Cone Algorithm of Identification for Adaptive Robust Control

Copyright (l;) IFAC System Identification, Copellhagen, Denmark, 1994

Cone Algorithm of Identification for Adaptive Robust Control V.F. SOKOLOV Univer"ity of Syktyvkar, Department of Mathematics, 55, Oktyabr"kil pro"pekt, Syktyvkar, 167001, Ru""ia.

In this paper a new set estimation algorithm, namely, the cone algorithm is proposed. This algorithm constructs the estimates of unknown parameters through the minimization of a preference function which is determined by a global control performance criterion. The additive disturbance is assumed to be bounded but no upper bound of its absolute value is known. It is supposed also that for the systems with the known parameters some parameter dependent regulators are given that ensure performance guarantee which is proportional to the least upper bound of the additive disturbance. The function of the cone algorithm is to ensure the same performance guarantee in adaptive case. It is also demonstrated that the cone algorithm can ensure the optimal performance of adaptive robust control by taking a first order system as an example.

Abstract. Study of performance guarantees and their improvement are among the main topics in theory of ~aptive robust control. Systems subjected to multiplicative and additive deterministic disturbances are considered in this paper. A new cone algorithm is proposed for identification under indirect adaptive control. The cone algorithm uses set estimates of unknown parameters and allows to improve performance guarantees of adaptive control. It is supposed that the robust control of systems with known parameters is given.

Key words. Adaptive control; robust control; estimation; performance guarantees.

1

Introduction

The problem of adaptive control of discrete linear system subjected to multiplicative and additive deterministic disturbances is considered. The multiplicative disturbance can be generated by parameter perturbations and/or by unmodeled dynamics. The additive disturbance can be generated by external noise and/or by measurement disturbances. In view of poor a priori information about disturbances the system is not identifiable.

2

Problem statement

We consider a single-input single-output system

a(q)y(t)

= qkb(q)u(t) + w(t) + v(t), tEN,

(1)

where y(t), u(t), w(t), and v(t) are output, control and disturbances, respectively, q - the backward shift operator (qz(t) = z(t - I)), k - the known time delay,

Until recently most attention has been concentrated on ensuring the stability of the closed-loop system. Some performance guarantees of adaptive controls were given in Lozano-Leal (1989), Praly et al. (1992), Radenkovic and Michel (1992), Sun and Ioannou (1992), and Ydstie (1992). Although gradient and least-squares algorithms are the most commonly used estimators, performance guarantees of these are in general an unsolved problem.

b( q)

= ba + b q + ... + bm qm . l

The unknown parameter vector of system (I)

belongs to the known set

e,

8 EeC R n +m + l Alternative set-membership approach to system identification (Bayard et al., 1992; Kosut et al., 1992; Smith and Doyle, 1992) remains clearly descriptive to date and there are no strict results of its applications to control design.

.

(2)

A priori information about the additive disturbance v(t) is in the inequality p := sup t

575

Iv(t)j < +00,

(3)

information. In view of these two aspects the stated problem is unsolved in adaptive control literature. The property of the form (9) of nonadaptive LQ regulators can be found in Sun and Ioannou (1992) (lemma 2.1)' where it was noted that this property has no analogy in adaptive case. It is in contrast to the stated problem.

where scalar p is unknown. A priori information about the multiplicative disturbance w(t) is in the inequality

(4)

Iw(t)1 :5 61l/J(t - 1)1 where

l/J(t - 1)

= (-y(t -

1), .. . ,-y(t - n), Remark 2.2. In fact, regulator (7) ensures a stronger form of (9) with I< = J«O, 6),

u(t - k), ... ,u(t - k - m)f, Il/J(t - 1) I denotes the Euclidean norm of l/J( t - 1), and 6 is minimal possible and unknown. The disturbance w(t) describes both parameter perturbations and unmodeled dynamics. A control of system (1) is realized by regulator

o-r(q)u(t)

= I'dq)y(t),

J«B, 6)

(5)

lim sup ly(t)1

1-00

=

+ rlu(t)l,

3

y(t)

r(t)

lim sup

ly(t)1

+ rlu(t)l:5

< +00.

= y(t) -

(10)

OT l/J(t - 1)

(7)

(11)

At time instant t new information about the unknown vector ~ consists in the inequality

(8)

]{p

+ w(t) + v(t).

