Simrock 3

Properties of Non-Linear Systems
Some properties of non-linear dynamic systems are:
They do not follow the principle of superposition (linearity and homogeneity)

They may have multiple isolated equilibrium points (linear systems can have only one)
They may exhibit properties such as limit-cycle, bifurcation, chaos
Finite escape time: The state of an unstable nonlinear system can go to infinity in finite time
For a sinusoidal input, the output signal may contain many harmonics and sub-harmonics with
various amplitudes and phase differences
(a linear system's output will only contain the sinusoid at the output)

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CAS-DSP, Sigtuna 2007-Control Theory-S.Simrock

Analysis and Control of Non -Linear Systems
There are several well-developed techniques for analyzing nonlinear feedback systems:
Describing function method
Phase Plane method
Lyapunov stability analysis
Singular Perturbation method
Popov criterion (described in The Lur'e Problem below)
Center manifold theorem
Small-gain theorem
Passivity analysis

2

Control Design of Non-Linear Systems
Control design techniques for non-linear systems also exist. These can be subdivided into
techniques which attempt to treat the system as a linear system in a limited range of operation and
use (well-known) linear design techniques for each region:
Gain Scheduling
Adaptive control
Those that attempt to introduce auxiliary nonlinear feedback in such a way that the system can
be treated as linear for purposes of control design:

Feedback linearization
Lyapunov based methods
Lyapunov Redesign
Back-stepping
Sliding mode control

3

Describing Function Method
Procedure for analyzing Non-Linear Control problems based on quasi-linearization
Replacement of the non-linear system by a system that is linear except for a dependence on the
amplitude of the input waveform
An example might be the family of sine-wave inputs; in this case the system would be characterized
by an SIDF or sine input describing function H (A, ω) giving the system response to an input
consisting of a sine wave of amplitude A and frequency ω.
(This SIDF is a generalization of the transfer function H (ω) used to characterize linear systems)
Other types of describing functions that have been used are DF's for level inputs and for Gaussian
Noise inputs. The DF's often suffice to answer specific questions about control and stability.

F (y)

Example piecewise linear function:

Slope

s

s3
y

Slope 2
Slope

s1


4

Phase Plane Method
Refers to graphically determining the existence of limit cycles
The Phase Plane, applicable for second order systems only, is a plot with axes being the values of
the two state variables, x2 vs x1
Vectors representing the derivatives [ f 2 (x1,x2 ), f1 ( x1,x2 )] at representative points are drawn
( With enough of these arrows in place the system behaviour over the entire plane can be
visualized and limit cycles can be easily identified )

x2
unstable

0.5
0.5

-0.5

-0.5

x1

stable

5

Lyapunov Stability
Lyapunov stability occurs in the study of dynamical systems
In simple terms, if all solutions of the dynamical system that start out near an equilibrium point
xe stay near xe forever, then xe is Lyapunov stable
More strongly, if all solutions that start out near xe converge to xe , then xe is
asymptotically stable
The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how
quickly the solutions converge
The idea of Lyapunov stability can be extended to infinite-dimensional
manifolds, where it is known as Structural Stability, which concerns the behaviour of different
but "nearby" solutions to differential equations
x
2

x1

6

The Lur'e problem
Control systems have a forward path that is linear and time-invariant, and a feedback path that
contains a memory-less, possibly time-varying, static non-linearity

u

y

Linear Part H(s)

Non-Linear
Part

e

_
+

The linear part can be characterized by four matrices ( A,B,C,D ), while the non-linear part is φ( y)
with a sector non-linearity
By A.I. Lur'e

7

Popov Stability Criteria
Absolute stability

u

u = ke
e

The marked region in figure is called sector

0 ≤ F (e ) ≤ ke for e > 0
ke ≤ F (e ) ≤ 0 for e < 0
or shorter

0≤

F (e )
≤k
e

The standard control loop is absolute stable in sector [0 ,k ] , if the closed loop has
only one global asymptotic stable equilibrium for any F (e ) lying in this sector

