Decentralized control and disturbance attenuation for large-scale nonlinear systems in generalized output-feedback canonical form

doi:10.1016/S0005-1098(03)00199-7

Automatica

Volume 39, Issue 11, November 2003, Pages 1923-1933

https://doi.org/10.1016/S0005-1098(03)00199-7 Get rights and content

Abstract

A global decentralized robust adaptive output-feedback dynamic compensator is proposed for stabilization, tracking, and disturbance attenuation of the decentralized generalized output-feedback canonical form. This represents the largest class for which decentralized robust adaptive output-feedback tracking and disturbance attenuation results are currently available. The system is allowed to contain unknown parameters multiplying output-dependent nonlinearities, and, also, unknown nonlinearities satisfying certain bounds. Under the assumption that a constant matrix can be found for each subsystem to achieve a certain property, it is shown that reduced-order observers and backstepping controllers can be designed to achieve practical stabilization of the tracking error in each subsystem in the presence of bounded disturbance inputs. Sufficient conditions under which asymptotic tracking and stabilization can be achieved are also obtained. Signal gains from disturbance inputs to tracking errors are presented in the input-to-output-practical-stability and integral-input-to-output-practical-stability frameworks. A particular case in which the standard $L_{2}$ -gain disturbance attenuation is achieved is also provided.

Introduction

The early results in decentralized control focused on linear systems (Šiljak, 1978; Jamshidi, 1983). Linearly bounded interconnections were addressed in Özgüner (1979), Khalil and Saberi (1982), Hammamed and Radouane (1983), Ioannou (1986), Gravel and Šiljak (1989), and Chen, Leitmann, and Xiong (1991). In Shi and Singh (1992), higher order (i.e., polynomial type) interconnections among the subsystems was considered for large-scale systems having the uncertainties and interconnections in the range space of control input (i.e., matching condition). Utilizing backstepping, global decentralized adaptive state-feedback (Jain & Khorrami, 1997a) and output-feedback (Jain & Khorrami, 1997b) controllers were designed for large-scale nonlinear systems of the output-feedback canonical form including uncertain parameters and unstructured uncertainties satisfying polynomial bounds. Decentralized output-feedback robust disturbance attenuation for a class of large-scale nonlinear systems comprising of subsystems in output-feedback canonical form with appended linear asymptotically stable dynamics with interconnections bounded by nonlinear functions of the outputs was addressed in Jiang, Khorrami, and Hill (1999), and Jiang, Repperger, and Hill (2001). The disturbance attenuation results in Jiang, Khorrami, & Hill 1999, Jiang, Repperger, & Hill 2001 were based on the earlier results in the centralized framework (Van Der Schaft, 1992; Pan & Basar, 1998; Marino & Tomei, 1999). Disturbance attenuation in the multi-input–multi-output (MIMO) context without requiring a decentralized control structure has also been considered (Liu, Zhou, & Gu, 1998; Lin & Qian, 2001; Isidori, 2003).

Recent advances in output-feedback design for centralized systems in the past few years has provided new tools for extending results in the decentralized framework. To this extent, we consider a class of large-scale systems whose mth subsystem is in the form $x ̇_{(i,m)} =φ_{(i,1,m)} (x_{(1,m)})+ ∑ j=2 n_{m} φ_{(i,j,m)} (x_{(1,m)})x_{(j,m)} + ψ_{(i,m)} (z,x,t,ϖ_{m} (t))+θ_{m}^{T} Ω_{(i,m)} (x_{(1,m)})+ μ_{(i−r_{m}+1,m)} (x_{(1,m)})u_{m}, i=1,…,n_{m},$ where μ_{(i−r_m+1,m)}=0 if i<r_m, and φ_(i,j,m)=0 for 1⩽i⩽r_m−1, j⩾i+2. $x_{m} =[x_{(1,m)},x_{(2,m)},…,x_{(n_{m},m)}]^{T} ∈ R^{n_{m}}$ is the state, $u_{m} ∈ R$ the input, y_m=x_(1,m) the output, and $ϖ_{m} ∈ R^{n_{ϖ_{m}}}$ the disturbance input of the mth subsystem. $z∈ R^{n_{z}}$ is the state of the appended dynamics: $z ̇ =q_{z} (z,x,ϖ,t),$ where x=[x₁^T,x₂^T,…,x_M^T]^T, ϖ=[ϖ₁^T,ϖ₂^T,…,ϖ_M^T]^T. M is the number of subsystems and $θ_{m} ∈ R^{n_{θ_{m}}}$ is a vector of constant unknown parameters. The functions, φ_(i,j,m) and $Ω_{(i,m)}$ , are smooth nonlinear functions of x_(1,m). μ_(i,m) are globally well-defined functions of their arguments. ψ_(i,m) represent the effect of unknown nonlinearities and disturbance inputs ϖ_m(t).

