Output Feedback-Based Output Tracking Control with Adaptive Output

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Output Feedback based Output Tracking Control with Adaptive Output Predictive Feedforward for MIMO Systems Ikuro Mizumoto, Seiya Fujii, and Hiroshi Mita Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.9b00107 • Publication Date (Web): 18 Feb 2019 Downloaded from http://pubs.acs.org on February 23, 2019

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Industrial & Engineering Chemistry Research

Output Feedback based Output Tracking Control with Adaptive Output Predictive Feedforward for MIMO Systems † ‡ Ikuro Mizumoto,

¶Faculty

∗,¶

Seiya Fujii,

§

and Hiroshi Mita

§

of Advanced Science and Technology, Kumamoto University, Kumamoto, JAPAN

§Department

of Mechanical Systems Engineering, Kumamoto University, Kumamoto, JAPAN E-mail: [email protected]

Phone: +81 (0)96 3423759. Fax: +81(0)96 3423729

Abstract This paper proposes a two-degree-of-freedom adaptive output feedback control system design scheme with an adaptive output predictive feedforward for MIMO LTI systems. In the proposed method, we rst present a novel simple adaptive output estimator design scheme by introducing a parallel feedforward compensator. Based on the designed adaptive output estimator, output predictor will be realized and then output predictive controller is designed for feedorwared control. The stability of the proposed method will be maintained by combining ASPR (almost strictly positive real) based output feedback i.e. by designing a two-degree-of-freedom control system. We validate the eectiveness of the proposed adaptive control through experiments on control of a three-thank process. ‡

The material in this paper was partially presented at the 20th IFAC World Congress, July 9-14, 2017,

Toulouse, France.

‡

This work was partially supported by JSPS KAKENHI Grant Number JP17K06501.

1

ACS Paragon Plus Environment

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Introduction A system is said to be àlmost strictly positive real (or ASPR)', if the resulting closed-loop system with a static output feedback is strictly positive real (or SPR) 1,2 . Under the ASPR condition on the controlled system, it has been well recognized that a simple output feedback control system can be designed easily and the obtained control system has strong robustness with respect to disturbances and system's uncertainties 38 . Therefore, the application of the ASPR based output feedback control strategy to practical system has attracted a great deal of attention due to its simple structure of control algorithm and ability to counter the control diculties on systems with some kind or another uncertainties and disturbances 2,3,7,912 . For discrete systems, even for SISO systems, the sucient conditions for the system to be ASPR has been provided as (1) the system is minimum-phase, (2) the system has a relative degree of 0, and (3) high frequency gain of the system is positive 2,13 . Unfortunately, however, since general systems do not have such ASPR property, the ASPR conditions imposed on the systems for designing simple output feedback have been a severe restriction for applying the ASPR based control method to practical systems. Especially, the restrictions of minimum-phase and relative degree are considerable restricted conditions in most practical systems. As an alleviation method for ASPR restrictions, the introduction of a parallel feedforward compensator (PFC) has been provided 2,6,1416 . The PFC will be introduced in parallel with the non-ASPR controlled system as illustrated in Fig. 1 in order to make the resulting augmented system ASPR. To the obtained ASPR augmented system, the ASPR based control strategy can be applied. However, since the control system is designed to the augmented system, if the gain of the PFC is relatively large to the one of the actual control system, a steady state error might be appeared on the actual output by the aect from the PFC output even if the augmented system's output ya was exactly track the given reference signal. To avoid this additional issue, a PFC design scheme in which one can design a PFC having small gain 15 and introduction of a feedforward input with augmented reference model 3,6 have 2


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u

y Plant

ya +

yf

+

PFC Figure 1: Introduction of a PFC been proposed. The PFC design method in Ref. 15 is a concrete PFC design scheme, however, the method is applicable only to non-ASPR but minimum-phase continuous systems. The introduction of feedforward input adopted in Ref. 3 and Ref. 6 is a powerful and simple way to attain output tacking and it has played very important role in the ASPR based output tracking control system design with a PFC 17,18 . In Ref. 19 and Ref. 17, adaptive NN feedforward control strategy has been proposed. On the other hand, in several practical industrial elds, `Predictive Control' including the model predictive control (MPC) and the generalized predictive control (GPC) has been well recognized as a powerful advanced control scheme 2023 , since the predictive control performs robustly compared with standard PID controls, and they have been widely used in industry in recent decades. However, in most cases, the performance of the resulting control system strongly depends on the accuracy of the given model in the predictive control design. Therefore, in order to obtain the accurate model, a number of experiments might be required and a lot of time and cost have to be spent. Furthermore, it is dicult to maintain the acceptable control performance if the controlled system has some kind or another uncertainty or if the controlled system is changing during the operation due to mismatch of the given model. Adaptive strategy is considered as one of the attractive ways to solve such a problem on system's uncertainties and adaptive type of predictive controls have been investigated 2427 . The development on the adaptive predictive controls are still continued and several kinds of adaptive predictive controls have been proposed in recent decade 2830 . Unfortunately, however, in order to obtain good parameter estimation for adaptive controls, generally the 3



