SIMULATION, IDENTIFICATION AND CONTROL OF DYNAMICAL SYSTEMS

Simulation, Identification and Control of Dynamical Systems



Simulation, Identification and Control of Dynamical Systems

Question 1

Part 1

Phase-locked loops (PLLs) are one of the basic building blocks in modern electronic systems. They have been widely used in communications, multimedia and many other applications. The theory and mathematical models used to describe PLLs are of two types: linear and nonlinear. Nonlinear theory is often complicated and difficult to deal with in real-world designs. Analog PLLs have been well modeled by linear control theory. Starting from a well-defined model in the continuous-time domain, this article introduces a modeling and design method for a digital PLL based on linear control theory (Muggleton & Lodhi 2010, 74-79).

It has been proved that a linear model is accurate enough for most electronic applications as long as certain conditions are met. The task of the PLLs in these devices is to recover the pixel clock based on input reference HS (horizontal sync). The PLL can be easily recognized as a feedback control system. This system consists of the following components.

Phase detector detects the phase difference between the input signal Fin(t) and the feedback signal feedback

Loop filter typically, a filter with low-pass characterization

VCO voltage-controlled oscillator whose output frequency is a function of its input voltage (Rovithakis & Christodoulou 2000, 35-38)

Part 2

Based on the condition that phase error is small, which can be expressed mathematically as sin(?) ˜ ?, a PLL can be accurately described by a linear model. ?in(t) is the phase of the input signal, and ?fd(t) is the phase of the feedback signal. Since the system is described in the continuous-time domain, the transfer functions of each component are given out in Laplace-transform format. Transfer function of loop filter:

Transfer function of VCO

Closed-loop transfer function of a PLL

Based on the closed-loop transfer function, one can see that this is a second-order system. In automatic control system theory, the transfer function of the second-order system often can be written as

Where ?n is defined as natural undamped frequency and ? is defined as damping ratio. This system is called a standard prototype second-order system. Based on the transfer function of a second-order prototype system, a characteristic equation of the system is defined as

By solving the roots of the characteristic equation, two poles of the system, S0 and S1, can be derived.

Where a is defined as damping factor and ? is defined as damped frequency. Based on Equations, as soon as ? and ?n of the system are given, the poles of a second-order prototype system can be determined. Those two parameters are usually used to specify performance requirements of a system. As a matter of fact, most transient-response performances of a system can be determined based on these two parameters. The following is a list of performance parameters defined based on ? and ?n (Severance 2001, 10-18).

Part 3

Damping factor a

Damped frequency ?

Settling time

Maximum overshoot time

Maximum overshoot

Maximum overshoot in percentage

Until this point, a second-order system has been defined in S-domain, and this system will meet ...