9

Example for Popov Criteria
The non-linearity is shown as a three-point characteristic, which already hosts
the amplification K of the linear subsystem
KLb

e

a -K b
L

-a

-

u

1
(1 + jω) (1 + j 2ω) (1 + j 3ω)

x

L( jω)

Popov frequency response locus Z=L(jω)

ω Im
1
−
Kp

g krit
0.1

Re
-0.3

-0.2

-0.1

0.1
-0.1
-0.2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Z = L ( jω)

The line g krit is tangential to the Popov frequency response locus curve and
intersects the real axis at 0.1

12

Example for Popov Criteria
The Popov sector is defined by the line K p e
u
K Lb
e
a

α
1 a

K Lb

e

To verify Popov criteria the gradient of three-point characteristic must be smaller the
calculated gradient
K b
tan α = L
a

K Lb
<1
a
If this is true for the given inequality, then is idle state global asymptotic stable
13

Gain Scheduling
It is an approach to control of non-linear systems that uses a family of linear controllers, each
of which provides satisfactory control for a different operating point of the system
One or more observable variables, called the scheduling variables, are used to determine what
operating region the system is currently in and to enable the appropriate linear controller
Aircraft example:
flight control system, the altitude and Mach number might be the scheduling variables, with
different linear controller parameters available for various combinations of these two variables.

Pg

Gain change Pg

t
14

Adaptive Control
Modifying the controller parameters to compensate slow time–varying behaviour or uncertainties

For example, as an aircraft flies, its mass will slowly decrease as a result of fuel consumption; we
need a control law that adapts itself to such changing conditions

Adaptive control does not need a priori information about the bounds on these uncertain or timevarying parameters

Robust control guarantees that if the changes are within given bounds the control law need not be
changed

While adaptive control is precisely concerned with control law changes

15

Feedback Linearization
Common approach used in controlling nonlinear systems
Transformation of the nonlinear system into an equivalent linear system through substitution of
variables and a suitable control input.
Feedback linearization may be applied to nonlinear systems of the following form:

&
x = f (x ) + g (x ) u
y = h( x )

(1)
(2 )

x state vector
u input vector
y output vector

The goal is to develop a control input u that renders either the input-output map linear,
or results in a linearization of the full state of the system

Existence of a state Fb control ?
Variable substitution ?

u = α (x ) + β (x ) y
z = T (x )
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Pendulum Example
Task: Stabilization of pendulum at the origin

&
x1 = x2
&
x2 = a [sin (x1 + δ) − sin (δ)] − bx2 + cu

Linear system

1. Choose u as follows to cancel nonlinear term:

a
v
u = [sin ( x1 + δ ) − sin (δ )] +
c
c
2. Stabilizing fb controller

v = − k1 x1 − k 2 x2

Closed loop

&
x1 = x2
&
x2 = −b ⋅ x2 + v
&
x1 = x2
&
x2 = k1 x1 − (k 2 + b ) x2

3. Re-substituting to or fb control law

1
⎛ a⎞
u = ⎜ ⎟ [sin (x1 +δ) − sin (δ)] − (k1x1 + k2x2 )
c
⎝c⎠
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Sliding Mode Control
In Control theory Sliding Mode Control is a type of variable control structure where the
dynamics of a nonlinear system is altered via application of a high-frequency switching control
Time varying state feedback control scheme
This control scheme involves following two steps:
Selection of a hypersurface or a manifold such that the system trajectory exhibits desirable
behaviour when confined to this manifold
Finding feed-back gains so that the system trajectory intersects and stays on the manifold
Advantages:
Controllers with switching elements are easy to realize
The controller outputs can be restricted to a certain region of operation
Can also be used inside the region of operation to enhance the closed loop dynamics

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Sliding Mode Control

Two types of switching controllers
Switching between constant values
Switching between state or output dependent values
Design of a switching controller therefore needs
The definition of a switching condition
The choice of the control law inside the areas separated by the switching law

