Closed-Loop Identification for Model Predictive

Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009

WeC14.3

Closed-loop Identification for Model Predictive Control: Direct Method Jun Yan, Eranda Harinath and Guy A. Dumont

Abstract— Model predictive control (MPC) is a widely used control scheme that handles constraints directly. In practice, the initial performance of MPC is usually satisfactory after a careful setup stage. However, over time, physical changes in the plant may invalidate the predictive model used in MPC and control performance degrades. Thus at least a model update is needed to restore the plant performance. Since the initial commissioning of MPC can be long and costly, serious attention should be given to closed-loop identification for MPC. This paper shows that if the MPC exhibited complex enough behavior during normal operation then one can obtain good model update based solely on the informative operation data. If needed, one can design an experiment aimed at increasing the complexity of MPC in order to avoid undesirable actuator saturations. The explicit piecewise affine solution of MPC is used to analyze both questions.

I. I NTRODUCTION Model predictive control (MPC), perhaps the most natural choice as a control scheme to handle constraints systematically, is still attracting attention from both academic and industrial sides. In practice, a stabilizing and well-performing MPC can be setup after a careful commissioning stage in open loop that can be long and costly. Over time (months to years), the performance of the closed-loop gradually deteriorates largely due to hardware wear and tear or other physical changes in the system. This indicates a need for a new model in order to improve the performance since the performance of MPC mainly depends on its ability to predict the future. This problem has been observed by our industrial partner on its mechanical pulping process control system. To avoid opening the loop and performing the lengthy commissioning again, the possibility of closed-loop identification should be studied, especially how to use the collected abundant operation data. Closed-loop identification approaches are usually divided into three categories ([1], [2]): •

•

direct identification uses the input-output data as in the open loop case without considering the effect of feedback; this approach, if it works, is the simplest and the most attractive one; indirect identification first identifies a closed-loop sensitivity then uses the knowledge of the linear feedback to determine the open-loop model; this approach heavily depends on linearity, while constrained MPC is usually nonlinear [3], therefore, the indirect approach cannot be applied to a closed-loop controlled by constrained MPC;

This work was supported by NSERC. J. Yan, E. Harinath, and G.A. Dumont are with Pulp and Paper Center, University of British Columbia, 2385 East Mall, Vancouver, BC V6T 1Z4, Canada {juny, eranda,

guyd}@ece.ubc.ca

978-1-4244-3872-3/09/$25.00 ©2009 IEEE

•

joint input-output identification regards the control and output as the outputs due to the independent excitation signal r and the process noise v as the inputs, first extracts the r component in the control signal, then uses it and the output data to fit a process model. With linearity, the two-stage method can be applied [4]. A more general approach is the projection method [5] in which the controller can be nonlinear. This approach can be applied to the data generated by MPC.

This paper focuses on applying the direct closed-loop prediction error method (PEM) to a linear time-invariant loop controlled by MPC. In [6], extensive case studies were presented and showed promising results of direct closed-loop identification for MPC. Here the focus is on the theoretical aspects to answer the following questions: Q1: under what conditions will direct closed-loop PEM give good model estimates based only on normal MPC operation data (i.e. no extra excitation)? Q2: if an extra experiment is needed, how to proceed? The success of PEM requires that: (1) the parameter estimates converge; (2) the experiment be informative enough; (3) the candidate model/parameter set contain the real system. Here (1) is assumed to be true, for the detailed convergence conditions see [2] and references therein. There is not much one can do about (3) except allowing high complexity in the model set. The second condition is usually paid the most attention. As shown in [2], the experiment for direct closed-loop PEM is informative if either the external excitation r(t) is persistently exciting (pe) or the feedback control law is sufficiently complex, meaning either a linear sufficiently high order controller or several linear regulators controlling the loop alternatively. Failing in both, PEM may give very strange and undesired estimates (such as a process model fitting the inverse of the controller). Note that if one considers actuator saturation which always exists in practice, then the effect of excitation in r may be diminished by the controller limits. [1] gave a thorough analysis on how complex the feedback law(s) should be so that PEM results in consistent model estimates. The sufficient complexity is guaranteed if a matrix generated by the linear regulators has full row rank almost everywhere. This paper follows the line of [1] to establish system identifiability based on the complexity of constrained MPC. The explicit feedback law of an MPC with quadratic cost criterion and linear constraints is equivalent to a piecewise affine feedback law [7] and online MPC switches between these different pieces. It is natural to check whether the piecewise affine law of a constrained MPC is complex