In the centralized framework, stabilization and tracking results for subclasses of system (1) without the z dynamics and the ψ terms are available under a variety of assumptions on φ_(i,j,m). The majority of available output-feedback results focus on the output-feedback canonical form (Marino & Tomei, 1995; Krstić, Kanellakopoulos, & Kokotović, 1995; Isidori, 2003) where φ_(i,j), j⩾2 are constants. Solutions were proposed in Pomet, Hirshorn, and Cebuhar (1993), Battilotti (1997) assuming existence of control Lyapunov functions of very specific structures and satisfying growth conditions. Recently, several assumptions inherent in the contributions above were removed for the nonadaptive case (θ known) in Praly and Kanellakopoulos (2000), Krishnamurthy, Khorrami, and Jiang (2002b) and for the adaptive case in Krishnamurthy and Khorrami (2003).

In this paper, we present, using the results in Krishnamurthy et al. (2002b), Krishnamurthy and Khorrami (2003), the first decentralized robust adaptive output-feedback tracking and disturbance attenuation results for the class of systems shown in (1) with the appended dynamics (2). This system structure is the largest class for which decentralized results are currently available. Furthermore, unlike previous results, the z dynamics are not required to be linear and the disturbance inputs are allowed to enter the system dynamics in a nonaffine manner. It is shown that if a matrix with the property in (A2) below can be found, a dynamic compensator can be designed to achieve practical stabilization of the tracking errors. It is also shown that under certain assumptions, asymptotic convergence of the tracking error to zero is obtained. Weaker conditions under which asymptotic stabilization to the origin can be achieved are also given. Gains from the disturbance inputs to the tracking errors are formulated in an input-to-output-practical-stability (IOpS) (Jiang, Teel, & Praly, 1994; Sontag, 2000) and an integral-input-to-output-practical-stability (iIOpS) (Sontag, 2000; Nesic & Dower, 2001) framework. A particular case in which the standard $L_{2}$ -gain disturbance attenuation is attained is also provided.

Section snippets

Problem statement

We consider systems that are globally diffeomorphic into the form , , in which unmeasured states occur linearly except for terms, ψ_(i,m). The control objective is to design a dynamic output-feedback control law $Λ ̇_{m} = ν ̄_{m} (Λ_{m},y_{m},y_{ref_{m}}, y ̇_{ref_{m}},y_{ref_{m}}^{(2)},…,y_{ref_{m}}^{(r_{m})}), u_{m} = μ ̄_{m} (Λ_{m},y_{m},y_{ref_{m}}, y ̇_{ref_{m}},y_{ref_{m}}^{(2)},…,y_{ref_{m}}^{(r_{m})}),$ where y_{ref_m}(t), m=1,…,M are r_m-times continuously differentiable bounded reference signals and y_{ref_m}^(k) denotes $d^{k} y_{ref_{m}} (t) d t^{k}$ with y_{ref_m}⁽⁰⁾=y_{ref_m} to achieve

