Lesson_28 - Digital Signal Processing Dr. Fred J. Taylor,...

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Digital Signal Processing Dr. Fred J. Taylor, Professor Lesson Title: State Variable Basics Lesson Number: [28] (Supplemental) State-Determined Systems The ability of form and manipulate transfer functions is fundamentally important to the analysis of continuous- and discrete-time at-rest LTI systems. A transfer function, however, only quantifies input-output behavior and does not describe the internal behavior of a system is defined by the system’s architecture . Architecture refers to how the basic system building blocks are assembled to implement a particular solution. What is desired is a representation methodology that is capable of producing information about both internal and external system behavior. This is the role of state variables. State Variables State variables, or simply states , represent the information are found within a dynamical system. For analog systems, the information is assume to be stored on analog devices having memory (capacitor (voltage) and inductor (current)). For discrete-time system, information is store in memory, or register for digital devices. In this context it is assumed that: 1. The future states of a state-determined system can be computed if, 2. all past state values are known, and 3. the mathematical relationship (sometimes called the bonds on interaction) between the state variables are known. This process is suggested in Figure 1. Figure 1: State variable information model. The general form of a single-input single-output continuous-time n-state state model (n information storage locations) is: Current States Past States State Model System Inputs Future States Memory STATE - 1
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Digital Signal Processing Dr. Fred J. Taylor, Professor ( 29 Equation Output t dv t c t y Equation tate S t v t dt t d T ) ( ) ( ; ) 0 ( ); ( ) ( ) ( 0 + = = + = x x x b Ax x 1. Here x (t) is a n -dimensional vector of states, v(t) is the input, and y(t) is the output. The other terms are A, an nxn matrix, b, an nx1 vector, c, an nx1 matrix, and d a scalar. The state- determined system is graphically interpreted in Figure 1. Figure 1: Single-input single-output state-determined continuous-time system. It is worth noting that any system defined by an n th -order ODE can be placed in the form shown in Equation 1 and Figure 1. To illustrate this point, consider the following simple example. Example Consider the RLC circuit shown in Figure 2. The system states are the inductor current and capacitor voltage. Figure 2: Simple RLC circuit. The state equations are produced in the following manner. ) ( ) ( 1 ) ( ) ( ) ( ) ( 0 t v dt t i C t v t v t Ri dt t di L C t L C L L = = + + Define x 1 (t)=i L (t) and x 2 (t)=v C (t). It then follows that the 2-state system is modeled as: x 0 A b x (t) c T d y(t) v(t) R L C v(t) i L v C STATE - 2
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Digital Signal Processing Dr. Fred J. Taylor, Professor ) ( 0 / 1 ) ( ) ( 0 / 1 / 1 / / ) ( / ) ( ) ( 2 1 2 1 t v L t x t x C L L R dt t dx dt t dx dt t d + -
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Lesson_28 - Digital Signal Processing Dr. Fred J. Taylor,...

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