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ASPR based Output Feedback Control with an Adaptive Predictive Feedfoward Input for MIMO Systems
Proceedings of the 20th World Congress The International Federation of Congress Automatic Control Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World Toulouse, France, July 2017 The International Federation of Automatic Control Available online at www.sciencedirect.com Toulouse, France,Federation July 9-14, 2017 The International of Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 5313–5318 ASPR based Output Feedback Control ASPR based Output Feedback Control ASPR based Output Feedback Control ASPR based Output Feedback Control with an Adaptive Predictive Feedfoward with an Adaptive Predictive Feedfoward with an Adaptive Predictive Feedfoward with anInput Adaptive Predictive Feedfoward for MIMO Systems Input for MIMO Systems Input for MIMO Systems Input for MIMO Systems ∗ ∗∗
Ikuro Mizumoto ∗ Seiya Fujii ∗∗ Ikuro Ikuro Mizumoto Mizumoto ∗∗ Seiya Seiya Fujii Fujii ∗∗ Ikuro Mizumoto Seiya Fujii ∗∗ ∗ ∗ Faculty of Advanced Science and Technology, Kumamoto University, ∗ Faculty of Advanced Science and Technology, Kumamoto University, of Advanced Science [email protected]). Technology, Kumamoto University, Kumamoto, Japan (e-mail: ∗ Faculty Japan (e-mail: [email protected]). Faculty of Advanced Science and Technology, KumamotoUniversity, University, ∗∗ Kumamoto, Kumamoto, Japan (e-mail: [email protected]). Department of Mechanical System Eng., Kumamoto ∗∗ of Mechanical System Eng., Kumamoto University, Kumamoto, Japan (e-mail: [email protected]). ∗∗ Department SystemJapan Eng., Kumamoto University, Kumamoto, ∗∗ Department of Mechanical Kumamoto, Japan Department of Mechanical SystemJapan Eng., Kumamoto University, Kumamoto, Kumamoto, Japan Abstract: Abstract: This This paper paper deals deals with with an an output output feedback feedback regulation regulation control control system system design design problem. problem. Abstract: This paper deals with an output feedback regulation control system design problem. An almost strictly positive real or ASPR based output feedback control system with the adaptive An almost strictly positive real or ASPR based output feedback control system with the adaptive Abstract: This paper deals with an output feedback regulation control system design problem. An almost feedforward strictly positive realwill or ASPR based output feedback control system with the adaptive predictive input be proposed for MIMO linear systems. The adaptive output predictive feedforward input will be proposed for MIMO linear systems. The adaptive output An almost strictly positive real or ASPR based output feedback control system with the adaptive predictiveproposed feedforward inputsystems will be will proposed for MIMO linear systems systems.and Thethe adaptive output predictor for SISO be expanded for MIMO stability of predictor proposed for SISO systems will be expanded for MIMO systems and the stability of the the predictive feedforward input will be proposed for MIMO linear systems. The adaptive output predictor control proposed for SISO systems will be expanded for MIMO systems and the control stabilitysystem of the resulting system will be maintained by designing two-degree of freedom resulting control system will be maintained by designing two-degree of freedom control system predictor proposed for SISO systems will be expanded for MIMO systems and the stability of the resulting control system feedback. will be maintained by designing two-degree of freedomsimple control system via based The proposed control strategy has relatively structure via ASPR ASPRcontrol based output output feedback. The proposed control strategy has relatively simple structure resulting system will output be maintained bycontrol designing two-degree of simple freedom control system via ASPR based output feedback. The proposed control strategy has relatively simple structure with a novel simple adaptive predictive and ASPR based output feedback. with a novel simple adaptive output predictive control and ASPR based simple output feedback. via ASPR based output feedback. The proposed control strategy has relatively simple structure with effectiveness a novel simple adaptive output predictive control and ASPR based simple output feedback. The of the method will be through numerical simulations. The effectiveness effectiveness ofadaptive the proposed proposed method will control be confirmed confirmed through numerical simulations. with a novel simple output predictive and ASPR based simple output feedback. The of the proposed method will be confirmed through numerical simulations. The effectiveness of the proposed method will beControl) confirmed through numerical © 2017, IFAC (International Federation of Automatic Hosting by Elsevier Ltd. Allsimulations. rights reserved. Keywords: Keywords: Adaptive Adaptive control, control, ASPR, ASPR, adaptive adaptive output output predictor, predictor, adaptive adaptive predictive predictive control, control, Keywords: Adaptive compensator control, ASPR, adaptive output predictor, adaptive predictive control, parallel feedforward parallel feedforward compensator Keywords: Adaptive control, ASPR, adaptive output predictor, adaptive predictive control, parallel feedforward compensator parallel feedforward compensator 1. In Kaufman et al. (1997); Mizumoto and Iwai (1996), 1. INTRODUCTION INTRODUCTION In et Mizumoto and (1996), 1. INTRODUCTION In Kaufman Kaufmanreference et al. al. (1997); (1997); Mizumoto and Iwai Iwai (1996), augmented model was introduced in order to augmented reference model was introduced in order to 1. INTRODUCTION In Kaufman et al. (1997); Mizumoto and Iwai (1996), augmented reference model wasoutput introduced in to order to alleviate the affects from PFC in order attain It has been well recognized that if the system satisfies alleviate the affects from PFC output in order to attain augmented reference model was introduced in order to It has been well recognized that if the system satisfies alleviate theregulation affects from PFC output in ordersystem. to attain the output for practical controlled In It has been wellpositive recognized that if the system satisfies ‘almost strictly real’ or ASPR condition, then the output regulation for practical controlled system. In alleviate the affects from PFC output in order to attain ‘almost strictly positive real’ or ASPR condition, then It has been well recognized if the with system satisfies output et regulation for Mizumoto practical controlled system. In Mizumoto al. (2009); and Tanaka (2010), ‘almost strictly positive real’that orsystems ASPR condition, then the one adaptive control very Mizumoto et al. (2009); and Tanaka (2010), the output regulation for Mizumoto practical controlled system. In one can can design design adaptive control systems with very simple simple ‘almost strictly positive real’ or ASPR condition, then Mizumoto et al. (2009); Mizumoto and Tanaka (2010), adaptive NN feedforward was proposed for feedforward one can design adaptive control systems with very simple structure through adaptive output feedback (Kaufman adaptive NN feedforward was proposed for feedforward Mizumoto et al. (2009); Mizumoto and Tanaka (2010), structure through adaptive output feedback (Kaufman one can design adaptive control systems with very simple adaptive NN feedforward was proposed for feedforward control. structure through adaptive output feedback (Kaufman et Barkana et 2014; and control. et al., al., 1997; 1997; Barkana et al., al., output 2014; Mizumoto Mizumoto and Iwai, Iwai, adaptive structure through adaptive feedback (Kaufman control. NN feedforward was proposed for feedforward et al., Fradkov 1997; Barkana et al., 2014; Mizumoto and Iwai, 1996; and Hill, 1998). The system is said to On the other hand, recently, a new control design scheme 1996; Fradkov and Hill, 1998). The system isand said to control. et al., Fradkov 1997; Barkana et al., 2014; Mizumoto Iwai, the other hand, recently, a control scheme 1996; and exists Hill, The system is said to On be if aa1998). static output feedback such On theon other hand, recently, a new new control control design design scheme based an adaptive predictive strategy with be ASPR ASPR if there there exists staticThe output feedback such 1996; Fradkov and Hill, 1998). system is said to based on an adaptive predictive control strategy with be ASPR if there closed exists loop a static output feedback such On the other hand, recently, a new control design scheme that the resulting system is strictly positive based on anoutput adaptive predictive control based strategy with an adaptive predictor and ASPR adaptive thatASPR the resulting closed loop system is strictly positive be if there exists a static output feedback such an adaptive output predictor and ASPR based adaptive that the resulting closed 1991). loop system is strictly however, positive based on an adaptive predictive control strategy with real (SPR) (Bar-Kana, Unfortunately, an adaptive output predictor and ASPR based adaptive output feedback has been proposed from a different viewreal (SPR) (Bar-Kana, 1991). Unfortunately, however, that (SPR) the resulting closed loop do system is strictly positive output feedback has been proposed from a different viewreal (Bar-Kana, 1991). Unfortunately, however, an adaptive output predictor and ASPR based adaptive since most practical system not satisfy the ASPR output feedback has been proposedPredictive from a different viewet al., 2015b,a). Control, insince (SPR) most practical system doUnfortunately, not satisfy thehowever, ASPR point(Mizumoto real (Bar-Kana, 1991).do et 2015b,a). Predictive Control, insince most the practical system not satisfy the ASPR point(Mizumoto output feedback has been proposed from aand different viewconditions, ASPR imposes severe restriction point(Mizumoto et al., al., 2015b,a). Predictive Control, including model predictive control (MPC) generalized conditions, the ASPR condition condition imposes severe the restriction since most practical system do not satisfy ASPR cluding model predictive control (MPC) and generalized conditions, the ASPR condition imposes severe restriction point(Mizumoto et al., 2015b,a). Predictive Control, into practical applications of the ASPR based adaptive model predictive control (MPC) and generalized predictive control (GPC), have attracted a great deal of to practicalthe applications of theimposes ASPRsevere basedrestriction adaptive cluding conditions, ASPR condition predictive control (GPC), have attracted a great deal to practical applications ofmethod the ASPR basedthe adaptive cluding model predictive control (MPC) and generalized controls. One of the simple to alleviate ASPR control (GPC), have attracted aresent great decades deal of of attention and widely used in industry for controls. One applications of the simpleofmethod to alleviate the ASPR predictive to practical the ASPR based adaptive attention and widely used in industry for controls. One of the simple method to alleviatefeedforward the ASPR predictive control (GPC), have attracted great decades deal of restriction is introduction of