State Space approach State Variables of a Dynamical System State Variable Equation Why State space approach Derive Transfer Function from State Space Equation Time Response and State Transition Matrix

Modern Control Systems State equation Dynamic equation Output equation State variable State space r- input p- output Modern Control Systems

Modern Control Systems Inner state variables C A D B + - Modern Control Systems

Modern Control Systems Motivation of state space approach Example 1 + - + noise Transfer function BIBO stable unstable Modern Control Systems

Modern Control Systems Example 2 BIBO stable, pole-zero cancellation -2 + - Modern Control Systems

Modern Control Systems then system stable State-space description Internal behavior description Modern Control Systems

Modern Control Systems Definition: The state of a system at time is the amount of information at that together with determines uniquely the behavior of the system for Example M Modern Control Systems

Modern Control Systems Example : Capacitor electric energy Input Example : Inductor Magnetic energy Modern Control Systems

Modern Control Systems Example M2 M1 B3 B1 B2 K Modern Control Systems

Modern Control Systems Example Armature circuit Field circuit Modern Control Systems

Modern Control Systems

Modern Control Systems Dynamical equation Transfer function Laplace transform matrix Transfer function Modern Control Systems

Modern Control Systems Example By Newton’s Law Modern Control Systems

Modern Control Systems State Space Equation Transfer Function Example: Transfer function of the Mass-damper-spring system Modern Control Systems

Modern Control Systems Example MIMO system Transfer function Modern Control Systems

Modern Control Systems Remark : the choice of states is not unique. + - exist a mapping Modern Control Systems

Modern Control Systems Different state equation description p is nonsingular Modern Control Systems

Modern Control Systems Definition : Two dynamical systems with are said to be equivalent. The nonsingular matrix p is called an equivalence transformation. & Theorem: two equivalent dynamical system have the same transfer function. Modern Control Systems

Modern Control Systems

Sampling of Continuous-Time Signals Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall Inc.

351M Digital Signal Processing Signal Types Analog signals: continuous in time and amplitude Example: voltage, current, temperature,… Digital signals: discrete both in time and amplitude Example: attendance of this class, digitizes analog signals,… Discrete-time signal: discrete in time, continuous in amplitude Example:hourly change of temperature in Austin Theory for digital signals would be too complicated Requires inclusion of nonlinearities into theory Theory is based on discrete-time continuous-amplitude signals Most convenient to develop theory Good enough approximation to practice with some care In practice we mostly process digital signals on processors Need to take into account finite precision effects Our text book is about the theory hence its title Discrete-Time Signal Processing Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing

Sample Data Control System Sampling Theorem Modern Control Systems

Sampling: Time Domain Many signals originate as continuous-time signals, e.g. conventional music or voice By sampling a continuous-time signal at isolated, equally-spaced points in time, we obtain a sequence of numbers n  {…, -2, -1, 0, 1, 2,…} Ts is the sampling period. Ts t Ts s(t) Sampled analog waveform impulse train

Modulation by cos(s t) Modulation by cos(2 s t) Replicates spectrum of continuous-time signal At offsets that are integer multiples of sampling frequency Fourier series of impulse train where ws = 2 p fs Modulation by cos(s t) Modulation by cos(2 s t) w F(w) 2pfmax -2pfmax w G(w) ws 2ws -2ws -ws

Shannon Sampling Theorem A continuous-time signal x(t) with frequencies no higher than fmax can be reconstructed from its samples x[n] = x(n Ts) if the samples are taken at a rate fs which is greater than 2 fmax. Nyquist rate = 2 fmax Nyquist frequency = fs/2. What happens if fs = 2fmax? Consider a sinusoid sin(2 p fmax t) Use a sampling period of Ts = 1/fs = 1/2fmax. Sketch: sinusoid with zeros at t = 0, 1/2fmax, 1/fmax, …

Shannon Sampling Theorem Assumption Continuous-time signal has no frequency content above fmax Sampling time is exactly the same between any two samples Sequence of numbers obtained by sampling is represented in exact precision Conversion of sequence to continuous time is ideal In Practice

Why 44.1 kHz for Audio CDs? Sound is audible in 20 Hz to 20 kHz range: fmax = 20 kHz and the Nyquist rate 2 fmax = 40 kHz What is the extra 10% of the bandwidth used? Rolloff from passband to stopband in the magnitude response of the anti-aliasing filter Okay, 44 kHz makes sense. Why 44.1 kHz? At the time the choice was made, only recorders capable of storing such high rates were VCRs. NTSC: 490 lines/frame, 3 samples/line, 30 frames/s = 44100 samples/s PAL: 588 lines/frame, 3 samples/line, 25 frames/s = 44100 samples/s

Sampling As sampling rate increases, sampled waveform looks more and more like the original Many applications (e.g. communication systems) care more about frequency content in the waveform and not its shape Zero crossings: frequency content of a sinusoid Distance between two zero crossings: one half period. With the sampling theorem satisfied, sampled sinusoid crosses zero at the right times even though its waveform shape may be difficult to recognize

Aliasing Analog sinusoid Sample at Ts = 1/fs x(t) = A cos(2pf0t + f) Sample at Ts = 1/fs x[n] = x(Ts n) = A cos(2p f0 Ts n + f) Keeping the sampling period same, sample y(t) = A cos(2p(f0 + lfs)t + f) where l is an integer y[n] = y(Ts n) = A cos(2p(f0 + lfs)Tsn + f) = A cos(2pf0Tsn + 2p lfsTsn + f) = A cos(2pf0Tsn + 2p l n + f) = A cos(2pf0Tsn + f) = x[n] Here, fsTs = 1 Since l is an integer, cos(x + 2pl) = cos(x) y[n] indistinguishable from x[n] Frequencies f0 + l fs for l  0 are aliases of frequency f0