Lecture 13 - Handling Nonlinearity
• Nonlinearity issues in control practice
• Setpoint scheduling/feedforward
– path planning replay - linear interpolation
• Nonlinear maps
–
–
–
–
B-splines
Multivariable interpolation: polynomials/splines/RBF
Neural Networks
Fuzzy logic
• Gain scheduling
• Local modeling
EE392m - Winter 2003
Control Engineering
13-1
Nonlinearity in control practice
Here are the nonlinearities we already looked into
• Constraints - saturation in control
– anti-windup in PID control
– MPC handles the constraints
• Control program, path planning
• Static optimization
• Nonlinear dynamics
– dynamic inversion
– nonlinear IMC
– nonlinear MPC
One additional nonlinearity in this lecture
• Controller gain scheduling
EE392m - Winter 2003
Control Engineering
13-2
Dealing with nonlinear functions
• Analytical expressions
– models are given by analytical formulas, computable as required
– rarely sufficient in practice
• Models are computable off line
– pre-compute simple approximation
– on-line approximation
• Models contain data identified in the experiments
– nonlinear maps
– interpolation or look-up tables
• Advanced approximation methods
– neural networks
EE392m - Winter 2003
Control Engineering
13-3
Path planning
• Real-time replay of a pre-computed reference trajectory
yd(t) or feedforward v(t)
• Reproduce a nonlinear function yd(t) in a control system
θ1 
 Y1 = yd (θ1 ) 
θ 
Y = y (θ ) 
Path planner,
d
2 
 2
t
yd(t) Y =  2
,
Θ
=
data arrays Y ,Θ
M
M


 


θ n 
Yn = yd (θ n )
Code:
1. Find j, such that θ j ≤ t ≤ θ j +1
2. Compute
θ j +1 − t
t −θ j
+ Y j +1
yd (t ) = Y j
θ j +1 − θ j
θ j +1 − θ j
EE392m - Winter 2003
Control Engineering
13-4
Linear interpolation vs. table look-up
• linear interpolation is more accurate
• requires less data storage
• simple computation
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Control Engineering
13-5
Empirical models
• Aerospace - most developed nonlinear approaches
– automotive and process control have second place
Example
• Aerodynamic tables
• Engine maps
– jet turbines
– automotive
• Process maps, e.g., in
semiconductor
manufacturing
• Empirical map for a
attenuation vs.
temperature in an
optical fiber
EE392m - Winter 2003
TEF=Trailing Edge Flap
Control Engineering
13-6
Approximation
• Interpolation:
– compute function that will provide given values Y j in the nodes θ j
– not concerned with accuracy in-between the nodes
• Approximation
– compute function that closely corresponds to given data, possibly
with some error
– might provide better accuracy throughout
EE392m - Winter 2003
Control Engineering
13-7
B-spline interpolation
• 1st-order
– look-up table, nearest neighbor
• 2nd-order
– linear interpolation
yd (t ) = ∑ Y j B j (t )
j
• n-th order:
– Piece-wise n-th order polynomials, matched n-2 derivatives
– zero outside a local support interval
– support interval extends to n nearest neighbors
EE392m - Winter 2003
Control Engineering
13-8
B-splines
• Accurate interpolation of smooth
functions with relative few nodes
• For 1-D function the gain from
using high-order B-splines is not
worth an added complexity
• Introduced and developed in CAD
for 2-D and 3-D curve and surface
data
• Are used for defining
multidimensional nonlinear maps
EE392m - Winter 2003
Control Engineering
13-9
Multivariable B-splines
• Regular grid in multiple variables
• Tensor product B-splines
• Used as a basis of finite-element models
y (u, v ) = ∑ w j ,k B j (u ) Bk ( v )
j ,k
EE392m - Winter 2003
Control Engineering
13-10
Linear regression for nonlinear map
• Linear regression
y ( x ) = ∑θ jϕ j ( x ) = θ T ⋅ φ ( x )
j
• Multidimensional B-splines
• Multivariate polynomials
ϕ j ( x1 ,K, xn ) = ( x1 ) k1 ⋅ K ⋅ ( xn ) kn
 x1 
x=M
 
