EE3CL4: Introduction to Linear Control Systems - …davidson/EE3CL4/slides/System... · control...

EE 3CL4, §21 / 97

Tim Davidson

ModellingphysicalsystemsTrans. Newton.Mech.

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Transfer fn ofDC motor

Our firstcontrol systemdesign

Block diagrammodelsBlock dia. transform.

EE3CL4:Introduction to Linear Control Systems

Section 2: System Models

Tim Davidson

McMaster University

Winter 2018

EE 3CL4, §22 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Outline1 Modelling physical systems

Translational Newtonian MechanicsRotational Newtonian Mechanics

2 Linearization

3 Laplace transforms

4 Laplace transforms in action

5 Transfer function

6 Step response

7 Transfer function of DC motor

8 Our first control system design

9 Block diagram modelsBlock diagram transformations

EE 3CL4, §24 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Differential equation models

• Most of the systems that we will deal with are dynamic• Differential equations provide a powerful way to

describe dynamic systems• Will form the basis of our models

• We saw differential equations for inductors andcapacitors in 2CI, 2CJ

• What about mechanical systems?both translational and rotational

EE 3CL4, §25 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Translational Spring

F (t): resultant force in direction xRecall free body diagrams and “action and reaction”

• Spring. k : spring constant, Lr : relaxed length of spring

F (t) = k([x2(t)− x1(t)]− Lr

EE 3CL4, §26 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Translational Damper

F (t): resultant force in direction x

• Viscous damper. b: viscous friction coefficient

F (t) = b(dx2(t)

dt− dx1(t)

(v2(t)− v1(t)

EE 3CL4, §27 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

F (t): resultant force in direction x

• Mass: M

F (t) = Md2xm(t)

dt2 = Mdvm(t)

dt= Mam(t)

EE 3CL4, §28 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Rotational spring

T (t): resultant torque in direction θ

• Rotational spring. k : rotational spring constant,φr : rotation of relaxed spring

T (t) = k([θ2(t)− θ1(t)]− φr

EE 3CL4, §29 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Rotational damper

• Rotational viscous damper.b: rotational viscous friction coefficient

T (t) = b(dθ2(t)

dt− dθ1(t)

(ω2(t)− ω1(t)

EE 3CL4, §210 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Rotational inertia

• Rotational inertia: J

T (t) = Jd2θm(t)

dt2 = Jdωm(t)

dt= Jαm(t)

EE 3CL4, §211 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Example system (translational)Horizontal. Origin for y : y = 0 when spring relaxed

• F = M dv(t)dt

• v(t) = dy(t)dt

• F (t) = r(t)− b dy(t)dt − ky(t)

Md2y(t)

dy(t)dt

+ ky(t) = r(t)

EE 3CL4, §212 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Example, continued

Md2y(t)

dy(t)dt

+ ky(t) = r(t)

Resembles equation for parallel RLC circuit.

EE 3CL4, §213 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Example, continued

• Stretch the spring a little and hold.• Assume an under-damped system.• What happens when we let it go?

EE 3CL4, §215 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Taylor’s series

• Nature does not have many linear systems• However, many systems behave approximately linearly

in the neighbourhood of a given point• Apply first-order Taylor’s Series at a given point• Obtain a locally linear model

• In this course we will focus on the case of a singlelinearized differential equation model for the system, inwhich the coefficients are constants

• e.g., in previous examples mass, viscosity and springconstant did not change with time, position, velocity,temperature, etc

EE 3CL4, §216 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Pendulum example

• Assume shaft is light with respect to M,and stiff with respect to gravitational forces

• Torque due to gravity: T (θ) = MgL sin(θ)• Apply Taylor’s series around θ = 0:

T (θ) = MgL(θ − θ3/3! + θ5/5!− θ7/7! + . . .

)• For small θ around θ = 0 we can build an approximate

model that is linear

T (θ) ≈ MgLθ

EE 3CL4, §218 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Laplace transform• Once we have a linearized differential equation (with

constant coefficients) we can take Laplace Transformsto obtain the transfer function

• We will consider the “one-sided” Laplace transform, forsignals that are zero to the left of the origin.

