Control theory

Kim Mathiassen

15.02.2011

Control theoryMass spring damper systemModelingOpen loop vs. closed loopSecond order systemStability

PID controlP - ProportionalI - IntegralD - Derivative

Optimal controlLQR

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Mass spring damper system

From Wikimedia Commons

x = displacement [m]f = force applied [kg ·m/s2]

m = mass of the block [kg ]B = damping constant [kg/s]k = spring constant [kg/s2]15.02.2011 3

Mass spring damper systemUsing Newton’s second law

∑fi = ma. We have three forces

I Spring force: f1 = −kxI Damping force: f2 = −f δxδt = −f xI External force: f3 = u

This gives the equation

mx = −kx − f x + u

Differential equation for mass spring damper system

x + fm x + k

mx = 1mu

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Modeling domains

Frequency domain (Transfer functions)

x(s)=h(s)u(s) h(s)=1m

s2+ fm s+ k

m

State space domain

x=Ax + Bu x1=x2

x2=− kmx1 − f

mx2 + 1mu

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Block diagrams

1m--

f

k

x2 x2 = x1 x1u

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SISO and MIMO

Single-Input Single-Output (SISO)The system has one input u and one output xMultiple-Input Multiple-Output (MIMO)The system has multiple input u and multipleoutput xSingle-Input Multiple-Output (SIMO)Can be regarded as several SISO systemsMultiple-Input Single-Output (MISO)Can be regarded as several SISO systems

Process

Process

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Open loop vs. closed loop

Open-loop

ProcessController xur

Closed-loop

ProcessController xue

Mesurements

-

r

y

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Second order systems

H(s) =1m

s2 + fm s + k

m

=1m

(s − λ1)(s − λ2)

SolutionThe generic solution gives three cases depending on poleplacemend. The three cases are called under-damped, over-dampedand critially damped

λ{1,2} = − f2m

(1±

√1− 4

kmf 2

)(1)

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Second order systems

Damping ratio

ζ = −(λ1+λ2)

2√λ1λ2

Over-damped, ζ > 1 (λ1 and λ2 real and distinct)

Slow system responce

Critically damped, ζ = 1 (λ1 = λ2)

Fastes system responce without oscillations

Under-damped, ζ < 1 (λ1 and λ2 complex conjugates)

Fast system responce, but with oscillations15.02.2011 10

Second order system responce

From Wikimedia Commons

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StabilityConsider the system y(s) = h(s)y0(s) where y0(s) has finite lengthand amplitude

Asymptotically stable

The system is asymptotically stable if y → 0 when t →∞

Marginally stable

The system is marginally stable if |y | <∞ for all t ≥ 0

UnstableIf the system is not stable, it is unstable

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PID controlWe want to make the system stable and controllable with acontroller. The PID controller is a simple controller that mayacheive this goal. The PID controller is often analyzed in thefrequency domain.

PID controller

u = Kpe + Ki

∫e(τ)dτ + Kd e

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Proportional

I A pure proportional controller will have a steady-state errorI Adding a integration term will remove the biasI High gain (Kp) will produce a fast systemI High gain may cause oscillations and may make the system

unstableI High gain reduces the steady-state error

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Proportional

From Wikimedia Commons

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Integral

I Removes steady-state errorI Increasing Ki accelerates the controllerI High Ki may give oscillationsI Increasing Ki will increase the settling time

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Integral

From Wikimedia Commons

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Derivative

I Larger Kd decreases oscillationsI Improves stability for low values of Kd

I May be highly sensitive to noise if one takes the derivative of anoisy error

I High noise leads to instability

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Derivative

From Wikimedia Commons

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PIDstop

From http://www.pidstop.com/demo

PID games

http://www.pidstop.com/demo (K1 = -110 K2 = 0.728)15.02.2011 20

Optimal control

I Optimal controll is another control approach than PIDI The idea is to specify a cost function and then find the

optimal inputI The Dynamics of the system is used to design the controllerI For non-linear system it is not always possible to find the

optimal solutionI A special case is for linear systems with a quadradic cost

functionI The optimal controller must have all states as inputI Most often used with an observer to estimate the states that

are not measured15.02.2011 21

Optimal control

ProcessController xue

Mesurements

-

r

yObservery

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Linear-quadratic regulator (LQR)

I The feedback is given as u = G 1x + G 2rI r is the reference functionI The matrix G 1 and G 2 is found based on the system dynamics

and the cost function using Pontryagin’s Maximum principleI When following a trajectory the function r(t) must be known

for all future timesteps in order to find the optimal solution

Cost function

J = 12

∫ ∞t

eTQe + uTPudt

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References

J. B. Balchen, T. Andresen, and B. A. Foss.Reguleringsteknikk.Institutt for teknisk kybernetikk, 2004.

PID controller.http://en.wikipedia.org/wiki/pid_controller, February 2011.

Damping.http://en.wikipedia.org/wiki/damping, February 2011.

O.A. Solheim and Norges tekniske høgskole Institutt for tekniskkybernetikk.Optimalregulering.Tapir, 1976.

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