A New Robust Weight Update For Cerebellar Model Articulation Controller Adaptive Control With Application To Transcritical Organic Rankine Cycles
A New Robust Weight Update For Cerebellar Model Articulation Controller Adaptive Control With Application To Transcritical Organic Rankine Cycles
A New Robust Weight Update For Cerebellar Model Articulation Controller Adaptive Control With Application To Transcritical Organic Rankine Cycles
Abstract
This work proposes modifications to the adaptive update law for a cerebellar model articulation controller (CMAC) and develops a model of a transcriti-
cal organic rankine cycle (ORC) to test it on. Owing to the local nature of its basis functions, the CMAC exhibits more weight drift (overlearning) than
other types of neural networks, and practical applications have been restricted to systems without persistent oscillations of the inputs. The proposed
solution to this problem here involves identifying a set of weights that is the best found so far in the training, and keeps the weights from drifting too far
from these best weights. The method results in uniformly ultimately bounded signals, established through Lyapunov analysis. To show the improved train-
ing algorithm now allows the CMAC to control more general systems, it is applied to the control of a transcritical ORC. Part of the contribution of this
paper also includes developing a model to describe the behaviour of a supercritical fluid in the ORC evaporator. The control method is compared with
proportional–integral control, where the controls have to provide robustness to fluctuations and step changes in heat source temperatures.
Keywords
Cerebellar model articulation controller, direct adaptive control, parameter drift, bursting, organic rankine cycle
bounded signals. In this way, CMAC can now be used for developed by James Albus in 1972 (Albus, 1975b); although
process control applications. it was the first type of neural network capable of nonlinear
In our simulation testing, the new method controls a tran- approximation, it did not become widely appreciated until
scritical organic rankine cycle (TORC). We develop a model shown to be useful for robotics applications (Miller et al.,
of a TORC because operation in the supercritical region in the 1987). Note that it can also be used to define membership-
evaporator can improve efficiency (Quoilin et al., 2011). Yet function domains in a fuzzy approximator (Jou, 1992; Nie
previous literature has only addressed subcritical ORCs; for and Linkens, 1994; Ozawa et al., 1992). Many contemporary
an exhaustive review of proposed strategies the reader is researchers still utilize CMAC for nonlinear adaptive control
referred to Zhang et al. (2018). Zhang et al. (2012) presented a in a wide variety of applications, including voice coil motors
multi-variable control strategy by incorporating a model- (Lin and Li, 2014), mobile robots (Wu et al., 2014), human–
based linear quadratic regulator (LQG) with a proportional– robot systems (Zhang et al., 2015), induction motor drives
integral (PI) controller, using a 4 3 4 multi-input, multi-output (Wang et al., 2015) and hysteresis compensation (Meng et al.,
(MIMO) strategy. Zhang et al. (2013) proposed a generalized 2016), to name a few. Here we briefly outline operation of
predictive control (GPC) strategy that does not require previ- CMAC, and refer the reader to Lewis et al. (1998) and Brown
ous knowledge of a linear state space model. Soon afterwards, and Harris (1994) for detailed descriptions of CMAC opera-
Zhang et al. (2014) designed a constrained model predictive tion and algorithmic implementation.
control (MPC) for a subcritical ORC that uses a 3 3 3 MIMO The CMAC local basis-function domains, or cells, are
strategy. Quoilin et al. (2011) used a simplified model without hypercubes indexed by input vector xin 2 Rn (Albus, 1975b).
multivariable interactions and then designed a decentralized The hybercube cells are arranged in M offset arrays
single-input, single-output (SISO) PI controller; our current (Figure 1). Like RBF networks the output is a weighted sum
work also follows this decentralized strategy, but with differ- of basis functions, and computational efficiency stems from
ent input–output pairings. We argue that our adaptive strat- only having to calculate the Mactivated weights
egy is more realistic and practical than model-based controls,
because ORCs are not only difficult to model but parameters X
M
will change over time. We also posit that our following-waste- ^f (xin ) = G i (xin )wi = G(xin )w ð1Þ
heat strategy is more realistic and practical than the following- i=1
Background
CMAC
The CMAC is a type of nonlinear approximator, capable of Figure 1. Example CMAC cell structure with two inputs and three
uniform approximation of nonlinear functions. It is was arrays.
