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Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form
European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form Yaprak Yalçın n Department of Control and Automation Engineering, Faculty of Electrical-Electronics Engineering, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey
art ic l e i nf o
a b s t r a c t
Article history: Received 4 April 2014 Received in revised form 12 October 2014 Accepted 12 May 2015 Recommended by Laura Menini
This paper presents a new method for the adaptive (tracking) regulation via partial state feedback for a class of discrete-time nonlinear systems in parametric strict-feedback form. A procedure is proposed based on immersion and invariance control approach where the parameter estimator and state observer designs are accomplished simultaneously. The algorithm utilizes discrete-time adaptive backstepping in the controller construction while providing a design without over parametrization. The proposed controller construction guarantees boundedness of the closed-loop trajectories and global (tracking) convergence to the origin of the state of the closed-loop system. The performance of the proposed method is illustrated by simulations. & 2015 European Control Association. Published by Elsevier Ltd. All rights reserved.
Keywords: Discrete-time nonlinear control Nonlinear partial feedback adaptive control Strict-feedback form Immersion and invariance control Backstepping
1. Introduction Adaptive control for continuous-time nonlinear systems is comprehensively studied and various techniques are presented in the literature [1,14,18]. Recent studies are mostly concentrated on time-delay systems, interconnected large-scale systems, hybrid systems and multi-agent systems mainly using neural networks and fuzzy control techniques [21,22,15,3,28]. For discrete-time systems there are relatively less but considerable number of studies. First Lyapunov-based designs are reported in [9,10]. A preliminary study proposing a “look-ahead” adaptive backstepping design for a class of discrete-time strict-feedback systems without using Lyapunov functions is given in [26], see also [17]. In [33,34] a recursive design scheme different from the standard backstepping has been proposed. In addition, [4] considers direct adaptive control for systems with matched uncertainties and [12] contains a periodic adaptive control approach. In [34,27], the output feedback adaptive control of nonlinear systems is considered. This problem is recently considered in [19] with a passivity based approach. Besides, discrete-time adaptive control for nonlinear systems with periodic parameters is studied in [11] with a lifting approach. On the other hand, most of the recent works deal with the robust adaptive control problems. Robust control problem of systems in strict-feedback form perturbed by a class of nonlinear uncertainties is studied/considered in [30], where local stability is proved without using Lyapunov functions. This method is improved in [31]. A robust backstepping adaptive controller design for nonlinear discrete-time systems in parametric-strict-feedback form without over parametrization is given [29,32]. In [25,6] robust asymptotic and output tracking adaptive control problems are considered for strict feedback SISO systems. Finally, in [16] some results on the robust control of first-order nonlinear systems with both parametric and non-parametric uncertainties are presented, and in [5,7,8,2] solutions to some discrete-time adaptive control problems for a class of strict feedback systems with unknown control directions are developed. This work is an extension of the previous study presented in [23,24] that presents an adaptive controller design via state-feedback for the adaptive regulation of linearly parametrized discrete-time systems in feedback form exploiting the idea introduced in [1,13] for continuoustime systems. In this paper, same problem is considered assuming the lack of measurement of some of the states and a partial feedback controller construction is given for the strict-feedback systems linear in unmeasurable states. A procedure is proposed based on immersion and invariance control approach where the parameter estimator and state observer designs are accomplished simultaneously. The discretetime state estimation (observer) error dynamics and parameter estimation error dynamics include a free function and a free dynamic
n
Tel.: þ 90 212 2856672. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.ejcon.2015.05.002 0947-3580/& 2015 European Control Association. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
Y. Yalçın / European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
weighting matrix which allow one to “shape” these estimation error dynamics. Note that, the algorithm in [23,24] was an overparametrized solution, in this paper an algorithm without over parametrization is provided assuming that the parameter estimate at time instance k is the constant parameter of the system and projecting the future evaluation of trajectories with this estimate in the controller construction. The rest of the paper is organized as follows. In Section 2 problem definition, preliminary results on the simultaneous observer and parameter estimator design for the class of nonlinear systems that are nonlinear in measurable states but linear in unmeasurable states are given and it is showed that this algorithm can be used for adaptive control design for the considered class of systems. The main results for systems in strict feedback form are presented in Section 3. Finally, in Section 4, the proposed method is applied to the course control problem for a ship. In the sequel, the notation ð:Þ þ ðkÞ ¼ ð:Þðk þ 1Þ, ð:Þ ðkÞ ¼ ð:Þðk 1Þ, namely x þ ðkÞ ¼ xðk þ 1Þ, x ðkÞ ¼ xðk 1Þ, y þ ðkÞ ¼ yðk þ 1Þ, y ðkÞ ¼ yðk 1Þ, η þ ðkÞ ¼ ηðk þ 1Þ, η ðkÞ ¼ ηðk 1Þ, z þ ðkÞ ¼ zðk þ 1Þ, z ðkÞ ¼ zðk 1Þ, θ þ ðkÞ ¼ θðk þ 1Þ, θ ðkÞ ¼ θðk 1Þ, is used. Note that when convenient the index k is omitted.
2. Preliminary results Consider the class of nonlinear systems described by the equation x
þ
¼ f ðx; u; θÞ ¼ f 0 ðxÞ þ f 1 ðxÞθ þ gðxÞu;
ð1Þ
where xðkÞ A Rn is the state vector, uðkÞ A Rm is the input vector, θðkÞ A Rp is a vector of parameters, f 0 ð0Þ ¼ 0 and f 1 ð0Þ ¼ 0, and stabilization of these systems via state feedback. If the parameters of the system are unknown (there is large uncertainties in the parameters of the system), in the perspective of the adaptive control, the following control problem needs to be considered, namely the problem of designing a discrete-time adaptive state feedback control law of the form þ ^ θ^ ¼ αðx; θÞ;
ð2Þ
^ u ¼ υðx; θÞ;
ð3Þ
such that all trajectories of the closed-loop system are bounded and lim xðkÞ ¼ 0:
ð4Þ
k-1
On the other hand, assume that parameters are known but some states are not available for measurement and consider a subclass of (1) which is bilinear in unmeasurable states as given below: y þ ¼ f y0 ðyÞ þ f y1 ðyÞη þ f y2 ðyÞθ þ g y1 ðyÞuy ðy; θÞ þ g y2 ðyÞuη ðη; uy ðy; θÞÞ; η þ ¼ f η0 ðyÞ þ f η1 ðyÞη þ f η2 ðyÞθ þ g η1 ðyÞuy ðy; θÞ þg η2 ðyÞuy ðη; uy ðy; θÞÞ;
ð5Þ
^ u^ ¼ diagðuy1 ðy; θÞ; …; uym ðy; θÞÞ and a A Rmxðn rÞ and in the compact form as follows: where uη ðη; uy Þðy; θÞÞ ¼ uaη, y þ ¼ f y ðy; η; u; θÞ; η þ ¼ f η ðy; η; u; θÞ;
ð6Þ
where measurable states are denoted as y A Rr , unmeasurable states are denoted as η A Rn r , here x ¼ ðy; ηÞ and u ¼ ½uy ; uη . In this case, the control problem that needs to be considered becomes designing a control law in the form of η^ þ ¼ wðy; η^ ; u; θÞ;
ð7Þ
u ¼ ϑðy; η^ ; θÞ;
ð8Þ
such that all trajectories of the closed loop system are bounded and x converges zero. If we consider both cases we need to deal with the following combined problem, namely the problem of designing a discrete-time adaptive partial-state feedback control law of the form ^ θÞ; ^ ^ θÞ; η^ þ ¼ wðy; η^ ; ϑðy; η;
ð9Þ
þ ^ θ^ ¼ αðy; η^ ; θÞ;
ð10Þ
^ u ¼ ϑðy; η^ ; θÞ;
ð11Þ
which achieves the control objectives, namely all trajectories of the closed loop system, ^ θÞ; ^ θÞ; η þ ¼ f η ðy; η; ϑðy; η; þ ^ θÞ; y ¼ f y ðy; η; ϑðy; η^ ; θÞ;
ð12Þ