attention and widely used in et industry for aresent resent decades Clarke et al. (1987); Garcia al. (1989); Mayne et al. restrictionOne is the the introduction of the thetoparallel parallel feedforward controls. of the simple method alleviate the ASPR Clarke et al. (1987); Garcia et al. (1989); Mayne et al. restriction is (PFC) the introduction of1987; the Iwai parallel feedforward attention and widely used in industry for resent decades compensator (Bar-kana, and Mizumoto, Clarke et al. (1987); Garcia et al. (1989); Mayne et al. Maciejowski (2002) due to its powerful control compensator (PFC) (Bar-kana,of1987; Iwai and Mizumoto, (2000); restriction is (PFC) the introduction theFradkov, parallel (2000); et Maciejowski due to its control compensator (Bar-kana, 1987; Iwai andfeedforward Mizumoto, Clarke al. However, (1987); (2002) Garcia et al. Mayne et al. 1994; and Iwai, 1996). The Maciejowski (2002) due to (1989); its powerful powerful control performance. the performance of the predictive 1994; Mizumoto Mizumoto and Iwai, 1996; 1996; Fradkov, 1996). The (2000); compensator (PFC) (Bar-kana, 1987; Iwai and Mizumoto, performance. However, the performance of the predictive 1994; Mizumoto and Iwai, 1996; Fradkov, 1996). The (2000); Maciejowski (2002) due to its powerful control PFC is introduced so as to render an ASPR augmented performance. However, the performance of the predictive depends significantly on accuracy of the given PFC isMizumoto introducedand so as to render an ASPR 1996). augmented 1994; Iwai, 1996; based Fradkov, The control control depends significantly on accuracy of the PFC is introduced so as to ASPR render an ASPR augmented performance. However, the performance of the predictive controlled system, the control scheme depends significantly on accuracy of the given given model of the controlled system. The method in Mizumoto controlled system, and and the ASPR based control scheme control PFC is introduced so as to render an ASPR augmented model of the system. The method in Mizumoto controlled system, andrealized the ASPR based control scheme control significantly on accuracy of thepredicgiven can to ASPR augmented system ofdepends the controlled controlled system. The method in the Mizumoto et al. (2015b,a) is one of the adaptive version of can be be applied applied to the the realized ASPR augmented system model controlled system, and the ASPR based control scheme et al. (2015b,a) is one of the adaptive version of the prediccan be applied to the realized ASPR augmented system model of the controlled system. The method in Mizumoto with a PFC. However, the control system is designed to al.control. (2015b,a) is one of the adaptive version of the predictive In the method, an adaptive output predictor withbe a PFC. However, the control system is designed to et can applied to the realized ASPR augmented system tive In the method, an adaptive output predictor with a PFC. However, the control system is PFC designed to et al.control. (2015b,a) is one of the adaptive version of the predicthe augmented system, the affects from the output tive control. In the method, an adaptive output predictor with a simple structure derived from output estimator the augmented system, the affects from the PFC output with a PFC. However, system designed to with a simple structure derived from output estimator the augmented system, the the control affects from theis PFC output tive control. In the method, an adaptive output predictor to the control performance of the practical system has a simplesystems structure derived et from output estimator for multi-rate (Mizumoto al., 2010b; Mizumoto to the control performance of the the from practical system has with the augmented system, the affects the PFC output for multi-rate (Mizumoto et al., 2010b; Mizumoto of practical system has to the control performance with a simplesystems structure derived from estimator been controversial in the the for multi-rate systems (Mizumoto et al., output 2010b; Mizumoto and Fujimoto, 2011) is utilized for designing the predictive been controversial in the the case caseofwhere where the gain gain of of the PFC PFC to the control performance the practical system has and Fujimoto, 2011) is utilized for designing the predictive been controversial inInthe casetowhere the gain of the issue, PFC for multi-rate systems (Mizumoto et al., 2010b; Mizumoto was relatively large. order solve this additional and Fujimoto, 2011) is utilized for designing the predictive control input. A parallel feedforward compensator (PFC) was relatively large.inInthe order to solve this additional issue, been controversial case the gain of the issue, PFC control input. A parallel feedforward compensator (PFC) was relatively large. In order towhere solveinput this additional Fujimoto, 2011) issoutilized for designing the predictive how to design aa feedforward control has played played very and control input. A parallel feedforward compensator (PFC) is again introduced as to render a minimum-phase how to design feedforward control input has very was relatively large. In order to solve this additional issue, again introduced so as to render a minimum-phase how to design a(Kaufman feedforward control input has played very is control input. A parallel feedforward compensator (PFC) important role et al., 1997; Mizumoto and Iwai, is again introduced so aasrelative to render a of minimum-phase augmented system with degree 1, and for reimportant rolea(Kaufman et al., 1997;input Mizumoto and Iwai, how to design feedforward control has played very augmented system with a relative degree of 1, and for important role (Kaufman et Mizumoto al., 1997; Mizumoto and2010). Iwai, is again introduced so as to render a minimum-phase 1996; Mizumoto et al., 2009; and Tanaka, augmented system with a relative degree of 1, and for rereduced first order system, output predictor with very simple 1996; Mizumoto et al., 2009; Mizumoto and Tanaka, 2010). important role (Kaufman et al., 1997; Mizumoto and Iwai, duced first order system, predictor with simple 1996; Mizumoto et al., 2009; Mizumoto and Tanaka, 2010). augmented system with aoutput relative degree of 1, very and control for reduced first order system, output predictor with very simple structure is designed. The stability of the resulting 1996; Mizumoto et al., 2009; Mizumoto and Tanaka, 2010). duced structure is designed. The stability of the first system, output predictor withsystem verycontrol simple work was supported by Council for Science, Technology and structure isorder designed. The stability of control the resulting resulting control This system maintained by switching the from This work was supported by Council for Science, Technology and system maintained by switching the control system from structure is designed. The stability of the resulting control Innovation Cross-ministerial Innovation PromoThis work(CSTI), was supported by CouncilStrategic for Science, Technology and system maintained by switching the control system from predictive control to ASPR based adaptive output feed Innovation (CSTI), Cross-ministerial Strategic Innovation PromoThis work(CSTI), was supported by Council for Science, Technology and predictive control to ASPR based adaptive output feedtion Program (SIP), “Innovative Combustion Technology” (Funding Innovation Cross-ministerial Strategic Innovation Promosystem maintained by switching the control system from predictive control tomethod. ASPR However, based adaptive output feedtion Program (SIP), Cross-ministerial “Innovative Combustion Technology” (Funding back control in this adopting switching Innovation (CSTI), Strategic Innovation Promoagency: JST) as a part of researchCombustion on “Control Technology” and System Modelling tion Program (SIP), “Innovative (Funding back control in this method. However, adopting switching predictive control to ASPR based adaptive output feedagency: JST) as a part of researchCombustion on “Control and System Modelling back control intothis method. However, adopting switching strategy, how set the switching timing is very important tion Program (SIP), “Innovative (Funding on Fuel Technology Innovation” agency: JST) as a part of research. on “Control Technology” and System Modelling strategy, how set the switching timing is very important back control into this method. However, adopting switching on Fuel JST) Technology Innovation” . on “Control and System Modelling strategy, how to set the switching timing is very important agency: as a part of research on Fuel Technology Innovation” . strategy, how to set the switching timing is very important on Fuel Technology Innovation” .
Copyright © 2017 IFAC 5493 Copyright 2017 IFAC 5493Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017, 2017 IFAC 5493 Peer review©under of International Federation of Automatic Copyright 2017 responsibility IFAC 5493Control. 10.1016/j.ifacol.2017.08.985
Proceedings of the 20th IFAC World Congress 5314 Ikuro Mizumoto et al. / IFAC PapersOnLine 50-1 (2017) 5313–5318 Toulouse, France, July 9-14, 2017
parameter to maintain good control performance and it sometimes be difficult and complex. In this paper, we consider adopting the adaptive predictive control as a feedforward input and designing an adaptive control for unknown MIMO systems. That is, we propose ASPR based output feedback control with an adaptive predictive feedforward input for MIMO systems. The adaptive output predictor proposed by Mizumoto et al. (2015b,a) will be expanded for MIMO systems and the stability of the resulting control system will be maintained by designing two-degree of freedom control system via ASPR based output feedback. The proposed control strategy has relatively simple structure with a novel simple adaptive output predictive control and ASPR based simple output feedback. The effectiveness of the proposed ASPR based output feedback control with the adaptive predictive feedfoward input will be confirmed through numerical simulations. 2. PROBLEM STATEMENT Let’s consider the following nth order uniformly sampled MIMO unknown system with m-input/m-output and sampling period of T : xk+1 = Axk + Buk (1) y k = Cxk where xk := x(kT ) and y k := y(kT ) are the state vector and the output vector of the system at a time instant kT , and uk := u(kT ) is the input vector updated at a time instant kT .
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)
with xa,k = [xTk , xTf,k ]T and Aa =
A 0 0 Af
C a = [ C Cf ] ,
, Ba =
B Bf
, (7)
is ASPR (or strongly ASPR(Mizumoto et al., 2007)), i.e. the augmented system is strictly minimum phase and has a relative degree of {0, 0, · · · , 0} with Df > 0. For the definition of relative degree of MIMO systems, refer Isidori (1995). The objective of this paper is to design stable two-degree of freedom ASPR based output feedback control system with adaptive predictive feedforward input under the given assumptions 1 and 2. 3. ADAPTIVE OUTPUT PREDICTIVE CONTROL DESIGN 3.1 Adaptive Output Estimator for MIMO System
Suppose that the considered system is basically uncertain, but a nominal (or a simple approximated) model is known, and there exist PFCs that satisfy the following assumptions for the considered whole set of the system (1).