structure and order of the system must be known and thus the adaptive predictive control algorithm became relatively complex for higher order controlled systems. With this in mind, a novel adaptive output predictor based adaptive predictive control strategy has been proposed from a dierent viewpoint 30,31 . In Ref. 30 and Ref. 31, an adaptive output predictor with a simple structure was derived from output estimator for designing the predictive control input. The initial version of the predictive control is the one proposed for multi-rate systems 32,33 . In the method, a parallel feedforward compensator (PFC) is again introduced so as to render a minimum-phase augmented system with a relative degree of 1, and the reduced rst order approximated model is utilized for designing the adaptive output estimator with a very simple structure and output predictor is designed by expanding the obtained output estimator. The adaptive predictive control is designed by using the predicted outputs simply. In this method, a switching strategy via ÀSPR input constraint' was adopted in order to maintain the stability of the obtained control system 30 . ASPR based output feedback and adaptive predictive control were switched based on the magnitude of the output tracking error in the method. However, the obtained control system might not be robust for unexpected disturbances and to determine the switching timing was sensitive to control performance, i.e. how to set the switching timing was very important but it was dicult. Taking this problem into consideration, a two-degree-of-freedom control with ASPR based output feedback was proposed for SISO systems 31 . By combining the adaptive predictive control as feedforward input and ASPR based output feedback with the PFC for stabilization, we can solve the stability problem on the adaptive predictive control and the output tracking problem on the ASPR based output feedback control with the PFC simultaneously. In this paper, we propose a two-degree-of-freedom adaptive output feedback control system design scheme with an adaptive output predictive feedforward for MIMO LTI systems. A novel simple adaptive output predictor design scheme by introducing a parallel feedforward compensator will be presented by expanding the adaptive output predictor proposed in Ref. 4


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30 and Ref. 31 for MIMO systems. The stability of the proposed method will be maintained by combining ASPR based output feedback and thus a two-degree-of-freedom control system with the ASPR based output feedback and with the proposed adaptive predictive control as a feedforward control will be proposed. The obtained two-degree-of-freedom control system has relatively simple structure with simple output feedback and simple output predictive controller. The eectiveness of the proposed ASPR based output feedback control with the adaptive predictive feedfoward input will be conrmed through experiments of a three-tank process control.

Problem Statement Let's suppose that the considered controlled system is expressed as the following uniformly sampled MIMO system of order n with m-input/m-output and sampling period of T :

xk+1 = Axk + Buk y k = Cxk

(1)

where xk := x(kT ), y k := y(kT ) and uk := u(kT ) represent the state vector, the output vector and the input vector of the system at a time instant kT , respectively. k is a step number of sampling. Further, suppose that the considered system is basically uncertain, but it satises the following assumptions.

Assumption 1

For the system (1), there exists a known stable parallel feedforward compen-

sator (PFC) with arbitrary order: xf p,k+1 = Af p xf p,k + Bf p uk y f p,k = Cf p xf p,k

5


(2)


Page 6 of 32

such that the resulting augmented system: xap,k+1 = Aap xap,k + Bap uk

(3)

y ap,k = Cap xap,k

with xap,k = [xTk , xTfp,k ]T and 







 A 0   B  Aap =  ,  , Bap =  0 Af p Bf p Cap = C Cf p ,

(4)

is strictly minimum phase, i.e. the transmission zeros of the system is asymptotically stable, and has a relative degree of {1, 1, · · · , 1} i.e. the relative McMillan degree of the system is (n − m)/n. Assumption 2

For the system (1), there exists a known stable parallel feedforward compen-

sator (PFC) with arbitrary order: xf,k+1 = Af xf,k + Bf uk y f,k = Cf xf,k + Df uk

(5)

such that the resulting augmented system: xa,k+1 = Aa xa,k + Ba uk y a,k = y k + y f k = Ca xa,k + Df uk

6


(6)

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with xa,k = [xTk , xTf,k ]T and 







 A 0   B  Aa =   , Ba =  , 0 Af Bf Ca = C Cf ,

(7)

is ASPR (or strongly ASPR 2 ), i.e. the augmented system is strictly minimum phase and has a relative degree of {0, 0, · · · , 0} with Df > 0. For the denition of relative degree of MIMO systems, refer Ref. 34. The main objective of this paper is to propose a design scheme of adaptive output predictive control for MIMO systems and to design a tow-degree-of-freedom output feedback control system with the designed predictive control as a feedforward input based on the ASPR property of the system in order to maintain the stability of the resulting control system under the imposed assumptions, Assumption 1 and 2.

Adaptive Output Predictive Control Design

Adaptive Output Estimator for MIMO System Consider the augmented system (3) with a PFC (2) satisfying Assumption 1 for the considered controlled system (1). We call this augmented system òutput estimator (OE) augmented system'. From Assumption 1, the OE augmented system has relative degree of

{1, 1, · · · , 1} and is minimum-phase. For the system (3) with relative degree of {1, 1, · · · , 1}, it has been claried that there exists a nonsingular transformation ξ k = [y Tap,k , η Tk ]T = Φxap,k such that the system can be transformed into the following canonical 34 :

y ap,k+1 = Ay y ap,k + By uk + Cη η k η k+1 = Aη η k + Bη y ap,k 7


(8)


Page 8 of 32

Moreover, the minimum-phase property of the system, the appeared zero dynamics of (8):

