Decentralized Adaptive Approximation Based Control With Safety Scheme Outside the Approximation Region Panagiotis Panagi and Marios M. Polycarpou Abstract— This paper presents a decentralized adaptive ap- proximation based control scheme for a class of interconnected nonlinear systems. The feedback control law consists of two schemes, an adaptive approximation controller operating inside a chosen approximation region and a decentralized safety scheme for outside the approximation region. Within the approximation region, linearly parameterized neural networks with a dead-zone modification are used to adaptively approxi- mate the unknown dynamics of each subsystem, as well as the unknown interconnections. Outside the approximation region, the decentralized safety control scheme is designed to steer back the trajectory by using an adaptive bounding approach. A rigorous stability analysis is presented and a simple simulation example is used to illustrate the decentralized adaptive control methodology. I. I NTRODUCTION Decentralized control of large-scale interconnected sys- tems has attracted significant attention during the last two decades. The problem of decentralized adaptive linear control was introduced by Ioannou [5], where weakly interconnected subsystems with relative degree one or two were studied. In [4] and [9] it was shown that stability of the decentralized system is ensured if there exists a positive definite M-matrix, which is related to the bound of the interconnections. Most of these approaches were focused on linear subsystems with possibly nonlinear interconnections. An alternative decentral- ized adaptive control method using the high gain approach was developed in [2], where a standard strict matching condition is assumed on the disturbances. A methodology for handling higher-order interconnections in a decentralized adaptive control framework was developed in [11]. One of the key challenges in decentralized control is the issue of dealing with uncertainty, both in the nonlinearities of the local subsystems as well as in the interconnections. A recent approach for dealing with uncertainty is based on the use of neural networks to approximate the unknown intercon- nections. In [13], [12], the authors developed a decentralized control design scheme for systems with interconnections that are bounded by first-order polynomials. In [3], the authors employ a composite Lyapunov function for handling both unknown nonlinear model dynamics and interconnections. The interconnections are assumed to be bounded by unknown smooth functions, which are indirectly approximated by neural networks. In [8], [7] and [6] it is assumed that the decentralized controllers share prior information about their P. Panagi and M. M. Polycarpou are with the Department of Electrical and Computer Engineering, University of Cyprus, Nicosia 1678, Cyprus (Email: ppanagi@ucy.ac.cy, mpolycar@ucy.ac.cy). This work is supported by the Research Promotion Foundation of Cyprus. reference models. Based on this assumption, it is then shown that the subsystems are able to asymptotically track their desired outputs. In this paper, we consider a system composed of nonlinear subsystems coupled by unknown nonlinear interconnections. We develop a decentralized adaptive approximation based control system [1] and derive stability results for the closed- loop system under certain assumptions. We consider both the case where the trajectory is inside the approximation region as well as the case where the trajectory leaves the approxi- mation region. In the latter case, we develop a decentralized safety control scheme based on the sliding mode control approach with adaptive bounding. The presented adaptive approximation based control scheme follows the general approach for decentralized systems developed in [12], [3] and [10]. The main contribution of this work is the synthesis and analysis of the dead-zone modification and the design of a stable decentralized safety control scheme for addressing the problem of the trajectory exiting the approximation region during the transient stage. The paper is organized as follows. In Section II, we design a decentralized feedback control law with a dead-zone modification in the adaptive laws to account for the residual approximation errors. Section III presents a decentralized safety control scheme for the case where the trajectories go outside the approximation region, while in Section IV a simulation example is used to illustrate the overall control methodology. Finally, Section V contains some concluding remarks. II. DECENTRALIZED ADAPTIVE CONTROL We consider a system comprised of n interconnected subsystems. The i-th subsystem, where i =1, 2,...,n, is described by ˙ x ij = x i(j+1) , j =1, 2,...,ρ i − 1 ˙ x iρi = f i (x i )+ g i (x i )u i +Δ i (x 1 ,x 2 , ..., x n ) y i = x i1 , where x i =[x i1 ,x i2 ,...,x iρi ] ⊤ ∈ℜ ρi is the state vector of the i-th subsystem, f i : ℜ ρi  →ℜ and g i : ℜ ρi  →ℜ are unknown smooth functions, Δ i : ℜ ρ  → ℜ (where ρ = ∑ n i=1 ρ i ) represents the interconnection effect between subsystems, u i ∈ℜ is the input and y i ∈ℜ is the output of the i-th subsystem. Our objective is to synthesize decentral- ized adaptive approximation based control laws u i such that each y i tracks a smooth bounded reference trajectory y di in the presence of the unknown interconnections Δ i , using only local measurements. Proceedings of the 47th IEEE Conference on Decision and Control Cancun, Mexico, Dec. 9-11, 2008 TuA03.4 978-1-4244-3124-3/08/$25.00 ©2008 IEEE 92