4SID Identification for Control

Copyright ID IFAC System Structure and Control, Bucharest, Romania, 1997

4SID IDENTIFICATION FOR CONTROL

B. Jora, C. Popeea, D. Popescu Politehnica University of Bucharest, Department of Automatic Control and Computers , 313 Splaiul Independentei , 77206, Bucharest 6, Romania , Fax: 4101044, Tel:+4.01.4100.400 , E-mail:[email protected]

Abstract: Many applications of control system design based on input-output measurements allow an iterative scheme of repeated identification and control with progressive refinements of the designed controller. In dealing with the interplay between identification and robust control design , a new alternative is given to aspects of the Subspace-based State Space System Identification (4SID) method under closed-loop experimental data for numerical robust control design. Copyright© 1998 IFAC Keywords: System identification; state space; interplay identification - control; numerical computing.

l.

Let us assume that an (approximate) state-space model Po for the plant is available from past (already known) data. At the current stage t of the adaptive procedure a new control u(t) is computed, e.g. by using a predictive LQ control algorithm and the corresponding actual plant output y(t) is measured. Now, by using these new inputoutput data, the plant model is updated (according to a specific sequential algorithm yet to be discovered) and the adaptive procedure is reiterated for the next (real-time) stage. Moreover, after this initial "training" phase , which is " computerdriven" as described above and hopefully produces "good" closed-loop results (in the user 's opinion), a state-space model for a conventional linear controller C may be computed and updated from the same available input-output data. Finally, if the controller's simulated outputs are close to the computer generated control inputs during a sufficiently long "testing" phase, then the controller is connected to the plant's terminals and the computer is removed . In this way a conventional control loop is obtained. The controller can be designed for an initial unknown plant during a learning process, centered on sequential 4SID and predictive LQ control algorithms.

INTRODUCTION

In the last few years Subspace-based State Space System IDentification (4SID) methods have been developed in order to compute reliable linear dynamic models for complex multivariable plants directly from input-output measurements. In contrast with the classical identification methods , e.g. prediction error method (PEM), 4SID methods do not need special (" canonical") parametrization and are non iterative ("one-shoot"). As a result , they do not suffer from typical disadvantages of non-linear optimization algorithms (divergence, local minima, sensitivity to initial approximations, etc.). On the other hand, the interplay between the identification and control has been deeply investigated and some iterative schemes have been proposed for solving a combined (closed-loop) identification and control design problem by using PEM and LQ or Hoo-type robustness criteria, respectively, see Astrom and Nilsson (1994); Van Den Hof and Schrama (1995); Gevers (1995); Zang et al. (1995). In this paper an adaptive procedure for solving the combined control-identification problem is presented which uses 4SID methods for plant identification. The required (closed loop) input-output data are real-time generated under computer's supervision as follows.

Our paper develops a 4SID method for iterative schemes dedicated to robust control design in closed-loop systems. We justify this option by the computing economy in the stage of identification

117

when the robust conditions are not guaranteed.

2.

po -

=I

Po

= P +~,

(1)

Po

= (1 + ~)p

(2)

I~

P~ 1<1,

1+ PC

(5)

\fwEIR

if the nominal system (p , C) is stable and ~ is stable as well. The relation (5) is similar to

~.

Let denote the real closed-loop system by (Po , C) and the nominal closed-loop system by (p, C). The controller must guarantee the achieved performance of the (Po, C) similar to the nominal performance of (p, C). In other words, for a criterion J (cost function), the values IJ (Po , C) I and IJ(P , Cll have to measure the same performance of the (Po , C) and (p, C) systems.

Po -,

I

P

p.

Pc,

1 +PC

I < 1,

\fw EIR.

(6)

One can notice that the two mismatches (4) and (5) differ through a multiplication term by the sensitivity function (1 + POC)-l From (4) results that a robust stability involves a robust performance through the plant sensitivity function.

Following the discussion in section 1, the problem is to identify a model P and design a controller C so that the controlled plant and the controlled model, both of which have relevant performances . Therefore, we are looking for close and small values of IJ(Po, C)I and IJ(P , C)I, where the specific control performance function J is given.

If the difference term from (4) is small, then highperformances are not guaranteed. Only if the left side of (6) is small the performances of the pairs (Po, C) and (p, C) are similar.