are bounded and x converges zero. In this study, the combined problem is considered. To solve this problem, let us define the state estimation error as zη ¼ η^ Σ η η þ β η ðy ; y; ηest Þ in which β η ðy ; y; ηest Þ ¼ βη ðy ; ηest Þy, ηest ¼ Σ η 1 ðη^ þβη yÞ where βη ¼ βη ðy ; ηest Þ and parameter estimation error as zθ ¼ θ^ Σ θ θ þ β θ ðy ; y; ηest Þ in which β θ ðy ; y; ηest Þ ¼ βθ ðy ; ηest Þy and θest ¼ Σ θ 1 ðθ^ þ βθ yÞ where βθ ¼ βθ ðy ; ηest Þ. Here, Σ θ and Σ η are dynamical full rank matrices defined by Σ ηþ ¼ κ η ðyÞΣ η and Σ θþ ¼ κθ ðyÞΣ θ where κ θ ðyÞ and κη ðyÞ are dynamic full rank matrices. Thus, the estimation error for simultaneous construction of update laws (8) and (9) is given below: # # " " # " Þy; η^ Σ η η þ βη ðy ; ηest η^ Σ η η þ β η ðy ; y; ηest Þ; zη z¼ ; ¼ ¼ zθ θ^ Σ θ θ þ βθ ðy ; η Þy; θ^ Σ θ θ þ β ðy ; y; η Þ; θ
est
est
^ Γ ¼ ½η; θ and β ¼ ½βη ; βθ . Note that z ¼ Γ^ ΣΓ þ βðy ; η Þy and z þ ¼ Γ^ þ Σ þ Γ þ þ βðy; ηest Þy þ where Now, let us define Γ^ ¼ ½η^ ; θ; est Σ ¼ ½Σ η 0ðn rÞxp ; 0pxðn rÞ Σ θ . Hence, Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
Y. Yalçın / European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎
z þ γz ¼ Γ^
þ
3
γ Γ^ ΔΓ þ βðy; ηest Þy þ γβðy ; ηest Þy;
where ΔΓ ¼ Σ þ Γ þ γΣΓ, 0 o γ o 1 and, 2 3 ^ ½f η0 ðyÞ þ ðf η1 ðyÞ þg η2 ðyÞuaÞη est þ f η2 ðyÞθ est þ g η1 ðyÞu 6 7 þ 1 1 ^ 7 ðf η1 ðyÞ þ g η2 ðyÞuaÞΣ z þ γz ¼ Γ^ γ Γ^ Σ þ 6 η zη f η2 ðyÞΣ θ zθ 4 5 θest Σ θ 1 zθ " # ηest Σ η 1 zη ^ þ γΣ þβðy; ηest Þðf y0 ðyÞ þ ðf y1 ðyÞ þ g y2 ðyÞuaÞη est þ f y2 ðyÞθ est θest Σ θ 1 zθ 1 1 ^ þ g y1 ðyÞu ðf y1 ðyÞ þ g y2 ðyÞuaÞΣ η zη f y2 ðyÞΣ θ zθ Þ γβðy ; ηest Þy:
Selecting the update law as " # " # ^ ηest f η0 ðyÞ þ ðf η1 ðyÞ þ g η2 ðyÞuaÞη þ est þ f η2 ðyÞθ est þ g η ðyÞu Γ^ ¼ γ Γ^ þΣ þ γΣ θest θest ^ Þy βðy; ηest Þðf y0 ðyÞ þ ðf y1 ðyÞ þg y2 ðyÞuaÞη þ γβðy ; ηest est þ f y2 ðyÞθ est þg y1 ðyÞuÞ
yields the estimation error dynamics: " # " 1 0 1 ^ ðf η1 ðyÞ þ g η2 ðyÞuaÞΣ f η2 ðyÞΣ θ 1 Ση η þ γI þ Σ γΣ B 1 B 0 Σ 0 þ pxðn rÞ θ z ¼B @ 1 1 ^ βðy; ηest Þ½ðf y1 ðyÞ þ g y2 ðyÞuaÞΣ η f y2 ðyÞΣ θ
0
#1
Σ θ 1
C C Cz A
This can be rewritten as " # 0 1 ^ f η1 ðyÞ þ g η2 ðyÞua f η2 ðyÞ þ I Σ 1 γΣΣ 1 C Bγ þ Σ Ip 0pxðn rÞ B C zþ ¼ B Cz: @ A ^ f y2 ðyÞΣ 1 βðy; ηest Þ½f y1 ðyÞ þ g y2 ðyÞua Note that if the dynamic matrix Σ and the function βðyÞ are such that σ ð½Σ þ F η Σ 1 βðy; ηest ÞF y Σ 1 Þ o 1;
ð13Þ
^ f η2 ðyÞ; 0pxðn rÞ I p and F y ¼ ½f y1 ðyÞ þ g y2 ðyÞua ^ f y2 ðyÞ, then the estimation error converges to zero. This result where F η ¼ ½f η1 ðyÞ þ g η2 ðyÞua motivates the following preliminary statement. Proposition 1. Consider the nonlinear system (5). Suppose there exists a control law u ¼ υðy; η; θÞ;
ð14Þ
such that the zero equilibrium of the closed-loop system, y þ ¼ f y0 ðyÞ þ f y1 ðyÞη þ f y2 ðyÞθ þ g y1 ðyÞυy ðy; η; θÞ þ g y2 ðyÞυη ðy; η; θÞ; η þ ¼ f η0 ðyÞ þ f η1 ðyÞη þ f η2 ðyÞθ þ g η1 ðyÞυy ðy; η; θÞ þ g η2 ðyÞυη ðy; η; θÞ; where υ ¼ ½υy ; υη , is globally asymptotically stable. Let βðyÞ and Σ ¼ ½Σ η 0ðn rÞxp ; 0pxðn rÞ Σ θ , which is invertible and defined by the dynamic expression Σ þ ¼ κðyÞΣ with κðyÞ ¼ ½κ η 0ðn rÞxp ; 0pxðn rÞ κθ , be such that σ ð½Σ þ F η Σ 1 βðy; ηest ÞF y Σ 1 Þ o 1;
ð15Þ
^ f η2 ðyÞ; 0pxðn rÞ I p and F y ¼ ½f y1 ðyÞ þ g y2 ðyÞua ^ f y2 ðyÞ, and such that the trajectories of the system F η ¼ ½f η1 ðyÞ þ g η2 ðyÞua z þ ¼ ½Σ þ F η Σ 1 βðy; ηest ÞF y Σ 1 z; where z ¼ ½zη zθ satisfy # 0 " 1 g y1 ðyÞ 1 1 ½υy ðy; η þ Σ η zη ; θ þ Σ θ zθ Þ υy ðy; η; θÞ C B g η1 ðyÞ B C B C " # lim B C¼0 ðyÞ g C k-1B y2 @ þ ½υη ðy; η þΣ η 1 zη ; θ þ Σ θ 1 zθ Þ υη ðy; η; θÞ A g η2 ðyÞ
ð16Þ
ð17Þ
for all y; η. Then all trajectories of the closed-loop system y þ ¼ f y0 ðyÞ þ f y1 ðyÞη þ f y2 ðyÞθ þ g y1 ðyÞυy ðy; ηest ; θest Þ þg y2 ðyÞυη ðy; ηest ; θest Þ; η þ ¼ f η0 ðyÞ þ f η1 ðyÞη þ f η2 ðyÞθ þ g η1 ðyÞυy ðy; ηest ; θest Þ þg η2 ðyÞυη ðy; ηest ; θest Þ; " # ^ f η0 ðyÞ þ ðf η1 ðyÞ þ g η2 ðyÞuaÞη þ est þ f η2 ðyÞθ est þ g η1 ðyÞu; þ ^ ^ Γ ¼ γ Γ þΣ θest " # ηest ^ γΣ Þy βðy; ηest Þðf y0 ðyÞ þðf y1 ðyÞ þ g y2 ðyÞuaÞη þ γβðy ; ηest est θest Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
Y. Yalçın / European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
þ f y2 ðyÞθest þ g y1 ðyÞuÞ
ð18Þ
are bounded and limk-1 xðkÞ ¼ 0 where x ¼ ðy; ηÞ; 0 o γ o 1. Proof. Rewrite system (5) with the control law u ¼ υðy; ηest ; θest Þ, i.e. # " " þ# " # g y1 ðyÞ f y0 ðyÞ þ f y1 ðyÞη þ f y2 ðyÞθ y þ ¼ υ ðy; η; θÞ g η1 ðyÞ y f η0 ðyÞ þ f η1 ðyÞη þ ðyÞf η2 ðyÞθ ηþ ! " # " # g y1 ðyÞ g y2 ðyÞ υðy; η þ Σ η 1 zη ; θ þ Σ θ 1 zθ Þ þ υη ðy; η; θÞ þ g η2 ðyÞ g η1 ðyÞ υy ðy; η; θÞ " # g y2 ðyÞ þ υη ðy; η þ Σ η 1 zη ; θ þ Σ θ 1 zθ Þ υη ðy; η; θÞ ; g η2 ðyÞ
ð19Þ
and consider the candidate Lyapunov function V ¼ zT z . Since, σ ð½Σ þ F η Σ 1 βðy; ηest ÞF y Σ 1 Þ o 1; then, ΔV ¼ Vðk þ 1Þ VðkÞ ¼ ðz þ ÞT z þ zT z; ¼ zT ð½Σ þ F η Σ 1 βðy; ηest ÞF y Σ 1 2 IÞz o 0; hence, J zðkÞ J is bounded and converges zero. Finally, by (17) and stability of the zero equilibrium of system (5), limk-1 xðkÞ ¼ 0. □ 3. Main results In this section following discrete-time system in strict feedback form is considered: y 1þ
¼ ψ 1 ðy 1 Þy 2 þ Ω1 ðy 1 Þy 1 þ Φ1 ðy 1 Þθ;
⋮
y iþ ¼ ψ i ðy 1 ,.., y i Þy i þ 1 þ Ωi ðy 1 ,.., y i Þy i þΦi ðy 1 ,.., y i Þθ; ⋮ y rþ ¼ ψ r ðy 1 ,.., y r Þη 1 þ Ωr ðy 1 ,.., y r Þy r þ Φr ðy 1 ,.., y r Þθ;
η 1þ ¼ ψ r þ 1 ðy 1 ,.., y r Þη 2 þΩr þ 1 ðy 1 ,.., y r Þη 1 þ Φr þ 1 ðy 1 ,.., y r Þθ; ⋮ η jþ ¼ ψ r þ j ðy 1 ,.., y r Þη j þ 1 þ Ωr þ j ðy 1 ,.., y r Þη j þΦr þ j ðy 1 ,.., y r Þθ; ⋮ η nþ r ¼ ψ n ðy 1 ,.., y r Þu þ Ωn ðy 1 ,.., y r Þη n r þ Φn ðy 1 ,.., y r Þθ þ d; ni
ð20Þ nr þ j
; η i ¼ ½ηi1 ; …; ηini ; j ¼ 1 ,.., ðn rÞ are the unmeasurable where the y i ðkÞ A R ; y i ¼ ½yi1 ; …; yini ; i ¼ 1 ,.., r are the measurable states, ηj ðkÞ A R states, uðkÞ A Rm is the control input, dðkÞ A Rm is the known or measurable disturbance input, ψ i A Rnð i þ 1Þ -Rni , i ¼ 1 ,.., n are full rank, Ωi A Rni -Rni , i ¼ 1 ,.., n, and Φi A Rp -Rni , i ¼ 1 ,.., n are such that Φi ð0Þ ¼ 0 and θ A Rp is a vector of unknown constant parameters. The objective is to design a discrete-time adaptive controller described by equations of the form (9)–(11) such that all closed-loop trajectories are bounded and lim ðy 1 ðkÞ y n1 ðkÞÞ ¼ 0;
k-1
ð21Þ
where y n1 ðkÞ is a given reference signal vector. This objective is achieved in two steps, as detailed hereafter. 3.1. Observer and estimator design Let reconsider the definitions given in Section 2 Þ ¼ βη ðy ; ηest Þy; ηest ¼ Σ η 1 ðη^ þ βη yÞ zη ¼ η^ Σ η η þ β η ðy ; yÞ; β η ðy ; y; ηest zθ ¼ θ^ Σ θ θ þ β θ ðy ; y; ηest Þ; β θ ðy ; y; ηest Þ ¼ βθ ðy ; ηest Þy; θest ¼ Σ θ 1 ðθ^ þ βθ yÞ
for system (20) with y ¼ ½y i ; …; y r ; η ¼ ½η 1 ; …; η n r , namely 2 3 " þ# " # ΦTy y y 0 R E 0 4 5θ þ ¼ þ þ þ T η η Φη U 0 S D where y ¼ ½y1 ,.., yr , η ¼ ½η1 ,.., ηn r , U ¼ ½0 ,.., 0; ψ n ðy 1 ,.., y r Þu, Φy ¼ ½Φ1 ,.., Φr T ; Φη ¼ ½Φr þ 1 ,.., Φn T ; D ¼ ½0 ,.., 0; d, 2 3 ψ 1 ðy 1 Þ 0n1 xn3 0n1 xn4 … 0n1 xnn Ω1 ðy 1 Þ 6 0n xn 7 0n2 xnn 6 2 1 Ω2 ðy 1 ; y 2 Þ ψ 2 ðy 1 ; y 2 Þ 0n1 xn4 … 7 6 7 6 7; 0n3 xn2 0n3 xn3 ⋱ ⋮ R ¼ 6 0n3 xn1 7 6 ⋮ 7 ⋮ ⋮ 4 5 0n2 xnn 0n3 xnn 0n4 xnn … Ωr ðy 1 ,.., yrÞ 0n1 xnn Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
Y. Yalçın / European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2 6 6 6 S¼6 6 6 4
Ωr þ 1 ðy 1
yr Þ
,..,
2 6 6 6 E¼6 6 6 4
ψ r þ 1 ðy 1 ,.., y r Þ
0
0 0
Ωr þ 2 ðy 1 ,.., y r Þ 0
⋮
⋮
⋮
0
0 3 0 07 7 7 ⋮ 7: 7 07 5 0
0
0
0
…
0
0
…
⋮
⋮
⋮
0
0 0
…
ψ r ðy 1
yr Þ
,..,
ψ r þ 2 ðy 1
yrÞ
,.., 0
5
0
…
0
0 ⋱
…
0 ⋮
0
…
Ωn ðy 1
3 7 7 7 7; 7 7 5 ,..,
yr Þ
^ Γ ¼ ½η; θ; β ¼ ½βη ; βθ ; z ¼ Γ^ ΣΓ þ βðy ; η Þy and z þ ¼ Γ^ þ Σ þ Γ þ þ βðy; ηest Þy þ where Σ ¼ ½Σ η 0; 0 Σ θ are ^ θ, Using the definitions Γ^ ¼ ½η; est obtained. Therefore, z þ γz ¼ Γ^
þ
γ Γ^ ΔΓ þ βðy; ηest Þy þ γβðy ; ηest Þy; " # T η Sη þΦη θ þ U þ D þ Þy Σ þ z þ γz ¼ Γ^ γ Γ^ γβðy ; ηest þ γΣ θ θ
þ βðy; ηest ÞðRy þEη þ ΦTy θÞ;
"
þ z γz ¼ Γ^ γ Γ^ γβðy ; ηest Þy Σ þ þ
" þ γΣ
ηest Σ η 1 zη
Sηest þ ΦTη θest þ SΣ η 1 zη Φη Σ θ 1 zθ þ U þ D θest Σ θ 1 zθ
#
θest Σ θ 1 zθ
#
þβðy; ηest ÞðRy þ Eηest þ ΦTy θest Þ
βðy; ηest ÞðEΣ η 1 zη þ ΦTy Σ θ 1 zθ Þ: Selecting the update law
"
þ Þy þ Σ þ Γ^ ¼ γ Γ^ þγβðy ; ηest
Sηest þ ΦTη θest þ U þ D
#
θest
" γΣ
ηest
#
θest
βðy; ηest ÞðRy þ Eηest þ ΦTy θest Þ
ð22Þ
yields " zþ ¼
γI þ Σ þ
SΣ η 1
ΦTη Σ θ 1
0
Σ θ 1
#
" γΣ
Σ η 1
0
0
Σ θ 1
!!