 xn 
y = θ 0 + θ1 x1 + θ 2 x2 + θ 3 ( x1 ) 2 + θ 4 x1 x2 + K
• RBF - Radial Basis Functions
(
)
ϕ j(x) = R x − cj = e
EE392m - Winter 2003
−a x −c j
2
Control Engineering
13-11
Linear regression approximation
[
Y = y (1) K y ( N )
• Nonlinear map data
– available at scattered nodal points
• Linear regression map
]
x (1) K x ( N )
[
]
Y = θ T ⋅ φ ( x (1) ) K φ ( x ( N ) ) = θ T Φ
• Linear regression approximation
– regularized least square estimate of the weight vector
−1
T
ˆ
θ = (ΦΦ + rI ) Φ Y T
• Works just the same for vector-valued data!
EE392m - Winter 2003
Control Engineering
13-12
Nonlinear map example - Epi
• Epitaxial growth (semiconductor process)
– process map for run-to-run control
EE392m - Winter 2003
Control Engineering
13-13
Linear regression for Epi map
• Linear regression model for epitaxial grouth
y = c0 x1 p1 ( x2 ) + c1 (1 − x1 ) p2 ( x2 )
p1 = w0 + w1 x2 + w3 ( x2 ) 2 + w4 ( x2 )3
c0 x1 p1 = w0c0 x1 + w1c0 x1 x2 + w3c0 x1 ( x2 ) 2 + w4c0 x1 ( x2 )3
{
{
{
{
θ1
θ2
θ3
θ4
c1 (1 − x1 ) p2 ( x2 ) =
v0c1 (1 − x1 ) + v1c1 (1 − x1 ) x2 + v3c0 (1 − x1 )( x2 ) 2 + w4 c0 (1 − x1 )( x2 )3
{
{
{
{
θ5
θ6
θ7
y ( x1 , x2 ) = ∑θ jϕ j ( x1 , x2 ) = θ T ⋅ φ ( x1 , x2 )
θ8
j
EE392m - Winter 2003
Control Engineering
13-14
Neural Networks
• Any nonlinear approximator might be called a Neural Network
–
–
–
–
RBF Neural Network
Polynomial Neural Network
B-spline Neural Network
Wavelet Neural Network
Linear in parameters
• MPL - Multilayered Perceptron
– Nonlinear in parameters
– Works for many inputs

1
y ( x ) = w1,0 + f  ∑ w1, j y j , y1j = w2,0 +
 j

1
f ( x) =
1 + e− x
EE392m - Winter 2003
y


f  ∑ w2, j x j 
 j

y=f(x)
x
Control Engineering
13-15
Multi-Layered Perceptrons
• Network parameter computation
– training data set
– parameter identification
[
Y = y (1) K y ( N )
]
x (1) K x ( N )
y ( x ) = F ( x ;θ )
• Noninear LS problem
V =∑ y
( j)
2
− F ( x ;θ ) → min
( j)
j
• Iterative NLS optimization
– Levenberg-Marquardt
• Backpropagation
– variation of a gradient descent
EE392m - Winter 2003
Control Engineering
13-16
Fuzzy Logic
• Function defined at nodes. Interpolation scheme
• Fuzzyfication/de-fuzzyfication = interpolation
• Linear interpolation in 1-D
∑ y µ ( x)
y( x) =
∑ µ ( x)
j
j
∑ µ ( x) = 1
j
j
j
j
j
• Marketing (communication) and social value
• Computer science: emphasis on interaction with a user
– EE - emphasis on mathematical computations
EE392m - Winter 2003
Control Engineering
13-17
Neural Net application
• Internal Combustion Engine
maps
• Experimental map:
– data collected in a steady state
regime for various combinations
spark
of parameters
advance
– 2-D table
RPM
• NN map
– approximation of the
experimental map
– MLP was used in this example
– works better for a smooth
surface
EE392m - Winter 2003
Control Engineering
13-18
Linear feedback in a nonlinear plant
• Simple example
y = f ( x ) + g ( x )u
yd
Controller
u = −k ( x )( y − yd ) + u ff ( x )
u
y
Plant
• Linearizing property of feedback for k ( x ) >> 1
−1
y = yd + (1 + k ( x ) g ( x ) ) ( f ( x ) + g ( x )u ff − yd )
Example:
• Control design requires
varying
k ( x ), u ff ( x ), yd ( x )
process
gain
• These variables are
scheduled on x
EE392m - Winter 2003
Control Engineering
13-19
Gain scheduling
Nonlinear
• Single out several
system
regimes - model
 vec( A) 
linearization or
 vec( B ) 
experiments

Y =
 vec(C ) 
• Design linear controllers


in these regimes:
 vec( D )
setpoint, feedback,
feedforward
• Approximate controller
dependence on the
Linear interpolation:
regime parameters
Y ( Θ) = ∑ Y jϕ j ( Θ)
j
EE392m - Winter 2003
Control Engineering
13-20
Gain scheduling - example
• Flight control
• Flight envelope
parameters are used
for scheduling
• Shown
– Approximation nodes
– Evaluation points
• Key assumption
– Attitude and Mach are
changing much slower
than time constant of
the flight control loop
EE392m - Winter 2003
Control Engineering
13-21
Local Modeling Based on Data
Heat Loads
• Data mining in the loop
• Honeywell product
Multidimensional
Data Cube
Relational
Database
Heat
demand
Outdoor
Query point
( What if ? ) temperature
EE392m - Winter 2003
Time
of day
Control Engineering
Forecasted
variable
Explanatory
variables
13-22