Toward Industrial Process Control Applications of

Toward Industrial Process Control Applications of the
Backstepping
Abder Rezak Benaskeur, Louis–Nicolas Paquin and André Desbiens
Groupe de Recherche sur les Applications de l’Informatique à l’Industrie
Minérale
Department of Electrical and Computer Engineering
Université Laval, Sainte-Foy (Québec), Canada G1K 7P4
Tel: (418) 656-2131 ext. 3408
Fax: (418) 656-3159
E-mail: [email protected]
ABSTRACT — This paper addresses practical aspects of the application of the backstepping method to the control of industrial processes. The limitations are emphased and
solutions are proposed. The steady-state performances of the backstepping-based controllers
are enhanced via the introduction of an integral action in the controller. In presence of non-zero
mean perturbations, this modification guarantees vanishing residual errors. The manipulated
variable behavior is also improved by the introduction of a new Lyapunov-based filter. In noisy
contexts, the later ensures a small variance of the control, without altering the closed-loop
stability. Finally, it is shown that an appropriate time scale selection is needed to obtain the
desired closed-loop dynamics specification. The proposed scheme is applied to the control of a
flotation column.
1
INTRODUCTION
Recent years have witnessed a rapid emergence of new methodologies for feedback control of
nonlinear systems. The backstepping (Krstić et al. 1995) is one of the most important results,
which provides a powerful design tool. The flexibility of the backstepping method allows it to
solve many design problems under conditions less restrictive than those encountered in other
methods. The backstepping has been successfully applied to a wide variety of nonlinear and
linear systems (Lin & Kanellakopoulos 1997, de Queiroz & Dawson 1996, Baicu et al. 1998,
Benaskeur & Desbiens 1998, Fossen & Grovlen 1998, Robertsson & Johansson 1998, Yanakiev
& Kanellakopoulos 1998). In the process control context, the success lags behind because of
the limitations the backstepping shows in the practice.
The purpose of this article is to present different practical aspects of the backstepping method.
Its limitations are pointed out and modifications are brought to render it more applicable to
actual industrial processes. Hence, the problem of the non-zero mean external perturbations
is considered in section 2. In such a case, the backstepping-based controllers fail to eliminate
the steady-state errors because of the absence of integral actions. For a practical use, such
an integral action must be introduced in the backstepping. This introduction needs a slight
modification of the design procedure and leads, in the case of linear plants, the well-known PID
controller scheme (Benaskeur 2000). To obtain the desired closed-loop dynamics, the time scale
must however be appriopriately chosen. This choice of the time scale is discussed in section 3.
The presence of the measurement noise constitutes another unavoidable factor, which must be
taken into account in almost every industrial control loop. Even though the noise amplitude
is weak, the use of derivative actions in the controllers may result in an unacceptable behavior of the manipulated variable, if an appropriate filtering is not applied. For the ”classical”
methods, the filter dynamics is either simply included in the plant transfer function or quantified as a phase margin reduction. Things go differently with the backstepping. Since this
method follows from the Lyapunov theory, every modification, i.e. addition and/or removal
of components, must be justified in terms of Lyapunov functions. For this kind of problems,
the observer backstepping (Krstić & Kokotović 1994) is still, due to the filtering effect of its
observer, the unique available solution in the context of Lyapunov theory. The second novelty
of this paper, described in section 4, lies then in the proposition of an alternative solution to
the observer backstepping method, in the case of small plants. Indeed, instead of an observer,
the proposed approach uses a new filter, whose design is also based on an augmented version of
the backstepping Lyapunov function. The application of the proposed solution to the control
of a flotation column is given in section 5, while the paper is concluded in section 6.
2
INTEGRAL ACTION DESIGN
In this section the tracking and disturbance rejection problem in the backstepping-based controllers is solved by introducing an integral action in the design procedure (Benaskeur 2000).
