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Fast Adaptive Control of Time-varying Systems
Copyright © IFAC 12th Triennial World Congress. Sydney, Australia, 1993
FAST ADAPTIVE CONTROL OF TIME-VARYING SYSTEMS Zbang Zbiyong*, Wang Sbifu*, Kang Jingli** and Fang Cbongzbi* *Department of Automation, Tsinghua University, Beijing 100084, PRC **Firsr Department, Beijing Institute o/Technology, Beijing 100081, PRC
Abstract.For time- varying sy.tems, the adaptive control algrithms became very complicated and the processing speed is a key problem. To speed up the computation of time- varying systems in which the parameters are usually described by time polynimials, this paper presents a new fast adaptive identification and control scheme. By means of auto regressive expansion of discrete time polynomial, the identification can be simplified to estimate coefficients of the highest power in stead of every coefficients. So that the amount of computations at each sample period is reduced greatly . Meanwhile, through the parallel processing technique, the simplified algorithm is divided into two parts which can be excuted concurrently to improve the control speed further. Finally, a simulation example which illustrates the efficiency of fast adaptive scheme is given . Keywords.Adaptive control; Time- varying systems; Identification; Autoregressive expansion; Parallel processing.
I.INTRODUCTION
the algorithms.Those modified forgetting factor methods mentioned above are, in essence, techniques for managing the covariance matrix of the estimator to maintain the alterness to time-varying parameters. But this is unfavourable to keep the identification stable and accurate.
An important requirement for controlling time-varying systems is the controller response speed . This is also considered in the adaptive control for time-varying systems. An adaptive control scheme is generally composed of parameter identification algorithms and control law. To make the standard recursive least square (RLS) estimator have the ability to track parameter changes, a lot of parameter tracking modifications based on so-called forgetting factors have been presented. Fortescue et al.(t981) suggested tuning the variable forgetting factor (VFF) according to the current prediction error. Kulhary (1984) and Hagglund (1983, 1985) developed the directional forgetting factor to perform a selective amplification of covariance matrix P(t). Canetti and Espana (t 989) studied the convergence of RLS algorithm with variable forgetting factor. Bittanti (t 990) analysed the convergence of directional forgetting algorithms. Goodwin (t 981) used the resetting technique to improve the algorithm alterness to parameter changes . J aneck (t 988) presented the time-varying but bounded gain matrix to guarantee the tracking ability and stability of
To solve this problem, a modified method is to expand the time - varying parameters to the discrete time functions, for example, Taylor, Chebyshev and Legendre polynomials (Chou and Horng, 1985; Hwang and Chen, 1985; Deng, 1988). The coefficients of those functions may vary more slowly than the original parameters, and it becomes easier to consider the tracking lbility and stability at the same time . But the expansion method brings a new problem. For seriously time - varying characteristics, the order of polynomials increases dramatically, and the computation load increases exponen tialy . To decrease the computation load at each sample period and enhance the speed of adaptive control, the new autoregressive expansion method of discrete - time polynomial for time - varying parameters was developed. By means of this method the moving identifica305
tion algorithm needs only to estimate the coefficients of the highest power of discrete time, which reduces the amount of calculations considerably, especially for rapidly time-varying terms. A modified variable forgetting factor is discussed which can ensure the tracking capability and convergence to the estimator simultaneously. Based on those and input-weighted minimum prediction error control law, a new adaptive controller was developed for time-varying systems in the presence of bounded disturbances.
Similar to the differentiation in continuous case, there is a concise form for polynomials of discrete time t. Let the discrete-time polynomial be T(/)=a o +a 1 I+ ••• +a N t
(5)
Lemma 2. The N-th order backward difference ofT(t) in (5) is (t) = N!a N
VN /
(6)
Proof. (omitted) From the lemmas we can construct the antoregressive expansion of T(t). Before the new theorem is presented, we consider few simple polynomials. For T(t) = ao+a 1t, N = I, then T(t)=T(t-l)+a 1• or T(t)=2T(t-1) -T(t-2). For T(t)=ao+a 1t+a 2t 2 • N=2. then T(t)=2T(t-l)-T(t-2)+2a2, or T(t)= 3T(t-1) - 3T(t-2)+T(t-3). For T(t) = aO+a 1t+a 2t 2+a 3t 3 • N = 3, then T(t) = 3T(t-I)-3T(t-2)+T(t-3) +6a 3• or T(t) = 4T(t-I)-6T(t-2)+4T(t- 3) -T(t-4).