# þ βðy; ηest Þ½EΣ η 1 ΦTy Σ θ 1
z
z þ ¼ ½Σ þ F η Σ 1 βðy; ηest ÞF y Σ 1 z; where " S Fη ¼ 0
ΦTη
ð23Þ
ð24Þ
# and
Ip
F y ¼ ½E ΦTy :
ð25Þ
The following proposition is given for the asymptotically stability of the system (24)–(25). Proposition 2. Consider the system (24)–(25). If Σ dynamics and β are selected as Σþ ¼
β¼
λ2 Σ 1 þ J Fη J Σλ1 F Ty
1 þ J F Ty F y J
ð26Þ
;
ð27Þ
where λ1 þ λ2 o 1 with λ1 ; λ2 4 0 A R and Σ 0 ¼ k0 I ðs þ pÞ ; s ¼ asymptotically stable, namely limk-1 zðkÞ ¼ 0.
Pn
i ¼ rþ1
ni ; k A R, then the zero equilibrium of the system (24)–(25) is
Proof. For (26) and (27), the estimation error system dynamics (24) becomes ! " # λ1 F Ty F y λ2 F η 1 zþ ¼ Σ Σ z: 1 þ J F η J 1 þ J F Ty F y J
ð28Þ
Now, consider the Lyapunov function V ¼ zT z. Then, T
ΔV ¼ Vðk þ 1Þ VðkÞ ¼ z þ z þ zT z; T
2
¼ z ðF IÞz
ð29Þ
Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
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where F ¼Σ
! λ1 F Ty F y λ2 F η Σ 1: 1 þ J F η J 1 þ J F Ty F y J
Note that for λ1 þ λ2 o1, the inequality ! " #! λ1 F Ty F y λ2 F η 1 Σ σ Σ o0 1 þ J F η J 1 þ J F Ty F y J is satisfied. Hence, ΔV o 0 for z a 0 is satisfied. Then the zero equilibrium of the state and parameter estimation error system (28) is asymptotically stable, namely, limk-1 V ¼ 0 and limk-1 z ¼ 0. □ 3.2. Controller design
Theorem 1. Consider the system (20). The control law, n X
n
u ¼ ∏ ðψ i 1 Þ þ ðn iÞ y 1n þ ðnÞ i¼1
n þ ðn iÞ þ ðn iÞ ∏ ðψ j 1 Þ þ ðn jÞ Ωiest xiest þ Φiest θest ψ n 1 d
ð30Þ
i¼1j¼i
where y n1 is the reference signal to be tracked, ηest ¼ Σ η 1 ðη^ þ βη yÞ and θest ¼ Σ θ 1 ðθ^ þβθ yÞ in which θ^ and η^ are obtained using the update law þ ðn jÞ þ ðn jÞ þ ðn jÞ (22), β ¼ ½βη ; βθ is given by (27), and Σ ¼ ½Σ η 0ðn rÞxp ; 0pxðn rÞ Σ θ is given by (26); ψ iest ¼ ψ i ðy 1est ,.., y jest Þ; þ ðn iÞ þ ðn iÞ þ ðn iÞ þ ðn iÞ þ ðn iÞ þ ðn iÞ þ ðn iÞ Ωiest ¼ Ωi ðy 1est ,.., y jest Þ and Φiest ¼ Φi ðy 1est ,.., y jest Þ; i ¼ 1 ,.., n; j ¼ 1 ,.., r with y jest ¼ y j and y jest , ðn iÞ 4 0 is ðn mÞ-step future values of y i obtained by recursively utilizing θest and ηest in Eqs. (20), renders all trajectories of the closed-loop system are bounded and, lim ðy 1 ðkÞ y n1 ðkÞÞ ¼ 0:
k-1
Proof. A backstepping procedure is pursued to show that the control law given in (30) achieves the control objective. Step 1 : Let y~ 1 ¼ y 1 y n1 , and note that y~ 1þ ¼ ψ 1 ðy 1 Þy 2 þ Ω1 ðy 1 Þy 1 þ Φ1 ðy 1 Þθ y 1n þ : Consider y 2 as a virtual control input and define y~ 2 ¼ y 2 y n2 ðy 1n þ Ω1 ðy 1 Þy 1 Φ1 ðy 1 Þθest Þ. Then the dynamics of y~ 1 can be rewritten as
ð31Þ where
n
y2
is
selected
as
n
y 2 ¼ ψ 1 1 ðy 1 Þ
y~ 1þ ¼ ψ 1 ðy 1 Þy~ 2 Φ1 ðy 1 Þzθ : P i ~ Therefore, if y~ 2 ¼ 0 and ni ¼ 1 Φ1i ðy 1 Þz is a l2 signal, then limk-1 y 1 ðkÞ ¼ 0. Step 2 : Note that y~ 2þ ¼ ψ 2 ðy 1 ; y 2 Þy 3 þ Ω2 ðy 1 ; y 2 Þy 2 þ Φ2 ðy 1 ; y 2 Þθ þ þ þ þ þ ψ 1 1 ðy 1est ÞðΩ1 ðy 1est Þy 1est þ Φ1 ðy 1est Þθest y 1n þ 2 Þ:
ð32Þ
ð33Þ
Here, (33) is obtained assuming that the parameter estimate at time k is the constant parameter of the system and projecting þ the future evaluation of trajectories with this estimate, namely θest ¼ θest . Following steps will be carried out with the same assumption. Now, consider y 3 as a virtual control input and define y~ 3 ¼ y 3 y n3 , where y n3 is selected as y n3 ¼ ψ 2 1 ðy 1 ; y 2 Þ þ þ þ þ þ ðψ 1 1 ðy 1est Þðy 1n þ 2 Ω1 ðy 1est Þy 1est Φ1 ðy 1est Þθest Þ Ω2 ðy 1 ; y 2 Þy 2 Φ2 ðy 1 ; y 2 Þθest Þ.As a result, þ y~ ¼ ψ ðy ; y Þy~ Φ2 ðy ; y Þzθ : ð34Þ 2
2
1
2
3
1
2
Step r : Applying the same procedure recursively yields y~ rþ ¼ ψ r ðy 1 ,.., y r Þη 1 þ Ωr ðy 1 ,.., y r Þy r þ Φr ðy 1 ,.., y r Þθ þ
rX 1
n þ ðn iÞ þ ðn iÞ ∏ ðψ i 1 Þ þ ðn iÞ Ωiest xiest þ Φiest θest
i¼1j¼i
ð35Þ ð36Þ
r
∏ ðψ i 1 Þ þ ðn iÞ y 1n þ r : i¼1
input and define η~ 1 ¼ η 1 η n1 , where η n1 is selected where x ¼ ½y; η. Consider η 1 as a virtual control Pr þ ðn iÞ þ ðn iÞ n r 1 þ ðn iÞ n þ r n 1 þ ðn iÞ η 1 ¼ ∏i ¼ 1 ðψ i Þ y 1 i ¼ 1 ∏j ¼ i ðψ i Þ Ωiest xiest þ Φiest θest . Then, dynamics of y~ r can be written as y~ rþ ¼ ψ r ðy 1 ,.., y r Þη~ 1 Φr zθ :
as ð37Þ
Step r þ 1 : Note that η~ 1þ ¼ ψ r þ 1 ðy 1 ,.., y r Þη 2 þ Ωr þ 1 ðy 1 ,.., y r Þη 1 þ Φr þ 1 ðy 1 ,.., y r Þθ r n X þ ðn iÞ þ ðn iÞ þ ∏ ðψ i 1 Þ þ ðn iÞ Ωiest xiest þ Φiest θest i¼1j¼i
ð38Þ
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rþ1
∏ ðψ i 1 Þ þ ðn iÞ y 1n þ ðr þ 1Þ :
ð39Þ
i¼1
η2 as a virtual control input and define η~ 2 ¼ η2 η n2 , where Consider Pr þ 1 n þ ðn iÞ þ ðn iÞ þ2 1 þ ðn iÞ n þ ðr þ 2Þ 1 þ ðn iÞ η n2 ¼ ∏ri ¼ ðψ Þ y ∏ ðψ Þ Ω x þ Φ θest . Thus, iest i¼1 i j¼i i 1 iest iest 1 η~ 1þ ¼ ψ r þ 1 ðy 1 ,.., y r Þη~ 2 Φr þ 1 zθ Ωr þ 1 zη1 :
η n2
is
selected
as ð40Þ
Step n : Applying the same procedure recursively yields η~ nþ r ¼ ψ n u þ n
∏
i¼1
n þ ðn iÞ þ ðn iÞ ∏ ðψ j 1 Þ þ ðn jÞ Ωiest xiest þ Φiest θest
n X
i¼1j¼i
ðψ i 1 Þ þ ðn iÞ y 1n þ ðnÞ þd:
ð41Þ
hence selecting u as in (30) yields η~ nþ r ¼ Ωn zηðn rÞ Φn zθ : Consequently, the closed-loop system in the x~ ¼ ðy~ ; η~ Þ and z co-ordinates is described by y~ 1þ ¼ ψ 1 ðy 1 Þy~ 2 Φ1 ðy 1 Þzθ ; y~ 2þ ¼ ψ 2 ðy 1 ; y 2 Þy~ 3 Φ2 ðy 1 ; y 2 Þzθ ; ⋮ y~ rþ ¼ ψ r ðy 1 ,.., y r Þη~ 1 Ωr zη1 Φr ðy 1 ,.., y r Þzθ ; η~ 1þ ¼ ψ Tr þ 1 ðy 1 ,.., y r Þη~ 2 Ωr þ 1 zη2 ΦTr þ 1 ðy 1 ,.., y r Þzθ ; ⋮ η~ nþ r ¼ Ωn zηðn rÞ Φn ðy 1 ,.., y r Þzθ ;
ð42Þ
~ ~ ¼ 0. In is bounded and moreover limk-1 xðkÞ together with system (27)–(28). Note that limk-1 zðkÞ ¼ 0 implies that xðkÞ order to show the local stability of the zero equilibrium of the closed-loop system, first consider the system y~ 1þ ¼ ψ 1 ðy 1 Þy~ 2 ; ⋮ y~ rþ ¼ ψ r ðy 1 ,.., y r Þη~ 1 ; η~ þ ¼ ψ r þ 1 ðy 1 ,.., y r Þη~ 2 ; 1
⋮ η~ nþ r ¼ 0
ð43Þ
and rewrite Eqs. (43) as x~ þ ¼ Ax~ ~ η~ Þ and where x~ ¼ ðy; 2 0 ψ1 0 … 6 6 0 0 ψ2 … A¼6 6⋮ ⋮ ⋮ ⋮ 4 0 0 0 …
0
3
7 0 7 7: ⋮ 7 5 ψn
Let V ¼ x~ T P x~ with P ¼ diagðp1 I n1 ; p2 I n2
,..,
ΔV ¼ x~ A PAx~ x~ P x~ o x~ μx~ for x~ a 0 2 p1 I n 1 0 0 T 6 I þ ψ p ψ 0 0 p 2 n2 6 1 1 1 T6 6 I þ ψ T2 p2 ψ 2 0 0 p ¼ x~ 6 3 n3 6 ⋮ ⋮ ⋮ 4 0 0 0 T
T
T
pn I nn Þ; pi 40, and note that
T
…
0
…
0
…
0
⋮ …
⋮ pn I nn þ ψ Tn 1 pn 1 ψ n 1
3 7 7 7 7x~ 7 7 5
o x~ T μx~ for x~ a 0: It is easily seen that this inequality is satisfied by any 0 op1 ; σ ðψ T1 p1 ψ 1 Þ o p2 ; σ ðψ T2 p2 ψ 2 Þ o p3 ; ……; σ ðψ Tn 1 pn 1 ψ n 1 Þ o pn . Now, consider the candidate Lyapunov function W ¼ x~ T P x~ þ δzT z ¼ x~ T P x~ þ δ1 zTθ zθ þ δ2 zTη zη ; P 4 0 A Rnn ;
δ ¼ diagðδ1 ; δ2 ÞÞ;
δ1 ; δ2 40 A R;
for the disturbed system (42). Then, ΔW ¼ ½P 1=2 ðAx~ ΦTx zθ Þ2 x~ T P x~ þ δððz þ Þ2 z2 Þ Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
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where Φx ¼ ½Φ1
Φn , and
,..,
ΔW ¼ x~ A PAx~ x~ P x~ 2x~ A PΦx zθ þ ½P 1=2 Φx zθ 2 T
T
T
T
T
þ δzT ðF 2 IÞz x~ T AT PAx~ þ ð1 þ ϵÞ½P 1=2 Φx zθ 2 þ δzT ðF 2 IÞz: ϵ For ϵ large μþ ζ AT PA=ϵ o μ=2 can be written, hence μ o x~ T x~ þð1 þ ϵÞζðPÞzTθ Φx ΦTx zθ þδzT ðF 2 IÞz 2 μ o x~ T x~ þzT ½ð1 þ ϵÞM þ δNz 2 o μx~ T x~ þ
where M ¼ ½0 0; 0 ζðPÞΦx ΦTx Z 0 and N ¼ F 2 I o0. Note now that for any compact set Ω there exists δ such that ð1 þ ϵÞM þ δN o 0. Hence the zero equilibrium of the closed-loop system is locally stable.□ þ Remark 1. The control law u given in (30) consists of ”look-ahead” values x iest of the state variable x i A x ¼ ½x 1 ; …x n ¼ þ ½y 1 ; …; y r ; η 1 ; …; η ðn rÞ . The ”look-ahead” values x iest are obtained substituting the estimated unmeasurable states ηest and estimated system parameters θest in system dynamics (20).
