SIMULTANEOUS SYSTEM IDENTIFICATION AND

DISTURBANCE ESTIMATION

FOR LINEAR TIME VARYING SYSTEM

WITH APPLICATION TO ACTIVE SUSPENSION SYSTEM

By

Dhruv Pramod Ghiya

A THESIS

Submitted to

Michigan State University

in partial fulﬁllment of the requirements

for the degree of

Mechanical Engineering-Master of Science

2019

ABSTRACT

SIMULTANEOUS SYSTEM IDENTIFICATION AND DISTURBANCE ESTIMATION

FOR LINEAR TIME VARYING SYSTEM

WITH APPLICATION TO ACTIVE SUSPENSION SYSTEM

By

Dhruv Pramod Ghiya

Various control techniques have been developed in order to improve the suspension performance

since late 20th century. Controllers have been developed to reject the road disturbance in case of

time invariant suspension systems. Although some of these researches give good results for non-

linear suspension systems with parameter uncertainties, there is not much research done which

treats a time varying system.

I propose here an approach based on output tracking state feedback controller with disturbance

rejection for linear time varying system to tackle this problem in case of a quarter car suspension

model. The proposed controller works even in case of completely unknown plants. This algorithm,

combined with a pole placement controller, can be used to ensure ride comfort and to avoid dam-

age to vehicle components. If a nominal model is known, it can also estimate time varying road

disturbances which can be written as a summation of sinusoids and this estimate can be used as a

preview to enhance the control performance of other vehicles. We compare the proposed algorithm

with LQR tracking algorithm in simulation and superior performance is demonstrated. Bounds on

the time varying parameters are derived in terms of norm of the states, within which the proposed

algorithm is guaranteed to work.

ACKNOWLEDGMENTS

Firstly, I would like to express my deepest thanks to my advisor, Dr Zhaojian Li, for his patience,

motivation and guidance throughout this work. He was always there to help me whenever I was

stuck on some problem. He has been a great mentor to me. I would also like to thank my committee

members, Prof Guoming Zhu and Prof Ranjan Mukherjee, for their valuable time and assistance. I

am really grateful to Prof Ranjan Mukherjee for giving me an amazing opportunity of o↵-campus

thesis during my undergraduate studies. If not for him, I would not even have started my studies

at MSU in the ﬁrst place.

Special thanks to Prof Hassan Khalil for sharing his knowledge with me and guiding me through-

out the project. His wisdom and expertise in this ﬁeld has been very inspiring and I am honored

to have been trained under him during my graduate studies.

Finally I would like to thank my Mom and Dad for their love and everlasting support in my

endeavors. I could not have done anything without you.

iii

TABLE OF CONTENTS

LIST OF FIGURES

KEY TO SYMBOLS

1 Introduction

1.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Motivation and approach towards the problem . . . . . . . . . . . . . . . . . . . . .
1.2.1 Active suspension system constraints . . . . . . . . . . . . . . . . . . . . . . .

2 Parameter Estimation

vi

ix

1
1
2
3

6

3 State Tracking State Feedback Controller

11
3.1 Without Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 Algorithm for Known Plant: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.2 Algorithm for Unknown Plant:
. . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.3 Controller for Suspension System . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 With Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Output Tracking State Feedback Algorithm

4.2 With Disturbance:

17
4.1 Without Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1.1 Algorithm for Known Plant: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
. . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1.2 Algorithm for Unknown Plant:
4.1.3 Controller for Suspension System:
. . . . . . . . . . . . . . . . . . . . . . . . 20
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2.1 Algorithm for Constant Disturbance . . . . . . . . . . . . . . . . . . . . . . . 23
4.2.2 Algorithm for Sinusoidal Disturbance
. . . . . . . . . . . . . . . . . . . . . . 24
4.2.3 Algorithm for Non-sinusoidal Disturbance . . . . . . . . . . . . . . . . . . . . 25
4.2.4 Known Plant and Unknown Disturbance . . . . . . . . . . . . . . . . . . . . . 25
4.2.5 Unknown Plant and Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Time Varying Case

29
5.1 State Tracking State Feedback Controller with Slowly Varying Parameters . . . . . . 29
5.2 Output Tracking State Feedback Controller with Slowly Varying Parameters
. . . . 31
5.3 Stability under Slowly Varying Parameters . . . . . . . . . . . . . . . . . . . . . . . . 33
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
. . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.3.1 Vanishing Perturbations:
5.3.2 Non-vanishing Perturbations:

6 Implementation of Pole Placement controller

38
6.1 Pole placement controller suspension system tracking a constant reference . . . . . . 39
6.2 Time Invariant System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2.1 Results in Absence of Disturbance . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2.2 Results in Presence of Disturbance . . . . . . . . . . . . . . . . . . . . . . . . 42
6.3 Time Variant System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.3.1 Results in Presence of Disturbance . . . . . . . . . . . . . . . . . . . . . . . . 43

iv

7 Tracking Comparison With LQR Tracking Controller

44
7.1 LQR Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8 Discussion and Conclusion

APPENDIX

REFERENCES

46

50

57

v

LIST OF FIGURES

Figure 1

Schematics illustrating the ﬂow of the state feedback algorithm. . . . . . . . . .

4

Figure 2
Image on the left hand side is a photo of the actual suspension system manufac-
tured by ‘Quanser Technologies’, while the image on the right hand side is the representation
of the system [1]. Parameters ms and mus being the masses for sprung mass and unsprung
mass, ks and kus the spring constants, bs and bus damping constants, Fc control force, x1,
x2 and zr the position vectors of sprung mass, unsprung mass and road proﬁle respectively.
Here sprung mass represents the vehicle body and unsprung mass is the wheel, ks and bs
represent the actual suspension parameters of the vehicle while kus and bus represent the
sti↵ness and damping in the tire.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Figure 3

The ﬁgure illustrates the error convergence for the parameter estimation of Eq. 11. 10

Figure 4
The ﬁgure illustrates the convergence of error for all the four states when state
tracking controller was used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

The ﬁgure on left illustrates the convergence of state 1 (zs   zus) with respect
Figure 5
to time steps (0.001 sec per step) which tracks the reference within 100 time steps. The
ﬁgure on right illustrates the convergence of state 2 ( ˙zs) which tracks the reference within
150 time steps. The blue signal is the plant state and yellow is the reference state.
. . . . .

The ﬁgure on left illustrates the convergence of state 3 (zus   zr) which tracks
Figure 6
the reference within 100 time steps. The ﬁgure on right illustrates the convergence of state
4 ( ˙zus) which tracks the reference within 150 time steps. The blue signal is the plant state
and yellow is the reference state.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 7
The ﬁgure on right illustrates the convergence of the parameter k1 which is
converged to [5.2348, -82.4405,0.0053,-0.4662] within 150 time steps. The right side ﬁgure
shows the convergence of the parameter k2 which is converged to 36.7169 within 150 time
steps.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The ﬁgure illustrates the output tracking controller performance for unknown
Figure 8
plant model when initial conditions for the adaptive law are chosen according to nominal
model. The plot on the left is the tracking error and the plot on right is the simulated plant
and reference output tracking.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 9
The ﬁgure illustrates signiﬁcance of knowing the nominal model of the system.
The results obtained here are when the initial conditions are taken to be 0 for all the
parameters. The plot on the left is the tracking error and the plot on right is the simulated
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
plant and reference output tracking.

14

15

15

21

22

Figure 10 The ﬁgure illustrates the output tracking controller performance in presence of
disturbance when the plant is assumed to be completely known. The plot on the left is the
tracking error and the plot on right is the simulated plant and reference output tracking.

.

26

vi

Figure 11 The ﬁgure illustrates the convergence of the disturbance parameters. As ex-
pected, the coe cients c1, c2, c3, c4 converged to 0.03, 0.05, 0.02, 0 respectively. The
convergence for all the parameters was within 400 time steps.
. . . . . . . . . . . . . . . .

Figure 12 The ﬁgure illustrates the disturbance estimation performance of the controller
when the plant is assumed to be completely known. The plot on the left is the error between
true and estimated disturbance and the plot on right is the true and estimated disturbance
proﬁle for a period of 10 time steps.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 13 The ﬁgure on the left illustrates the output tracking the given reference in
presence of disturbance. The ﬁgure on right illustrates the error convergence.
. . . . . . . .

Figure 14 The ﬁgure on left illustrates the convergence of the error for all the four states
when slowly varying system is controlled using the state tracking feedback control. The ﬁgure
on right illustrates the tracking performance of the controller for suspension deﬂection (x1).

Figure 15 The results plotted here are for lower variation in the system. The ﬁgure on left
illustrates the convergence of the error for all the four states when slowly varying system
is controlled using the state tracking feedback control. The ﬁgure on right illustrates the
tracking performance of the controller for suspension deﬂection (x1).
. . . . . . . . . . . . .

Figure 16 The results are for the case where A0 = 0.01,! = 5. The ﬁgure on the right
illustrates the tracking of the plant output to the reference output when slowly varying
system is controlled using the output tracking feedback control. The ﬁgure on left illustrates
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
the convergence of the output error.

Figure 17 The results are for the case where A0 = 0.05,! = 5. The ﬁgure on the right
illustrates the tracking of the plant output to the reference output when slowly varying
system is controlled using the output tracking feedback control. The ﬁgure on left illustrates
the convergence of the output error.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 18 The results are for the case where A0 = 0.005,! = 5. The ﬁgure on the right
illustrates the tracking of the plant output to the reference output when slowly varying
system is controlled using the output tracking feedback control. The ﬁgure on left illustrates
the convergence of the output error.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 19 The results are for the case where A0 = 0.005,! = 10. The ﬁgure on the
right illustrates the tracking of the plant output to the reference output when slowly varying
system is controlled using the output tracking feedback control. The ﬁgure on left illustrates
the convergence of the output error.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 20 The ﬁgure illustrates the performance of output tracking combined with pole
placement controller. The plot on the left is the tracking error and the plot on right is the
simulated plant and reference output tracking with the reference model’s output is made to
track a constant reference 0.05 using pole placement.
. . . . . . . . . . . . . . . . . . . . .