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WeC14.3 enough so that the normal closed-loop operation data is informative. The result in [7] was derived in state feedback form which certainly indicates the same for output feedback. Section 2 is dedicated to derive the piecewise affine output feedback solution of MPC. In Section 3, analysis of controller complexity is given using the same argument as in [1]. The extra constant terms induced by MPC do not change the complexity condition. Since controller saturation may mute persistent excitations and render the experiment data noninformative, with MPC in the loop, there might be some room to achieve informative experiment via involving more pieces of the MPC law while avoiding actuator saturation. This is the topic of Section 4. II. T HE P IECEWISE AFFINE SOLUTION OF OUTPUT FEEDBACK MPC First consider an MPC using the following scalar model: y(t) = G(z−1 )u(t) + d(t), G(z−1 ) = z−d d(t) = H(z−1 )v(t), H(z−1 ) =

B(z−1 ) A(z−1 )

C(z−1 ) D(z−1 )

,

(1) (2)

where v(t) ∼ N(0, σ 2 ) is a gaussian white noise and some process delay d ≥ 1 is enforced to avoid circulating dependence between y(t) and u(t). The degrees of the polynomials are deg(A) = deg(B) = n and deg(C) = deg(D) = m. The noise model H(z−1 ) = C(z−1 )/D(z−1 ) can be made monic via proper scaling of the noise covariance σ . The constrained optimization formulation of MPC is: N

∑ (yˆt+k|k − r(t + k))2 + λ u2 (t + k − 1|t),

(3)

k=1

subj. to: AU ≤ b,

(4)

where U , [u(t|t), . . . , u(t + N − 1|t)]T . For simplicity, the horizon is N without differentiating the control horizon and the prediction horizon. The control applied to the process is the first element of the optimal solution u(t) = u(t|t). Here nc linear constraints are considered, i.e. A ∈ R nc ×N and b ∈ R nc . To derive the explicit control law of the MPC, one needs to 1) obtain a proper output prediction that separates the past and the future; 2) transform the constrained optimization problem into a quadratic programming (QP) problem in terms of U; 3) solve the QP symbolically for each well-defined active constrained set for the feedback law and its active region. A. The output prediction The procedures to form the output prediction here is a brief version of that in [3], which requires solving two sets of Diophantine equations: d −1 C(z−1 ) d −1 −k Fk (z ) = E (z ) + z , k D(z−1 ) D(z−1 ) B(z−1 ) F (z−1 ) = Ek (z−1 ) + z−(k−d+1) k −1 . −1 A(z ) A(z )

v(t|t) ˆ = H −1 (z−1 )(y(t) − G(z−1 )u(t)) v(t ˆ + k|t) = 0, k ≥ 1. This leads to the minimum variance predictor: ˆ + k|t) = d(t

Fkd (z−1 ) v(t|t), ˆ D(z−1 )

y(t ˆ + k|t) = Ek (z−1 )u(t + k − d|t) +

h F d (z−1 )

k y(t) C(z−1 ) i z−1 Fk (z−1 )C(z−1 ) − z−d Fkd (z−1 )B(z−1 ) + u(t) . A(z−1 )C(z−1 )

(6)

Note that since the delay is d ≥ 1, in (6) for k ≥ d, the first term is the future which depends solely on the optimization variables in U and the second term is the past which is a combination of filtered past outputs y up to t and filtered past controls u up to (t − 1). If the model (1) does not contain the disturbance part H, then the output prediction requires solving only one set of Diophantine equations and the past in the prediction is a linear combination of past outputs and controls rather than a filtered version as in (6). B. Transforming the open-loop optimization into QP Define the following auxiliary variables: Y , [y(t ˆ + d|t) . . . y(t ˆ + N|t)]T ,U , [u(t|t) . . . u(t + N − 1|t)]T ,

U OPT = arg min J(y(t), N,U) J=

The degrees of the polynomials satisfy deg(Ek ) ≤ (k − d), deg(Fk ) = (n−1), deg(Ekd ) ≤ (k −1), and deg(Fkd ) = (m−1). These can be calculated via long division. Since v(t) is a white noise process, a common choice of vˆ is