A new identification algorithm which is more adequate to the stated problem may have the following reason. Define an extended vector ~ of unknown parameters of system (1) as

Iy(t) - OT l/J(t - 1)1:5 61l/J(t - 1)1 + p

where ~ : 0 >---> R is the known bounded function, the solutions of control system (1) and (7) satisfy the inequality 1-00

1)

they can not ensure inequality (9).

is given such that for 6, ~(O),

= OT l/J(t -

Gradient and least-squares algorithms are the most commonly used estimators. Based on the minimization in the proper sense of the residuals

Nonadaptive robust control. We assume that for system (1) with known parameter vector o E 0 the regulator

0:5 6 :5

J(o-(q, 0), I'(q, B))

Identification problem

r> 0,

= I'(q, O)y(t),

sup IV(I)I~l

For the purpose of identification we rewrite equation (1) in the form of linear regression

(6)

where y(t) and u(t) are the solutions of the control system (1) and (5) with some initial conditions. In what follows only the stable closed-loop systems will be designed. Then performance index (6) will not depend on the initial conditions that may be arbitrary. The values of index (6) depend on disturbance sequences wC), v(·) omitted in notation.

o-(q, O)u(t)

sup IW(I)I~61,p(I-l)1

The problem of ensuring the inequality (9) with parameter dependent constant I< in adaptive case is significantly more complex than the stated problem and is considered in Sokolov (1994).

whe.r.e UJe time-dependent polynomials in q, QI (.) and I'd·) are to be determined at each time instant. Performance of regulator (5) is determined by the performance index J (0- d .), 1'1 (-))

=

(12)

which is obtained from (3),(4). This inequality selects a nonconvex set in the space of parameters {. Without going into details, which can be found in Sokolov (1994), we emphasize that system (1) with arbitrary parameters 0 and 6 can produce the observed data y(t), y(t - 1), ... and u(t - 1), u(t - 2), ... due to appropriate additive disturbance v(-) or, in other words, due to large enough value of the parameter p.

(9)

Problem formulation. Find regulator (5) such that the solutions y(t) and u(t) of closed-loop system (1) and (5) satisfy inequality (9) in spite of unknown vector 0 and unknown scalars 6 and p.

Remark 2.1. Two aspects of the problem are noteworthy. First, no upper bound of the additive disturbance is assumed to be known. Second, the adaptive regulator must ensure inequality (9) with the same constant J( as nonadaptive robust regulators given complete parameter

The main idea of the new identification algorithm is to choose such an estimate {(t) which ensures the minimal admissible value of p(t). To get a recursive algorithm we .make some simplifications. First, we weaken information (12) 576

Otherwise

as follows. Let < (t - 1) be the estimate of the actual vector < at time instant t - 1 and

j E arg

7J(t) = sign(y(t) - «t - l)T t/J(t - 1)).

(20)

f[T(t)+

min {l:ld(I)T{. {t»O}

~dt)(e(t) - T(tf d(t))/d(t)T~I:(t)),

From (12) it follows

(y(t) - OT t/J(t - I))7J(t) ~ e51t/J(t - 1)1

+ p.

~dt + 1)

(13)

={

~dt), if k C

"I:

= j,

(t) _ c.(t)d{trh{t) if k "J

d{IJT{J{t) ,

-J.

J'

r,

Defining an extended regression vector 4J(t - 1) as

(21 ) T(t+I) = T(t)+~j (t)(e(t)-T(tf d(t))/d(t)T~j(t).

= (7J(t)t/J(t - If, It/J(t -

If T(t + 1) E U, then the step of the cone algorithm at the time instant t + 1 is finished. Otherwise the procedure (19)-(22) is repeated cyclically on i E {l, ... r} with the substitution off i ,1'i instead ofd(t),e(t), respectively, until T(t + 1) E U.

r/J(t - 1)

(22)

1)1, If,

we can rewrite (13) in the form

e r/J(t -

1) ~ y(t)7J(t).

(14)

The second idea of the new identification algorithm IS to use set estimates of unknown parameters. Namely, at each time instant t we remember the cone C(t) composed of exactly dim< = n + m + 3 of inequalities (14). Strict description of the algorithm is given in section 4.

