Bishop, Robert H. - The Mechatronics Handbook [CRC Press 2002] Part 13 pot

We will see that for a closed-loop system, the polar plot of the loop transfer function is useful in determining the stability of the system. The polar plots of some simple systems are shown in Fig. 27.9. 27.4 Log-Magnitude Versus Phase Plots Another approach to presenting the frequency response of a system by a single graph is to plot its logarithmic magnitude versus the phase angle over a frequency range of interest. The resulting curve is a function of the frequency ω . Such log-magnitude versus phase plots are called Nichols charts. Advantages of the Nichols chart are that the relative stability of the closed-loop system can be deter- mined quickly and that the process of closed-loop compensation can be carried out easily. The Nichols charts of the systems in Fig. 27.9 are depicted in Fig. 27.10 for comparison. Figure 27.11 displays three different frequency-response curves of the second-order system FIGURE 27.7 Polar plots of system with various system types as ω → 0. FIGURE 27.8 Polar plots of system with various relative degrees as ω → ∞. Gs() ω n 2 s 2 2Vw n s ω n 2 ++ = 0066_frame_C27 Page 8 Wednesday, January 9, 2002 7:10 PM ©2002 CRC Press LLC We will see that for a closed-loop system, the polar plot of the loop transfer function is useful in determining the stability of the system. The polar plots of some simple systems are shown in Fig. 27.9. 27.4 Log-Magnitude Versus Phase Plots Another approach to presenting the frequency response of a system by a single graph is to plot its logarithmic magnitude versus the phase angle over a frequency range of interest. The resulting curve is a function of the frequency ω . Such log-magnitude versus phase plots are called Nichols charts. Advantages of the Nichols chart are that the relative stability of the closed-loop system can be deter- mined quickly and that the process of closed-loop compensation can be carried out easily. The Nichols charts of the systems in Fig. 27.9 are depicted in Fig. 27.10 for comparison. Figure 27.11 displays three different frequency-response curves of the second-order system FIGURE 27.7 Polar plots of system with various system types as ω → 0. FIGURE 27.8 Polar plots of system with various relative degrees as ω → ∞. Gs() ω n 2 s 2 2Vw n s ω n 2 ++ = 0066_frame_C27 Page 8 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 29 Digital Signal Processing for Mechatronic Applications 29.1 Introduction 29.2 Signal Processing Fundamentals Continuous-Time Signals • Discrete-Time Signals 29.3 Continuous-Time to Discrete-Time Mappings Discretization • s -Plane to z -Plane Mappings • Frequency Domain Mappings 29.4 Digital Filter Design IIR Filter Design • FIR Filter Design • Computer-Aided Design of Digital Filters • Filtering Examples 29.5 Digital Control Design Digital Control Example 29.1 Introduction Most engineers work in the world of mechatronics as there are relatively few systems that are purely mechanical or electronic. There are a variety of means by which electrical systems augment mechanical systems and vise versa. For example, most microprocessors found in a computer today have some sort of heat sink and perhaps a fan attached to them to keep them within their operational temperature zone. Electrical systems are widely employed to monitor and control a wide variety of mechanical systems. With the advent of inexpensive digital processing chips, digital filtering and digital control for mechanical systems is becoming commonplace. Examples of this can be seen in every automobile and most household appliances. For example, sensor signals used in monitoring and controlling of mechanical systems require some form of signal processing. This signal processing can range from simply “cleaning-up” the signal using a low pass filter to more advanced analyses such as torque and power monitoring in a DC servo motor. This chapter presents a brief overview of digital signal processing methods suitable for mechanical systems. Since this chapter is limited in space, it does not give any derivation or details of analysis. For a more detailed discussion, see references [1,2]. 29.2 Signal Processing Fundamentals A few fundamental concepts on signal processing must be introduced before a discussion of filtering or control can be undertaken. Bonnie S. Heck Georgia Institute of Technology Thomas R. Kurfess Georgia Institute of Technology ©2002 CRC Press LLC 30 Control System Design Via H 2 Optimization 30.1 Introduction 30.2 General Control System Design Framework Central Idea: Design Via Optimization • The Signals • General H 2 Optimization Problem • Generalized Plant • Closed Loop Transfer Function Matrices • Overview of H 2 Optimization Problems to Be Considered 30.3 H 2 Output Feedback Problem Hamiltonian Matrices 30.4 H 2 State Feedback Problem Generalized Plant Structure for State Feedback • State Feedback Assumptions 30.5 H 2 Output Injection Problem Generalized Plant Structure for Output Injection • Output Injection Assumptions 30.6 Summary 30.1 Introduction This chapter addresses control system design via H 2 (quadratic) optimization. A unifying framework based on the concept of a generalized plant and weighted optimization permits designers to address state feedback, state estimation, dynamic output feedback, and more general structures in a similar fashion. The framework permits one to easily incorporate design parameters and/or weighting functions that may be used to influence the outcome of the optimization, satisfy desired design specifications, and systematize the design process. Optimal solutions are obtained via well-known Riccati equations; e.g., Control Algebraic Riccati Equation (CARE) and Filter Algebraic Riccati Equation (FARE). While dynamic weight- ing functions increase the dimension of the Riccati equations being solved, solutions are readily obtained using today’s computer-aided design software (e.g., MATLAB, robust control toolbox, µ -synthesis tool- box, etc.). In short, H 2 optimization generalizes all of the well-known quadratic control and filter design methodologies: • Linear Quadratic Regulator (LQR) design methodology [7,11], • Kalman–Bucy Filter (KBF) design methodology [5,6], • Linear Quadratic Gaussian (LQG) design methodology [4,10,11]. H 2 optimization may be used to systematically design constant gain state feedback control laws, state estimators, dynamic output controllers, and much more. Armando A. Rodriguez Arizona State University 0066_Frame_C30 Page 1 Thursday, January 10, 2002 4:43 PM ©2002 CRC Press LLC . interest. The resulting curve is a function of the frequency ω . Such log-magnitude versus phase plots are called Nichols charts. Advantages of the Nichols chart are that the relative stability of the. chart are that the relative stability of the closed-loop system can be deter- mined quickly and that the process of closed-loop compensation can be carried out easily. The Nichols charts of the. 2002 7:10 PM ©2002 CRC Press LLC We will see that for a closed-loop system, the polar plot of the loop transfer function is useful in determining the stability of the system. The polar plots of some