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Self-Tuning Control of Nonlinear Systems Characterised by Hammerstein Models
Cop yright © I FAC Contro l Scicnl'c ann Tt'ch ll ()l()g~ (Rth T riennial \\'orld COllgn's:-. ) Kyoto. J apan . IYHI
SELF-TUNING CONTROL OF NONLINEAR SYSTEMS CHARACTERISED BY HAMMERSTEIN MODELS K. Anbuma ni* , I. G. Sarma** and L. M. Patnaik** 'Departmen t of Electrical Engineering, GOt'emment Co llege of Technology, Coimbatore 641013, Tamilnadu, India "School oj".-i. lItumation , I ndian I nstitu t e of Science, Bangalore 560012, I ndia
Abstract. The concept of self-tuning is applied to the control of Hammerstein models. It is also shown that these models are not amenable to self-tuning if the performance index includes control weighting. KeYW0rds.Nonlinea r systems; selftun1ng control INTRODUCTION
explicit manner. Further, the paper shows that self-tuning is not feasible in Hammerstein models if control weighting is included in the performance criterion.
The concept of self-tuning has been developed for the optimal control of systems whose parameters are unknown and pos s ibly time-varying slowly. Self-tuning control has been successfully applied to a variety of linear systems (Astrom, and co-workers 1977) •
THEORY OF SELF-TUNING CONTROL OF HA~~RSTEIN ~ODELS The Hammerstein model, depicted in Fig.l, of the stochastic nonlinear system consists of a static-nonlinearity block (1), a linear-dynamics block (2), and a block (3) characterizing the random disturbance.
Keviczky and co-workers (1979) have applied self-tuning to a Hammerstein model with a second order nonlinearity, their control objective being to keep the output at the extremum point of the above second order nonlinearity.
r------- -------- - - -- - --- ---
Any operating point nonlinearity actually pr esent in a system is reflected as a ti ~e-va rying parameter which the self-tune. can handle as it is para meter-ad a ptive. Thus linear self-tuners can, albeit subopti mally, deal with mild operating point nonlinearities in an implicit manner. On the other hand , it has been observed, ' basic self-tuners are not amenable to actuator nonlinearities' (Zanker and Wellstead, 1979). They have given and ad hoc technique for the special Case of ideal saturation in the actuator.
:
i
l .1
J
I
I~ C I Z-')-
: II
3
J
:
LA"(ZTl
,I
I
.-
i
1
I
--
2
;
u'+{;rJ "-tz~;~-:~-i
--i--
Y,
I
I
,I . ------- - - - - - ~ - - - -- -
- -- -- - -
-
--'
Fig.l. Hammerstein model of nonlinear system with disturbance. The difference equation description of this model is
In this paper self-tuning is appli-
ed to the control of any nonlinear stochastic system that can be adequately represented by the Hammerstein model wherein the static nonlinearity part can be any odd order polynomial. ~he method presented deals with the nonlinearity in an
(1) ( 2) t -
7':;
0,1,2, •••
(3)
80
K. Anbum;} !1i , 1. G.
where A( Z
-1
)
-
l+al
Z
-1
-n
l+~ Z
-1
+ ••• +cn z
~1.
P;lln a ik
we have J Yt+ k/ t - Yr , t ) 2 + V
+ ••• +an z
= b o+ bl Z -1+ ••• + bn Z -n ..
Sarma and L.
::I
(
This is minimized to V if
-n
Yt+ k/ t
-1
and Z is the backward shift operator; Yt ' the output; Ut' the control input; et' white noise of unit variance; v t ' white noise of variance A2; A, noise gain; k, the time delaY ( ~l). Henceforth the argument z- l in the polynomials is omitted for simplici ty. The process output is required to track a reference signal Yr, t with minimum variance. So the performance criterion is J :z [(Yt+k - Yr.t)2] (4) where E denotes the expectation operation,given data upto time t.
