OBSERVERBASED QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS

OBSERVER-BASED QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng. , Univ. of Illinois at Urbana-Champaign IFAC World Congress, Seoul, Korea, July 2008 1 of 11

QUANTIZED OUTPUT FEEDBACK PLANT QUANTIZER CONTROLLER Motivation: • limited communication between sensor and actuator • trade-off between communication and computation Objectives: • analyze effect of quantization on system stability • design controllers robust to quantization errors 2 of 11

QUANTIZER Encoder Decoder QUANTIZER finite set Output space is divided into quantization regions Assume such that: 1. 2. is the range, For is the quantization error bound , the quantizer saturates 3 of 11

LINEAR SYSTEM [Brockett-L] Plant: Luenberger observer-based controller: quantization error Closed-loop system: or in short where is Hurwitz if and are Hurwitz 4 of 11

LINEAR SYSTEM (continued) Hurwitz For we have Recall: level sets of V Solutions go from the larger level set to the smaller one 5 of 11

INPUT-TO-STATE STABILITY (ISS) is of class function [Sontag] if • for each fixed • as for each Example: ISS: where Equivalent Lyapunov characterization: when for some 6 of 11

NONLINEAR SYSTEM Plant: Dynamic controller: Closed-loop system: quantization error or in short Assume: this is ISS w. r. t. quantization error (so in particular, should have GAS when ) 7 of 11

NONLINEAR SYSTEM (continued) ISS Lyap. function and class function s. t. level sets of V Solutions go from the larger level set to the smaller one Can recover GAS using dynamic quantization 8 of 11

ISS ASSUMPTION: CLOSER LOOK Closed-loop system is ISS if for some we have 1. and 2. Reason: cascade argument Can extend this via a small-gain argument (need ) 9 of 11

ISS CONTROLLER DESIGN Closed-loop system: 1. This is ISS property of control law w. r. t. observation errors: • Not always possible to achieve [Freeman ’ 95, Fah ’ 99] • Results exist for classes of systems [Freeman & Kokotovic ’ 93, ’ 96, Freeman ’ 97, Fah ’ 99, Jiang et al. ’ 99, Sanfelice & Teel ’ 05, Ebenbauer, Raff & Allgower ’ 07, ’ 08] • ISS assumption is fundamental in quantized control of nonlinear systems [L ’ 03] 10 of 11

ISS OBSERVER DESIGN Closed-loop system: 2. This is ISS property of observer w. r. t. additive output errors • This property can be achieved for with detectable and globally Lipschitz, very restrictive • Almost no results on design of such ISS observers exist, except recent work of H. Shim, J. H. Seo, A. R. Teel, J. S. Kim More research on this problem is needed 11 of 11