F (s) =∫ ∞

0−f (t)e−st dt

• What does∫∞ mean? limT→∞

∫ T .

• Does this limit exist?

• If |f (t)| < Meαt , then exists for all Re(s) > α.Includes all physically realizable signals

• Note: When multiplying transfer function by Laplace of input, outputis only valid for values of s in intersection of regions of convergence

EE 3CL4, §219 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Poles and zeros• In this course, most Laplace transforms will be rational

functions, that is, a ratio of two polynomials in s; i.e.,

F (s) =nF (s)dF (s)

where nF (s) and dF (s) are polynomials

• Definitions:• Poles of F (s) are the roots of dF (s)• Zeros of F (s) are the roots of nF (s)

• Hence,

F (s) =KF∏M

i=1(s + zi)∏nj=1(s + pj)

∏Mi=1 zi∏n

j=1 pj

) ∏Mi=1(s/zi + 1)∏nj=1(s/pj + 1)

where −zi are the zeros and −pj are the poles

EE 3CL4, §220 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Visualizing poles and zeros

• Consider the simple Laplace transform F (s) = s(s+3)s2+2s+5 .

• zeros: 0, −3; poles: −1 + j2, −1− j2• Pole-zero plot (left) and magnitude of F (s) (right)

EE 3CL4, §221 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Visualizing poles and zeros

• F (s) = s(s+3)s2+2s+5 ; zeros: 0, −3; poles: −1 + j2, −1− j2

• |F (s)| from above (left) and prev. view of |F (s)| (right)

EE 3CL4, §222 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Laplace transform pairs

• Simple ones can be computed analytically;often available in tables; see Tab. 2.3 in 12th ed. of text

• For more complicated ones, one can typically obtainthe inverse Laplace transform by• identifying poles• constructing partial fraction expansion• using of properties and some simple pairs to invert

each component of partial fraction expansion

EE 3CL4, §223 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Some Laplace transform pairs

Recall that complex poles come in conjugate pairs.

EE 3CL4, §224 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Key properties

Linearity

df (t)dt

←→ sF (s)− f (0−)

−∞f (x)dx ←→ F (s)

∫ 0−

−∞f (x)dx

EE 3CL4, §225 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Final value theorem

Can we avoid having to do an inverse Laplace transform?Sometimes.

Consider the case when we only want to find the final valueof f (t), namely limt→∞ f (t).

• If F (s) has all its poles in the left half plane, except,perhaps, for a single pole at the origin, then

limt→∞

f (t) = lims→0

sF (s)

Common application: Steady state value of step response

What if there are poles in RHP, or on the jω-axis and not atthe origin?

EE 3CL4, §227 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Mass-spring-damper system

• Horizontal (no gravity)• Set origin of y where spring is “relaxed”• F = M dv(t)

• v(t) = dy(t)dt

• F (t) = r(t)− b dy(t)dt − ky(t)

Md2y(t)

dy(t)dt

+ ky(t) = r(t)

EE 3CL4, §228 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

MSD system

Md2y(t)

dy(t)dt

+ ky(t) = r(t)

Consider t ≥ 0 and take Laplace transform

s2Y (s)−sy(0−)− dy(t)dt

∣∣∣∣t=0−

)+b(sY (s)−y(0−)

)+kY (s) = R(s)

Y (s) =1/M

s2 + (b/M)s + k/MR(s)

+(s + b/M)

s2 + (b/M)s + k/My(0−)

s2 + (b/M)s + k/Mdy(t)

∣∣∣∣t=0−

Note that linearity yields superposition

EE 3CL4, §229 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Response to static init. cond.

