Prototype 2nd order system: target

Type 1: Settling time: = (3 or 4 or 5)/s For 5%, 2%, 1% tol

Pole location determines transient

All closed-loop poles must be strictly in the left half planes Transient dies away Dominant poles: those which contribute the most to the transient Typically have dominant pole pair (complex conjugate) Closest to jω-axis (i.e. the least negative) Slowest to die away

Typical design specifications Steady-state: ess to step ≤ # % ts ≤ · · · Speed (responsiveness) tr ≤ · · · td ≤ · · · Relative stability Mp ≤ · · · %

These specs translate into requirements on ζ, ωn or on closed-loop pole location : Find ranges for ζ and ωn so that all 3 are satisfied.

Find conditions on σ and ωd.

In the complex plane : Any pole on the ray have the same z p=-s + jwd Mp < … or z > … corresponds to a conic region about the neg real axis j -s

Constant σ : vertical lines σ > # is half plane Any poles on the same vertical line have the same s, and the same settling time ts s=1 or ts=5 s>2 or ts=2.5 ts < … corresponds to a half plane to the left of a vertical line

Constant ωd : horizontal line ωd < · · · is a band ωd > · · · is the plane excluding band Any poles on the same horizontal line have the same wd, and the same oscillation frequency A centered horizontal band corresponds to oscillation frequency < … The plane excluding a centered horizontal band corresponds to oscillation frequency > …

ωn < · · · inside of a circle ωn > · · · outside of a circle Constant ωn : circles ωn < · · · inside of a circle ωn > · · · outside of a circle Any poles on the same circle have the same wn, and similar rise time/delay time Inside circle corresponds to tr> … Outside a circle corresponds to tr < …

Constant ζ : φ = cos-1ζ constant Constant ζ = ray from the origin ζ > · · · is the cone ζ < · · · is the other part

If more than one requirement, get the common (overlapped) area e.g. ζ > 0.5, σ > 2, ωn > 3 gives Sometimes meeting two will also meet the third, but not always.

Try to remember these:

When given unit step input, the output looks like: Example: + - When given unit step input, the output looks like: Q: estimate k and τ.

When given unit step input, the output looks like: Example: + - When given unit step input, the output looks like: Q: estimate k and τ. Yss = ____; is it prototype? Mp = ____; tp = ____; 5% ts = ____

Solve for J and t

Effects of additional zeros Suppose we originally have: i.e. step response Now introduce a zero at s = -z The 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 be avoided in design. i.e. Do not introduce such a zero in your controller.

Effects of additional pole Suppose, instead of a zero, we introduce a pole at s = -p, i.e.

L.P.F. has smoothing effect, or averaging effect Effects: Slower, Reduced overshoot, May increase or decrease ts