R , [r(t + 1), . . . , r(t + N)]T , f y (t) ,

F d (z−1 ) [ 1 −1 , C(z )

(7)

F2d (z−1 ) ,..., C(z−1 )

FNd (z−1 ) T ] y(t), C(z−1 )

z−1 F1 (z−1 )C(z−1 )−z−d F1d (z−1 )B(z−1 ) A(z−1 )C(z−1 )  



 f u (t) ,  



.. .

z−1 FN (z−1 )C(z−1 )−z−d FNd (z−1 )B(z−1 ) A(z−1 )C(z−1 )

Clearly from (6), Y = LU + f y (t) + f u (t),  e0 0 . . . 0 e0 e1 . . . 0  L= . .. .. . .  .. . . . e0

−1

e1

−1

Ek (z ) = e0 + e1 z

...

eN−d

  u(t). 

0 0 .. . 0

 ... 0 . . . 0  ..  , .. . . ... 0

+ . . . + ek−d z−(k−d) .

Then the MPC constrained optimization problem can be written as a QP problem in terms of U min U T MU + 2 f T (t)LU U

(8)

subj. to: AU ≤ b,

(5)

where M , (LT L + λ I), f T (t) , ( f y (t) + f u (t) − R)T L. At each sample time t, the above QP problem can be solved numerically to generate the control u(t). However,

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WeC14.3 at this point we are interested in expressing U and its first portion u(t) = u(t|t) as a closed-form function of y(t) and r(t). C. The explicit solution of MPC The explicit solution of the QP (8) can be obtained via solving its KKT conditions for each well-defined set of active constraints. The KKT conditions are  T  MU + f + A α = 0, (9) AU ≤ b,   T α (AU − b) = 0, αi ≥ 0, i = 1, . . . , nc , where α = (α1 , . . . , αnc )T is the vector of Lagrange multipliers. Denote p the index set of l active constraints, clearly p , {p1 , p2 , . . . , pl } ⊆ {1, . . . , nc }, then denote I p the matrix formed by stacking the p1 th ∼ pl th rows of the identity matrix Inc and similarly I pc by stacking the remaining rows of Inc . Then I p α shows all the nonzero (positive) αi s or equivalently α = I pT I p α . The solution of the KKT equations with I p is given by µ ¶· ¸ · ¸ M AT I pT U −f = . (10) Ip α Ip b Ip A 0

Hence the optimal solution (I p α ,U pOPT ) can be obtained. The detailed formula is straightforward from (10) but somewhat complicated and has little value for the identification problem discussed in this paper, hence is omitted. The regional constraint for p being the active constraint set can be obtained from ( I pc AU pOPT < I pc b, (11) I p α > 0, which is a condition on f (t). Note that in [7], the piecewise solution was derived for state feedback and the regional conditions were given on the current states. In the above output feedback settings, the conditions for each feedback law are given on f (t) which is really a condition on the unforced response ( f y (t) + f u (t)), an interesting alternative to the current states. The final step is to retrieve the expression of u(t) = u(t|t) as the closed-form solution of the MPC which is given by £ ¤ u(t) = u(t|t) = 1 0 . . . 0 U pOPT Therefore, if (11) holds, the effective feedback control law takes the following form u(t) = C p + Fpy (z−1 )y(t) + Fpr (z−1 )r(t),

(12)

Though complicated, by specifying every active constraint set, we may obtain u(t) as a piecewise affine feedback law (12) and their regional conditions on f (t) (11). Remarks: The above derivation presents the basic steps towards the explicit MPC law. Although it was derived for SISO LTI system with linear input constraints, the extension to MIMO LTI system with linear input and output constraints can be done with little change to the process. Therefore, in general, if an MPC controller based on an LTI

input-output model can be written into a QP problem, then its explicit feedback law takes the piecewise affine form of (12). This is the underpinning of establishing identifiability condition under which the direct PEM may succeed based solely on the operation data of a loop with such MPC controller. Before moving onto the identifiability analysis, let us see an example of the explicit control law. An example: Consider the following model: 0.5z−1 1 + 0.5z−1 u(t) + v(k). 1 − 0.9z−1 1 − z−1 The MPC problem is y(t) =

N

∑ u(t+k|t) min

k=1

¤ £ (y(t ˆ + k|t) − r(t + k))2 + λ u2 (t + k − 1)

subj. to: u(t + k − 1|t) ≤ umax .