- Y r ,t
= O.
(8)
Mu.1tiplyin~ (8)
bl C and substituting for Cy from (5) we obtain
(9 ) GY t + FBx t - Cy r,t = 0 Equation (9) gives the optimal control law.
Now defining an auxiliary variable (Clarke and Gawthrop, 1975) ~t+k
= Yt+k
- Yr,t' the k-step
" prediction of'VdefinedbY't'is "
1\
't' t+k/ t '" Yt+k/t
- Yr
,t
='+'t+ k -w t+ k
(10) Multiplying (10) by C and rearranging we get ~
To derive the optimal control law, the linear-dynamics equation (2) is rearranged into a prediction model (Astrom, 1970; Clarke and Gawthrop, 1975) C Yt+k/t .. GY t + FBx t (5)
" from (5) we Substituting for C 'Y have '+' t+k" Gy t + FBx t - ey r, t+w~k( 11)
where Yt+k/t .. Yt+k - wt + k
where
(6)
is the k-step ahead prediction of Yt and wt+k ::a Fv t+k is the prediction error. The polynomials P and G satisfy (Astrom, 1970) C = AF + z -kG (7) Using (6) in (4) J = B[
(y
t+k/t + "t+k - Yr ,t)2]
= (Y t+ k/ t
- Yr. t ) 2
+ 2E[ wt+ k
(y t+ k/ t
1\
't' t+k/t
"
= CYt+k/t-CYr,t+(l-C)'Yt+k/t
A
wi+k - wt+k + (l-e)~ t+k/t
(12) is the noise term. Equation (11) is an estimation model of the process. Substituting from (1) for x in (11) we get '-Vt+k - Yt+k - Yr,t -Gy t+PB(do+d l u t +· .+dmU:)-CY r, t+wt..k
- Yr , t ) ]+ Ew ~+ k
The expectation operation is omitted from the first term since it is deterministic at t, and the second term equals zero as wt+k is uncorrelated with Yt+k/t and Yr,t (Clarke and Gaw throp, 1975). Letting V - Bw~ • ,\2 (l+f{+ •••+f~_l) where fi are the coefficients of P,
-Gy t+ PBd 0+ PBd l u t +· .+PBdmU:-CY r,t+"t+k =
Self - Tunin g Control of
(13) where n a , n~, and ny are the degrees of a = G, ~ ~ FB, and y = -C, respectively. 6 = F{l) B{l) do and ~ij .. d i
Pj
•
Similarly, the control law (9) becomes (Pa oYt+alYt- l+···+ana Yt -n +0 a
+~lOUt+~llu t _l +·· .+Pln~ u t _
np
Comparison of (13) and (14) shows that the estimator and the controller have the same parameters and hence the Hammerstein model readily lends itself to fast self-tuning control. Unlike in linear systems, what results from (14) is not Ut but a polynomial in Ut with known coefficients .< :1 m (15) O=q+~lOUt+t-' 2')u t +·· .+P moUt where q stands for all the other terms of (14). However, (15) can be solved using any suitable root-sol ving algorithm. To avoid the possibility of all the roots turning out complex we restrict m to be odd, in which case there will be at least one real root. If there are more real roots the one with the lea st magnitude is selected for Ut. The expression for closed loop output can be obtained using (6) in (8) as Y t .. Y r, t-k + Fv t ( 17) Equation (18) showe that the mean steady state error ie zero even if there is a d.c. term do. That self-tuning of Hammerstein models ie not feasible i f control costing is included in the perfo~ mance criterion is shown in the appendix.