Recently, parallel processing technique received considerable attention (Irwin, 199 I) which is considered to be an efficient way to cope with processing speed. In this paper, The above algorithms are formed in parallel structure including two parts: Part I : parameters innovation and control law computation; and Part 2: covariance computation. Through parallelism, the adaptive control scheme can be speeded up nearly two times.
It is easy to see that: (i) Autoregressive expansion is simple and easy for recursive calculation and (ii) The number of unknown coefficients has been reduced.
The contents of this paper is arranged as follows. The autoregressive expansion of polynomials is given in Section 2. Described in Section 3 is the fast algorithms of identification and adaptive control. The parallel structure of the fast adaptive control is given in Section 4. A simulation example is presented in Section 5, and the concluding remarks follow in Section 6. Finally Section 7 is the references.
The number of unknown coefficients in the autoregressive expansion is called the degree of freedom. The expression containing the unknown coefficient of the highest power term is called the autoregressive expansion of freedom degree one and the expression without the coefficients of the highest power term is called the autoregressive expansion of freedom degree zero. Theorem 1. N -th order discrete time polynomial function T(t) can be expressed as the autoregressive expansion of freedom degree one
2. AUTOREGRESSIVE EXPANSION OF POLYNOMIAL The definition of backward difference of general discrete-time function f(t) and some lemmas are given first. Definition 1. The first order backward difference of f(t) is
N
T(t) =
\1 f (t) = f (t) - f (t - 1) (I) the second order backward difference is 2 \1 f (t) = \1 f (t) - \1 / (I - I) (2) and the m-th order backward difference is \l"'/(t)=\l",-I/(/)-\1",-I/{t_1)
N
k~1
+ N!
(-
I)H 1
k!(:~ k)!
T(t - k) (7)
• aN
where aN is the coefficient of the highest power N term t of T( t} ,or the antoregressive expansion of freedom degree zero
(3)
N+ 1
T(t) =
L (- I)H
(N + 1)! T(t - k) (8) k!(N + 1 - k)! Proof. Equation (7) can be obtained from Eqns. (4) and (6). From Eqn. (7) we have 1
k-I
Lemma 1. The m -th order backward difference of f(t) has the following form
\1'" / (t) =
kto (- I)h k!(mm~ k)! /
N
(t - k)
T(t - I) =
(4)
L (- 1)H
1
k-I
- k - 1)
Proof. (omitted) 306
+ N! •
aN
N! T(I k!(N-k)! (9)
Subtracting (9) from (7) yields
coefficients of the highest power terms. Then the problem of identifying time-varying parameters aj(l). bJt) is reduced to the problem of identifying time-invariant parameters (Xj.
T(t) - T(t - 1) = NT(t - 1) N-I
_ L(-1)HI
Pj
+ N! ] N! k!(N - k)! • (k + 1)!(N - k - 1)! • T(t - k - 1) - ( - 1) N + I T( T - N - 1) Equation (8) follows.
.
[
Define the (m+n+ I)-dimensional parameter vector to be estimated (J=(-o: l' -0: 2' •• ' ,- 0 :
(10)
11'
T
PO,P I , •••• Pm) (18) and (m+n+ I)-dimensional measurable information vector
3. FAST IDENTIFICATION AND CONTROL ALGORITHMS
= [Yet). yet - 1) ••••• yet - n + I).
qJ(t) T
u(t). u(t - 1) ••••• u(t - m)]
Consider the time-varying system
+ 1) = B(t.
