On Some Advanced Control Techniques

Copyright © IFAC Advances in Automotive Control. Ascona. Switzerland. 1995

ON SOME ADVANCED CONTROL TECHNIQUES M. M'SAAD", L. DUGARD", R . RAMIREZ-MENDOZA"t and V. CERVENKAt Laborl1toire d'AutomAtique de Gren06/e (INPGICNRS), ENSIEG ·BP 46 • 38402 SAint· MArtin d'Here~, FrAnce. " They Are A180 with the GDR CNRS AutomAtique t He i. npported 6y CONACYT·Merico t He i~ A/~o with the FAculty of ElectricAl Engineering, Slo'flAk TechnicAl Uni'fler~ity. Abstract.The motivation of this paper is to present the partial state reference model (PSRM) adaptive control approach and investigate its applicability to diesel engines which are widely used as power sources. The control objective consists in well damped and offset-free response to speed commands in spite of the well known complexity of the engine dynamics. This particularly allows to improve the fuel efficiency. Keywords.Partial state reference model control (PSRMC), Linear quadratic control design, Gener· alized predictive control design, Robustness, Parameter Adaptation, Diesel Engines

1. INTRODUCTION

problem represents a challenging opportunity to investigate the advanced control theory. Research in control theory has reached a reasonable level of maturity : many of the challenging control problems involved in coarsely known plants can be now accomodated . This is mainly a result of many years of effort devoted to the understanding of the control theory. Several books and comprehensive surveys about the principles as well as the applications of theses controllers are available. Of fundamental interest, the rapid and revolutionary progress in the computer technology makes the implementation of the underlying control systems simpler and cheaper. In this paper one aims at presenting the PSRM adaptive control approach proposed in M'Saad et al (1990, 1993) and evaluates its effectiveness to deal with the control problems involved in diesel engines. The PSRM control has shown to be a potential approach to deal with the main control problems, namely relatively important time delay, inversely unstable systems, suitable tracking, offset-free performance, stability and performance robustness and time varying systems. Several real time applications have been already reported, they involve fermentation processes (Queinnec et ai, 1993), flexible mechanical systems (M'Saad and Hejda, 1994 and M'Saad et ai, 1993) and chemical reactors (Miclovicova et ai, 1994). The control of diesel engines is mainly motivated by efficiency requirements. The underlying control

The applicability of the PSRM control approach is evaluated in realistic simulation framework involving a set of diesel engine input-output models that have been identified by Jiang (1994) . These models have been used to design a PID gain scheduling controller. The design is carried out using an appropriate linear quadratic control approach design with and without parameter adaptation (M'Saadet ai, 1990). This allows to demonstrate the robustness of the considered control design as well as the interest of the parameter adaptation. Of particular importance, the analysis of the performances is made in the spirit of the robust control theory (Green and Limbeer, 1995).

2. THE PSRMC APPROACH

In the following, we will precise the main components of the PSRMC approach we are concerned with . 2.1 The Plant Model

We will consider the class of plants which inputoutput behaviors may be appropriately approximated by the following backward shift operator

227

A(q-l)y(t) D(q-l )V(t)

B(q-l)U(t - d - 1) + v(t) =

+ w(t)

C(q-l)~(t)

This allows to restate the ideal deadbeat PSRMC objective as follows:

(1)

(9)

with

ey(t) = y(t) - f3B(q-l)y.(t)

with

A(q-I) = B(q-l) = C(q-l) = D(q-l)

1 + alq-l

+ ... + anaq-na bo + b1q-l + ... + bnbq-nb 1 + Clq-l + ... + cneq-ne 1 + d1q-l + ... + dndq-nd

eu(i) = u(i) - f3A(q-l)y.(t

PSRMC Objective

For the sake of simplicity, let us first consider the ideal case, i.e. w(t) = ~(t) = 0 for all t. The plant model may be hence given the following partial state representation .