26

27

28

30

30

31

31

32

32

41

vii

Figure 21 The results are for the case where r(t) = 0.05 and disturbance as 0.1sin(25t).
The ﬁgure on the right illustrates the tracking of the plant output to the reference output
when system is controlled using the output tracking feedback control with pole placement.
The ﬁgure on left illustrates the convergence of the output error.
. . . . . . . . . . . . . . .

Figure 22 The result is for the case where r(t) = 0.05 and disturbance as 0.1sin(25t). The
ﬁgure illustrates the convergence of the estimation error. . . . . . . . . . . . . . . . . . . . .

42

42

Figure 23 The results are for the case where A0 = 0.01,! = 5, r(t) = 0.05 and disturbance
as 0.1sin(25t). The ﬁgure on the right illustrates the tracking of the plant output to the
reference output when slowly varying system is controlled using the output tracking feedback
control with pole placement. The ﬁgure on left illustrates the convergence of the output error. 43

Figure 24 The results are for the case where A0 = 0.01,! = 5, r(t) = 0.05 and disturbance
as 0.1sin25t. The ﬁgure on the right illustrates the tracking of the true and estimated road
proﬁles when slowly varying system is controlled using the output tracking feedback control
with pole placement. The ﬁgure on left illustrates the convergence of the estimation error. .

Figure 25 The comparison between output tracking and LQR tracking algorithms for the
case where A0 = 0.0001,! = 5, r(t) = 0.01. The ﬁgure on the on the left is the comparison
for the constant disturbance of 0.01 while ﬁgure on the right is for time varying disturbance
of 0.01sin10t.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 26 The results are for the case where A0 = 0.0001,! = 5, r(t) = 0.01. The ﬁgure
on the on the left is for the constant disturbance of 0.01 while ﬁgure on the right is for time
varying disturbance of 0.01sin10t.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

44

45

viii

KEY TO SYMBOLS

ms : sprung mass

mus : unsprung mass

ks : spring constant for sprung mass

kus : spring constant for unsprung mass

bs : damping constant for sprung mass

kus : damping constant for unsprung mass

Fc : control force

x : states

y : output

u : control

zr : road disturbance

zs : change of variable for position vector of sprung mass

zus : change of variable for position vector of unsprung mass

✓ : vector of parameters

  : regressor vector

⇤ : Hurwitz polynomial deﬁned as per the algorithm in use

Pr : parameter projection

  : scalar gain

  : vector gain

✏ : error vector

r : reference signal

k1, k2 : parameters used to deﬁne matching condition in state tracking controller

d : disturbance

du : disturbance in input signal

dy : disturbance in output signal

ci : coe cients of the disturbance signal

G(s) : transfer function

Z(s) : zeros of a transfer function

ix

P (s) : poles of a transfer function

⇢, !, ⇠, ⇣, m : parameters used to deﬁne the output tracking adaptive law

t : time

g(t, x) : perturbation

  : bound on perturbation

V : Lyapunov function

Rn : vector space of n dimension

|| : norm
| : mod

||
|
m : model reference parameter

⇤ : true parameter

ˆ: estimated parameter

˜: di↵erence in true and estimated parameter

x

1

Introduction

1.1 Literature review

Automotive suspension systems play an important role in ride comfort and are the reason behind a

smooth drive. As such, many researchers and practitioners have been working on new designs and

control methodologies to improve the performance. Automotive suspension systems can be cate-

gorized as passive [2], semi-active [3–5] and active suspension systems [6–17]. Passive systems are

those without any control implementation. In semi-active systems the suspension parameters (e.g.

spring or damping constants) are varied according to the requirement, while active systems involve

separate actuators to implement the control. Due its obvious advantages over passive suspension

systems, active and semi-active suspension systems have attracted much research attention since

late 20th century. In the initial period, more straightforward, simpler controllers were used. For

example linear parameter varying control combined with non-linear backstepping technique [18], lin-

ear quadratic regulator (LQR) [6], optimal linear state feedback based [19] controllers were studied.

Studies have been going on to enhance these control techniques and develop new, novel controllers

in order to improve the suspension performance. In recent years more advanced control techniques

like ADRC (active disturbance rejection control), H1 based, neural network based [17] robust con-
trol and variations of sliding mode, backstepping controls [7,9,11,14,15] have been used to improve

ride comfort in complex, non-linear situations like half [20] or full car [21] suspension systems in

presence of road disturbance. In [22], sliding mode control is used to estimate car body mass for

a quarter car system. In [12], an electromagnetic active suspension system is proposed which can

provide stability during cornering, braking and in presence of road disturbances. Although such a

suspension system is costly to make and has more volume as compared to the conventional systems.

In [17], the neural network based controller gives good performance in adapting to random road

disturbance but it assumes that the system model is fully known and time invariant. [23] deals

with a quarter car system with time varying loadings when the nominal loading is known. [24]

proposes a Disturbance observer (DOB) based sliding mode controller which works under various

loading conditions and road proﬁles. Even though it provides a robust control over the system,

it does not guarantee the convergence (of either suspension deﬂection or body acceleration) if the

load has changed signiﬁcantly. Even in [25], a DOB based optimal control is designed for a quarter

1

car model. Both these models work well in presence of road disturbances, although they don’t

seem to handle model uncertainties or time varying parameters quite well. In [26], the authors

have developed a control law based on feedback linearizion which adapts to the parameters whose

values change based on state space region within a given bound. However, the algorithm requires

an accurate model of the suspension system. In [27], the authors use adaptive sliding mode based

controller with Takagi-Sugeno Fuzzy approach which takes care of the uncertain parameters and

non-linearity for half-car model. They also proposed a bound on the unknown non-linear parameter

uncertainty which ensures asymptotic stability. In [13], authors have proposed an adaptive control

for nonlinear uncertain suspension system which seemed to handle the model uncertainties quite

well even in presence of unknown disturbances, but the control is only on the vertical suspension

displacement. They did not consider controlling the body acceleration which in fact is highly im-

portant in order to make the drive comfortable. The adaptive tracking control designed in [10] was

probably the best controller I came across which dealt with all the problems discussed above. But

then what if we can design a controller which can not only overcome all the above stated issues and

also estimate the road proﬁle online? With the road proﬁle now known, it can be used to enhance

suspension control and enable comfort-based route planning [28]. Controllers have been developed

to estimate the road disturbance [28, 29] but they are for the case of time invariant systems and

don’t tackle some of the issues mentioned earlier.

1.2 Motivation and approach towards the problem

For a highly nonlinear system (e.g. a full car suspension model), neural network based control or

other approximation based techniques might work. When there is model uncertainty or when the

parameters are time varying, some of the adaptive controllers might work. Robust controllers like

DOB might work when we need good accuracy in presence of road disturbance. But to satisfy

all the constraints, we need a better design. For instance, a combination of the above mentioned

algorithms or an adaptive controller with disturbance rejection will do the job if it could estimate

the road disturbance while dealing with model uncertainties and time varying parameters.

For this reason, we ﬁrst developed a DOB based controller which estimates road disturbance.

We assumed the road disturbance as a summation of sinusoids for this. The simulations gave really

good results and we were able to estimate a highly nonlinear road disturbance for a time invariant

2

system when an accurate model of the system was known. But the results were pretty bad when

we performed experiments. The estimation were not accurate, and some times used to blow up.

Even after trying a bunch of alternate methods to tackle this issue, we could not estimate the road

disturbance. This might have been due to the uncertainties in parameters of the suspension system

under consideration. Due to this, we switched to adaptive control considering it would work for

the given system and realized that a simple adaptive state feedback law with disturbance rejection

could do the job. These laws, while dealing with the uncertainties and rejecting the disturbance,

can even estimate this disturbance when certain conditions are satisﬁed. Speciﬁcally we designed

‘State and Output Tracking’ algorithm for the given suspension system which deals with controlling

states and output, mainly suspension deﬂection and body acceleration.

1.2.1 Active suspension system constraints

According to [14] for the full car model of an active vehicle suspension system, following four

constraints should be satisﬁed,

1. Suspension Deﬂection: In order to avoid damaging vehicle components, the suspension space

must less than the allowable upperbounds on deﬂection.

2. Road Holding: Dynamic tire loads should be less than static tire loads for four wheels to ensure

ﬁrm uninterrupted contact of wheels with the road

3. Actuator Saturation: Saturating the control at the maximum amplitude of the actuators.

4. Ride Comfort: Stabilizing roll, pitch, heave motions and avoiding disturbance forces transmitted

to passengers, which could be possible when the body acceleration is minimum

In this report, I designed a State and Output tracking state feedback law for a quarter car model,

which can satisfy all theses constraints and also estimate the road disturbance. The controllers were

designed to work in case of completely unknown systems. I showed that with the use of nominal

model, the performance of these controllers can be improved. These controllers can actually ﬁnd

the the nominal model of the system, but for this we need to make sure that the Persistently

Excitation (PE) conditions of the regressor vector is satisﬁed [30]. There are certain rules to select

the reference signal for the controllers which can make sure that the PE conditions are satisﬁed.

Although, research has been done only in case of linear parametric models to deﬁne these rules,

there is no general rule which can be applied straight away to the state feedback laws described

3

in the report. For this purpose, I designed a parameter estimation algorithm to ﬁnd the nominal

model of the system. This algorithm uses a linear parametric model while designing the adaptive

law and that’s why we can ensure the true convergence of the parameters with the appropriate

reference signal.
I. 

 

ur(t)

Road Disturbance

u(t)

Reference 

Model

Plant

Adaptive

Law

Estimated Disturbance

i.c.
State Feedback Control

9

 
 
II. 

 

ur(t)

Road Disturbance

u(t)

Pole 

Placement

Reference 

Model

Plant

Adaptive

Law

Estimated Disturbance

i.c.
State Feedback Control

Reference 
Plant

Tracking

Nominal 
Model

Parameter 
Estimation

System Identification

Desired reference signal
Reference 
Plant

Tracking

Nominal 
Model

Parameter 
Estimation

System Identification

 

 

11
Figure 1 Schematics illustrating the ﬂow of the state feedback algorithm.

The approach towards addressing this problem of satisfying the ‘four constraints’ was simple.