A(t. q -I)y(t
q -1)U(t)
+ ~(t)
By using the RLS with variable forgetting factor to estimate (J(t). then the recursive identification algorithm is
(1 I)
where yet). u(t) and ~(t) are system input. output and bounded disturbances. respectively. A(t. q -1) and B(t. q - 1) are polynomials in the one-step backward-shifting operator and discrete time t. and -I
A (t. q
B(t. q
I-I
)=bo(t)+bl(t) q
-\
f
j(k)Qj(t - k»)y(t - i
k-I
+1)
- £(i:
(12)
•
= yet + 1) - qJ(t) T O(t - I)
- I (i:
+ a 2 (t)q
+···+a (t)q-' -I
t(t)
-1-2
) = 1 + a I (t)q
(19)
J- •
+ ...
r(t) (13)
+b (t) q-'"
= qJ(t) T pet - 1)qJ(t)
(21)
K(t) = pet -( 1)qJ(t) + r(t) pet) O(t) = O(t - 1) + K(t)t(t) pet) = pet - I) pet - 1)qJ(t)qJ(t) T pet - I)
'"
Take assumptions for system (1 I): (A I) The upper bounds of the orders n and m are known. (A2) sup 1~(t)I
pet) pet) = 0 •
{
:f
-(
+ r{t)
It(OI <
(22)
(23) (24)
2L\
t(t)} . max { po.l - r(t)r • Po e(o. 1). otherwise (25)
where pet) is the time variable forgetting factor. is a prespecified constant. Qj (t). 6 j(t) are calculated according to Eqns. (6) and (7). Po=P(-I)=oI. 0>0. I is the identity matrix of dimension m+n+ 1.
According to Theorem 1, aj(t). bj(t) can be rewritten as autoregressive expansion of freedom degree one: a.(t)= ~ !,(k)a.(t-k)+(X. I L..., I I
g /k)6P - k»)u(t - j)(20)
k-I
r
(14)
k -I
L
b/t)=
g/k)b/t-k)+P j
Given the desired output y. (t) at time t. the adaptive control law can be obtained by minimizing the following cost function (Goodwin. 1984): 1 • 2 J(t + I) = 2 [y(t + 1) - y (t + 1)]
(15)
k-I
where
(X . I
= N .! •
p.
aN'
I
I
)
= M .! • b
AI
)
I
•
= 1.2.···.n. j= Q.l.···.m are unknown and
HI
f;
Nj! k!(N . _ k)!
A + 2 [u(t) -
(16)
I
are
known.
a N,
'
b AI are
the
u(t - 1)]
2
(26)
where A is the input weighting coefficient and yet + 1) is the predicted output of (3).
unknown
I
307
According to the minimum prediction error control law, the new adaptive control scheme is •
T
(28) Sp = t, / tp Ep is defined as the average utilisation of the p processors expressed in percentage : (29) E p = (S p / p) * 100%
~
Y (t + 1) = cp (t)e(t - 1)
+ i(
I
I (~
+
j- 0
f
/k)fi/t-k))y(t-i+ 1)
5. SIMULA nON EXAMPLE
g }.(k)b(t - k))u{t - J) }
k - 1
+ ).(u(t) - u(t - 1)) where fi(t), bet) and I j
e(t -
The simulated plant is similar to the example of Deng (1988)
(27) 1) are given in
yet
(14)-(17).
+ 1) = -
a I (t)y(t) - a /t)y(t - 2)
+ b 0 (t)u(t)
The amount of calculations required in the fast algorithms is much reduced from O(p+n+m)3 0 f the original algorithms to O(n+m)3.
+ D(t)
(30)
The real parameters and disturbance are a I (t) = 2
= - 4 - 0.15t + O.0013t b o(!) = - 3 - 0.02t
a 2(t)
4. PARALLELISM OF FAST ADAPTIVE CONTROL
2
(31)
ID(!)I ~ 0.05
The reference input is a square wave of amplitude 2.5 and of period 10. In the adaptive control algorithm (20)-(25) and (27), the structure parameters are n = 2, m = 0, NI = 0, N 2 = 2, Mo = 1. The simulation result is shown as in Fig 2. It can be seen that the approximation error is small. But the amount of calculations required is much reduced.