A(q-l )D(q-l )x(t) = D(q-l )u(t - d - 1) y(t) = B(q-l)x(t)

J(t) = E

(3)

for ch~i
with

Wyd(q-l)ey/(t) Wud(q-l)eu/(t)

(14)

=

Wyn (q-l )e y(i) (15) D( q-l) Wun (q-l )e u (t)

u

W. 4 (Z-I)

wr .. ( z -') d t ·fi d . an d W y ( z -I) = W,,4(Z-' ) eno e user speCl e Input and output frequency weighting, respectively. The frequency weightings are mainly motivated by stability and performance robustness considerations and are such that all the polynomials Wr.r(q-l) are Hurwitz . The underlying control problem will be handled using the generalized predictive control approach (Clarke ei ai, 1987) or a linear quadratic control approach as in M'Saad et al (1990) .

where ~ ~ denotes the transfer function corresponding to the partial state reference model to be considered ; {u·(t)} is a bounded set-point sequence and f3 is a scalar gain which is introduced to get a unitary closed loop static gain, i.e. a_I 1m) .

/J -

Substituting the control objective (5)-(6) into the plant model (2)-(3) yields

d -1) (7) 2. 3 The PSRMC Performances

y(t) = B(q-l)f3y·(t)

(13)

where ph, ch and sh represent the prediction, control and starting horizons, respectively, p is a non negative scalar and W (z-l) = D (z -')Wu(z-' )

B·(q-l)u·(t) (6)

=D(q-l)u(i -

- ,h))'}

or

(5)

A(q-l)D(q-l)f3y·(t)

(',j (1+ j))' + p( 'oj (t +i

eu/(i+i)=O

x·(t) = f3y.(t)

=

{t.,

subject to

where {x·(t)} represents the desired partial state reference sequence. The latter may be specified as the output of an asymptotically stable system as follows:

A·(q-I)y.(t + d + 1)

(11)

(12)

(2)

where x(t) denotes the partial state. This allows to consider the following ideal deadbeat PSRMC objective x(t) - x·(t) = 0 (4)

with

+ d + 1)

The desired performances are hence completely defined by the sequences {ey (i)} and {D( q-l )e u (i)} that can be viewed as suitable performance quantifiers for systems with arbitrary zeros which are more a rule than an exception within digital control contexts. In the real world life, the modelling error sequence {w(i)} as well as the disturbances model input sequence {~(i)} are by no means identically zero. It is therefore more advisable to rela.x the ideal PSRMC objective in order to use a more adequate design approach . One can particularly consider the control objective that consists in minimizing, in a receding horizon sense, one of the following linear quadratic cost functions .

where u(t) is the control variable, y(t) is the measured plant output, d denotes the minimum plant model delay in sampling periods, v(t) and w(t) represent the external disturbances and modelling errors, respectively and {~(t)} is assumed to be a sequence of widely spread pulses of unknown magnitude or independent random variables with zero mean values and finite variances.

e.e The

(10)

(8)

228

The control design is carried out from the following plant parametrization, where the modelling errors are not taken into account.

error shaping) . The latter is closely related to the characteristic polynomial Pc(8, q-I) which can be facto red as follows (19)

with

(16)

where p/(q-I) and PO(q-l) denote the characteristic polynomial of the underlying linear quadratic control and the state observer respectively. Notice that the observer dynamics could be chosen bearing in mind the optimal estimation theory. The stability robustness as well as the nominal performances of the underlying control system can be evaluated from its various sensitivity functions, namely

A(q-I) = A(q-I )D(q-I )WI/d(q-I )Wun(q-I) B(q-I) = B(q-I)Wl/n(q-I)Wud(q-l) C(q-I)

= C(q-I)Wl/n(q-I)Wun(q-l)

Whatever is the control design, the resulting controller may be given the following linear form.

• The transfer function between the output disturbances and the output given by

where the polynomials S( q-I) and R( q-I) depend on the plant model as well as the design parameters . The input-output behavior of the control system composed by the plant model (16) in closed loop with the PSRM controller (17) is described by PC(q-l)el//(t)

S(q-I )C(q-I )~I//(t)

pc(q-I)eu/(t)

-R(q-I )C(q-I )~u/(t)

with with

which is commonly called sensitivity function. The latter provides the modulus margin which is more usefull than the standard gain margin and the attenuation bandwith.

(17) PC(q-l) =

A(q-I)S(q-l)

• The opposite of the transfer function between the noise measurement and the output given by

+

q-d-l B(q-I )R(q-I)

-I

To(z

where ~I// (t) and ~u/ (t) denote the input of the disturbances' model filtered by the frequency weightings WI/(z-I) and Wu(z-I), respectively. The controller may be rewritten as follows

)=

z-d-l B(z-I )R(z-I) Pc(q-I)

which is commonly called complementary sensitivity function. Such an appellation is motivated by the fact that SO(z-l) +

TO(z-I)=l.