For the system (1), consider the augmented system (3) with a PFC (2) satisfying Assumption 1. We call this augmented system ‘output estimator augmented system’. Since the output estimator augmented system has relative degree of {1, 1, · · · , 1} from Assumption 1, there exists a nonsingular transformation ξk = [y Tap,k , η Tk ]T = Φxap,k such that the augmented system (3) can be transformed into the following canonical form (Isidori, 1995):
Assumption 1. For the system (1), there exists a known stable parallel feedforward compensator (PFC) with arbitrary order:
y ap,k+1 = Ay y ap,k + By uk + Cη η k η k+1 = Aη η k + Bη y ap,k
xf p,k+1 = Af p xf p,k + Bf p uk y f p,k = Cf p xf p,k such that the resulting augmented system: xap,k+1 = Aap xap,k + Bap uk y ap,k = Cap xap,k
(2)
(3)
It should be noted that since the output estimator augmented system is minimum phase from Assumption 1, the zero dynamics of (8): η k+1 = Aη η k is stable. Taking into account the fact that the zero dynamics in (8), we consider designing the adaptive output estimator by the following form:
with xap,k = [xTk , xTfp,k ]T and Aap =
ˆyk uk−1 ˆ apk = Aˆyk y apk−1 + B y
A 0 B , Bap = , 0 Af p Bf p
(4) Cap = [ C Cf p ] , 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 compensator (PFC) with arbitrary order: xf,k+1 = Af xf,k + Bf uk y f,k = Cf xf,k + Df uk
(5)
(8)
(9)
by disregarding the affect from the zero dynamics. Where, ˆyk are identified matrices of Ay and By respecAˆyk and B tively, and they are adaptively adjusting by the following parameter adjusting laws: Aˆy,k = σ ¯ Aˆy,k−1 − σ ¯ k y Tap,k−1 ΓA + PA,k ˆy,k−1 − σ ˆy,k = σ ¯B ¯ k uTk−1 ΓB + PB,k B
(10)
(11) 1 , σ>0 ΓA , ΓB > 0 , σ ¯= 1+σ ˆ apk − y apk is the output estimated error, where, k = y and PA,k and PB,k are parameter projections which keep adjusted parameters in the known parameter existing range.
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In this paper, for the brevity of discussion, we consider the case that [1]
[m]
ΓA = diag[γa[1] · · · γa[m] ], ΓB = diag[γb · · · γb ] were set. In this case, the parameter adjusting laws for [ij] ˆy,k = [ˆb[ij] ] are given as follows: Aˆy,k = [ˆ ay,k ] and B y,k [ij]
[ij]
[j]
[i]
[ij]
¯a ˆyk−1 − σ ¯ γa[j] yapk−1 k + pak a ˆyk = σ
[ij]
upper bound and lower bound of parameters ay are known as [ij]
parameter projections [ij] pak
[ij] pbk
= =
[ij]
[ij]
[ij]
[ij]
where (14)
ˆk − Best v ¯ ak(20) = Aˆayk y apk − Aest xf p,k + Aˆk B
Aˆy,k ¯ k(0) v ˆ . .. ¯ a,k = v , Aay,k = .. , . np ¯ k(np −1) v Aˆy,k I 0 ··· 0 Cf p Af p ˆ . . .. Cf p A2f p Ay,k I . . ˆ , Aest = , Ak = .. . . . .. .. 0 . .. n np −1 Cf p Af pp Aˆy,k · · · Aˆy,k I ˆy,k 0 · · · 0 B . 0 B ˆy,k . . . .. ˆ , Bk = . .. .. .. . . 0 ˆy,k 0 ··· 0 B Cf p B f 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) (16)
[i]
[i] [ij] [ij] [j] [j] ¯ γb uk−1 k fbk = σ ¯ˆbyk−1 − σ
ˆ apk − y apk Note that for the output estimated error k = y is not directly available because of causality problem, but this can be equivalently obtained by ˆyk−1 uk−1 − y apk ¯B σ ¯ Aˆyk−1 y apk−1 + σ 1+σ ¯ y Tapk−1 ΓA y apk−1 + σ ¯ uTk−1 ΓB uk−1
j=1
¯a ˆyk−1 − σ ¯ γa[j] yapk−1 k fak = σ
k =
¯ k(j−1) (19) Cf p Ai−j f p Bf p v
T ˆ k(np ) ˆ k(1) · · · y ˆ pk = y y
[ij]
[j]
i
Consequently, 1-step to np -step predicted outputs are obtained as
[ij]
0 (b[ij] y ≤ fbk ≤ by ) [j] [j] [i] σ ¯ γb uk−1 k otherwise
ˆ ¯ k(j−1) Aˆi−j y,k By,k v
−Cf p Aif p xf pk −
can be designed by
[ij] 0 (a[ij] y ≤ fak ≤ ay ) [j] [i] σ ¯ γa[j] yapk−1 k otherwise
i j=1
and by
[ij] [ij] [ij] ay[ij] ≤ a[ij] y ≤ ay , by ≤ by ≤ by [ij] pbk
ˆ k(i) = Aˆiy,k y ap,k + y
(13)
[ij] [ij] [ij] [ij] where a ˆy,k , ˆby,k and pak , pbk are the (i, j) element of ˆy,k and PA,k , PB,k , and [j] , y [j] , u[j] are j-th Aˆy,k , B k apk k element of k , y apk , uk , respectively. In the case where
[ij] pak ,
¯ k(i) . Finally, the predicted output can be represented v as follows by using available signals and predicting input ¯ k(j) , (j = 0, 1, · · · , i). v
(12)
[ij] [j] [j] [i] [ij] ˆb[ij] = σ ¯ˆbyk−1 − σ ¯ γb uk−1 k + pbk yk
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(17)
using available signals. All the signals in the designed adaptive output estimator are bounded with bounded input and bounded output. This can be shown using the same analysis strategy given in Mizumoto et al. (2015b).
3.3 Output Predictive Control Input
3.2 Adaptive Output Predictor for MIMO System
Using the designed output predictor (20), the desired control input is determined so as to minimize the following performance function.
Let’s denote i-step future value of the output signal y from kth sampling instant by y k(i) with y k(0) = y k . From the output estimator designed in (9) for the output estimator augmented system, consider designing i-step output predictor 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) (18) = Aˆy,k y
(i = 1, 2, · · · ) ˆ k(0) = y k (0) = y k . Where, y ˆ k(i) is the predicted with y ¯ k(i) is an i-step i-step future value of output y k(i) and v ˆ f p,k(i) is the i-step future input to be determined later. y future output of the PFC given in (2) with the input
Jk =
np i=1
ˆ k(i) − y r,k(i) y
2
¯ k(i−1) ¯ Tk(i−1) Λi v +v
T ˆ p,k − y ¯ Ta,k Λ¯ ¯ r,k ¯ r,k + v ˆ p,k − y y v a,k = y with Λ = diag Λ0 · · · Λnp −1 , Λi > 0. Where
(21)
T ¯ rk = y r,k(1) · · · y r,k(np ) y
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 The optimal input v performance function Jk is obtained by using the least squares method as follows:
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We now consider designing the control system as shown in the Fig. 1, i.e., the control input is designed by
Feedforward +
yr − Θp
−
ue +
v y
u Plant
ya
yf +
+
Fig. 1. Closed loop system −1 T ¯ ak = − WkT Wk + Λ Wk xvk v
(22)
ˆk − Best and where Wk = Aˆk B ¯ rk xvk = Aˆayk y apk − Aest xf pk − y The output predictive control input at kth sampling in¯ k(0) in the stant is then designed by using the input v ¯ a,k . obtained optimal control input v 4. OUTPUT FEEDBACK CONTROL SYSTEM DESIGN In practice, the stability of the control system with adap¯ k(0) as the control input is not nective predictive control v essarily guaranteed as it is. We consider designing ASPR based output feedback control in parallel with adaptive predictive control in order to maintain the stability of the obtained control system. 4.1 ASPR based Output Feedback Control with Predictive feedforward Input
(27)
vk =
(23)
It should be noted that the ideal of the PFC in the feedback control system must be given as follows: x∗f,k+1 = Af x∗f,k y ∗f,k = Cf x∗f,k = 0
where x∗k B ¯ x∗f,k , B = 0
¯ k(0) ( ¯ v v k(0) < v max ) v max ( ¯ v k(0) ≥ v max )
(29)
Note that Θp is a feedback gain and since the augmented system with the PFC is ASPR there exist Θ∗p such that the resulting closed-loop system is SPR or stable for any Θp ≥ Θ∗p .
Moreover, it should be noted that the designed feedback control input ue,k is not able to directly realize due to causality problem. However, fortunately, the control input can be realized using available signals as follows by taking into consideration that the PFC input is also given by ue,k . uek = −Θp {y k + y f,k − y rk }
= −Θp {y k + Cf xf,k + Df uek − y rk }
(30)
and thus we have
˜ pe ˜ak uek = −Θ −1
(31)
˜ak = y k +Cf xf k −y rk . Θp and e
Concerning the boundedness of all the 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 (26) designed by (27) and (29) are bounded, and if v k ≡ v ∗k were achieved, then ek = y k − y r,k converges to zero as k → ∞. Proof. Define X k := xak − x∗ak and ∆v k := v k − v ∗k . Then we have the following error system.