4. Example The proposed methodology is applied to adaptive control problem of a synchronous generator which is a 2 2 MIMO system. Continuous time system dynamics are considered as below in [20]: δ_ ¼ ω ω_ ¼ p1 sin ð2δÞ p2 ω p3 ψ f sin ðδÞ p4 P m þ p4 P v ψ_f ¼ p5 cos ðδÞ p6 ψ f p7 Efd Efd ke E_fd ¼ þ ue Te Te P m þP g P_m ¼ Tt Pg 1 _ P g ¼ þ ug Tg Tg The aim of the control is to regulate the electrical generated power and the terminal voltage. Electrical generated power P e ¼ f 2 ðδ; ψ f ; θÞ and terminal voltage V t ¼ f 2 ðδ; ψ f ; θÞ are functions of δ; ψ f and system parameters θ as follows: p3 ψ f sin ðδÞ p1 sin ð2δÞ p4 #2 2 " 0 xaf xt ψ f Vxq sin ðδÞ Vxd cos ðδÞ 2 2 þ þ V f ¼ f 2 ðδ; ψ f ; θÞ ¼ xt þ xq xt þ x0d ðxt þ x0d Þxf
Pe ¼
Therefore y1 ¼ δ and y2 ¼ ψ f are selected as the outputs. To be able to apply the proposed technique it is need to formulate the system dynamic in convenient block strict form as in the sequel. For the purpose the dynamics of y1 ¼ δ; y2 ¼ ψ f ; η1 ¼ ω; η2 ¼ Efd are captured in the following MIMO system in block strict feedback form considering P m also as an input designating it as upm : y_ ¼ ψ 1 η þ Φ1 ðyÞθ; η_ ¼ ψ 2 u þ Ω2 η þ Φ2 ðy 1 ; y 2 Þθ þ d where y ¼ ½y1 ; y2 T ; η ¼ ½η1 ; η2 T ; u ¼ ½upm ; ue T ; upm ¼ Pm; θ ¼ ½p1 ; p2 ; p3 ; p5 ; p6 T and, " # " # 1 0 0 0 0 0 0 ψ1 ¼ ; Φ1 ¼ ; 0 p7 0 0 0 cos ðx11 Þ x12 " # 0 0 sin ðx11 Þ x21 x12 sin ðx11 Þ 0 0 ; Ω1 ¼ Φ2 ¼ ; 0 T1e 0 0 0 0 0 " # 0 p4 p4 P v : d¼ ψ2 ¼ ke ; 0 Te 0 The dynamics of P m and P g is considered as another system in the strict feedback form given in (42). The proposed method is applied to the first MIMO system to construct real input ue and the fictional input upm . Afterwards, ug is obtained applying backstepping design to the
Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
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second system considering pnm ¼ upm as the desired reference: ξ_1 ¼ ψ ξ1 ξ2 þ Φξ1 ðξ1 Þ; ξ_2 ¼ ψ ξ2 ug þ Φξ2 ðξ1 ; ξ2 Þ where ξ1 ¼ P m ; ξ2 ¼ P g ; ψ ξ1 ¼ 1=T t ; ψ ξ2 ¼ 1=T g ; Φξ1 ðξ1 Þ ¼ 1=T t P m , and Φξ2 ðξ1 ; ξ2 Þ ¼ 1=T g P g . Note that, this system does not include any unmeasurable states or unknown parameters, hence the backstepping design can be directly applied. The discrete time dynamics are obtained by Euler method as follows: y 1þ ¼ y 1 þTψ 1 η 1 þ TΦ1 ðy 1 Þθ; η 1þ ¼ η 1 þ Tψ 2 u þ TΩ2 η 1 þTΦ2 ðy 1 ; y 2 Þθ þ Td and, ξ1þ ¼ ξ1 þ Tψ ξ1 ξ2 þ TΦξ1 ðξ1 Þ;
ξ2þ ¼ ξ2 þ Tψ ξ2 ug þTΦξ2 ðξ1 ; ξ2 Þ where T is the sampling period used in the discretization. The obtained discrete time systems are not in strict feedback form therefore the following coordinate transformations is considered: y^ 1 ¼ y η^ 1 ¼ y 1 þ Tψ 1 η and, ξ^1 ¼ ξ1 ; ξ^2 ¼ ξ1 þ Tψ ξ1 ξ2 : This coordinate transformation yields below discrete-time systems in strict feedback form: ^ 1 ðy^ 1 Þθ; y^ 1þ ¼ η^ 1 þ Φ
^ 2 η^ 1 þ Φ ^ 2 ðy^ 1 Þθ þ d^ η^ 1þ ¼ ψ^ 2 u þ Ω ^ 2 ¼ I 2 þ ψðI 2 þ TΩ2 Þψ 1 ; d^s ¼ ψðI 2 þ TΩ2 Þψ 1 þ T 2 ψ 1 ds and, ^ 1 ¼ TΦ1 ; Φ ^ 2 ¼ TðTψ 1 Φ2 þ Φ1 Þθ; Ω where ψ^ 1 ¼ I 2 ; ψ^ 2 ¼ T 2 ψ 1 ψ 2 ; Φ þ ^ ξ1 ðξ^1 Þ; ξ^1 ¼ ξ^2 þ Φ þ ^ ξ2 ðξ^1 ; ξ^2 Þ: ξ^ ¼ ψ^ ξ2 ug þ Φ 2
The parameters in (42) are in terms of real system parameters and the values of real system parameters considered as considered in [20] are given below. T ¼ 0:005 is considered in the discretization of the system dynamics: ω0 V 2 ðxq x0 dÞ ; r f ¼ 0:0012 p:u ., V ¼ 1 p:u: 4Hðxt þ x0 dÞðxt xq Þ ω0 d ; xd ¼ 1:75 p:u ., d ¼ 0:006d p2 ¼ 2H ω0 Vxaf ; xt ¼ 1:665 p:u ., f ¼ 60 Hz p3 ¼ 2Hxf ðxt þx0d Þ ω0 ; xq ¼ 1:68 p:u ., ke ¼ 25 p4 ¼ 2H ω0 r f Vxaf ; xf ¼ 1:665 p:u ., T e ¼ 0:04 s p5 ¼ xf ðxt þ x0d Þ ω0 r f ðxt þ x0d Þ ; xaf ¼ 1:56 p:u ., T t ¼ 0:3 s p6 ¼ xf ðxt þ x0d Þ ω0 r f p7 ¼ ; x0d ¼ 0:285 p:u ., T g ¼ 0:08 s xf p1 ¼
x0d ¼ xd
x2af xf
;
ω0 ¼ 2πf ;
H ¼ 3:82 s
Note that, in this example ω0 ; H and r f =xf , namely, p4 and p7 are assumed to be known and others unknown parameters. Here, V is the voltage on infinity bus and xt is the transmission line reactance. The design parameters of the estimator are selected as λ1 ¼ 0:1; λ2 ¼ 0:2 and γ ¼ 0:1. The initial conditions for the state and parameter ^ estimator have been set to η^ ð0Þ ¼ ½0:001; 0:001; θð0Þ ¼ ½0:1; 0:1; 0:1; 0:1; 0:1; κ θ ¼ I 5 ; κ η ¼ I 2 ; Σð0Þ ¼ I 7 , and the initial state has been selected as xð0Þ ¼ ½00:001; 0:001; 0:001; 0:001; 0:001; 0:001. Fig. 1 shows the time histories of state and parameter estimation error z ¼ ½z1 ; z2 ; z3 ; z4 ; z5 ; z6 ; z7 T which converge to zero, while Fig. 2 displays the time history of the measurable states y1 and y2 . Finally, Fig. 3 demonstrates the time history of true and estimated values of unmeasurable states η1 and η2 . In Fig. 4 the evaluation of the control signals Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
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2
1.5
z1,z2,z3, z4,z5,z6,z7
1
0.5
0
0
1
2
3
4
5
Time[k] Fig. 1. Time histories of the estimation error variable z ¼ ½z1 ; z2 ; z3 ; z4 ; z5 ; z6 ; z7 .
2.5
2
1.5
y1, y2 1
0.5
0
0
1
2
3
4
5
6
7
Time[k] Fig. 2. Time histories of the states y1 and y2 . The dashed line displays the time history of the states y1 and y2 when the known-parameter and partial state-feedback controller are used.
2.5
2
1.5
η1, η2
1
0.5
0
0
1
2
3
4
5
6
7
Time[k] Fig. 3. Time histories of the states η1 and η2 . The dashed line displays the time history of the states η1 and η2 when the known-parameter and partial state-feedback controller are used.
is shown. The simulation results demonstrate that the proposed adaptive controller renders the closed-loop system approximately achieving the performance that the true parameter controller yields. 5. Conclusions In this paper a novel partial-feedback adaptive controller design method for the adaptive regulation of a class of linearly parameterized discrete-time systems in strict-feedback form that is linear in unmeasurable states has been developed. A procedure without over Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
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11
x 106
1.5 1 0.5
u1,u2
0 −0.5 −1 −1.5
0
1
2
3
4
5
6
7
Time[k] Fig. 4. Time histories of control signals u1 and u2 . The dashed line displays the time history of the control signals u1 and u2 of the known-parameter and partial statefeedback controller.
parametrization that provides simultaneous construction of update laws for both state estimation and parameter estimation error dynamics is presented. State and parameter estimation error dynamics include a free function and a dynamic matrix that makes possible to shape the transient dynamics of the estimation error system. It is shown that the presented combined update law renders the zero equilibrium of the estimation error system stable. The controller construction is performed via a back-stepping procedure with immersion and invariance control approach and utilizing presented update laws. Assuming that the parameter estimate at time instance k is the constant parameter of the system and projecting the future evaluation of trajectories with this estimate in the back stepping controller construction has been enabled to use the presented algorithm without over parametrization. The procedure is utilized for the adaptive control problem of a synchronous generator and corresponding simulation results are presented.
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[25] C. Yang, S. Dai, S. Ge, T. Lee, Adaptive asymptotic tracking control of a class of discrete-time nonlinear systems with parametric and nonparametric uncertainties, in: Proceedings of the American Control Conference, Hyatt Regency Riverfront, St. Louis, MO, USA, 2009. [26] P.-C. Yeh, P. Kokotovic, Adaptive control of a class of nonlinear discrete-time systems, Int. J. Control 62 (1995) 303–324. [27] P. Yeh, P.V. Kokotovic, Y. Soh, Adaptive output-feedback design for a class of nonlinear discrete-time systems, IEEE Trans. Autom. Control 40 (9) (1995) 1663–1668. [28] S.J. Yoo, L. Han, C. Wen, L. Xu, Distributed adaptive containment control of uncertain nonlinear multi-agent systems in strict-feedback form, Automatica 49 (7) (2013) 2145–2153. [29] Y. Zhang, W.H. Chen, Y. Soh, Improved robust backstepping adaptive control for nonlinear discrete-time systems without overparameterization, Automatica 33 (2008) 864–867. [30] Y. Zhang, C. Wen, Y. Soh, Robust adaptive control of uncertain discrete-time systems, Automatica 35 (1999) 321–329. [31] Y. Zhang, C. Wen, Y. Soh, Discrete-time robust backstepping adaptive control for nonlinear time-varying systems, IEEE Trans. Autom. Control 45 (9) (2000) 1749–1755. [32] Y. Zhang, C. Wen, Y. Soh, Robust adaptive control of nonlinear discrete-time systems by backstepping without overparameterization, Automatica 37 (2001) 551–558. [33] J. Zhao, I. Kanellakopoulos, Adaptive control of discrete-time strict-feedback nonlinear systems, in: Proceedings of the American Control Conference, Albuquerque, New Mexico, 1997. [34] J. Zhao, I. Kanellakopoulos, Discrete-time adaptive control of output-feedback nonlinear systems, in: Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA, 1997, pp. 4326–4331.
Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
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European Journal of Control journal homepage: www.elsevier.com/locate/ejcon
Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form Yaprak Yalçın n Department of Control and Automation Engineering, Faculty of Electrical-Electronics Engineering, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey
art ic l e i nf o
a b s t r a c t
Article history: Received 4 April 2014 Received in revised form 12 October 2014 Accepted 12 May 2015 Recommended by Laura Menini
This paper presents a new method for the adaptive (tracking) regulation via partial state feedback for a class of discrete-time nonlinear systems in parametric strict-feedback form. A procedure is proposed based on immersion and invariance control approach where the parameter estimator and state observer designs are accomplished simultaneously. The algorithm utilizes discrete-time adaptive backstepping in the controller construction while providing a design without over parametrization. The proposed controller construction guarantees boundedness of the closed-loop trajectories and global (tracking) convergence to the origin of the state of the closed-loop system. The performance of the proposed method is illustrated by simulations. & 2015 European Control Association. Published by Elsevier Ltd. All rights reserved.
Keywords: Discrete-time nonlinear control Nonlinear partial feedback adaptive control Strict-feedback form Immersion and invariance control Backstepping
1. Introduction Adaptive control for continuous-time nonlinear systems is comprehensively studied and various techniques are presented in the literature [1,14,18]. Recent studies are mostly concentrated on time-delay systems, interconnected large-scale systems, hybrid systems and multi-agent systems mainly using neural networks and fuzzy control techniques [21,22,15,3,28]. For discrete-time systems there are relatively less but considerable number of studies. First Lyapunov-based designs are reported in [9,10]. A preliminary study proposing a “look-ahead” adaptive backstepping design for a class of discrete-time strict-feedback systems without using Lyapunov functions is given in [26], see also [17]. In [33,34] a recursive design scheme different from the standard backstepping has been proposed. In addition, [4] considers direct adaptive control for systems with matched uncertainties and [12] contains a periodic adaptive control approach. In [34,27], the output feedback adaptive control of nonlinear systems is considered. This problem is recently considered in [19] with a passivity based approach. Besides, discrete-time adaptive control for nonlinear systems with periodic parameters is studied in [11] with a lifting approach. On the other hand, most of the recent works deal with the robust adaptive control problems. Robust control problem of systems in strict-feedback form perturbed by a class of nonlinear uncertainties is studied/considered in [30], where local stability is proved without using Lyapunov functions. This method is improved in [31]. A robust backstepping adaptive controller design for nonlinear discrete-time systems in parametric-strict-feedback form without over parametrization is given [29,32]. In [25,6] robust asymptotic and output tracking adaptive control problems are considered for strict feedback SISO systems. Finally, in [16] some results on the robust control of first-order nonlinear systems with both parametric and non-parametric uncertainties are presented, and in [5,7,8,2] solutions to some discrete-time adaptive control problems for a class of strict feedback systems with unknown control directions are developed. This work is an extension of the previous study presented in [23,24] that presents an adaptive controller design via state-feedback for the adaptive regulation of linearly parametrized discrete-time systems in feedback form exploiting the idea introduced in [1,13] for continuoustime systems. In this paper, same problem is considered assuming the lack of measurement of some of the states and a partial feedback controller construction is given for the strict-feedback systems linear in unmeasurable states. A procedure is proposed based on immersion and invariance control approach where the parameter estimator and state observer designs are accomplished simultaneously. The discretetime state estimation (observer) error dynamics and parameter estimation error dynamics include a free function and a free dynamic
n
Tel.: þ 90 212 2856672. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.ejcon.2015.05.002 0947-3580/& 2015 European Control Association. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
Y. Yalçın / European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
weighting matrix which allow one to “shape” these estimation error dynamics. Note that, the algorithm in [23,24] was an overparametrized solution, in this paper an algorithm without over parametrization is provided assuming that the parameter estimate at time instance k is the constant parameter of the system and projecting the future evaluation of trajectories with this estimate in the controller construction. The rest of the paper is organized as follows. In Section 2 problem definition, preliminary results on the simultaneous observer and parameter estimator design for the class of nonlinear systems that are nonlinear in measurable states but linear in unmeasurable states are given and it is showed that this algorithm can be used for adaptive control design for the considered class of systems. The main results for systems in strict feedback form are presented in Section 3. Finally, in Section 4, the proposed method is applied to the course control problem for a ship. In the sequel, the notation ð:Þ þ ðkÞ ¼ ð:Þðk þ 1Þ, ð:Þ ðkÞ ¼ ð:Þðk 1Þ, namely x þ ðkÞ ¼ xðk þ 1Þ, x ðkÞ ¼ xðk 1Þ, y þ ðkÞ ¼ yðk þ 1Þ, y ðkÞ ¼ yðk 1Þ, η þ ðkÞ ¼ ηðk þ 1Þ, η ðkÞ ¼ ηðk 1Þ, z þ ðkÞ ¼ zðk þ 1Þ, z ðkÞ ¼ zðk 1Þ, θ þ ðkÞ ¼ θðk þ 1Þ, θ ðkÞ ¼ θðk 1Þ, is used. Note that when convenient the index k is omitted.
2. Preliminary results Consider the class of nonlinear systems described by the equation x
þ
¼ f ðx; u; θÞ ¼ f 0 ðxÞ þ f 1 ðxÞθ þ gðxÞu;
ð1Þ
where xðkÞ A Rn is the state vector, uðkÞ A Rm is the input vector, θðkÞ A Rp is a vector of parameters, f 0 ð0Þ ¼ 0 and f 1 ð0Þ ¼ 0, and stabilization of these systems via state feedback. If the parameters of the system are unknown (there is large uncertainties in the parameters of the system), in the perspective of the adaptive control, the following control problem needs to be considered, namely the problem of designing a discrete-time adaptive state feedback control law of the form þ ^ θ^ ¼ αðx; θÞ;
ð2Þ
^ u ¼ υðx; θÞ;
ð3Þ
such that all trajectories of the closed-loop system are bounded and lim xðkÞ ¼ 0:
ð4Þ
k-1
On the other hand, assume that parameters are known but some states are not available for measurement and consider a subclass of (1) which is bilinear in unmeasurable states as given below: y þ ¼ f y0 ðyÞ þ f y1 ðyÞη þ f y2 ðyÞθ þ g y1 ðyÞuy ðy; θÞ þ g y2 ðyÞuη ðη; uy ðy; θÞÞ; η þ ¼ f η0 ðyÞ þ f η1 ðyÞη þ f η2 ðyÞθ þ g η1 ðyÞuy ðy; θÞ þg η2 ðyÞuy ðη; uy ðy; θÞÞ;
ð5Þ
^ u^ ¼ diagðuy1 ðy; θÞ; …; uym ðy; θÞÞ and a A Rmxðn rÞ and in the compact form as follows: where uη ðη; uy Þðy; θÞÞ ¼ uaη, y þ ¼ f y ðy; η; u; θÞ; η þ ¼ f η ðy; η; u; θÞ;
ð6Þ
where measurable states are denoted as y A Rr , unmeasurable states are denoted as η A Rn r , here x ¼ ðy; ηÞ and u ¼ ½uy ; uη . In this case, the control problem that needs to be considered becomes designing a control law in the form of η^ þ ¼ wðy; η^ ; u; θÞ;
ð7Þ
u ¼ ϑðy; η^ ; θÞ;
ð8Þ
such that all trajectories of the closed loop system are bounded and x converges zero. If we consider both cases we need to deal with the following combined problem, namely the problem of designing a discrete-time adaptive partial-state feedback control law of the form ^ θÞ; ^ ^ θÞ; η^ þ ¼ wðy; η^ ; ϑðy; η;
ð9Þ
þ ^ θ^ ¼ αðy; η^ ; θÞ;
ð10Þ
^ u ¼ ϑðy; η^ ; θÞ;
ð11Þ
which achieves the control objectives, namely all trajectories of the closed loop system, ^ θÞ; ^ θÞ; η þ ¼ f η ðy; η; ϑðy; η; þ ^ θÞ; y ¼ f y ðy; η; ϑðy; η^ ; θÞ;
ð12Þ