η k+1 = Aη η k must be stable. Taking notice of the form in (8) and considering the fact that the zero dynamics of (8) is stable, we propose designing the adaptive output estimator as follows:

ˆy,k uk−1 ˆ ap,k = Aˆy,k y ap,k−1 + B y

(9)

by adaptively adjusting unknown parameter matrices Ay and By and disregarding the af-

ˆyk are identied matrices of Ay and By fect from the zero dynamics. Where, Aˆyk and B respectively, and they are adjusted by the following adaptive parameter adjusting laws:

Aˆy,k = σ ¯ Aˆy,k−1 − σ ¯ k y Tap,k−1 ΓA + PA,k

(10)

ˆy,k = σ ˆy,k−1 − σ B ¯B ¯ k uTk−1 ΓB + PB,k

(11)

ΓA , ΓB > 0 , σ ¯=

1 , σ>0 1+σ

ˆ ap,k − y ap,k is the output estimated error, and PA,k and PB,k are parameter where, k = y projections which keep adjusted parameters in the known parameter existing range. [ij] ˆy,k = [ˆb[ij] ] are expressed by Each (i, j) element of Aˆy,k = [ˆ ay,k ] and B y,k

[ij] a ˆy,k

=

[ij] σ ¯a ˆy,k−1

−σ ¯

[ij] ˆb[ij] = σ ¯ˆby,k−1 − σ ¯ y,k

m X l=1 m X

[i] [l]

[ij]

k yapk−1 γa[lj] + pak [i] [l]

[lj]

[ij]

k uk−1 γb + pbk

(12) (13)

l=1 [ij] [ij] [ij] [ij] ˆy,k and PA,k , PB,k , and where a ˆy,k , ˆby,k and pa,k , pb,k are the (i, j ) element of Aˆy,k , B [j]

[j]

[j]

k , yap,k , uk are j -th element of k , y apk , uk , respectively. The parameter projection [ij]

[ij]

PA,k = [pa,k ] and PB,k = [pb,k ] are designed as follows using the given information about

8


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[ij]

[ij]

upper bound and lower bound of parameters ay and by such as [ij]

[ij] [ij] [ij] [ij] a[ij] y ≤ ay ≤ ay , by ≤ by ≤ by

[ij]

(14)

[ij]

[Parameter projections PA,k = [pa,k ] and PB,k = [pb,k ]]

[ij] pa,k

[ij]

pb,k

=

   0

[ij]

[ij]

(ay ≤ fa,k ≤ a[ij] y )

P [i] [l] [lj]   σ otherwise ¯ m l=1 k yap,k−1 γa  [ij]  [ij]  0 (b[ij] y ≤ fb,k ≤ by ) = P [i] [l] [lj]   σ otherwise ¯ m l=1 k uk−1 γb [ij] fa,k

=

[ij] σ ¯a ˆy,k−1

−σ ¯

[ij] [ij] fb,k = σ ¯ˆby,k−1 − σ ¯

m X j=1 m X

(15)

(16)

[i] [l]

k yap,k−1 γa[lj] [i] [l]

[lj]

k uk−1 γb

j=1

It should be noted that taking the form in (10) and (11) into consideration, it turns out

ˆ ap,k − y ap,k cannot be utilized directly from the that the output estimated error signal k = y causality problem. However fortunately, the output estimated error k can be equivalently obtained from available signals by

k =

ˆy,k−1 uk−1 − y ap,k σ ¯ Aˆy,k−1 y ap,k−1 + σ ¯B 1+σ ¯ y Tap,k−1 ΓA y ap,k−1 + σ ¯ uTk−1 ΓB uk−1

(17)

As for the boundedness of the signals in the obtained adaptive output estimator, we have the following lemma. Lemma 1

All signals in the designed adaptive output estimator are bounded with a bounded

input and a bounded output.

9



Proof.

Page 10 of 32

Let's consider the following positive denite function: (18)

Va,k = Va1,k + Va2,k + Va3,k + Va4,k Va1,k = ρ1 Tk P k , Va2,k = ρ2 η Tk Pη η k o n o n −1 T T Va3,k = σ ¯ tr ∆Ay,k Γ−1 ∆A , V = σ ¯ tr ∆B Γ ∆B a4,k y,k y,k y,k A B

ˆy,k − By . ρ1 , ρ2 > 0 are any positive constants and with ∆Ay,k = Aˆy,k − Ay , ∆By,k = B P = PT > 0 is any positive denite matrix. Pη is also positive denite matrix satisfying the following Lyapunov equation for any positive denite matrix Qη :

ATη Pη Aη − Pη = −Qη

(19)

Since the zero dynamics of the augmented system is stable from Assumption 1, such positive denite matrices exist. Dene the deference of Va,k by ∆Va,k = Va,k − Va,k−1 . The deference of Va1,k can be evaluated as

∆Va1,k ≤ ρ1 α11 kk k2 − ρ1 α12 kk−1 k2

(20)

α11 = λmax [P ], α12 = λmin [P ] The deference of Va2,k is evaluated by

( ∆Va2,k ≤ −{ρ2 α21 − δ2 }kη k−1 k2 + ρ2

) 2 ρ2 α23 ky ap,k−1 k2 α22 + δ2

(21)