(24) where
when the perfect output tracking is achieved, and thus the ideal state x∗a,k of the augmented system is represented as ¯ ∗ x∗a,k+1 = Aa x∗a,k + Bv k ∗ ∗ y a,k = Ca xak = y ∗k = y rk
4.2 Analysis of the Obtained Control System
Now, impose the following additional assumption on the controlled system. Assumption 3. There exist ideal state x∗k and ideal input v ∗k uch 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 rk
(28)
and y f,k is the output of PFC (6) with the input ue,k . v k is the feedforward input which designed using the predictive ¯ k(0) with an input constrain v max as follows: control v
˜ p = {I +Θp Df } where Θ
xa,k+1 = Aa xa,k + Ba uk y a,k = Ca xa,k + Df uk
=
ue,k = −Θp ea,k , Θp > 0
ea,k = y a,k − y r,k , y a,k = y k + y f,k
Under assumption 2, consider the ASPR augmented controlled system with the PFC as in (6).
(26)
where
PFC
x∗a,k
uk = ue,k + v k
+
¯f ∆v k X k+1 = Aac X k + Ba ∆v k − B eak = Cac X k + Df ∆v k − Df ∆v k
(32)
˜ p Ca , Cac = I − Df Θ ˜ p Ca , Aac = Aa − Ba Θ ¯f = [0 · · · 0 BfT ]T B
(25)
Since the closed-loop system (Aac , Ba , Cac , Df ) is SPR from Assumption 2, there exist positive definite matrices P = P T > 0, Q = QT > 0 and appropriate matrices L and W such that the following Kalman-Yakubovich-Popov Lemma is satisfied. 5496
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Ikuro Mizumoto et al. / IFAC PapersOnLine 50-1 (2017) 5313–5318
ATac P Aac − P = −Q − LLT T ATac P Ba = Cac − LW
BaT P Ba
= Df +
DfT
−z 2 + 0.8z + 0.9 − 2.7z 2 + 2.8z − 0.95 500 G∗22 (z) = z − 0.98 G∗21 (z) =
(33) T
−W W
Consider the positive definite function: Vk = X Tk P X k (34) using positive definite matrix P in (33). The deference ∆Vk = Vk − Vk−1 can be evaluated by ∆Vk ≤ −ρ λmin Q − δλmax P T Cac ] X k−1 2 − δ1 λmax [Cac 1 + ρ + ρλmax [Df + DfT ] δ1 1 ¯T P B ¯f ] ∆v max (35) +ρ λmax [B f δ 1 > 0 and δ1 > 0 are some where 1 > δ > 0, ρ = 1−δ positive constants. Considering positive constants δ and T δ1 such that λmin Q − δλmax P − δ1 λmax [Cac Cac ] > 0 and from the fact that ∆v max = v max −v k is bounded, it follows that X k is bounded and then y k and uk are also bounden. From the boundedness of y k and uk , the signals in the adaptive output predictor is also bounded. Thus one can conclude that all the signals in the control system are bounded. Moreover, in the case that ∆v k ≡ 0, we obtain that X 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 → ∞.
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3z 3
Using the nominal model, we design PFCs Hest (z) and Haspr (z) for output prediction and output feedback as follows according to model-based PFC design scheme(Mizumoto et al., 2010a). Hest (z) = Gest (z) − G∗ (z) ∗
Haspr (z) = Gaspr (z) − G (z)
(39) (40)
where Gest (z) and Gaspr (z) are given and desired augmented models satisfying Assumptions 1 and 2 for output prediction and output feedback, respectively, and designed as follows in this simulation.
1 1 Gest (z) = z −10.99 z −10.99 z − 0.99 z − 0.99 450 0 Gaspr (z) = z − 0.98 650 0 z − 0.98
(41)
(42)
The design parameters in the designed adaptive output predictor were given by ΓA = diag[0.1 0.1], ΓB = diag[10 10] σ = 1.0 × 10−7 , Λi = diag[1 1], (i = 1, 2, ..., 10)
5. VALIDATION THROUGH NUMERICAL SIMULATION
T
np = 10, v max = [ 30 30 ] , Θp = diag[10 10]
The effectiveness of the proposed method is confirmed through numerical simulations. We consider a two-input/ two-output system as the system to be controlled. The controlled system in the simulation is given by G11 (z) G12 (z) G(z) = (36) G21 (z) G22 (z) 122.2z − 106.7 G11 (z) = 2 z − 1.929z + 0.9297 72.85z − 61.53 G12 (z) = 2 z − 1.888z + 0.8869 −0.2528z 2 + 0.4344z + 0.4424 G21 (z) = 3 z − 2.847z 2 + 2.701z − 0.8538 381 G22 (z) = z − 0.9765 We assume that the system (36) is uncertain but a nominal (or approximated) model are known as follows: ∗ G11 (z) G∗12 (z) G∗ (z) = (37) G∗21 (z) G∗22 (z) 100z − 90 G∗11 (z) = 2 z − 1.8z + 0.81 180z − 75.77 ∗ G12 (z) = 2 z − 1.9z + 0.91 (38)
Figs. 2, 3, 4, 5 show the simulation results for the unknown system (36). Even though the system is not exactly known and the accuracy of the given nominal model is not so good, by designing adaptive output predictor and assuring the stability of the resulting control system by ASPR based output feedback, the control performance shows very good results as in Fig. 2. 6. CONCLUSION In this paper, ASPR based output feedback control with an adaptive predictive feedforward input was proposed for MIMO systems. The adaptive output predictor based feedforward input design scheme for MIMO systems was provided, and by designing two-degree of freedom control system via ASPR based output feedback, the stability of the resulting control system was guaranteed. The effectiveness of the proposed ASPR based output feedback control with the adaptive predictive feedfoward input was confirmed through numerical simulation. ACKNOWLEDGEMENTS We give our sincerely thanks to Mr. J. Tsunematsu, who is former student in our Lab., for his dedicated support to this research.
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Proceedings of the 20th IFAC World Congress 5318 Ikuro Mizumoto et al. / IFAC PapersOnLine 50-1 (2017) 5313–5318 Toulouse, France, July 9-14, 2017
12
Output1
10 8 6 4 Tracking model of y1
2
Output y1
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8
Output2
6
4
2 Traking model of y
2
Output y
2
0
800
900
k
Fig. 2. Simulation results by the proposed method: Output tracking results −3
4
x 10
3
Input1
2 1 0 −1 −2 −3
0
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−3
2
x 10
1.5
Input2
1 0.5 0 −0.5 −1
0
k
Fig. 3. Control inputs of the system a ˆyk
[11]
1.2 1 0.8 0.6
0
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a ˆyk
[12]
0.2 0 −0.2 −0.4
a ˆyk
[21]
0.1 0
a ˆyk
[22]
−0.1
1
0.8
k
Fig. 4. Estimated parameters Aˆyk ˆb[11] yk
0.01 0 −0.01
ˆb[12] yk
−0.02
0
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0 −0.02 −0.04 0
ˆb[21] yk
−3
2 0 −2 −4 −6 −8
x 10
ˆb[22] yk
0.01 0 −0.01 −0.02
k
ˆyk Fig. 5. Estimated parameters B REFERENCES Bar-kana, I. (1987). Parallel feedforward and simplified adaptive control. International Journal of Adaptive Control and Signal Processing, 1(2), 95–109. Bar-Kana, I. (1991). Positive realness in multivariable continuous-time systems. Journal of the Franklin Institute, 328(4), 403–418. Barkana, I., Rusnak, I., and Weiss, H. (2014). Almost passivity and simple adaptive control in discrete-time systems. Asian Journal of Control, 16(4), 1–12. Clarke, D., Mohtadi, C., and Tuffs, P. (1987). Generalized predictive control - part 1. the basic algorithm. Auto-
matica, 23(2), 137–148. Fradkov, A.L. (1996). Shunt output feedback adaptive controller for nonlinear plants. Proc. of 13th IFAC World Congress, San-Francisco, July, K, 367–372. Fradkov, A.L. and Hill, D.J. (1998). Exponentially feedback passivity and stabilizability of nonlinear systems. Automatica, 34(6), 697–703. Garcia, C., Prett, D., and Morari, M. (1989). Model predictive control: Theory and practice -a survey. Automatica, 25(3), 335–348. Isidori, A. (1995). Nonlinear Control Systems. Springer, 3rd edition. Iwai, Z. and Mizumoto, I. (1994). Realization of simple adaptive control by using parallel feedforward compensator. Int. J. of Control, 59(6), 1543–1565. Kaufman, H., Barkana, I., and Sobel, K. (1997). Direct Adaptive Control Algorithms. Springer, 2nd edition. Maciejowski, J. (2002). Predictive Control with constraintss. Prentic Hall. Mayne, D., Rawlings, J., Rao, C., and Scokaert, P. (2000). Constrained model predictive control: Stability and optimality. Automatica, 36(6), 789–814. Mizumoto, I., Chen, T., Ohdaira, S., Kumon, M., and Iwai, Z. (2007). Adaptive output feedback control of general mimo systems using multirate sampling and its application to a cart-crane system. Automatica, 43(12), 2077–2085. Mizumoto, I., Fujii, S., and Ikejiri, M. (2015a). Control of a magnetic levitation system via output feedback based two dof control with an adaptive predictive feedforward input. Proc. of 2015 IEEE Multi-Conference on Systems and Control, September, Sydney, Australia, 71–76. Mizumoto, I. and Fujimoto, Y. (2011). Fast-rate adaptive output feedback control with adaptive output estimator for non-uniformly sampled multirate systems. Proc. of IEEE CDC-ECC 2011, 8297–8303. Mizumoto, I., Fujimoto, Y., and Ikejiri, M. (2015b). Adaptive output predictor based adaptive predictive control with aspr constraint. Automatica, 57, 152–163. Mizumoto, I., Ikeda, D., Hirahata, T., and Iwai, Z. (2010a). Design of discrete time adaptive pid control systems with parallel feedforward compensator. Control Engineering Practice, 18(2), 168–176. Mizumoto, I. and Iwai, Z. (1996). Simplified adaptive model output following control for plants with unmodelled dynamics. Int. J. of Control, 64(1), 61–80. Mizumoto, I., Ohdaira, S., Watanabe, N., Tanaka, H., Harada, H., Fujimoto, Y., Kinoshita, H., and Iwai, Z. (2010b). Output feedback control of multirate sampled systems with an adaptive output estimator and its application to a liquid level process control. Journal of System Design and Dynamics, 4(2), 314–330. Mizumoto, I., Okamatsu, Y., Tanaka, H., and Iwai, Z. (2009). Output regulation of nonlinear systems based on adaptive output feedback with adaptive nn feedforward control. Int. J. of Innovative Computing, Information and Control (IJICIC), 5(10), 3527–3539. Mizumoto, I. and Tanaka, N. (2010). Adaptive output regulation of a class of discrete-time nonlinear systems based on output feedback and nn feedforward control. Proc. of 48th IEEE Conference on Decision and Control, Atlanta, USA, 4631 – 4636.