are bounded and x converges zero. In this study, the combined problem is considered. To solve this problem, let us define the state estimation error as zη ¼ η^ Σ η η þ β η ðy ; y; ηest Þ in which β η ðy ; y; ηest Þ ¼ βη ðy ; ηest Þy, ηest ¼ Σ η 1 ðη^ þβη yÞ where βη ¼ βη ðy ; ηest Þ and parameter estimation error as zθ ¼ θ^ Σ θ θ þ β θ ðy ; y; ηest Þ in which β θ ðy ; y; ηest Þ ¼ βθ ðy ; ηest Þy and θest ¼ Σ θ 1 ðθ^ þ βθ yÞ where βθ ¼ βθ ðy ; ηest Þ. Here, Σ θ and Σ η are dynamical full rank matrices defined by Σ ηþ ¼ κ η ðyÞΣ η and Σ θþ ¼ κθ ðyÞΣ θ where κ θ ðyÞ and κη ðyÞ are dynamic full rank matrices. Thus, the estimation error for simultaneous construction of update laws (8) and (9) is given below: # # " " # " Þy; η^ Σ η η þ βη ðy ; ηest η^ Σ η η þ β η ðy ; y; ηest Þ; zη z¼ ; ¼ ¼ zθ θ^ Σ θ θ þ βθ ðy ; η Þy; θ^ Σ θ θ þ β ðy ; y; η Þ; θ
est
est
^ Γ ¼ ½η; θ and β ¼ ½βη ; βθ . Note that z ¼ Γ^ ΣΓ þ βðy ; η Þy and z þ ¼ Γ^ þ Σ þ Γ þ þ βðy; ηest Þy þ where Now, let us define Γ^ ¼ ½η^ ; θ; est Σ ¼ ½Σ η 0ðn rÞxp ; 0pxðn rÞ Σ θ . Hence, Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
Y. Yalçın / European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎
z þ γz ¼ Γ^
þ
3
γ Γ^ ΔΓ þ βðy; ηest Þy þ γβðy ; ηest Þy;
where ΔΓ ¼ Σ þ Γ þ γΣΓ, 0 o γ o 1 and, 2 3 ^ ½f η0 ðyÞ þ ðf η1 ðyÞ þg η2 ðyÞuaÞη est þ f η2 ðyÞθ est þ g η1 ðyÞu 6 7 þ 1 1 ^ 7 ðf η1 ðyÞ þ g η2 ðyÞuaÞΣ z þ γz ¼ Γ^ γ Γ^ Σ þ 6 η zη f η2 ðyÞΣ θ zθ 4 5 θest Σ θ 1 zθ " # ηest Σ η 1 zη ^ þ γΣ þβðy; ηest Þðf y0 ðyÞ þ ðf y1 ðyÞ þ g y2 ðyÞuaÞη est þ f y2 ðyÞθ est θest Σ θ 1 zθ 1 1 ^ þ g y1 ðyÞu ðf y1 ðyÞ þ g y2 ðyÞuaÞΣ η zη f y2 ðyÞΣ θ zθ Þ γβðy ; ηest Þy:
Selecting the update law as " # " # ^ ηest f η0 ðyÞ þ ðf η1 ðyÞ þ g η2 ðyÞuaÞη þ est þ f η2 ðyÞθ est þ g η ðyÞu Γ^ ¼ γ Γ^ þΣ þ γΣ θest θest ^ Þy βðy; ηest Þðf y0 ðyÞ þ ðf y1 ðyÞ þg y2 ðyÞuaÞη þ γβðy ; ηest est þ f y2 ðyÞθ est þg y1 ðyÞuÞ
yields the estimation error dynamics: " # " 1 0 1 ^ ðf η1 ðyÞ þ g η2 ðyÞuaÞΣ f η2 ðyÞΣ θ 1 Ση η þ γI þ Σ γΣ B 1 B 0 Σ 0 þ pxðn rÞ θ z ¼B @ 1 1 ^ βðy; ηest Þ½ðf y1 ðyÞ þ g y2 ðyÞuaÞΣ η f y2 ðyÞΣ θ
0
#1
Σ θ 1
C C Cz A
This can be rewritten as " # 0 1 ^ f η1 ðyÞ þ g η2 ðyÞua f η2 ðyÞ þ I Σ 1 γΣΣ 1 C Bγ þ Σ Ip 0pxðn rÞ B C zþ ¼ B Cz: @ A ^ f y2 ðyÞΣ 1 βðy; ηest Þ½f y1 ðyÞ þ g y2 ðyÞua Note that if the dynamic matrix Σ and the function βðyÞ are such that σ ð½Σ þ F η Σ 1 βðy; ηest ÞF y Σ 1 Þ o 1;
ð13Þ
^ f η2 ðyÞ; 0pxðn rÞ I p and F y ¼ ½f y1 ðyÞ þ g y2 ðyÞua ^ f y2 ðyÞ, then the estimation error converges to zero. This result where F η ¼ ½f η1 ðyÞ þ g η2 ðyÞua motivates the following preliminary statement. Proposition 1. Consider the nonlinear system (5). Suppose there exists a control law u ¼ υðy; η; θÞ;
ð14Þ
such that the zero equilibrium of the closed-loop system, y þ ¼ f y0 ðyÞ þ f y1 ðyÞη þ f y2 ðyÞθ þ g y1 ðyÞυy ðy; η; θÞ þ g y2 ðyÞυη ðy; η; θÞ; η þ ¼ f η0 ðyÞ þ f η1 ðyÞη þ f η2 ðyÞθ þ g η1 ðyÞυy ðy; η; θÞ þ g η2 ðyÞυη ðy; η; θÞ; where υ ¼ ½υy ; υη , is globally asymptotically stable. Let βðyÞ and Σ ¼ ½Σ η 0ðn rÞxp ; 0pxðn rÞ Σ θ , which is invertible and defined by the dynamic expression Σ þ ¼ κðyÞΣ with κðyÞ ¼ ½κ η 0ðn rÞxp ; 0pxðn rÞ κθ , be such that σ ð½Σ þ F η Σ 1 βðy; ηest ÞF y Σ 1 Þ o 1;
ð15Þ
^ f η2 ðyÞ; 0pxðn rÞ I p and F y ¼ ½f y1 ðyÞ þ g y2 ðyÞua ^ f y2 ðyÞ, and such that the trajectories of the system F η ¼ ½f η1 ðyÞ þ g η2 ðyÞua z þ ¼ ½Σ þ F η Σ 1 βðy; ηest ÞF y Σ 1 z; where z ¼ ½zη zθ satisfy # 0 " 1 g y1 ðyÞ 1 1 ½υy ðy; η þ Σ η zη ; θ þ Σ θ zθ Þ υy ðy; η; θÞ C B g η1 ðyÞ B C B C " # lim B C¼0 ðyÞ g C k-1B y2 @ þ ½υη ðy; η þΣ η 1 zη ; θ þ Σ θ 1 zθ Þ υη ðy; η; θÞ A g η2 ðyÞ
ð16Þ
ð17Þ
for all y; η. Then all trajectories of the closed-loop system y þ ¼ f y0 ðyÞ þ f y1 ðyÞη þ f y2 ðyÞθ þ g y1 ðyÞυy ðy; ηest ; θest Þ þg y2 ðyÞυη ðy; ηest ; θest Þ; η þ ¼ f η0 ðyÞ þ f η1 ðyÞη þ f η2 ðyÞθ þ g η1 ðyÞυy ðy; ηest ; θest Þ þg η2 ðyÞυη ðy; ηest ; θest Þ; " # ^ f η0 ðyÞ þ ðf η1 ðyÞ þ g η2 ðyÞuaÞη þ est þ f η2 ðyÞθ est þ g η1 ðyÞu; þ ^ ^ Γ ¼ γ Γ þΣ θest " # ηest ^ γΣ Þy βðy; ηest Þðf y0 ðyÞ þðf y1 ðyÞ þ g y2 ðyÞuaÞη þ γβðy ; ηest est θest Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
Y. Yalçın / European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
þ f y2 ðyÞθest þ g y1 ðyÞuÞ
ð18Þ
are bounded and limk-1 xðkÞ ¼ 0 where x ¼ ðy; ηÞ; 0 o γ o 1. Proof. Rewrite system (5) with the control law u ¼ υðy; ηest ; θest Þ, i.e. # " " þ# " # g y1 ðyÞ f y0 ðyÞ þ f y1 ðyÞη þ f y2 ðyÞθ y þ ¼ υ ðy; η; θÞ g η1 ðyÞ y f η0 ðyÞ þ f η1 ðyÞη þ ðyÞf η2 ðyÞθ ηþ ! " # " # g y1 ðyÞ g y2 ðyÞ υðy; η þ Σ η 1 zη ; θ þ Σ θ 1 zθ Þ þ υη ðy; η; θÞ þ g η2 ðyÞ g η1 ðyÞ υy ðy; η; θÞ " # g y2 ðyÞ þ υη ðy; η þ Σ η 1 zη ; θ þ Σ θ 1 zθ Þ υη ðy; η; θÞ ; g η2 ðyÞ
ð19Þ
and consider the candidate Lyapunov function V ¼ zT z . Since, σ ð½Σ þ F η Σ 1 βðy; ηest ÞF y Σ 1 Þ o 1; then, ΔV ¼ Vðk þ 1Þ VðkÞ ¼ ðz þ ÞT z þ zT z; ¼ zT ð½Σ þ F η Σ 1 βðy; ηest ÞF y Σ 1 2 IÞz o 0; hence, J zðkÞ J is bounded and converges zero. Finally, by (17) and stability of the zero equilibrium of system (5), limk-1 xðkÞ ¼ 0. □ 3. Main results In this section following discrete-time system in strict feedback form is considered: y 1þ
¼ ψ 1 ðy 1 Þy 2 þ Ω1 ðy 1 Þy 1 þ Φ1 ðy 1 Þθ;
⋮
y iþ ¼ ψ i ðy 1 ,.., y i Þy i þ 1 þ Ωi ðy 1 ,.., y i Þy i þΦi ðy 1 ,.., y i Þθ; ⋮ y rþ ¼ ψ r ðy 1 ,.., y r Þη 1 þ Ωr ðy 1 ,.., y r Þy r þ Φr ðy 1 ,.., y r Þθ;
η 1þ ¼ ψ r þ 1 ðy 1 ,.., y r Þη 2 þΩr þ 1 ðy 1 ,.., y r Þη 1 þ Φr þ 1 ðy 1 ,.., y r Þθ; ⋮ η jþ ¼ ψ r þ j ðy 1 ,.., y r Þη j þ 1 þ Ωr þ j ðy 1 ,.., y r Þη j þΦr þ j ðy 1 ,.., y r Þθ; ⋮ η nþ r ¼ ψ n ðy 1 ,.., y r Þu þ Ωn ðy 1 ,.., y r Þη n r þ Φn ðy 1 ,.., y r Þθ þ d; ni
ð20Þ nr þ j
; η i ¼ ½ηi1 ; …; ηini ; j ¼ 1 ,.., ðn rÞ are the unmeasurable where the y i ðkÞ A R ; y i ¼ ½yi1 ; …; yini ; i ¼ 1 ,.., r are the measurable states, ηj ðkÞ A R states, uðkÞ A Rm is the control input, dðkÞ A Rm is the known or measurable disturbance input, ψ i A Rnð i þ 1Þ -Rni , i ¼ 1 ,.., n are full rank, Ωi A Rni -Rni , i ¼ 1 ,.., n, and Φi A Rp -Rni , i ¼ 1 ,.., n are such that Φi ð0Þ ¼ 0 and θ A Rp is a vector of unknown constant parameters. The objective is to design a discrete-time adaptive controller described by equations of the form (9)–(11) such that all closed-loop trajectories are bounded and lim ðy 1 ðkÞ y n1 ðkÞÞ ¼ 0;
k-1
ð21Þ
where y n1 ðkÞ is a given reference signal vector. This objective is achieved in two steps, as detailed hereafter. 3.1. Observer and estimator design Let reconsider the definitions given in Section 2 Þ ¼ βη ðy ; ηest Þy; ηest ¼ Σ η 1 ðη^ þ βη yÞ zη ¼ η^ Σ η η þ β η ðy ; yÞ; β η ðy ; y; ηest zθ ¼ θ^ Σ θ θ þ β θ ðy ; y; ηest Þ; β θ ðy ; y; ηest Þ ¼ βθ ðy ; ηest Þy; θest ¼ Σ θ 1 ðθ^ þ βθ yÞ
for system (20) with y ¼ ½y i ; …; y r ; η ¼ ½η 1 ; …; η n r , namely 2 3 " þ# " # ΦTy y y 0 R E 0 4 5θ þ ¼ þ þ þ T η η Φη U 0 S D where y ¼ ½y1 ,.., yr , η ¼ ½η1 ,.., ηn r , U ¼ ½0 ,.., 0; ψ n ðy 1 ,.., y r Þu, Φy ¼ ½Φ1 ,.., Φr T ; Φη ¼ ½Φr þ 1 ,.., Φn T ; D ¼ ½0 ,.., 0; d, 2 3 ψ 1 ðy 1 Þ 0n1 xn3 0n1 xn4 … 0n1 xnn Ω1 ðy 1 Þ 6 0n xn 7 0n2 xnn 6 2 1 Ω2 ðy 1 ; y 2 Þ ψ 2 ðy 1 ; y 2 Þ 0n1 xn4 … 7 6 7 6 7; 0n3 xn2 0n3 xn3 ⋱ ⋮ R ¼ 6 0n3 xn1 7 6 ⋮ 7 ⋮ ⋮ 4 5 0n2 xnn 0n3 xnn 0n4 xnn … Ωr ðy 1 ,.., yrÞ 0n1 xnn Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
Y. Yalçın / European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2 6 6 6 S¼6 6 6 4
Ωr þ 1 ðy 1
yr Þ
,..,
2 6 6 6 E¼6 6 6 4
ψ r þ 1 ðy 1 ,.., y r Þ
0
0 0
Ωr þ 2 ðy 1 ,.., y r Þ 0
⋮
⋮
⋮
0
0 3 0 07 7 7 ⋮ 7: 7 07 5 0
0
0
0
…
0
0
…
⋮
⋮
⋮
0
0 0
…
ψ r ðy 1
yr Þ
,..,
ψ r þ 2 ðy 1
yrÞ
,.., 0
5
0
…
0
0 ⋱
…
0 ⋮
0
…
Ωn ðy 1
3 7 7 7 7; 7 7 5 ,..,
yr Þ
^ Γ ¼ ½η; θ; β ¼ ½βη ; βθ ; z ¼ Γ^ ΣΓ þ βðy ; η Þy and z þ ¼ Γ^ þ Σ þ Γ þ þ βðy; ηest Þy þ where Σ ¼ ½Σ η 0; 0 Σ θ are ^ θ, Using the definitions Γ^ ¼ ½η; est obtained. Therefore, z þ γz ¼ Γ^
þ
γ Γ^ ΔΓ þ βðy; ηest Þy þ γβðy ; ηest Þy; " # T η Sη þΦη θ þ U þ D þ Þy Σ þ z þ γz ¼ Γ^ γ Γ^ γβðy ; ηest þ γΣ θ θ
þ βðy; ηest ÞðRy þEη þ ΦTy θÞ;
"
þ z γz ¼ Γ^ γ Γ^ γβðy ; ηest Þy Σ þ þ
" þ γΣ
ηest Σ η 1 zη
Sηest þ ΦTη θest þ SΣ η 1 zη Φη Σ θ 1 zθ þ U þ D θest Σ θ 1 zθ
#
θest Σ θ 1 zθ
#
þβðy; ηest ÞðRy þ Eηest þ ΦTy θest Þ
βðy; ηest ÞðEΣ η 1 zη þ ΦTy Σ θ 1 zθ Þ: Selecting the update law
"
þ Þy þ Σ þ Γ^ ¼ γ Γ^ þγβðy ; ηest
Sηest þ ΦTη θest þ U þ D
#
θest
" γΣ
ηest
#
θest
βðy; ηest ÞðRy þ Eηest þ ΦTy θest Þ
ð22Þ
yields " zþ ¼
γI þ Σ þ
SΣ η 1
ΦTη Σ θ 1
0
Σ θ 1
#
" γΣ
Σ η 1
0
0
Σ θ 1
!!
