Control theory
Kim Mathiassen
15.02.2011
Control theory
Mass spring damper system
Modeling
Open loop vs. closed loop
Second order system
Stability
PID control
P - Proportional
I - Integral
D - Derivative
Optimal control
LQR
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Mass spring damper system
From Wikimedia Commons
x = displacement [m]
f = force applied [kg · m/s 2 ]
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m = mass of the block [kg ]
B = damping constant [kg /s]
k = spring constant [kg /s 2 ]
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Mass spring damper system
Using Newton’s second law
P
I
Spring force: f1 = −kx
I
Damping force: f2 = −f
I
External force: f3 = u
fi = ma. We have three forces
δx
δt
= −f ẋ
This gives the equation
mẍ = −kx − f ẋ + u
Differential equation for mass spring damper system
ẍ +
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f
m ẋ
+
k
mx
=
1
mu
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Modeling domains
Frequency domain (Transfer functions)
x(s)=h(s)u(s)
h(s)=
1
m
f
k
s2+ m
s+ m
State space domain
ẋ=Ax + Bu
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ẋ1 =x2
k
ẋ2 =− m
x1 −
f
m x2
+
1
mu
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Block diagrams
u
-
-
1
m
ẋ2
x2 = ẋ1
x1
f
k
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SISO and MIMO
Single-Input Single-Output (SISO)
The system has one input u and one output x
Multiple-Input Multiple-Output (MIMO)
The system has multiple input u and multiple
output x
Single-Input Multiple-Output (SIMO)
Can be regarded as several SISO systems
Multiple-Input Single-Output (MISO)
Can be regarded as several SISO systems
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Process
Process
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Open loop vs. closed loop
Open-loop
r
Controller
u
Process
x
Closed-loop
r
e
Controller
u
Process
x
y
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Mesurements
8
Second order systems
H(s) =
s2 +
1
m
f
ms
+
k
m
=
1
m
(s − λ1 )(s − λ2 )
Solution
The generic solution gives three cases depending on pole
placemend. The three cases are called under-damped, over-damped
and critially damped
!
r
f
km
λ{1,2} = −
1± 1−4 2
(1)
2m
f
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Second order systems
Damping ratio
ζ=
−(λ1 +λ2 )
√
2 λ1 λ2
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 oscillations
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Second order system responce
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From Wikimedia Commons
Stability
Consider the system y (s) = h(s)y0 (s) where y0 (s) has finite length
and 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
Unstable
If the system is not stable, it is unstable
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PID control
We want to make the system stable and controllable with a
controller. The PID controller is a simple controller that may
acheive this goal. The PID controller is often analyzed in the
frequency domain.
PID controller
Z
u = Kp e + Ki
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e(τ )d τ + Kd ė
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Proportional
I
A pure proportional controller will have a steady-state error
I
Adding a integration term will remove the bias
I
High gain (Kp ) will produce a fast system
I
High gain may cause oscillations and may make the system
unstable
I
High gain reduces the steady-state error
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Proportional
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From Wikimedia Commons
Integral
I
Removes steady-state error
I
Increasing Ki accelerates the controller
I
High Ki may give oscillations
I
Increasing Ki will increase the settling time
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Integral
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From Wikimedia Commons
Derivative
I
Larger Kd decreases oscillations
I
Improves stability for low values of Kd
I
May be highly sensitive to noise if one takes the derivative of a
noisy error
I
High noise leads to instability
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Derivative
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From Wikimedia Commons
PIDstop
From http://www.pidstop.com/demo
PID games
http://www.pidstop.com/demo
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(K1 = -110 K2 = 0.728)
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Optimal control
I
Optimal controll is another control approach than PID
I
The idea is to specify a cost function and then find the
optimal input
I
The Dynamics of the system is used to design the controller
I
For non-linear system it is not always possible to find the
optimal solution
I
A special case is for linear systems with a quadradic cost
function
I
The optimal controller must have all states as input
I
Most often used with an observer to estimate the states that
are not measured
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Optimal control
r
u
ê Controller
Process
x
ŷ
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Observer
y
Mesurements
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Linear-quadratic regulator (LQR)
I
The feedback is given as u = G 1 x + G 2 r
I
r is the reference function
I
The matrix G 1 and G 2 is found based on the system dynamics
and the cost function using Pontryagin’s Maximum principle
I
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=
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1
2
Z
∞
e T Qe + u T Pudt
t
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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 teknisk
kybernetikk.
Optimalregulering.
Tapir, 1976.
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