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IFAC PapersOnLine 50-1 (2017) 5313–5318 ASPR based Output Feedback Control ASPR based Output Feedback Control ASPR based Output Feedback Control ASPR based Output Feedback Control with an Adaptive Predictive Feedfoward with an Adaptive Predictive Feedfoward with an Adaptive Predictive Feedfoward with anInput Adaptive Predictive Feedfoward for MIMO Systems Input for MIMO Systems Input for MIMO Systems Input for MIMO Systems ∗ ∗∗
Ikuro Mizumoto ∗ Seiya Fujii ∗∗ Ikuro Ikuro Mizumoto Mizumoto ∗∗ Seiya Seiya Fujii Fujii ∗∗ Ikuro Mizumoto Seiya Fujii ∗∗ ∗ ∗ Faculty of Advanced Science and Technology, Kumamoto University, ∗ Faculty of Advanced Science and Technology, Kumamoto University, of Advanced Science [email protected]). Technology, Kumamoto University, Kumamoto, Japan (e-mail: ∗ Faculty Japan (e-mail: [email protected]). Faculty of Advanced Science and Technology, KumamotoUniversity, University, ∗∗ Kumamoto, Kumamoto, Japan (e-mail: [email protected]). Department of Mechanical System Eng., Kumamoto ∗∗ of Mechanical System Eng., Kumamoto University, Kumamoto, Japan (e-mail: [email protected]). ∗∗ Department SystemJapan Eng., Kumamoto University, Kumamoto, ∗∗ Department of Mechanical Kumamoto, Japan Department of Mechanical SystemJapan Eng., Kumamoto University, Kumamoto, Kumamoto, Japan Abstract: Abstract: This This paper paper deals deals with with an an output output feedback feedback regulation regulation control control system system design design problem. problem. Abstract: This paper deals with an output feedback regulation control system design problem. An almost strictly positive real or ASPR based output feedback control system with the adaptive An almost strictly positive real or ASPR based output feedback control system with the adaptive Abstract: This paper deals with an output feedback regulation control system design problem. An almost feedforward strictly positive realwill or ASPR based output feedback control system with the adaptive predictive input be proposed for MIMO linear systems. The adaptive output predictive feedforward input will be proposed for MIMO linear systems. The adaptive output An almost strictly positive real or ASPR based output feedback control system with the adaptive predictiveproposed feedforward inputsystems will be will proposed for MIMO linear systems systems.and Thethe adaptive output predictor for SISO be expanded for MIMO stability of predictor proposed for SISO systems will be expanded for MIMO systems and the stability of the the predictive feedforward input will be proposed for MIMO linear systems. The adaptive output predictor control proposed for SISO systems will be expanded for MIMO systems and the control stabilitysystem of the resulting system will be maintained by designing two-degree of freedom resulting control system will be maintained by designing two-degree of freedom control system predictor proposed for SISO systems will be expanded for MIMO systems and the stability of the resulting control system feedback. will be maintained by designing two-degree of freedomsimple control system via based The proposed control strategy has relatively structure via ASPR ASPRcontrol based output output feedback. The proposed control strategy has relatively simple structure resulting system will output be maintained bycontrol designing two-degree of simple freedom control system via ASPR based output feedback. The proposed control strategy has relatively simple structure with a novel simple adaptive predictive and ASPR based output feedback. with a novel simple adaptive output predictive control and ASPR based simple output feedback. via ASPR based output feedback. The proposed control strategy has relatively simple structure with effectiveness a novel simple adaptive output predictive control and ASPR based simple output feedback. The of the method will be through numerical simulations. The effectiveness effectiveness ofadaptive the proposed proposed method will control be confirmed confirmed through numerical simulations. with a novel simple output predictive and ASPR based simple output feedback. The of the proposed method will be confirmed through numerical simulations. The effectiveness of the proposed method will beControl) confirmed through numerical © 2017, IFAC (International Federation of Automatic Hosting by Elsevier Ltd. Allsimulations. rights reserved. Keywords: Keywords: Adaptive Adaptive control, control, ASPR, ASPR, adaptive adaptive output output predictor, predictor, adaptive adaptive predictive predictive control, control, Keywords: Adaptive compensator control, ASPR, adaptive output predictor, adaptive predictive control, parallel feedforward parallel feedforward compensator Keywords: Adaptive control, ASPR, adaptive output predictor, adaptive predictive control, parallel feedforward compensator parallel feedforward compensator 1. In Kaufman et al. (1997); Mizumoto and Iwai (1996), 1. INTRODUCTION INTRODUCTION In et Mizumoto and (1996), 1. INTRODUCTION In Kaufman Kaufmanreference et al. al. (1997); (1997); Mizumoto and Iwai Iwai (1996), augmented model was introduced in order to augmented reference model was introduced in order to 1. INTRODUCTION In Kaufman et al. (1997); Mizumoto and Iwai (1996), augmented reference model wasoutput introduced in to order to alleviate the affects from PFC in order attain It has been well recognized that if the system satisfies alleviate the affects from PFC output in order to attain augmented reference model was introduced in order to It has been well recognized that if the system satisfies alleviate theregulation affects from PFC output in ordersystem. to attain the output for practical controlled In It has been wellpositive recognized that if the system satisfies ‘almost strictly real’ or ASPR condition, then the output regulation for practical controlled system. In alleviate the affects from PFC output in order to attain ‘almost strictly positive real’ or ASPR condition, then It has been well recognized if the with system satisfies output et regulation for Mizumoto practical controlled system. In Mizumoto al. (2009); and Tanaka (2010), ‘almost strictly positive real’that orsystems ASPR condition, then the one adaptive control very Mizumoto et al. (2009); and Tanaka (2010), the output regulation for Mizumoto practical controlled system. In one can can design design adaptive control systems with very simple simple ‘almost strictly positive real’ or ASPR condition, then Mizumoto et al. (2009); Mizumoto and Tanaka (2010), adaptive NN feedforward was proposed for feedforward one can design adaptive control systems with very simple structure through adaptive output feedback (Kaufman adaptive NN feedforward was proposed for feedforward Mizumoto et al. (2009); Mizumoto and Tanaka (2010), structure through adaptive output feedback (Kaufman one can design adaptive control systems with very simple adaptive NN feedforward was proposed for feedforward control. structure through adaptive output feedback (Kaufman et Barkana et 2014; and control. et al., al., 1997; 1997; Barkana et al., al., output 2014; Mizumoto Mizumoto and Iwai, Iwai, adaptive structure through adaptive feedback (Kaufman control. NN feedforward was proposed for feedforward et al., Fradkov 1997; Barkana et al., 2014; Mizumoto and Iwai, 1996; and Hill, 1998). The system is said to On the other hand, recently, a new control design scheme 1996; Fradkov and Hill, 1998). The system isand said to control. et al., Fradkov 1997; Barkana et al., 2014; Mizumoto Iwai, the other hand, recently, a control scheme 1996; and exists Hill, The system is said to On be if aa1998). static output feedback such On theon other hand, recently, a new new control control design design scheme based an adaptive predictive strategy with be ASPR ASPR if there there exists staticThe output feedback such 1996; Fradkov and Hill, 1998). system is said to based on an adaptive predictive control strategy with be ASPR if there closed exists loop a static output feedback such On the other hand, recently, a new control design scheme that the resulting system is strictly positive based on anoutput adaptive predictive control based strategy with an adaptive predictor and ASPR adaptive thatASPR the resulting closed loop system is strictly positive be if there exists a static output feedback such an adaptive output predictor and ASPR based adaptive that the resulting closed 1991). loop system is strictly however, positive based on an adaptive predictive control strategy with real (SPR) (Bar-Kana, Unfortunately, an adaptive output predictor and ASPR based adaptive output feedback has been proposed from a different viewreal (SPR) (Bar-Kana, 1991). Unfortunately, however, that (SPR) the resulting closed loop do system is strictly positive output feedback has been proposed from a different viewreal (Bar-Kana, 1991). Unfortunately, however, an adaptive output predictor and ASPR based adaptive since most practical system not satisfy the ASPR output feedback has been proposedPredictive from a different viewet al., 2015b,a). Control, insince (SPR) most practical system doUnfortunately, not satisfy thehowever, ASPR point(Mizumoto real (Bar-Kana, 1991).do et 2015b,a). Predictive Control, insince most the practical system not satisfy the ASPR point(Mizumoto output feedback has been proposed from aand different viewconditions, ASPR imposes severe restriction point(Mizumoto et al., al., 2015b,a). Predictive Control, including model predictive control (MPC) generalized conditions, the ASPR condition condition imposes severe the restriction since most practical system do not satisfy ASPR cluding model predictive