Chapter 2: Time-Domain Representations of Linear Time-Invariant Systems. Chih-Wei Liu
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1 Chapter : Time-Domain Representations of Linear Time-Invariant Systems Chih-Wei Liu
2 Outline Characteristics of Systems Described by Differential and Difference Equations Block Diagram Representations State-Variable Descriptions of LTI Systems Exploring Concepts with MATLAB Summary Lec - cwliu@twins.ee.nctu.edu.tw
3 Block Diagram Representations A block diagram is a graphical representation of the elementary operations acting on the input signal. Scalar multiplication: y(t) = cx(t), and y[n] = cx[n]. Addition: y(t) = x(t) + w(t), and y[n] = x[n] + w[n] t 3. Integration and time shift: and y[n] = x[n ]. yt () x( ) d t yt () x( ) d (a) (b) Figure.3 (c) (a) Scalar multiplication. (b) Addition. (c) Integration for continuous-time systems and time shifting for discrete-time systems.
4 Discrete-Time Block Diagram Representation Example Cascade Form (Direct Form I) y[n] ay[n ] ay[n ] b0x[n] bx[n ] bx[n ] Feed-forward path Feedback path Second-order 4 storage elements w[n] b0x[n] bx[n ] bx[n ] y[n] w[n] ay[n ] ay[n ]
5 Discrete-Time Block Diagram Representation Example Interchange the order of Direct Form I. Canonical Form (Direct Form II) y[n] ay[n ] ay[n ] b0x[n] bx[n ] bx[n ] Feedback path Feed-forward path Second-order storage elements f[n] af[n ] af[n ] x[n] y[n] b0f[n] bf[n] bf[n]
6 Continuous-Time Block Diagram Representation d dt n n t t t n, 0 and,, 3,... n t n t d n,,, 3,... () () () () Ex. Second-order system: y(t) ay (t) a y (t) b x(t) b x (t) b x (t) 0 0 Cascade Form (Direct Form I) Canonical Form (Direct Form II) 6 Lec - cwliu@twins.ee.nctu.edu.tw
7 State-Variable Description of LTI Systems The state-variable description consist of A series of first-order differential or difference equations that describe how the state of the system evolves An equation that relates the output of the system to the current state variables and input q [n+] State variables (The output of the S operation) The next value of the state (The input to the S operation) the state of a system may be defined as a minimal set of signals that represents the system s entire memory of the past. canonical Form
8 State-Variable Description Example q [n+]. Start from the canonical form (i.e. direct form II) of the system. Choose state variables: q [n] and q [n]. 3. State equation: 3. Matrix Form of state equation: q[n ] aq[n] aq[n] x[n] q[n ] a aq[n] q[n ] q[n] x[n] q[n ] 0 q[n] 0 4. Output equation: 4. Matrix form of output equation: y[n] x[n] aq [n] aq [n] bq [n] bq [n], q[n] y[n] (ba )q [n] (b a )q [n] x[n] y[n] [ba b a ] []x[n] q[n]
9 State-Variable Description Example (conti.) Define state vector as the column vector q[n] q[n] q[n] We can rewrite equations as q[n ] Aq[n] bx[n] y[n] cq[n] Dx[n] where matrix A, vectors b and c, and scalar D are given by A a a 0 b 0 c b a b a D q [n+]. Except block diagram, the state-variable describes the internal structure of the system. The state-variable description is used in any problem in which the internal system structure needs to be considered 3. For N-th order difference equation, the state vector q[n] is N, and matrices A is NN, b is N, c is N. 9 Lec - cwliu@twins.ee.nctu.edu.tw
10 Example.8 Find the state-variable description corresponding to the system depicted in Fig..40 by choosing the state variable to be the outputs of the unit delays. <Sol.>. State equation: q n q n x n 0 q n q n q n x n 3. Define state vector as q n n q n q q[n ] Aq[n] bx[n] y[n] cq[n] Dx[n]. Output equation: 0 A c yn q n q n b D
11 State-Variable Description of Continuous-Time LTI System Recall discrete-time case q[n ] Aq[n] bx[n] y[n] cq[n] Dx[n] For continuous-time LTI system d q (t) Aq (t) b x(t) dt y(t) cq(t) Dx(t) States in continuous-time systems? In electrical (continuous-time) systems, the (energy) storage devices are the capacitor and the inductor. Lec - cwliu@twins.ee.nctu.edu.tw
12 Example.9 Derive a state-variable description of the system if the input is the applied voltage x(t) and the output is the current y(t) through the resistor R. <Sol.>. State variables: the voltage across each capacitor.. KVL Eq. for the loop involving x(t), R, and C : xt y t Rq t KVL Eq. for the loop involving C, R, and C : q t Ri() t q() t y(t) q (t) x(t) R R Output eq. 3. Current Eq: d y ti i t C () tit q t d q t q () t q () t dt dt CR CR d i ( t) y( t) i ( t) i t C q t dt d dt R q ( t) R x( t) R q ( t) R q ( t) q ( t) ( ) q( t) q( t) x( t ) C R R C R C R Lec - cwliu@twins.ee.nctu.edu.tw
13 Example.30 Determine the state-variable description corresponding to the block diagram. <Sol.> d q t q t q dt t x t d q q dt 3 y t q t q t Output eq. 3 Lec - cwliu@twins.ee.nctu.edu.tw
14 State Transformation The transformation of states can be considered as a new set of state variables that are a weighted sum of the original ones. After state variables transformation, the IO characteristics of the system (or the system s behavior) is not changed.. Original state-variable description: q Aq bx y cq Dx. Transformation: q = Tq q = T q 3. New state-variable description: ) State equation: q TAq Tbx. q TAT q Tbx. ) Output equation: y cq Dx y ct q' Dx then q ' A' b' x and A TAT, btb, cct, and D D y c' q' D' x q = Tq 4 Lec - cwliu@twins.ee.nctu.edu.tw
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