, especially for complex systems.

SFG Terminology

Node: Represents a system variable (e.g., R(s), C(s)).
Branch: A directed line segment joining two nodes. Each branch has an associated gain (transmittance).
Input (Source) Node: A node with only outgoing branches.
Output (Sink) Node: A node with only incoming branches.
Mixed Node: A node with both incoming and outgoing branches.
Path: A continuous traversal of connected branches in the direction of the arrows.
Forward Path: A path from an input node to an output node that does not traverse any node more than once.
Loop (Feedback Loop): A path that starts and ends at the same node.
Path Gain: The product of the branch gains along a path.
Loop Gain: The product of the branch gains along a loop.
Non-touching Loops: Two or more loops are non-touching if they do not share any common nodes.

Converting Block Diagrams to SFGs

Represent every variable (inputs, outputs, signals at summing points and take-off points) as a node.
If a signal Xj is connected to a block G to produce Xk, draw a branch from node Xj to node Xk with gain G.
For a summing point, draw branches from the input signal nodes to the output signal node. For a signal Y = X1 - X2, branches from X1 to Y have gain +1, and from X2 to Y have gain -1.

Example:
Canonical feedback loop: R(s) -> O -> [G] -> C(s), with feedback [H] from C(s) to the summing point.

Nodes: R(s), E(s), C(s)
Equations:
- E(s) = R(s) - H(s) * C(s)
- C(s) = G(s) * E(s)

SFG:

TEXT

      1         G(s)
    R(s) O----->O E(s)----->O C(s)
         <-----------------
              -H(s)

Mason's Gain Formula

This formula provides a direct method to find the overall transfer function (gain) of a system from its signal flow graph.

The formula is:

TEXT

T = C(s) / R(s) = (1 / Δ) * Σ [Pk * Δk]

Where:

T: The overall gain of the system.
k: The number of forward paths from input to output.
Pk: The gain of the k-th forward path.
Δ: The determinant of the graph, calculated as:
Δ = 1 - (Sum of all individual loop gains) + (Sum of gain products of all possible combinations of two non-touching loops) - (Sum of gain products of all possible combinations of three non-touching loops) + ...
Δk: The value of Δ for the part of the graph that does not touch the k-th forward path. (To find Δk, simply remove the k-th forward path and all its nodes from the graph and recalculate Δ for the remaining part).

Example Application

Consider the SFG:

TEXT

        G1      G2      G3
  R -> O -> O -> O -> O -> O -> C
       ^  \ /         \ /  ^
       |   X           X   |
       |  / \         / \  |
       <--- -H1 <-----> -H2 ---
               G4

Forward Paths (Pk):
- P1 = G1 * G2 * G3 (The straight path)
- P2 = G4 (The path through the bottom)
Individual Loops (Li):
- L1 = -G1 * H1
- L2 = -G2 * H2
- L3 = G4 * H2 * G2 * H1 (The big outer loop)
Pairs of Non-touching Loops:
- Loops L1 and L2 do not share any nodes. So, L1 and L2 are non-touching.
- Gain product: (L1)(L2) = (-G1 * H1) * (-G2 * H2) = G1 * G2 * H1 * H2
Calculate Δ:
Δ = 1 - (L1 + L2 + L3) + (L1 * L2)
Δ = 1 - (-G1*H1 - G2*H2 + G4*H2*G2*H1) + (G1*G2*H1*H2)
Δ = 1 + G1*H1 + G2*H2 - G4*G2*H1*H2 + G1*G2*H1*H2
Calculate Δk:
- Δ1: For forward path P1. Remove P1 from the graph. No loops remain. Therefore, Δ1 = 1.
- Δ2: For forward path P2. Remove P2 from the graph. No loops remain. Therefore, Δ2 = 1.
Apply Mason's Gain Formula:
T = (P1*Δ1 + P2*Δ2) / Δ
T = (G1*G2*G3 * 1 + G4 * 1) / (1 + G1*H1 + G2*H2 - G4*G2*H1*H2 + G1*G2*H1*H2)
T = (G1*G2*G3 + G4) / (1 + G1*H1 + G2*H2 - G4*G2*H1*H2 + G1*G2*H1*H2)

Concept of Poles and Zeros

The transfer function T(s) of a linear time-invariant (LTI) system is a rational function of the complex variable s:

TEXT

T(s) = N(s) / D(s) = K * [(s-z1)(s-z2)...(s-zm)] / [(s-p1)(s-p2)...(s-pn)]

where n >= m for a physically realizable system.

Zeros

Zeros of a system are the values of s (real or complex) for which the transfer function T(s) becomes zero. They are the roots of the numerator polynomial N(s).

Mathematical Definition: s = z_i such that T(z_i) = 0.
Physical Significance: At a frequency corresponding to a zero (s = jω), the system's output is zero. Zeros can be used to "block" or attenuate specific frequencies.

Poles

Poles of a system are the values of s (real or complex) for which the transfer function T(s) becomes infinite. They are the roots of the denominator polynomial D(s). The equation D(s) = 0 is called the characteristic equation of the system.

Mathematical Definition: s = p_i such that T(p_i) -> ∞.
Physical Significance: The poles determine the natural response of the system (the transient behavior). The location of the poles in the complex s-plane is directly related to the stability and performance of the system.

The s-Plane and System Stability

The location of the poles in the complex s-plane dictates the stability of the system. The s-plane is a complex plane with a real axis (σ) and an imaginary axis (jω).