Spring stretched to a point y0, held, then let go at time t = 0

Hence, r(t) = 0 and dy(t)dt

∣∣∣t=0−

Hence,

Y (s) =(s + b/M)

s2 + (b/M)s + k/My0

What can we learn about this response without having toinvert Y (s)

EE 3CL4, §230 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Standard form

Y (s) =(s + b/M)

s2 + (b/M)s + k/My0

=(s + 2ζωn)

s2 + 2ζωns + ω2n

where ωn =√

k/M and ζ = b2√

Poles: s1, s2 = −ζωn ± ωn√ζ2 − 1

• ζ > 1 (equiv. b > 2√

kM): distinct real roots, overdamped

• ζ = 1 (equiv. b = 2√

kM): equal real roots, critically damped

• ζ < 1 (equiv. b < 2√

kM): complex conj. roots, underdamped

EE 3CL4, §231 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Overdamped case

• s1, s2 = −ζωn ± ωn√ζ2 − 1

• Overdamped response: ζ > 1 (equiv. b > 2√

kM)• y(t) = c1es1t + c2es2t

• y(0) = y0 =⇒ c1 + c2 = y0

• dy(t)dt

∣∣∣t=0

= 0 =⇒ s1c1 + s2c2 = 0

• What does this look like when strongly overdamped• s2 is large and negative, s1 is small and negative• Hence es2t decays much faster than es1t

• Also, c2 = −c1s1/s2. Hence, small• Hence y(t) ≈ c1es1t

• Looks like a first order system!

EE 3CL4, §232 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Critically damped case

• s1 = s2 = −ωn

• y(t) = c1e−ωnt + c2te−ωnt

• y(0) = y0 =⇒ c1 = y0

• dy(t)dt

∣∣∣t=0

= 0 =⇒ −c1ωn + c2 = 0

EE 3CL4, §233 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Underdamped case

• s1, s2 = −ζωn ± jωn√

1− ζ2

• Therefore, |si | = ωn: poles lies on a circle• Angle to negative real axis is cos−1(ζ).

EE 3CL4, §234 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Underdamped case

• Define σ = ζωn, ωd = ωn√

1− ζ2. Response is:

y(t) = c1e−σt cos(ωd t) + c2e−σt sin(ωd t)

= Ae−σt cos(ωd t + φ)

• Homework: Relate A and φ to c1 and c2.

• Homework: Write the initial conditions y(0) = y0 anddy(t)

∣∣∣t=0

= 0 in terms of c1 and c2, and in terms of A and φ

EE 3CL4, §235 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Numerical examples

• Y (s) = (s+2ζωn)

s2+2ζωns+ω2n

y0, where ωn =√

k/M, ζ = b2√

• Poles: s1, s2 = −ζωn ± ωn√ζ2 − 1

• ζ > 1: overdamped; ζ < 1: underdamped

• Consider the case of M = 1, k = 1. Hence, ωn = 1,

• b = 3→ 0. Hence, ζ = 1.5→ 0

• Initial conds: y0 = 1, dy(t)dt

∣∣∣t=0

EE 3CL4, §236 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Poles and transient response,b = 3

EE 3CL4, §237 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Poles and transient response,b = 2.75

EE 3CL4, §238 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §239 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §240 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §241 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §242 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §243 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §244 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §245 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §246 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §247 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §248 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §249 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §251 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Transfer functionDefinition: Laplace transform of output over Laplacetransform of input when initial conditions are zero

• Most of the transfer functions in this course will beratios of polynomials in s.

• Hence, poles and zeros of transfer functions havenatural definitions

Example: Recall the mass-spring-damper system,

EE 3CL4, §252 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Transfer function, MSD system

For the mass-spring-damper system,

Y (s) =1/M

s2 + (b/M)s + k/MR(s)

+(s + b/M)

s2 + (b/M)s + k/My(0−)

s2 + (b/M)s + k/Mdy(t)

∣∣∣∣t=0−

Therefore, transfer function is:

1/Ms2 + (b/M)s + k/M

Ms2 + bs + k

EE 3CL4, §254 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

• Recall that u(t)←→ 1s

• Therefore, for transfer function G(s), the step responseis:

L −1{G(s)

}• For the mass-spring-damper system, step response is

L −1{ 1

s(Ms2 + bs + k)

• What is the final position for a step input?Recall final value theorem. Final position is 1/k .

• What about the complete step response?

