Understanding Control Systems

• Control involves gauging a system's controlled variable value and using a control signal to adjust or limit deviations from the desired value.
• Controlling something can mean changing it from one state to another, keeping it consistent, or achieving a particular response within a certain time frame.
• Systems consist of components working together to achieve a specific goal, not necessarily physical, with "system" encompassing physical, biological, and economic contexts.

Key Components of a Control System

• Setpoint (input) represents the desired value for the controlled variable, acting as the system's target.
• Error signal indicates the difference between the setpoint and the actual controlled variable value, reflecting system deviation from its intended state.
• Controller is the control system's decision-maker, receiving state information (error signal) and determining actions (actuating signal).
• Plant refers to equipment or machine parts performing a specific operation like mechanical devices, heating furnaces, chemical reactors, or spacecraft.
• Controlled variable stands for the specific quantity or characteristic regulated within a system, aiming to maintain it at a desired value, such as voltage in electrical systems, speed in mechanical, or pressure in chemical.
• Control signal serves as the controller's output and instructs the actuator to adjust the controlled variable to reach the desired value, such as a PWM or digital signal.
• Feedback element measures the actual controlled variable value and sends the data back to the controller in closed-loop systems.
• Feedback signal is the feedback element's output in closed-loop control, giving the controller critical information about the system's current condition.
• Disturbances are signals that negatively affect the output value, classified as internal if generated within the system, and external if originating outside.

Types of Control Systems

• Open Loop Control System does not use output feedback to modify control action, operating without knowledge of its output; examples are toasters and traffic lights.
• Open Loop Systems are simple to design and implement, less expensive due to simpler hardware, and work well when the input-output relationship is predictable.
• Open Loop Systems are prone to errors, cannot adapt to changes, and are unsuitable where precise control/adaptability is needed.
• Close Loop Control System monitors the output continuously and feeds it back to adjust control, using its output to enhance performance, such as cruise control and robot arms.
• Close Loop Systems achieve higher precision, are more stable, and can adapt to environmental or system changes.
• Close Loop Systems are more complex to design and implement and are prone to instability if feedback loops are improperly designed.

Transfer Function

• A transfer function represents a system's response to an input signal mathematically and defines the input/output relationship in the frequency domain.

• Transfer functions are powerful analysis/design tools, providing a compact system behavior representation to help designers understand/manipulate dynamics.

• The transfer function is the ratio of the Laplace transform of output to input, assuming zero initial conditions: $$T(s) = \frac{Y(s)}{X(s)}$$

Laplace Transform (LT)

• The Laplace Transform can solve a differential equation. $$\mathcal{L}{f(t)} = F(s) = \int_{0}^{\infty} f(t) \cdot e^{-st} , dt$$

• The inverse Laplace Transform is: $$\mathcal{L}^{-1}{F(s)} = \frac{1}{2\pi j} \int_{\sigma - j\omega}^{\sigma + j\omega} F(s) \cdot e^{st} , ds = f(t)u(t)$$ Where: - F(s) is the Laplace transform of f(t). - s is a complex frequency variable. - f(t) is a function of time.

• The following theorem includes the nth derivative transform $$\mathcal{L}\left{\frac{d^n f}{dt^n}\right} = s^n F(s) - s^{n-1}f(0) - s^{n-2}f'(0) - \dots - s^0 f^{(n-1)}(0)$$

• Examples of Laplace and Inverse Transforms:

  • Knowing the functions and using various transforms

Standard Test Signals

• Standard test signals, like impulse, step, ramp, and parabolic, help determine control system performance with output time responses.
• Impulse input is infinite at t=0 and zero elsewhere, with a unit impulse area of 1, defined as δ(t) = 0 for t≠0 and ∫δ(t)dt = 1.
• Step input covers all positive 't' values, including zero, with a value of one during this period, defined as u(t) = 1 for t≥0 and u(t) = 0 for t<0; R(s) = 1/s.
• Ramp input: a unit ramp signal r(t) is defined as, r(t) = t for t≥0 and = 0 for t<0.
• Parabolic input: a unit parabolic signal p(t) is defined as, p(t) = ((t^2)/2) for t≥0, and =0 for t<0.

State-Space Representation

• State-space representation models physical systems mathematically as functions of input, output, and related state variables using equations.

• State variables (x) represent the minimum set required to fully describe system state, including physical amounts or abstract parameters representing a system's internal configuration.

• Input variables (u) correspond to control inputs or forces applied to the system, which designers or controllers use to manipulate system behavior.

• Output variables (y) signify measurable variables useful for the designer or controller and furnish data on the system's response to inputs and state.

• State equations represent a series of first-order differential equations with 'n' variables, where these variables are state variables to be solved.

• Output equations express system output variables as linear combinations of input and state variables.

• General state-space representations are described through equations: $$ \dot{x} = Ax + Bu \ y = Cx + Du $$

• Where, x is state vector, y is output vector, u is control vector, A is system matrix, B is input matrix, C is output matrix and D is feedforward matrix.

• Steps to convert a transfer function to state-space:

  • Identify the Transfer Function.
  • Write the State Equations.
  • Write the Output Equations.
  • Convert the Transfer Function to State Space.

Note

• Poles and Zeros: The transfer function's characteristics, such as stability and system dynamics, are determined by the location of its poles and zeros in the complex "s" plane.
  • Poles are values of "s" that make the denominator of the transfer function equal to zero, while zeros are values that make the numerator equal to zero.
  • Poles influence the system's natural response and stability.
  • Zeros can affect system dynamics and frequency response.
• Frequency Response: Transfer functions provide insight into the frequency response of a system. By analyzing the behavior of the transfer function as a function of "s," you can determine how the system responds to different frequencies in the input signal.