According to conditional adaptive control algorithms, (20)-(27) are executed sequentially . Now reconstruct them as following two parts executed concurrently in Fig.l. In Fig. I, the two parts need to exchange neccessary information. (j(t - 1) is calculated in Part I, then tranformed to Part 11.
4
Part 11
Part I
y(t+ I), cp(t) , e(t - I) cp(t) ,P(t-I) r y(t+I),cp(t),(1(t - I). P(t- 1) t (20) (20)
t
....
~
I
?
,>:0++-I+r--l~+~I--..H--.4Ih4l-~1-r-4h
100
t
(23)
(25)
t
~
(27)
(24)
-4
Fig.1 Parallel Algorithms Driven Scheduling 10
The parallel algorithms divide adaptive control into two parts--the part of parameters innovation and input computation and the part of covariance computation. The excution time in two microprocessors are arranged very near. Thus, the processing speed of adaptive control can be improved nearly two times.
-10 Fig.2. The Curves of Input and Output of the Time-varying System
In evaluating the performance of a parallel system, the measures of particular interest are the speed up ratio Sp and efficiency Ep . Sp is defined as the ratio of the execution time ts on a single processor to the execution time 1p on p processor:
The experiments show that the parallel adaptive control for a controlled plant with 4th regression model spands 22ms compared that conditional sequential execution time is 42ms . Sp = 42 / 22 = 1.9999 ~ 2, Ep = 99 .99%. It is il308
lustrated that the parallel system has good performances.
Control, 41, 135-144. Deng, Li Zheng (1988) . Discrete-time adaptive control for time - varying systems subject to unknown fast time - varying deterministic disturbances, lEE Proc ., Pt. D , 135, 445-450 . Fortescue, T . R., L. S., Kershenbaum, and B. E., Y datie (198 I) . Implementation of self-tuning regulators with variable forgetting factors, Automatica, 17,831-835 . Goodwin, G .c. , H .G., ElliotO 981), Deterministic convergence of a self- tuning regulator with covariance resetting, lEE Proc., Pt. D., 126, 19-23. Goodwin, G. c., K . S. Sin (1984) . Adaptive Filtering, Prediction and Control, Englewood Cliffs NJ:Prentice- Hall. Hagglund, T .O 983) . The problem of forgetting old data in recursive estimation, Proc. of IFAC Workshop on Adaptive S ystems in Control and Signal Processing, San Fransisco, CA. Hagglund, T.(1985)·. Recursive estimation of slowly time-varying parameters, Proc. of IFAC Symp. on Identification and System Parameters Estimation, York . Hwang, C., and M . Chen (1985). Analysis and optimal control of time-varying systems via shifted Legendre polynomials, lnt. J. Control, 41,1317-1330. Irwin, G.W. (199 I). Parallel processing for real-time control, lEE Proc., Pt. D, 138, 177-178. Kulhavy, R. and M . Karny (1984). Tracking of slowly varying parameters by directional forgetting, 9th IFAC World Congress , Budapest, 77-83 .
6. CONCLUSION
The fast adaptive control algorithm for time- varying systems subject to bounded disturbances is presented. The autoregressive expansion is used to approximate the time-varying parameters and disturbances. Then the amount of calculations, which cost much computing time with direct use of polynomials, is reduced to a great extent. The parallelism structure is used to speed up the fast adaptive control further and can be realised easily with two processors. The parallel processing technique by use of multi - processors more than two or VLSI array processor applied to adaptive control is an important subject to study in the future.
7. REFERENCES Bittanti, S.(1990) . Convergence and exponential convergence of identification algorithms with directional forgetting factors, Automatica, 26, 929-932. Canetti, R.M. (1989) . Convergence analysis of the least squares identification with a variable forgetting factor for time-varying linear systems, A utomatica, 26, 609-612. Chou,J., and I. Horng (1985). Application of Chebyshev polynomials to the optimal control of time- varying linear systems, Int . J .
