12.4 Stability Analysis of Discrete-Time Systems

Stability analysis of discrete-time systems is crucial for understanding sampled-data systems. It focuses on determining whether a system's response will remain bounded over time, using tools like pole locations and the Jury stability test.

This analysis bridges the gap between continuous and discrete systems, showing how sampling affects stability. It's essential for designing stable discrete-time controllers, which are key components in digital control systems.

Pole Locations for Stability

Stability Conditions

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• A discrete-time system is stable if and only if all of its poles lie within the unit circle in the complex z-plane
• The region of convergence (ROC) for a stable system includes the unit circle and extends outward to infinity
• If any pole lies on or outside the unit circle, the system is unstable
• For a causal system, the ROC must include the region outside the outermost pole

Determining Pole Locations

• The location of the poles can be determined by finding the roots of the characteristic equation, which is obtained from the system's transfer function
  • Example: For a transfer function $H(z) = \frac{1}{z^2 - 0.5z + 0.1}$, the characteristic equation is $z^2 - 0.5z + 0.1 = 0$
  • Solving this equation yields the pole locations ($z = 0.25 \pm 0.2i$)
• The stability of a system can also be determined by examining its impulse response
  • If the impulse response decays to zero as time approaches infinity, the system is stable
  • Example: A system with impulse response $h[n] = (0.8)^n u[n]$ is stable because $(0.8)^n$ approaches zero as $n$ approaches infinity

Jury Stability Test

Jury Table Construction

• The Jury stability test is an algebraic method for determining the stability of a discrete-time system without explicitly finding the roots of the characteristic equation
• The test involves constructing a Jury table using the coefficients of the characteristic equation
  • The first row of the Jury table consists of the coefficients of the characteristic equation in descending order of z-powers
  • Subsequent rows are formed by performing a series of cross-multiplications and subtractions using the elements from the previous two rows
  • Example: For a characteristic equation $z^3 + 2z^2 - 3z + 1 = 0$, the first row of the Jury table would be [1, 2, -3, 1]

Stability Conditions

• The system is stable if and only if the following conditions are satisfied:
  • The first element in the first row is positive
  • The elements in the first column alternate in sign, starting with a positive sign
  • The absolute value of the last element in the first column is less than 1
• If any of these conditions are violated, the system is unstable
  • Example: A Jury table with a first column of [1, -2, 0.5, 0.75] indicates a stable system because it satisfies all three conditions
  • Example: A Jury table with a first column of [1, 2, -0.5, 1.2] indicates an unstable system because the last element (1.2) has an absolute value greater than 1

Continuous vs Discrete Stability

Sampling and Stability

• A continuous-time system can be converted to an equivalent discrete-time system through the process of sampling
• The stability of the discrete-time system depends on the sampling period and the location of the poles of the continuous-time system
  • If a continuous-time system has poles in the left-half of the complex s-plane, the corresponding discrete-time system will have poles inside the unit circle in the z-plane, provided that the sampling period is sufficiently small
  • As the sampling period increases, the poles of the discrete-time system move closer to the unit circle
  • If the sampling period is too large, the poles may cross the unit circle, resulting in an unstable discrete-time system

Mapping between s-plane and z-plane

• The mapping between the s-plane and the z-plane is given by the relationship $z = e^{sT}$, where $T$ is the sampling period
  • Poles in the left-half of the s-plane map to the interior of the unit circle in the z-plane
  • Poles in the right-half of the s-plane map to the exterior of the unit circle
  • Example: A continuous-time system with poles at $s = -1$ and $s = -2$ will have corresponding discrete-time poles at $z = e^{-T}$ and $z = e^{-2T}$, respectively, which lie inside the unit circle for any positive sampling period $T$

Stable Discrete-Time Controllers

Controller Design Process

• Sampled-data systems involve the control of continuous-time plants using discrete-time controllers
• The design of stable discrete-time controllers requires consideration of both the continuous-time plant dynamics and the sampling period
  • The first step in designing a stable discrete-time controller is to obtain a discrete-time model of the continuous-time plant using techniques such as the zero-order hold (ZOH) or the bilinear transformation
  • The discrete-time controller is then designed based on the discrete-time plant model, ensuring that the closed-loop system poles lie within the unit circle

Design Techniques and Considerations

• Common design techniques for discrete-time controllers include pole placement, LQR (Linear Quadratic Regulator), and LQG (Linear Quadratic Gaussian) control
  • Example: Pole placement involves selecting desired closed-loop pole locations and designing the controller gains to achieve those pole locations
• The controller design must also consider the effects of sampling, such as aliasing and the introduction of time delays
• The stability of the closed-loop sampled-data system can be assessed using techniques such as the Nyquist criterion or the Bode plot, adapted for discrete-time systems
  • Example: The Nyquist criterion for discrete-time systems states that the closed-loop system is stable if the Nyquist plot of the open-loop transfer function does not encircle the point $(-1, 0)$
• If the continuous-time plant is unstable or has poles close to the imaginary axis, the sampling period must be chosen carefully to ensure that the discrete-time controller can stabilize the system