As shown in the block digram in Fig. 1, the state feedback algorithm is based on a speciﬁc reference

model which we choose such that the state (and output) tracking controller makes the plant state

4

(and output) track the reference model state (and output). Now, we can use a pole placement

controller to make this reference model (and in turn our plant) track the desired signal. With

this combined ‘State/ Output Tracking and Pole Placement Controller’, we can be make the body

acceleration track a zero reference, in turn making the drive smooth.

It can also be used in

restricting the states within a certain bounds which ensures the safety of the suspension system

(by controlling the suspension deﬂection given by zs   zus) and also satisﬁes road holding condition
(by restricting zus   zr which is the third state in my model). By saturating the control signal
according the requirements, we can make sure that the actuators use can provide the desired control.

Although, I never tested this in my results as I did not have the actuator information. Also, I dealt

with a quarter car model, there was no need to consider roll, pitch motions and therefore the ride

could be made comfortable just by controlling the body acceleration using the tracking algorithm.

Then with the addition of time varying parameters (as a perturbation from the nominal model)

and time varying disturbance, I used the same algorithm with some modiﬁcations to estimate the

disturbance and reject it to give desired tracking of states (and output). I also gave a bound on this

perturbation by considering it ﬁrst as vanishing perturbation, then for non vanishing perturbation.

In the end, results were simulated and compared with LQR tracking to conﬁrm that the proposed

algorithm performs better.

5

2 Parameter Estimation

The following suspension system was used in the project;

Figure 2 Image on the left hand side is a photo of the actual suspension system manufactured by
‘Quanser Technologies’, while the image on the right hand side is the representation of the system
[1]. Parameters ms and mus being the masses for sprung mass and unsprung mass, ks and kus the
spring constants, bs and bus damping constants, Fc control force, x1, x2 and zr the position vectors
of sprung mass, unsprung mass and road proﬁle respectively. Here sprung mass represents the
vehicle body and unsprung mass is the wheel, ks and bs represent the actual suspension parameters
of the vehicle while kus and bus represent the sti↵ness and damping in the tire.

Before designing the controller we need to ﬁrst identify our system. For this, we need equations

of motion for the system to derive a state space model. These equations were derived using the

free body diagrams of the system and are shown in Eq.1 and Eq.2.

¨x1 =  g +

Fc
ms

+

¨x2 =  g  

Fc
mus  

(bs + bus) ˙x1

mus

+

bs ˙x1
ms  
bs ˙x2
mus

+

bs ˙x2
ms

+

bus ˙zr
mus  

ksx2
ms

ksx1
ms  
(kus + ks)x1

mus

+

ksx2
mus

+

kuszr
mus

(1)

(2)

By taking zs and zus (refer Eq.3) such that the gravity terms form the equations of motion are

removed we get Eq.4.

zs = x2 +

msg
ks

+

g(ms + mus)

kus

, zus = x1 +

g(ms + mus)

kus

(3)

6

    mus¨zus =  (bs + bus) ˙zus   Fc + bs ˙zs + bus ˙zr   (zus + zs)ks   (zus   zr)kus

ms¨zs = bs ˙zus + Fc   bs ˙zs   (zs   zus)ks

The state space model is given by Eq.5.

˙x = Ax + Bu,

y = Cx + Du

where A, B, C, D matrices are given by

A =

266666664

0

1

 ks/ms  bs/ms

0

0

0

0

0

 1
bs/ms

1

ks/mus

bs/mus  kus/mus  (bs + bus)/ms

C =264

1

0

0

0

 ks/ms  bs/ms 0 bs/ms

and the states are

266666664

0

0

 1

0

1/ms

0

bus/mus  1/mus

377777775

, B =

377777775
375 , D =264

0

0

0 1/ms

375

x =

266666664

zs   zus

˙zs

zus   zr

˙zus

,

377777775

u =264

˙zr

Fc

375 ,

y =264

zs   zus

¨zs

375

(4)

(5)

(6)

(7)

Now, with zero road disturbance, we can use parameter projection method described in the

appendix (refer section 8). The transfer function from Fc to suspension deﬂection (zs   zus) is
given by the following equation,

(zs   zus)(s)

Fc(s)

=✓ 1

msmus◆ (mus + ms)s2 + sbus + kus

P (s)

(8)

7

where

P (s) = s4 +

(msbus + bsmus + bsms)

msmus

s3 +

(kusms + bsbus + ksmus + ksms)

s2+

msmus

(ksbus + bskus)

msmus

(9)

s +

kskus
msmus

which implies its a 4th order system. Therefore taking ⇤(s) as (s + 1)4 we get the linear parametric

model as y(t) = ✓⇤ (t) where

✓⇤ =

,  (t) =

(10)

where Z0 to Z2 are the coe cients of numerator and P0 to P3 are of the denominator of the

transfer function given by Eq.8. Similarly we get a linear parametric model for the Fc to body

acceleration (¨zs) case as shown in Eq.11.

¨zs(s)
Fc(s)

=✓ 1

msmus◆ s2(muss2 + buss + kus)

P (s)

✓⇤ =

(11)

2666666666666666664
26666666666666666666666664

u(t)
⇤(s)
su(t)
⇤(s)
s2u(t)
⇤(s)
y(t)
⇤(s)
sy(t)
⇤(s)
s2y(t)
⇤(s)
s3y(t)
⇤(s)

u(t)
⇤(s)
su(t)
⇤(s)
s2u(t)
⇤(s)
s3u(t)
⇤(s)
s4u(t)
⇤(s)
y(t)
⇤(s)
sy(t)
⇤(s)
s2y(t)
⇤(s)
s3y(t)
⇤(s)

3777777777777777775
37777777777777777777777775

2666666666666666664
26666666666666666666666664

Z0

Z1

Z2
1   P0
4   P1
6   P2
4   P3

0

0

Z2

Z3

Z4
1   P0
4   P1
6   P2
4   P3

3777777777777777775
37777777777777777777777775

,  (t) =

8

where Z2 to Z4 are now the coe cients of numerator of the transfer function given in Eq.11.

Now using the above linear parametric models we can formulate adaptive laws to estimate

required parameters. The manufacturer of the suspension system, ‘Quanser Technologies’ has

given a range within which all the physical parameters are bounded. This range can be utilized to

develop parameter projection algorithm. The derivation of this algorithm is given in Sec. 8. The

gradient algorithm for the suspension system was developed based on this derivation and is given

by,

˙✓i =   iPri✏ i,

1  i  number of parameters

with ✏(t) = y⇤(t)   y(t) and

(12)

(13)

Pri =

1 + ✓b
i ✓i
 
1 + ✓i ✓a
 

i

if ✓i >✓ b

i and ✏ i < 0

if ✓i <✓ a

i and ✏ i > 0

1

otherwise

8>>>>>><>>>>>>:

and   &  i are decided according to the parameter bounds, the need of accuracy and convergence

speed of the respective parameters.

Now, with the input of richness order 8, we can guarantee that we will get true parameter

convergence. For a linear parametric model, the PE condition can be found in [30], according to

which the input signal should at least be rich of order 1 more than that of the total number of

unknown coe cients of the transfer function in order to achieve true parameter convergence. From

Eq. 8 and 11, as there are 7 unknown coe cients, we need input signal of richness order 8.

9

Figure 3 The ﬁgure illustrates the error convergence for the parameter estimation of Eq. 11.

10

 3 State Tracking State Feedback Controller

3.1 Without Disturbance

Plant:

Reference Model:

˙x = Ax + Bu,

x 2 Rn, u 2 R

˙xm = Amxm + Bmr,

xm 2 Rn, r 2 R

(14)

(15)

Here Am is Hurwitz and r(t) is piecewise continuous and bounded.

3.1.1 Algorithm for Known Plant:

Take k⇤1 2 Rn, k⇤2 2 R such that

A + B(k⇤1) = Am,

B(k⇤2) = Bm

(16)

with u = (k⇤1)T x + k⇤2r and e = x   xmwe get

˙x = [A + B(k⇤1)T ]x + B(k⇤2)r

= Amx + Bmr

) ˙e = Ame

As Am is Hurwitz limt!1 e(t) = 0.

3.1.2 Algorithm for Unknown Plant:

Let k1(t), k2(t) be the estimates of the true parameters. Take Q = QT > 0. We have P = P T > 0

as the solution of the Lyapunov equation

P Am + AT

mP =  Q, Q = QT > 0

(17)

11

with u = kT

1 (t)x(t) + kT

2 (t)r(t) and k⇤1 = k1   ˜k1, k⇤2 = k2   ˜k2 we get

with the Lyapunov function

and the adaptive law

˙e = Ame + Bm

1
k⇤2

(˜kT

1 x + ˜k2r)

V = eT P e +

˜kT
1   1˜k1 +

1
|k⇤2|

˜k2
2

1
 |k⇤2|

˙k1 =   xeT P Bm
˙k2 =   reT P Bm

(18)

(19)

we get ˙V =  eT Qe < 0 and hence global asymptotic stability such that limt!1 e(t) = 0.

3.1.3 Controller for Suspension System

Our plant is already in the form given in Eq. 14, so we can choose the reference model in the

form given in Eq. 15. Assuming we have access to all the states, we can design the adaptive

state feedback controller designed here will be such that all closed loop signals are bounded and

limt!1[x(t)  xm(t)] = 0 [30]. (Although if the we don’t have access to the states, we can use high
gain observers to estimate the states ﬁrst and then use the estimates to design the feedback law.

In this case we might have to face other issues like peaking phenomenon caused due to the high

gain, but with saturating the control and some additional modiﬁcation in the control law this issue

can be resolved [31].)