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Sliding Mode Control (Cnt’d)
Consider a linear system

The control law

&& = u
y

u = −ky

with k > 0

It is clear that controller is not able to stabilize the semi stable system that has two open loop
eigen values of zero. The root locus plot with two branches along the imaginary axis
Switching function:

&
&
s ( y, y ) = my + y

With a scalar m> 0 , then a variabale structure control law

&
⎧ − 1 s ( y,y ) > 0
u=⎨
&
⎩ 1 s ( y, y ) < 0

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Phase Portrait with Example
&
Consider a second order system x1 = x 2

&
x 2 = h (x ) + g (x ) u

g(x) and h(x) are unknown non-linear functions
On the manifold s the motion is
independent of g( x) and h( x)

Goal: state FB
stabilize the origin.

&
x1 = −ax1

s = a1 x1 + x2 = 0

s =0

x2

x1

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Chattering on Sliding Manifold
Caused by delays
(no ideal sliding on manifold)
Region

s <0

trajectory reverses

s

Delay leads to crossing again

s >0
s <0
Disadvantages:
Low control accuracy
Power losses
Excitation of unmodeled dynamics
Can result in instabilities

23

What is System Identification?
Using experimental data obtained from input/output relations to model dynamic systems

disturbances
input

System

output

Different approaches to system identification depending on model class
Linear/Non-linear
Parametric/Non-parametric
Non-parametric methods try to estimate a generic model
(step responses, impulse responses, frequency responses)
Parametric methods estimate parameters in a user-specified model
(transfer functions, state-space matrices)

1

Identification Steps
Experiment design :
Its purpose is to obtain good experimental data, and it
includes the choice of the measured variables and of
the character of the input signals

Experimental design

Data collection

Selection of model structure :
A suitable model structure is chosen using prior
knowledge and trial and error

Data pre-filtering

Model structure selection

Choice of the criterion to fit :
A suitable cost function is chosen, which reflects how
well the model fits the experimental data

Parameter estimation

Parameter estimation :
An optimization problem is solved to obtain the
numerical values of the model parameters

Model validation

Model ok?

Model validation :
The model is tested in order to reveal any inadequacies

No
Yes
2


Different Mathematical Models
Model description:
Transfer functions
State-space models
Block diagrams (e.g. Simulink)

Notation for continuous time models
Complex Laplace variable ‘s’ and differential operator ‘p’

∂x(t )
= p x(t )
∂t

&
x(t ) =
Notation for discrete time models

Complex z-transform variable and shift operator ‘q’

x(k + 1) = q x(k )
3

Experiments and Data Collection

Often good to use a two-stage approach

Preliminary experiments
Step/impulse response tests to get basic understanding of system dynamics
Linearity, static gains, time delays constants, sampling interval

Data collection for model estimation
Carefully designed experiment to enable good model fit
Operating point, input signal type, number of data points to collect

4

Preliminary Experiments- Step Response
Useful for obtaining qualitative information about the system
Dead time (delay)
Static gain
Time constants (rise time)
Resonance frequency

Sample time can be determined from time constants
Rule-of-thumb: 4-10 sample points over the rise time

5

Design Experiment for Model Estimation
Input signal should excite all relevant frequencies

Estimated model more accurate in frequency ranges where input has high energy
Pseudo-Random Binary Sequence (PRBS) is usually a good choice

Trade-off in selection of signal amplitude

Large amplitude gives high signal-to-noise ratio (SNR), low parameter variance
Most systems are non-linear for large input amplitudes

Big difference between estimation of system under closed loop control or not!

6

Collecting Data
w(t )
u (t )

y (t )
System

+
Lu

u (k )

e(t )

LY

y (k )

Sampling time selection and anti-alias filtering are central!