+ R(q-I)y(t) = T( q- I )y. (t + d + 1)

S(q-I)D(q-I)U(t)

with

• The transfer function between the input disturbances and the output given by (18)

S ( -I) = z-d-I B(z-I) S ( -1) pZ

= S(q-I)Wl/d(q-I)Wun(q-l) R(q-I) = R(q-I)Wl/n(q-I)Wud(q-l) S(q-I)

A(z-I)

OZ

which is nothing but the product of the plant model transfer function by the sensitivity function.

T(q-I) = ,8pc(q-I)Wl/n(q-l)Wud(q-l)

The corresponding tracking and regulation dynamics are then respectively given by

• The opposite of the transfer function between the output disturbances or noise measurement and the input given by

and R(Z-I)

S(Z-I )D(Z-I) So(z This clearly shows that the PSRMC allows to separate the desired tracking behavior (i .e. the partial state reference model choice) and the desired regulation dynamics (i .e. the resulting tracking

-I

)

which is nothing but the product of the regulator transfer function by the sensitivity function.

229

The shapes of the sensitivity functions may be refined by properly specifying the involved design parameters. To do so, an iterative procedure is needed and hence a useful CACSD software package. 2.4 The PSRM Adaptive Controller

of the PSRMC to diesel engines. These simulations have been carried out using the CACSD package SIMART that has been developed with an engineering perspective in mind (Hejda and M'Saad, 1994) . To this end , we will particulary consider the diesel engine control problem formulated by J iang (1994) . The analysis made there points out that diesel engines are highly nonlinear systems to be described by only one linearized model for all operating conditions. That is why the range of the engine operation has been divided into several zones according to engine speed and power output. The control design has been carried out using a PlO based gain scheduling technique. Table 1 shows the engine models, relating the speed to the throttle position, that have been identified at different speeds and power outputs with the sampling period of 0.1 s. Of particular interest, the input-output behaviors exhibit time delays of two sampling periods when the engine speed is 1300 or 1500 rpm, and three sampling periods when the engine speed is 1000 rpm.

A remarkable research activity has been devoted to the question of designing adaptive controllers that would perform well in the presence of state disturbances, plant model parameter variations and unmodelled dynamics. The comprehensive survey of Ortega and Tang (1989) covers the major contributions from stability perspective point of view. The key issues to get a robust adaptive controller are suitable parameter estimation plant model and a robust parameter adaptation algorithm . The former can be simply obtained from the plant model by filtering and normalizing the data, while the latter has to incorporate an adequate parameter adaptation alertness and freezing. These robustness features can be achieved by the regularized constant trace algorithm with conditionnal parameter adaptation proposed in M'Saad et al (1990) . More specifically, the parameter adaptation is frozen whenever the available information is not likely to improve the parameter estimation process. The adaptive control law is obtained by simply invoking the certainty equivalence principle. This consists in replacing the plant model parameters 8 by its admissible estimate 80 (t) when deriving the PSRMC law. More specifically, the adaptive control system is implemented as follows :

The main purpose of this experimental evaluation consists in designing a robust PSRM controller that performs reasonably well for all engine operating conditions. Furthemore, it is shown that the performance could be improved using a suitable parameter adaptation . The performance requirement consists in well damped and precise response to speed commands in spite of the dynamics complexity of the engine. In the following, we will describe how the design parameters have been specified to deal with such a control objective and present the experimental results that have been obtained. Figures 1,2 and 3 show the Bode diagrams of the various models when the speed is 1000, 1300 and 1500 rpm , respectively. These frequency responses suggest to consider only one model for each nominal engine speed. More specifically, the models corresponding to the loads 2 kW, lkW and 0.7 kW are likely to be these models for nominal speeds 1000, 1300 and 1500 rpm , respectively. Such a choice has been confirmed by the control performance that has been obtained around the considered nominal speeds. The underlying results will not be reported for the sake of space limitation . Figure 4 shows the input-output performance of the robust PSRM controller for all the engine speed configurations . The latter has been designed using the infinite horizon linear quadratic control approach with the nominal plant corresponding to 1300 rpm . The diesel engine inputoutput behavior has been simulated using the nominal models corresponding to the speeds 1300 , 1500 and 1000 over the time intervals [Os ,9051 , [ 90s,180s 1and [ 1805, 260s 1,respectively An input