For a prediction horizon N = 2, the prediction polynomials are given by E1d = 1, E2d = 1 + 1.5z−1 , F1d = 1.5, F2d = 1.5, E1 = 0.5, E2 = 0.5 + 0.45z−1 , F1 = 0.45, F2 = 0.405. Set umax = 1 and λ = 0.01 then the closed-loop control law for Nµ= 2 is ¶ · ¸ 1 −3.7343 3.2316 , then f< • if 1 3.2316 −6.6427 y

u(t) = F1 (z−1 )y(t) + F1r (z−1 )r(t),

−0.754 + 0.6786z−1 , 1 − 0.4054z−1 − 0.2642z−2 2 −3.232z + 5.027z − 0.0395 − 1.68z−1 , F1r (z−1 ) = 1 − 0.4054z−1 − 0.2642z−2 y

F1 (z−1 ) =

(13)

thisµcorresponds to the ¶ case of· no ¸active constraints; −2.1622 0 1 • if f< , then −3.2316 6.6427 −1 y

u(t) = F2 (z−1 )y(t) + F2r (z−1 )r(t) + 1.0465,

−3.081 + 2.773z−1 , 1 − 1.06z−1 − 0.00973z−2 2 0.973z + 0.6919z − 0.8703 − 0.4865z−1 , F2r (z−1 ) = 1 − 1.06z−1 − 0.00973z−2 y

F2 (z−1 ) =

(14)

this corresponds to the case of only the second constraint being active; • otherwise, u(t) = umax = 1, this corresponds to whenever the first constraint is active. III. C ONSISTENT D IRECT C LOSED - LOOP I DENTIFICATION FOR C ONSTRAINED MPC Section 2 indicates that in general the feedback control law of a QP MPC is piecewise affine. We thus assume that the closed-loop data {u(t), y(t), t = 1, . . . , T } was generated by an explicit MPC control law with q pieces of active feedback laws: ui (t) = Ci + Fiy (z−1 )yi (t) + Fir (z−1 )r(t), i = 1, . . . , q, (15) which is not necessarily SISO. And denote γi the portion of time when the ith feedback law was effective, note that q ∑i=1 γi = 1.

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WeC14.3 The scenario is very similar to (Section 10.3, [1]), in which the controller is switching between several linear feedback regulators, i.e. (15) without the constant term. System identifiability can be established if a matrix generated by {Fir , Fiy , i = 1, . . . , q} has full row rank. Hence applying PEM to {u(t), y(t), t = 1, . . . , T } can give consistent estimates of the system (G0 , H0 ) if T is sufficiently large. The philosophy behind the need for system identifiability is that under mild conditions PEM estimates can converge to a global minimum of the asymptotic loss function, say T V = tr(R∞ ) where R∞ = limT →∞ T1 ∑t=1 ε (t)ε T (t) and ε (t) is the prediction error. Thus a system identifiability condition is to make sure that the desired solution G = G0 , H = H0 is the unique global minimum. Following the same procedure, a slightly extended identifiability condition can be established when (15) is in the loop. Suppose that the prediction error corresponding to (15) is

εi (t) = H −1 (yi (t) − Gui (t)), i = 1 ∼ q,

(16)

where the closed-loop expressions of yi and ui are yi (t) = (1 − G0 Fiy )−1 G0 (Ci + Fir r(t)) + (1 − G0 Fiy )−1 H0 v(t),

(17)

ui (t) = [I + Fiy (1 − G0 Fiy )−1 G0 ](Ci + Fir r(t)) + Fiy (1 − G0 Fiy )−1 H0 v(t).

(18)

Substituting (17) and (18) into (16),

εi (t) = H −1 [(1 − G0 Fiy )−1 G0 − G(I + Fiy (I − G0 Fiy )−1 G0 )] (Ci + Fir r(t)) + H −1 (1 − GFiy )(1 − G0 Fiy )−1 H0 v(t).

q

R∞ = ∑ γi E{εi (t)εiT (t)} ≥ E{v(t)vT (t)}. i=1

Clearly, the global asymptotic minimum cost is bounded from below V ≥ trE{v(t)vT (t)} and (G = G0 , H = H0 ) is a minimum such that εi (t) = v(t), R∞ = E{v(t)vT (t)} and V = trE{v(t)vT (t)}. Now we need to show under what condition (G = G0 , H = H0 ) is the unique solution of (20)