α21 = λmin [Qη ], α22 = kPη kkBη k2 , α23 = kAη kkPη kkBη k with any positive constant δ2 . Moreover, taking the fact that

∆Ay,k−1 = σ ¯ −1 ∆Ay,k + k y Tap,k−1 ΓA + σAy − σ ¯ −1 PA,k

10


(22)

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into consideration, the deference ∆Va3,k can be obtained by

n o n o −1 T T ∆Va3,k = σ ¯ tr ∆Ay,k Γ−1 ∆A − σ ¯ tr ∆A Γ ∆A y,k−1 A y,k y,k−1 A n o n o T T =− σ ¯ −1 − σ ¯ tr ∆Ay,k Γ−1 ∆A − 2tr ∆A y y,k ap,k−1 k y,k A n o n o T −¯ σ tr k y Tap,k−1 ΓA y ap,k−1 Tk − σ ¯ σ 2 tr Ay Γ−1 A y A o n o n T −2σtr ∆Ay,k Γ−1 − 2¯ σ σtr Ay y ap,k−1 Tk a Ay n o n o −1 T T −1 +2tr σ ¯ −1 ∆Ay,k + k y Tap,k−1 ΓA + σAy Γ−1 P − σ ¯ tr P Γ P A,k A A,k A,k (23) A [ij]

At rst, consider the case where ay

[ij]

for all (i, j) i.e. PA,k = 0. The ≤ fak ≤ a[ij] y

deference of ∆Va3k is evaluated by

o n o n T T − 2tr ∆A y ∆Va3,k ≤ − σ ¯ −1 − σ ¯ tr ∆Ay,k Γ−1 ∆A y,k ap,k−1 k y,k A n o n o 1 n o −1 T 2 T T σ tr A Γ A − 2¯ +δ3 tr ∆Ay,k Γ−1 ∆A + σ σtr A y y y ap,k−1 k y y,k A A δ3 m X [i] ≤ − σ ¯ −1 − σ ¯ − δ3 λmin [Γ−1 ] k∆ay,k k2 − 2Tk ∆Ay,k y ap,k−1 A i=1

+

[i]

1 2 σ λmax [Γ−1 A ] δ3

m X

2 2 ka[i] y k + δ3 kk k +

i=1

1 2 2 σ ¯ σ kAy k2 ky ap,k−1 k2 δ3

(24)

[i]

where ∆ay,k and ay are the i-th row vectors of ∆Ay,k and Ay respectively. δ3 is any positive constant. [ij]

[ij]

Similarly, for the case where b[ij] y ≤ fbk ≤ by for all (i, j) i.e. PB,k = 0, we have

[i] 2 T ∆Va4,k ≤ − σ ¯ −1 − σ ¯ − δ3 λmin [Γ−1 B ]k∆by,k k − 2k ∆By,k uk−1 +

1 2 1 2 2 [i] 2 2 σ λmax [Γ−1 σ ¯ σ kBy k2 kuk−1 k2 B ]kby k + δ3 kk k + δ3 δ3

Therefore, by considering the fact that

k = ∆Ay,k y ap,k−1 + ∆By,k uk−1 − Cη η k−1

11


(25)


Page 12 of 32

we obtain in this case that

∆Va,k

1 2 ≤ − (2 − ρ1 α11 − 2δ3 − δη ) kk k − ρ1 α12 kk−1 k − ρ2 α21 − δ2 − kCη k kη k−1 k2 δη ( ) m m X X [i] [i] − σ ¯ −1 − σ ¯ − δ3 λmin [Γ−1 k∆ay,k k2 + λmin [Γ−1 k∆by,k k2 A ] B ] 2

2

(

2

i=1 m X

i=1

)

m X

σ −1 2 2 λmax [Γ−1 ka[i] kb[i] y k + λmax [ΓB ] y k A ] δ3 i=1 ) i=1 ( ) ( 2 1 2 2 1 ρ2 α23 + σ ¯ σ kAy k2 ky ap,k−1 k2 + σ ¯ 2 σ 2 kBy k2 kuk−1 k2 (26) + ρ2 α22 + δ2 δ3 δ3 +

[ij]

[ij]

On the other hand, for the case where fak < ay

[ij]

< fak for all (i, j) i.e. and a[ij] y

Pa,k = σ ¯ k y Tap,k−1 ΓA , since we have n T o T 2tr σ ¯ −1 ∆Ay,k + k y Tap,k−1 ΓA + σAy Γ−1 σ ¯ y Γ k ap,k−1 A A n T o T −¯ σ −1 tr σ ¯ k y Tap,k−1 ΓA Γ−1 σ ¯ y Γ k ap,k−1 A A o n T T T T ≤ 2tr ∆Ay,k y ap,k−1 k + σ ¯ k y ap,k−1 ΓA y ap,k−1 k + σ¯ σ Ay y ap,k−1 k o n o n T T T = 2 k ∆Ay,k y ap,k−1 + σ¯ σ k Ay y ap,k−1 + σ ¯ tr k y ap,k−1 ΓA y ap,k−1 k n o ≤ 2 Aˆy,k − Ay + σ¯ σ Ay kk kky ap,k−1 k + σ ¯ tr k y Tap,k−1 ΓA y ap,k−1 Tk o n ≤ 2 kA˜y k + σ ¯ kAy k kk kky ap,k−1 k + σ ¯ tr k y Tap,k−1 ΓA y ap,k−1 Tk