1. Wait for the clock pulse and sample the plant

output. 2. Update the plant model parameters using the considered parameter estimator. 3. Construct the admissible estimated plant model as follows : if O( t) is stabilizable otherwise where O(t) is the estimated plant model provided by the considered parameter adaptation algorithm . 4. Evaluate the adaptive control law using the admissible estimated plant model 80 (t) . 5. Implement the control signal and go to 1. 3. THE EXPERIMENTAL EVALUATION In the following we will present a set of simulations conducted to demonstrate the applicability

230

Figure 1: Bode graphs at 1000 rpm

Figure 3: Bode graphs at 1500 rpm 6. REFERENCES Clarke, D.V., C. Mohtadi, and P.S. Tuffs (1987) . Generalized predictive control. Automatica, 23, 137-160. Green,M and D.J .N. Limebeer (1995) . Linear robust control. Prentice Hall. Englewood Cliffs, New Jersey. Hejda, I., and M. M'Saad (1994) . The CACSD package SIMART: description and case study. Proc. of the IEEE/IFAC Joint Symposium on CACSD, Tucson, Arizona. Jiang, J . (1994) . Optimal Gain Scheduling Controller for a Diesel Engine. IEEE Control Systems, 14, 42-48. Miclovicova, E, V. Cervenka, I. Hejda, M. M'Saad and A.M. Latifi (1994). Partial state reference model control using the delta operator: a case study. Proc. of the first IFAC Workshop on New Trends in Design of Control Systems, Smolenice, Slovakia. M'Saad, M., L. Dugard, and Sh. Hammad (1993) . A suitable generalized predictive adaptive controller case study: control of a flexible arm," Automatica, 29, 589-608 . M'Saad , M., and I. Hejda (1994) . On the adaptive control of flexible transmission system. Control Engineering Practice,2, 629-639. M'Saad, M.,LD. Landau, and M. Samaan (1990) . Further evaluation of partial state model reference adaptive design. Int. J. Adapt. Control and Signal Processing, 4, 133-148. Ortega, R., and Y. Tang (1989) . Robustness of adaptive controllers - a survey. Automatica , 25, 651-677. Queinnec, I. , B. Dahhou and M. M'Saad (1993) . On the adaptive control of fedbatch fermentation processes. Int. J. Adapt. Control and Signal Processing, 6, 521-536 .

--.

--D~~--~--~--~'~~.---.~~~~--~--~

Figure 2: Bode graphs at 1300 rpm

step like disturbance followed by an output step like disturbance has been introduced during each time interval to emphasize the regulation behavior of the control system. An iterative procedure has been used to obtain the desired shaping of the sensitivity functions. Notice that the performances are quite acceptable in both tracking and regulation cases except when the nominal speed is 1000 rpm. Theses results are confirmed by the sensibility functions plots given in Figure 5. Table 2 provides the resulting modulus, gain, phase and delay margins (MM, GM, PM and DM) together with the attenuation bandwidth (AB) for all the nominal models. The input-output behavior of the adaptive version of the robust PSRM controller as well as the time evolution of the parameter estimates are respectively given in Figures 6 and 7. Notice that the parameter adaptation allows to improve the performances up to some transient period.

4. CONCLUSION The motivation of this paper was to present the PSRMC approach and investigate its applicability to diesel engine control The fundamental design features of the proposed control approach are suitables tracking capability, offset-free performance, stability robustness and parameter adaptation alertness. A great attention should however be paid to the choice of design parameters that has been determinated using an iterative procedure. The CACSD package SIMART has been revealed to be a powerfull tool to deal with the involved control design .