The state model of the plant for the active suspension system without any disturbance is as

follows,

x =

266666664

zs   zus

˙zs

zus   zr

˙zus

377777775

, u =Fc  , y = x

12

(20)

A =

266666664

0

1

 ks/ms  bs/ms

0

0

0

0

0

 1
bs/ms

1

ks/mus

bs/mus  kus/mus  (bs + bus)/ms

, B =

377777775

266666664

0

1/ms

0

 1/mus

377777775

C = eye(4), D = 0

(21)

(22)

Using the information of the nominal model of given suspension system, the reference model is

chosen such that Am is Hurwitz and the matching condition can be satisﬁed with some k1 and k2.

Am =

266666664

0

1

0

 100  10  100
0
0

0

0

10

1

245

24.5  2255  29.5

377777775

, Bm =

266666664

Matching condition is given by,

Cm = eye(4), Dm = 0

A + B(k⇤1)T = Am, Bk⇤2 = Bm

0

4.0816

377777775

0

 10

(23)

(24)

(25)

where k⇤1 and k⇤2 are the true parameters which are to be estimated. With k1(t) and k2(t) being

the estimates take u such that

we get from Eq. 14

u = k1(t)T x + k2(t)r

˙x = Amx + Bmr

Now, as described in the algorithm in the previous section, with e = x   xm

˙e = Ame

13

(26)

(27)

(28)

As Am is Hurwitz, limt!1[x(t) xm(t)] = 0. With P = P T > 0 being the solution of the Lyapunov
equation given in 17, the adaptive law is given by (refer to Sec. 3.1 for detailed proof)

˙k1 =   xeT P Bm ˙k2 =   reT P Bm

(29)

The convergence results are as follows,

Figure 4 The ﬁgure illustrates the convergence of error for all the four states when state tracking
controller was used.

Figure 5 The ﬁgure on left illustrates the convergence of state 1 (zs   zus) with respect to time
steps (0.001 sec per step) which tracks the reference within 100 time steps. The ﬁgure on right
illustrates the convergence of state 2 ( ˙zs) which tracks the reference within 150 time steps. The
blue signal is the plant state and yellow is the reference state.

14

      Figure 6 The ﬁgure on left illustrates the convergence of state 3 (zus zr) which tracks the reference
within 100 time steps. The ﬁgure on right illustrates the convergence of state 4 ( ˙zus) which tracks
the reference within 150 time steps. The blue signal is the plant state and yellow is the reference
state.

Figure 7 The ﬁgure on right illustrates the convergence of the parameter k1 which is converged to
[5.2348, -82.4405,0.0053,-0.4662] within 150 time steps. The right side ﬁgure shows the convergence
of the parameter k2 which is converged to 36.7169 within 150 time steps.

15

      3.2 With Disturbance

Plant:

˙x = Ax + B(u + d),

d(t) = d0 +

[djfj(t)]

qXj=1

(30)

Where fj are known bounded continuous functions and d0 & dj are the unknown part of the

disturbance. In case of the given suspension system, we can not satisfy the necessary matching

condition B = Bd for the disturbance and the equation is rather of the form ˙x = Ax + Bu + Bdd.

Therefore a di↵erent method is required for the estimation of disturbance. In the next section,

Output Tracking State Feedback controller is described.

16

4 Output Tracking State Feedback Algorithm

4.1 Without Disturbance

4.1.1 Algorithm for Known Plant:

Plant:

˙x = Ax + Bu,

y = Cx,

x 2 Rn, u, y 2 R

with the transfer function given by

G(s) = C(sI   A) 1B = kp

Z(s)
P (s)

,

kp 6= 0

(31)

(32)

P (s) = det(sI   A), Z(s) is monic and Hurwitz, deg(P ) = n, deg(Z) = m < n. A, B, C stabilizable
and detectable.

Reference Model:

˙x = Amxm + Bmr,

ym = Cmxm

(33)

such that, Gm(s) = 1
Pm
u = (k⇤1)T x + k⇤2r gives the close loop transfer function as

i.e. ym = 1
Pm

r and degPm = n   m and Pm is monic and Hurwitz. With

G1(s) = C(sI   A   B(k⇤1)T ) 1Bk⇤2

(34)

As state feedback does not change the numerator polynomial of the transfer function we get

from Eq. 34 and 32 we get

(35)

G1(s) = [C(sI   A   B(k⇤1)T ) 1B]k⇤2

= k⇤2

kpZ(s)

det[sI   A   B(k⇤1)T ]

17

choose k⇤1, k⇤2 to satisfy

det[sI   A   B(k⇤1)T ] = Pm(s)Z(s),

k⇤2 =

1
kp

such that G1(s) =

1

Pm(s)

(36)

4.1.2 Algorithm for Unknown Plant:

Assumptions: deg(Z) and sign(kp) is known.

Let k1(t), k2(t) be the estimates of the true parameters. With ˜k1 = k1   k⇤1, ˜k2 = k2   k⇤2 and
u = kT

1 (t)x + k2(t)r our system becomes;

˙x = [A + B(k⇤1)T ]x + Bk⇤2r + B(˜k1)T x + B˜k2r

(37)

The solution of this, given the initial condition at t = 0 as x(0), and therefore y(t) is given by

exp(A+B(k⇤1 )T )(t ⌧ )[Bk⇤2r + B(˜k1)T x + B˜k2r](⌧ ) d⌧

0

y(t) = Cx(t)

x(t) = exp(A+B(k⇤1 )T )t x(0) +Z t
= C exp(A+B(k⇤1 )T )t x(0) +Z t
= C exp(A+B(k⇤1 )T )t x(0) +Z t

C exp(A+B(k⇤1 )T )(t ⌧ )[Bk⇤2r + B(˜k1)T x + B˜k2r](⌧ ) d⌧

C exp(A+B(k⇤1 )T )(t ⌧ ) Bk⇤2[r + kp(˜k1)T x + kp˜k2r](⌧ ) d⌧

0

0

Now,

L(Z t

0

C exp(A+B(k⇤1 )T )(t ⌧ ) Bk⇤2v(⌧ ) d⌧) = C[sI   A   B(k⇤1)T ] 1Bk⇤2v(s)

=

1

Pm(s)

v(s)

in hybrid notation we can write

y(t) = C exp(A+B(k⇤1 )T )t x(0) +

1

Pm(s)

[r + kp(˜k1)T x + kp˜k2r](t)

(38)

18

with ym = 1

Pm(s) r, e(t) = y(t)   ym(t) and ⇢⇤ = kp we get

e(t) =

⇢⇤

Pm(s)⇥˜kT

1 x + ˜k2r(t)⇤ + C exp(A+B(k⇤1 )T )t x(0)

From Eq. 36 we know that A + B(k⇤1)T is Hurwitz therefore limt!1 C exp(A+B(k⇤1 )T )t x(0) = 0.
Now we design the adaptive law to ensure that the remaining terms of e(t) go to 0. For that deﬁne
!, ✓, ✓⇤ and ˜✓ as

!(t) =264

x(t)

r(t)

375 ,✓

(t) =264

k1(t)

k2(t)

375 ,✓

⇤(t) =264

k⇤1
k⇤2

375 ,

˜✓(t) = ✓   ✓⇤ =264

˜k1
˜k2

375

(39)

Therefore,

e(t) =

⇢⇤

Pm(s)

˜✓T (t)!(t)

⇢⇤

⇢⇤

⇢⇤

Pm(s)⇥✓T (t)!(t)   (✓⇤)T !(t)⇤
✓T (t)!(t)   ⇢⇤(✓⇤)T
✓T (t)!(t)   ⇢⇤✓T (t)

Pm(s)

Pm(s)

1

1

Pm(s)

Pm(s)

!(t)

(since ✓⇤ is constant)

!(t)   ⇢⇤ ˜✓T

1

Pm(s)

!(t)

=

=

=

Let,

⇣(t) =

1

Pm(s)

!(t)

1

✓T (t)!(t)

⇠(t) = ✓T (t)⇣(t)  
Pm(s)
˜⇢ = ⇢   ⇢⇤
⇢(t) = kp(t),
e(t) = ⇢⇤ ˜✓T (t)⇣(t)   ⇢⇤⇠(t)
✏(t) = e(t) + ⇢(t)⇠(t)

(40)

with a positive deﬁnite function V = |⇢⇤|˜✓T   1 ˜✓ + 1
as,

  ˜⇢2 we make ˙V < 0 by taking the adaptive law

˙✓ =  

sign(⇢⇤) ⇣✏

m2

,

˙⇢ =  

 ⇠✏
m2 , m2 = 1 + ⇣T ⇣ + ⇠2, u = ✓T !

(41)

With this we make ensure that ✓(t) 2 L1,⇢ (t) 2 L1, ✏(t)

m(t) 2 L2 \ L1, ˙✓(t) 2 L2 \ L1 and ˙⇢(t) 2

19

L2 \ L1 which in turn gives limt!1 e(t) = 0 as shown in [30].

4.1.3 Controller for Suspension System:

As shown in the previous section, take the plant as

with

˙x = Ax + Bu,

y = Cx,

x 2 Rn, u 2 R

G(s) = C(sI   A) 1B = kp

Z(s)
P (s)

,

kp 6= 0

(42)

(43)

with monic Z(s). From Eq. 21 we know that {A, B, C} is controllable and observable and Z(s) is
Hurwitz, therefore we can design output tracking controller using the method given in Sec. 4.1.2.

We choose the reference model according to the conditions given in the Sec. 4.1.2. In case of

the given suspension system, we chose Pm = s2 + 69s + 69 which gives the reference model transfer

function as shown below,

Gm(s) =

1

s2 + 69s + 69

and therefore the reference model can be designed to be the following.