In order to illustrate the design method, the following second order plant is considered
Gp (s) =
1
c(s2 + bs + a)
(1)
To render possible the above-mentioned modification, a virtual integrator is first introduced in
the plant transfer function. This integrator will later be slid into the controller function. The
design procedure, which is still identical to the original backstepping method, is performed on
the augmented plant transfer function. In the state space, such a manipulation is equivalent to
using a non-minimal state representation of the plant. Both representations, i.e. state space
and transfer function, lead to the derivative of the actual control. Sliding the added integrator
from the plant to the controller equation yields the final expression of the control law, which
will contain the desired integral action. It is worth noting that for the linear plant (1) this
approach leads to the well known PID controller parallel structure, whose equation is given by
Ti u = c s2 + bs + a yr + Td s + Kc +
(yr − y)
s
(2)
where y is the plant output and yr is the reference trajectory. The controller parameters are
tuned as follows
Td = c(k1 + k2 + k3 − b)
Kc = c(2 + k1 k2 + k1 k3 + k2 k3 − a)
Ti = c(k1 + k3 + k1 k2 k3 )
where k1 , k2 and k3 are positive design parameters, which define the desired closed-loop regulation dynamics. For higher order plants, this approach leads, obviously, to higher order PID
controllers. Along the trajectories of the plant equation (1), the control (2) renders negative
the derivative of the following function of Lyapunov
V3 =
1 2 1 2 1 2
ε + ε + ε
2 1 2 2 2 3
where (for design details, see (Benaskeur 2000))
ε1 = y − yr
ε2 = ẏ − ẏr + k1 ε1
ε3 = ÿ − ÿr + (1 − k12 )ε1 + (k1 + k2 )ε2
With the above-given choices, the derivative of (3) will be reduced to
V̇3 = −k1 ε21 − k2 ε22 − k3 ε23 ≤ 0
(3)
3
SCALE TIME CHOICE
The closed-loop dynamics of the above system is given by
  
 
 
ε̇1
−k1
1
0
ε1
ε1
ε̇2  =  −1 −k2
1  ε2  = A0 ε2 
ε̇3
0
−1 −k3
ε3
ε3
which can be rewritten as D(s)ε1 = 0, where
D(s) = Is − A0 = s3 + (k1 + k2 + k3 )s2 + (2 + k1 k2 + k1 k3 + k2 k3 )s + k1 + k3 + k1 k2 k3
To simplify, one may take k1 = k2 = k3 = k, which leads to
D(s) = s3 + 3ks2 + (2 + 3k 2 )s + 2k + k 3
If one gives as a desired dynamics specification
Dd (s) = (s + a)3 = s3 + 3as2 + 3a2 s + a3
where −a is the closed-loop desired triple pole, the parameter k must be chosen such that
3a = 3k
3a2 = 2 + 3k 2
a3 = k(k 2 + 2)
It is clear that no exact solution exists for the above system. To obtain an approximate solution,
one has to take k = a, and choose the time scale to guarantee that k 2 >> 2. This condition can
easily be justified if one calculates the poles of the closed-loop transfer function. These poles
are given by
√
√
p1 = −k,
p2 = −k − i 2
and
p3 = −k + i 2
where it is clear that to obtain a triple
pole at s = −k, the design parameter k must verify the
√
above given condition , i.e. k >> 2.
4
LYAPUNOV BASED FILTERING
To implement the above state feedback controller as an output feedback, one has to either use
the observer backstepping approach or directly estimate the derivative of the output signal. In
presence of a measurement noise, this direct estimation leads to an unacceptable behavior of the
manipulated variable, unless an appropriate filtering is applied on the measured variable. In the
Lyapunov theory context, such a filtering is not so straightforward. The stability proofs must be
established in terms of positivity and no approximation is tolerated, unless it is justified with a
Lyapunov function. Hence, for a practical implementation, the derivative action is substituted,
in the above PID controller, with a new Lyapunov-based realizable approximation. The later,
besides preserving the overall stability of the closed-loop system, achieves a smoother control.
Using this approximation, the plant output will be the only needed signal, which leads to a
totally output feedback control structure. Instead of directly applying the filter to ẏ, the later
ˆ To
is applied to ÿ. A simple integration will then allow to obtain the desired estimate, i. e. ẏ.
simplify the subsequent equations, the following notation will be used
x = ÿ
The introduction of the filter on x consists then in replacing it with the estimate
x̂ = x + x̃
Due to the non-zero estimation error x̃, the stability cannot be guaranteed. The expression of
the modified control v ∗ , which uses x̂ instead of x, is given by
cv ∗ = c s3 + bs2 + as yr + Kc s + Ti (yr − y) − Td x̂ − ÿr
= cv − Td x̃
(4)
where v is the original control, which uses the non-filtered version of x. With the new control
(4), the function of Lyapunov (3) has as a derivative
V̇3 = −k1 ε21 − k2 ε22 − k3 ε23 −
Td
ε3 x̃
c
Since this derivative is not negative, the stability of the loop can not be established. To remove
the uncertain term, one has to define the new Lyapunov function
V3∗ =
1 2 1 2 1 2
1 2
ε + ε + ε +
x̃
2 1 2 2 2 3 2γ
Its derivative along the plant trajectories is given by
Td
1
ε3 x̃ + x̃x̃˙
c
γ
1
γTd
= −k1 ε21 − k2 ε22 − k3 ε3 + x̃ x̃˙ −
ε3
γ
c
