the amplitude spectral density gives a continuous display of the amplitude density spectrum, which in this case is in the form of impulses, rather than just a number. Similarly the Fourier transform of a train of impulses of the form (23.27) is given by (23.28) Energy and Power Spectral Density 6 Suppose x(t) is an aperiodic signal with a Fourier transform X( f ), then its energy is given by (23.29) This is Parseval’s theorem and it shows that the principle of conservation of energy in the time and frequency domains holds. The amplitude spectrum X( f) can be expressed as TABLE 23.5 Properties of the Fourier Transform Property Signal Description Fourier Transform Linearity ax + by ; a, b constants aX + bY Evenness and oddness Time shift e −j2 π f τ X Time scale x Time reversal x(−t) X ∗ Duality Xx Time convolution x ∗ yXY Frequency convolution xy X ∗ Y Modulation X Time differentiation Frequency differentiation t n x Integration Correlation YX Parseval’s theorem t() t() f() f() xt–() xt()= xt–() x– t()= Xf() 2 xt() 2 π ft()cos td 0 ∞ ∫ = Xf() 2– xt() 2 π ft()sin td 0 ∞ ∫ = xt t–() f() at() 1 a X f a f() t() f–() t() t() f() f() t() t() f() f() xt()e j2 π f 0 t ff 0 –() d n dt n xt() j2pf() n Xf() t() j 2p n d n df n Xf() x τ () τ d ∞– ∞ ∫ 1 j2 π f Xf() 1 2 X 0() δ f()+ R xy τ () yt()xt τ –()td ∞– ∞ ∫ = f–() f() xt() 2 td ∞– ∞ ∫ Xf() 2 fd ∞– ∞ ∫ pt() δ tnT–() n=∞– ∞ ∑ = Pf() 1 T δ fkF s –(), F s k=∞– ∞ ∑ 1 T == ER xx 0() xt() 2 td ∞– ∞ ∫ Xf() 2 fd ∞– ∞ ∫ == = Xf() Xf() Xf()∠= 0066_Frame_C23 Page 15 Wednesday, January 9, 2002 1:53 PM ©2002 CRC Press LLC the amplitude spectral density gives a continuous display of the amplitude density spectrum, which in this case is in the form of impulses, rather than just a number. Similarly the Fourier transform of a train of impulses of the form (23.27) is given by (23.28) Energy and Power Spectral Density 6 Suppose x(t) is an aperiodic signal with a Fourier transform X( f ), then its energy is given by (23.29) This is Parseval’s theorem and it shows that the principle of conservation of energy in the time and frequency domains holds. The amplitude spectrum X( f ) can be expressed as TABLE 23.5 Properties of the Fourier Transform Property Signal Description Fourier Transform Linearity ax + by ; a, b constants aX + bY Evenness and oddness Time shift e −j2 π f τ X Time scale x Time reversal x(−t) X ∗ Duality Xx Time convolution x ∗ yXY Frequency convolution xy X ∗ Y Modulation X Time differentiation Frequency differentiation t n x Integration Correlation YX Parseval’s theorem t() t() f() f() xt–() xt()= xt–() x– t()= Xf() 2 xt() 2 π ft()cos td 0 ∞ ∫ = Xf() 2– xt() 2 π ft()sin td 0 ∞ ∫ = xt t–() f() at() 1 a X f a f() t() f–() t() t() f() f() t() t() f() f() xt()e j2 π f 0 t ff 0 –() d n dt n xt() j2pf() n Xf() t() j 2p n d n df n Xf() x τ () τ d ∞– ∞ ∫ 1 j2 π f Xf() 1 2 X 0() δ f()+ R xy τ () yt()xt τ –()td ∞– ∞ ∫ = f–() f() xt() 2 td ∞– ∞ ∫ Xf() 2 fd ∞– ∞ ∫ pt() δ tnT–() n=∞– ∞ ∑ = Pf() 1 T δ fkF s –(), F s k=∞– ∞ ∑ 1 T == ER xx 0() xt() 2 td ∞– ∞ ∫ Xf() 2 fd ∞– ∞ ∫ == = Xf() Xf() Xf()∠= 0066_Frame_C23 Page 15 Wednesday, January 9, 2002 1:53 PM ©2002 CRC Press LLC The Discrete Fourier Transform Consider a finite length sequence that is zero outside the interval 0 ≤ k ≤ N − 1. Evaluation of the z transform X ( z ) at N equally spaced points on the unit circle z = exp( i ω k T ) = exp[ i (2 π / NT ) kT ] for k = 0, 1,…, N − 1 defines the discrete Fourier transform (DFT) of a signal x with a sampling period h and N measurements: (23.60) Notice that the discrete Fourier transform is only defined at the discrete frequency points (23.61) In fact, the discrete Fourier transform adapts the Fourier transform and the z transform to the practical requirements of finite measurements. Similar properties hold for the discrete Laplace transform with z = exp( sT ), where s is the Laplace transform variable. The Transfer Function Consider the following discrete-time linear system with input sequence { u k } (stimulus) and