309
FAST ADAPTIVE CONTROL OF TIME-VARYING SYSTEMS Zbang Zbiyong*, Wang Sbifu*, Kang Jingli** and Fang Cbongzbi* *Department of Automation, Tsinghua University, Beijing 100084, PRC **Firsr Department, Beijing Institute o/Technology, Beijing 100081, PRC
Abstract.For time- varying sy.tems, the adaptive control algrithms became very complicated and the processing speed is a key problem. To speed up the computation of time- varying systems in which the parameters are usually described by time polynimials, this paper presents a new fast adaptive identification and control scheme. By means of auto regressive expansion of discrete time polynomial, the identification can be simplified to estimate coefficients of the highest power in stead of every coefficients. So that the amount of computations at each sample period is reduced greatly . Meanwhile, through the parallel processing technique, the simplified algorithm is divided into two parts which can be excuted concurrently to improve the control speed further. Finally, a simulation example which illustrates the efficiency of fast adaptive scheme is given . Keywords.Adaptive control; Time- varying systems; Identification; Autoregressive expansion; Parallel processing.
I.INTRODUCTION
the algorithms.Those modified forgetting factor methods mentioned above are, in essence, techniques for managing the covariance matrix of the estimator to maintain the alterness to time-varying parameters. But this is unfavourable to keep the identification stable and accurate.
An important requirement for controlling time-varying systems is the controller response speed . This is also considered in the adaptive control for time-varying systems. An adaptive control scheme is generally composed of parameter identification algorithms and control law. To make the standard recursive least square (RLS) estimator have the ability to track parameter changes, a lot of parameter tracking modifications based on so-called forgetting factors have been presented. Fortescue et al.(t981) suggested tuning the variable forgetting factor (VFF) according to the current prediction error. Kulhary (1984) and Hagglund (1983, 1985) developed the directional forgetting factor to perform a selective amplification of covariance matrix P(t). Canetti and Espana (t 989) studied the convergence of RLS algorithm with variable forgetting factor. Bittanti (t 990) analysed the convergence of directional forgetting algorithms. Goodwin (t 981) used the resetting technique to improve the algorithm alterness to parameter changes . J aneck (t 988) presented the time-varying but bounded gain matrix to guarantee the tracking ability and stability of
To solve this problem, a modified method is to expand the time - varying parameters to the discrete time functions, for example, Taylor, Chebyshev and Legendre polynomials (Chou and Horng, 1985; Hwang and Chen, 1985; Deng, 1988). The coefficients of those functions may vary more slowly than the original parameters, and it becomes easier to consider the tracking lbility and stability at the same time . But the expansion method brings a new problem. For seriously time - varying characteristics, the order of polynomials increases dramatically, and the computation load increases exponen tialy . To decrease the computation load at each sample period and enhance the speed of adaptive control, the new autoregressive expansion method of discrete - time polynomial for time - varying parameters was developed. By means of this method the moving identifica305
tion algorithm needs only to estimate the coefficients of the highest power of discrete time, which reduces the amount of calculations considerably, especially for rapidly time-varying terms. A modified variable forgetting factor is discussed which can ensure the tracking capability and convergence to the estimator simultaneously. Based on those and input-weighted minimum prediction error control law, a new adaptive controller was developed for time-varying systems in the presence of bounded disturbances.
Similar to the differentiation in continuous case, there is a concise form for polynomials of discrete time t. Let the discrete-time polynomial be T(/)=a o +a 1 I+ ••• +a N t
(5)
Lemma 2. The N-th order backward difference ofT(t) in (5) is (t) = N!a N
VN /
(6)
Proof. (omitted) From the lemmas we can construct the antoregressive expansion of T(t). Before the new theorem is presented, we consider few simple polynomials. For T(t) = ao+a 1t, N = I, then T(t)=T(t-l)+a 1• or T(t)=2T(t-1) -T(t-2). For T(t)=ao+a 1t+a 2t 2 • N=2. then T(t)=2T(t-l)-T(t-2)+2a2, or T(t)= 3T(t-1) - 3T(t-2)+T(t-3). For T(t) = aO+a 1t+a 2t 2+a 3t 3 • N = 3, then T(t) = 3T(t-I)-3T(t-2)+T(t-3) +6a 3• or T(t) = 4T(t-I)-6T(t-2)+4T(t- 3) -T(t-4).