Samiuddin et al. 3
coming out from the expander passes through the other side
of the recuperator where it is precooled (6–7). The condenser
utilizes a fan for further cooling of the fluid into liquid (7–1).
Note that the recuperator reduces the load on the evaporator
and the condenser by mutual heat transfer within the cycle,
increasing efficiency.
We will utilize dynamic models of the heat exchangers
(evaporator, condenser and recuperator) and static models
for the compressor, expander and valve. The dynamic model
allows control design and/or testing. A static version of the
model, with all time derivatives set to zero, enables identifica-
tion of an appropriate setpoint.
Table 1. Symbols.
Figure 6. Schematic of the evaporator. where Dmax is a positive constant. Note that the other terms
(in addition to D(t)) depend on measured variables and can be
u = ½m, Xpp , Nfan T ð8Þ accounted for by the CMAC adaptive control.
The output error is
where 0 ł m ł 1 is the opening of the valve, 0 ł Xpp ł 1 is the
capacity fraction of the pump and Nfan is the speed of the con- zj = yj yj, sp ð18Þ
densing fan in revolutions per minute. The state-space equa-
where yj, sp is the jth value of the current set point. A control
tion is
that can drive the ORC system towards its desired working
x_ = f(x, Tsf ) + g(x, Tsf )u ð9Þ condition is
where f() and g() contain nonlinear terms. The measured out-
uj = uj, n + Duj ð19Þ
puts are
y = ½Pev , Tout, ev , Tout, c 0 ð10Þ where Duj is the CMAC-based control
where Pev is the pressure in the evaporator, Tout, ev is the outlet Duj = Gj (q)^
wj + Kj zj ð20Þ
temperature of the evaporator and Tout, c is the output of the
condenser. where Kj is a positive constant control gain. Note the CMAC
is capable of approximating all unknown linear and nonlinear
CMAC adaptive control terms that depend on measurable signals in a region D. Thus,
the CMAC inputs must be
The control strategy uses three decentralized SISO loops with
three control inputs (manipulated variables) matched to three qj = ½ Pev Tout, ev Tout, c Tsf m(t T ) Xpp (t T ) Nfan (t T) 0
outputs (controlled variables) as
ð21Þ
m ! Pev ð11Þ
To make the control adaptive, one can apply the e-modifica-
Xpp ! Tout, ev ð12Þ tion weight update law
Nfan ! Tout, c ð13Þ
^_ j = bj (G0j zj njzj j^
w wj ) ð22Þ
which is justified for the particular model in this paper using
a relative gain array (RGA) analysis (described later). Three where (typically) the weights begin at zero.
CMAC adaptive controls will be applied to the three control Theorem 1. For the system described by (16) and (18), under
loops. Because stability will be established for each control condition (17), if the region of CMAC uniform approximation
loop independently, technically this would be considered a D is chosen large enough, then applying CMAC control (19)
decentralized SISO approach. However, each CMAC will be and (20) and CMAC weight updates (22) results in semi-
given full state information, so that in practice the CMACs globally uniformly ultimately bounded (SGUUB) signals (zj
will end up compensating for much of the dynamic cross- ^ j ) in D.
and w
coupling in the MIMO system. The proof of Theorem 1 appearing in Appendix A uses an
The steady-state inputs uj, n and corresponding steady-state adaptive control Lyapunov function
outputs yj, n , for j = 1, 2, 3, define a nominal setpoint.
For control design purposes, we assume g(x, Tsf ) from (9) 1 2 1 T
is constant (in practise it changes very little). In this case, ele- Vj = zj + ~ w
w ~j ð23Þ
2gj 2bj j
ments of g are replaced by constants gi, j and the dynamics of
the jth SISO subsystem becomes where w~ j = wj w
^ j is the weight estimation error (wj is a set
X of unknown ideal constant weights).
y_ j = f1, j (y, Tsf, n ) + f2, j (x, y, Tsf, n ) gj, j uj gi, j ui ð14Þ Using e-modification (22) is not the only method for pro-
i6¼j viding robustness and preventing bursting in adaptive control.