7

Data Pre-Filtering
Remove:
Transients needed to reach desired operating point
Mean values of input and output values, i.e. work with

1
Δu[t ] = u[t ] −
N
1
Δy[t ] = y[t ] −
N

N

∑ u[t ]
t =1

N

∑ y[t ]
t =1

Trends (use ‘detrend’ in MATLAB)
Outliers (obvious erroneous data points)

8

General Model Structure
Systems described in terms of differential equations and transfer functions
Provides insight into the system physics and a compact model structure
General-linear polynomial model or the general-linear model

( )

( )

y (k ) = q − k G q −1 u (k ) + H q −1 e(k )

u (k ) and y (k ) are the input and output of the system
e(k ) is zero-mean white noise, or the disturbance of the system

( )
H (q )

G q −1

−1

is the transfer function of the deterministic part of the system
is the transfer function of the stochastic part of the system

e(k )
C (q )
D(q )

General-linear model structure:

u (k )

B(q )
F (q )

+

1
A(q )

y (k )
9


Model Structures
Model structures based on input-output:

Model
ARX
ARMAX
Box-Jenkins
Output Error
FIR

A(q) y[k ] =

~ (q )
p
B(q )
A(q )
B(q )
A(q )
B(q )
F (q )
B(q )
F (q )
B(q )

~ (q )
pe
1
A(q )
C (q)
A(q )
C (q)
D(q)

1
1

B(q)
C (q )
p
p
u[k ] +
y[k ] or y[k ] = ~ (q ) u[k ] + ~e (q ) e[k ]
D(q )
F(q)

Provides flexibility for system and stochastic dynamics. Simpler models that are a subset
of the General Linear model structure
10

Parametric Models
Each of these methods has their own advantages and
disadvantages and is commonly used in real-world
applications
AR Model
Choice of the model structure to use depends on the
dynamics and the noise characteristics of the system

ARX Model
ARMAX Model

Using a model with more freedom or parameters
is not always better as it can result in the modeling
of nonexistent dynamics and noise characteristics

Box-Jenkins Model
Output-Error Model
State-Space Model

This is where physical insight into a system is helpful

11

AR Model
Process model where outputs are only dependent on previous outputs
No system inputs or disturbances are used in the modelling
Class of solvable problems is limited, for signals not for systems
Time series analyses, such as linear prediction coding commonly use the AR model

y[k ] =

1
e[k ]
A(q )

e(k )

1
A(q )

y (k )

AR Model Structure
12

ARX Model
Advantages:
Is the most efficient of the polynomial estimation methods
solving linear regression equations in analytic form.
Unique solution, that satisfies the global minimum of the loss function
Preferable, especially when the model order is high
Disadvantages:
Disturbances are part of the system dynamics.
The transfer function of the deterministic part G (q −1 ) of the system and the transfer function
of the stochastic part H q −1 of the system have the same set of poles

( )

This coupling can be unrealistic
The system dynamics and stochastic dynamics of the system do not share the same set of poles
all the time
(Can bias the estimation of the ARX model)
However, you can reduce this disadvantage if you have a good signal-to-noise ratio
13

ARX Model (Cnt‘d)
Set the model order higher than the actual model order to minimize the equation error,
especially when the signal-to-noise ratio is low
However, increasing the model order can change some dynamic characteristics of the
model, such as the stability of the model

y[k ] =

B(q )
1
u[k ] +
e[k ]
A(q )
A(q )
e(k )
u (k )

B(q )

+

1
A(q )

y (k )

ARX Model Structure
14

ARMAX Model
Includes disturbance dynamics!
ARMAX models are useful when you have dominating disturbances that enter early in the
process, such as at the input
i.e. a wind gust affecting an aircraft is a dominating disturbance early in the process
More flexibility in the handling of disturbance modelling

y[k ] =

B(q )
C (q )
u[k ] +
e[k ]
A(q )
A(q )

e(k )

C (q )

u (k )

B(q )

+

1
A(q )

y (k )