1

0

375 , Bm =264
Am =264
Cm =1 0  , Dm = 0

 69  69

0

1

375

(44)

(45)

This reference model is chosen according to those conditions and the nominal model of the

suspension system along with the parameter values given by the manufacturer of the suspension

system (in general we don’t need to consider the parameter values ahead of time but if we have

a general idea about the parameter range, we can locally restrict our estimation near the true

parameters, this improves our convergence). It is chosen such that there exist parameters k⇤1 and
k⇤2 which satisfy the equations given in Eq. 36. By taking !, ✓, ⇢, ⇠, ⇣, ✏, m according to the

20

deﬁnitions given in the previous section, we can develop adaptive law as follows,

˙✓ =  

sign(⇢⇤) ⇣✏

m2

,

˙⇢ =  

 ⇠✏
m2

(46)

The ✓ and ⇢ values were calculated analytically using the given parameter values and this

solution was used to decide the initial conditions for the adaptive law. The results based on this

can be seen the following ﬁgures.

Figure 8 The ﬁgure illustrates the output tracking controller performance for unknown plant model
when initial conditions for the adaptive law are chosen according to nominal model. The plot on
the left is the tracking error and the plot on right is the simulated plant and reference output
tracking.

Making use of the nominal model to analytically solve for the unknown parameters gives a faster

convergence as the least square solution does not get stuck in some local optima. To illustrate this,

we can take the initial conditions as 0 for all the unknown parameters. The result based on this

is shown in ﬁg. 9. Compared to the convergence results obtained above, these results showed that

even after a signiﬁcant amount of time, the solution did not converge completely. Even though the

error was small in this case, the solution was not the optimal solution.

21

   Figure 9 The ﬁgure illustrates signiﬁcance of knowing the nominal model of the system. The results
obtained here are when the initial conditions are taken to be 0 for all the parameters. The plot
on the left is the tracking error and the plot on right is the simulated plant and reference output
tracking.

22

   4.2 With Disturbance:

4.2.1 Algorithm for Constant Disturbance

Plant:

˙x = Ax + Bu + Bddu,

y = Cx + dy,

G(s) = C(sIA) 1B = kP

x 2 Rn, u, y, du, dy 2 R

Z(s)
P (s)

, kp 6= 0

(47)

P (s) = det(sI   A)

In case of known plant and disturbance, disturbance rejection algorithm for constant disturbance

du(t) = du0 and dy(t) = dy0 is designed as follows,

u(t) = (k⇤1)T x(t) + k⇤2r(t) + k⇤3

˙x = [A + B(k⇤1)T ]x + Bk⇤2r + Bk⇤3 + Bddu

choose k⇤1 and k⇤2 as before such that (refer Eq. 35)

C(sI   A   B(k⇤1)T ) 1Bk⇤2 =

1

Pm(s)

(48)

(49)

Now using the adaptive law designed in section 4.1.1 the Eq.38 will get modiﬁed due to the distur-

bance as follows

y(s) =

1

Pm(s)

r(s) + C[sI   A   B(k⇤1)T ) 1B

k⇤3
s

+ C[sI   A   B(k⇤1)T ) 1Bd

du0
s

+ dy0

(50)

Let  (s) = C[sI   A   B(k⇤1)T ) 1B k⇤3
det[sI   A   B(k⇤1)T ] are in LHS of the real axis, applying ﬁnal value theorem [30] we get

s + C[sI   A   B(k⇤1)T ) 1Bd

du0
s + dy0. Since all zeros of

lim
t!1

 (t) = lim
s!1

s (s) = c1k⇤3 + c2 + dy0

c1

def= C[ A   B(k⇤1)T ] 1B 6= 0,

c2

def= C[ A   B(k⇤1)T ] 1Bddu0

Now choose k⇤3 as

k⇤3 =

c2 + dy0

c1

so that limt!1  (t) = 0 and therefore limt!1⇥y(t)   ym(t)⇤ = 0.

23

(51)

(52)

For unknown plant and disturbance we take the estimates of true parameters as k1(t), k2(t), k3(t)

with ˜k1 = k1   k⇤1, ˜k2 = k2   k⇤2 , ˜k3 = k3   k⇤3 and u = kT

1 (t)x + k2(t)r + k3(t) our system becomes,

˙x = [A + B(k⇤1)T ]x + Bk⇤2r + Bk⇤3 + Bddu0 + B⇥(˜k1)T x + ˜k2r + ˜k3⇤

y(t) = Cx(t) + dy0

Now adaptive law can be deﬁned as shown in section 4.1.2 with

!(t) =266664

x(t)

r(t)

1

377775

,✓

(t) =266664

k1(t)

k2(t)

k3(t)

377775

,✓

The adaptive law is given by Eq. 40 and Eq. 41.

⇤(t) =266664

k⇤1
k⇤2
k⇤3

377775

,

˜✓(t) = ✓   ✓⇤ =266664

˜k1
˜k2
˜k3

377775

(53)

(54)

4.2.2 Algorithm for Sinusoidal Disturbance

Similar to constant disturbance case, adaptive law can also be designed for sinusoidal disturbances

with known frequency components as given in [30].

For the disturbances of the form

du(t) = du0 +

dy(t) = dy0 +

quXj=1
qyXj=1

(↵ujsin!ujt +  ujcos!ujt)

(↵yjsin!yjt +  yjcos!yjt)

(55)

where all the frequencies are known and amplitudes are unknown. Taking k3 of the form

k3(t) = k30 +

quXj=1

(k3↵jsin!ujt + k3 j cos!ujt) +

qyXj=1

(k3↵(qu+j)sin!yjt + k3 (qu+j)cos!yjt)

(56)

The addition of new parameters will alter the deﬁnition of !, ✓ and ✓⇤ (which can be found in [30])

and the adaptive law will again be deﬁned as shown in Eq. 40 and Eq. 41.

24

4.2.3 Algorithm for Non-sinusoidal Disturbance

If the disturbance is of the form

du(t) = d0 +

djfj(t)

qXj=1

dy(t) = 0, B = Bd

where fj are known bounded continuous functions and d0&dj are the unknown part of the distur-

bance, then we can deﬁne the adaptive law as shown in [30]. Although, in the suspension system

case, the matching condition B = Bd is not satisﬁed so we can’t use this algorithm.

4.2.4 Known Plant and Unknown Disturbance

For simplicity, the disturbance case was ﬁrst studied when the plant was known (nominal plant

case). In the next section, we will deal with the case where the system is time varying. The adaptive

law was designed as shown in Sec. 4.2.

For the suspension system, we assume that there is no noise or disturbance in the output signal.

With this we can take dy(t) = 0. The unknown road disturbance was taken to be

du(t) = 0.03 + 0.05sin(6t) + 0.02sin(10t)

(57)

The algorithm described in Sec. 4.2 requires the disturbance frequencies to be known. In our case

it is not possible to know the road disturbance frequencies in advance. For this, if we assume the

road disturbance to be a summation of sinusoids with a known range of frequencies, we can take

a set of frequencies in the given range and estimate the disturbance by using the adaptive law for

unknown disturbance. The accuracy of this estimation can be increased by increasing the number

of frequencies taken. For example, if the frequency range is given as t to 10t, we can assume the
a set {t, 2t, ..., 10t} and take ˆd = c1 + c2sin(t) + c3sin(2t) + ... + c11sin(10t) and use the adaptive
law to estimate all the coe cients c1 to c11.

25

In the simulations, for simplicity the set of frequencies was taken to be {5t, 6t, 10t} so that the
estimated disturbance was ˆd = c1 + c2sin(6t) + c3sin(10t) + c4sin(5t). The result of the simulations

is shown in following ﬁgures.

Figure 10 The ﬁgure illustrates the output tracking controller performance in presence of distur-
bance when the plant is assumed to be completely known. The plot on the left is the tracking error
and the plot on right is the simulated plant and reference output tracking.

Figure 11 The ﬁgure illustrates the convergence of the disturbance parameters. As expected, the
coe cients c1, c2, c3, c4 converged to 0.03, 0.05, 0.02, 0 respectively. The convergence for all the
parameters was within 400 time steps.

26

    Figure 12 The ﬁgure illustrates the disturbance estimation performance of the controller when the
plant is assumed to be completely known. The plot on the left is the error between true and
estimated disturbance and the plot on right is the true and estimated disturbance proﬁle for a
period of 10 time steps.

4.2.5 Unknown Plant and Disturbance

Now, let’s consider the case when the plant is completely unknown. In this case, even though the

above designed algorithm gives good tracking of the outputs (see Fig. 13), in order to get accurate

disturbance estimation we need to ensure the persistently excitation (PE) condition is satisﬁed.

This condition states that if the regressor vector is persistently excited (i.e. all its modes are

excited), then we can guarantee that all unknown parameters are going to converge to their true

values. For a linear parametric model, this condition is equivalent to saying that the reference signal

is su ciently rich [30]. Although there is no such rule of checking the PE condition for a general

system. In case of our system, if we know that with some reference signal, the PE conditions is

satisﬁed for the regressor vector, we can ensure that we will get true convergence for the unknown

parameters and in turn we can estimate the true disturbance.

Here although the output is tracking the reference signal, as the PE conditions have not been

satisﬁed we can’t estimate the true road disturbance. For solving such a problem, if we know

the nominal model of the system, we can make use of it to estimate the disturbance with some

variance. Instead of considering the entire system as unknown, we can take the nominal model and

add a time varying component to it and then use this model to estimate the disturbance. This is

done in the following section where this time varying component is assumed to be a perturbation

from the nominal system and then error bounds are estimated for the vanishing and non-vanishing

27

     Figure 13 The ﬁgure on the left illustrates the output tracking the given reference in presence of
disturbance. The ﬁgure on right illustrates the error convergence.

disturbances.

28

   5 Time Varying Case

Both the state feedback controllers are designed only for the time invariant systems. As shown

in Sec. 3 and 4), when we show that the derivative of the Lyapunov function is less than zero by

designing the adaptive laws such that the non-negative term in the equation will become zero, we

make use of the fact that the system is time invariant. This simpliﬁes the derivation and makes

it easy to deﬁne those adaptive laws. Although, as shown by the author in [32], there is a way

we can prove that the same adaptive laws can work for time varying systems with some additional

constraints. If we assume that the time varying parameters (✓(t)) of the system are slowly changing

with respect to time and their norms are bounded, we can prove that the controller developed for the

time invariant system can still work in for the time varying case if ✓(t) is continuously di↵erentiable

and ||

˙✓(t)|| is su ciently small [32].
Before checking if these assumptions work in our case, ﬁrst we will check if our controllers will

work for the time varying system when the change in system is small with respect to time.