(1)
global boundedness of all

Observer design

The unmeasured states of the mth subsystem, namely x_(2,m),…,x_{(n_m,m)}, are estimated using an observer with the filter dynamics $x ̂ ̇_{(i,1,m)} = ∑ j=i+1 n_{m} φ_{(i,j,m)} x ̂_{(j,1,m)} +l_{(i,1,m)} +μ_{(i−r_{m}+1,m)} u_{m},$ $x ̂ ̇_{(i,2,m)} =l_{(i,2,m)}, 2⩽i⩽n_{m},$ where $x ̂_{(i,1,m)} ∈ R$ , $x ̂_{(i,2,m)} ∈ R^{n_{θ_{m}}}$ , $f_{(i,m)} (x_{(1,m)})= ∫ 0 x_{(1,m)} φ_{(i,2,m)} (s)+g_{(i,m)} (s) φ_{(1,2,m)} (s) d s,$ g_(2,m),…,g_{(n_m,m)} are chosen to satisfy assumption (A2), and $l_{(i,1,m)} =− φ_{(i,2,m)} +g_{(i,m)} φ_{(1,2,m)} (φ_{(1,1,m)} +φ_{(1,2,m)} [x ̂_{(2,1,m)} + f_{(2,m)}])+φ_{(i,1,m)} + ∑ j=2 i φ_{(i,j,m)} [x ̂_{(j,1,m)} +f_{(j,m)}]+ ∑ j=i+1 n_{m} φ_{(i,j,m)} f_{(j,m)},$ $l_{}$

Controller design

In this section, the controller for the mth subsystem is designed using the observer backstepping technique (Krstić et al., 1995) applied to the subsystem comprising of the states $(x_{(1,m)}, x ̂_{(2,1,m)},…, x ̂_{(r_{m},1,m)})$ whose dynamics are given by (18) and $x ̇_{(1,m)} =φ_{(1,1,m)} +φ_{(1,2,m)} [x ̂_{(2,1,m)} +θ_{m}^{T} x ̂_{(2,2,m)} +f_{(2,m)}]− φ_{(1,2,m)} e_{(2,m)} +ψ_{(1,m)} +θ_{m}^{T} Ω_{(1,m)} .$

Step 1: Differentiating and upper-bounding the Lyapunov function V_(1,m)=η_m(ξ_(1,m)²)/2, where $ξ_{(1,m)} =x_{(1,m)} −y_{ref_{m}}$ and η_m is a smooth class K_∞ function which will be

Stability analysis and main results

Defining the composite Lyapunov function $V_{x_{m}} =V_{(r_{m},m)} +2 (r_{m} +κ)ε_{(1,m)} ε_{m} V_{om} + 12 (θ ̂_{m_{f}} −θ_{m_{f}})^{T} Γ_{(1,m)}^{−1} (θ ̂_{m_{f}} −θ_{m_{f}})$ and using , , $V ̇_{x_{m}} ⩽−η_{m} ′ξ_{(1,m)} ν_{m} − ∑ j=2 r_{m} γ_{(j,m)} ξ_{(j,m)}^{2} − κε_{(1,m)} |e_{m} |^{2} − 12 γ_{gm} λ_{min} (Γ_{(3,m)})| θ ̂_{m_{f}} −θ_{m_{f}} |^{2} + d ̄_{m} + γ_{gm} λ_{max}^{2} (Γ_{(3,m)}) 2λ_{min} (Γ_{(3,m)}) |θ_{m_{f}} |^{2},$ where $d ̄_{m} =r_{m} ε_{(1,m)} ψ_{(1,m)}^{2} + 4(r_{m} +κ)ε_{(1,m)} λ_{max}^{2} (P_{m})|d_{m} |^{2} /ε_{m}^{2} .$ $∑_{m=1}^{M} d ̄_{m}$ can be bounded as $∑ m=1 M d ̄_{m} ⩽π_{0} +ρ_{1} (|z|)+ ∑ m=1 M ρ_{(2,m)} (|x_{(1,m)} |)+ d_{ψ}^{2} ρ_{(3,m)} (|x_{(1,m)} |) max (c^{2},d_{ψ}^{2}) + ρ_{(4,m)} (|x_{(1,m)} |) ∑ k=1 M q_{k}^{2} (|ϖ_{k} |),$ where $π_{0} =[d_{ψ}^{2} max (c^{2},d_{ψ}^{2})] (1+∑_{m=1}^{M} ∑_{i=2}^{n_{m}} f_{(i,m)} ′(0)^{2})$ and ρ₁,ρ_(i,m), i=2,3,4