EE 3CL4, §255 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Step response• Step response: L −1

{G(s)1

}• Hence poles of Laplace transform of step response are

poles of G(s), plus an additional pole at s = 0.

• For the mass-spring-damper system, using partialfractions, step response is:

L −1{ 1

s(Ms2 + bs + k)

}= L −1

}− 1

kL −1

{ Ms + bMs2 + bs + k

u(t)− 1k

L −1{ Ms + b

Ms2 + bs + k

• Consider again the case of M = k = 1, b = 3→ 0.ωn = 1, ζ = 1.5→ 0.

EE 3CL4, §256 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Poles and step response, b = 3

EE 3CL4, §257 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Poles and step resp., b = 2.75

EE 3CL4, §258 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §259 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §260 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Poles and step resp., b = 2

EE 3CL4, §261 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §262 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §263 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §264 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §265 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §266 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §267 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §268 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §269 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

EE 3CL4, §271 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

A DC motor

• We will consider linearized model for each component• Flux in the air gap: φ(t) = Kf if (t) (Magnetic cct, 2CJ4)• Torque: Tm(t) = K1φ(t)ia(t) = K1Kf if (t)ia(t).• Is that linear?• Only if one of if (t) or ia(t) is constant• We will consider “armature control”: if (t) constant

EE 3CL4, §272 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Armature controlled DC motor

• if (t) will be constant (to set up magnetic field), if (t) = If• Torque: Tm(t) = K1Kf If ia(t) = Kmia(t)• Will control motor using armature voltage Va(t)• What is the transfer function from Va(s) to angular

position θ(s)?• Origin?

EE 3CL4, §273 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Towards transfer function

• Tm(t) = Kmia(t) ←→ Tm(s) = KmIa(s)• KVL: Va(s) = (Ra + sLa)Ia(s) + Vb(s)• Vb(s) is back-emf voltage, due to Faraday’s Law• Vb(s) = Kbω(s), where ω(s) = sθ(s) is rot. velocity• Remember: transfer function implies zero init. conds

EE 3CL4, §274 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

• Torque on load: TL(s) = Tm(s)− Td(s)• Td(s): disturbance. Often small, unknown• Load torque and load angle (Newton plus friction):

TL(s) = Js2θ(s) + bsθ(s)

• Now put it all together

EE 3CL4, §275 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

• Tm(s) = KmIa(s) = Km

(Va(s)−Vb(s)

Ra+sLa

)• Vb(s) = Kbω(s)• TL(s) = Tm(s)− Td(s)• TL(s) = Js2θ(s) + bsθ(s) = Jsω(s) + bω(s)• Hence ω(s) = TL(s)

Js+b• θ(s) = ω(s)/s

EE 3CL4, §276 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Block diagram

• Tm(s) = KmIa(s) = Km

(Va(s)−Vb(s)

Ra+sLa

)• Vb(s) = Kbω(s)• TL(s) = Tm(s)− Td(s)• TL(s) = Js2θ(s) + bsθ(s) = Jsω(s) + bω(s)

• Hence ω(s) = TL(s)Js+b

• θ(s) = ω(s)/s

EE 3CL4, §277 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Transfer function

• Set Td(s) = 0 and solve (you MUST do this yourself)

G(s) =θ(s)

Va(s)=

s[(Ra + sLa)(Js + b) + KbKm

s(s2 + 2ζωns + ω2n)

• Third order :(

EE 3CL4, §278 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Second-order approximation

G(s) =θ(s)

Va(s)=

s[(Ra + sLa)(Js + b) + KbKm

]• Sometimes armature time constant, τa = La/Ra, is

negligible• Hence (you MUST derive this yourself)

G(s) ≈ Km

s[Ra(Js + b) + KbKm

] = Km/(Rab + KbKm)

s(τ1s + 1)

where τ1 = RaJ/(Rab + KbKm)

EE 3CL4, §279 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Model for a disk drive readsystem

• Uses a permanent magnet DC motor• Can be modelled using arm. contr. model with Kb = 0• Hence, motor transfer function:

