The true dynamic system is described by a general first-order, ordinary differential equation (28.2) where f is the nonlinear dynamics function that incorporates all significant deterministic effects of the environment, αα αα is a vector of parameters used in the model, and w ( t ) is a random process that accounts for the noise present from mismodeling in f or from the quantum uncertainty of the universe, depending on the accuracy of the deterministic model in use. With these general models available, a linear Kalman filter (LKF) may be derived in a discrete-time formulation. The dynamics and measurement functions are linearized about a known reference state, ( t ), which is related to the true environment state, X ( t ), via (28.3) The LKF state estimate is related to the true difference by (28.4) where the “” denotes the state estimate (or filter state), is the estimation error, and “ ± ” indicates whether the estimate and error are evaluated instantaneously before ( − ) or after ( + ) measurement update at discrete time t k . It is important to emphasize that the LKF filter state is the estimate of the difference between the environment and the reference state. The LKF mode of operation will therefore carry along a reference state and the filter state between measurement updates. Only the filter state is at the time of measurement update. Figure 28.1 illustrates the generalized relationship between the true, reference, and filter states in an LKF estimating a two-dimensional trajectory. Linearization of Dynamic and Measurement System Models The dynamics and measurement functions may be linearized about the known reference state, ( t ), according to (28.5) (28.6) FIGURE 28.1 LKF tracking of a two-dimensional trajectory. Reference Trajectory True Trajectory State Estimate Estimation Error X ˙ t() f X t(), αα αα ,t()w t()+= X ˜ X ˜ t() x t()+ X t()= x ˆ k ±() x k dx k ±() += ˆ dx k ±() X ˜ f X, αα αα , t() Ӎ f X ˜ t(), αα αα , t()FX ˜ t(), αα αα , t()x t() w t()++ h X, αα αα , t() Ӎ h X ˜ t(), ββ ββ , t()HX ˜ t(), ββ ββ , t()x t() v t()++ 0066_Frame_C28 Page 2 Wednesday, January 9, 2002 7:19 PM ©2002 CRC Press LLC The true dynamic system is described by a general first-order, ordinary differential equation (28.2) where f is the nonlinear dynamics function that incorporates all significant deterministic effects of the environment, αα αα is a vector of parameters used in the model, and w ( t ) is a random process that accounts for the noise present from mismodeling in f or from the quantum uncertainty of the universe, depending on the accuracy of the deterministic model in use. With these general models available, a linear Kalman filter (LKF) may be derived in a discrete-time formulation. The dynamics and measurement functions are linearized about a known reference state, ( t ), which is related to the true environment state, X ( t ), via (28.3) The LKF state estimate is related to the true difference by (28.4) where the “” denotes the state estimate (or filter state), is the estimation error, and “ ± ” indicates whether the estimate and error are evaluated instantaneously before ( − ) or after ( + ) measurement update at discrete time t k . It is important to emphasize that the LKF filter state is the estimate of the difference between the environment and the reference state. The LKF mode of operation will therefore carry along a reference state and the filter state between measurement updates. Only the filter state is at the time of measurement update. Figure 28.1 illustrates the generalized relationship between the true, reference, and filter states in an LKF estimating a two-dimensional trajectory. Linearization of Dynamic and Measurement System Models The dynamics and measurement functions may be linearized about the known reference state, ( t ), according to (28.5) (28.6) FIGURE 28.1 LKF tracking of a two-dimensional trajectory. Reference Trajectory True Trajectory State Estimate Estimation Error X ˙ t() f X t(), αα αα ,t()w t()+= X ˜ X ˜ t() x t()+ X t()= x ˆ k ±() x k dx k ±() += ˆ dx k ±() X ˜ f X, αα αα , t() Ӎ f X ˜ t(), αα αα , t()FX ˜ t(), αα αα , t()x t() w t()++ h X, αα αα , t() Ӎ h X ˜ t(), ββ ββ , t()HX ˜ t(), ββ ββ , t()x t() v t()++ 0066_Frame_C28 Page 2 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 . 0066_Frame_C28 Page 2 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 . t()x t() v t()++ 0066_Frame_C28 Page 2 Wednesday, January 9, 2002 7: 19 PM 2002 CRC Press LLC The true dynamic system is described by a general first-order, ordinary differential equation (28.2) where. 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