4

For solution of the problem stated in section 2 we specify the preference function f as

f(T)

= TT d.,

d. E RI

(23)

Algorithm (19)-(23) has a simple interpretation. Proposition 4.1. The vector T( t + 1) is a solution of linear programming

Cone algorithm

Consider an infinite system of linear inequalities

TT d(t) ~ eft), tEN.

(24)

(15 )

At

=

C(t

=

= [fl, ... ,fr)T is an r

x I matrix and

( 17)

5

AI:
= n{TIT T d(t)

un 5

#- 0,

~ eft)};

4) det[6(O), ... ,~I(O)];f O. Then for each c > 0 the sequence of the cones C(t) defined by (19)-(23) has the following properties: i) "It EN V C C(t) ; ii) "It E N T(t) E U ; iii) the sequence {J(T(t))} is non decreasing and for all t f(T(t)) ~ minf(T); (26)

(18) Here T(t) is the apex and ~dt), ... ,~I(t) are the edges of the cone C(t). Cone algorithm. Given C(O), for all tEN

= C(t),

(25)

2) U C C(O) and T(O) E U; 3) cI!'~I:(O) ~ 0 for k E {I, ... ,l};

1:=1

+ 1)

+ 1)),

tEN

I

C(t

T(t

Theorem 4.2. Suppose that: I) the set U is bounded and V := where

Algorithm described below determines recursively a sequence of cones C(t) of the form

+L

-

In view of proposition 4.1, algorithm (19)-(23) was called in Sokolov (1993a) as recursive linear programming (RLP) algorithm. The following theorem shows the purpose of the algorithm.

be a given function called preference one. Finally, let c be some positive constant called the parameter of the cone algorithm.

C(t) = {T E RIIT = T(t)

+ 1) = conv(A t

where conv(A) denotes the convex hull of a set A

Let

f : U t----+ R

nu

~ eft)}

and

with respect to vector T (TI •... ,Td T . Here d(t) E RI and eft) E R. Let U be a polyhedron U {T E R'I fT ~ 1'}, (16) where f

= C(t) n {TIT T d(t)

if T(tf d(t) ~ eft) - cld(t)l. (19)

rEV

577

iv) if the vectors (dt),t E N,k E {I, ... ,l} are uniformly bounded, then for large enough t

C(t+l)

= C(t) and r~d(t) 2: e(t)-cld(t)1.

5) the initial cone C(O) is chosen in accordance with conditions 2)-4) of theorem 4.2; 6) the vectors (1:(t), tEN, k E {I, ... , I} are uniformly bounded. Then for each c S c./V2 the solutions of adaptive system (1),(19)-(23) and (28)-(30) satisfy the inequality

(27)

where r 00 is the apex of the limit cone Coo. The theorem can be proven analogously to Sokolov (1993b).

lim sup ly(t)1

1_+00

Remark 4.3. Inequality (26) is the key for solution of the problem stated in section 2. The use of preference function (23) with d.

= (0, ...0, If

(28)

allows us to guarantee the inequality p(t) S p for the magnitudes of additive disturbances of the B1
1](t)

O'(q, Ooo)u(t)

lim sup

To use the cone algorithm for estimation we set / = n + m + 3,

1-+00

+ rlu(t)1 S J«poo + c).

(33)

Remark 5.2. A decrease of the parameter c: brings inequality (31) closer to inequality (9) but makes the convergence of estimates «t) worse. Remark 5.3. Condition 2) of theorem 5.1 is not restrictive and means that control systems (1) and (7) have some known uniform reserve of stability. Remark 5.4. More restrictive condition 1) of theorem 5.1 can be replaced by the conventional assumption of convexity of the set n. We have avoided it in order to simplify the presentation of the cone algorithm.

I) at time instant t a control of system (1) with ,he extended parameter vector ~ = (OT, 6, pf is 'ealized by the regulator

= ,B(q,~(t))y(t);

Iy(t)/

V2c < c.

<

= n x [0,+00);

O'(q,~(t))u(t)

= f3(q, Ooo)y(t).

Since the actual vector
(29)

Theorem 5.1. Suppose that: 1) the set n = {(O, 6)10 E e, 0 S 6 S ~(O)} is a bounded polyhedron; 2) there exists c. > 0 such that nonadaptive robust regulator (7) ensures inequality (9) for each 6 E [0, ~(O) + c.]; 3) the estimate (( t) is defined as the apex of the ;one C(t) determined by (19)-(23) with param~ters (28),(29), and

U

- ~~ 1/J(t - 1)),

From condition 2 and the inequality we have

Adaptive control

= y(t + 1)1](t + 1).