control (MPC) and generalized conditions, the ASPR condition imposes severe restriction point(Mizumoto et al., 2015b,a). Predictive Control, into practical applications of the ASPR based adaptive model predictive control (MPC) and generalized predictive control (GPC), have attracted a great deal of to practicalthe applications of theimposes ASPRsevere basedrestriction adaptive cluding conditions, ASPR condition predictive control (GPC), have attracted a great deal to practical applications ofmethod the ASPR basedthe adaptive cluding model predictive control (MPC) and generalized controls. One of the simple to alleviate ASPR control (GPC), have attracted aresent great decades deal of of attention and widely used in industry for controls. One applications of the simpleofmethod to alleviate the ASPR predictive to practical the ASPR based adaptive attention and widely used in industry for controls. One of the simple method to alleviatefeedforward the ASPR predictive control (GPC), have attracted great decades deal of restriction is introduction of attention and widely used in et industry for aresent resent decades Clarke et al. (1987); Garcia al. (1989); Mayne et al. restrictionOne is the the introduction of the thetoparallel parallel feedforward controls. of the simple method alleviate the ASPR Clarke et al. (1987); Garcia et al. (1989); Mayne et al. restriction is (PFC) the introduction of1987; the Iwai parallel feedforward attention and widely used in industry for resent decades compensator (Bar-kana, and Mizumoto, Clarke et al. (1987); Garcia et al. (1989); Mayne et al. Maciejowski (2002) due to its powerful control compensator (PFC) (Bar-kana,of1987; Iwai and Mizumoto, (2000); restriction is (PFC) the introduction theFradkov, parallel (2000); et Maciejowski due to its control compensator (Bar-kana, 1987; Iwai andfeedforward Mizumoto, Clarke al. However, (1987); (2002) Garcia et al. Mayne et al. 1994; and Iwai, 1996). The Maciejowski (2002) due to (1989); its powerful powerful control performance. the performance of the predictive 1994; Mizumoto Mizumoto and Iwai, 1996; 1996; Fradkov, 1996). The (2000); compensator (PFC) (Bar-kana, 1987; Iwai and Mizumoto, performance. However, the performance of the predictive 1994; Mizumoto and Iwai, 1996; Fradkov, 1996). The (2000); Maciejowski (2002) due to its powerful control PFC is introduced so as to render an ASPR augmented performance. However, the performance of the predictive depends significantly on accuracy of the given PFC isMizumoto introducedand so as to render an ASPR 1996). augmented 1994; Iwai, 1996; based Fradkov, The control control depends significantly on accuracy of the PFC is introduced so as to ASPR render an ASPR augmented performance. However, the performance of the predictive controlled system, the control scheme depends significantly on accuracy of the given given model of the controlled system. The method in Mizumoto controlled system, and and the ASPR based control scheme control PFC is introduced so as to render an ASPR augmented model of the system. The method in Mizumoto controlled system, andrealized the ASPR based control scheme control significantly on accuracy of thepredicgiven can to ASPR augmented system ofdepends the controlled controlled system. The method in the Mizumoto et al. (2015b,a) is one of the adaptive version of can be be applied applied to the the realized ASPR augmented system model controlled system, and the ASPR based control scheme et al. (2015b,a) is one of the adaptive version of the prediccan be applied to the realized ASPR augmented system model of the controlled system. The method in Mizumoto with a PFC. However, the control system is designed to al.control. (2015b,a) is one of the adaptive version of the predictive In the method, an adaptive output predictor withbe a PFC. However, the control system is designed to et can applied to the realized ASPR augmented system tive In the method, an adaptive output predictor with a PFC. However, the control system is PFC designed to et al.control. (2015b,a) is one of the adaptive version of the predicthe augmented system, the affects from the output tive control. In the method, an adaptive output predictor with a simple structure derived from output estimator the augmented system, the affects from the PFC output with a PFC. However, system designed to with a simple structure derived from output estimator the augmented system, the the control affects from theis PFC output tive control. In the method, an adaptive output predictor to the control performance of the practical system has a simplesystems structure derived et from output estimator for multi-rate (Mizumoto al., 2010b; Mizumoto to the control performance of the the from practical system has with the augmented system, the affects the PFC output for multi-rate (Mizumoto et al., 2010b; Mizumoto of practical system has to the control performance with a simplesystems structure derived from estimator been controversial in the the for multi-rate systems (Mizumoto et al., output 2010b; Mizumoto and Fujimoto, 2011) is utilized for designing the predictive been controversial in the the case caseofwhere where the gain gain of of the PFC PFC to the control performance the practical system has and Fujimoto, 2011) is utilized for designing the predictive been controversial inInthe casetowhere the gain of the issue, PFC for multi-rate systems (Mizumoto et al., 2010b; Mizumoto was relatively large. order solve this additional and Fujimoto, 2011) is utilized for designing the predictive control input. A parallel feedforward compensator (PFC) was relatively large.inInthe order to solve this additional issue, been controversial case the gain of the issue, PFC control input. A parallel feedforward compensator (PFC) was relatively large. In order towhere solveinput this additional Fujimoto, 2011) issoutilized for designing the predictive how to design aa feedforward control has played played very and control input. A parallel feedforward compensator (PFC) is again introduced as to render a minimum-phase how to design feedforward control input has very was relatively large. In order to solve this additional issue, again introduced so as to render a minimum-phase how to design a(Kaufman feedforward control input has played very is control input. A parallel feedforward compensator (PFC) important role et al., 1997; Mizumoto and Iwai, is again introduced so aasrelative to render a of minimum-phase augmented system with degree 1, and for reimportant rolea(Kaufman et al., 1997;input Mizumoto and Iwai, how to design feedforward control has played very augmented system with a relative degree of 1, and for important role (Kaufman et Mizumoto al., 1997; Mizumoto and2010). Iwai, is again introduced so as to render a minimum-phase 1996; Mizumoto et al., 2009; and Tanaka, augmented system with a relative degree of 1, and for rereduced first order system, output predictor with very simple 1996; Mizumoto et al., 2009; Mizumoto and Tanaka, 2010). important role (Kaufman et al., 1997; Mizumoto and Iwai, duced first order system, predictor with simple 1996; Mizumoto et al., 2009; Mizumoto and Tanaka, 2010). augmented system with aoutput relative degree of 1, very and control for reduced first order system, output predictor with very simple structure is designed. The stability of the resulting 1996; Mizumoto et al., 2009; Mizumoto and Tanaka, 2010). duced structure is designed. The stability of the first system, output predictor withsystem verycontrol simple work was supported by Council for Science, Technology and structure isorder designed. The stability of control the resulting resulting control This system maintained by switching the from This work was supported by Council for Science, Technology and system maintained by switching the control system from structure is designed. The stability of the resulting control Innovation Cross-ministerial Innovation PromoThis work(CSTI), was supported by CouncilStrategic for Science, Technology and system maintained by switching the control system from predictive control to ASPR based adaptive output feed Innovation (CSTI), Cross-ministerial Strategic Innovation PromoThis work(CSTI), was supported by Council for Science, Technology and predictive control to ASPR based adaptive output feedtion Program (SIP), “Innovative Combustion Technology” (Funding Innovation Cross-ministerial Strategic Innovation Promosystem maintained by switching the control system from predictive control tomethod. ASPR However, based adaptive output feedtion Program (SIP), Cross-ministerial “Innovative Combustion Technology” (Funding back control in this adopting switching Innovation (CSTI), Strategic Innovation Promoagency: JST) as a part of researchCombustion on “Control Technology” and System Modelling tion Program (SIP), “Innovative (Funding back control in this method. However, adopting switching predictive control to ASPR based adaptive output feedagency: JST) as a part of researchCombustion on “Control and System Modelling back control intothis method. However, adopting switching strategy, how set the switching timing is very important tion Program (SIP), “Innovative (Funding on Fuel Technology Innovation” agency: JST) as a part of research. on “Control Technology” and System Modelling strategy, how set the switching timing is very important back control into this method. However, adopting switching on Fuel JST) Technology Innovation” . on “Control and System Modelling strategy, how to set the switching timing is very important agency: as a part of research on Fuel Technology Innovation” . strategy, how to set the switching timing is very important on Fuel Technology Innovation” .
