Discretization of continuous time state space equations

Introduction

Discretization is an important concept in digital control systems. It involves converting continuous time state space equations into discrete time equations. Before diving into the discretization process, let's first understand the fundamentals of continuous time state space equations.

Fundamentals of Continuous Time State Space Equations

Continuous time state space equations are mathematical models used to describe the behavior of dynamic systems. They consist of a set of first-order differential equations that relate the system's inputs, outputs, and states. The general form of a continuous time state space equation is given as:

$$\dot{x}(t) = Ax(t) + Bu(t)$$

$$y(t) = Cx(t) + Du(t)$$

Where:

• $x(t)$ represents the state vector of the system at time $t$
• $u(t)$ represents the input vector at time $t$
• $y(t)$ represents the output vector at time $t$
• $A$, $B$, $C$, and $D$ are matrices that define the system's dynamics and input-output relationships.

Discretization of Continuous Time State Space Equations

Discretization is the process of converting continuous time state space equations into discrete time equations. This is necessary when implementing control algorithms on digital systems, as they operate in discrete time.

Discretization Methods

There are several methods available for discretizing continuous time state space equations. Some commonly used methods include:

1. Zero-order hold (ZOH)
2. First-order hold (FOH)
3. Tustin's method
4. Impulse invariance method

Let's take a closer look at each of these methods.

Zero-order hold (ZOH)

The ZOH method is a simple and commonly used discretization method. It assumes that the input to the system remains constant over each sampling interval. The ZOH method approximates the continuous time state space equations by replacing the continuous input signal with a sequence of discrete values that are held constant over each sampling interval.

First-order hold (FOH)

The FOH method is an improvement over the ZOH method. It takes into account the slope of the input signal within each sampling interval. Instead of assuming a constant input, the FOH method linearly interpolates the input signal between two consecutive sampling points.

Tustin's method

Tustin's method, also known as the bilinear transform, is a widely used discretization method. It approximates the continuous time state space equations by mapping the s-plane (continuous time domain) to the z-plane (discrete time domain) using a bilinear transformation.

Impulse invariance method

The impulse invariance method is another discretization method that approximates the continuous time state space equations by matching the impulse response of the continuous time system to the impulse response of the discrete time system. This method preserves the frequency response characteristics of the continuous time system.

Discretization using the ZOH method

The ZOH method is a straightforward discretization method. Here's a step-by-step procedure for discretizing continuous time state space equations using the ZOH method:

1. Determine the sampling time interval, denoted as $T_s$.
2. Discretize the state equation using the forward Euler method:

$$x(k+1) = x(k) + T_s(Ax(k) + Bu(k))$$

1. Discretize the output equation using the ZOH method:

$$y(k) = Cx(k) + Du(k)$$

Discretization using Tustin's method

Tustin's method is a more accurate discretization method compared to the ZOH method. Here's a step-by-step procedure for discretizing continuous time state space equations using Tustin's method:

1. Determine the sampling time interval, denoted as $T_s$.
2. Calculate the prewarping frequency, denoted as $\omega_p$, using the formula:

$$\omega_p = \frac{2}{T_s} \tan\left(\frac{\omega_c T_s}{2}\right)$$

Where $\omega_c$ is the desired cutoff frequency of the discrete time system.

1. Discretize the state equation using the following formula:

$$x(k+1) = e^{AT_s}x(k) + \int_0^{T_s} e^{A(t-T_s)}Bu(t)dt$$

1. Discretize the output equation using the following formula:

$$y(k) = Cx(k) + Du(k)$$

Discretization using the impulse invariance method

The impulse invariance method is another widely used discretization method. Here's a step-by-step procedure for discretizing continuous time state space equations using the impulse invariance method:

1. Determine the sampling time interval, denoted as $T_s$.
2. Calculate the impulse response of the continuous time system, denoted as $h(t)$.
3. Discretize the state equation using the following formula:

$$x(k+1) = e^{AT_s}x(k) + \int_0^{T_s} e^{A(t-T_s)}Bu(t)dt$$

1. Discretize the output equation using the following formula:

$$y(k) = Cx(k) + Du(k)$$

Comparison of discretization methods

Each discretization method has its own advantages and disadvantages. Some factors to consider when choosing a discretization method include:

• Accuracy: How closely does the discretized system approximate the continuous time system?
• Computational complexity: How computationally intensive is the discretization method?
• Stability: Does the discretized system remain stable?

Examples and Applications

Let's explore some examples and applications of discretization.

Example problem: Discretization of a continuous time state space equation using the ZOH method

Consider the following continuous time state space equation:

$$\dot{x}(t) = Ax(t) + Bu(t)$$

$$y(t) = Cx(t) + Du(t)$$

We want to discretize this equation using the ZOH method. Here's a step-by-step solution:

1. Determine the sampling time interval, denoted as $T_s$.
2. Discretize the state equation using the forward Euler method:

$$x(k+1) = x(k) + T_s(Ax(k) + Bu(k))$$

1. Discretize the output equation using the ZOH method:

$$y(k) = Cx(k) + Du(k)$$

Real-world application: Discretization of a continuous time state space equation for a control system

Consider a control system that operates in continuous time. To implement this control system on a digital platform, we need to discretize its continuous time state space equation. This allows us to design and implement digital control algorithms that can achieve the desired control objectives.

The discretization process involves converting the continuous time state space equation of the control system into a discrete time equation. This enables us to implement the control system using digital hardware or software.

The benefits of discretization in a control system include:

• Compatibility with digital platforms: Discretization allows the control system to be implemented on digital hardware or software platforms.
• Improved performance: Digital control algorithms can be designed and implemented to achieve better control performance compared to analog control systems.
• Flexibility: Discrete time control systems offer more flexibility in terms of parameter tuning and control algorithm design.

Conclusion

Discretization is a crucial step in implementing control systems on digital platforms. It involves converting continuous time state space equations into discrete time equations. Several discretization methods, such as the ZOH method, FOH method, Tustin's method, and impulse invariance method, are available for this purpose. Each method has its own advantages and disadvantages, and the choice of method depends on factors such as accuracy, computational complexity, and stability. Discretization enables the implementation of control systems on digital platforms, offering improved performance and flexibility.

In conclusion, understanding the process of discretization and its applications is essential for digital control system design and implementation.

Summary

Discretization is the process of converting continuous time state space equations into discrete time equations. It is an important step in implementing control systems on digital platforms. There are several methods available for discretizing continuous time state space equations, including the zero-order hold (ZOH) method, first-order hold (FOH) method, Tustin's method, and impulse invariance method. Each method has its own advantages and disadvantages, and the choice of method depends on factors such as accuracy, computational complexity, and stability. Discretization enables the implementation of control systems on digital platforms, offering improved performance and flexibility.

Analogy

Imagine you have a continuous video recording and you want to analyze it frame by frame. To do this, you need to discretize the video by capturing individual frames at regular intervals. Similarly, in digital control systems, continuous time state space equations need to be discretized to analyze and control the system at discrete time intervals.