When the nominal system has been derived well, then a solution based on the interplay between the identification and control can be proposed, by using the triangle inequalities:

Robust performance for the computed controller.

The mismatch between Po and P leads to the difference between the closed-loop transfer functions (in SISO terminology)

Pc I 1 + Pc

(4)

Robust stability for the computed controller. The controller C is designed on the nominal model but this controller stabilizes the (unknown) plant. From the "small gain theorem" and from the multiplicative uncertainty we realise that for the robust stability we can use the condition

Let us consider the model P, identified by the "training" with 4SID method, using past available data. This model is to be considered a nominal one and can be only an approximation of the real plant model Po. The mismatch between the plant Po and the nominal model P can be expressed in additional or multiplicative terms:

PoC 1 + PoC I

Pc 1 I . 1 + Pc . 1 + PoC .

P

The robust performance imposes a small difference in (4) and this difference makes sense only if the controller C stabilizes both Po and P.

INTERPLAY BETWEEN IDENTIFICATION AND CONTROL DESIGN PROBLEMS

by the uncertainty structure

P

IJ(P, C) -IJ(Po, C) - J(p, C)II ~ J(Po, C) ~ ~ J(p, C)

(3)

+ IJ(Po, C) -

J(p, C)I,

(7)

where the terms have the following significances: The aim is to find a nominal model P so that the pairs (Po, C) and (p, C) should have identical performances.

J (Po, C) - the measure achieved performance,

J(p, C) - the measure of the nominal performance,

The relation (3) can be formulated as

PoC 1 + PoC

of the

Pc 1 + Pc

IJ(Po, C) - J(p, C)I- the measure of the degradation.

118

where x(i) E rn,n is the state, uti) E rn,m is the (sufficiently persistently exciting) (control) input, y(i) E rn,1 is the measured output and ~(i), 1J(i) are unmeasurable external disturbances and measurement noise, respectively. (For more details see Van Overschee and De Moor (1994) ; Verhaegen (1994); Viberg (1994), (1995)).

In order to obtain the high performance m the plant , two requirements are necessary : J(P, G) - small,

IJ(Po , G) - J(F, G)I

(8)

«

J(F , G),

(9)

l ~l

Let us define the data block-Hankel matrices: where the first belongs to a nominal performance by minimizing the control function cost and the second represents the demand of the robust performance . If (g) is satisfied then the difference between the J(Po , G)-function and J(F, G)-function is considered small , see Van Den Hof and Schrama (1995) ; Gevers (1995)).

=

y

=

u(PI u(,B+1)

u(O')

u(O' + 1)

u(0'+',B-1)

y(l) y(2)

y(2) y(3)

y(,B) y(,B+1)

y( 0')

y( 0' + 1)

...

argmm J(Fi+I ' G) ,

G

1

y( 0' + ,B - 1)

1 (14)

argmm IJ(Po, C;) - J(P, Gdl (10) P

where positive integers 0', ,B satisfy 0' ~ Vo + 1, f~ f.I def d were h . . . . I . fJ ~ fJO = O'm an Vo IS an ll11tla estImation for the observability index of the pair (G, A). Define 0' + ,B = t , where t :-::: to and to = 0' + ,Ba is the initial stage of the 4SID procedure, when enough data are available to compute the plant order n and the corresponding first approximate plant matrices P = (A , B , G, D).

and G,+I

u(2) u(3)

U=

In optimization terms , the nominal model P and the controller G are obtained in an iterative procedure Pi + 1

u(l) u(2)

( 11)

where F and G vary over appropiate modelcontroller classes.

The input-output matrix equation for the deterministic part of (13) is

The iterative procedure develops by satisfying the robust condition

Y

= SX + TU

(15 )

where X = [x( 1) x(2) . .. x(,B) 1is the initial state matrix and In case robustness is not guaranteed we make use the 4SID procedure to identify a more accurate nominal model.

(16)

Now , for the iterative procedure, the 4SID sequential identification method is proposed.

o D

3.

