Time domain response specificationsDefined based on unit step response with i.c. = 0Defined for closed-loop system
Prototype 2nd order system:target
Prototype 2nd order system:
Settling time:Remember:
+-Example:When given unit step input, the output looks like:Q: estimate k and .
Effects of additional zerosSuppose we originally have:i.e. step responseNow introduce a zero at s = -zThe new step response:
Effects:Increased speed,Larger overshoot,Might increase ts
When z < 0, the zero s = -z is > 0,is in the right half plane.
Such a zero is called a nonminimum phase zero.
A system with nonminimum phase zeros is called a nonminimum phase system.Nonminimum phase zero should beavoided in design.i.e. Do not introduce such a zero in your controller.
Effects of additional poleSuppose, instead of a zero, we introducea pole at s = -p, i.e.
L.P.F. has smoothing effect, oraveraging effectEffects:Slower,Reduced overshoot,May increase or decrease ts
StabilityBIBO-stable:Def: A system is BIBO-stable if any bounded input produces bounded output. Otherwise its not BIBO-stable.
Asymptotically StableA system is asymptotically stable if for any arbitrary initial conditions, all variables in the system converge to 0 as t when input=0.A system is marginally stable if for all initial conditions, all variables in the system remain finite, but for some initial conditions, some variable does not converge to 0 as t.A system is unstable if there are initial conditions that can cause some variables in the system to diverge to infinity.
A.S., M.S. and unstable are mutually exclusive.
Asymptotically Stable
Asymptotically Stable vs BIBO-stableThm: If a system is A.S., then it is BIBO-stable
If a system is not BIBO-stable, then it cannot be A.S., it has to be either M.S. or unstable.
But BIBO-stable does not guarantee A.S. in general.
If there is no pole/zero cancellation, then BIBO-stable Asymp Stable
Characteristic polynomialsThree types of models:Assume no p/z cancellationSystem characteristic polynomial is:
A polynomial
is said to be Hurwitz or stable if all of its roots are in O.L.H.P
A system is stable if its char. polynomial is HurwitzA nxn matrix is called Hurwitz or stableif its char. poly det(sI-A) is Hurwitz, orif all eigenvalues have real parts
Routh-Hurwitz MethodFrom now on, when we say stability we mean A.S. / M.S. or unstable.
We assume no pole/zero cancellation,A.S.BIBO stableM.S./unstable not BIBO stable
Since stability is determined by denominator, so just work with d(s)
Routh Table
Repeat the process until s0 row
Stability criterion:d(s) is A.S. iff 1st col have same signthe # of sign changes in 1st col = # of roots in right half plane
Note: if highest coeff in d(s) is 1,A.S. 1st col >0
If all roots of d(s) are
Example:has roots:3,2,-1
(1*3-2*5)/1=-7(1*10-2*0)/1=10(-7*5-1*10)/-7
Remember this
Remember this
e.g.
Routh CriteriaRegular case: (1) A.S. 1st col. all same sign(2)#sign changes in 1st col. =#roots with Re(.)>0
Special case 1: one whole row=0Solution: 1) use prev. row to form aux. eq. A(s)=0 2) get:
3) use coeff of to replace 0-row 4) continue as usual
Examplewhole row=0
Replace by e
Useful case: parameter in d(s)How to use: 1) form table as usual2) set 1st col. >03) solve for parameter range for A.S.2) set one in 1st col=03) solve for parameter that leads to M.S. or leads to sustained oscillation
Examples+3s(s+2)(s+1) Kp+
Q: find region of stability in K-a plane. aK
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