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 functio
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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 G0(s) = P0(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.
−5
−4
−3
−2
−1 0 1 2 3 4 5
Real Axis
Complete Root Locus for −∞ < K < +∞
∞
-=
-=
=
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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 G0(s) = P0(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.
−5
−4
−3
−2
−1 0 1 2 3 4 5
Real Axis
Complete Root Locus for −∞ < K < +∞
∞
-=
-=
=
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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 jωt= cosωt+j sinωt, 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 U0cosωt, while the second term, without the imaginary number j, is the Laplace transform of U0 sinωt
Suppose that the transfer function G(s) can be written as
(27.3)
- U0s+jw
- U0s
- j U0w
-+
Jyh-Jong Sheen
National Taiwan Ocean University
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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, ββββis a vector of modeling parameters, and v(t) is a random process accounting for measurement noise
Timothy P Crain II
NASA Johnson Space Center
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