-
STATE DEPENDENT PARAMETER, PROPORTIONAL-INTEGRAL-PLUS
(SDP-PIP)
CONTROL OF A NONLINEAR ROBOT DIGGER ARM E.M. Shaban*, C.J.
Taylor* and A. Chotai+
*Engineering Department, +Environmental Science Department,
Lancaster University, UK. Email: [email protected] Fax: +44
(0) 1524381707
Keywords: nonlinear excavator; state dependent parameters;
proportional-integral-plus; model-based control.
Abstract
This paper develops a State Dependent Parameter,
Proportional-Integral-Plus (SDP-PIP) control system for a 1/5th
scale representation of the Lancaster University Computerised
Intelligent Excavator (LUCIE). To the authors knowledge, this
represents the first time that SDP-PIP methods have been applied to
a practical nonlinear system. Here, the parameters of the SDP model
are functionally dependent on other variables in the system. The
implementation results suggest that the new algorithm offers
considerable improvement over existing linear PIP designs, so
providing smoother, more accurate movement of the excavator
bucket.
1. Introduction
Previous papers have introduced the linear
Proportional-Integral-Plus (PIP) controller [5, 10], in which
Non-Minimal State Space (NMSS) models are formulated so that full
state variable feedback control can be implemented directly from
the measured input and output signals of the controlled process,
without resort to the design and implementation of a deterministic
state reconstructor or a stochastic Kalman filter. Over the last
few years, such PIP control systems have been successfully employed
in a range of practical applications, e.g. [2, 6]. To date,
however, any inherent nonlinearities in the system have been
accounted for in a rather ad hoc manner at the design stage,
sometimes leading to reduced control performance when applied to
particularly difficult, highly nonlinear systems.
One novel research area currently being investigated in order to
improve PIP control in such cases, is based on the State Dependent
Parameter (SDP) system identification methodology. Here, the
nonlinear system is modelled using a quasi-linear model structure
in which the parameters vary as functions of the state variables
[11-12]. For the closely related time variable parameter (TVP)
model, the parametric rate of change is slow when compared with
that of the state variables [11]. In contrast, SDP models are
descriptions of
true nonlinear systems, where the parameters are functionally
dependent on other variables in the system. In this manner, SDP
models can provide a description of a widely applicable class of
nonlinear systems that even includes chaotic processes and systems
that have previously been modelled using a bilinear approach
[1].
The linear-like, ‘affine’ structure of the SDP model means that,
at each sampling instant, it can be considered as a ‘frozen’ linear
system. This formulation is then used to design an SDP-PIP control
law using linear system design strategies such as pole assignment
or suboptimal Linear Quadratic (LQ) design. This yields SDP-PIP
control systems in which the state feedback gains are themselves
state dependent. Earlier simulation studies have demonstrated the
feasibility of the approach [3-4]. However, the present paper
applies these methods to a practical nonlinear system for the first
time, namely a hydraulically controlled robot arm.
The civil and construction industries currently deploy a large
number of manually controlled plants for a wide variety of tasks
within the construction process. The excavation of foundations,
general earthworks and earth removal tasks are activities which
involve the machine operator in a series of repetitive operations.
The automation of the earth removal process is likely to provide a
number of benefits such as a reduced dependence on operator skills
and a lower operator work load, both of which might be expected to
contribute to improvements in quality. Automation also allows for
the removal of the need for a human operator when working in
hazardous environments.
In this regard, the present paper considers control of a digger
arm, a 1/5th scale representation of the more widely known
Lancaster University Computerised Intelligent Excavator (LUCIE),
which is being developed to dig foundation trenches on a building
site [2]. The 1/5th excavator provides a valuable insight into the
full scale system and is used here to develop the new SDP-PIP
methodology, with the aim of providing smoother, more accurate
movement of the bucket. In particular, the paper compares the
nonlinear SDP-PIP approach with a conventional optimal linear PIP
design, through a series of robustness and disturbance tests.
Control 2004, University of Bath, UK, September 2004 ID-059
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2. Linear PIP Control Design
In order to develop a linear PIP control algorithm, a linearised
representation of the system is required. Here, the small
perturbation behaviour is approximated by a simple linearised
transfer function model. For the purposes of the present paper, the
discussion is limited to the following discrete-time system,
kknn
mm
k uzAzBu
zazazbzby
)()(
1 11
11
11
−
−
−−
−−
=+++
++=
L
L (1)
where ky is the angle of the boom joint (degrees) and ku is the
applied voltage, expressed as a percentage in the range -1000 to
+1000. Positive and negative inputs open or close the boom joint
respectively. Here, )( 1−zA and )( 1−zB are appropriately defined
polynomials in the backward shift operator ikk
i yyz −− = . For convenience, any pure time delay
of 1>δ samples can be accounted for by setting the 1−δ
leading parameters of the )( 1−zB polynomial to zero, i.e.