BASIC 4SID METHOD

GB

Consider the input-output data (u(i), y(i)) , i 1 : t - I , generated by a (MIMO) plant having an unknown discrete state-space model x(i + 1) {

y(i)

= Ax(i) + Bu(i) + ~(i) = Gx(i) + Du(i) + 1J(i)

o o

il (17)

are the (highly structured) extended observability and block-Toeplitz matrices associated with (13).

(13)

119

1. The matrix V need not be stored. From (18) it results UV2 = 0, hence (15) gives

In order to compute approximations for the unknown plant matrices from (15), at the initial stage t to, i.e. with (3 (30, the basic (modified) I 4-SID method described in Viberg (1994) proceeds as follows

=

=

(22) This formula allow us to extract the (extended) observability matrix S by using only the (compressed) block Y z as in (20).

1. Compute the LQ-decomposition

2. As the pair (C, A) is observable, S has maximum column rank, hence n rankY2 is the (computed) model order . We assume this n is the true process order and we will keep it unmodified at subsequent stages.

=

where LI E IRamxam, Y z E IRalxal are square lower triangular matrices and V = [VI Vz ]E IR,6 x (am+al) has orthogonal columns.

3. S is uniquely defined up to a (nonsingular) similarity transformation , by choosing N. We fix the (first) leading n columns of RI as N- I RIpT = XVz , i.e. we fix some basis in the state space IR n of our model. This basis must be the same at all subsequent stages of our updating algorithm, see below.

2. Perform the (truncated) QR-decomposition with column pivoting Y z = QRpT, i.e.

where Q = [QI Qz] is an othogonal matrix, P is a (column) permutation matrix and RI is upper trapezoidal with a leading n x n non-

Remark 1. In the same way we can adapt the other, more reliable 4SID methods , i.e. the MOESP scheme robustified by incorporating past inputs and/or past outputs as instrumental variables used by Verhaegen (1994), the N4SID algorithms described in Van Overschee and De Moor (1994) , etc. The details are omitted.

singular block, where n ~f rankY2 . 3. We can take (20)

4.

or, equivalently,

Assuming the (first approximation) plant model (A, B, C, D) is available and the (predicted) state, at the current time step t to, to + 1, .. . , x(t) is known, the new control input u(t) is computed 2 and the corresponding plant output y(t) is measured. By using these new data the updating algorithm is applied in order to obtain new approximation for the plant model (A, B, C, D) and the predicted state x(t + 1).

=

where N E IR nxn is any nonsingular matrix; at the initial stage let N = In. 4. Next, the plant matrices are determined as usual, i.e. the pair (C, A) is obtained by exploiting the special structure of S given by (20) and matrices Band D are computed by solving an overdetermined system of linear equations deduced from

The main features of our updating procedure are • it propagates a compressed data array of constant order 0'( m + I) and

(21 ) see (15) and (18), where to (19) and (20).

BASIC UPDATING 4SID PROCEDURE

QI S = 0 according

• it preserves a fixed state-space basis in order to deliver subsequent estimates for the same (unknown) plant model matrices at every stage t = to, to + 1, ....

Comments. Consider the past data array, constructed from the available input-output data (u(i), y(i», i =

1 By "modified" we mean the use of the QR-decomposition with column pivoting instead of the truncated SVD (which is everywhere recommended) in order to maintain unmodified state-space basis in the updating algorithm to follow at the subsequent stages.

2By using any convenient model-based control design procedure.

120

1 : t - 1, having the triangular form as in the right hand side of (18). The basic steps of our updating algorithm are the following:

• Second, y. is cancelled in a similar way, i.e . [ L;

y. 1

A . Update the LQ-decomposition

0

o ][

Y2

y.

lam

-T V32

0

YI (+I)

Yi3 +1

=

=

(28)