5.1 State Tracking State Feedback Controller with Slowly Varying Parameters

The system was analyzed by adding a time dependent component ✓(t) it such that the {A, B}
matrices will change as shown below,

377777777775

(58)

A =

266666666664

 1

bs
ms

+ A0sin(!t)

1

  A0sin(!t)

0

1

 ks
ms   A0sin(!t)  bs

ms   A0sin(!t)

0

0

0

0

0

ks
mus

+ A0sin(!t)

bs
mus

+ A0sin(!t)  kus

mus   A0sin(!t)  (bs+bus)

ms

B =

266666666664

377777777775

0

1
ms

+ A0sin(!t)

0

 1
mus   A0sin(!t)

29

Here the amplitude A0 and frequency ! were chosen according the nominal values of the parameters

so that the variation in the parameter is comparable to its nominal value. Value of ! depends on

how slow we want the system to vary with respect to time. For the results shown below, ! was taken

to be 5. As ✓(t) here can be considered as a non vanishing perturbation in the system due to which

the convergence is ultimately bounded by a certain bound which depends on the upperbound of

the norm of the perturbation. This bound can be found by Lyapunov analysis shown in [32] and [31].

Figure 14 The ﬁgure on left illustrates the convergence of the error for all the four states when
slowly varying system is controlled using the state tracking feedback control. The ﬁgure on right
illustrates the tracking performance of the controller for suspension deﬂection (x1).

Figure 15 The results plotted here are for lower variation in the system. The ﬁgure on left illustrates
the convergence of the error for all the four states when slowly varying system is controlled using
the state tracking feedback control. The ﬁgure on right illustrates the tracking performance of the
controller for suspension deﬂection (x1).

30

      5.2 Output Tracking State Feedback Controller with Slowly Varying Parame-

ters

The controller designed in Sec. 4 for the time invariant system was used for the time varying system

described in the previous section. The results obtained from simulations are shown below.

Figure 16 The results are for the case where A0 = 0.01,! = 5. The ﬁgure on the right illustrates
the tracking of the plant output to the reference output when slowly varying system is controlled
using the output tracking feedback control. The ﬁgure on left illustrates the convergence of the
output error.

Figure 17 The results are for the case where A0 = 0.05,! = 5. The ﬁgure on the right illustrates
the tracking of the plant output to the reference output when slowly varying system is controlled
using the output tracking feedback control. The ﬁgure on left illustrates the convergence of the
output error.

31

      Figure 18 The results are for the case where A0 = 0.005,! = 5. The ﬁgure on the right illustrates
the tracking of the plant output to the reference output when slowly varying system is controlled
using the output tracking feedback control. The ﬁgure on left illustrates the convergence of the
output error.

Figure 19 The results are for the case where A0 = 0.005,! = 10. The ﬁgure on the right illustrates
the tracking of the plant output to the reference output when slowly varying system is controlled
using the output tracking feedback control. The ﬁgure on left illustrates the convergence of the
output error.

From the results shown in Fig. 16 to Fig. 19, we can see that as the amplitude or the frequency

is increased, the upper bound on output tracking error is also increasing. When the frequency is

increased from 5 to 10, the upper bound on error has also doubled. When the amplitude is increase

from 0.005 to 0.05, the upper bound is not increasing by the same proportion, but the tracking

is not as smooth as that of the lower amplitude case. There is obviously a relation between these

two parameters and the upper bound on the error. In the next section, we will discuss about this

relationship.

32

      5.3 Stability under Slowly Varying Parameters

The time varying parameters ✓(t) can be considered to be a perturbation from the nominal system.

For example, let us consider a system

˙x = f (t, x) + g(t, x)

(59)

where functions f and g are in the domain [0,1] X D ! Rn are piecewise continuous in t and locally
Lipschitz in x in the given domain with D ⇢ Rn is a domain that contains the origin. Such a system
can be considered to be perturbed from the nominal model ˙x = f (t, x) with perturbation g(t, x).

This perturbation term can be a result of disturbance, model uncertainty, modeling error [32]. In

our case, we consider g(t, x) = ✓(t). These perturbations can either be vanishing or non-vanishing.

For both the cases, there is a way we can ﬁnd bounds on ||g(x, t)|| as described in [32].

5.3.1 Vanishing Perturbations:

For vanishing perturbations where the perturbations vanish at the origin x = 0, we have g(t, 0) = 0.

For such case, if g(t, x) is locally Lipschitz in x and piecewise continuous in t, in a bounded

neighborhood of origin, it satisﬁes linear growth bound [32] such that,

||g(t, x)||   ||x||,

8t   0, 8x 2 D,     0

(60)

Let the origin of the system be an exponentially stable equilibrium point of the nominal system

˙x = f (t, x). By the Converse Lyapunov Theorem [31], we can ﬁnd a Lyapunov function, V (t, x)

such that,

(61)

c1||x||2  V (t, x)  c2||x||2
@V
@V
f (t, x)   c3||x||2
@x
@t
@V

+

        

@x          c4||x||

33

For such a system, we can show that

f (t, x) +

g(t, x)

@V
@x

˙V =

@V
@t

+

@V
@x

  c3||x||2 +        

  c3||x||2 + c4 ||x||2

@V

@x         ||g(t, x)||

If   is small enough,

 <

c3
c4

(62)

(63)

we get ˙V   (c3    c4)||x||2 making the system exponentially stable in presence of the perturba-
tions.

In our case, since it is a linear system and as A matrix is Hurwitz, the origin is exponentially

stable. Now, we can design the Lyapunov function by solving the Lyapunov equation as similar

to Eq. 17. With Q =  I4X4 we get a positive deﬁnite matrix by using the matlab command
P = lyap(A0, Q). With this P we deﬁne the Lyapunov function as V = xT P x. As this is a

quadratic in x, with this choice of Lyapunov function we can ﬁnd out all the coe cients c1 to c4

as shown below

therefore

Now,

(64)

(65)

(66)

 min(P )||x||2  V   max(P )||x||2

c1 =  min(P ) = 0.0360

c2 =  max(P ) = 201.0587

@V
@x

= 2xT P

 2 max(P )||x||

) c4 = 2 max(P ) = 402.1174

34

Finding c3 is a little tedious process. But with some calculations we can show that

@V
@t

+

@V
@x

f (t, x)   || x||2

with this we get c3 = 1. Therefor the bound on g(t, x) can be given by

g(t, x)   ||x||, <

c3
c4

= 0.0025

(67)

(68)

This bound can be improved with a better choice of Lyapunov function which satisﬁes all the

conditions.

5.3.2 Non-vanishing Perturbations:

For a non-vanishing perturbation, we cannot guarantee that the origin will be a stable equilibrium

point. We cannot also say that the states will converge to 0 as t ! 1.
ﬁnd, if there exist, a small bound within which the states will be ultimately bounded when the

In this case, we can

perturbations themselves are bounded by certain bound. In [32], the author has shown this for an

exponentially stable nominal system. As our system is also exponentially stable in absence of any

disturbance, this formulation applies to our system as well.

If we know that the states x belong to a domain C = x 2 Rn   ||x|| < r  we can ﬁnd an upper

bound on g(t, x) such that

||g(t, x)||   <

✓r,

8t   0

(69)

c3

c4r c1

c2

where ✓ is some positive constant less than 1. For such system, for all initial condition such that

||x(t0)|| <q c1

c2

r, the solution satisﬁes the following condition

||x(t)|| r c2

c1

exp[  (t   t0)]||x(t0)||,

8t0  t  t0 + T

and the ultimate bound on states is given by

||x(t)|| 

c4

c3r c2

c1

 
✓

,

8t   t0 + T

35

(70)

(71)

for some ﬁnite T and   = (1 ✓)c3

2c2

. This is proved in [32].

For our suspension system, by choosing the same Lyapunov function as before, we can get the

bound as

||g(t, x)||   < 3.3276 ⇥ 10 5r✓

and the ultimate bound on the states is given by

||x(t)||  3.0051 ⇥ 104  

✓

(72)

(73)

Again, with the choice of a better Lyapunov function, we can estimate more tightened bounds.

With these bounds we can ﬁnd the bounds on error. In state feedback controller,

therefore

e(t) = x   xm

||e(t)|| = ||x||   ||xm||

and in case of output feedback controller we have from [30]

e(t) = y   ym
1

= ⇢⇤

Pm(s)

˜✓!

therefore

||e(t)|| = |⇢⇤|

1
Pm||˜✓|| ||!||

(74)

(75)

(76)

(77)

Now when we known the bound on x for the system, we can ﬁnd all these bounds in terms of

it.

For ﬁnding the relation between the frequency of the time varying parameters and the error, we

need to consider the method given in [32]. According to it, if u(t) is the time varying parameter,

we can prove that if it is continuously di↵erentiable and if it is slowly varying (i.e || ˙u|| is su ciently
small), then the system can have similar properties as that of time invariant system under certain

conditions. Although in our system, as all the parameters are considered to be time varying, this

method proved to be very tedious. More research has to be done in this area to ﬁnd the exact

36

relation between the time varying frequency and the error when multiple parameters are changing

with respect to time.

37

6

Implementation of Pole Placement controller

Till now, we have seen how our system will track the signals (either states or output) form the

reference system. With the controllers described till now, as such we don’t have straightforward

control over the tracking.

It’s true that we have the freedom of choosing our reference model

according to our needs, but what if we need to track a particular signal. One way to tackle this is

we ﬁrst make sure that our reference model tracks the desired signal and then use the controllers

from previous sections to make our plant track the reference model.

This can be done in multiple ways. For example we can use LQR tracking for reference model

tracking. Although with this controller, we can only track constant references. Pole placement

controller can be another option. In case of ‘Polynomial Approach of Pole Placement’ controller,

we can track non-linear references as well. This was the main motivation behind choosing this

particular controller here. Using this controller, along with the output tracking controller, we

can track any desired signal in presence of disturbance and time varying system. (For detailed

explanation of pole placement controller, see sec.8.)