Copyright © 2017 IFAC 5493 Copyright 2017 IFAC 5493Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017, 2017 IFAC 5493 Peer review©under of International Federation of Automatic Copyright 2017 responsibility IFAC 5493Control. 10.1016/j.ifacol.2017.08.985
Proceedings of the 20th IFAC World Congress 5314 Ikuro Mizumoto et al. / IFAC PapersOnLine 50-1 (2017) 5313–5318 Toulouse, France, July 9-14, 2017
parameter to maintain good control performance and it sometimes be difficult and complex. In this paper, we consider adopting the adaptive predictive control as a feedforward input and designing an adaptive control for unknown MIMO systems. That is, we propose ASPR based output feedback control with an adaptive predictive feedforward input for MIMO systems. The adaptive output predictor proposed by Mizumoto et al. (2015b,a) will be expanded for MIMO systems and the stability of the resulting control system will be maintained by designing two-degree of freedom control system via ASPR based output feedback. The proposed control strategy has relatively simple structure with a novel simple adaptive output predictive control and ASPR based simple output feedback. The effectiveness of the proposed ASPR based output feedback control with the adaptive predictive feedfoward input will be confirmed through numerical simulations. 2. PROBLEM STATEMENT Let’s consider the following nth order uniformly sampled MIMO unknown system with m-input/m-output and sampling period of T : xk+1 = Axk + Buk (1) y k = Cxk where xk := x(kT ) and y k := y(kT ) are the state vector and the output vector of the system at a time instant kT , and uk := u(kT ) is the input vector updated at a time instant kT .
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)
with xa,k = [xTk , xTf,k ]T and Aa =
A 0 0 Af
C a = [ C Cf ] ,
, Ba =
B Bf
, (7)
is ASPR (or strongly ASPR(Mizumoto et al., 2007)), i.e. the augmented system is strictly minimum phase and has a relative degree of {0, 0, · · · , 0} with Df > 0. For the definition of relative degree of MIMO systems, refer Isidori (1995). The objective of this paper is to design stable two-degree of freedom ASPR based output feedback control system with adaptive predictive feedforward input under the given assumptions 1 and 2. 3. ADAPTIVE OUTPUT PREDICTIVE CONTROL DESIGN 3.1 Adaptive Output Estimator for MIMO System
Suppose that the considered system is basically uncertain, but a nominal (or a simple approximated) model is known, and there exist PFCs that satisfy the following assumptions for the considered whole set of the system (1).
For the system (1), consider the augmented system (3) with a PFC (2) satisfying Assumption 1. We call this augmented system ‘output estimator augmented system’. Since the output estimator augmented system has relative degree of {1, 1, · · · , 1} from Assumption 1, there exists a nonsingular transformation ξk = [y Tap,k , η Tk ]T = Φxap,k such that the augmented system (3) can be transformed into the following canonical form (Isidori, 1995):
Assumption 1. For the system (1), there exists a known stable parallel feedforward compensator (PFC) with arbitrary order:
y ap,k+1 = Ay y ap,k + By uk + Cη η k η k+1 = Aη η k + Bη y ap,k
xf p,k+1 = Af p xf p,k + Bf p uk y f p,k = Cf p xf p,k such that the resulting augmented system: xap,k+1 = Aap xap,k + Bap uk y ap,k = Cap xap,k
(2)
(3)
It should be noted that since the output estimator augmented system is minimum phase from Assumption 1, the zero dynamics of (8): η k+1 = Aη η k is stable. Taking into account the fact that the zero dynamics in (8), we consider designing the adaptive output estimator by the following form:
with xap,k = [xTk , xTfp,k ]T and Aap =
ˆyk uk−1 ˆ apk = Aˆyk y apk−1 + B y
A 0 B , Bap = , 0 Af p Bf p
(4) Cap = [ C Cf p ] , 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 compensator (PFC) with arbitrary order: xf,k+1 = Af xf,k + Bf uk y f,k = Cf xf,k + Df uk
(5)
(8)
(9)
by disregarding the affect from the zero dynamics. Where, ˆyk are identified matrices of Ay and By respecAˆyk and B tively, and they are adaptively adjusting by the following parameter adjusting laws: Aˆy,k = σ ¯ Aˆy,k−1 − σ ¯ k y Tap,k−1 ΓA + PA,k ˆy,k−1 − σ ˆy,k = σ ¯B ¯ k uTk−1 ΓB + PB,k B
(10)
(11) 1 , σ>0 ΓA , ΓB > 0 , σ ¯= 1+σ ˆ apk − y apk is the output estimated error, where, k = y and PA,k and PB,k are parameter projections which keep adjusted parameters in the known parameter existing range.
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In this paper, for the brevity of discussion, we consider the case that [1]
[m]
ΓA = diag[γa[1] · · · γa[m] ], ΓB = diag[γb · · · γb ] were set. In this case, the parameter adjusting laws for [ij] ˆy,k = [ˆb[ij] ] are given as follows: Aˆy,k = [ˆ ay,k ] and B y,k [ij]
[ij]
[j]
[i]
[ij]
¯a ˆyk−1 − σ ¯ γa[j] yapk−1 k + pak a ˆyk = σ
[ij]
upper bound and lower bound of parameters ay are known as [ij]
parameter projections [ij] pak
[ij] pbk
= =
[ij]
[ij]
[ij]
[ij]
where (14)
ˆk − Best v ¯ ak(20) = Aˆayk y apk − Aest xf p,k + Aˆk B
Aˆy,k ¯ k(0) v ˆ . .. ¯ a,k = v , Aay,k = .. , . np ¯ k(np −1) v Aˆy,k I 0 ··· 0 Cf p Af p ˆ . . .. Cf p A2f p Ay,k I . . ˆ , Aest = , Ak = .. . . . .. .. 0 . .. n np −1 Cf p Af pp Aˆy,k · · · Aˆy,k I ˆy,k 0 · · · 0 B . 0 B ˆy,k . . . .. ˆ , Bk = . .. .. .. . . 0 ˆy,k 0 ··· 0 B Cf p B f 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) (16)
[i]
[i] [ij] [ij] [j] [j] ¯ γb uk−1 k fbk = σ ¯ˆbyk−1 − σ
ˆ apk − y apk Note that for the output estimated error k = y is not directly available because of causality problem, but this can be equivalently obtained by ˆyk−1 uk−1 − y apk ¯B σ ¯ Aˆyk−1 y apk−1 + σ 1+σ ¯ y Tapk−1 ΓA y apk−1 + σ ¯ uTk−1 ΓB uk−1
j=1
¯a ˆyk−1 − σ ¯ γa[j] yapk−1 k fak = σ
k =
¯ k(j−1) (19) Cf p Ai−j f p Bf p v
T ˆ k(np ) ˆ k(1) · · · y ˆ pk = y y
[ij]
[j]
i
Consequently, 1-step to np -step predicted outputs are obtained as
[ij]
0 (b[ij] y ≤ fbk ≤ by ) [j] [j] [i] σ ¯ γb uk−1 k otherwise
ˆ ¯ k(j−1) Aˆi−j y,k By,k v
−Cf p Aif p xf pk −
can be designed by
[ij] 0 (a[ij] y ≤ fak ≤ ay ) [j] [i] σ ¯ γa[j] yapk−1 k otherwise
i j=1
and by
[ij] [ij] [ij] ay[ij] ≤ a[ij] y ≤ ay , by ≤ by ≤ by [ij] pbk
ˆ k(i) = Aˆiy,k y ap,k + y
(13)
[ij] [ij] [ij] [ij] where a ˆy,k , ˆby,k and pak , pbk are the (i, j) element of ˆy,k and PA,k , PB,k , and [j] , y [j] , u[j] are j-th Aˆy,k , B k apk k element of k , y apk , uk , respectively. In the case where
[ij] pak ,
¯ k(i) . Finally, the predicted output can be represented v as follows by using available signals and predicting input ¯ k(j) , (j = 0, 1, · · · , i). v
(12)
[ij] [j] [j] [i] [ij] ˆb[ij] = σ ¯ˆbyk−1 − σ ¯ γb uk−1 k + pbk yk
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(17)
using available signals. All the signals in the designed adaptive output estimator are bounded with bounded input and bounded output. This can be shown using the same analysis strategy given in Mizumoto et al. (2015b).
3.3 Output Predictive Control Input
3.2 Adaptive Output Predictor for MIMO System
Using the designed output predictor (20), the desired control input is determined so as to minimize the following performance function.
Let’s denote i-step future value of the output signal y from kth sampling instant by y k(i) with y k(0) = y k . From the output estimator designed in (9) for the output estimator augmented system, consider designing i-step output predictor 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) (18) = Aˆy,k y
(i = 1, 2, · · · ) ˆ k(0) = y k (0) = y k . Where, y ˆ k(i) is the predicted with y ¯ k(i) is an i-step i-step future value of output y k(i) and v ˆ f p,k(i) is the i-step future input to be determined later. y future output of the PFC given in (2) with the input
Jk =
np i=1
ˆ k(i) − y r,k(i) y
2
¯ k(i−1) ¯ Tk(i−1) Λi v +v
T ˆ p,k − y ¯ Ta,k Λ¯ ¯ r,k ¯ r,k + v ˆ p,k − y y v a,k = y with Λ = diag Λ0 · · · Λnp −1 , Λi > 0. Where
(21)
T ¯ rk = y r,k(1) · · · y r,k(np ) y
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 The optimal input v performance function Jk is obtained by using the least squares method as follows:
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We now consider designing the control system as shown in the Fig. 1, i.e., the control input is designed by
Feedforward +
yr − Θp
−
ue +
v y
u Plant
ya
yf +
+
Fig. 1. Closed loop system −1 T ¯ ak = − WkT Wk + Λ Wk xvk v
(22)
ˆk − Best and where Wk = Aˆk B ¯ rk xvk = Aˆayk y apk − Aest xf pk − y The output predictive control input at kth sampling in¯ k(0) in the stant is then designed by using the input v ¯ a,k . obtained optimal control input v 4. OUTPUT FEEDBACK CONTROL SYSTEM DESIGN In practice, the stability of the control system with adap¯ k(0) as the control input is not nective predictive control v essarily guaranteed as it is. We consider designing ASPR based output feedback control in parallel with adaptive predictive control in order to maintain the stability of the obtained control system. 4.1 ASPR based Output Feedback Control with Predictive feedforward Input
(27)
vk =
(23)
It should be noted that the ideal of the PFC in the feedback control system must be given as follows: x∗f,k+1 = Af x∗f,k y ∗f,k = Cf x∗f,k = 0
where x∗k B ¯ x∗f,k , B = 0
¯ k(0) ( ¯ v v k(0) < v max ) v max ( ¯ v k(0) ≥ v max )
(29)
Note that Θp is a feedback gain and since the augmented system with the PFC is ASPR there exist Θ∗p such that the resulting closed-loop system is SPR or stable for any Θp ≥ Θ∗p .