231

2

,..rerenc;:e afgnaI

Output and

h

r-

10

v ' v fl ~

r-

0

o

-10

l1. ~

'-'

~

-1

-

~

~

t'----'

~

-20 ~

-2

o

so

lOO

200

250

~

-40

0

..,

20

110

110

lOO

1..,

120

1110

1110

time(MC)

~r_----~------~------r-----~------~-----,

200

''''

lOO

so

,

o

I

I I

l

L

I

-so

I

-~0L------SO~----~I00-------ISO~----~200~-----25O~----~~ -lOO

Iime(-)

o

20

eo

110

lOO

120

1.0

1110

1110

200

Iime(-)

Figure 4: Input-Output Performance (For the three models ) ~Iary

S_1MIy func:Iion

10

..,.;oMty IuncIIon

estimAted parwMt.,.

~r_------------,

1~

1~

~ "-~----1

1-

Figure 6: Performance of the PSRM adaptive controller

0

3

1000

10

f_~

f

2

...

.t21

1500

~0~--~1~0----~~~--~~·

~0~--~10~--~~~--~~ ~.-)

hwquency(-"-)

o

ReguIeto< • SenoitiviIy Junction

r-

a(3) -1

~

5

'§

0

i

8(1)

'h

-2

W

-51000 . .. . ... ......... . . -1oo'------1-o-----~----.l~

-4

o

hwquency(-"-)

~oLoad

1.0 2 .5 3.0 3 .7 Parameters Loads (kW) No load 0 .7 1.85 3.1 44

al

I

a2

-1.7732 -1.6952 -1.7490 -1:r9 2 -1.8665 al

0 .7077 0 .6663 0 .7618 0 :r693 1.0419 a2

-2.0867 -2.0707 -1.9778 -2 .0399 -2.0169 al

1.3388 1.3438 1.2188 1 .3569 1.2293 a2

-1.9115 -1.8508 -1.7552 -1.7032 -1.6594

0.8867 0 .8761 0 .7881 0 .6956 0 .8301

I

mo~el

a~ I I a5 I 6. I °1 Engine Nominal Speed at 1000 rpm 0 .2112 -0 .2244 0 .0071 -0 .0009 0 .0916 0 .2459 -0 .3750 0 .0104 0 .0035 0 .1709 0 .1367 0.0990 0.0096 0.0051 -0 .2331 0 .1586 -0.1655 0 .0507 0 .0089 0 .0023 -0 .1164 -0.1049 0 .0643 0.0091 0 .0044 b. a3 a~ a5 °1 Engine Nominal Speed at 1300 rpm -0 .2887 0 .0803 -0.0376 0 .0135 -0 .0019 -0 .2270 -0 .0586 0.0136 0.0010 0 .0180 -0.2724 0 .0139 0 .0041 0 .0760 -0 .0403 0 .0040 -0.3537 0 .0755 -0.0341 0 .0126 -0.1377 0.0036 -0 .0756 0 .0055 0 .0135 0. a3 a~ a5 bl Engine Nominal Speed at 1500 rpm 0.0872 -0 .0009 0 .0168 -0 .0008 -0.0533 0 .0055 0 .0168 0 .0030 0 .0308 -0 .0555 0 .0167 0 .0060 -0 .1014 0 .1812 -0 .1069 0.0052 0 .0732 -0 .0089 0 .0167 -0 .0486 0 .0083 0.0156 -0 .1701 -0 .0501 00575

a3

Table 2

dH

GM (dH

t'M (d

UM

-5.5856 -4.8759 -4.9086

8 .0302 8 .7347 8 .5738

39 .516 61 .694 43 .214

0.5711 1.14620 0 .6471

MM

120

1'"

1110

Figure 7: Parameters adaptation I

Table 1 The engme models at different speed and power

at 1300 at 1000 at 1500

lOO Iima(-)

Figure 5: Sensitivity functions Parameters Loads (kW) No Load 0 .7 0 .85 1.2 2 .0 Parameters Loads (kW)

eo

110

~

•

AB 0 .9031 0 .9691 0 .9032

Input-output performance of the robust PSRMC

232

°2 -0 .0006 0 .0011 0 .0002 O.OO!J:'! 0 .00009

I

03

°2

0 .0022 0.0025 0 .0017 -0.0007 0.0022 03

-0.0053 -0.0010 0.0026 0.0012 0 .0009 02

0 .0029 0 .0029 0 .0014 0 .0034 0.0030 63

-0 .0089 -0 .0050 -0.0010 0.0018 0 .0049

0 .0018 0.0029 0 .0017 0 .0049 0 .0070

1110

200