~onlineJr
Sys Lems
81
SHIOLAT10N dE:3ULTS Details of the example systems used in the digital siwulations are given in the appendix. Three defferent non linearities (NL1, NL2, and NL3) are used in the static part of the Hammerstein model. The highest degrees of NLl and NL2 are odd while that of NL3 is even. NL2 is comparatively strong, the relative magnitude of its coefficients being larger. Recursive Least Squares algorithm is employed for parameters estimation. Zero is the starting value of all the paraIDeters excepting P which mo is given a small non-zero value of 0.01 to prevent break down of root solving. The initial covariance matrix is 101 and the forgetting factor is 0.995 for the first 200 samples and 1.0 afterwards. The reference input is zero during the first 100 samples and increased to 10 thereafter. Since the parameter estimates are initially bad, the output deviations are large initially. To avoid these large excursions, the system is initially (for the first 15 samples) controlled by a simulated relay controller. The self tuning controller takes over from the sixteenth sample. However, parameter estimation is allowed to proceed from the start. The relay has a dead zone of 2.0, an output magnitude of 5.0, and a rate feedback constant of 5.0. The system response with the nonlinearity NLl, controlled by the nonlinear self-tuner is depicted in Fig. 2.
';
l().O
!,.:
'.J
'T ..' .<
Y"
" ,,\
. .)
ell, • .
-.,..
o~ --.- ~-' :. ~.--_- -'- ~- . I
.10.o r·
I
I
'.-,
:' - ...
,. 1
i "',_, '"'."-
l
u(4!'n ... " Iop 4!' )
Fig. 2 Performance of Nonlinear Self-tuner with NLl. There is no steady state mean error and the actual performance cost averaged over the last 100 samples is
82
K. Anbuman i. I . C. Sarma .]nu L.
0.036 which compares favourably with the theoretically optimal cost of ,\ a "" 0.028 obtainable on final con vergence. The response of the nonlinear self tuner with the stronger non linearity NL2 is shown 10 Fig. 3.
1
1
1
200
)00
.'-PC
.
1
. " - u (en\l e lope '
~\.
Patnaik
and 3 with Fi ~ . 4 reveals that linear self-tuners do not perform acceptably with nonlinear systems while the nonlinear self-tuners of the paper yield optimal performance. Next, the system with an even de~ree nonlinearity (NL3 of degree four) is controlled by the n cnlinea r selftuner which has been designed assuming the degree of the plant nonlinearity to be three, to make it odd. 'r he perf crlll8llce is displayed in Fig. 5 •
Fig. 3. Performance of Nonlinear Self-tuner with NL2. Here too there is no steady state mean error. The average cost is 0.037. In order to test t h e adequacy of linear self-tuners for nonlinear systems, the above two example systems are then controlled by a linear self-tuner which ign ores the presence of any nonl1oearity in the syste~ Curve NLl of Fig. 4. shows the response of the system with the nonlinearity NLl, controlled by the linear self-tuner.
Fig. 5. Performance of Nonlinear Self-tuner with NL3. In this case also the mean steady
state error is zero and the performance cost is 0.064 which is good though slightly suboptimal. CONCLUSIONS
loot 100 , I
o1-IOO~
i
- 1 00~
Fig. 4. Performance of Linear Self-tuner. ~ild fluctuations occur upto about the fiftieth sample even with different choices of starting values. ihen relay control is ell1ployed these large deviations appear on transition of control to the self-tuner. There is a mean steady state error of 22 percent and the cost is consequently very high. A limit of ~ 50 has been put on the control in the case of Fig. 4. The performance with the stronger nonlinearity NL2. indicated by the dotted envelope NL2, is unacceptable. Comparision of Figs.2
This paper applies self-tuning to the control of nonlinear stochastic plants which can be adequately represented by Hammerstein models with unknown parameters of both the plant and the noise. The method takes the nonlinearity into account explicitl~ The a priori data needed are the same as for linear self-tuners, viz, the system order m and the delay k, in addition the polynomial degree m. Simulations with other sets of nonlinearity coefficients all demonstrated the applicablli ty of self-tuning to Hammerstein models. The paper also brings out the superiority of nonlinear self-tuners over linear self-tuners even in the case of weakly nonlinear systems.
