1. EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Stability

2. 8 - 2 Stability Many possible definitions Two key issues for practical systems System response to zero input: internal stability System response to non- zero but finite amplitude (bounded) input: bounded input bounded output (BIBO) stability For zero-input response If a system remains in a particular state (or condition) indefinitely, then state is an equilibrium state of system System’s output due to nonzero initial conditions should approach 0 as t  System’s output generated by initial conditions is made up of characteristic modes

3. 8 - 3 Stability Three cases for zero-input response A system is stable if and only if all characteristic modes go to 0 as t   A system is unstable if and only if at least one of the characteristic modes grows without bound as t   A system is marginally stable if and only if the zero-input response remains bounded (e.g. oscillates between lower and upper bounds) as t  

4. 8 - 4 Characteristic Modes Distinct characteristic roots 1, 2, …, n Where =  + j  in Cartesian form Units of  are in radians/second StableUnstable Im{ } Re{ } Marginally Stable Left-hand plane (LHP) Right-hand plane (RHP)

5. 8 - 5 Characteristic Modes Repeated roots For r repeated roots of value. For positive k Decaying exponential decays faster than t k increases for any value of k One can see this by using the Taylor Series approximation for e t about t = 0:

6. 8 - 6 Stability Conditions An LTIC system is asymptotically stable if and only if all characteristic roots are in LHP. The roots may be simple (not repeated) or repeated. An LTIC system is unstable if and only if either one or both of the following conditions exist: (i) at least one root is in the right-hand plane (RHP) (ii) there are repeated roots on the imaginary axis. An LTIC system is marginally stable if and only if there are no roots in the RHP, and there are no repeated roots on imaginary axis

7. 8 - 7 Response to Bounded Inputs Stable system: a bounded input (in amplitude) should give a bounded response (in amplitude) Test for linear-time-invariant (LTI) systems Bounded-Input Bounded-Output (BIBO) stable h(t)h(t) y(t)y(t)f(t)f(t)

8. 8 - 8 Impact of Characteristic Modes Zero-input response consists of the system’s characteristic modes Stable system  characteristic modes decay exponentially and eventually vanish If input has the form of a characteristic mode, then the system will respond strongly If input is very different from the characteristic modes, then the response will be weak

9. 8 - 9 Impact of Characteristic Modes Example: First-order system with characteristic mode e  t Three cases

10. 8 - 10  e -1 /RC 1/RC t h(t)h(t) System Time Constant When an input is applied to a system, a certain amount of time elapses before the system fully responds to that input Time lag or response time is the system time constant No single mathematical definition for all cases Special case: RC filter Time constant is  = RC Instant of time at which h(t) decays to e -1  0.367 of its maximum value

11. 8 - 11 t0t0 thth t h(t)h(t) h(t0)h(t0) h(t)h(t) ĥ(t)ĥ(t) System Time Constant General case: Effective duration is t h seconds where area under ĥ(t) C is an arbitrary constant between 0 and 1 Choose t h to satisfy this inequality General case applied to RC time constant:

12. 8 - 12 h(t)h(t) y(t)y(t)u(t)u(t) t u(t)u(t) 1 t h(t)h(t) A t y(t)y(t) A t h trtr Step Response y(t) = h(t) * u(t) Here, t r is the rise time of the system How does the rise time t r relate to the system time constant of the impulse response? A system generally does not respond to an input instantaneously trtr

13. 8 - 13 13 Filtering and Time Constant Impulse response Response to high frequency input Response to low frequency input

14. 8 - 14 Filtering A system cannot effectively respond to periodic signals with periods shorter than t h This is equivalent to a filter that passes frequencies from 0 to 1/t h Hz and attenuates frequencies greater than 1/t h Hz (lowpass filter) 1/t h is called the cutoff frequency 1/t r is called the system’s bandwidth (t r = t h ) Bandwidth is the width of the band of positive frequencies that are passed “unchanged” from input to output

15. 8 - 15 Transmission of Pulses Transmission of pulses through a system (e.g. communication channel) increases the pulse duration (a.k.a. spreading or dispersion) If the impulse response of the system has duration t h and pulse had duration t p seconds, then the output will have duration t h + t p Refer to slides 5-2, 5-3 and 5-4