Conclusion

In this paper, we have proposed a global decentralized robust adaptive output-feedback dynamic compensator for stabilization, tracking, and disturbance attenuation of a class of large-scale systems that are globally diffeomorphic into systems which are interconnections of subsystems in generalized output-feedback canonical form along with appended ISpS dynamics. This class of systems which includes as subclasses the various structures considered previously in the literature represents the

Prashanth Krishnamurthy received a B.Tech degree (1999) in electrical engineering from Indian Institute of Technology, Chennai, and a M.S degree (2002) in electrical engineering from Polytechnic University, Brooklyn, NY, where he is currently working towards his Ph.D. He is the co-author of 18 journal and conference papers, and the book mentioned in the biography of the second author. His research interests include robust and adaptive nonlinear control with applications.

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Farshad Khorrami was born on January 22, 1962 in Iran. He received his B.S. degrees in Mathematics and electrical engineering at The Ohio State University, Columbus, in 1982 and 1984, respectively. He received the M.S. degree in Mathematics in 1984 and the Ph.D. degree in electrical engineering in 1988, also from The Ohio State University. He is currently a professor of electrical and computer engineering at Polytechnic University in Brooklyn, NY where he joined as an assistant professor in September 1988. His research interests include adaptive and nonlinear control, large scale systems and decentralized control, unmmaned autonomous vehicles, smart structures, robotics and high speed positioning application, and microprocessor based control and instrumentation. He has published more than 140 refereed journal and conference papers and currently holds ten U.S. patents and two more are pending. He has developed and directed the Control/Robotics Research Laboratory (CRRL) at Polytechnic University. Dr. Khorrami has served on the program committees of several conferences and has been a member of the Conference Editorial Board of the Control Systems Society. Dr. Khorrami is also an author of a recently published book entitled “Modeling and Adaptive Nonlinear Control of Electric Motors” published by Springer Verlag.

^☆: This work is supported in part by the NSF under grant ECS-9977693. An earlier version of this paper was presented at the IEEE Conference on Decision and Control, Orlando, FL, December 2001. This paper was recommended for publication in revised form by Associate Editor Alessandro Astolfi under the direction of Editor H. K. Khalil.

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Brief PaperDecentralized control and disturbance attenuation for large-scale nonlinear systems in generalized output-feedback canonical form☆

Abstract

Introduction

Section snippets

Problem statement

Observer design

Controller design

Stability analysis and main results

Conclusion

Systems and Control Letters

Automatica

Robust control design for interconnected systems with time-varying uncertainties

International Journal of Control

Decentralized adaptive controlStructural conditions for stability

IEEE Transactions on Automatic Control

Decentralized nonlinear adaptive feedback stabilization of large-scale interconnected systems

IEE Proceedings

Decentralized adaptive control of interconnected systems

IEEE Transactions on Automatic Control

Nonlinear control systems

Decentralized adaptive control of a class of large-scale interconnected nonlinear systems

IEEE Transactions on Automatic Control

Decentralized adaptive output feedback design for large-scale nonlinear systems

IEEE Transactions on Automatic Control

Large-scale systems: Modeling and control

Decentralized nonlinear output-feedback stabilization with disturbance attenuation

IEEE Transactions on Automatic Control

Small-gain theorem for ISS systems and applications

Mathematics of Control, Signals and Systems

Decentralized stabilization of nonlinear interconnected systems using high-gain feedback

IEEE Transactions on Automatic Control

Brief Paper
Decentralized control and disturbance attenuation for large-scale nonlinear systems in generalized output-feedback canonical form☆