APPENDIX Self-tuning with Control Weighting The performance criterion with weight R on control is
Se lf- Tunin g Contro l of Konlin e ar Systems
J2
:s
2 E[ ( y t+ k _y)2]+Ru r,t t
:s
(y t+k/t
_y
r,t
83
slope renders the control weight inconstant. Due to these reasons selftuning of Hammerstein models w1th control weighting is not feasible in general.
)2+Ru 2 +V t
To minimize J 2 6J 2 6u t
:s
Partirn]]ats of Example Systems
0
=(1 t+k/t
_ y
)
r,t 6/\
6"
Y t+k/t
bU
Ytk/t
+ Rut (18)
t
dx
nu
(19)
The simulated nonlinear processes are described by Static nonlinearity: x = do+dlu+d2u2+d3u3 Linear dynamics:
The first derivative on the right hand side is obtained from (5) as equal to b o which is the leading coefficient of the polynomial B. Hence the optimality cond1tion (18) becomes ( 20) l'JUltiply1n~
(20)bl C and subst1tuting for cy from l5) we obtain the control law Cy r, t 0 Gy + PBx + --------- - Cy r,t=(21) t t bo(dx/du) Attempting to derive an estimation model having the same parameters as the optimal control equation (21) in the case of control weighting, we define the auxiliary variable as (Clarke and Gawthrop, 1975)
Proceeding along the lines of derivation of (11) from (10) we obtain the estimation model. (22) Equations (21) and (22) will have the same parameters if the control weight satisfies the condition R = b o (dx/du)
(23)
In general, the slope of the nonlinearity may change its sign in the operating region. This will make the control weight negative which is unacceptable. Even i f the slope is of the same sign, variation of the
yes) = Hl (s)x(s)+H 2 (s)e(s) NLl: do:::ZO.S, d1=1.0, d 2..-O.5,d =0.1 3 NL2: d 0 =1.0, dl-l.O, d 2==-l.0,d =0.2 3 NL3 : d o=0.5, dl-l.O, d 2- -0.5 ,d 3=0.1, d -O.Ol 4
2 10 H (s)=-------------, H2 (S)=- -----1 (1+8s)(1+15s) (1+12s) For digital simulation, these are discretized with a sampling time of one unit. The use of a zero order hold between the digital cowputer and the plant is presumed. Estimation
~wdel
84
K. Anbuma ni , I . C . Sa rm a wn d L.
n =n-l, Cl
n~=n+k-2, ...
n an Y
Equation (13)becomes Yr,t-k=
q>~9
- Yr,t-k+Wt
Hence As the controller converges, w' cont verges to wt ' the prediction error (Clarke and Gawthrop, 1975). REFERENCES Astrom, K.J. (1970). Introduction to stochastic Control Theory. Academic Press, New York.
~!.
Pa tn a i k
Astrom, K.J.,U. Borisson, L.Ljung, and B. Wittenmark (1977). Theory and applications of self-tuning regulators. Automatioa, U, 457-476. Clarke, n.w., and P.J. Gawthrop (1975). Self-tuning controller. ProctIEE, ~, 929-934. Keviczky, L.,I.Vajk, and Hetthessy (1979). A self-tuning extremal controller for the generalized Hammerstein model. 5th IFAC Svmpt on System Identification, 1147-1151. Zanker, P.M.! and p. E.Wellstead (1979). ~ractical features of self-tuning. IEE cont. an Trends in on-line Computer Control Systems, 160-164. Discussion to Paper 3 . 4 H. Kimura (Japan): Usually , high gain feed back results in non - interaction . What can be e xpected if , in your simulation result de s cribed in Fig . 3 , only the first command signal is step function and the second one is kept zero? B . Porter ,fEngland) : The command input vec t nr { l, O} E Cv and i s therefore fa it hf¥lly tracked by the plant output vector / y . ' Y2 ' ; the interac:t i on between y , (t) and Y2 ca n be made as small as des i red by i ncreasi n g t he gain parameter .
d.)