1

ly(t)-O~1/J(t-l)1 S (6 00 +nl2)I1/J(t-l)I+Poo+c (32) Now inequality (32) can be interpreted as follows. For large enough t the sequences y(.) and u(-) may be considered as the solutions of the control system (1) with the extended parameter vector
The theorem can be proven analogously to Sokolov (1993b).

e(t)

= sign(y(t)

S V211/J(t)1 +

the above inequality can be rewritten in the form

Theorem 4.5. Given conditions 1)-4) of theorem 4.4 and / S 3, algorithm (19)-(23) ensures (27) for large enough t.

= rjJ(t),

Proof. We may treat the set U in condition 3) as bounded considering only those its elements the last components of which are not greater than the actual value of p. It is possible since this inequality is ensured for the estimates ~(t) due to property iii) from theorem 4.2 and determination (28). Using now the property iv) from theorem 4.2, we have ~(t) = ~oo := (O'!:o,6 00 ,Poof and

for large enough t. Since IrjJ(t)/ and

u.

d(t)

(31)

~~rjJ(t - 1) 2: y(t)1](t) - clrjJ(t - 1)1·

Remark 4.4. The finite convergence of the cone algorithm is proven under undesirable additional assumption of uniform boundness of the vectors (dt). This assumption can be substituted for uniform boundness of the vectors r(t) including intermediate calculations of vectors r(t) rt. Most likely both of these assumptions are only technical. The following theorem confirms partially this supposition.

5

+ rlu(t)1 S K(p + c).

(30) 578

6 Other possibilities (example of suboptimal adaptive robust control)

and the identification algorithm calculates recursively only 4 vectors of dimension 3 (3 edges and the apex of cone). The identification process is clearly illustrated by the picture in the subspace of parameters (o,p). The line L on the picture is the boundary of the half-plane (14). The bold lines represent the cones C(t - I) and C(t) with the apexes ~ (t - 1) and €(t), respecti vely. The constants Ci determine the level curves of function (38).

In remark 2.2 it was noted that the constant K from (9) really depends on parameters 9 and 6 and is very sensitive with respect to the latter. To improve performance of adaptive control it is necessary to take this fact into consideration. Although gradient and least-squares algorithms do it by the indirect route , performance guarantees of these is still an open question. In this section we give a simple example of adaptive robust control which ensures the best performance using the cone algorithm. Consider the system

y(t)+ ay(t - 1)

= u(t -

1)

+ w(t) + v(t),

tEN (34)

with

Iw(t)1

:s 6ly(t -

1)1. sup Iv(t)1 t

= P:S +CXl.

(35)

The parameter vector ~ = (a, 6, pf is unknown. Let a performance index be J =

lim sup

Iy(t)l.

1-+00

Proposition 6.1. Nonadaptive robust regulator u(t) = ay(t) (36)

6.

ensures the inequality

J < -p- 1- 6

From the picture one can see that a small output y(t-l) may only increase p(t) and decrease o(t), whereas a large output y(t - 1) causes the opposite. Indeed, the vector l/J(t - 1) is orthogonal to the line L. Then the line L is nearly orthogonal to p-axis in the case of small y(t -1) and is nearly parallel to the p-axis in the case of large y( t - 1). The apex of the new cone, that is, the new estimate ~(t), has therefore a larger p-component and a smaller 6-component than ~(t - 1) in the first case and the opposite in the second case.

(37)

:s

for each a E R, p ~ 0, and 0 6 < 1. This regulator is optimal in that the right-hand side of (37) is minimal possible for fixed values of a, 6, and p. The proof of the proposition can be established by the direct presentation of solutions of the time-variant system

y(t)

1 6

The formal description of the adaptive control law is following. Let E E (0,1) be a chosen parameter of the cone algorithm,

= o(t)y(t - 1) + v(t) 10(t)1 :s 0 and Iv(t)1 :s

with arbitrary p. So, the constant K(9, 0) from remark 2.2 is equal to

1/(1 - 0). Remark 6.2. Note that the level surfaces of the function p (38) /(0 1 - 0

7)(t)

= are the planes p + Co = C, C > O.