Combine (20) with (19) and note that r(t) and v(t) are independent, then any solution (G, H) of (20) should satisfy y

y

y

H −1 [(1 − G0 Fi )−1 G0 − G(I + Fi (I − G0 Fi )−1 G0 )] (Ci + Fir r(t)) = 0, H

−1

y y (1 − GFi )(1 − G0 Fi )−1 H0

≡ I,

¢ H −1 − H0−1 H −1 G−1 − H0−1 G−1 0 µ ¶ I ... I 0 ... 0 ≡ 0. y y r r −F1 . . . −Fq F1 . . . Fq

¡

Therefore, if the matrix F,

(21) (22)

The equation (22) is exactly the same as in [1] and (21) is slightly different for the term (Ci + Fir r(t)). Note that Ci is a constant, then (Ci + Fir r(t)) = Fir (Ci /Fir (1) + r(t)). As in [1]: assuming that r(t) is persistently exciting (pe), so

µ

I y −F1

... ...

I y −Fq

0 F1r

... ...

0 Fqr

¶

(23)

has full row rank almost everywhere, then ¡ −1 ¢ ≡ 0 or (G = G0 , H = H − H0−1 H −1 G−1 − H0−1 G−1 0 H0 ) almost everywhere. The assumption that r(t) is pe is not a strong one, in practice the set-point r(t) may change due to the production requirement and can provide certain excitation. In case that r(t) is not pe, it can be viewed as a filtered version of a pe signal r(t) = Lr (z−1 )˜r and the Fir s in (23) should be replaced by Fir Lr , see [1]. This rank condition (23) establishes system identifiability defined in [1] which depends on the model structure, identification method, and experiment condition. In a different setting, this condition can be viewed as informative data resulting from complex enough controllers [2]. Therefore, before applying PEM to the closed-loop operation data {u(t), y(t), t = 1, . . . , T }, condition (23) should be checked. If it passes and T is sufficiently large, without any further experiments consistent model estimates (G, H) should be expected provided that the model structure contains the true system (G0 , H0 ).

(19)

Although the overall prediction error ε (t) is not a stationary process, each εi (t) is a stationary process. Hence the asymptotic prediction error covariance is

εi (t) = v(t).

is (Ci /Fir (1) + r(t)), then (21) and (22) for i = 1, . . . , q) are equivalent to

IV. E XPERIMENT DESIGN FOR MPC The conditions such that direct closed-loop PEM can give consistent model estimates are 1) PEM converges; 2) the operation data set is informative enough; 3) the model structure contains the true system. The first condition is commonly assumed to be true. The third condition ensures that the desired solution is obtainable, although it can never be checked in practice, a guideline on the choice of model structure can be given: in [2] it is shown that direct closed-loop PEM has a bias term B added on the estimate of the system model G where B = (H0 − H)Φvu Φ−1 u , Φu is the spectrum of u and Φvu is the cross spectrum between u and v. Only the exact disturbance model H = H0 makes the exact process model G = G0 possible. Therefore, the disturbance model should be independently parameterized and given enough complexity, such as a highorder Box-Jenkins (BJ) model. A small Φvu Φ−1 u term can also help, how to achieve this via choosing r with constrained MPC in the loop is not a trivial matter but bears future interests. The second condition guarantees the uniqueness of the global minimum of PEM and it attracts the most attention. As shown in (Chapter 13, [2]) with a controller switching between several linear regulators, informative data can be achieved if

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WeC14.3 •

V. A N E XAMPLE

•

In this section, the example MPC shown in Section 2 is applied to the following true process