SELF-TUNING CONTROL OF NONLINEAR SYSTEMS CHARACTERISED BY HAMMERSTEIN MODELS K. Anbuma ni* , I. G. Sarma** and L. M. Patnaik** 'Departmen t of Electrical Engineering, GOt'emment Co llege of Technology, Coimbatore 641013, Tamilnadu, India "School oj".-i. lItumation , I ndian I nstitu t e of Science, Bangalore 560012, I ndia
Abstract. The concept of self-tuning is applied to the control of Hammerstein models. It is also shown that these models are not amenable to self-tuning if the performance index includes control weighting. KeYW0rds.Nonlinea r systems; selftun1ng control INTRODUCTION
explicit manner. Further, the paper shows that self-tuning is not feasible in Hammerstein models if control weighting is included in the performance criterion.
The concept of self-tuning has been developed for the optimal control of systems whose parameters are unknown and pos s ibly time-varying slowly. Self-tuning control has been successfully applied to a variety of linear systems (Astrom, and co-workers 1977) •
THEORY OF SELF-TUNING CONTROL OF HA~~RSTEIN ~ODELS The Hammerstein model, depicted in Fig.l, of the stochastic nonlinear system consists of a static-nonlinearity block (1), a linear-dynamics block (2), and a block (3) characterizing the random disturbance.
Keviczky and co-workers (1979) have applied self-tuning to a Hammerstein model with a second order nonlinearity, their control objective being to keep the output at the extremum point of the above second order nonlinearity.
r------- -------- - - -- - --- ---
Any operating point nonlinearity actually pr esent in a system is reflected as a ti ~e-va rying parameter which the self-tune. can handle as it is para meter-ad a ptive. Thus linear self-tuners can, albeit subopti mally, deal with mild operating point nonlinearities in an implicit manner. On the other hand , it has been observed, ' basic self-tuners are not amenable to actuator nonlinearities' (Zanker and Wellstead, 1979). They have given and ad hoc technique for the special Case of ideal saturation in the actuator.
:
i
l .1
J
I
I~ C I Z-')-
: II
3
J
:
LA"(ZTl
,I
I
.-
i
1
I
--
2
;
u'+{;rJ "-tz~;~-:~-i
--i--
Y,
I
I
,I . ------- - - - - - ~ - - - -- -
- -- -- - -
-
--'
Fig.l. Hammerstein model of nonlinear system with disturbance. The difference equation description of this model is
In this paper self-tuning is appli-
ed to the control of any nonlinear stochastic system that can be adequately represented by the Hammerstein model wherein the static nonlinearity part can be any odd order polynomial. ~he method presented deals with the nonlinearity in an
(1) ( 2) t -
7':;
0,1,2, •••
(3)
80
K. Anbum;} !1i , 1. G.
where A( Z
-1
)
-
l+al
Z
-1
-n
l+~ Z
-1
+ ••• +cn z
~1.
P;lln a ik
we have J Yt+ k/ t - Yr , t ) 2 + V
+ ••• +an z
= b o+ bl Z -1+ ••• + bn Z -n ..
Sarma and L.
::I
(
This is minimized to V if
-n
Yt+ k/ t
-1
and Z is the backward shift operator; Yt ' the output; Ut' the control input; et' white noise of unit variance; v t ' white noise of variance A2; A, noise gain; k, the time delaY ( ~l). Henceforth the argument z- l in the polynomials is omitted for simplici ty. The process output is required to track a reference signal Yr, t with minimum variance. So the performance criterion is J :z [(Yt+k - Yr.t)2] (4) where E denotes the expectation operation,given data upto time t.
- Y r ,t
= O.
(8)
Mu.1tiplyin~ (8)
bl C and substituting for Cy from (5) we obtain
(9 ) GY t + FBx t - Cy r,t = 0 Equation (9) gives the optimal control law.