G(s) =θ(s)

Va(s)=

s(Ra + sLa)(Js + b)

• Assume for now that the arm is stiff

EE 3CL4, §280 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Typical values

G(s) =θ(s)

Va(s)=

s(Ra + sLa)(Js + b)

G(s) =5000

s(s + 20)(s + 1000)

EE 3CL4, §281 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Time constants

• Initial model

G(s) =5000

s(s + 20)(s + 1000)

• Motor time constant = 1/20 = 50ms• Armature time constant = 1/1000 = 1ms• Hence

G(s) ≈ G(s) =5

s(s + 20)

EE 3CL4, §283 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

A simple feedback controllerNow that we have a model, how to control?

Simple idea: Apply voltage to motor that is proportional toerror between where we are and where we want to be.

Here, V (s) = Va(s) and Y (s) = θ(s).

EE 3CL4, §284 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Simplified block diagram

• What is the transfer function from command toposition? Derive this yourself

Y (s)R(s)

=KaG(s)

1 + KaG(s)

• Using second-order approx. G(s) ≈ G(s) = 5s(s+20) ,

Y (s) ≈ 5Ka

s2 + 20s + 5KaR(s)

• For 0 < Ka < 20: overdamped;for Ka > 20: underdamped

EE 3CL4, §285 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Response to r(t) = 0.1u(t);Ka = 10

Poles in s-plane Response

Slow. Slower than IBMs first drive from late 1950’s.Disks in the 1970’s had 25ms seek times; now < 10msPerhaps increase Ka?That would result in a “bigger” input to the motor for a givenerror

EE 3CL4, §286 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Response to r(t) = 0.1u(t);Ka = 10,15

Changing Ka changes the position of the closed-loop polesHence, step response changes

EE 3CL4, §287 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Response to r(t) = 0.1u(t);Ka = 10,15,20

Changing Ka changes the position of the closed-loop polesHence, step response changes (now critically damped)

EE 3CL4, §288 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Response to r(t) = 0.1u(t);Ka = 10,15,20,40

Changing Ka changes the position of the closed-loop polesHence, step response changes (now underdamped)

EE 3CL4, §289 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Response to r(t) = 0.1u(t);Ka = 10,15,20,40,60

Changing Ka changes the position of the closed-loop polesHence, step response changes (now more underdamped)

EE 3CL4, §290 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Response to r(t) = 0.1u(t);Ka = 10,15,20,40,60,80

What is happening to the settling time of the underdampedcases?Only just beats IBM’s first driveWhat else could we do with the controller? Prediction?

EE 3CL4, §292 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Bock diagram models

• As we have just seen, a convenient way to represent atransfer function is via a block diagram

• In this case, U(s) = Gc(s)R(s) and Y (s) = G(s)U(s)• Hence, Y (s) = G(s)Gc(s)R(s)• Consistent with the engineering procedure of breaking

things up into little bits, studying the little bits, and thenput them together

EE 3CL4, §293 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Simple example

• Y1(s) = G11(s)R1(s) + G12(s)R2(s)• Y2(s) = G21(s)R1(s) + G22(s)R2(s)

EE 3CL4, §294 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Example: Loop transfer function

• Ea(s) = R(s)− B(s) = R(s)− H(s)Y (s)• Y (s) = G(s)U(s) = G(s)Ga(s)Z (s)• Y (s) = G(s)Ga(s)Gc(s)Ea(s)

• Y (s) = G(s)Ga(s)Gc(s)(

R(s)− H(s)Y (s))

Y (s)R(s)

=G(s)Ga(s)Gc(s)

1 + G(s)Ga(s)Gc(s)H(s)

• Each transfer function is a ratio of polynomials in s• What is Ea(s)/R(s)?

EE 3CL4, §295 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Block diagram transformations

EE 3CL4, §296 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Using block diagramtransformations

EE 3CL4, §297 / 97

Tim Davidson

Rot. Newton. Mech.

Linearization

Laplacetransforms

Laplace inaction

Transferfunction

Step response

Using block diagramtransformations

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