the external excitation r(t) is persistent; or, the controller is complex enough. Checking whether the experiment data is informative plus adding persistent r may seem to be the simplest. In practice, actuator saturation may happen and renders part of the experiment useless for identification. MPC can address the existence of such saturation as constraints and the analysis in Section 3 points out a way to avoid actuator saturation while making the experiment informative via increasing the complexity of the effective part of the MPC controller, i.e. designing identification experiments such that the rank condition (23) holds. Suppose that the normal operation data set {u(t), y(t)} under MPC is not informative enough or the matrix F is rank deficient in (23), therefore additional experiment should be designed to activate serval other pieces of control law (12) to make F full row rank. Obviously, such excitation signal r can not be designed off-line but online via the regional condition (11). Regard the MPC law as: the control law (15) is active if Ri ( f y (t) + f u (t) − R) < bi , a simplified version of (11). Assume that at time t, the normal reference signal r(t) (or R as defined in (7)) suggests that the current active region is R1 ( f y (t) + f u (t) − R) < b1 , while the target of the identification experiment is to activate another piece corresponding to R2 ( f y (t) + f u (t) − R) < b2 . Therefore, consider adding ∆r(t) to r(t) such that ( R1 ( f y (t) + f u (t) − R) < b1 , or, R2 ( f y (t) + f u (t) − R − ∆R) < b2 , R2 ∆R > (R1 + R2 )( f y (t) + f u (t) − R) − (b1 + b2 ),

y(t) = G0 (z−1 )u(t) + H0 (z−1 )v(t),

(25)

0.6z−1

G0 (z−1 ) =

, 1 − 1.6z−1 + 0.7z−2 1 + 0.8z−1 H0 (z−1 ) = , 1 − 1.2z−1 + 0.2z−2

where v ∼ N(0, 0.01). Several cases will be studied to show how the complexity of MPC would affect the direct closedloop PEM estimates: i.e. when F has full row rank 2 consistent estimates are expected, otherwise the result can be problematic. Data {u(t), y(t)} were collected for 24000 samples in order to have converged identification result in each simulation. The model is chosen to be Box-Jenkins and the estimates were calculated via MATLAB routine bj. The order/structural information of the true process is never known a priori, however for illustrative purposes, here the orders of the polynomials in G0 and H0 are assumed to be exact except that of the numerator of G0 which is assumed to be 2. The bode plots of the true and estimated process models are shown in Figure 1 and the bode plots of the disturbance models are shown in Figure 2 Bode Diagram 30

G0

25

G1

20

G2

15

G3 G4

10

(24)

5

where ∆R = [∆r(t + 1), . . . , ∆r(t + N)]T . Furthermore, additional excitation in closed-loop means product degradation. Thus, one may design ∆r via (24) for several candidate pieces and then choose the least damaging one. The idea that one should identify models based the operation data first, then if necessary, perform the least disturbing experiment in the closed-loop has been studied under the name of least costly identification [8]. In [9], necessary and sufficient conditions on getting just sufficiently rich data are given for common model structures in both open loop and closed-loop under linear feedback settings. The framework used identifiability and informative data definitions given in [2] rather than the system identifiability definition in [1] and in this paper. However, extending the results in [9] to include constrained MPC may bear value beyond this paper, because the rank condition (23) is a sufficient but not necessary condition on informative data. It guarantees informativity solely via the number of effective affine control laws, but not the switching effect between them which may also contribute. To illustrate the lack of necessity consider this case: if the effective control laws are u = umax and u = umin , then the rank condition fails but a particular switching behavior may render u(t) a PRBS like signal which may give informative experiment data. The difficulty lies in evaluating the informativity condition for a switched closedloop system.

−10

0 −5

2596

−15 −20 180

90

0

−90

−180 −3 10

Fig. 1.

−2

−1

10

0

10 Frequency (rad/sec)

1

10

10

Bode plots of the true G0 and estimates process models Gi s.

Bode Diagram 150 H0 100

H1 H2 H3

50

H4

0

−50 90

0

−90

−180 −5 10

Fig. 2.

−4

10

−3

10

−2

10 Frequency (rad/sec)

−1

10

0

10

1

10

Bode plots of the true H0 and estimates process models Hi s.

WeC14.3 A. r(t) = 0, unconstrained control In this case, the effective controller is u(t) = F1y (z−1 )y(t), the complexity condition (23) has become F = [1, F1y ]T whose row rank is always 1. The identified BJ model is G1 =

−2.369z−1 +2.014z−2 , 1−0.1165z−1 −0.8026z−2

H1 =

Consider the following two cases: (1). MPC is unconstrained u(t) = F1y (z−1 )y(t)·+ F1r (z−1¸)r(t) I 0 has with r ∼ N(0, 0.01), the corresponding F = y F1 F1r full row rank almost everywhere; the identified models are consistent with the true process: H2 =