011 =−δbb K .
The present research utilises the Simplified Refined
Instrumental Variable (SRIV) algorithm to estimate the model
parameters [8-9]. However, an appropriate model structure first
needs to be identified, i.e. the most appropriate values for the
triad [ δ,, mn ]. The two main statistical measures employed to
help determine these values are the coefficient of determination
2TR , based on the response error; and YIC (Young’s Information
Criterion), which provides a combined measure of model fit and
parametric efficiency, with large negative values indicating a
model which explains the output data well, without
over-parameterisation [8-9].
It is easy to show that the model (1) can be represented by the
following linear Non-Minimal State Space (NMSS) equations,
kk
kdkkky
yuhx
dgFxx=
++= −− ,11 (2)
where, F, g, d and h are defined by [5, 10]. The n+m dimensional
non-minimal state vector kx , consists of the present and past
sampled values of the input and output variables, i.e.,
T
kmkk
nkkkk zuu
yyy
=
+−−
+−−
11
11
L
Lx (3)
Here, }{ ,1 kkdkk yyzz −+= − is the integral-of-error between
the reference or command input kdy , and the sampled output ky .
Inherent type 1 servomechanism performance is introduced by means
of the integral-of-error state kz . If the closed-loop system is
stable, then this ensures that steady-state tracking of the command
level is inherent in the basic design. The control law associated
with the NMSS model (2) takes the usual State Variable Feedback
(SVF) form,
kku xk−= (4)
B ( z −1 )A ( z −1 )
K I1 − z −1
1G (z − 1 )
F ( z −1 )
-+
-+
Figure 1. PIP control block diagram.
where ][ 11110 Imn Kggfff −= −− LLk is the SVF control gain
vector. In more conventional block-diagram terms, the SVF
controller (4) can be implemented as shown in Figure 1, where it is
clear that it can be considered as one particular extension of the
ubiquitous PI controller, where the PI action is, in general,
enhanced by the higher order forward path and feedback compensators
)(1 1−zG and )( 1−zF ,
)1(1
11
1
)1(1
110
1
1)()(
−−−
−−
−−−
−−
+++=+++=
mm
nn
zgzgzGzfzffzF
LL (5)
However, because it exploits fully the power of SVF within the
NMSS setting, PIP control is inherently much more flexible and
sophisticated, allowing for well-known SVF strategies such as
closed loop pole assignment, with decoupling control in the
multivariable case; or optimisation in terms of a Linear-Quadratic
(LQ) cost function of the form,
{ }∑ += ∞=0
2T21
iiii urJ xQx (6)
where ][ 111 mnmnnn qqqqqdiag +−++= KKQ is a diagonal state
weighting matrix and r is an additional scalar weight on the input.
The resulting SVF gains are obtained recursively from the well
known Algebraic Riccati Equation (ARE) – see equation (10)
below.
3. Nonlinear SDP-PIP Control Design
The research is based on the identification of stochastic models
with state dependent parameters (SDP). The idea of using such
models to represent nonlinear dynamic systems goes back to Young
[7], who showed how the forced logistic growth equation could be
represented, identified and estimated in SDP form. However, the
practical development of these ideas is of a more recent origin,
see e.g. [11-12].
The state dependent parameters are estimated using recursive
filtering and fixed interval smoothing algorithms, coupled with
Maximum Likelihood (ML) estimation of the hyperparameters. For
brevity, the present paper does not review these approaches: see
[11] and the references therein for details. Note, however, that
all these statistical tools and associated estimation algorithms
have been assembled as the CAPTAIN toolbox within the Matlab®
software environment (www.es.lancs.ac.uk/cres/captain). The 2nd
author can be contacted for further details about this toolbox.
For the purposes of the present research, the SDP model takes a
similar structural form to equation (1), but with the following
state dependent polynomials,
Control 2004, University of Bath, UK, September 2004 ID-059
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( ) ( ) ( )( ) ( ) ( ) mkmkkk
nknkkk
zbzbzB
zazazA−−−
−−−
++=
+++=
χχχ
χχχ
K
K1
11
11
1
,
1, (7)
where ( )1, −zA kk χ and ( )1, −zB kk χ are the time varying
equivalents of )( 1−zA and )( 1−zB . Note, however, that the
variation of the parameters is assumed to arise due to their
dependency on the states. In particular, the notation indicates
that the parameters are nonlinear functions of the vector χ k
where, in general, χ k is defined in terms of any measured
variables. The application of such SDP models within a control
context, leads naturally to controller gains that are also state
dependent. The main advantage of such a system is its applicability
to general linear systems theory, so that conventional SVF design
procedures can be exploited in the design of the SDP-PIP
controller.
