Frequency domain analysis of signals and control systems

Aleksei Tepljakov

Lecture outline

• Fourier Series and Transform
• Laplace Transform
• Frequency Domain Analysis
• Stability Margins and Robust Control
• PID Control Case Studies (live demo)

Periodic functions

In what follows, we will consider the representation of functions $f(x)$ of period $2\pi$.

Fundamental trigonometric functions

The phase difference between $\sin(x)$ and $\cos(x)$ is $\frac{\pi}{2}$ rad.

Fourier series

Fourier coefficients

Spectrograms for common signals

Importance of Fourier Analysis

• Every periodic function (or signal) has a corresponding Fourier series representation.
• Constant signals can be seen as periodic functions with relatively long periods.
• Therefore, every signal $u(t)$ has a Fourier series representation, that is, every signal can be seen as a combination of fundamental harmonic functions.

Interactive Fourier series demo

Fourier Transform

• The general Fourier transform is defined as
  \[ F(k) = \int_{-\infty}^{\infty} f(t) {\rm e}^{-2\pi j k t}{\rm d}t. \]
• Notice that integration is from $-\infty$ to $\infty$, i.e., we are assumed to have knowledge of $f(t)$ for $t\in(-\infty, \infty)$.
• In practice, we rarely have this knowledge for signals and operate on a limited time interval (window) of $t\in[t_a, t_b]$.
• Therefore, we can use Fast Fourier Tranform (FFT) and Discrete Fourier Tranform (DFT) for practical purposes.

Fourier Transform for identification

When the FFT of the input signal $u(t)$ is known, measure the output signal $y(t)$, and process it with FFT. Then, use the difference in input/output signal spectrum to figure out the system model. Above: process identification for PID tuning.

Laplace Transform

• A function $F(s)$ of the complex variable $s=\sigma + j\omega$ is called the Laplace transform of the original function $f(t)$ and defined as
  \[ F(s)=\int_0^{\infty} {\rm e}^{-s t}f(t){\rm d} t. \]
• Notice that the definition is similar to Fourier transform, except the limits of integration are now $0\leqslant t < \infty$.
• It is usual to use the Fourier transform to study signals, and the Laplace transform to study systems.

Differential equations and transfer functions

Frequency domain analysis

Recall that for a function $f(x)$ the Fourier series is given by

\begin{equation} f(x) = a_0 + \sum_{n=1}^{\infty}(a_n \cos nx + b_n \sin nx), \end{equation}

which means that any signal $u(t)$ can be represented by a Fourier series, i.e., as an infinite sum of weighted harmonic functions.

Now, consider an input signal $u(t)=U\sin(\omega t)$ passed through an asymptotically stable transfer function $H(s)$ such that

\[ \stackrel{u(t)}{\longrightarrow}\boxed{H(s)}\stackrel{y(t)}{\longrightarrow} \]

Steady-state output will be $y(t)=U \left|H(j\omega)\right|\sin\left(\omega t+\angle H(j\omega)\right)$.

Frequency domain analysis (continued)

To construct the magnitude and phase responses of a system represented by a transfer function $G(s)$ with a transport delay $L$, substitute $s=j\omega$. Then, for

at a particular frequency $\omega_k$ we have

where $|\cdot|$ denotes the absolute value, and $\arg(\cdot)$—the argument (or angle, in radians) of the complex value $G(j\omega_k)$.

Recall: polar form of complex numbers

Frequency domain analysis (continued)

Frequency domain characteristics completely describe the behavior of a linear, time-invariant system. Frequency response is graphically represented in the following ways:

• Bode plot—two separate graphs for magnitude and phase against frequency, usually on logarithmic scales;
• Nyquist plot—a single graph depicting real vs. imaginary parts of the response covering the full frequency range;
• Nichols plot—a single graph depicting magnitude vs. argument.

Since it is possible to assess qualitative properties of the linear system under study (e.g., relative stability margins), frequency domain analysis is essential in control design.

Bode plot

• The Bode plot (named after Hendrik Wade Bode) is a graph of the absolute value $|G(j\omega)|$ and phase shift $\angle G(j\omega)$ of a transfer function $G(s)$ evaluated in $s=j\omega$.
• The Bode plot shows the system frequency response as a function of frequency $\omega$, for all $\omega>0$.
• The frequency axis $\omega$ is in logarithmic scale.
• The absolute value is expressed as decibel (dB):
  \[ |G(j\omega)|_{dB}=20 \log_{10} |G(j\omega)|. \]
• Unity gain: $|G(j\omega)|=1$ we have $|G(j\omega)|_{dB}=20\log_{10}1=0$ dB.

Bode plot (continued)

Bode frequency analysis

To study the frequency response of a system it is useful to rewrite the transfer function $G(s)$ in Bode form:

where $K$ is the Bode gain; $h$ is the type of system, i.e., the number of poles in $s=0$; $T_j$ (for real $T_j>0$) is a time constant; $\xi_j$ is a damping ratio, with $-1<\xi_j<1$; and $\omega_{nj}$ is a natural frequency of the system.

Analysis using Bode form

Nyquist (polar) plot

• The Nyquist plot (named after Harry Nyquist), also called polar plot, is the graph in polar coordinates of $G(j\omega)$ for $\omega\in[0,+\infty)$ in the complex plane.
• The Nyquist plot combines the Bode magnitude and phase plot on a single plot and therefore provides a more compact representation.
• Stability assessment: For an open-loop asymptotically stable system $G(s)$, the closed-loop system $W(s)$ is asymptotically stable if and only if the Nyquist plot $G(j\omega)$ does not encircle clockwise the critical point $-1+j0$.

Relative stability margins

• Assume $N(s)=C(s)G(s)$ is open-loop asymptotically stable
  \[ \stackrel{r(t)}{\longrightarrow}\boxed{C(s)}\stackrel{u(t)}{\longrightarrow}\boxed{G(s)}\stackrel{y(t)}{\longrightarrow} \]
• Let $\omega_c$ be the gain crossover frequency: $|N(j\omega_c)|=1$. The phase margin is the quantity
  \[ \varphi_m = \angle N(j\omega_c)-(-\pi). \]
• Let $\omega_p$ be the phase crossover frequency: $\angle N(j\omega_p)=-\pi$. The gain margin is the quantity
  \[ G_m = 20 \log_{10} \frac{1}{|N(j\omega_p)|}. \]

Robust control

• The characteristics of real-life systems may change during operation, and frequency characteristics may get shifted from the nominal case potentially causing instability.
• For robustness of stability, the values of the gain and phase margins should be as large as possible.
• The phase margin is linked to damping of the system.
• Gain and phase margins should be chosen such that a balance is achieved between ensuring stability, fast reference tracking, and disturbance rejection.

Further reading

1. K. Ogata, Modern Control Engineering, 5th edition, Prentice Hall, 2009.
2. E. Kreyszig, Advanced Engineering Mathematics, 10th edition, John Wiley & Sons, 2010.
3. N. Nise, Control Systems Engineering, 6th edition, John Wiley & Sons, 2011.