B. Compute the QR-decomposition with column pivoting Y 2(+I) = Q(+I)R(+I )pT(+I) , i.e.

~~~.:;~ 1'

[ u(.8 +ex)

[

:1

Y 2(+I)

We stress that the matrix factors of Y2 are known from the previous stage, i.e. Y2 = QRpT

(23)

ui3+1

-b

V33

0

[ L,(+I)

where

v~, 1~

0 1122

Y 2(+I)

~i~ :;~ 1

= [QI(+I)

Q2(+1) 1 [

Rl~+l)

] .

(29) where the matri ces have the same meaning as in (20).

y( j3 +ex)

=

ar~ the new input-outpu t data at time t ex + 13. As in (18) , L1 (+1) E rn.0mxom, Y2 ( + 1) E rn. 01 x 01 are sq uare lower tri angu lar matrices and the transformation matrix has orthogonal columns (x-s denote uninteresting blocks).

Comment B. Now, from (25) we have

(30) where XV2 = N- I RIPI' (see Comment 3.) and , similar to (20), let us denote

Comment A. The square blo ck V22 of order exl must be computed and stored. The updating step A can be descri bed as follows :

S( +1)

• First , the elements of Ui3+ 1 are cancelled by

= Q( +1)

1

[ N(t ) ] ,

(31 )

using a sequence of Givens rotations, i.e .

[ L, YI

0

Ui3+1

Y2

][ Vi'

Yi3 +1

V31

[ L;

0

Y·1

Y2

0 101

v" o

0

V33

-a

or, equivalently,

1=

0

1

(24)

where Y2 remaInS unmodified. have

Hence, we

y'

[Y2 Y' 1 = S( +1) [XV2

From (28) and (29) it results

[ QI(+I)R I (+1)p T (+1)

x·

1 . [ N(t ) ] [ N- I RIPT

o 1 = Q(+1) · X

• 1 [V22 V23 ], -T - b v32

v33

hence , taking into account the orthogonality of = [QI( +1) Q2( +1)], we must have

1' (25)

Q( +1 )

where S( + 1) is the updated observability matrix and

[

(26)

(27)

121

N(~ 1)

] [ N-I RIPT

x·

1=

or

REFERENCES Astrom, K. and J. Nilsson (1994). Scheme of Iterated Identification and Control. Proc. SYSID'94, lO-th IFAC Symposium on System Identification. volume Vol. 2 171-176 . Gevers, M. (1995). Identification for Control. Proc. IFAC Conference . 1-12. Van Den Hof, P. and Schrama R.J .P. (1995). Identification and Control - Closed-loop Issues. A utomatica 31 , 1751-1770. Van Overschee, P. and B. De Moor (1994). N4SID: Subspace Algorithms for the Identification of Combined Deterministic- Stochastic Systems. A utomatica 30, 75-93. Verhaegen , M. (1994). Identification of the Deterministic Part of MIMO State Space Models given in Innovations Form from InputOutput Data. Automatica 30, 61-74. Viberg, M. (1994). Subspace Methods in System Identification. Proc. SYSID'94 , lO-th IFAC Symposium on System Identification. volumeVoI.11 - 12. Vib erg, M. (1995). Subspace-based Methods for the Identification of Linear Time-invariant Systems. Automatica 31 , 1835- 1851. Zang , Z. , R.R. Bitmead and M. Gevers (1995) . Iterative Weighted Least-Squares Identification and Weighted LQG Control Design. Automatica 31 , 1577- 1594.

(33)

C. Compute N( +1) from the first n columns of

(33): N(+l) =::::(: , 1: n)(R J (: , 1 : n))-l N (35)

and then compute 5(+1) from (31).

Comment C. From (27) and (34) we have

which can be used to compute x({3 + 1). So , we can obtain the predicted state x( t + 1) (where t = Cl' + (3 ) which is nedeed for the next control input computation. D. Finally, the new approximations A(+l) , B( +1) , C( +1), D( +1) are computed as in the basic 4SID algorithm (step 4). From (30) and (31) the condition QI(+1)5(+1) = 0 still holds. Remark 2. Something similar to statements from Remark 1 holds but many details remain to be worked out, especially concerning "the best" step B above. The main question is how the past decomposition Y2 = QRpT may be updated to obtain the new one Y 2 ( + 1) Q(+l)R(+l)pT(+l) in the simplest way?

5.

CONCLUSIONS

In this paper, the interplay between identification and robust model-based control design was discussed. The purpose of the identification - control design is to obtain relevant performances of the controlled process. The 4SID method is used in the identification stage of the iterative scheme of control design for the initialization and when the robustness conditions are not fulfilled. The alternative 4SID is justified by the efficiency of the numerical computation, especially for the case of MIMO systems, and for the guaranty of the stability and robustness in real-time closed-loop systems.

122