6 Transactions of the Institute of Measurement and Control 00(0)
The method of parameter projection is often used (Egardt, cell has been activated, Ti , is less than a pre-identified
1979), and we introduce it here because our proposed method critical time Tc ;
will use a similar technique. A simple form of projection for C. applies a modified weight update if the weight is
neural networks operates on each weight individually within the imposed bound and Ti øTc , where the mod-
8 ified weight update keeps the weight close to a best
<0 if zj . 0 and w
^ j = wmax weight (bi ) identified so far;
^_ j = 0
w if zj \0 and w^ j = wmax ð24Þ
: b G0 z otherwise
j j j or, mathematically,
where wmax is a known maximum value of the ideal weight, 8
<0 if k ^p k = pmax
which would have to be identified using either model knowl-
^p_ i = bGi z if k ^p k \pmax and Ti \Tc
edge or extensive pretraining. The main idea is that w
^ j = wmax :
b½Gi z + hjzi j(bi ^pi ) otherwise
implies w~ ł 0 and w ^ j = wmax implies w~ ø0. Although a
complete proof is beyond the scope of this paper, note that ð27Þ
applying (19), (20) and (24) in the disturbance-free, ideal-
neural-network case ensures the time derivative of (23) is where h is a positive constant.
The best weight identified so far (bi ) is that weight that has
8
> 2
~ j =gj Kj z2j ł 0 if zj . 0 and w
^ j = wmax resulted in the lowest average error previously in the training,
< zj w
_ 2 2 as measured over the cell’s domain plus three subsequent cell
V = zj w~ j =gj Kj zj ł 0 if zj \0 and w ^ j = wmax ð25Þ
>
: K z2 ł 0 domains on the same CMAC layer. The rationale for (bi )
j j otherwise
based on four cell activations on the same layer is that a sinu-
soidal oscillation of highly related states will occur in a planar
resulting in a negative-semi-definite derivative and a guaran-
ellipse even in high-dimensional space; a small oscillation
tee of stability. The practical difficulty of applying parameter-
should occur through, at most, four cells. This means the
projection is that persistent disturbances usually result in the
average error should be measured through a full period of
weights reaching their imposed bound, making the update
oscillation of the states. Therefore, the best weight found so
bj G0j zj irrelevant to the ultimate performance.
far results in the lowest average error through a full period of
oscillation (and avoids rewarding a weight for reducing the
Novel solution for weight update design to prevent error in a partial oscillation at the cost of increasing overall
error).
bursting The main ideas behind this strategy are as follows.
We find the update (22) leads to a large trade-off between sta-
bility and performance owing to the problem of overlearning A. In the initial training with ^p_ i = bGi z, the performance
in the CMAC, i.e. choosing n large enough to prevent weight CMAC is capable of reducing the error to small val-
drift leads to worse performance than with PI control. The ues even as the robust CMAC is training with a large
other popular robust modification of deadzone requires n in (22).
weight updates to stop in the region jzj j\dj, max =Kj , but B. The performance-CMAC weights are bounded using
because dj, max includes the very large disturbances in the heat pmax to guarantee bounded signals.
source the deadzone would be extraordinarily large. Thus, we C. The performance-CMAC weight update is trying to
propose a novel method for preventing weight update that ensure that the algorithm:
achieves high performance while halting weight drift. (a) will capture the best performance found during
The proposed method uses two CMACs in parallel, a the training by stopping weight drift when the
robust CMAC Gj (q)^ wj and a performance CMAC Gj (q)^pj and performance has reached its peak;
the control output becomes (b) will try to prevent the weights from ever reach-
ing pmax .
Duj = Gj (q)^
wj + Gj (q)^pj + Kj zj ð26Þ
Note that applying condition (A) above to bound the
The subscript j is omitted in the rest of this section without
weights differs from traditional parameter projection in two
loss of generality. The robust CMAC updates with e-modifi-
important ways: (1) the bound pmax does not have to be cho-
cation (22) at all times, with a conservative value of n, i.e. a
sen larger than k p k to guarantee stability, i.e. knowledge of
value of n that is large enough to prevent weight drift but
p is not required; and (2) a persistent disturbance would not
would reduce performance if only the robust CMAC con-
be expected to drive k ^p k to pmax .
trolled the system. The update for an individual weight in the
performance CMAC, on the other hand, operates according Theorem 2. For the system described by (16), (18), under con-
to these conditions: dition (17), if the region of robust-CMAC and performance-
CMAC uniform approximation D is chosen large enough then
A. applies zero update if the weight norm has reached applying CMAC control (19), (26), robust-CMAC weight
some imposed bound, pmax ; updates (22), and performance-CMAC weight updates (27)
B. applies an unmodified weight update, if the weight is results in SGUUB signals (zj and w^ j ) in D.
within the imposed bound, pmax , and the total time the The proof appears in Appendix B.
