Block Diagram Algebra, Signal Flow Graphs, and Stability

Flashcards covering Block Diagram Algebra, Signal Flow Graphs, Mason's Rule, and System Stability as presented by Engr. John R. Dela Cruz.

What is the definition of a Block Diagram?

A shorthand, pictorial representation of the cause-and-effect relationship between the input and output of a physical system.

In the canonical form of a feedback control system, what do the symbols $G$, $H$, and $GH$ represent?

$G$ is the direct transfer function, $H$ is the feedback transfer function, and $GH$ is the open-loop transfer function.

In canonical form, what represent the ratios $\frac{C}{R}$, $\frac{E}{R}$, and $\frac{B}{R}$?

$\frac{C}{R}$ is the closed-loop transfer function, $\frac{E}{R}$ is the actuating signal ratio (error ratio), and $\frac{B}{R}$ is the primary feedback ratio.

What is the equation for the closed-loop transfer function in canonical form?

$$\frac{C}{R} = \frac{G}{1 \pm GH}$$

What is the equation for the actuating signal ratio (error ratio)?

$$\frac{E}{R} = \frac{1}{1 \pm GH}$$

What is the equation for the primary feedback ratio?

$$\frac{B}{R} = \frac{GH}{1 \pm GH}$$

What is the resulting equation when combining blocks $P_1$ and $P_2$ in cascade?

$$Y = (P_1 P_2) X$$

What is the resulting equation when combining blocks $P_1$ and $P_2$ in parallel?

$$Y = P_1 X ± P_2 X$$

What are the first three steps in reducing complicated block diagrams?

Step 1: Combine all cascade blocks; Step 2: Combine all parallel blocks; Step 3: Eliminate all minor feedback loops.

In Step 4 of block diagram reduction, where should summing points and takeoff points be shifted relative to the major loop?

Shift summing points to the left and take-off points to the right of the major loop.

What is a Signal Flow Graph?

A pictorial representation of the simultaneous equations describing a system, displaying the transmission of signals through the system.

In Signal Flow Graph terminology, what is a 'path'?

A continuous unidirectional succession of branches along which no node is passed more than once.

Define an input node (source) and an output node (sink) in a Signal Flow Graph.

An input node (source) is a node with only outgoing branches, while an output node (sink) is a node with only incoming branches.

What is a self-loop?

A feedback loop consisting of a single branch.

Define path gain and loop gain.

Path gain is the product of the branch gains encountered in traversing a path; loop gain is the product of the branch gains of the loop.

State the formula for Mason's Rule used to find the transfer function.

$$G(s) = \frac{C(s)}{R(s)} = \frac{\sum T_k \Delta_k}{\Delta}$$

In Mason's Rule, how is the determinant $\Delta$ defined?

$$\Delta = 1 - \sum (\text{loop gains}) + \sum (\text{nontouching loop gains taken two at a time}) - \sum (\text{nontouching loop gains taken three at a time}) + \dots$$

What two conditions imply stability in Linear Time-Invariant (LTI) systems?

(1) Only forced response remains as the natural response approaches zero; (2) Every bounded input yields a bounded output.

When is a system considered 'Linear'?

If the relationship between input and output satisfies the superposition property and scaling: $H\{\alpha x_1(t) + β x_2(t)\} = α H\{x_1(t)\} + β H\{x_2(t)\}$.

How is 'Stable' defined in terms of natural response?

The natural response approaches $0$ as time approaches infinity.

How is 'Marginally Stable' defined in terms of natural response?

The natural response neither decays nor grows but remains constant or oscillates as time approaches infinity.

In terms of poles, when is a closed-loop transfer function considered 'Unstable'?

When it has at least one pole in the right half-plane and/or poles of multiplicity greater than $1$ on the imaginary axis.

What does the Routh-Hurwitz Criteria determine about system poles?

It tells how many closed-loop system poles are in the left half-plane (LHP), right half-plane (RHP), and on the imaginary axis ($j\omega$), rather than their exact locations.

According to the Routh-Hurwitz Criteria, how is the number of roots in the right half-plane determined from the Routh table?

It is equal to the number of sign changes in the first column of the Routh table.

What causes a 'row of zeros' in a Routh Table and what does it indicate?

It is caused by a factor of an even polynomial and indicates the existence of roots on the $j\omega$ axis.