(27)

[ij] [ij] [ij] ˜y = max[|ay , a[ij] with A˜y = [˜ ay ], a y ], ∆Va3,k can be evaluated in this case as

¯ ∆Va3,k ≤ − σ

−1

−σ ¯−

δ3 λmin [Γ−1 A ]

m X

[i]

k∆ay,k k2

i=1 m

−2Tk ∆Ay,k y ap,k−1

X 1 2 + σ 2 λmax [Γ−1 ] ka[i] y k A δ3 i=1

1 2 2 ¯ σ kAy k2 ky ap,k−1 k2 +δ3 kk k2 + σ δ3 ˜ +2 kAy k + σ ¯ kAy k kk kky ap,k−1 k

12


(28)

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[ij]

[ij]

[ij]

Similarly, we obtain for fbk < b[ij] ¯ k y Tbp,k−1 ΓB that y and by < fbk for all (i, j) with PB,k = σ

∆Va4,k ≤ − σ ¯

−1

−σ ¯−

δ3 λmin [Γ−1 B ]

m X

[i]

k∆by,k k2

i=1 m

−2Tk ∆By,k uk−1

X 1 2 ] kb[i] + σ 2 λmax [Γ−1 y k B δ3 i=1

1 2 2 +δ3 kk k2 + σ ¯ σ kBy k2 kuk−1 k2 δ3 ˜ ¯ kBy k kk kkuk−1 k +2 kBy k + σ

(29)

[ij] [ij] ˜[ij] ˜y = [˜b[ij] where B y ], by = max[|by , by ].

Thus ∆Va,k can be evaluated in this case that

∆Va,k ≤ − (2 − ρ1 α11 − 2δ3 − 2δa − δη ) kk k2 − ρ1 α12 kk−1 k2 1 2 − ρ2 α21 − δ2 − kCη k kη k−1 k2 δη ) ( m m X X [i] [i] k∆by,k k2 − σ ¯ −1 − σ ¯ − δ3 λmin [Γ−1 k∆ay,k k2 + λmin [Γ−1 B ] A ] i=1 m X

m X

!

i=1

σ2 −1 2 2 λmax [Γ−1 ka[i] kb[i] y k + λmax [ΓB ] y k A ] δ3 i=1 i=1 ( ) 2 2 1 2 2 1 ρ2 α23 + σ ¯ σ kAy k2 + kA˜y k + σ ¯ kAy k ky ap,k−1 k2 + ρ2 α22 + δ2 δ3 δa ( ) 2 1 2 2 1 ˜y k + σ + ¯ kBy k kuk−1 k2 (30) σ ¯ σ kBy k2 + kB δ3 δa

+

with any positive constants δi . Using the similar way, it is also easy to conrm that the deference of ∆Va,k for other set of PA,k and PB,k can also be evaluated as in (30). Finally, by setting appropriate constants

13



Page 14 of 32

δ2 , δ3 , δa , δη , ρ1 , ρ2 such that 2 − ρ1 α11 − 2δ3 − 2δa − δη > 0 ρ2 α21 − δ2 −

1 kCη k2 > 0 δη

σ ¯ −1 − σ ¯ − δ3 > 0 we can conclude that all signals in the output estimator are bounded provided that ky ap,k k and kuk k are bounded.

Adaptive Output Predictor for MIMO System Let's denote i-step future value of the output signal y from k th sampling instant by y k(i) with y k(0) = y k . Since the output estimator (9) is designed for the augmented system with a PFC, the estimated output is the one of the augmented system by necessity. Taking this into account,

i-step output predictor for the practical system is designed by the following form: ˆ k(i) = y ˆ ap,k(i) − y ˆ f p,k(i) y ˆy,k v ˆ ap,k(i−1) + B ¯ k(i−1) − y ˆ f p,k(i) (i = 1, 2, · · · ) = Aˆy,k y

(31)

ˆ k(0) = y k (0) = y k . Where, y ˆ k(i) is the predicted i-step future value of output y k(i) and with y ¯ k(i) is an i-step future input to be determined later. y ˆ f p,k(i) is the i-step future output of v ¯ k(i) . Consequently, using the available signals at k th the PFC given in (2) with the input v ¯ k(j) , (j = 0, 1, · · · , i), it follows that the predicted sampling instant and the predicted input v output can be represented by

ˆ k(i) y

= Aîy,k y ap,k +

i X

ˆ ¯ k(j−1) − Cf p Aif p xf p,k − Aî−j y,k By,k v

i X j=1

j=1

14


¯ k(j−1) Cf p Ai−j f p Bf p v

(32)

Page 15 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Finally, 1-step to np -step predicted outputs are obtained as

ˆ pk = y

T ˆ k(1) · · · y

ˆ k(np ) y

ˆ ˆ ˆ ¯ ak = Aay,k y ap,k − Aest xf p,k + Ak Bk − Best v

(33)

where

 ¯ a,k v

  =   

Aest



 ¯ k(0) v .. . ¯ k(np −1) v

 Cf p Af p   Cf p A2 fp  = ..  .   n Cf p Af pp

     , Aây,k =      



 Aˆy,k .. . np Aˆy,k

  ,   

0 ··· 0    I   ..  ..   Aˆy,k . I .   ˆ   , A =  k  , .. .. ..   . . 0  .       np −1 ˆ ˆ Ay,k · · · Ay,k I