1+0.7978z−1 . 1−1.196z−1 +0.1965z−2

(2). MPC is constrained u ≤ 1 with r = 0, then the effective controllers is switching between· ui = Fiy¸y + Ci , i = 1, 2 and I I has full row rank u = 1. The corresponding F = y F1 F2y almost everywhere; 27.6% of the simulated data corresponds to u1 , 8.9% corresponds to u2 , and 63.5% from u = 1. The estimated models are consistent with the true process: G3 =

0.6001z−1 −0.0003913z−2 1−1.602z−1 +0.7014z−2

, H3 =

1+0.8014z−1 . 1−1.198z−1 +0.1978z−2

C. Full MPC in the loop In this case, the setpoint r is switching between different values from time to time as shown in Figure 3 and the constraint is u ≤ 1. The corresponding F has full row rank almost everywhere.

0

0.5

1

1.5

2

H4 =

1+0.8007z−1 . 1−1.193z−1 +0.1927z−2

This paper has shown that a constrained MPC can be a complex enough controller making normal operation data informative so that the direct closed-loop identification may succeed. The least complexity is measured by the row rank of a transfer function matrix that is generated from the piecewise affine solution of the constrained MPC. In the case that the normal operation is not informative enough, an extra experiment can be designed through the knowledge of the piecewise affine solution (the regional conditions) of MPC. Such designed experiments may have the advantage of avoiding non-informative data due to actuator saturation. Future study should include the accuracy aspect of the closed-loop identification utilizing the explicit solution of MPC and the contribution of the switching effect of MPC to the informativity of the data. R EFERENCES [1] T. S¨oderstr¨om and P. Stoica, System Identification. Prentice Hall International, 1989. [2] L. Ljung, System Identification: Theory for the User, 2nd ed. Prentice Hall, 1999. [3] J. Maciejowski, Predictive Control with Constraints. Prentice-Hall, 2002. [4] P. V. den Hof and R. Schrama, “An indirect method for transfer function estimation from closed loop data,” Automatica, vol. 29, pp. 1523–1527, 1993. [5] U. Forssell and L. Ljung, “A projection method for closed-loop identification,” IEEE Transactions on Automatic Control, vol. 45, no. 11, pp. 2101–2106, November 2000. [6] S. Amjad, “Closed-loop identification for model predictive control: A case study,” Master’s thesis, King Fahd University of Petroleum and Minerals, October 2003. [7] A. Bemporad, M. Morari, V. Dua, and E. Pistikopoulos, “The explicit linear quadratic regulator for constrained systems,” Automatica, vol. 38, pp. 3–20, 2002. [8] X. Bombois, G. Scorletti, M. Gevers, P. V. den Hof, and R. Hildebrand, “Least costly identification experiment for control,” Automatica, vol. 42, pp. 1651–1662, 2006. [9] M. Gevers, A. Bazanella, and L. Miˇskovi´c, “Informative data: how to get just sufficiently rich?” in Proceedings of the 47th IEEE Conference on Decision & Control, Cancun, Mexico, 2007, pp. 1962–1967.

2.5 4

x 10

Fig. 3.

0.6002z−1 −0.0003044z−2 , 1−1.601z−1 +0.7012z−2

VI. C ONCLUDING R EMARKS

B. Only partial MPC is effective

0.6023z−1 +0.001607z−2 , 1−1.601z−1 +0.6986z−2

G4 =

1−0.6437z−1 . −0.4329z−1 −0.5408z−2

As shown in Figure 1 and Figure 2, the estimates are quite biased. Note that if the model structure is restricted to the true structure (in this simulation, the order of G1 ’s numerator is one), consistent estimates can be obtained. This means even though the rank condition (23) failed, by choosing specific ‘small’ model set that contains unique global minimum, good estimates can still be expected. Mathematically, with r = 0 and only one linear output feedback F1y in the loop, PEM is converging to a solution of (22) which in general has multiple solutions. Only if certain structural/parametric constraints that is compatible with the complexity of the feedback are imposed upon G and H, the solution of (22) is unique, thus one can have convergent identification results. On the other hand, if the rank condition (23) holds, then PEM can converge to the desired solution over an arbitrarily large model set as long as it contains the true process.

G2 =

The simulation data contains 52.2% from u1 , 4.5% from u2 , and 43.3% from u = 1. The model estimates are consistent with the true process:

The setpoint change.

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