Recently, parallel processing technique received considerable attention (Irwin, 199 I) which is considered to be an efficient way to cope with processing speed. In this paper, The above algorithms are formed in parallel structure including two parts: Part I : parameters innovation and control law computation; and Part 2: covariance computation. Through parallelism, the adaptive control scheme can be speeded up nearly two times.
It is easy to see that: (i) Autoregressive expansion is simple and easy for recursive calculation and (ii) The number of unknown coefficients has been reduced.
The contents of this paper is arranged as follows. The autoregressive expansion of polynomials is given in Section 2. Described in Section 3 is the fast algorithms of identification and adaptive control. The parallel structure of the fast adaptive control is given in Section 4. A simulation example is presented in Section 5, and the concluding remarks follow in Section 6. Finally Section 7 is the references.
The number of unknown coefficients in the autoregressive expansion is called the degree of freedom. The expression containing the unknown coefficient of the highest power term is called the autoregressive expansion of freedom degree one and the expression without the coefficients of the highest power term is called the autoregressive expansion of freedom degree zero. Theorem 1. N -th order discrete time polynomial function T(t) can be expressed as the autoregressive expansion of freedom degree one
2. AUTOREGRESSIVE EXPANSION OF POLYNOMIAL The definition of backward difference of general discrete-time function f(t) and some lemmas are given first. Definition 1. The first order backward difference of f(t) is
N
T(t) =
\1 f (t) = f (t) - f (t - 1) (I) the second order backward difference is 2 \1 f (t) = \1 f (t) - \1 / (I - I) (2) and the m-th order backward difference is \l"'/(t)=\l",-I/(/)-\1",-I/{t_1)
N
k~1
+ N!
(-
I)H 1
k!(:~ k)!
T(t - k) (7)
• aN
where aN is the coefficient of the highest power N term t of T( t} ,or the antoregressive expansion of freedom degree zero
(3)
N+ 1
T(t) =
L (- I)H
(N + 1)! T(t - k) (8) k!(N + 1 - k)! Proof. Equation (7) can be obtained from Eqns. (4) and (6). From Eqn. (7) we have 1
k-I
Lemma 1. The m -th order backward difference of f(t) has the following form
\1'" / (t) =
kto (- I)h k!(mm~ k)! /
N
(t - k)
T(t - I) =
(4)
L (- 1)H
1
k-I
- k - 1)
Proof. (omitted) 306
+ N! •
aN
N! T(I k!(N-k)! (9)
Subtracting (9) from (7) yields
coefficients of the highest power terms. Then the problem of identifying time-varying parameters aj(l). bJt) is reduced to the problem of identifying time-invariant parameters (Xj.
T(t) - T(t - 1) = NT(t - 1) N-I
_ L(-1)HI
Pj
+ N! ] N! k!(N - k)! • (k + 1)!(N - k - 1)! • T(t - k - 1) - ( - 1) N + I T( T - N - 1) Equation (8) follows.
.
[
Define the (m+n+ I)-dimensional parameter vector to be estimated (J=(-o: l' -0: 2' •• ' ,- 0 :
(10)
11'
T
PO,P I , •••• Pm) (18) and (m+n+ I)-dimensional measurable information vector
3. FAST IDENTIFICATION AND CONTROL ALGORITHMS
= [Yet). yet - 1) ••••• yet - n + I).
qJ(t) T
u(t). u(t - 1) ••••• u(t - m)]
Consider the time-varying system
+ 1) = B(t.
