data was then found with the MATLAB software. 13 The transfer-function between the applied input voltage V x (t) and the output of the inductive sensor y(t) was found to be (23.134) with units of V/V. Equation (23.135) was scaled by the inductive sensor gain (30 Å/V) and the transfer- function between the applied voltage V x (t) and the actual displacement of the piezo-tube x p (t) is given by (23.135) with units of Å/V. Time Scaling of a Transfer-Function Model We present below an approach for rescaling time for G 2 (s) from seconds [s] to milliseconds [ms]. We briefly recall the time scaling property of the Laplace transform presented in [1, Chapter 3, section 1.4]. Let F(s) be the Laplace transform of f(t), i.e., (23.136) where L denotes the Laplace transform operator. Now, consider a new time scale defined as , where a is a constant. The Laplace transform of (at) is given by (23.137) Using relation (23.137), we can reduce the coefficients of G 2 (s) by changing the time units of both the input signal u(t) and output signal y(t) as follows: (23.138) Therefore, to rescale time for G 2 (s) from seconds [s] to millisecond [ms], we choose and the new rescaled transfer (s) becomes (23.139) 13 The MATLAB command invfreqs gives real numerator and denominator coefficients of experimentally determined frequency response data. G 1 s() Ys() V x s() = 5.544 10 5 s 4 7.528 10 9 s 3 1.476 10 15 × s 2 4.571 10 18 s 9.415 10 23 ×+×–+×–× s 6 1.255 10 4 s 5 1.632 10 9 s 4 1.855 10 13 s 3 × 6.5 10 17 s 2 6.25 10 21 s 1.378 10 25 ×+×+×++×+×+ = G 2 s() X p s() V x s() = 1.663 10 7 s 4 2.258 10 11 s 3 4.427 10 16 × s 2 1.371 10 20 s 2.825 10 25 ×+×–+×–× s 6 1.255 10 4 s 5 1.632 10 9 s 4 1.855 10 13 s 3 × 6.5 10 17 s 2 6.25 10 21 s 1.378 10 25 ×+×+×++×+×+ = L ft() Fs()→ t ˆ at= f(t ˆ ) f= L ft ˆ () fat() 1 a F s a → F ˆ s()== G ˆ s() Y ˆ s() U ˆ s() Ys/a()/a Us/a()/a Ys/a() Us/a() == G s a == t ˆ at 0.001t== G ˆ 2 G ˆ 2 s() G 2 s a a=0.001 = G 2 1000s()= G ˆ 2 s() 16.63s 4 225.8s 3 4.427 10 4 s 2 1.371 10 5 s×– 2.825 10 7 ×+×+– s 6 12.55s 5 1.632+ 10 3 s 4 1.855 10 4 s 3 6.5 10 5 s 2 6.25 10 6 s 1.378 10 7 ×+×+×+×+×+ = 0066_Frame_C23 Page 50 Wednesday, January 9, 2002 1:56 PM ©2002 CRC Press LLC data was then found with the MATLAB software. 13 The transfer-function between the applied input voltage V x (t) and the output of the inductive sensor y(t) was found to be (23.134) with units of V/V. Equation (23.135) was scaled by the inductive sensor gain (30 Å/V) and the transfer- function between the applied voltage V x (t) and the actual displacement of the piezo-tube x p (t) is given by (23.135) with units of Å/V. Time Scaling of a Transfer-Function Model We present below an approach for rescaling time for G 2 (s) from seconds [s] to milliseconds [ms]. We briefly recall the time scaling property of the Laplace transform presented in [1, Chapter 3, section 1.4]. Let F(s) be the Laplace transform of f(t), i.e., (23.136) where L denotes the Laplace transform operator. Now, consider a new time scale defined as , where a is a constant. The Laplace transform of (at) is given by (23.137) Using relation (23.137), we can reduce the coefficients of G 2 (s) by changing the time units of both the input signal u(t) and output signal y(t) as follows: (23.138) Therefore, to rescale time for G 2 (s) from seconds [s] to millisecond [ms], we choose and the new rescaled transfer (s) becomes (23.139) 13 The MATLAB command invfreqs gives real numerator and denominator coefficients of experimentally determined frequency response data. G 1 s() Ys() V x s() = 5.544 10 5 s 4 7.528 10 9 s 3 1.476 10 15 × s 2 4.571 10 18 s 9.415 10 23 ×+×–+×–× s 6 1.255 10 4 s 5 1.632 10 9 s 4 1.855 10 13 s 3 × 6.5 10 17 s 2 6.25 10 21 s 1.378 10 25 ×+×+×++×+×+ = G 2 s() X p s() V x s() = 1.663 10 7 s 4 2.258 10 11 s 3 4.427 10 16 × s 2 1.371 10 20 s 2.825 10 25 ×+×–+×–× s 6 1.255 10 4 s 5 1.632 10 9 s 4 1.855 10 13 s 3 × 6.5 10 17 s 2 6.25 10 21 s 1.378 10 25 ×+×+×++×+×+ = L ft() Fs()→ t ˆ at= f(t ˆ ) f= L ft ˆ () fat() 1 a F s a → F ˆ s()== G ˆ s() Y ˆ s() U ˆ s() Ys/a()/a Us/a()/a Ys/a() Us/a() == G s a == t ˆ at 0.001t== G ˆ 2 G ˆ 2 s() G 2 s a a=0.001 = G 2 1000s()= G ˆ 2 s() 16.63s 4 225.8s 3 4.427 10 4 s 2 1.371 10 5 s×– 2.825 10 7 ×+×+– s 6 12.55s 5 1.632+ 10 3 s 4 1.855 10 4 s 3 6.5 10 5 s 2 6.25 10 6 s 1.378 10 7 ×+×+×+×+×+ = 0066_Frame_C23 Page 50 Wednesday, January 9, 2002 1:56 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 . a=0.001 = G 2 1000s()= G ˆ 2 s() 16.63s 4 2 25. 8s 3 4.427 10 4 s 2 1 .37 1 10 5 s×– 2.8 25 10 7 ×+×+– s 6 12 .55 s 5 1. 632 + 10 3 s 4 1. 855 10 4 s 3 6 .5 10 5 s 2 6. 25 10 6 s 1 .37 8 10 7 ×+×+×+×+×+ = 0066_Frame_C 23 Page 50 Wednesday,. a=0.001 = G 2 1000s()= G ˆ 2 s() 16.63s 4 2 25. 8s 3 4.427 10 4 s 2 1 .37 1 10 5 s×– 2.8 25 10 7 ×+×+– s 6 12 .55 s 5 1. 632 + 10 3 s 4 1. 855 10 4 s 3 6 .5 10 5 s 2 6. 25 10 6 s 1 .37 8 10 7 ×+×+×+×+×+ = 0066_Frame_C 23 Page 50 Wednesday,. = G 2 s() X p s() V x s() = 1.6 63 10 7 s 4 2. 258 10 11 s 3 4.427 10 16 × s 2 1 .37 1 10 20 s 2.8 25 10 25 ×+×–+×–× s 6 1. 255 10 4 s 5 1. 632 10 9 s 4 1. 855 10 13 s 3 × 6 .5 10 17 s 2 6. 25 10 21 s 1 .37 8 10 25 ×+×+×++×+×+