 ˆ 0 ··· 0   By,k  ..   0 B ˆy,k . . . .   ˆk =  B  . , .. ..  .. . . 0      ˆ 0 ··· 0 By,k  Cf p Bf p 0 ··· 0   .. ..  Cf p Af p Bf p . . . . .  Best =  .. .. ..  . . . 0   n −1 Cf p Af pp Bf p · · · Cf p Af p Bf p Cf p Bf p

15


        


Page 16 of 32

Output Predictive Control Input Using the designed output predictor (33), the desired control input is determined so as to minimize the following performance function.

Jk =

np n X

ˆ k(i) − y r,k(i) y

i=1

ˆ p,k − y ¯ r,k = y

T

with Λ = diag

T

o ¯ Tk(i−1) Λi−1 v ¯ k(i−1) ˆ k(i) − y r,k(i) + v y

ˆ p,k − y ¯ r,k + v ¯ Ta,k Λ¯ y v a,k

(34)

Λ0 · · ·

Λnp −1

, Λi > 0 (i = 0, 1, · · · , np − 1). Where

¯ r,k = y

T y r,k(1) · · ·

y r,k(np )

and y r,k is reference signal (or desired trajectory) for which the output y k is required to follow.

¯ a,k so as to minimize the given performance function Jk is obtained The optimal input v by using the least squares method as follows:

−1 T ¯ a,k = − WkT Wk + Λ v Wk xv,k

(35)

where

ˆk − Best Wk = Aˆk B ¯ r,k xvk = Aây,k y ap,k − Aest xf pk − y The output predictive control input at k th sampling instant is then designed by using

¯ k(0) in the obtained optimal control input v ¯ a,k . the input v

16


Page 17 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Two-Degree-Of-Freedom Output Feedback Control System ¯ k(0) only as the control In the case where we adopt the designed adaptive predictive control v input, the stability of the obtained control system is not necessarily guaranteed as it is. In order to maintain the boundedness of all signals in the obtained control system, we introduce ASPR based output feedback control in addition to the adaptive predictive control and then construct two-degree-of-freedom control system.

ASPR based Output Feedback Control with Predictive feedforward Input Under assumption 2, suppose that the controlled system can be rendered ASPR with the PFC as in (6). The ASPR augmented system is then represented by

xa,k+1 = Aa xa,k + Ba uk y a,k = Ca xa,k + Df uk Now, impose the following additional assumption on the controlled system. Assumption 3

There exist ideal state x∗k and ideal input v ∗k such that the following perfect

output tracking is attained for a given reference signal y rk . x∗k+1 = Ax∗k + Bv ∗k y ∗k

=

Cx∗k

= y r,k

(36)

If the perfect output tracking is achieved, the ideal state of the PFC in the feedback control system must satisfy the following relation:

x∗f,k+1 = Af x∗f,k y ∗f,k

=

Cf x∗f,k

17

=0


(37)


Page 18 of 32

Feedforward +

yr − Θp

−

ue +

v y

u Plant

ya

+

yf +

+ PFC

Figure 2: Closed loop system Therefore, in this case, the ideal state x∗a,k of the augmented system is expressed as

¯ ∗ x∗a,k+1 = Aa x∗a,k + Bv k y ∗a,k = Ca x∗ak = y ∗k = y rk

(38)

where

  x∗a,k = 

 x∗k x∗f,k





 ¯  B  , B =   0

With these preparation, the control system is constructed as shown in Fig. 2 with twodegree-of-freedom control form, i.e. the control input is designed by

uk = ue,k + v k

(39)

where

ue,k = −Θp ea,k , Θp > 0

(40)

ea,k = y a,k − y r,k , y a,k = y k + y f,k

(41)

and y f,k is the output of the PFC (6) with the input ue,k . v k is the feedforward input which

18


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¯ k(0) with the following input constrain: is set by the predictive control v

vk =

   v ¯ k(0) (k¯ v k(0) k < v max )   v max (k¯ v k(0) k ≥ v max )

(42)

From the ASPR property of the augmented system, it is apparent that there exists an ideal feedback gain Θ∗p such that the resulting closed-loop system is SPR or stable, and the augmented system is stabilized with any feedback gain Θp ≥ Θ∗p . Note that the designed feedback control input ue,k is not available directly due to causality problem. However, fortunately, taking into consideration that the PFC input is also given by ue,k , we can realize the equivalent control input using available signals as follows:

ue,k = −Θp {y k + y f,k − y r,k } = −Θp {y k + Cf xf,k + Df ue,k − y r,k }

(43)

and thus we have

˜ pe ˜ ak uek = −Θ

(44)

˜ p = {I + Θp Df }−1 Θp and e ˜ a,k = y k + Cf xf,k − y r,k . where Θ Concerning the boundedness of all signals in the control system, we have the following theorem. Theorem 1

Under Assumptions 1, 2 and 3, all the signals in the resulting control system

with the control input (39) designed by (40) and (42) are bounded, and if v k ≡ v ∗k were achieved, then ek = y k − y r,k converges to zero as k → ∞. Proof.

