EE 380. Linear Control Systems. Lecture 10

Transcription

1 EE 380 Linear Control Systems Lecture 10 Professor Jeffrey Schiano Department of Electrical Engineering Lecture 10. 1

2 Lecture 10 Topics Stability Definitions Methods for Determining Stability Lecture 10. 2

3 Types of Stability Internal Stability Describes behavior of state variables Determined by Roots of the characteristic equation Eigenvalues of the system matrix External Stability Describes input-output behavior Determined by Impulse response function Transfer function poles Lecture 10. 3

4 Systems of Interest Linear Time-Invariant Causal (LTIC) Systems ODE Representation n n 1 m m 1 d y d y d u d u a 1 () 1 1 () n n aoy t b n m b m m b m 1 ou t dt dt dt dt State-Space Representation x Ax Bu y Cx Du Transfer Function Representation (SISO) Y s bs b s b C si A B D U s s a s a m m 1 1 m m 1 0 n n 1 n 1 0 Lecture 10. 4

5 Characteristic Roots Consider a LTIC system represented by the ODE n n 1 m m 1 d y d y d u d u n n 1 n 1 o m m m 1 m 1 o a a y() t b b b u() t dt dt dt dt The characteristic equation is n Q( ) a a a 0 n 1 n 1 1 The roots of the characteristic equation are identical to the eigenvalues of the system matrix A o The roots of the characteristic equation determine the natural response of the system Lecture 10. 5

6 Internal Stability Definitions A LTIC system is asymptotically stable if, and only if, all the characteristic roots are in the left-half plane (LHP). The roots may be simple or repeated. A LTIC system is unstable if, and only if, any of the following conditions hold At least one root in the right-half plane (RHP) Repeated roots on the imaginary axis A LTIC system is marginally stable if, and only if, there are no RHP roots, and there are some unrepeated (simple) poles on the imaginary axis Lecture 10. 6

7 External Stability Definition A system is said to be bounded-input boundedoutput (BIBO) stable if, for zero initial conditions, every bounded input yields a bounded output ut () G () p s yt () BIBO stability means that if u(t) C 1 < for all t, then y(t) C 2 < for all t, where C 1 and C 2 are finite constants Lecture 10. 7

8 Conditions for BIBO Stability A LTIC system represented by a proper transfer function G p (s) is BIBO stable if and only if all the poles of G p (s) lie strictly in the left-half plane A LTIC system represented by the impulse response function h(t) is BIBO stable if and only 0 h( ) d C 3 for some finite constant C 3 Lecture 10. 8

9 Example 1 Determine the internal and external stability of the following systems dy (1) ut ( ) dt 2 d y (2) ut ( ) 2 dt Y() s (3) s 2 2 U() s s 2s (4) x x u y 1 3 (5) y y 2y f f x Lecture 10. 9

10 Example 1 Solution Lecture

11 Example 1 Solution Lecture

12 Example 1 Solution Lecture

13 Example 1 Solution Lecture

14 Example 1 Solution Lecture

15 Methods for Determining Stability Factor the characteristic equation by hand to determine the characteristic roots Use a CAD tool to determine the characteristic roots Use the Routh-Hurwitz criterion to determine how many roots do not lie in the strict LHP Lecture

16 EE 380 Linear Control Systems Lecture 10 Professor Jeffrey Schiano Department of Electrical Engineering Lecture 10. 1

17 Lecture 10 Topics Stability Definitions Methods for Determining Stability Lecture 10. 2

18 Types of Stability Internal Stability Describes behavior of state variables Determined by Roots of the characteristic equation Eigenvalues of the system matrix External Stability Describes input-output behavior Determined by Impulse response function Transfer function poles Lecture 10. 3

19 Systems of Interest Linear Time-Invariant Causal (LTIC) Systems ODE Representation State-Space Representation Transfer Function Representation (SISO) Lecture 10. 4

20 Characteristic Roots Consider a LTIC system represented by the ODE The characteristic equation is The roots of the characteristic equation are identical to the eigenvalues of the system matrix A The roots of the characteristic equation determine the natural response of the system Lecture 10. 5

21 Internal Stability Definitions A LTIC system is asymptotically stable if, and only if, all the characteristic roots are in the left-half plane (LHP). The roots may be simple or repeated. A LTIC system is unstable if, and only if, any of the following conditions hold At least one root in the right-half plane (RHP) Repeated roots on the imaginary axis A LTIC system is marginally stable if, and only if, there are no RHP roots, and there are some unrepeated (simple) poles on the imaginary axis Lecture 10. 6

22 External Stability Definition A system is said to be bounded-input boundedoutput (BIBO) stable if, for zero initial conditions, every bounded input yields a bounded output BIBO stability means that if u(t) C 1 < for all t, then y(t) C 2 < for all t, where C 1 and C 2 are finite constants Lecture 10. 7

23 Conditions for BIBO Stability A LTIC system represented by a proper transfer function G p (s) is BIBO stable if and only if all the poles of G p (s) lie strictly in the left-half plane A LTIC system represented by the impulse response function h(t) is BIBO stable if and only for some finite constant C 3 Lecture 10. 8

24 Example 1 Determine the internal and external stability of the following systems Lecture 10. 9

25 Example 1 Solution Lecture

26 Example 1 Solution Lecture

27 Example 1 Solution Lecture

28 Example 1 Solution Lecture

29 Example 1 Solution Lecture

30 Methods for Determining Stability Factor the characteristic equation by hand to determine the characteristic roots Use a CAD tool to determine the characteristic roots Use the Routh-Hurwitz criterion to determine how many roots do not lie in the strict LHP Lecture