# þ βðy; ηest Þ½EΣ η 1 ΦTy Σ θ 1
z
z þ ¼ ½Σ þ F η Σ 1 βðy; ηest ÞF y Σ 1 z; where " S Fη ¼ 0
ΦTη
ð23Þ
ð24Þ
# and
Ip
F y ¼ ½E ΦTy :
ð25Þ
The following proposition is given for the asymptotically stability of the system (24)–(25). Proposition 2. Consider the system (24)–(25). If Σ dynamics and β are selected as Σþ ¼
β¼
λ2 Σ 1 þ J Fη J Σλ1 F Ty
1 þ J F Ty F y J
ð26Þ
;
ð27Þ
where λ1 þ λ2 o 1 with λ1 ; λ2 4 0 A R and Σ 0 ¼ k0 I ðs þ pÞ ; s ¼ asymptotically stable, namely limk-1 zðkÞ ¼ 0.
Pn
i ¼ rþ1
ni ; k A R, then the zero equilibrium of the system (24)–(25) is
Proof. For (26) and (27), the estimation error system dynamics (24) becomes ! " # λ1 F Ty F y λ2 F η 1 zþ ¼ Σ Σ z: 1 þ J F η J 1 þ J F Ty F y J
ð28Þ
Now, consider the Lyapunov function V ¼ zT z. Then, T
ΔV ¼ Vðk þ 1Þ VðkÞ ¼ z þ z þ zT z; T
2
¼ z ðF IÞz
ð29Þ
Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
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where F ¼Σ
! λ1 F Ty F y λ2 F η Σ 1: 1 þ J F η J 1 þ J F Ty F y J
Note that for λ1 þ λ2 o1, the inequality ! " #! λ1 F Ty F y λ2 F η 1 Σ σ Σ o0 1 þ J F η J 1 þ J F Ty F y J is satisfied. Hence, ΔV o 0 for z a 0 is satisfied. Then the zero equilibrium of the state and parameter estimation error system (28) is asymptotically stable, namely, limk-1 V ¼ 0 and limk-1 z ¼ 0. □ 3.2. Controller design
Theorem 1. Consider the system (20). The control law, n X
n
u ¼ ∏ ðψ i 1 Þ þ ðn iÞ y 1n þ ðnÞ i¼1
n þ ðn iÞ þ ðn iÞ ∏ ðψ j 1 Þ þ ðn jÞ Ωiest xiest þ Φiest θest ψ n 1 d
ð30Þ
i¼1j¼i
where y n1 is the reference signal to be tracked, ηest ¼ Σ η 1 ðη^ þ βη yÞ and θest ¼ Σ θ 1 ðθ^ þβθ yÞ in which θ^ and η^ are obtained using the update law þ ðn jÞ þ ðn jÞ þ ðn jÞ (22), β ¼ ½βη ; βθ is given by (27), and Σ ¼ ½Σ η 0ðn rÞxp ; 0pxðn rÞ Σ θ is given by (26); ψ iest ¼ ψ i ðy 1est ,.., y jest Þ; þ ðn iÞ þ ðn iÞ þ ðn iÞ þ ðn iÞ þ ðn iÞ þ ðn iÞ þ ðn iÞ Ωiest ¼ Ωi ðy 1est ,.., y jest Þ and Φiest ¼ Φi ðy 1est ,.., y jest Þ; i ¼ 1 ,.., n; j ¼ 1 ,.., r with y jest ¼ y j and y jest , ðn iÞ 4 0 is ðn mÞ-step future values of y i obtained by recursively utilizing θest and ηest in Eqs. (20), renders all trajectories of the closed-loop system are bounded and, lim ðy 1 ðkÞ y n1 ðkÞÞ ¼ 0:
k-1
Proof. A backstepping procedure is pursued to show that the control law given in (30) achieves the control objective. Step 1 : Let y~ 1 ¼ y 1 y n1 , and note that y~ 1þ ¼ ψ 1 ðy 1 Þy 2 þ Ω1 ðy 1 Þy 1 þ Φ1 ðy 1 Þθ y 1n þ : Consider y 2 as a virtual control input and define y~ 2 ¼ y 2 y n2 ðy 1n þ Ω1 ðy 1 Þy 1 Φ1 ðy 1 Þθest Þ. Then the dynamics of y~ 1 can be rewritten as
ð31Þ where
n
y2
is
selected
as
n
y 2 ¼ ψ 1 1 ðy 1 Þ
y~ 1þ ¼ ψ 1 ðy 1 Þy~ 2 Φ1 ðy 1 Þzθ : P i ~ Therefore, if y~ 2 ¼ 0 and ni ¼ 1 Φ1i ðy 1 Þz is a l2 signal, then limk-1 y 1 ðkÞ ¼ 0. Step 2 : Note that y~ 2þ ¼ ψ 2 ðy 1 ; y 2 Þy 3 þ Ω2 ðy 1 ; y 2 Þy 2 þ Φ2 ðy 1 ; y 2 Þθ þ þ þ þ þ ψ 1 1 ðy 1est ÞðΩ1 ðy 1est Þy 1est þ Φ1 ðy 1est Þθest y 1n þ 2 Þ:
ð32Þ
ð33Þ
Here, (33) is obtained assuming that the parameter estimate at time k is the constant parameter of the system and projecting þ the future evaluation of trajectories with this estimate, namely θest ¼ θest . Following steps will be carried out with the same assumption. Now, consider y 3 as a virtual control input and define y~ 3 ¼ y 3 y n3 , where y n3 is selected as y n3 ¼ ψ 2 1 ðy 1 ; y 2 Þ þ þ þ þ þ ðψ 1 1 ðy 1est Þðy 1n þ 2 Ω1 ðy 1est Þy 1est Φ1 ðy 1est Þθest Þ Ω2 ðy 1 ; y 2 Þy 2 Φ2 ðy 1 ; y 2 Þθest Þ.As a result, þ y~ ¼ ψ ðy ; y Þy~ Φ2 ðy ; y Þzθ : ð34Þ 2
2
1
2
3
1
2
Step r : Applying the same procedure recursively yields y~ rþ ¼ ψ r ðy 1 ,.., y r Þη 1 þ Ωr ðy 1 ,.., y r Þy r þ Φr ðy 1 ,.., y r Þθ þ
rX 1
n þ ðn iÞ þ ðn iÞ ∏ ðψ i 1 Þ þ ðn iÞ Ωiest xiest þ Φiest θest
i¼1j¼i
ð35Þ ð36Þ
r
∏ ðψ i 1 Þ þ ðn iÞ y 1n þ r : i¼1
input and define η~ 1 ¼ η 1 η n1 , where η n1 is selected where x ¼ ½y; η. Consider η 1 as a virtual control Pr þ ðn iÞ þ ðn iÞ n r 1 þ ðn iÞ n þ r n 1 þ ðn iÞ η 1 ¼ ∏i ¼ 1 ðψ i Þ y 1 i ¼ 1 ∏j ¼ i ðψ i Þ Ωiest xiest þ Φiest θest . Then, dynamics of y~ r can be written as y~ rþ ¼ ψ r ðy 1 ,.., y r Þη~ 1 Φr zθ :
as ð37Þ
Step r þ 1 : Note that η~ 1þ ¼ ψ r þ 1 ðy 1 ,.., y r Þη 2 þ Ωr þ 1 ðy 1 ,.., y r Þη 1 þ Φr þ 1 ðy 1 ,.., y r Þθ r n X þ ðn iÞ þ ðn iÞ þ ∏ ðψ i 1 Þ þ ðn iÞ Ωiest xiest þ Φiest θest i¼1j¼i
ð38Þ
Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
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rþ1
∏ ðψ i 1 Þ þ ðn iÞ y 1n þ ðr þ 1Þ :
ð39Þ
i¼1
η2 as a virtual control input and define η~ 2 ¼ η2 η n2 , where Consider Pr þ 1 n þ ðn iÞ þ ðn iÞ þ2 1 þ ðn iÞ n þ ðr þ 2Þ 1 þ ðn iÞ η n2 ¼ ∏ri ¼ ðψ Þ y ∏ ðψ Þ Ω x þ Φ θest . Thus, iest i¼1 i j¼i i 1 iest iest 1 η~ 1þ ¼ ψ r þ 1 ðy 1 ,.., y r Þη~ 2 Φr þ 1 zθ Ωr þ 1 zη1 :
η n2
is
selected
as ð40Þ
Step n : Applying the same procedure recursively yields η~ nþ r ¼ ψ n u þ n
∏
i¼1
n þ ðn iÞ þ ðn iÞ ∏ ðψ j 1 Þ þ ðn jÞ Ωiest xiest þ Φiest θest
n X
i¼1j¼i
ðψ i 1 Þ þ ðn iÞ y 1n þ ðnÞ þd:
ð41Þ
hence selecting u as in (30) yields η~ nþ r ¼ Ωn zηðn rÞ Φn zθ : Consequently, the closed-loop system in the x~ ¼ ðy~ ; η~ Þ and z co-ordinates is described by y~ 1þ ¼ ψ 1 ðy 1 Þy~ 2 Φ1 ðy 1 Þzθ ; y~ 2þ ¼ ψ 2 ðy 1 ; y 2 Þy~ 3 Φ2 ðy 1 ; y 2 Þzθ ; ⋮ y~ rþ ¼ ψ r ðy 1 ,.., y r Þη~ 1 Ωr zη1 Φr ðy 1 ,.., y r Þzθ ; η~ 1þ ¼ ψ Tr þ 1 ðy 1 ,.., y r Þη~ 2 Ωr þ 1 zη2 ΦTr þ 1 ðy 1 ,.., y r Þzθ ; ⋮ η~ nþ r ¼ Ωn zηðn rÞ Φn ðy 1 ,.., y r Þzθ ;
ð42Þ
~ ~ ¼ 0. In is bounded and moreover limk-1 xðkÞ together with system (27)–(28). Note that limk-1 zðkÞ ¼ 0 implies that xðkÞ order to show the local stability of the zero equilibrium of the closed-loop system, first consider the system y~ 1þ ¼ ψ 1 ðy 1 Þy~ 2 ; ⋮ y~ rþ ¼ ψ r ðy 1 ,.., y r Þη~ 1 ; η~ þ ¼ ψ r þ 1 ðy 1 ,.., y r Þη~ 2 ; 1
⋮ η~ nþ r ¼ 0
ð43Þ
and rewrite Eqs. (43) as x~ þ ¼ Ax~ ~ η~ Þ and where x~ ¼ ðy; 2 0 ψ1 0 … 6 6 0 0 ψ2 … A¼6 6⋮ ⋮ ⋮ ⋮ 4 0 0 0 …
0
3
7 0 7 7: ⋮ 7 5 ψn
Let V ¼ x~ T P x~ with P ¼ diagðp1 I n1 ; p2 I n2
,..,
ΔV ¼ x~ A PAx~ x~ P x~ o x~ μx~ for x~ a 0 2 p1 I n 1 0 0 T 6 I þ ψ p ψ 0 0 p 2 n2 6 1 1 1 T6 6 I þ ψ T2 p2 ψ 2 0 0 p ¼ x~ 6 3 n3 6 ⋮ ⋮ ⋮ 4 0 0 0 T
T
T
pn I nn Þ; pi 40, and note that
T
…
0
…
0
…
0
⋮ …
⋮ pn I nn þ ψ Tn 1 pn 1 ψ n 1
3 7 7 7 7x~ 7 7 5
o x~ T μx~ for x~ a 0: It is easily seen that this inequality is satisfied by any 0 op1 ; σ ðψ T1 p1 ψ 1 Þ o p2 ; σ ðψ T2 p2 ψ 2 Þ o p3 ; ……; σ ðψ Tn 1 pn 1 ψ n 1 Þ o pn . Now, consider the candidate Lyapunov function W ¼ x~ T P x~ þ δzT z ¼ x~ T P x~ þ δ1 zTθ zθ þ δ2 zTη zη ; P 4 0 A Rnn ;
δ ¼ diagðδ1 ; δ2 ÞÞ;
δ1 ; δ2 40 A R;
for the disturbed system (42). Then, ΔW ¼ ½P 1=2 ðAx~ ΦTx zθ Þ2 x~ T P x~ þ δððz þ Þ2 z2 Þ Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
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where Φx ¼ ½Φ1
Φn , and
,..,
ΔW ¼ x~ A PAx~ x~ P x~ 2x~ A PΦx zθ þ ½P 1=2 Φx zθ 2 T
T
T
T
T
þ δzT ðF 2 IÞz x~ T AT PAx~ þ ð1 þ ϵÞ½P 1=2 Φx zθ 2 þ δzT ðF 2 IÞz: ϵ For ϵ large μþ ζ AT PA=ϵ o μ=2 can be written, hence μ o x~ T x~ þð1 þ ϵÞζðPÞzTθ Φx ΦTx zθ þδzT ðF 2 IÞz 2 μ o x~ T x~ þzT ½ð1 þ ϵÞM þ δNz 2 o μx~ T x~ þ
where M ¼ ½0 0; 0 ζðPÞΦx ΦTx Z 0 and N ¼ F 2 I o0. Note now that for any compact set Ω there exists δ such that ð1 þ ϵÞM þ δN o 0. Hence the zero equilibrium of the closed-loop system is locally stable.□ þ Remark 1. The control law u given in (30) consists of ”look-ahead” values x iest of the state variable x i A x ¼ ½x 1 ; …x n ¼ þ ½y 1 ; …; y r ; η 1 ; …; η ðn rÞ . The ”look-ahead” values x iest are obtained substituting the estimated unmeasurable states ηest and estimated system parameters θest in system dynamics (20).