See Appendix.

19



Figure 3: Two-tank system

Experimental Validation Through Three-Tank System We conrmed the eectiveness of the proposed method through experiments on a liquid level control of a practical three-tank process. The structure of the liquid level three-tank process is shown in Fig. 3 with the picture of the equipment. The outputs of the three-tank process are the liquid level of the tank 1 and tank 3, and they are measured using a pressure sensor which has the sensor resolution of the 0.6 [mm]. The control inputs are the voltage of pumps applied to tank 2 and tank 3. The pump supplies water to tank 2 and tank 3 at a ow rate proportional to the voltage. Input restriction of the pump is 0.25 - 7 [V]. In this experiments, we set a sampling period of T = 10[s].

20


Page 20 of 32

Page 21 of 32

50

35

45

30

40 25

Output[mm]

Output[mm]

35 30 25 20 15 10

20 15 10 5

5 0

Plant approximate response

0 -5


-5 0

1000

2000

3000

4000

5000

6000

0

1000

2000

Time[s]

3000

4000

5000

6000

Time[s]

(a) Step response of G∗11

(b) Step response of G∗12 50

40

45

35

40 30 35 25

Output[mm]

Output[mm]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


20 15 10

30 25 20 15 10

5 5


0


0

-5

-5 0

1000

2000

3000

4000

5000

6000

0

1000

Time[s]

2000

3000

4000

5000

6000

Time[s]

(c) Step response of G∗21

(d) Step response of G∗22

Figure 4: Step responses of the three-tank process and the nominal model In order to design PFCs, we set a nominal model G∗p (z) as follows:

  G∗ (z) = 

 G∗11 (z)

G∗12 (z)

G∗21 (z) G∗22 (z)

 

(45)

7.551×10−3 z 3 + 1.279×10−2 z 2 − 1.69×10−2 z − 2.997×10−3 z 4 − 3.078z 3 + 3.314z 2 − 1.393z + 0.1564 4.245×10−4 z 4 + 2.874×10−3 z 3 − 9.285×10−4 z 2 − 2.007×10−3 z − 1.254×10−4 G∗12 (z) = z 5 − 3.775z 4 + 5.49z 3 − 3.788z 2 + 1.207z − 0.1338 −3 3 6.459×10 z + 1.669×10−2 z 2 − 1.785×10−2 z − 5.166×10−3 G∗21 (z) = z 4 − 3.565z 3 + 4.775z 2 − 2.852z + 0.6425 3.108×10−2 z 4 + 5.822×10−2 z 3 − 7.566×10−2 z 2 − 1.235×10−2 z − 2.87×10−4 G∗22 (z) = z 5 − 2.767z 4 + 2.489z 3 − 0.6744z 2 + 4.706×10−2 z − 7.406×10−4 G∗11 (z) =

The nominal model of the three-tank system was obtained by applying Prnoy's exponential analysis method 35 from a step response input/output data. Fig. 4 shows step responses of the three-tank process and the nominal model. It seems a good nominal model is obtained. However, if the properties of the tank system was changed from the one of the nominal 21



50

35

45

30

40 25

Output[mm]

Output[mm]

35 30 25 20 15 10

20 15 10 5

5

Plant new Plant old

0

Plant new Plant old

0

-5

-5 0

1000

2000

3000

4000

5000

6000

0

1000

2000

Time[s]

3000

4000

5000

6000

Time[s]

(a) Step responses of G11

(b) Step responses of G12 50

40

45

35

40 30 35 25

Output[mm]

Output[mm]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 32

20 15 10

30 25 20 15 10

5 5

Plant new Plant old

0

Plant new Plant old

0

-5

-5 0

1000

2000

3000

4000

5000

6000

0

1000

2000

Time[s]

3000

4000

5000

6000

Time[s]

(c) Step responses of G21

(d) Step responses of G22

Figure 5: Dierence of step responses' data of the three-tank process collected on dierent days model obtained before implementing the control, the control performance with the controller designed based on only the nominal model will degrade. The gure 5 shows the dierence of the step responses which data sets were collected on dierent days. It is unrealistic to identify the system every time when we do control the system. That is why the adaptive control is required to adjust the change of the system properties. The PFCs for designing a control system are designed by the model-based scheme and thus they are designed by

Hest (z) = Gest (z) − G∗ (z)

(46)

Haspr (z) = Gaspr (z) − G∗ (z)

(47)

22


Page 23 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


with the following ideal model satisfying Assumptions 1 and 2:

  Gest (z) =    Gaspr (z) = 

1 z − 0.99 0.1 z − 0.99 50z z − 0.9 0

 0.1 z − 0.99   1 z − 0.99  0  50z  z − 0.9

(48)

(49)

The design parameters in the controller are set as

ΓA = diag[0.1 0.1], ΓB = diag[1 1] σ = 1.0 × 10−7 , Λi = diag[7.7 7.7], (i = 0, 1, ..., np − 1) T np = 20, v max = 30 30 , Θp = diag[1 × 105 1 × 105 ] It should be noted that nominal parameters Ay and By in (8) are obtained from (48) as









