Control Systems: Open Loop and Closed Loop

Introduction to Control Systems

• Control systems manage, direct, or regulate devices/systems using control loops
• Control systems range from simple home heating to complex industrial systems
• A control system provides a desired output response by controlling the input

Examples of control systems

• Traffic lights
• Washing machines

Classification of Control Systems

• Can be classified based on various parameters
• Classified as continuous-time or discrete-time based on the type of signal used
• Continuous-time control systems: all signals are continuous
• Discrete-time control systems: have discrete-time signals
• Divided into SISO and MIMO based on the number of inputs and outputs present
• SISO: Single Input Single Output systems have one input and one output
• MIMO: Multiple Input Multiple Output systems have more than one input and one output
• Classified as open loop or closed loop based on the feedback path

Open Loop Control Systems

• Output is not fed back to the input
• Control action is independent of the desired output

Closed Loop Control Systems

• Output is fed back to the input
• Control action depends on the desired output

Error Detection and Correction

• Error detectors produce an error signal
• The error signal represents the difference between the input and feedback signal
• Feedback signals come from the block utilizing the overall system's output
• Controllers use the error signal as input, rather than the direct input
• The controller produces an actuating signal adjusting the plant

Automatic Control Systems

• Output automatically adjusts until desired response is achieved
• Closed loop control systems are also known as automatic control systems
• Traffic lights with sensors are an example of closed loop automatic control

Basic Terminologies in Control Systems

• System: A combination of components achieving a goal
• Control: Action to command/direct/regulate a system
• Plant/Process: The part of a system to be controlled
• Input: The signal/excitation applied to a control system
• Output: The actual response from the control system
• Controller: The part of a system controlling the plant
• Disturbances: Signals adversely affecting performance
• Control system: A system that directs/regulates itself or another system to a specific goal
• Automation: Control of a process by automatic means
• Control System: An interconnection of components forming a system configuration that will provide a desired response.
• Actuator: Device causing the process to provide output and motive power
• Design: Conceiving/inventing system forms/parts for a purpose
• Simulation: Model to investigate system behavior using real input signals
• Optimization: Adjusting parameters for the most favorable design
• Feedback Signal: Output measure used for feedback control
• Block Diagrams: Unidirectional blocks representing element transfer functions
• Signal Flow Graph (SFG): Diagram with nodes/branches graphically representing linear relations
• Specifications: Statements that define device/product requirements and performance criteria
• Regulator: System with fixed outputs, focused on disturbance rejection
• Servo System: Control system for mechanical quantities (acceleration, velocity, position)
• Stability: System can follow input commands
• A system is unstable if output goes out of control or increases unbounded
• Multivariable Control System: System with >1 input or output variable
• Trade-off: Compromise between conflicting criteria
• Sensitivity: How much system output changes in response to input/parameter variations
• Robustness: System's ability to maintain stability/performance despite uncertainties; a robust system is less sensitive to changes.

Feedback

• Output, or a portion of it returning to the input side to influence the system
• Feedback helps to improve control system performance.

Types of Feedback

• Positive Feedback: Adds reference input (R(s)) and feedback output
• Negative Feedback: Reduces error between reference input (R(s)) and system output.

Transfer Functions

• T represents the transfer function.
• G signifies open loop gain (function of frequency).
• H is the gain of feedback path (function of frequency).

Effects of Feedback on Overall Gain (Negative Feedback)

• The gain may increase or decrease as measured by G / (1+GH).
• If (1+GH) is less than 1, the overall gain increases, when 'GH' is negative
• If (1+GH) is greater than 1, the overall gain decreases, when 'GH' is positive
• Feedback increases overall gain in one frequency range, decreasing it in another

Effect of Feedback on Sensitivity

• Closed loop control system sensitivity, T, to variations in open loop gain, G
• Reciprocal of (1+GH)
• Sensitivity may increase or decrease depending on (1+GH)
• If (1+GH) < 1, sensitivity increases; 'GH' is negative
• If (1+GH) > 1, sensitivity decreases; 'GH' is positive

Effect of Feedback on Stability

• System is stable if output is controllable; otherwise, it is unstable
• If the denominator in Equation 2 is zero (GH = -1), then the system becomes unstable.
• Feedback must be properly deployed to ensure a stable control system.

Effect of Feedback on Noise

• Using transfer function relations, the effect of noise can be compared with and without feedback
• Mathematical Models for Control Systems
• Useful for system analysis and design.
• Mathematical models used: differential equation, transfer function, and state space models.

Transfer Function Representation: Block Diagrams

• Block diagrams represent control systems, and consist of single/combined blocks

Basic Elements

• Block
• Summing Point
• Take-off Point

Component Representation

• Represented by blocks and has a single input and single output

Summing Point

• Represented by a circle with a cross
• The output represents the algebraic sum of the inputs
• Summation, subtraction, or a combination is performed based on the inputs

Take-off Point

• Input signal is passed/duplicated through one or more branches to other blocks or summing points

Block Diagram Algebra

• Governs basic block diagram elements
• Pictorial representation of algebraic equations

Block Connections

• Three basic types: series, parallel, and feedback

Series Connection (Cascade)

• Two series blocks have transfer functions G1(s) and G2(s)
• Resulting with overall transfer function: G(s) = G1(s)G2(s).

Parallel Connection

• Parallell blocks with transfer functions G1(s) and G2(s) with the same input
• Their outputs are connected to a summing point with and overall transfer function of: G(s) = G1(s) + G2(s).

Feedback Connection

• Using negative feedback to achieve a closed loop transfer function of: G(s) / [1+G(s)H(s)].

Block Diagram Algebra for Summing Points

• Two possibilities for shifting the placement of summing points with respect to blocks:
  • Before the block
  • After the block

Block Diagram Reduction Rules

• Simplify in series
• Simplify in parallel
• Simplify feedback loops
• Shift the take-off point
• Iterate until a simplified single block is achieved

Notes on Simplification

• Overall block diagram transfer function.
• The transfer function can be calculated with multiple inputs
• Overcome the drawback of drawing partially-simplified in every step by using signal flow graphs.

Signal Flow Graph

• Signal Flow Graph (SFG) is graphical representation of algebraic equations

Key Elements

• Nodes and branches make up a signal flow graph
• Nodes- Represent variables/signals have types; input, output, mixed
  • Input Node- Outgoing branches only -Output Node- Incoming branches only
  • Mixed Node- Incoming and outgoing branches
• Branch-Line segment from 2 nodes which has gain and direction