1.3 System Representations and Block Diagrams

System representations and block diagrams are key tools for understanding dynamic systems. They help us visualize and analyze how different parts of a system interact, showing us how inputs affect outputs over time. These methods are crucial for predicting system behavior and designing effective controls.

Transfer functions, state-space models, and block diagrams each offer unique insights into system dynamics. By mastering these representations, we can better grasp system stability, response characteristics, and overall performance. This knowledge forms the foundation for advanced system analysis and control design.

System Representations

Transfer Functions

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• Transfer functions mathematically represent the relationship between the input and output of a linear, time-invariant system in the frequency domain
• Expressed as the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming zero initial conditions
• Provide insights into a system's frequency response, stability, and pole-zero locations
• The poles of a transfer function determine the system's stability and transient response (e.g., stable, unstable, or marginally stable)
• The zeros of a transfer function affect the system's steady-state response and can introduce phase lead or lag (e.g., minimum phase or non-minimum phase systems)

State-Space Models

• State-space models mathematically represent a dynamic system using a set of first-order differential equations in the time domain
• Consist of state variables, input variables, and output variables
• The state variables represent the minimum set of variables required to completely describe the system's behavior at any given time
• The state-space model consists of two equations:
  • The state equation describes the evolution of the state variables over time
  • The output equation relates the state variables to the system's output
• The state matrix (A) represents the system's dynamics, the input matrix (B) represents the effect of inputs on the state variables, the output matrix (C) relates the state variables to the output, and the feedthrough matrix (D) represents the direct influence of inputs on the output
• Provide information about the system's internal dynamics, controllability, and observability
• The eigenvalues of the state matrix determine the system's stability and natural response (e.g., stable, unstable, or marginally stable)
• The eigenvectors of the state matrix determine the modes of the system (e.g., oscillatory or exponential modes)

Block Diagrams for System Visualization

Components and Interactions

• Block diagrams graphically represent the functional relationships between the components of a dynamic system
• Consist of blocks representing system components, signals representing inputs and outputs, and arrows indicating the direction of signal flow
• Each block represents a transfer function or a mathematical operation (addition, subtraction, multiplication, or division)
• Summing junctions combine multiple input signals, while take-off points distribute a signal to multiple blocks
• Feedback loops connect the output of a block back to its input or to the input of a previous block in the system
  • Negative feedback is denoted by a minus sign at the summing junction
  • Positive feedback has no sign at the summing junction

Interpreting Block Diagrams

• Block diagrams visually represent the relationships between system components and the flow of signals
• Help identify feedback loops, which can affect system stability and performance (e.g., negative feedback can improve stability, while positive feedback can lead to instability)
• The presence of integrators or differentiators in a block diagram indicates the system's ability to track ramp inputs or respond to high-frequency inputs, respectively
• By examining the structure of a block diagram, one can identify potential sources of instability (positive feedback loops) and devise strategies for improving system performance (adding compensators or filters)

Block Diagram Simplification

Reduction Techniques

• Block diagram reduction techniques simplify complex block diagrams by combining blocks and eliminating intermediate variables
• Result in a single transfer function that represents the entire system
• The series reduction technique is used when two or more blocks are connected in series, with the output of one block serving as the input to the next
  • The overall transfer function is the product of the individual transfer functions
• The parallel reduction technique is used when two or more blocks are connected in parallel, with their outputs summed together
  • The overall transfer function is the sum of the individual transfer functions
• The feedback reduction technique is used when a portion of the output signal is fed back to the input of the system
  • The overall transfer function is determined by applying the feedback formula, which involves the forward path transfer function and the feedback path transfer function

Rearranging Block Diagrams

• The moving a summing point technique is used to move a summing point ahead of or behind a block, adjusting the input signals accordingly
  • Useful for rearranging the block diagram to facilitate further simplification
• The moving a take-off point technique is used to move a take-off point ahead of or behind a block, adjusting the output signals accordingly
  • Also useful for rearranging the block diagram to facilitate further simplification

Interpreting System Representations

Transfer Functions and Frequency Response

• Transfer functions provide insights into a system's frequency response, stability, and pole-zero locations
• The poles of a transfer function determine the system's stability and transient response
  • Stable systems have poles in the left-half of the complex plane
  • Unstable systems have poles in the right-half of the complex plane
  • Marginally stable systems have poles on the imaginary axis
• The zeros of a transfer function affect the system's steady-state response and can introduce phase lead or lag
  • Minimum phase systems have zeros in the left-half of the complex plane
  • Non-minimum phase systems have zeros in the right-half of the complex plane

State-Space Models and System Dynamics

• State-space models provide information about the system's internal dynamics, controllability, and observability
• The eigenvalues of the state matrix determine the system's stability and natural response
  • Stable systems have eigenvalues with negative real parts
  • Unstable systems have eigenvalues with positive real parts
  • Marginally stable systems have eigenvalues with zero real parts
• The eigenvectors of the state matrix determine the modes of the system
  • Oscillatory modes correspond to complex conjugate eigenvalue pairs
  • Exponential modes correspond to real eigenvalues
• Controllability indicates whether the system's states can be influenced by the input, while observability indicates whether the system's states can be inferred from the output

Overall System Representation

• The overall transfer function obtained from block diagram reduction provides a concise representation of the system's input-output relationship
• Can be used to analyze the system's performance, stability, and frequency response
• By examining the structure of a block diagram, one can identify potential sources of instability (positive feedback loops) and devise strategies for improving system performance (adding compensators or filters)