Samiuddin et al. 7
in
out n m Xpp Nfan
Nominal
value
RGA analysis
The TORC is a MIMO system, yet MIMO control designs
typically require a model of the system. On the other hand
model information is not required for SISO controls. RGA
analysis identifies the degree of process interactions between
the control inputs and outputs (Bristol, 1966; Chen and
Seborg, 2002), indicating whether a decentralized control
using several SISO loops is appropriate. The resulting RGA
matrix shows diagonal dominance, justifying our decentra-
lized control strategy (Table 4).
Note that although the PI control and adaptive-CMAC
control in the simulations are designed using a decentralized
strategy, the testing of both PI controls and adaptive-CMAC
controls takes place with the full MIMO system model.
Figure 9. Step change in Nfan from 300 to 500rpm.
Figure 11. Net power output for disturbance rejection test. Figure 13. Setpoint tracking under the influence of disturbance.
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Appendix A
Quoilin S (2011) Sustainable Energy Conversion Through the Use of
Organic Rankine Cycles for Waste Heat Recovery and Solar Appli- Proof of Theorem 1. Consider the (positive-definite) direct
cations. PhD Thesis, University of Liege (Belgium). adaptive control Lyapunov function
Quoilin S, Aumann R, Grill A, Schuster A, Lemort V and Spliethoff
H (2011) Dynamic modeling and optimal control strategy of 1 2 1 T
waste heat recovery organic rankine cycles. Applied Energy 88(6): Vj = z + ~ w
w ~j ð28Þ
2gj j 2bj j
2183–2190.
Rasmussen BP and Alleyne AG (2004) Control-oriented modeling of
transcritical vapor compression systems. Journal of Dynamic Sys- ~ j = wj w
where w ^ j is the weight estimation error (wj is a set
tems, Measurement, and Control 126(1): 54–64. of unknown ideal constant weights).
Rodrı́guez FO, de Jesús Rubio J, Gaspar CRM, Tovar JC and Without loss of generality, for the rest of this proof we
Armendáriz MAM (2013) Hierarchical fuzzy CMAC control for drop the single subscript j and assume that y_ sp = 0. The
nonlinear systems. Neural Computing and Applications 23(1): Lyapunov time derivative becomes
323–331.
Samiuddin et al. 13
" #
X n k wk2
_V = 1 z f1 (y, Tsf ) gi, j ui (t T ) + D gj (un + Du) dz =
dmax
+ ð36Þ
gj i6¼j
K 4K
1 0_ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
~w
w ^ ð29Þ kwk dmax k wk2
b dw = + + ð37Þ
P 2 n 4
f1 (y, Tsf ) i6¼j gi, j ui (t T )
=z + D un G^
w Kz + d(t) Thus, if D has been chosen larger than (includes) Lyapunov
gj
1 ~0 _ surface V (z, k w
~ k ) = V (dz , dw ), the signals (jzj,k w
~ k) are
ww ^ SGUUB. The uniform ultimate bound given is given by
b
Lyapunov surface V (z, k w~ k ) = V (dz , dw ). h
ð30Þ
^_ =b)
= Kz2 + z(e + D) + w~0 (G0 z w ð33Þ ~ + hjzj~p0 B(^p b)
V_ = Gz2 + z(e + D) + njzjw~0 w njzjw~0 w
ð39Þ
Using the e-modification update law from equation (22)
gives the following where B has elements Bi along the diagonal. The bound is
where dmax is a positive constant that bounds je + Dj based Thus, all signals are SGUUB (in region D) since a bound on
on (17) and (31). By completing the square in (35) one can k ~p k is imposed and V_ \0 if either jzj or k w
~ k grows large
show that V_ \0 in region D when either jzj . dz or k w
~ k . dw enough. h
where