What damping ratio range do we typically want?___0.4~1____ For 5% Mp, what damping ratio do we need:___0.7___ For 10% Mp, what damping ratio do we need:__0.6____ For 15% Mp, what damping ratio do we need:___0.5___ Overshoot formula in terms of damping ratio is: Mp =___e^(-pi*zeta/rt(1-zeta^2)) For +-1% settling the approximate settling time is: ts = _5/sigma__ For +-5% settling the approximate settling time is: ts = _3/sigma_ A slightly more accurate formula for settling time is ts = _-ln(tol*rt(1-zeta^2))/sigma_ A very crude formula for rise time is tr = __1.8/wn, or 1.5 to 2.2 / wn____

The root locus technique Obtain closed-loop TF and char eq d(s) = 0 Re-arrange to get Mark zeros with “o” and poles with “x” High light segments of x-axis and put arrows Decide #asymptotes, their angles, and x-axis meeting place: Determine jw-axis crossing using Routh table Compute breakaway: Departure/arrival angle:

Effects of additional pole One additional R.L. branch shoots out It increases # asymp. by one More asymptotes go towards +Re-axis More likely to be unstable Poles tend to push R.L. away from them Don’t introduce poles unless required by other concerns

Examples:

Effects of additional zero It sinks one branch of R.L. It reduces the # asymp. by one Asymptotes move more towards –Re-axis More likely to be stable Zeros attract R.L. Each zero attracts one branch If > 1 branches nearby, they go to Re-axis & split, the one branch goes to zero Never have >= 2 branches go to a zero

Examples:

If we put that additional zero near (0,0): The dominant pole pair are more negative But there is one pole (real) close to s = 0, which will settle very slowly (sluggish settling)

Controller design by R.L. Typical setup: C(s) G(s)

This is the R.L. eq. With no zi, pi, controller design means to pick good K for R.L. Those zi, pi means to pick additional poles / zeros to R.L.

Proportional control design Draw R.L. with given Pick a point on R.L. to be desired c.l. pole: PD Compute

When to use: What is that region: If R.L. of G(s) goes through the desired region for c.l. poles What is that region: From design specs, get desired Mp, ts, tr, etc. Use formulae for 2nd order system to get desired ωn , ζ, σ, ωd Identify / plot these in s-plane

Example: When C(s) = 1, things are okay But we want initial response speed as fast as possible; yet we can only tolerate 10% overshoot. Sol: From the above, we need that means: C(s) G(s)

This is a cone around –Re axis with ±60° area We also want tr to be as small as possible. i.e. : want ωn as large as possible i.e. : want pd to be as far away from s = 0 as possible Draw R.L. Pick pd on R.L., in cone & | pd | max

Example: Want: , as fast as possible Sol: Draw R.L. for Draw cone ±45° about –Re axis Pick pd as the cross point of the ζ = 0.7 line & R.L.

Controller tuning: First design typically may not work Identify trends of specs changes as K is increased. e.g.: as KP , pole Perform closed-loop step response Adjust K to improve specs e.g. If MP too much, the 2. says reduce KP

PD controller design This is introducing an additional zero to the R.L. for G(s) Use this if the dominant pole pair branches of G(s) do not pass through the desired region

Design steps: From specs, draw desired region for pole. Pick from region. Compute Select Select:

Example: Want: Sol: (pd not on R.L.) (Need a zero to attract R.L. to pd)

Closed-loop step response simulation: » ng = [1] ; » dg = [1 2 0] ; » nc = [Kp, Kd] » n = conv(nc, ng) ; » d = dg + n ; » stepspec(n, d) ; Tuning: for fixed z: Q: What’s the effect of tuning z?

Drawbacks of PD Not proper : deg of num > deg of den High frequency gain → ∞: High gain for noise Saturated circuits cannot be implemented physically

Lead Controller Approximation to PD Same usefulness as PD It contributes a lead angle:

Lead Design: Draw R.L. for G From specs draw region for desired c.l. poles Select pd from region Let Pick –z somewhere below pd on –Re axis Let Select

There are many choices of z, p More neg. (–z) & (–p) → more close to PD & more sensitive to noise, and worse steady-state error But if –z is > Re(pd), pd may not dominate

Example: Lead Design MP is fine, but too slow. Want: Don’t increase MP but double the resp. speed Sol: Original system: C(s) = 1 Since MP is a function of ζ, speed is proportional to ωn

Draw R.L. & desired region Pick pd right at the vertex: (Could pick pd a little inside the region to allow “flex”)

Clearly, R. L. does not pass through pd, nor the desired region Clearly, R.L. does not pass through pd, nor the desired region. need PD or Lead to “bend” the R.L. into region. (Note our choice may be the easiest to achieve) Let’s do Lead:

Pick –z to the left of pd