For example, an SDP-NMSS representation is based on equation
(2), but with the constant parameters replaced by their state
dependent equivalents, i.e.,
kk
kkkkkk
xyruxx
hdgF
=++= −−−− 1111 (8)
Assuming that this SDP-NMSS system ],,,[ hdgF kk is globally
defined and fully controllable for all k , then there exists a
valid vector of state dependent feedback gains kk , that will form
the following SVF control law,
kk xu kk−= (9) Here, the control vector kk is itself state
dependent and is obtained recursively at each sample using the ARE.
For the sake of simplicity, the ‘frozen parameter’ system [ kF′ ,
kg′ ], defined as a single sample member of the family of [ kF , kg
] is utilised in this analysis [4]. In this case, the matrix P is a
time invariant symmetrical positive-definite matrix solution of the
ARE,
( )[ ]kiTkkiTkkkkk
iTk
i
r FPggPggFM
MPFQP
′′+′′′−′=
′+=+−+
+
)1(1)1(
)1()(
(10)
Here, the initial value n)(i+P is set equal to Q. Once the value
of the positive-definite matrix P is obtained from (10), the value
of state dependent gains are determined from,
[ ] kTkkTkk r FPggPgk 1−+= (11) with ]..........[ ,,1,1,1,1,
kIkmkknkkok Kggfff −= −−k . The state dependent SVF controller
described by (10) and (11) is implemented on-line in a similar
manner to Figure 1, but with time variable feedback terms.
4. Application to the Robot Arm
The SRIV algorithm combined with the YIC and 2TR identification
criteria discussed above, reveal that a first order model with two
samples time delay provides the best estimated model and most
optimum fit to the data across a wide range of operating
conditions, i.e.,
kk uzazby 11
22
1 −−
+= (12)
where ky is the angle of the 1/5th excavator boom joint and
ku is the applied voltage scaled from -1000 to 1000. For
example, in a typical ‘boom opening’ experiment with
500=ku , the SRIV algorithm yields 11 −=a and 015.02 =b ; while
in a ‘boom closing’ experiment with 500−=ku , the SRIV algorithm
gives 0233.02 =b . Here, it
should be pointed out that the SRIV algorithm is utilised with
1a fixed a priori at -1, so that only the numerator parameter
is estimated. This is because the arm always acts like an
integrator, i.e. there is no movement when the input is zero. This
assumption is supported by an initial modelling study, which
usually returns 1a close to unity. Furthermore,
01915.02 =b yields the best fit for the full range of positive
and negative applied voltages. In other words, the linear model
(12) represents the ‘average’ behaviour of the nonlinear system,
and yields the most robust linear PIP control performance found to
date.
In this case, the linear NMSS equations are given by (2) with [
]Tkkkk zuyx 1−= . Finally, trial and error experimentation
at this operating point suggests that setting ]50013000[diag=Q
and 1.0=r yields a suitably fast
PIP-LQ algorithm. At this juncture, it is important to stress
that the linear model (12) is only applicable for small
perturbations close to a given operating point. Since the present
example is effectively an integrator, such ‘small perturbations’
actually refer to an angular velocity, i.e. the rate of change of
the boom angle, and the associated input voltage. In practice,
larger applied voltages yield a disproportionate (nonlinear) change
in the velocity and different responses are obtained when the booms
opens or closes.
In fact, SDP analysis of this system suggests that a more
appropriate nonlinear model form is instead given by,
( )( )
( )( ) kkk
kkk
kkk u
zuazub
uzuAzuB
y1
21
222
12
12
1,,
−−
−−
−−
−−
+== (13)
Here, the polynomials are both functions of the lagged input
variable 2−ku , as illustrated in Figure 2. In particular, the
state dependent parameters are defined by,
01898.0108459.5)(
110238.0)(
26
22
22
621
+×−=
−×=
−−
−
−−
−
kk
kk
uub
uua (14)
Note that Figure 2 is utilised to identify the shape of the
parameter state dependencies. The final stage of the analysis is
usually to numerically optimise the coefficients of the nonlinear
model directly from measured data [11]. The coefficients (14) are
obtained from such an analysis. Figure 3 shows the simulation fit
of the optimised SDP model, where
=2TR 0.892. By contrast, with 11 −=a and 01915.02 =b , the
linear model (12) does not return meaningful results.
