⏳Intro to Dynamic Systems Unit 12 – Sampled-Data Systems & Z-Transform

Key Concepts and Definitions

• Sampled-data systems involve the interaction between continuous-time and discrete-time signals and systems
• Sampling converts a continuous-time signal into a discrete-time signal by taking samples at regular intervals (sampling period)
• Nyquist theorem states that the sampling frequency must be at least twice the highest frequency component in the signal to avoid aliasing
• Z-transform is a mathematical tool used to analyze and design discrete-time systems
  • Converts a discrete-time signal or sequence into a complex frequency-domain representation
• Transfer functions in the z-domain describe the input-output relationship of a discrete-time system
• Stability of discrete systems is determined by the location of poles in the z-plane
  • A system is stable if all poles lie within the unit circle
• Digital control design techniques include pole placement, deadbeat control, and optimal control methods

Continuous vs. Discrete-Time Systems

• Continuous-time systems have signals that are defined for all values of time and are typically described by differential equations
  • Examples include analog circuits and mechanical systems
• Discrete-time systems have signals that are defined only at specific time instants (usually equally spaced) and are described by difference equations
  • Examples include digital filters and computer-controlled systems
• Sampling is the process of converting a continuous-time signal into a discrete-time signal
• Reconstruction is the process of converting a discrete-time signal back into a continuous-time signal
• Aliasing occurs when the sampling frequency is too low, causing high-frequency components to be misinterpreted as low-frequency components
• Quantization is the process of representing a continuous-amplitude signal with a finite set of discrete values
  • Introduces quantization noise in the system

Sampling Theory and Nyquist Theorem

• Sampling theory deals with the process of converting continuous-time signals into discrete-time signals
• Nyquist theorem (also known as the sampling theorem) states that a band-limited signal can be perfectly reconstructed from its samples if the sampling frequency is at least twice the highest frequency component in the signal
  • The minimum sampling frequency required is called the Nyquist rate
• Undersampling occurs when the sampling frequency is lower than the Nyquist rate, leading to aliasing
• Oversampling occurs when the sampling frequency is higher than the Nyquist rate, providing better signal representation and allowing for simpler anti-aliasing filters
• Anti-aliasing filters are low-pass filters used before the sampling process to limit the bandwidth of the signal and prevent aliasing
• Sample and hold circuits are used to maintain the value of the sampled signal between sampling instants
• Reconstruction filters (low-pass filters) are used to smooth the discrete-time signal and recover the original continuous-time signal

Z-Transform: Basics and Properties

• The z-transform is a mathematical tool used to analyze and design discrete-time systems
• It converts a discrete-time signal or sequence $x[n]$ into a complex frequency-domain representation $X(z)$
  • Defined as $X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}$
• The region of convergence (ROC) is the set of values in the complex plane for which the z-transform converges
  • Provides information about the stability and causality of the system
• Z-transform properties include linearity, time-shifting, scaling, convolution, and differentiation
• The inverse z-transform is used to convert the z-domain representation back into the time-domain signal
  • Can be performed using partial fraction expansion, power series expansion, or contour integration
• The unilateral z-transform is used for causal systems, while the bilateral z-transform is used for non-causal systems

Transfer Functions in Z-Domain

• Transfer functions in the z-domain describe the input-output relationship of a discrete-time system
• Defined as the ratio of the z-transform of the output to the z-transform of the input, assuming zero initial conditions
  • $H(z) = \frac{Y(z)}{X(z)}$
• Can be represented in pole-zero form, factored form, or polynomial form
• Poles and zeros of the transfer function determine the system's stability, transient response, and steady-state behavior
• The order of the transfer function is determined by the highest power of z in the denominator polynomial
• Discrete-time systems can be classified as FIR (finite impulse response) or IIR (infinite impulse response) based on their transfer functions
  • FIR systems have only zeros and a finite-length impulse response
  • IIR systems have both poles and zeros and an infinite-length impulse response

Stability Analysis of Discrete Systems

• Stability is a crucial property of discrete-time systems, determining whether the system's output remains bounded for bounded inputs
• A discrete-time system is stable if all poles of its transfer function lie within the unit circle in the z-plane
  • Poles outside the unit circle indicate instability
  • Poles on the unit circle indicate marginal stability
• The bounded-input, bounded-output (BIBO) stability criterion is commonly used for discrete-time systems
• Jury's stability test is an algebraic method for determining the stability of a discrete-time system based on its characteristic equation
• The Routh-Hurwitz criterion can be adapted for discrete-time systems to assess stability
• Stability margins, such as gain margin and phase margin, provide a measure of the system's robustness to parameter variations
• The root locus technique can be used to analyze the stability and transient response of discrete-time systems as a parameter (usually the gain) varies

Digital Control Design Techniques

• Digital control design involves the synthesis of discrete-time controllers for continuous-time systems
• Pole placement is a design technique where the controller's poles are placed at desired locations in the z-plane to achieve specific performance objectives
  • Requires the system to be controllable
• Deadbeat control aims to achieve the fastest possible settling time by placing all closed-loop poles at the origin of the z-plane
  • Results in a finite settling time but may require large control efforts
• Optimal control methods, such as linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) control, minimize a cost function that balances performance and control effort
• Discrete-time PID controllers are widely used in industrial applications and can be designed using various methods, such as Ziegler-Nichols tuning or pole placement
• Feedforward control can be used to improve the system's response to known disturbances or reference inputs
• Digital filters (FIR and IIR) can be incorporated into the control loop to shape the frequency response and improve performance

Real-World Applications and Examples

• Digital audio processing: Sampling and quantization of audio signals, digital filters for equalization and noise reduction
• Digital image processing: Sampling and quantization of images, digital filters for enhancement and compression
• Control systems: Discrete-time controllers for industrial processes, robotics, and automotive systems
  • Examples include temperature control, motor speed control, and autonomous vehicle navigation
• Digital communication systems: Sampling and quantization of analog signals, digital modulation and demodulation techniques
• Biomedical signal processing: Sampling and analysis of physiological signals, such as ECG and EEG
• Financial systems: Discrete-time models for stock prices, interest rates, and economic indicators
• Aerospace and defense: Digital control systems for aircraft, spacecraft, and missile guidance
• Consumer electronics: Digital control in home appliances, such as thermostats and washing machines