2. QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign 1 of 11 Mediterranean Control Conference, Athens, Greece, Jun 2007

3. 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

4. QUANTIZER is the range, is the quantization error bound Assume such that: 1.1. 2.2. EncoderDecoder finite set QUANTIZER Output space is divided into quantization regions For, the quantizer saturates 3 of 11

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

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

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

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

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

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

11. ISS CONTROLLER DESIGN 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 ] ISS assumption is fundamental in quantized control of nonlinear systems [ L ’03 ] This is ISS property of control law w.r.t. observation errors: 10 of 11

12. ISS OBSERVER DESIGN Closed-loop system: 2. This is ISS property of observer w.r.t. additive output errors No results on design of such ISS observers exist This property can be achieved for with detectable and globally Lipschitz, very restrictive More research on this problem is needed! 11 of 11