V̇3∗ = −k1 ε21 − k2 ε22 − k3 ε23 −
With this notation, an interesting choice for the filter becomes more obvious. It suffices now to
take
γTd
x̃˙ =
ε3
c
to guarantee the negativity of the derivative. Since x is not constant, the dynamics of x̃ and x̂
are quite different. Indeed, the later is given by
γTd
x̂˙ = ẋ + x̃˙ = v ∗ − aẏ − bx +
ε3
c
where γ is a positive design parameter. The error ε3 is given by
ε3 = x − ÿr + 1 − k12 ε1 + k1 + k2 ε2
Substituting it in the expression of x̂˙ gives
γTd
γTd
x̂˙ = v ∗ − aẏ +
−b x+
c
c
1−
k12
ε1 + k1 + k2 ε2 − ÿr
and
Td
Td
x̂ + yr(3) +
+ b ÿr +
v = aẏ −
c
c
∗
Kc
Ti
+a s+
(yr − y)
c
c
Replacing v ∗ in the expression of x̂˙ allows to rewrite the later as
γTd
Td
Td
− b x + s3 +
+ b s2 yr
x̂˙ = − x̂ +
c
c
c
Ti
γTd
Kc
+a s+
ε1 +
1 − k12 ε1 + k1 + k2 ε2 − ÿr
−
c
c
c
Since x̂ is the estimate of x = ÿ, to have a unit gain between the two variables, the parameter
γ must verify
Td
γTd
=
−b
c
c
which can be rewritten as
γ=
Td + cb
k1 + k2 + k3
=
Td
k1 + k2 + k3 − b
Furthermore, to guarantee the positivity of γ, the parameters k1 , k2 et k3 must be chosen such
that k1 + k2 + k3 > b. These two conditions ensure that the designed filter will be stable and
have a unit gain. This filter is given by
˙x̂ = Td (x − x̂) + s3 yr − ( Kc + a)s + Ti ε1 + γTd (1 − k 2 )ε1 + (k1 + k2 )ε2
1
c
c
c
c
which, once simplified, can be rewritten as (with ÿ = x)
ÿˆ =
s3
k 2 + k22 + k1 k2 − 2
k3 + k12 k2 + k1 k22
Td s2
y+
yr + 1
ε̇1 +
ε1
cs + Td
cs + Td
cs + Td
cs + Td
(5)
The design of the above filter and the estimation of ÿ may seem superfluous, since this variable
does not appear in the expression (2) of the final controller u. Indeed, ÿˆ will not be used, but
instead, it will yield an estimate of ẏ, which does appear in the expression of u. The integration
of the equation (5) leads to the desired estimate
ẏˆ =
5
Td s
s2
2 − k12 − k22 − k1 k2
k3 + k12 k2 + k1 k22
y+
yr +
(yr − y) −
(yr − y)
cs + Td
cs + Td
cs + Td
s(cs + Td )
APPLICATION
From the above-given developments, it is clear that the backstepping leads to continuous-time
controllers. For practical considerations and on computer implementation, discrete-time versions are preferable. Since the discrete time counterpart of the backstepping method reduces
to a very simple ”look ahead” control (Zhao & Kanellakopoulos 1997), an alternative solution
to obtain discrete-time controllers with the backstepping consists in the discretization (Ioannou
& Kokotović 1996) of the final resulting continuous-time control law. Note that almost all the
continuous-time/discrete-time conversion methods can be used to perform this discretization,
with no noticeable difference in the loop performances. Different methods have been tested and
have yielded to very similar results.
The proposed scheme was successfully applied to a pilot flotation column, which is described
in details in (Desbiens et al. 1998). A flotation column is a device used in the mineral industry
to separate the valuable minerals from the carrier gangue. The column is fed with the ground
ore and water, thus forming a pulp. Addition of air and adequate chemical reagents makes
the valuable mineral particules rise at the top of the column, dividing it into two zones with
very different percentages of air contents. The level of the interface between the zones is the
variable to be controlled by manipulating the gangue pulp flow rate. On figure (b), the effect
of the integral action is clearly visible. The level follows closely the reference trajectory and
the disturbance, i.e. change in the feed flow rate shown in figure (a), occuring at t = 12 min is
rapidly eliminated. Figure (a) also shows the variations of the manipulated variable, which are
acceptable even if the level measurement is noisy.
6
CONCLUSION
The backstepping method is considered and its limitations in the process control context are
highlighted. The major problem of the disturbance rejection is first resolved by the introduction of an integral action in the design procedure. The steady-state performances of the
backstepping-based controllers are thus enhanced. Also is considered the unavoidable problem
caused by the presence, in the control loop, of the measurement noise. The behavior of the
obtained controllers is improved via the introduction of new Lyapunov-based filter. The later,
while preserving the overall stability of the loop, yields a smoother and small control effort. To
illustrate the efficiency of the proposed scheme, the later is applied to the control of a flotation
column. The results are very conclusive. The obtained performances can even be improved
by exploiting the adaptive version of the backstepping, which can compensate for the plant
nonlinearities. These aspects are currently under investigation.
1
50
Feed flow rate
Gangue pulp flow rate
0.9
48
46
44
0.8
42
0.7
40
38
0.6
36
34
0.5
32
0.4
0
2
4
6
8
10
Temp[min]
12
14
(a) Manipulated variable and disturbance
16
30
0
Reference trajectory
Interface level
2
4
6
8
10
Temp[min]
12
14
16
(b) Controlled variable
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