A(t. q -I)y(t
q -1)U(t)
+ ~(t)
By using the RLS with variable forgetting factor to estimate (J(t). then the recursive identification algorithm is
(1 I)
where yet). u(t) and ~(t) are system input. output and bounded disturbances. respectively. A(t. q -1) and B(t. q - 1) are polynomials in the one-step backward-shifting operator and discrete time t. and -I
A (t. q
B(t. q
I-I
)=bo(t)+bl(t) q
-\
f
j(k)Qj(t - k»)y(t - i
k-I
+1)
- £(i:
(12)
•
= yet + 1) - qJ(t) T O(t - I)
- I (i:
+ a 2 (t)q
+···+a (t)q-' -I
t(t)
-1-2
) = 1 + a I (t)q
(19)
J- •
+ ...
r(t) (13)
+b (t) q-'"
= qJ(t) T pet - 1)qJ(t)
(21)
K(t) = pet -( 1)qJ(t) + r(t) pet) O(t) = O(t - 1) + K(t)t(t) pet) = pet - I) pet - 1)qJ(t)qJ(t) T pet - I)
'"
Take assumptions for system (1 I): (A I) The upper bounds of the orders n and m are known. (A2) sup 1~(t)I
pet) pet) = 0 •
{
:f
-(
+ r{t)
It(OI <
(22)
(23) (24)
2L\
t(t)} . max { po.l - r(t)r • Po e(o. 1). otherwise (25)
where pet) is the time variable forgetting factor. is a prespecified constant. Qj (t). 6 j(t) are calculated according to Eqns. (6) and (7). Po=P(-I)=oI. 0>0. I is the identity matrix of dimension m+n+ 1.
According to Theorem 1, aj(t). bj(t) can be rewritten as autoregressive expansion of freedom degree one: a.(t)= ~ !,(k)a.(t-k)+(X. I L..., I I
g /k)6P - k»)u(t - j)(20)
k-I
r
(14)
k -I
L
b/t)=
g/k)b/t-k)+P j
Given the desired output y. (t) at time t. the adaptive control law can be obtained by minimizing the following cost function (Goodwin. 1984): 1 • 2 J(t + I) = 2 [y(t + 1) - y (t + 1)]
(15)
k-I
where
(X . I
= N .! •
p.
aN'
I
I
)
= M .! • b
AI
)
I
•
= 1.2.···.n. j= Q.l.···.m are unknown and
HI
f;
Nj! k!(N . _ k)!
A + 2 [u(t) -
(16)
I
are
known.
a N,
'
b AI are
the
u(t - 1)]
2
(26)
where A is the input weighting coefficient and yet + 1) is the predicted output of (3).
unknown
I
307
According to the minimum prediction error control law, the new adaptive control scheme is •
T
(28) Sp = t, / tp Ep is defined as the average utilisation of the p processors expressed in percentage : (29) E p = (S p / p) * 100%
~
Y (t + 1) = cp (t)e(t - 1)
+ i(
I
I (~
+
j- 0
f
/k)fi/t-k))y(t-i+ 1)
5. SIMULA nON EXAMPLE
g }.(k)b(t - k))u{t - J) }
k - 1
+ ).(u(t) - u(t - 1)) where fi(t), bet) and I j
e(t -
The simulated plant is similar to the example of Deng (1988)
(27) 1) are given in
yet
(14)-(17).
+ 1) = -
a I (t)y(t) - a /t)y(t - 2)
+ b 0 (t)u(t)
The amount of calculations required in the fast algorithms is much reduced from O(p+n+m)3 0 f the original algorithms to O(n+m)3.
+ D(t)
(30)
The real parameters and disturbance are a I (t) = 2
= - 4 - 0.15t + O.0013t b o(!) = - 3 - 0.02t
a 2(t)
4. PARALLELISM OF FAST ADAPTIVE CONTROL
2
(31)
ID(!)I ~ 0.05
The reference input is a square wave of amplitude 2.5 and of period 10. In the adaptive control algorithm (20)-(25) and (27), the structure parameters are n = 2, m = 0, NI = 0, N 2 = 2, Mo = 1. The simulation result is shown as in Fig 2. It can be seen that the approximation error is small. But the amount of calculations required is much reduced.