Moreover, it should be noted that the designed feedback control input ue,k is not able to directly realize due to causality problem. However, fortunately, the control input can be realized using available signals as follows by taking into consideration that the PFC input is also given by ue,k . uek = −Θp {y k + y f,k − y rk }
= −Θp {y k + Cf xf,k + Df uek − y rk }
(30)
and thus we have
˜ pe ˜ak uek = −Θ −1
(31)
˜ak = y k +Cf xf k −y rk . Θp and e
Concerning the boundedness of all the 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 (26) designed by (27) and (29) are bounded, and if v k ≡ v ∗k were achieved, then ek = y k − y r,k converges to zero as k → ∞. Proof. Define X k := xak − x∗ak and ∆v k := v k − v ∗k . Then we have the following error system.
(24) where
when the perfect output tracking is achieved, and thus the ideal state x∗a,k of the augmented system is represented as ¯ ∗ x∗a,k+1 = Aa x∗a,k + Bv k ∗ ∗ y a,k = Ca xak = y ∗k = y rk
4.2 Analysis of the Obtained Control System
Now, impose the following additional assumption on the controlled system. Assumption 3. There exist ideal state x∗k and ideal input v ∗k uch 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 rk
(28)
and y f,k is the output of PFC (6) with the input ue,k . v k is the feedforward input which designed using the predictive ¯ k(0) with an input constrain v max as follows: control v
˜ p = {I +Θp Df } where Θ
xa,k+1 = Aa xa,k + Ba uk y a,k = Ca xa,k + Df uk
=
ue,k = −Θp ea,k , Θp > 0
ea,k = y a,k − y r,k , y a,k = y k + y f,k
Under assumption 2, consider the ASPR augmented controlled system with the PFC as in (6).
(26)
where
PFC
x∗a,k
uk = ue,k + v k
+
¯f ∆v k X k+1 = Aac X k + Ba ∆v k − B eak = Cac X k + Df ∆v k − Df ∆v k
(32)
˜ p Ca , Cac = I − Df Θ ˜ p Ca , Aac = Aa − Ba Θ ¯f = [0 · · · 0 BfT ]T B
(25)
Since the closed-loop system (Aac , Ba , Cac , Df ) is SPR from Assumption 2, there exist positive definite matrices P = P T > 0, Q = QT > 0 and appropriate matrices L and W such that the following Kalman-Yakubovich-Popov Lemma is satisfied. 5496
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ATac P Aac − P = −Q − LLT T ATac P Ba = Cac − LW
BaT P Ba
= Df +
DfT
−z 2 + 0.8z + 0.9 − 2.7z 2 + 2.8z − 0.95 500 G∗22 (z) = z − 0.98 G∗21 (z) =
(33) T
−W W
Consider the positive definite function: Vk = X Tk P X k (34) using positive definite matrix P in (33). The deference ∆Vk = Vk − Vk−1 can be evaluated by ∆Vk ≤ −ρ λmin Q − δλmax P T Cac ] X k−1 2 − δ1 λmax [Cac 1 + ρ + ρλmax [Df + DfT ] δ1 1 ¯T P B ¯f ] ∆v max (35) +ρ λmax [B f δ 1 > 0 and δ1 > 0 are some where 1 > δ > 0, ρ = 1−δ positive constants. Considering positive constants δ and T δ1 such that λmin Q − δλmax P − δ1 λmax [Cac Cac ] > 0 and from the fact that ∆v max = v max −v k is bounded, it follows that X k is bounded and then y k and uk are also bounden. From the boundedness of y k and uk , the signals in the adaptive output predictor is also bounded. Thus one can conclude that all the signals in the control system are bounded. Moreover, in the case that ∆v k ≡ 0, we obtain that X 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 → ∞.
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3z 3
Using the nominal model, we design PFCs Hest (z) and Haspr (z) for output prediction and output feedback as follows according to model-based PFC design scheme(Mizumoto et al., 2010a). Hest (z) = Gest (z) − G∗ (z) ∗
Haspr (z) = Gaspr (z) − G (z)
(39) (40)
where Gest (z) and Gaspr (z) are given and desired augmented models satisfying Assumptions 1 and 2 for output prediction and output feedback, respectively, and designed as follows in this simulation.
1 1 Gest (z) = z −10.99 z −10.99 z − 0.99 z − 0.99 450 0 Gaspr (z) = z − 0.98 650 0 z − 0.98
(41)
(42)
The design parameters in the designed adaptive output predictor were given by ΓA = diag[0.1 0.1], ΓB = diag[10 10] σ = 1.0 × 10−7 , Λi = diag[1 1], (i = 1, 2, ..., 10)
5. VALIDATION THROUGH NUMERICAL SIMULATION
T
np = 10, v max = [ 30 30 ] , Θp = diag[10 10]
The effectiveness of the proposed method is confirmed through numerical simulations. We consider a two-input/ two-output system as the system to be controlled. The controlled system in the simulation is given by G11 (z) G12 (z) G(z) = (36) G21 (z) G22 (z) 122.2z − 106.7 G11 (z) = 2 z − 1.929z + 0.9297 72.85z − 61.53 G12 (z) = 2 z − 1.888z + 0.8869 −0.2528z 2 + 0.4344z + 0.4424 G21 (z) = 3 z − 2.847z 2 + 2.701z − 0.8538 381 G22 (z) = z − 0.9765 We assume that the system (36) is uncertain but a nominal (or approximated) model are known as follows: ∗ G11 (z) G∗12 (z) G∗ (z) = (37) G∗21 (z) G∗22 (z) 100z − 90 G∗11 (z) = 2 z − 1.8z + 0.81 180z − 75.77 ∗ G12 (z) = 2 z − 1.9z + 0.91 (38)
Figs. 2, 3, 4, 5 show the simulation results for the unknown system (36). Even though the system is not exactly known and the accuracy of the given nominal model is not so good, by designing adaptive output predictor and assuring the stability of the resulting control system by ASPR based output feedback, the control performance shows very good results as in Fig. 2. 6. CONCLUSION In this paper, ASPR based output feedback control with an adaptive predictive feedforward input was proposed for MIMO systems. The adaptive output predictor based feedforward input design scheme for MIMO systems was provided, and by designing two-degree of freedom control system via ASPR based output feedback, the stability of the resulting control system was guaranteed. The effectiveness of the proposed ASPR based output feedback control with the adaptive predictive feedfoward input was confirmed through numerical simulation. ACKNOWLEDGEMENTS We give our sincerely thanks to Mr. J. Tsunematsu, who is former student in our Lab., for his dedicated support to this research.
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12
Output1
10 8 6 4 Tracking model of y1
2
Output y1
0 0
100
200
300
400
500
600
700
0
100
200
300
400
500
600
700
800
900
8
Output2
6
4
2 Traking model of y
2
Output y
2
0
800
900
k
Fig. 2. Simulation results by the proposed method: Output tracking results −3
4
x 10
3
Input1
2 1 0 −1 −2 −3
0
100
200
300
400
100
200
300
400
500
600
700
800
900
500
600
700
800
900
−3
2
x 10
1.5
Input2
1 0.5 0 −0.5 −1
0
k
Fig. 3. Control inputs of the system a ˆyk
[11]
1.2 1 0.8 0.6
0
100
200
300
400
500
600
700
800
900
0
100
200
300
400
500
600
700
800
900
0
100
200
300
400
500
600
700
800
900
0
100
200
300
400
500
600
700
800
900
a ˆyk
[12]
0.2 0 −0.2 −0.4
a ˆyk
[21]
0.1 0
a ˆyk
[22]
−0.1
1
0.8
k
Fig. 4. Estimated parameters Aˆyk ˆb[11] yk
0.01 0 −0.01
ˆb[12] yk
−0.02
0
100
200
300
400
500
600
700
800
900
100
200
300
400
500
600
700
800
900
0
100
200
300
400
500
600
700
800
900
0
100
200
300
400
500
600
700
800
900
0 −0.02 −0.04 0
ˆb[21] yk
−3
2 0 −2 −4 −6 −8
x 10
ˆb[22] yk
0.01 0 −0.01 −0.02
k
ˆyk Fig. 5. Estimated parameters B REFERENCES Bar-kana, I. (1987). Parallel feedforward and simplified adaptive control. International Journal of Adaptive Control and Signal Processing, 1(2), 95–109. Bar-Kana, I. (1991). Positive realness in multivariable continuous-time systems. Journal of the Franklin Institute, 328(4), 403–418. Barkana, I., Rusnak, I., and Weiss, H. (2014). Almost passivity and simple adaptive control in discrete-time systems. Asian Journal of Control, 16(4), 1–12. Clarke, D., Mohtadi, C., and Tuffs, P. (1987). Generalized predictive control - part 1. the basic algorithm. Auto-
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