CSE 314 S YSTEMS A NALYSIS Lecture 10

CSE 314: S YSTEMS A NALYSIS Lecture 10: Linear Dynamical Systems Dr. Ahmed Mahmoud, 29/12/2020

Linear Dynamical System Response 29/12/2020 CSE 314: Systems Analysis, Lec 10 2

DC Gain 29/12/2020 CSE 314: Systems Analysis, Lec 10 3

Continuous-Time Steady-State Response … (1) 29/12/2020 CSE 314: Systems Analysis, Lec 10 4

Continuous-Time Steady-State Response … (2) 29/12/2020 CSE 314: Systems Analysis, Lec 10 5

Remarks on System Response … (1) 29/12/2020 CSE 314: Systems Analysis, Lec 10 6

Discrete-Time Steady-State Response 29/12/2020 CSE 314: Systems Analysis, Lec 10 7

Initial Value Problems • Physical systems are often not initially at rest. • Dealing with non-zero initial conditions introduces some complexity in the analysis. • Mathematicians call such systems with non-zero initial conditions initial value problems. • We can adapt our methods to deal with initial conditions. 29/12/2020 CSE 314: Systems Analysis, Lec 10 8

LTI Difference Equations … (1) 29/12/2020 CSE 314: Systems Analysis, Lec 10 9

LTI Difference Equations … (2) • By Applying Z transform: • By rearranging: • Then, 29/12/2020 CSE 314: Systems Analysis, Lec 10 10

LTI Difference Equations … (3) 29/12/2020 CSE 314: Systems Analysis, Lec 10 11

Example on LTI Difference Equations … • Consider the difference equation: • Taking the Z transform: • Therefore, • The transfer function: 29/12/2020 CSE 314: Systems Analysis, Lec 10 12

Example on LTI Difference Equations: Zero-State Impulse Response 29/12/2020 CSE 314: Systems Analysis, Lec 10 13

Example on LTI Difference Equations: Zero-Input + Zero-State Responses 29/12/2020 CSE 314: Systems Analysis, Lec 10 14

Example on LTI Difference Equations: Total Response • This can also be expressed as the sum of the steady-state and the transient response: • The decomposition of the response into the sum of the zero-state and zero-input responses is different from its decomposition into the steady-state and transient responses. 29/12/2020 CSE 314: Systems Analysis, Lec 10 15

LTI Differential Equations … (1) 29/12/2020 CSE 314: Systems Analysis, Lec 10 16

LTI Differential Equations … (2) • By Applying Laplace transform: • By rearranging: • Then, 29/12/2020 CSE 314: Systems Analysis, Lec 10 17

LTI Differential Equations … (3) 29/12/2020 CSE 314: Systems Analysis, Lec 10 18

Example on LTI Differential Equations … 29/12/2020 CSE 314: Systems Analysis, Lec 10 19

Example on LTI Differential Equations: Total Response • Taking the inverse Laplace transform: • As in the case of difference equations, the decomposition of the response into zero-state and zero-input responses is different from the decomposition into transient and steadystate responses. • The steady-state response does not exist if the system is unstable, whereas the former decomposition always exists. 29/12/2020 CSE 314: Systems Analysis, Lec 10 20

Discrete-Time State Space Models 29/12/2020 CSE 314: Systems Analysis, Lec 10 21

Discrete-Time State Space Models • The discrete-time SISO state-space model response can also be written as: • By defining the following Z Transforms: • Then, 29/12/2020 CSE 314: Systems Analysis, Lec 10 22

Continuous-Time State Space Models 29/12/2020 CSE 314: Systems Analysis, Lec 10 23

Continuous-Time State Space Models • The continuous-time SISO state-space model response can also be written as: • By defining the following Laplace Transforms: • Then, 29/12/2020 CSE 314: Systems Analysis, Lec 10 24

Canonical State Space Form 29/12/2020 CSE 314: Systems Analysis, Lec 10 25

LTI System Representations • Difference or differential equations, used to describe many physical systems. • Transfer functions used for frequency-domain analysis, and in feedback design. • State-space models, used in modern control theory. • More recent approaches use finite state machines !!. • Visit this link: https: //www. linkedin. com/posts/mahmood-khaled_stochastic-verifiedamytiss-activity-6690510327928393728 -haz. L 29/12/2020 CSE 314: Systems Analysis, Lec 10 26

Reading • Read carefully sections 13. 6 to 13. 8 from Chapter 13. • Also, check this online course for further information: https: //see. stanford. edu/Course/EE 263 29/12/2020 CSE 314: Systems Analysis, Lec 10 27