Control 2004, University of Bath, UK, September 2004 ID-059
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0 100 200 300 400-1
-0.95
-0.9
-0.85
-0.8
-0.75
samples
Den
omin
ator
[a1]
0 100 200 300 4000.01
0.015
0.02
0.025
samples
Num
erat
or [b
2]
0 2 4 6 8 10
x 105
-1
-0.95
-0.9
-0.85
-0.8
-0.75
uk-22
-1000 -500 0 500 10000.01
0.015
0.02
0.025
uk-2 Figure 2. Left hand side: SDP model parameters plotted
against sample number. Right hand side: estimated )( 21 −kua
(top) and )( 22 −kub (bottom) plotted
against the state (lagged input variable).
0 50 100 150 200 250 300 350 400 450-100
-50
0
50
100
samples
Out
put A
ngle
0 50 100 150 200 250 300 350 400 450-1000
-500
0
500
1000
samples
Inpu
t Dem
and
Figure 3. SDP modelling of the 1/5th digger arm. Top:
experimental data (dots) and SDP model simulation
response (thick). Bottom: input voltage.
0 10 20 30 40 50 60 70 80 90 1000
5
10
15
20
25
samples
PIP
Res
pons
e
0 10 20 30 40 50 60 70 80 90 1000
5
10
15
20
25
Samples
SD
P-P
IP R
espo
nse
Figure 4. Monte Carlo simulation comparing linear
PIP (top) and nonlinear SDP-PIP (bottom).
The associated SDP-PIP gain vector (11) is given by,
+×−+×−
= −−
−−
082.1194.0109.2
53.49101789.12
4
22
5
k
kTk u
uk (15)
An initial simulation study, based on the identified models,
reveals little difference between these linear PIP and nonlinear
SDP-PIP algorithms in the case of disturbance rejection and basic
command tracking. However, Monte Carlo analysis does suggest that
the SDP-PIP design is more robust to model uncertainty, as
illustrated in Figure 4. Here,
2b is chosen randomly from a probability distribution in the
range 75% to 180% of its linear value. As would be expected, the
SDP-PIP design performs better in this case, since the algorithm
automatically updates the control parameters (15) as a function of
the current state.
5. Implementation
Typical implementation results for the boom arm are illustrated
in Figure 5, where it is clear that the SDP-PIP design is more
robust than the linear PIP design to large steps in the command
level and yields a considerably smoother control input signal. The
control gains are shown in Figure 6, which compares the time
varying (state dependent) gains in the SDP-PIP case with the fixed
gain, linear PIP design.
6. Conclusions
The present paper has developed a State Dependent Parameter,
Proportional-Integral-Plus (SDP-PIP) boom angle control system for
a 1/5th scale representation of the Lancaster University
Computerised Intelligent Excavator (LUCIE). To the authors
knowledge, this represents the first practical implementation of a
controller based on SDP nonlinear system identification methods.
Here, the nonlinear behaviour of the system is built into the
SDP-PIP algorithm at the design stage.
Implementation results suggests that the new nonlinear SDP-PIP
algorithm offers considerable improvement over existing linear
designs, so providing smoother, more accurate movement of the
excavator bucket. The complete bucket position control system,
based on solving the system kinematics and also controlling the
dipper joint, is presently being developed by the authors.
The SDP methodology applies to a very wide class of nonlinear
and chaotic systems [11], presaging the likely utility of the
approach for other problems.
Control 2004, University of Bath, UK, September 2004 ID-059
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0 20 40 60 80 100 120 140 160 180 200-20
0
20
40
60
samples
Res
pons
e
0 20 40 60 80 100 120 140 160 180 200-1000
-500
0
500
1000
samples
App
lied
Inpu
t Dem
and
Figure 5. On-line control of the robot digger arm. Top:
Linear
PIP (thin trace), nonlinear SDP-PIP (thick trace) and boom angle
command input (dashed). Bottom: control input, where the linear
controller is again shown as the thin
(oscillatory) trace.
0 20 40 60 80 100 120 140 160 180 20042
44
46
48
50
samples
Pro
porti
onal
gai
n [f o
]
0 20 40 60 80 100 120 140 160 180 2000.7
0.8
0.9
1
1.1
samples
Inpu
t gai
n [g 1
]
Figure 6. Linear PIP (horizontal trace) and SDP-PIP (time
varying traces) control gains utilised in Fig. 5.
Acknowledgements
The authors are grateful for the support of the Engineering and
Physical Sciences Research Council.
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