Discretization

all of the above are discretization methods used to convert a continuous-time signal into a discrete-time signal.

ZOH:

involves holding each sampled value of the continuous-time signal constant until the next value gets sampled to obtain a discrete-time signal. This produces a kind of staircase signal. Although the ZOH method is simple, it can introduce errors and distortions, especially for high-frequency signals, which can be reduced using higher-order hold methods such as linear interpolation or spline interpolation.

FOH:

involves approximating the continuous-time signal with a first-order polynomial function between to adjacent sample points. Basically connects two nearby sample points with a line. The FOH method is more accurate than the ZOH method, especially for high-frequency signals, but it can still introduce errors and distortions for rapidly changing signals. To reduce these errors, higher-order hold methods such as cubic interpolation or spline interpolation can be used.

impulse invariant:

involves sampling the impulse response of the continuous-time system and using it to derive the coefficients of the discrete-time system. The assumption is that the continuous-time system is linear and time-invariant and that the sampling period is small enough to capture the system's essential dynamics. However, the method can introduce errors in the discretization process, especially for systems with high-frequency components, so careful consideration must be given to the choice of sampling period and the limitations of the method.

Tustin method

involves replacing "s" in the continuous-time transfer function with a formula that includes the discrete-time variable "z". The resulting expression is simplified, and the terms are rearranged to obtain the discrete-time transfer function. The Tustin method is preferred because it preserves the stability of the continuous-time system and approximates the integral term more accurately than the impulse invariant method. However, like any discretization method, it introduces errors, especially at high frequencies, so the choice of the sampling period must be carefully considered.

matched poles and zeros method:

involves modifying the poles and zeros of the continuous-time transfer function by replacing them with their discrete-time counterparts. This preserves the frequency response and stability properties of the continuous-time system, and the resulting transfer function can be used to implement the discrete-time system. However, the method may not be feasible for systems with complex conjugate poles or zeros that lie close to the imaginary axis, as the transformation may cause the poles and zeros to move outside the unit circle in the z-plane, leading to an unstable discrete-time system.

• Replace each pole "p" with "exp(p*T)", where "T" is the sampling period.

• Replace each zero "z" with "exp(z*T)".

If we convert the s-domain transferfunction into its equivalent z-domain (discretization) the following happens:

• Left half s‐plane is transformed into the unit circle 0 < Z <1 whereat the right half s‐plane is transformed to the outside Z > 1 of the unit circle

• All poles of the Z‐transfer function shall be inside the unit circle to have a stable system

• The jω‐axes of the s‐plane corresponds to the unit circle of the z‐plane and will runned trough infintely

• Lines of constant damping (σ = const) within the s‐plane are transformed to circles around the origing of the z‐plane

• Lines of constant frequency (ω = const) within the s‐plane are convert to beams in the z‐plane starting at the origin of the z‐plane with an constant angle of ϕ = ωT

  • Increasing the frequency also increases the angles of those beams

With Foward Euler:

• It is possible that a stable continous time system is

  mapped into an unstable discrete‐time system

With Backward Euler:

• A stable continous system, will always give a stable

  discrete‐time system

With Tustin:

• Advantage that the left half‐s‐plane is transformed

  into the unit circle

• Stable continous‐time systems are therefore

  transformed into stable sampled systems

• Unstable continous‐time systems are transformed into

  unstable discrete‐time‐systems

Discretization is the process of converting a continuous-time signal or system into a discrete-time signal or system. It involves sampling the continuous-time signal at regular intervals and quantizing its amplitude values to a finite number of levels. The result is a sequence of discrete-time values that can be stored and processed using digital systems.

There are several reasons why discretization is necessary. First, many real-world signals and systems are inherently discrete in nature, such as digital audio or video signals. Second, digital systems are often more efficient and cost-effective than analog systems, making them a popular choice for signal processing applications. Finally, discretization allows for the use of digital signal processing techniques, which can provide greater accuracy and flexibility than analog techniques.

There are two main paths of discretization:

Overall, discretization is a crucial process in digital signal processing, allowing us to convert real-world signals into a format that can be easily processed by digital systems.

Advantages:

• using digital signal processing methods i.e. with the use of microcontrollers and computers

• creating computerbased models for simulation and testing of a real word system

• cheaper to design

• easier to modify while developing

Disatvantages:

• loss of information —> values only at discrete rate

• adding delay (measurements already old) —> since we have to sample data and quantize it using algorithm befor we can use the data.

• Quatisation error —> can be decreased by increasing the sampling frequency but that means more computing power is needed

The following three methods can be used:

• Too low sampling frequency may loose so much information that the control performance degraded, and the system dynamics is lost

• Too high sampling rate increases the burden of the processor; also, it may lead to discrete representation with bad numerical properties

• For oscillating systems, the sampling interval is often tied to the frequency of the dominating oscillation. For damped systems the sampling interval is usually chosen to be in relation to the time constant