 1 0.1   0.99 0  Ay =    , By =  0.1 1 0 0.99 Remark 1

The design guiding principles for Gest (z) and Gaspr (z) are provided as follows:

1. The ideal model: Gest (z) for output estimation, Gest (z) is designed such that Gest (z)−1 has relatively small gain in order to make the eect from unmodelled parts small. See Ref. 31 for the basic design guiding principle of Gest (z). 2. The ideal ASPR model: Gaspr (z) for output feedback, the simple ASPR model with relatively large gain is recommended to set in order to maintain ASPR-ness for large operating range. We rst tried to control the three-tank process by using a controller with nominal parameters without adaptation in the predictive feedforward input. Figure 6 show experimental 23



50

Page 24 of 32

8

40 6 30 20

4

10 2 0 -10

0 0

1000

2000

3000

4000

5000

6000

80

0

1000

2000

0

1000

2000

3000

4000

5000

6000

3000

4000

5000

6000

8

60

6

40 4 20 2

0 -20

0 0

1000

2000

3000

4000

5000

6000

Time[s]

Time[s]

(a) Output tracking results

(b) Control inputs

Figure 6: Experimental results with a nominal model results with nominal parameters. Even though the obtained model seems accurate, the resulting control performance were not acceptable. Especially, the output (liquid level) of the tank 3 varied violently since the input (liquid ow) is directly injected to the tank 3 and the output of tank 2 has o-set errors. Figure 7 shows the results with the proposed method. By adaptively adjusting the output estimator for obtaining the predictive output for predictive feedforward input, the variance of the inputs and outputs are eectively reduced and pretty good control performance is obtained.

Conclusion In this paper, a design scheme of a novel output predictive control based on adaptive output estimator with a simple structure was proposed for MIMO LTI systems. Moreover, twodegree-of-freedom output feedback control with the proposed adaptive predictive control as the feedforward input was considered in order to maintain the stability of the resulting control system. Furthermore, the eectiveness of the proposed two-degree-of-freedom output feedback control with the adaptive predictive feedfoward input was conrmed through experiments on liquid level control of a three-thank process.

24


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8 50

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40 30

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6

60 40

4

20

2 0

0

-20 0

1000

2000

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6000

Time[s]

Time[s]

(a) Output tracking results

(b) Control inputs

1.5

2

1

1

0.5

0 0

1000

2000

3000

4000

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6000

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0

1000

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0

0

-0.5

-1 0

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6000

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1

0

0

-0.5

-1 0

1000

2000

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4000

5000

6000 10

1.5 1

5 0.5 0

0 0

1000

2000

3000

4000

5000

6000

Time[s]

Time[s]

(c) Estimated results of Ay

(d) Estimated results of By

Figure 7: Experimental results with the proposed method

Appendix: Proof of Theorem 1 Dene X k := xak − x∗ak and ∆v k := v k − v ∗k . Then the following error system is obtained:

¯f ∆v k X k+1 = Aac X k + Ba ∆v k − B eak = Cac X k + Df ∆v k − Df ∆v k

(50)

where

Aac

˜ ˜ = Aa − Ba Θp Ca , Cac = I − Df Θp Ca ,

¯f = [0 · · · 0 BfT ]T B From Assumption 2, for the system with the feedback gain Θp ≥ Θ∗p , the closed-loop system (Aac , Ba , Cac , Df ) is SPR, and thus there exist positive denite matrices P = 25



Page 26 of 32

P T > 0, Q = QT > 0 and appropriate matrices L and W such that the following KalmanYakubovich-Popov Lemma is satised.

ATac P Aac − P = −Q − LLT T ATac P Ba = Cac − LW

(51)

BaT P Ba = Df + DfT − W T W As a candidate of the Lyapunov function, the following positive denite function with the positive denite matrix P in (51) is considered:

Vk = X Tk P X k

(52)

The dierence ∆Vk = Vk − Vk−1 can be evaluated by

∆Vk

n o T ≤ −ρ λmin kQk − δλmax kP k − δ1 λmax [Cac Cac ] kX k−1 k2 1 1 T T ¯ ¯ + ρ + ρλmax [Df + Df ] + ρ λmax [Bf P Bf ] k∆v max k δ1 δ

where 1 > δ > 0, ρ =

1 1−δ

(53)

> 0 and δ1 > 0 are any positive constants. Consider positive

constants δ and δ1 so as to satisfy the following relation: T λmin kQk − δλmax kP k − δ1 λmax [Cac Cac ] > 0

From the fact that k∆v max k = kv max − v k k is bounded, it follows that kX k k is bounded and then y k and uk are also bounden. Thus it follows from the boundedness of y k and uk that the signals in the adaptive output predictor are also bounded. Consequently, the boundedness all the signals in the control system can be conrmed. Moreover, in the case that ∆v k ≡ 0, we obtain that kX k k tends to zero and thus conclude that if v k ≡ v ∗k were achieved, then

ek = y k − y r,k converges to zero as k → ∞ in order to stay zero 36,37 . 26


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Adaptive output predictive control

Page 32 of 32

yˆ a

+

−

v yr −

ue

+

u

Plant PFC ASPR

ASPR based output feedback


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y

+ yf

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Output Feedback-Based Output Tracking Control with Adaptive Output

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