CSE 416 DIGITAL CONTROL Lecture 07 Testing Stability

CSE 416: DIGITAL CONTROL Lecture 07: Testing Stability for Discrete-time Systems Dr. Ahmed Mahmoud, 25/11/2019

Characteristic Equation (1) • The feedback system has the following transfer function: • Using partial fractions, the system can be expressed as: 25/11/2019 CSE 416: DIGITAL CONTROL 2

Characteristic Equation (2) 25/11/2019 CSE 416: DIGITAL CONTROL 3

Stability for Discrete-time Systems • Many analysis and design techniques for continuous-time LTI systems, such as the Routh–Hurwitz criterion and Bode techniques, are based on the property that in the s-plane the stability boundary is the imaginary axis. • These techniques cannot be applied to LTI discrete-time systems in the z-plane since the stability boundary is the unit circle. • Either: 1. Transform the discrete-time system back to continuous system (Bilinear transform). 2. Use tools for discrete-time systems (Jury’s test). 25/11/2019 CSE 416: DIGITAL CONTROL 4

Bilinear Transform 25/11/2019 CSE 416: DIGITAL CONTROL 5

Review on Routh-Hurwitz Criterion (1) 1. For the characteristic equation: Form the Routh array as follows: 25/11/2019 CSE 416: DIGITAL CONTROL 6

Review on Routh-Hurwitz Criterion (2) 2. The remaining rows are calculated as follows: 3. Once the array has been formed, the Routh–Hurwitz criterion states that the number of roots of the characteristic equation with positive real parts is equal to the number of sign changes of the coefficients in the first column of the array. 25/11/2019 CSE 416: DIGITAL CONTROL 7

Example on Routh-Hurwitz with Bilinear … 25/11/2019 CSE 416: DIGITAL CONTROL 8

Example on Routh-Hurwitz with Bilinear … 25/11/2019 CSE 416: DIGITAL CONTROL 9

Example on Routh-Hurwitz with Bilinear … 25/11/2019 CSE 416: DIGITAL CONTROL 10

Jury’s Stability Test (1) 25/11/2019 CSE 416: DIGITAL CONTROL 11

Jury’s Stability Test (1) 25/11/2019 CSE 416: DIGITAL CONTROL 12

Example on Jury’s Stability Test • Investigate the stability for the following characteristic equation: • Test the first conditions: • Then, construct the Jury’s array as follows and test the remaining conditions: • The system is stable. 25/11/2019 CSE 416: DIGITAL CONTROL 13