output sequence { y k } (response). The dependency of the output of a linear system is characterized by the convolution- type equation and its z transform, (23.62) where the sequence { v k } represents some external input of errors and disturbances and with Y ( z ) = ᐆ { y }, U ( z ) = ᐆ { u }, V ( z ) = ᐆ { v } as output and inputs. The weighting function h ( kT ) = , which is zero for negative k and for reasons of causality is sometimes called pulse response of the digital system (compare impulse response of continuous-time systems). The pulse response and its z transform, the pulse transfer function , (23.63) determine the system’s response to an input U ( z ); see Fig. 23.18. The pulse transfer function H ( z ) is obtained as the ratio (23.64) FIGURE 23.18 Block diagram with an assumed transfer function relationship H ( z ) between input U ( z ), disturbance V ( z ), intermediate X ( z ), and output Y ( z ). {x k } k=0 N−1 X k DFT xkT(){} x l iw k lT–()exp l=0 N−1 ∑ Xe iw k T ()== = {X k } k=0 N−1 w k 2p NT k, for k 0, 1,…, N 1–== y k h m u k−m v k + m=0 ∞ ∑ h k−m u m v k , k+ m=−∞ k ∑ …, −1, 0, 1, 2,…== = Yz() Hz()Uz() Vz()+= {h k } k=0 ∞ Hz() ᐆ hkT(){}h k z k– k=0 ∞ ∑ == Hz() Xz() Uz() = U(z) X(z) Y(z) V(z) Σ H(z) 0066_Frame_C23 Page 33 Wednesday, January 9, 2002 1:55 PM ©2002 CRC Press LLC 24 State Space Analysis and System Properties 24.1 Models: Fundamental Concepts 24.2 State Variables: Basic Concepts Introduction • Basic State Space Models • Signals and State Space Description 24.3 State Space Description for Continuous-Time Systems Linearization • Linear State Space models • State Similarity Transformation • State Space and Transfer Functions 24.4 State Space Description for Discrete-Time and Sampled Data Systems Linearization of Discrete-Time Systems • Sampled Data Systems • Linear State Space Models • State Similarity Transformation • State Space and Transfer Functions 24.5 State Space Models for Interconnected Systems 24.6 System Properties Controllability, Reachability, and Stabilizability • Observability, Reconstructibility, and Detectability • Canonical Decomposition • PBH Test 24.7 State Observers Basic Concepts • Observer Dynamics • Observers and Measurement Noise 24.8 State Feedback Basic Concepts • Feedback Dynamics • Optimal State Feedback. The Optimal Regulator 24.9 Observed State Feedback Separation Strategy • Transfer Function Interpretation for the Single-Input Single-Output Case 24.1 Models: Fundamental Concepts An essential connection between an engineer/scientist and a system relies on his/her ability to describe the system in a way which is useful to understand and to quantify its behavior. Any description supporting that connection is a model . In system theory, models play a fundamental role, since they are needed to analyze, to synthesize, and to design systems of all imaginable sorts. There is not a unique model for a given system. Firstly, the need for a model may obey different purposes. For instance, when dealing with an electric motor, we might be interested in the electro- mechanical energy conversion process, alternatively, we might be interested in modelling the motor either as a thermal system, or as a mechanical system to study vibrations, the strength of the materials, and so on. Mario E. Salgado Universidad Técnica Federico Santa María Juan I. Yuz Universidad Técnica Federico Santa María ©2002 CRC Press LLC . CRC Press LLC the amplitude spectral density gives a continuous display of the amplitude density spectrum, which in this case is in the form of impulses, rather than just a number. Similarly the. Parseval’s theorem and it shows that the principle of conservation of energy in the time and frequency domains holds. The amplitude spectrum X( f) can be expressed as TABLE 23.5 Properties of the Fourier. the amplitude spectral density gives a continuous display of the amplitude density spectrum, which in this case is in the form of impulses, rather than just a number. Similarly the Fourier