1. 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 1of 11

2. 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

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

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

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

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

7. NONLINEAR SYSTEM 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 ) Plant: 7of 11

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

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

10. ISS CONTROLLER DESIGN This is ISS property of control law w.r.t. observation errors: Closed-loop system: 1. • 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] 10of 11

11. ISS OBSERVER DESIGN • This property can be achieved for with detectable and globally Lipschitz, very restrictive Closed-loop system: 2. This is ISS property of observer w.r.t. additive output errors • 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 11of 11