Now defining an auxiliary variable (Clarke and Gawthrop, 1975) ~t+k
= Yt+k
- Yr,t' the k-step
" prediction of'VdefinedbY't'is "
1\
't' t+k/ t '" Yt+k/t
- Yr
,t
='+'t+ k -w t+ k
(10) Multiplying (10) by C and rearranging we get ~
To derive the optimal control law, the linear-dynamics equation (2) is rearranged into a prediction model (Astrom, 1970; Clarke and Gawthrop, 1975) C Yt+k/t .. GY t + FBx t (5)
" from (5) we Substituting for C 'Y have '+' t+k" Gy t + FBx t - ey r, t+w~k( 11)
where Yt+k/t .. Yt+k - wt + k
where
(6)
is the k-step ahead prediction of Yt and wt+k ::a Fv t+k is the prediction error. The polynomials P and G satisfy (Astrom, 1970) C = AF + z -kG (7) Using (6) in (4) J = B[
(y
t+k/t + "t+k - Yr ,t)2]
= (Y t+ k/ t
- Yr. t ) 2
+ 2E[ wt+ k
(y t+ k/ t
1\
't' t+k/t
"
= CYt+k/t-CYr,t+(l-C)'Yt+k/t
A
wi+k - wt+k + (l-e)~ t+k/t
(12) is the noise term. Equation (11) is an estimation model of the process. Substituting from (1) for x in (11) we get '-Vt+k - Yt+k - Yr,t -Gy t+PB(do+d l u t +· .+dmU:)-CY r, t+wt..k
- Yr , t ) ]+ Ew ~+ k
The expectation operation is omitted from the first term since it is deterministic at t, and the second term equals zero as wt+k is uncorrelated with Yt+k/t and Yr,t (Clarke and Gaw throp, 1975). Letting V - Bw~ • ,\2 (l+f{+ •••+f~_l) where fi are the coefficients of P,
-Gy t+ PBd 0+ PBd l u t +· .+PBdmU:-CY r,t+"t+k =
Self - Tunin g Control of
(13) where n a , n~, and ny are the degrees of a = G, ~ ~ FB, and y = -C, respectively. 6 = F{l) B{l) do and ~ij .. d i
Pj
•
Similarly, the control law (9) becomes (Pa oYt+alYt- l+···+ana Yt -n +0 a
+~lOUt+~llu t _l +·· .+Pln~ u t _
np
Comparison of (13) and (14) shows that the estimator and the controller have the same parameters and hence the Hammerstein model readily lends itself to fast self-tuning control. Unlike in linear systems, what results from (14) is not Ut but a polynomial in Ut with known coefficients .< :1 m (15) O=q+~lOUt+t-' 2')u t +·· .+P moUt where q stands for all the other terms of (14). However, (15) can be solved using any suitable root-sol ving algorithm. To avoid the possibility of all the roots turning out complex we restrict m to be odd, in which case there will be at least one real root. If there are more real roots the one with the lea st magnitude is selected for Ut. The expression for closed loop output can be obtained using (6) in (8) as Y t .. Y r, t-k + Fv t ( 17) Equation (18) showe that the mean steady state error ie zero even if there is a d.c. term do. That self-tuning of Hammerstein models ie not feasible i f control costing is included in the perfo~ mance criterion is shown in the appendix.