4>(t - 1)

This fact allows one to use the cone algorithm with the prefdim~ 3 erence function (38). In this case I

=

= sign(y(t) -

u(t - 1)

= (-y(t -

+ e(t

- l)y(t - 1)),

1)7)(t), Iy(t - 1)1.1f.

Then

=

u(t) 579

= e(t)y(t),

(39)

~(t)

={

~(t - 1), if ~(t - I)T ,p(t - 1)

8

>

(y(t) - u(t - 1))fJ(t) - €I,p(t - 1)1, ~(t -1) + Ajdj(t -1), otherwise, (40)

) = arg {kld.(I_I)T mm -I »O} Tk = ~(t - 1) + Akdk(t - 1), k E {I, 2,3}, <1>(1

=

ddt - 1), if k = j or dk(t - I f,p(t - 1) 0, ddt - 1) - dj(t -1)dk(t - Ifx ,p(t - 1)/dj (t - I f,p(t - 1), . otherwise.

=

Lozano- Leal, R. (1989). Robust Adaptive Regulation Without Persistent Excitation. IEEE Trnns. Aut. Contr., AC-34, 12601267.

1 Tt

Here is the ith component of the vector Tk and d.(t)(k = 1,2,3) are the edges and ~(t) is the ap~x of the cone C(t). The initial values are

= (-a.,O,Of,ddO) = (1,o,of, d2 (0) = (0,1, O)T, d3 (0) = (0,0,1 f.

~(O)

f/. u

= {~llal:S a.,O:S o:s o.,p ~ O},

the vector ~(t) is to be modified as it was indicated in section 4.

:s

:s o.

Theorem 6.3. Given lal a. and 0 < 1 with known constants a. and 0., the adaptive

control law (39) ,( 40) with arbitrary positive € < 1ensures the inequality

o.

lim sup 1-+00

ly(t)l:S

-p-

1- 0

+ M€,

Smith, R.S., and J .C. Doyle (1992) Model Validation: A Connection Between Robust Control and Identification. IEEE Trans. Aut. Contr., AC-37, 942-952. Sokolov, V.F. (1993a). Recursive linear programming in adaptive minmax control problems. Izvestlya Akad. Nauk, Tekhn. I
where M is some €-independent constant, for the output of system (34). The theorem can be proven similarly to theorem 5.1. We emphasize that theorem 6.3 has no need of assumption 6) of theorem 5.1 due to theorem 5.2. So, the cone algorithm adjusts the estimates of the magnitudes of additive and multiplicative disturbances optimally for control objective.

7

Praly, 1., S.-F.Lin, and P.R.Kumar (1989) A robust adaptive minimum variance controller. SIAM 1. Control and OptimIzatIOn, 27,235-266. Radenkovic, M.S., and A.N.Michel (1992). Robust Adaptive Systems and Self Stabilization. IEEE Trans. Aut. Contr., AC-37, 1355-1369.

If in (40)

W)

Bayard, D.S., Y.Yans, and E.Mettler (1992). A Criterion for Joint Optimization of Identification and Robust Control. IEEE Trans. Aut. Contr., AC-37, 986-991. Kosut, R.1., M.K.Lau, and S.P.Boyd (1992). Set-Membership Identification of Systems with Parametric and Nonparametric Uncertainty. IEEE Truns. Aut. Contr., AC-37, 929-941.

).. _ (y(t) - u(t - 1))fJ(t) - ~(t - I f,p(t - 1) k dk(t _ I)T,p(t _ 1) ,

ddt)

References

Sokolov, V.F. (1994). Closed-loop Identification for the Best Asymptotic Performance of Adaptive Robust Control. (Proceedmgs of the present symposIUm). Sun, J., and P.Ioannou (1992). Robust Adaptive LQ Control Schemes. IEEE Trans. Aut. Contr., AC-37, 100-106.

Conclusion

Ydstie, B.E. (1992). Transient Performance and Robustness of Direct Adaptive Control (1992). IEEE Trnns. Aut. Contr., AC-37, 1091-1105.

The new cone algorithm for robust system identification have been proposed. The main problem was to adapt to the unknown least upper bound of additive disturbance. It was also shown by a simple example that the adaptive robust control based on the new estimation algorithm can ensure the control performance which is arbitrarily close to the optimal one given complete parameter information. The cone algorithm allows one strictly to substantiate performance guarantees of adaptive controls. 580