Now, to compare the results of this combined controller with LQR tracking controller, we

consider a case of constant reference tracking. The next sections will describe how to design the

controller for our system.

38

6.1 Pole placement controller suspension system tracking a constant reference

For a constant reference r(s), we deﬁne Qm(s) = s. The reference model transfer function is given

by Eq. 44 which gives

P (s) = s2 + 69s + 69, Z(s) = 1

(78)

Therefore deg(A⇤(s)) = 4, deg(C(s)) = 1 and deg(D(s)) = 2. Assuming A⇤(s) = s4 +70s3 +140s2 +

70s + 1 we can ﬁnd out the polynomials C(s) and D(s) by solving the Diophantine as shown in

Sec. 8 which results into the following,

C(s) = s + 1, D(s) = 2s2 + s + 1

With this if we take the control as shown in Eq. 98, we get

u =  

2s2 + s + 1

s2 + s

e

(79)

(80)

which is not stable because zeros of the denominator are on the imaginary axis. For this reason we

modify the control so as to make it stable.

For this, assume a monic Hurwitz polynomial ⇤c(s) of degree same as that of deg(D(s)). From

Eq. 80 we have,

Qm(s)C(s)u =  D(s)e

0 =  Qm(s)C(s)u   D(s)e

⇤c(s)u =⇤ c(s)u   Qm(s)C(s)u   D(s)e
⇤c(s)u = [⇤ c(s)   Qm(s)C(s)]u   D(s)e

Which gives us the control as shown in Eq. 82.

u =h ⇤c(s) + Qm(s)C(s)

⇤c(s)

iu  

D(s)
⇤c(s)

e

(81)

(82)

Now since here the denominator is Hurwitz and both the transfer functions are proper, our

control is stable.

39

With this control, we can track the output of the model system given in Eq. 45 and as our

original plant tracks the output of this model, it in turn tracks the constant reference r(t).

40

6.2 Time Invariant System

6.2.1 Results in Absence of Disturbance

Figure 20 The ﬁgure illustrates the performance of output tracking combined with pole placement
controller. The plot on the left is the tracking error and the plot on right is the simulated plant and
reference output tracking with the reference model’s output is made to track a constant reference
0.05 using pole placement.

41

   6.2.2 Results in Presence of Disturbance

Figure 21 The results are for the case where r(t) = 0.05 and disturbance as 0.1sin(25t). The
ﬁgure on the right illustrates the tracking of the plant output to the reference output when system
is controlled using the output tracking feedback control with pole placement. The ﬁgure on left
illustrates the convergence of the output error.

Figure 22 The result is for the case where r(t) = 0.05 and disturbance as 0.1sin(25t). The ﬁgure
illustrates the convergence of the estimation error.

As it can be seen in the ﬁgures above, the road disturbance was estimated perfectly when time

invariant system was considered.

42

      6.3 Time Variant System

6.3.1 Results in Presence of Disturbance

Figure 23 The results are for the case where A0 = 0.01,! = 5, r(t) = 0.05 and disturbance as
0.1sin(25t). The ﬁgure on the right illustrates the tracking of the plant output to the reference
output when slowly varying system is controlled using the output tracking feedback control with
pole placement. The ﬁgure on left illustrates the convergence of the output error.

Figure 24 The results are for the case where A0 = 0.01,! = 5, r(t) = 0.05 and disturbance as
0.1sin25t. The ﬁgure on the right illustrates the tracking of the true and estimated road proﬁles
when slowly varying system is controlled using the output tracking feedback control with pole
placement. The ﬁgure on left illustrates the convergence of the estimation error.

43

      7 Tracking Comparison With LQR Tracking Controller

7.1 LQR Algorithm

The LQR tracking algorithm can be designed as shown in Sec. 8. With

D

B

375

(83)

(84)

0 C

0 A

˜A =264
375 ,
Q =264

˜B =264
375

0

Q1 0

0

where Q1 = 1000 and R = 1, K can be found in matlab using the ‘lqr’ command with N = 0.

With this, for the nominal model of our system we get,

K = [31.6228, 0.2665, 0.0867, 0.0021, 0.0001]

(85)

Using this we can deﬁne the control as shown in Sec. 8 to get the following results.

Figure 25 The comparison between output tracking and LQR tracking algorithms for the case where
A0 = 0.0001,! = 5, r(t) = 0.01. The ﬁgure on the on the left is the comparison for the constant
disturbance of 0.01 while ﬁgure on the right is for time varying disturbance of 0.01sin10t.

44

   Figure 26 The results are for the case where A0 = 0.0001,! = 5, r(t) = 0.01. The ﬁgure on
the on the left is for the constant disturbance of 0.01 while ﬁgure on the right is for time varying
disturbance of 0.01sin10t.

The comparison between Output Tracking algorithm and LQR Tracking algorithm for the given

suspension system is shown in the ﬁgures above. Results were compared for time varying and time

invariant system tracking a constant reference (as LQR only works for constant reference case) in

presence and absence of disturbance. In case of time varying system, when the norm of disturbance

is high, output tracking performed better than LQR tracking in every aspect. Otherwise when

disturbance is of order 0.01, as shown in ﬁg.25, for constant case LQR performed better and for

sinusoid case output tracking performed better. The road disturbance estimation results for the

later algorithm can be seen in Fig. 26. In general, the later algorithm will work better as the road

disturbances are highly nonlinear.

45

   8 Discussion and Conclusion

An adaptive state feedback algorithm was proposed in this report to identify a linear time varying

system and estimate the disturbance. An application to this algorithm is ‘Active Suspension Sys-

tem’ which can estimate the road disturbance using an output tracking controller described here.

As the suspension system developed by ‘Quanser Technologies’ has two outputs - suspension de-

ﬂection and body acceleration - we can use the output tracking algorithm to ensure that the body

acceleration tracks a 0 reference while making sure that the suspension deﬂection is also within the

required bounds.

In Sec. 3, a state tracking algorithm is designed. This algorithm will help in restricting the

states within a certain bounds which ensures the safety of the suspension system. For example if

the suspension deﬂection (x1) or velocity of the sprung (x3) or unsprung mass (x4) is too high it

can wear out the suspension system and also damage other vehicle components. From Fig. 4 to

6 it is clear that the states of the unknown suspension system are tracking the reference model

states within 150 time steps. Although this algorithm could not be used in case of disturbance as

the road disturbance in suspension system case does not satisfy the required matching condition.

In Sec. 5, from Fig. 14 and 15 it can be seen that the state tracking algorithm works even in case

of time varying parameters and as expected, the tracking performance is better when the norm of

the perturbations is lower. In Fig. 14, it can be seen that even though the perturbation magnitude

is 10% of the nominal parameter value, the suspension deﬂection tracking performance was really

good.

In Sec. 4, an algorithm is developed for output tracking with and without disturbance. This

algorithm can be used in restricting the output within the desired bound for example restricting

the suspension deﬂection output (y1), or regulating the body acceleration (y2) to a zero reference in

case of our suspension system. The Fig. 8 to 13 show the performance of this algorithm. Although

the error converged to zero faster in case of output tracking state feedback controller as compared

to state tracking controller, there is no such evidence to prove that in general the former converges

better. This might have happened since the initial conditions were closer to the global minima in

case of output tracking controller. From Fig. 8 and Fig. 9 it can be said that the algorithm performed

better when a nominal model was known since the least square solution will not get stuck in some

46

local minima. The disturbance rejection algorithm works even for completely unknown plant and

the plant output tracks the reference output within 500 time steps (see Fig. 13). Although the road

disturbance can only be estimated when the nominal model is known as shown in Fig. 10. This is

because of the fact that this estimation is based on the true value of system parameters. If the PE

condition can be satisﬁed for this case, one can estimate the road disturbance even for the unknown

plant. In Sec. 5, from Fig. 16 to 19 it can be seen that the tracking performance was good when

the time varying parameters were changing with a lower frequency and when the magnitude of the

perturbation was lower, also the tracking error was always bounded by a small value. In case of
5% magnitude change with ! = 5 (see Fig. 17), the convergence error was of the order 8 ⇥ 10 3
but there was some chattering in the tracking. For the same frequency, when the magnitude was

decreased to A0 = 0.005 the chattering vanished and the upper bound on error was also reduced

(Fig. 18). For the same A0 even when the frequency was doubled (Fig. 19) there was no chattering

but the upper bound on error went up to the order of 10 2. With A0 = 0.01 and ! = 5, the

tracking seemed to be working better than any other cases shown here as seen in Fig. 16. Due to

the variation in the results according to the involved perturbation, the stability analysis was done

in order to deﬁne an upper bound on the perturbations which would guarantee an ultimate bound

on the convergence error. In Sec. 7, the road disturbance estimation results were given in case of

time varying parameters. The estimation seemed to work really good as seen in Fig. 26.

In Sec. 6 a pole placement controller was used to make the suspension system track the desired

reference signal. Simulation results were shown in presence and absence of time varying road

disturbance for both time invariant and time varying systems. From Fig. 22 we can see that

the combined algorithm described in this section could perfectly track the desired output and it

even estimated the road disturbance perfectly. This is due to the fact that the nominal system

was considered to be fully known in this case.

In the time varying case, obviously due to the

non vanishing perturbations the tracking was not perfect (Fig. 24) although the disturbance was

estimated with a good accuracy Fig. 23.

In the next section, the proposed algorithm was compared to the well known LQR tracking

algorithm. It was seen from Fig. 25 that the proposed algorithm worked better in case of time

varying disturbances. This is because of the fact that LQR works only for constant disturbances

while the proposed controller works even in case of highly nonlinear disturbances. Also, due to

47

the use of adaptive laws and since the frequency of disturbance was taken into consideration while

designing this algorithm, its performance is good even in case of time varying systems. The only

problem in estimating the road disturbance would be when the time varying parameter frequency

is same as that of the road frequency. In such case, it was seen that the disturbance estimation

did not work. But it can be assumed that the road disturbance is going to vary really fast as

compared to the time varying parameters. This is a valid assumption and with this the disturbance

estimation can be performed with a good accuracy.

