Crc Press Mechatronics Handbook 2002 By Laxxuss Episode 3 Part 8 doc

Example 4 (revisited) In this example, if K increases from to , the closed-loop system poles move along the comple- mentary root locus, and then the usual root locus, as illustrated in Fig. 26.17. 26.5 Root Locus for Systems with Time Delays The standard feedback control system considered in this section is shown in Fig. 26.18, where the controller C and plant P are in the form and with ( N c , D c ) and ( N p , D p ) being coprime pairs of polynomials with real coefficients. 2 The term e − hs is the transfer function of a pure delay element (in Fig. 26.18 the plant input is delayed by h seconds). In general, time delays enter into the plant model when there is • a sensor (or actuator) processing delay, and/or • a software delay in the controller, and/or • a transport delay in the process. In this case the open-loop transfer function is where G 0 ( s ) = P 0 ( s ) C ( s ) corresponds to the no delay case, h = 0. Note that magnitude and phase of G ( j w ) are determined from the identities (26.18) (26.19) FIGURE 26.17 Complementary and usual root loci for Example 4. 2 A pair of polynomials is said to be coprime pair if they do not have common roots. −4 −2 0 2 4 6 8 −5 −4 −3 −2 −1 0 1 2 3 4 5 Real Axis Imag Axis Complete Root Locus for −∞ < K < +∞ ∞– +∞ Cs() N c s() D c s() = Ps() e hs– P 0 s() where P 0 s() N p s() D p s() == Gs() G h s() e hs– G 0 s()== Gjw() G 0 jw()= Gjw()∠ hw – G 0 jw()∠+= 066_Frame_C26 Page 17 Wednesday, January 9, 2002 1:59 PM ©2002 CRC Press LLC Example 4 (revisited) In this example, if K increases from to , the closed-loop system poles move along the comple- mentary root locus, and then the usual root locus, as illustrated in Fig. 26.17. 26.5 Root Locus for Systems with Time Delays The standard feedback control system considered in this section is shown in Fig. 26.18, where the controller C and plant P are in the form and with ( N c , D c ) and ( N p , D p ) being coprime pairs of polynomials with real coefficients. 2 The term e − hs is the transfer function of a pure delay element (in Fig. 26.18 the plant input is delayed by h seconds). In general, time delays enter into the plant model when there is • a sensor (or actuator) processing delay, and/or • a software delay in the controller, and/or • a transport delay in the process. In this case the open-loop transfer function is where G 0 ( s ) = P 0 ( s ) C ( s ) corresponds to the no delay case, h = 0. Note that magnitude and phase of G ( j w ) are determined from the identities (26.18) (26.19) FIGURE 26.17 Complementary and usual root loci for Example 4. 2 A pair of polynomials is said to be coprime pair if they do not have common roots. −4 −2 0 2 4 6 8 −5 −4 −3 −2 −1 0 1 2 3 4 5 Real Axis Imag Axis Complete Root Locus for −∞ < K < +∞ ∞– +∞ Cs() N c s() D c s() = Ps() e hs– P 0 s() where P 0 s() N p s() D p s() == Gs() G h s() e hs– G 0 s()== Gjw() G 0 jw()= Gjw()∠ hw – G 0 jw()∠+= 066_Frame_C26 Page 17 Wednesday, January 9, 2002 1:59 PM ©2002 CRC Press LLC 27 Frequency Response Methods 27.1 Introduction 27.2 Bode Plots 27.3 Polar Plots 27.4 Log-Magnitude Versus Phase plots 27.5 Experimental Determination of Transfer Functions 27.6 The Nyquist Stability Criterion 27.7 Relative Stability 27.1 Introduction The analysis and design of industrial control systems are often accomplished utilizing frequency response methods. By the term frequency response, we mean the steady-state response of a linear constant coefficient system to a sinusoidal input test signal. We will see that the response of the system to a sinusoidal input signal is also a sinusoidal output signal at the same frequency as the input. However, the magnitude and phase of the output signal differ from those of the input signal, and the amount of difference is a function of the input frequency. Thus, we will be investigating the relationship between the transfer function and the frequency response of linear stable systems. Consider a stable linear constant coefficient system shown in Fig. 27.1. Using Euler’s formula, e j ω t = cos ω t + j sin ω t , let us assume that the input sinusoidal signal is given by (27.1) Taking the Laplace transform of u ( t ) gives (27.2) The first term in Eq. (27.2) is the Laplace transform of U 0 cos ω t , while the second term, without the imaginary number j , is the Laplace transform of U 0 sin ω t . Suppose that the transfer function G ( s ) can be written as (27.3) ut() U 0 e jwt U 0 wtcos jU 0 wtsin+== Us() U 0 sjw– U 0 sjw+ s 2 w 2 + U 0 s s 2 w 2 + j U 0 w s 2 w 2 + +== = Gs() ns() ds() ns() sp 1 +()sp 2 +() … sp n +() == Jyh-Jong Sheen National Taiwan Ocean University 0066_frame_C27 Page 1 Wednesday, January 9, 2002 7:10 PM ©2002 CRC Press LLC 28 Kalman Filters as Dynamic System State Observers 28.1 The Discrete-Time Linear Kalman Filter Linearization of Dynamic and Measurement System Models • Linear Kalman Filter Error Covariance Propagation • Linear Kalman Filter Update 28.2 Other Kalman Filter Formulations The Continuous–Discrete Linear Kalman Filter • The Continuous–Discrete Extended Kalman Filter 28.3 Formulation Summary and Review 28.4 Implementation Considerations 28.1 The Discrete-Time Linear Kalman Filter Distilled to its most fundamental elements, the Kalman filter [1] is a predictor-corrector estimation algorithm that uses a dynamic system model to predict state values and a measurement model to correct this prediction. However, the Kalman filter is capable of a great deal more than just state observation in such a manner. By making certain stochastic assumptions, the Kalman filter carries along an internal metric of the statistical confidence of the estimate of individual state elements in the form of a covariance matrix. The essential properties of the Kalman filter are derived from the requirements that the state estimate be • a linear combination of the previous state estimate and current measurement information • unbiased with respect to the true state • and optimal in terms of having minimum variance with respect to the true state. Starting with these basic requirements an elegant and efficient formulation for the implementation of the Kalman filter may be derived. The Kalman filter processes a time series of measurements to update the estimate of the system state and utilizes a dynamic model to propagate the state estimate between measurements. The observed measurement is assumed to be a function of the system state and can be represented via (28.1) where Y ( t ) is an m dimensional observable, h is the nonlinear measurement model, X ( t ) is the n dimensional system state, ββ ββ is a vector of modeling parameters, and v ( t ) is a random process accounting for measurement noise. Y t() h X t(), ββ ββ ,t()v t()+= Timothy P. Crain II NASA Johnson Space Center 0066_Frame_C28 Page 1 Wednesday, January 9, 2002 7:19 PM ©2002 CRC Press LLC . University 0066_frame_C27 Page 1 Wednesday, January 9, 2002 7:10 PM 2002 CRC Press LLC 28 Kalman Filters as Dynamic System State Observers 28. 1 The Discrete-Time Linear Kalman Filter Linearization. 066_Frame_C26 Page 17 Wednesday, January 9, 2002 1:59 PM 2002 CRC Press LLC 27 Frequency Response Methods 27.1 Introduction 27.2 Bode Plots 27 .3 Polar Plots 27.4 Log-Magnitude Versus. t()+= Timothy P. Crain II NASA Johnson Space Center 0066_Frame_C 28 Page 1 Wednesday, January 9, 2002 7:19 PM 2002 CRC Press LLC