4. Example The proposed methodology is applied to adaptive control problem of a synchronous generator which is a 2 2 MIMO system. Continuous time system dynamics are considered as below in [20]: δ_ ¼ ω ω_ ¼ p1 sin ð2δÞ p2 ω p3 ψ f sin ðδÞ p4 P m þ p4 P v ψ_f ¼ p5 cos ðδÞ p6 ψ f p7 Efd Efd ke E_fd ¼ þ ue Te Te P m þP g P_m ¼ Tt Pg 1 _ P g ¼ þ ug Tg Tg The aim of the control is to regulate the electrical generated power and the terminal voltage. Electrical generated power P e ¼ f 2 ðδ; ψ f ; θÞ and terminal voltage V t ¼ f 2 ðδ; ψ f ; θÞ are functions of δ; ψ f and system parameters θ as follows: p3 ψ f sin ðδÞ p1 sin ð2δÞ p4 #2 2 " 0 xaf xt ψ f Vxq sin ðδÞ Vxd cos ðδÞ 2 2 þ þ V f ¼ f 2 ðδ; ψ f ; θÞ ¼ xt þ xq xt þ x0d ðxt þ x0d Þxf
Pe ¼
Therefore y1 ¼ δ and y2 ¼ ψ f are selected as the outputs. To be able to apply the proposed technique it is need to formulate the system dynamic in convenient block strict form as in the sequel. For the purpose the dynamics of y1 ¼ δ; y2 ¼ ψ f ; η1 ¼ ω; η2 ¼ Efd are captured in the following MIMO system in block strict feedback form considering P m also as an input designating it as upm : y_ ¼ ψ 1 η þ Φ1 ðyÞθ; η_ ¼ ψ 2 u þ Ω2 η þ Φ2 ðy 1 ; y 2 Þθ þ d where y ¼ ½y1 ; y2 T ; η ¼ ½η1 ; η2 T ; u ¼ ½upm ; ue T ; upm ¼ Pm; θ ¼ ½p1 ; p2 ; p3 ; p5 ; p6 T and, " # " # 1 0 0 0 0 0 0 ψ1 ¼ ; Φ1 ¼ ; 0 p7 0 0 0 cos ðx11 Þ x12 " # 0 0 sin ðx11 Þ x21 x12 sin ðx11 Þ 0 0 ; Ω1 ¼ Φ2 ¼ ; 0 T1e 0 0 0 0 0 " # 0 p4 p4 P v : d¼ ψ2 ¼ ke ; 0 Te 0 The dynamics of P m and P g is considered as another system in the strict feedback form given in (42). The proposed method is applied to the first MIMO system to construct real input ue and the fictional input upm . Afterwards, ug is obtained applying backstepping design to the
Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
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second system considering pnm ¼ upm as the desired reference: ξ_1 ¼ ψ ξ1 ξ2 þ Φξ1 ðξ1 Þ; ξ_2 ¼ ψ ξ2 ug þ Φξ2 ðξ1 ; ξ2 Þ where ξ1 ¼ P m ; ξ2 ¼ P g ; ψ ξ1 ¼ 1=T t ; ψ ξ2 ¼ 1=T g ; Φξ1 ðξ1 Þ ¼ 1=T t P m , and Φξ2 ðξ1 ; ξ2 Þ ¼ 1=T g P g . Note that, this system does not include any unmeasurable states or unknown parameters, hence the backstepping design can be directly applied. The discrete time dynamics are obtained by Euler method as follows: y 1þ ¼ y 1 þTψ 1 η 1 þ TΦ1 ðy 1 Þθ; η 1þ ¼ η 1 þ Tψ 2 u þ TΩ2 η 1 þTΦ2 ðy 1 ; y 2 Þθ þ Td and, ξ1þ ¼ ξ1 þ Tψ ξ1 ξ2 þ TΦξ1 ðξ1 Þ;
ξ2þ ¼ ξ2 þ Tψ ξ2 ug þTΦξ2 ðξ1 ; ξ2 Þ where T is the sampling period used in the discretization. The obtained discrete time systems are not in strict feedback form therefore the following coordinate transformations is considered: y^ 1 ¼ y η^ 1 ¼ y 1 þ Tψ 1 η and, ξ^1 ¼ ξ1 ; ξ^2 ¼ ξ1 þ Tψ ξ1 ξ2 : This coordinate transformation yields below discrete-time systems in strict feedback form: ^ 1 ðy^ 1 Þθ; y^ 1þ ¼ η^ 1 þ Φ
^ 2 η^ 1 þ Φ ^ 2 ðy^ 1 Þθ þ d^ η^ 1þ ¼ ψ^ 2 u þ Ω ^ 2 ¼ I 2 þ ψðI 2 þ TΩ2 Þψ 1 ; d^s ¼ ψðI 2 þ TΩ2 Þψ 1 þ T 2 ψ 1 ds and, ^ 1 ¼ TΦ1 ; Φ ^ 2 ¼ TðTψ 1 Φ2 þ Φ1 Þθ; Ω where ψ^ 1 ¼ I 2 ; ψ^ 2 ¼ T 2 ψ 1 ψ 2 ; Φ þ ^ ξ1 ðξ^1 Þ; ξ^1 ¼ ξ^2 þ Φ þ ^ ξ2 ðξ^1 ; ξ^2 Þ: ξ^ ¼ ψ^ ξ2 ug þ Φ 2
The parameters in (42) are in terms of real system parameters and the values of real system parameters considered as considered in [20] are given below. T ¼ 0:005 is considered in the discretization of the system dynamics: ω0 V 2 ðxq x0 dÞ ; r f ¼ 0:0012 p:u ., V ¼ 1 p:u: 4Hðxt þ x0 dÞðxt xq Þ ω0 d ; xd ¼ 1:75 p:u ., d ¼ 0:006d p2 ¼ 2H ω0 Vxaf ; xt ¼ 1:665 p:u ., f ¼ 60 Hz p3 ¼ 2Hxf ðxt þx0d Þ ω0 ; xq ¼ 1:68 p:u ., ke ¼ 25 p4 ¼ 2H ω0 r f Vxaf ; xf ¼ 1:665 p:u ., T e ¼ 0:04 s p5 ¼ xf ðxt þ x0d Þ ω0 r f ðxt þ x0d Þ ; xaf ¼ 1:56 p:u ., T t ¼ 0:3 s p6 ¼ xf ðxt þ x0d Þ ω0 r f p7 ¼ ; x0d ¼ 0:285 p:u ., T g ¼ 0:08 s xf p1 ¼
x0d ¼ xd
x2af xf
;
ω0 ¼ 2πf ;
H ¼ 3:82 s
Note that, in this example ω0 ; H and r f =xf , namely, p4 and p7 are assumed to be known and others unknown parameters. Here, V is the voltage on infinity bus and xt is the transmission line reactance. The design parameters of the estimator are selected as λ1 ¼ 0:1; λ2 ¼ 0:2 and γ ¼ 0:1. The initial conditions for the state and parameter ^ estimator have been set to η^ ð0Þ ¼ ½0:001; 0:001; θð0Þ ¼ ½0:1; 0:1; 0:1; 0:1; 0:1; κ θ ¼ I 5 ; κ η ¼ I 2 ; Σð0Þ ¼ I 7 , and the initial state has been selected as xð0Þ ¼ ½00:001; 0:001; 0:001; 0:001; 0:001; 0:001. Fig. 1 shows the time histories of state and parameter estimation error z ¼ ½z1 ; z2 ; z3 ; z4 ; z5 ; z6 ; z7 T which converge to zero, while Fig. 2 displays the time history of the measurable states y1 and y2 . Finally, Fig. 3 demonstrates the time history of true and estimated values of unmeasurable states η1 and η2 . In Fig. 4 the evaluation of the control signals Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
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2
1.5
z1,z2,z3, z4,z5,z6,z7
1
0.5
0
0
1
2
3
4
5
Time[k] Fig. 1. Time histories of the estimation error variable z ¼ ½z1 ; z2 ; z3 ; z4 ; z5 ; z6 ; z7 .
2.5
2
1.5
y1, y2 1
0.5
0
0
1
2
3
4
5
6
7
Time[k] Fig. 2. Time histories of the states y1 and y2 . The dashed line displays the time history of the states y1 and y2 when the known-parameter and partial state-feedback controller are used.
2.5
2
1.5
η1, η2
1
0.5
0
0
1
2
3
4
5
6
7
Time[k] Fig. 3. Time histories of the states η1 and η2 . The dashed line displays the time history of the states η1 and η2 when the known-parameter and partial state-feedback controller are used.
is shown. The simulation results demonstrate that the proposed adaptive controller renders the closed-loop system approximately achieving the performance that the true parameter controller yields. 5. Conclusions In this paper a novel partial-feedback adaptive controller design method for the adaptive regulation of a class of linearly parameterized discrete-time systems in strict-feedback form that is linear in unmeasurable states has been developed. A procedure without over Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i
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2
11
x 106
1.5 1 0.5
u1,u2
0 −0.5 −1 −1.5
0
1
2
3
4
5
6
7
Time[k] Fig. 4. Time histories of control signals u1 and u2 . The dashed line displays the time history of the control signals u1 and u2 of the known-parameter and partial statefeedback controller.
parametrization that provides simultaneous construction of update laws for both state estimation and parameter estimation error dynamics is presented. State and parameter estimation error dynamics include a free function and a dynamic matrix that makes possible to shape the transient dynamics of the estimation error system. It is shown that the presented combined update law renders the zero equilibrium of the estimation error system stable. The controller construction is performed via a back-stepping procedure with immersion and invariance control approach and utilizing presented update laws. Assuming that the parameter estimate at time instance k is the constant parameter of the system and projecting the future evaluation of trajectories with this estimate in the back stepping controller construction has been enabled to use the presented algorithm without over parametrization. The procedure is utilized for the adaptive control problem of a synchronous generator and corresponding simulation results are presented.
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Please cite this article as: Y. Yalçın, Discrete time immersion and invariance adaptive control via partial state feedback for systems in block strict feedback form, European Journal of Control (2015), http://dx.doi.org/10.1016/j.ejcon.2015.05.002i