Left Half-Plane (LHP, Re[s] < 0): Poles in the LHP correspond to decaying exponential terms in the time response. The system is stable.
Right Half-Plane (RHP, Re[s] > 0): Poles in the RHP correspond to growing exponential terms. The system is unstable.
Imaginary Axis (jw-axis, Re[s] = 0):
- A single pole at the origin (s=0) or a single pair of conjugate poles on the jw-axis results in a marginally stable system (sustained oscillations or constant output).
- Repeated (multiple) poles on the jw-axis result in an unstable system (oscillations with growing amplitude).

Pole Location	Time Domain Response Contribution
Negative real axis (`s = -σ`)	Decaying exponential (`e^(-σt)`)
Positive real axis (`s = +σ`)	Growing exponential (`e^(σt)`)
Origin (`s = 0`)	Constant (step)
Imaginary axis (`s = ±jω`)	Sustained oscillation (`cos(ωt)`)
LHP complex pair (`s = -σ±jω`)	Decaying oscillation (`e^(-σt)cos(ωt)`)
RHP complex pair (`s = +σ±jω`)	Growing oscillation (`e^(σt)cos(ωt)`)

Pole-Zero Cancellation: If a pole and a zero have the same location, they cancel each other out, and the corresponding mode will not be present in the output response. However, if the cancelled pole is in the RHP, the system is still internally unstable.

Effect of Feedback

Feedback is the process of taking a fraction of the output signal and feeding it back to the input to be compared with the reference input. Negative feedback is predominantly used in control systems.

Consider the closed-loop transfer function T(s) = G(s) / (1 + G(s)H(s)).

1. Effect on Overall Gain

Feedback generally reduces the overall gain of the system.
If the loop gain G(s)H(s) >> 1, then 1 + G(s)H(s) ≈ G(s)H(s).
The transfer function becomes:
T(s) ≈ G(s) / (G(s)H(s)) = 1 / H(s)
This means the overall gain becomes independent of the forward path G(s) and is determined primarily by the feedback element H(s), which can be designed with high precision.

2. Effect on Stability

Feedback can improve stability but can also cause instability.

The stability of a system is determined by the location of its closed-loop poles, which are the roots of the characteristic equation 1 + G(s)H(s) = 0.
The open-loop poles are the roots of the denominator of G(s)H(s).
Feedback changes the location of the poles. A properly designed negative feedback system can move unstable open-loop poles from the RHP to the stable LHP.
However, improper design (e.g., high gain, significant phase shift) can move stable open-loop poles into the RHP, making the closed-loop system unstable.

3. Effect on Sensitivity

Feedback reduces the system's sensitivity to variations in its parameters (e.g., due to aging or environmental changes).
The sensitivity of the overall transfer function T with respect to changes in the forward path gain G is defined as:
S_G^T = (∂T/T) / (∂G/G) = (∂T/∂G) * (G/T)

For the closed-loop system:
∂T/∂G = [(1 + GH)*1 - G*H] / (1 + GH)^2 = 1 / (1 + GH)^2
S_G^T = [1 / (1 + GH)^2] * [G / (G / (1 + GH))] = 1 / (1 + GH)

TEXT

S_G^T = 1 / (1 + G(s)H(s))

If the loop gain |G(s)H(s)| is large, the sensitivity S_G^T becomes very small. This means that even large variations in G will have a minimal effect on the overall system transfer function T.

4. Effect on External Disturbances (Noise)

Feedback helps in reducing the effect of external disturbances or noise.
Consider a disturbance D(s) added between two blocks G1(s) and G2(s) in the forward path.

TEXT

       R(s)      E(s)          D(s)
Input ----- O ----->| G1(s) |---+---->| G2(s) |-----> C(s) Output
          ^ -       -------   |     -------
          |                   v
          |_________________[ H(s) ]_________________|

The output C(s) has two components, one from R(s) and one from D(s). Using superposition:

Output due to R(s) (let D(s)=0): C_R(s) = [G1(s)G2(s) / (1 + G1G2H)] * R(s)
Output due to D(s) (let R(s)=0): C_D(s) = [G2(s) / (1 + G1G2H)] * D(s)

The total output is C(s) = C_R(s) + C_D(s).
The effect of the disturbance on the output is C_D(s). The term 1 + G1G2H in the denominator reduces the effect of D(s). If |1 + G1G2H| >> 1, the disturbance is significantly attenuated.

5. Effect on Bandwidth

Bandwidth is the range of frequencies over which the magnitude of the system's frequency response is equal to or greater than 1/√2 (or -3dB) of its low-frequency value.

Generally, negative feedback increases the bandwidth of a system.
This improvement in bandwidth comes at the cost of reduced gain. The gain-bandwidth product of a system tends to remain constant.
A larger bandwidth means the system can respond faithfully to faster-changing inputs.

Unit 1

Unit 3

Unit 2 - Notes

Table of Contents

Unit 2: Modelling and Representations of Control Systems

Block Diagram Representation and Reduction Techniques

Basic Components

Canonical Form of a Feedback Control System

Block Diagram Reduction Rules

Signal Flow Graphs (SFG)

SFG Terminology

Converting Block Diagrams to SFGs

Mason's Gain Formula

Example Application

Concept of Poles and Zeros

Zeros

Poles

The s-Plane and System Stability

Effect of Feedback

1. Effect on Overall Gain

2. Effect on Stability

3. Effect on Sensitivity

4. Effect on External Disturbances (Noise)

5. Effect on Bandwidth