~onlineJr
Sys Lems
81
SHIOLAT10N dE:3ULTS Details of the example systems used in the digital siwulations are given in the appendix. Three defferent non linearities (NL1, NL2, and NL3) are used in the static part of the Hammerstein model. The highest degrees of NLl and NL2 are odd while that of NL3 is even. NL2 is comparatively strong, the relative magnitude of its coefficients being larger. Recursive Least Squares algorithm is employed for parameters estimation. Zero is the starting value of all the paraIDeters excepting P which mo is given a small non-zero value of 0.01 to prevent break down of root solving. The initial covariance matrix is 101 and the forgetting factor is 0.995 for the first 200 samples and 1.0 afterwards. The reference input is zero during the first 100 samples and increased to 10 thereafter. Since the parameter estimates are initially bad, the output deviations are large initially. To avoid these large excursions, the system is initially (for the first 15 samples) controlled by a simulated relay controller. The self tuning controller takes over from the sixteenth sample. However, parameter estimation is allowed to proceed from the start. The relay has a dead zone of 2.0, an output magnitude of 5.0, and a rate feedback constant of 5.0. The system response with the nonlinearity NLl, controlled by the nonlinear self-tuner is depicted in Fig. 2.
';
l().O
!,.:
'.J
'T ..' .<
Y"
" ,,\
. .)
ell, • .
-.,..
o~ --.- ~-' :. ~.--_- -'- ~- . I
.10.o r·
I
I
'.-,
:' - ...
,. 1
i "',_, '"'."-
l
u(4!'n ... " Iop 4!' )
Fig. 2 Performance of Nonlinear Self-tuner with NLl. There is no steady state mean error and the actual performance cost averaged over the last 100 samples is
82
K. Anbuman i. I . C. Sarma .]nu L.
0.036 which compares favourably with the theoretically optimal cost of ,\ a "" 0.028 obtainable on final con vergence. The response of the nonlinear self tuner with the stronger non linearity NL2 is shown 10 Fig. 3.
1
1
1
200
)00
.'-PC
.
1
. " - u (en\l e lope '
~\.
Patnaik
and 3 with Fi ~ . 4 reveals that linear self-tuners do not perform acceptably with nonlinear systems while the nonlinear self-tuners of the paper yield optimal performance. Next, the system with an even de~ree nonlinearity (NL3 of degree four) is controlled by the n cnlinea r selftuner which has been designed assuming the degree of the plant nonlinearity to be three, to make it odd. 'r he perf crlll8llce is displayed in Fig. 5 •
Fig. 3. Performance of Nonlinear Self-tuner with NL2. Here too there is no steady state mean error. The average cost is 0.037. In order to test t h e adequacy of linear self-tuners for nonlinear systems, the above two example systems are then controlled by a linear self-tuner which ign ores the presence of any nonl1oearity in the syste~ Curve NLl of Fig. 4. shows the response of the system with the nonlinearity NLl, controlled by the linear self-tuner.
Fig. 5. Performance of Nonlinear Self-tuner with NL3. In this case also the mean steady
state error is zero and the performance cost is 0.064 which is good though slightly suboptimal. CONCLUSIONS
loot 100 , I
o1-IOO~
i
- 1 00~
Fig. 4. Performance of Linear Self-tuner. ~ild fluctuations occur upto about the fiftieth sample even with different choices of starting values. ihen relay control is ell1ployed these large deviations appear on transition of control to the self-tuner. There is a mean steady state error of 22 percent and the cost is consequently very high. A limit of ~ 50 has been put on the control in the case of Fig. 4. The performance with the stronger nonlinearity NL2. indicated by the dotted envelope NL2, is unacceptable. Comparision of Figs.2
This paper applies self-tuning to the control of nonlinear stochastic plants which can be adequately represented by Hammerstein models with unknown parameters of both the plant and the noise. The method takes the nonlinearity into account explicitl~ The a priori data needed are the same as for linear self-tuners, viz, the system order m and the delay k, in addition the polynomial degree m. Simulations with other sets of nonlinearity coefficients all demonstrated the applicablli ty of self-tuning to Hammerstein models. The paper also brings out the superiority of nonlinear self-tuners over linear self-tuners even in the case of weakly nonlinear systems.