One issue here is that the algorithm will estimate road disturbance as long as it is assumed to

be a summation of sinusoids with a known frequency range. There is an algorithm deﬁned in [30]

which can estimate non sinusoidal disturbances but for this algorithm to work, the road disturbance

input should have had satisﬁed the matching condition B = Bd as shown in Sec. 3.2 for the state

tracking algorithm with disturbance case and in Sec. 4.2.3 in case of output tracking algorithm. As

this condition is not satisﬁed, we can only deal with sinusoidal road disturbance cases.

Another issue with the proposed algorithm is that I assumed all the sates of the system are

accessible. Although, even if they are not, one can just design an observer to estimate the unknown

states and then use these estimates in our state feedback algorithm. The best option would be to

use a high gain observers as they work in case of unknown non linearity as well. In this case, when

we make ✏ su ciently small (i.e. gain su ciently large) we can recover three important properties

of the state feedback control [31],

1. Asymptotic Stability

2. State Trajectories

3. Region of Attraction

and when all these properties are recovered, we can guarantee that our algorithm will work with

the use of estimated states (this requires the system to be linear and when a quarter car model

is considered, the system is in fact linear). Even though these observers give good results, there

are some issues related to them. Firstly, due to peaking phenomenon, we can’t really make ✏

inﬁnitesimally small after a certain bound. Also, in presence of noise, if ✏ is taken too small the

e↵ect of noise increases. There is actually a trade-o↵ given in [31] which deﬁnes an optimal ✏ which

can be used to solve this issue. For this we need the norm of the noise to be less than a certain

value with which we can make sure that the ✏ can be made small enough to give accurate estimates.

48

Future work would be to consider the full car model and deal with the nonlinearity, roll and

pitch motions, estimation of states in nonlinear case which recovers true trajectories. The estimated

bound on the time varying parameters was based on a particular choice of Lyapunov function for

the system. With a better choice of Lyapunov function, these bounds can be tightened. Also, the

road disturbance estimation algorithm designed here would only work when the nominal model of

the system is known. But if the PE condition for the output tracking controller is derived, one

can use this condition to make sure that the road disturbance could be estimated even in case of

unknown system.

In short, the Output Tracking State Feedback Controller with Disturbance rejection can be used

combined with a Pole Placement Controller to give highly accurate tracking for unknown systems

and if the nominal model is known a good estimate of the road disturbance can also be achieved

for a linear time varying system.

49

APPENDIX

50

APPENDIX

Parameter Estimation

For a general time-invariant system given by

G(s) =

y(s)
u(s)

=

Z(s)
P (s)

=

zmsm + zm 1sm 1 + ... + z1s + z0
sn + pn 1sn 1 + ... + p1s + p0

(86)

Taking ⇤(s) = sn +  n 1sn 1 + ... +  1s +  0 as Hurwitz and by rearranging terms we get

P (s)
⇤(s)

y(s) =

) y(s) =

=

Z(s)
⇤(s)
Z(s)
⇤(s)
Z(s)
⇤(s)

u(s)

u(s) + y(s)  

P (s)
⇤(s)

y(s)

u(s) + ⇤(s)   P (s)

⇤(s)

 y(s)

in hybrid notation we can write the linear parametric model as

y(t) = (✓⇤)T  (t),

where ✓⇤ =

26666666666666666666666666664

z0

z1

:

zm 1
zm

 0   p0
 1   p1

:

 n 2   pn 2
 n 1   pn 1

37777777777777777777777777775

,  (t) =

26666666666666666666666666664

1
⇤(s) u(t)
s
⇤(s) u(t)

:

sm 1
⇤(s) u(t)
sm
⇤(s) u(t)
1
⇤(s) y(t)
s
⇤(s) y(t)

:

sn 2
⇤(s) y(t)
sn 1
⇤(s) y(t)

37777777777777777777777777775

(87)

For such model, the adaptive law is given by Ref. [30] as

˙✓(t) =   ✏(t) (t),✏

(t) = ✓T (t) (t)   y(t)

(88)

The convergence of ✓ is guaranteed when the regressor vector   is persistently excited. For the

51

given linear system, when P (s) is Hurwitz and Z(s), P (s) are coprime, input signal of richness

order n + m + 1 gives persistently excited regressor vector. Although when this condition is not

satisﬁed we can use robust adaptive laws like  -Modiﬁcation, Switching  -Modiﬁcation, Dead-Zone

Modiﬁcation or Parameter Projection.

Parameter Projection:

Given a range for ✓⇤ we can use this algorithm to make sure the parameter does not leave the given

range which in turn improves the performance of the adaptive law.

i  ✓⇤  ✓b

For ✓a
adaptive law we can restrict the parameters in the set ⌦  = {✓ 2 Rp | ✓a

i , (1  i  p) where p is the number of parameters with the following change in the
i + },   > 0

i     ✓⇤  ✓b

where

˙✓i =   iPri✏ i,

1  i  p

1 + ✓b
i ✓i
 
1 + ✓i ✓a
 

i

if ✓i >✓ b

i and ✏ i < 0

if ✓i <✓ a

i and ✏ i > 0

1

otherwise

Pri =

8>>>>>><>>>>>>:

(89)

(90)

52

Polynomial Approach for Pole Placement Control

This method comes under one of the indirect adaptive control algorithms, where plant parameters

are ﬁrst estimated which are used to ‘indirectly’ estimate the control parameters. As I have only

used this method for a known plant, I have only described a part of the adaptive control which

deals with known plants. For the complete algorithm refer [30].

Consider a linear time-invariant plant given by,

G(s) =

Z(s)
P (s)

(91)

such that P (s) = sn + pn 1sn 1 + ... + p1s + p0 and Z(s) = zmsm + zm 1sm 1 + ... + z1s + z0, and
m < n. The goal is to design feedback control such that limt!1(y(t)   ym(t)) = 0. For this, we
take a monic polynomial Qm(s) that has zeros on the imaginary axis such that the reference signal

satisﬁes Qm(s)ym(s) = 0. For example, if the reference signal is ym(s) = 0 or a constant, we take

Qm(s) to be 1 or s respectively.

For the regulation case that is when ym(s) = 0 & Qm(s) = 1, let the feedback control be

u =  

D(s)
C(s)

y

where C(s) is monic. With this control we have

y =

Z(s)
P (s)

u =  

D(s)Z(s)
C(s)P (s)

y

(92)

(93)

that gives us [C(s)P (s) + D(s)Z(s)]y = 0. So when the closed-loop characteristic polynomial

C(s)P (s) + D(s)Z(s) has roots with poles on the left half of the plane our output will track zero

reference.

53

Now for the case where ym(s) 6= 0 that is Qm(s) 6= 1 we do the following,

Qm(s)ym = 0

) Qm(s)P (s)ym = 0

also sinceP (s)y = Z(s)u

(94)

Qm(s)P (s)y = Z(s)Qm(s)u

) Qm(s)P (s)(y   ym) = Z(s)Qm(s)u

with e = y   ym and ¯u = Qm(s)u we can convert this into regulation problem given by

Qm(s)P (s)e = Z(s)¯u

(95)

With this, we can make limt!1 e(t) = 0 by taking the control as shown in Eq. 92.
For implementing this algorithm, we use the polynomial approach as described in [30]. We ﬁrst

choose a monic polynomial A⇤(s) such that

deg(A⇤) = deg(P ) + deg(Qm) + deg(C)

such that its roots are in left half plane. Then we solve the Diophantine equation

C(s)Qm(S)P (S) + D(s)Z(s) = A⇤(s)

(96)

(97)

for monic polynomial C(s) with degree n 1 and polynomial D(s) such that it follows the condition

 deg(D) = deg(P ) + deg(Qm)   1 . With this the controller is given by

u =  

e

(98)

D(s)

C(s)Qm(s)

54

Linear Quadratic Regulator Tracking

For a linear dynamic system given by

˙x = Ax + Bu

(99)

one of the ways to ﬁnd the optimal control is by minimizing the cost function given by

J = K(x(tf ), tf ) +Z tf

t0

L(x(t), u(t))dt

(100)

where K is the terminal cost, L is the incremental cost, t0, tf the initial and terminal time
respectively. Suppose the initial time is 0 and ﬁnal time is 1. Also, the terminal cost is zero and
the incremental cost is a quadratic function in x and u given by

L = xT Qx + uT Ru + 2xT N u

In this case the cost function would be

J =Z 1

0

[xT Qx + uT Ru + 2xT N u]dt

(101)

(102)

This can be minimized if pair (A, B) is stabilizable, R is positive deﬁnite and Q   N R 1N T is
positive semideﬁnite. For this we use a state feedback control u =  Kx where the gain K is given
by

K = R 1(BT S + N T )

where S is the solution of the Riccati equation given by

AT S + SA   (SB + N )R 1(BT S + N T ) + Q = 0

Now, let say the output of the system is given by

y = Cx(t) + Du(t)

(103)

(104)

(105)

and we want it to track a constant reference r(t) = c. For this we deﬁne e(t) = y(t)   r and

55

z = [e

˙x]T . Therefore we have

where

With the choice of Q, R and N as given below

(106)

(107)

(108)

(109)

(110)

0 C

0 A

0 C

0 A

e

˙x

B

D

D

375 ˙u = ˜Az + ˜Bµ
375 ,

375
264
375 +264
375 ,
˜B =264
375 , Q1   0, R > 0, N = 0

˙u = µ

B

˙e

¨x

˙z =264
375 =264
˜A =264
Q =264
J =Z 1

0

0

Q1 0

0

we can get the cost function as

[eT (t)Q1e(t) + µT (t)Rµ(t)]dt.

With this we use control

µ =  KT z =  K1e   K2 ˙x

to get limt!1 e(t) = 0, which will make the output of the system track the constant reference.

56

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