According to conditional adaptive control algorithms, (20)-(27) are executed sequentially . Now reconstruct them as following two parts executed concurrently in Fig.l. In Fig. I, the two parts need to exchange neccessary information. (j(t - 1) is calculated in Part I, then tranformed to Part 11.
4
Part 11
Part I
y(t+ I), cp(t) , e(t - I) cp(t) ,P(t-I) r y(t+I),cp(t),(1(t - I). P(t- 1) t (20) (20)
t
....
~
I
?
,>:0++-I+r--l~+~I--..H--.4Ih4l-~1-r-4h
100
t
(23)
(25)
t
~
(27)
(24)
-4
Fig.1 Parallel Algorithms Driven Scheduling 10
The parallel algorithms divide adaptive control into two parts--the part of parameters innovation and input computation and the part of covariance computation. The excution time in two microprocessors are arranged very near. Thus, the processing speed of adaptive control can be improved nearly two times.
-10 Fig.2. The Curves of Input and Output of the Time-varying System
In evaluating the performance of a parallel system, the measures of particular interest are the speed up ratio Sp and efficiency Ep . Sp is defined as the ratio of the execution time ts on a single processor to the execution time 1p on p processor:
The experiments show that the parallel adaptive control for a controlled plant with 4th regression model spands 22ms compared that conditional sequential execution time is 42ms . Sp = 42 / 22 = 1.9999 ~ 2, Ep = 99 .99%. It is il308
lustrated that the parallel system has good performances.
Control, 41, 135-144. Deng, Li Zheng (1988) . Discrete-time adaptive control for time - varying systems subject to unknown fast time - varying deterministic disturbances, lEE Proc ., Pt. D , 135, 445-450 . Fortescue, T . R., L. S., Kershenbaum, and B. E., Y datie (198 I) . Implementation of self-tuning regulators with variable forgetting factors, Automatica, 17,831-835 . Goodwin, G .c. , H .G., ElliotO 981), Deterministic convergence of a self- tuning regulator with covariance resetting, lEE Proc., Pt. D., 126, 19-23. Goodwin, G. c., K . S. Sin (1984) . Adaptive Filtering, Prediction and Control, Englewood Cliffs NJ:Prentice- Hall. Hagglund, T .O 983) . The problem of forgetting old data in recursive estimation, Proc. of IFAC Workshop on Adaptive S ystems in Control and Signal Processing, San Fransisco, CA. Hagglund, T.(1985)·. Recursive estimation of slowly time-varying parameters, Proc. of IFAC Symp. on Identification and System Parameters Estimation, York . Hwang, C., and M . Chen (1985). Analysis and optimal control of time-varying systems via shifted Legendre polynomials, lnt. J. Control, 41,1317-1330. Irwin, G.W. (199 I). Parallel processing for real-time control, lEE Proc., Pt. D, 138, 177-178. Kulhavy, R. and M . Karny (1984). Tracking of slowly varying parameters by directional forgetting, 9th IFAC World Congress , Budapest, 77-83 .
6. CONCLUSION
The fast adaptive control algorithm for time- varying systems subject to bounded disturbances is presented. The autoregressive expansion is used to approximate the time-varying parameters and disturbances. Then the amount of calculations, which cost much computing time with direct use of polynomials, is reduced to a great extent. The parallelism structure is used to speed up the fast adaptive control further and can be realised easily with two processors. The parallel processing technique by use of multi - processors more than two or VLSI array processor applied to adaptive control is an important subject to study in the future.
7. REFERENCES Bittanti, S.(1990) . Convergence and exponential convergence of identification algorithms with directional forgetting factors, Automatica, 26, 929-932. Canetti, R.M. (1989) . Convergence analysis of the least squares identification with a variable forgetting factor for time-varying linear systems, A utomatica, 26, 609-612. Chou,J., and I. Horng (1985). Application of Chebyshev polynomials to the optimal control of time- varying linear systems, Int . J .
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