APPENDIX Self-tuning with Control Weighting The performance criterion with weight R on control is
Se lf- Tunin g Contro l of Konlin e ar Systems
J2
:s
2 E[ ( y t+ k _y)2]+Ru r,t t
:s
(y t+k/t
_y
r,t
83
slope renders the control weight inconstant. Due to these reasons selftuning of Hammerstein models w1th control weighting is not feasible in general.
)2+Ru 2 +V t
To minimize J 2 6J 2 6u t
:s
Partirn]]ats of Example Systems
0
=(1 t+k/t
_ y
)
r,t 6/\
6"
Y t+k/t
bU
Ytk/t
+ Rut (18)
t
dx
nu
(19)
The simulated nonlinear processes are described by Static nonlinearity: x = do+dlu+d2u2+d3u3 Linear dynamics:
The first derivative on the right hand side is obtained from (5) as equal to b o which is the leading coefficient of the polynomial B. Hence the optimality cond1tion (18) becomes ( 20) l'JUltiply1n~
(20)bl C and subst1tuting for cy from l5) we obtain the control law Cy r, t 0 Gy + PBx + --------- - Cy r,t=(21) t t bo(dx/du) Attempting to derive an estimation model having the same parameters as the optimal control equation (21) in the case of control weighting, we define the auxiliary variable as (Clarke and Gawthrop, 1975)
Proceeding along the lines of derivation of (11) from (10) we obtain the estimation model. (22) Equations (21) and (22) will have the same parameters if the control weight satisfies the condition R = b o (dx/du)
(23)
In general, the slope of the nonlinearity may change its sign in the operating region. This will make the control weight negative which is unacceptable. Even i f the slope is of the same sign, variation of the
yes) = Hl (s)x(s)+H 2 (s)e(s) NLl: do:::ZO.S, d1=1.0, d 2..-O.5,d =0.1 3 NL2: d 0 =1.0, dl-l.O, d 2==-l.0,d =0.2 3 NL3 : d o=0.5, dl-l.O, d 2- -0.5 ,d 3=0.1, d -O.Ol 4
2 10 H (s)=-------------, H2 (S)=- -----1 (1+8s)(1+15s) (1+12s) For digital simulation, these are discretized with a sampling time of one unit. The use of a zero order hold between the digital cowputer and the plant is presumed. Estimation
~wdel
84
K. Anbuma ni , I . C . Sa rm a wn d L.
n =n-l, Cl
n~=n+k-2, ...
n an Y
Equation (13)becomes Yr,t-k=
q>~9
- Yr,t-k+Wt
Hence As the controller converges, w' cont verges to wt ' the prediction error (Clarke and Gawthrop, 1975). REFERENCES Astrom, K.J. (1970). Introduction to stochastic Control Theory. Academic Press, New York.
~!.
Pa tn a i k
Astrom, K.J.,U. Borisson, L.Ljung, and B. Wittenmark (1977). Theory and applications of self-tuning regulators. Automatioa, U, 457-476. Clarke, n.w., and P.J. Gawthrop (1975). Self-tuning controller. ProctIEE, ~, 929-934. Keviczky, L.,I.Vajk, and Hetthessy (1979). A self-tuning extremal controller for the generalized Hammerstein model. 5th IFAC Svmpt on System Identification, 1147-1151. Zanker, P.M.! and p. E.Wellstead (1979). ~ractical features of self-tuning. IEE cont. an Trends in on-line Computer Control Systems, 160-164. Discussion to Paper 3 . 4 H. Kimura (Japan): Usually , high gain feed back results in non - interaction . What can be e xpected if , in your simulation result de s cribed in Fig . 3 , only the first command signal is step function and the second one is kept zero? B . Porter ,fEngland) : The command input vec t nr { l, O} E Cv and i s therefore fa it hf¥lly tracked by the plant output vector / y . ' Y2 ' ; the interac:t i on between y , (t) and Y2 ca n be made as small as des i red by i ncreasi n g t he gain parameter .
d.)