ARMAX Model Structure
15

Box Jenkins Model
Provides a complete model with disturbance properties modelled separately
from system dynamics
The Box-Jenkins model is useful when you have disturbances that enter late in the process
For example, measurement noise on the output is a disturbance late in the process
( not output error model)

y[k ] =

B( q )
C( q )
u[k ] +
e[k ]
F( q )
D( q )

e(k )
C (q )
D(q )

u (k )

B(q )
F (q )

+

y (k )

Box-Jenkins Model Structure
16

Output Error Model
Describes the system dynamics separately
No parameters are used for modelling the disturbance characteristics

y[k ] =

B(q )
u[k ] + e[k ]
F (q )
e(k )
u (k )

B(q )
F (q )

+

y (k )

OE Model Structure
17

State Space Model
All previous methods are based on minimizing a performance function
For complex high order systems the classical methods can suffer from several problems
Find many local minima in the performance function and don’t converge to global minima
The user will need to specify complicated parameterization. They also may suffer potential
problems with numerical instability and excessive computation time to execute the iterative
numerical minimization methods needed
In addition, modern control methods require a state-space model of the system
For cases such as these the State-Space (SS) identification method is the appropriate model
structure

Hint:

When the model order is high, use an ARX model because the algorithm involved in ARX model estimation is fast
and efficient when the number of data points is very large. The state-space model estimation with a large number of
data points is slow and requires a large amount of memory. If you must use a state-space model, for example in
modern control methods, reduce the sampling rate of the signal in case the sampling rate is unnecessarily high.

18

State Space (Cnt‘d)
x(k ) state vector

x(k + 1) = A x(k ) + B u (k ) + K e(k )
y (k ) = C x(k ) + D u (k ) + e(k )

y (k ) system output
u (k ) system input
e(k ) stochastic error
A,B,C,D, and K: system matrices.

In general, they provide a more complete representation of the system, especially for complex
MIMO systems
The identification procedure does not involve nonlinear optimization so the estimation reaches
a solution regardless of the initial guess
Parameter settings for the state-space model are simpler
You need to select only the order / states, of the model
Prior knowledge of the system or by analyzing the singular values of A

19

Determine Parameters
Determining the delay and model order for the prediction error methods,
ARMAX, BJ, and OE, is typically a trial-and-error process
The following is a useful set of steps that can lead to a suitable model
(this is not the only methodology you can use, nor is this a comprehensive procedure)
(1) Obtain useful information about the model order by observing the number of resonance peaks
in the nonparametric frequency response function. Normally, the number of peaks in the
magnitude response equals half the order of A(q ) , F (q )
(2) Obtain a reasonable estimate of delay using correlation analysis and/or by testing reasonable
values in a medium size ARX model. Choose the delay that provides the best model fit based
on prediction error or other fit criterion
(3) Test various ARX model orders with this delay choosing those that provide the best fit
(4) Since the ARX model describes both the system dynamics and noise properties using the same
set of poles, the resulting model may be unnecessarily high in order
(5) By plotting the zeros and poles and looking for cancellations you can reduce the model order

20

Determine Parameters (Cnt’d)
(5) Are there additional signals? Measurements can be incorporated as extra input signals!
(6) If you cannot obtain a suitable model following these steps additional physical insight into
the problem might be necessary.
( Compensating for nonlinear sensors or actuators and handling of important physical
non - linearities are often necessary in addition to using a ready-made model )
The higher the order of the model is, the better the model fits the data because the model
has more degrees of freedom.
(but you need more computation time and memory)
Choosing the model with the smallest degree of freedom, or number of parameters,
if all the models fit the data well and pass the verification test.

21

Conclusion
Variety of model structures available
Choice is based upon an understanding of the system identification method and algorithm
System and disturbance are important
Handling a wide range of system dynamics without knowing system physics
Reduction of engineering effort
Identification Tools (Matlab, Matrix or LabVIEW ) are available for developing, prototyping
and deploying control algorithms

22

Matlab System Identification Tool

23

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