Lyapunov Design for Controls

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3 F.L. Lewis, NAI Moncrief-O Donnell Chair, UTA Research Institute (UTARI) The University of Texas at Arlington, USA and Foreign Professor, Chongqing University, China Supported by : China Qian Ren Program, NEU China Education Ministry Project 111 (No.B08015) NSF, ONR Lyapunov Design for Controls Talk available online at

4 F.L. Lewis Moncrief-O Donnell Endowed Chair Head, Controls & Sensors Group Supported by : NSF - PAUL WERBOS ARO RANDY ZACHERY Automation & Robotics Research Institute (ARRI) The University of Texas at Arlington Nonlinear Network Structures for Feedback Control: Part I- Adaptive, Robust, & Neural Net Control Talk available online at

5 David Song Yongduan

6 Meng Tz 500 BC He who exerts his mind to the utmost knows nature s pattern. The way of learning is none other than finding the lost mind. Man s task is to understand patterns in nature and society. Mencius

7 Importance of Feedback Control Darwin FB and natural selection Vito Volterra FB and fish population balance Adam Smith FB and international economy James Watt FB and the steam engine FB and cell homeostasis The resources available to most species for their survival are meager and limited Nature uses Optimal control

8 Feedback Control Systems Aircraft autopilots Car engine controls Ship controllers Compute Hard disk drive controllers Industry process control chemical, manufacturing Robot control Industrial Revolution Windmill control, British millwrights s Steam engine and prime movers James Watt 1769 Steamship Steam Locomotive boiler control Sputnik 1957 Aerospace systems

9 Lagrange Dynamical systems Newton s Law F ma mx F(t) m v(t) x F( t) m u( t) x(t) Lagrange s Eqs. Of Motion Industrial Process and Motion Systems (Vehicles, Robots) M( q)q+v m (q,q)q+g(q)+f(q)+τ d B( q) τ Control Input Actuator problems inertia Coriolis/centripetal force gravity friction disturbances

10 Dynamical System Models Continuous-Time Systems Discrete-Time Systems x f ( x) g( x) u y h( x) Nonlinear system x y k 1 k f ( x h( x k ) k ) g( x k ) u k x y Ax Cx Bu Linear system x y k 1 k Ax Cx k k B k Control Inputs u g(x) x Internal States z -1 x 1/s h(x) Measured Outputs y f(x)

11 Issues in Feedback Control Disturbances Sensor noise Desired trajectories Feedforward controller Control inputs system Measured outputs Feedback controller Stability Tracking Boundedness Robustness to disturbances to unknown dynamics Unknown Process dynamics Process Nonlinearities Unknown Disturbances

12 Definitions of System Stability x(t) x(t) B(d) x f ( x) x f ( x k 1 k ) t d t Asymptotic Stability Marginal or Bounded Stability - Stable in the Sense of Lyapunov (SISL) x(t) Const Bound B x e +B x e x e -B t 0 t 0 + T t T Uniform Ultimate Boundedness

13 Clips are from Nonlinear Control Systems book by Slotine and Li.

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25 Example 1. Linear System Feedback Linearization y 1 2 s as 1 a2 u y a y a y u 1 2 desired to track a reference input yd () t Tracking error e y y d Sliding variable e y a y a yu d r e e 1 2 ree y ea ya yu d 1 2 Auxiliary input u v y e d Error dynamics ra ya yv 1 2 Of the form r f( x) v Unknown parameters Unknown function y T f ( x) a1y a2y a1 a2 W ( x) y Known Regression Vector

26 Example 2. Nonlinear Lagrange System yd( y, y ) k( y) u unknown nonlinear friction unknown nonlinear damping term desired to track a reference input y () t Tracking error Sliding variable Auxiliary input e y y Error dynamics r f ( x) v d e y d( y, y ) k( y) u d r e e u v y e d d Feedback Linearization Lagrangian System Appears in: Process control Mechanical systems Robots with f ( x) d( y, y ) k( y) Assume Linear in the Parameters (LIP) d ( y, y ) f x D K W x k1( y, y) 1 T ( ) ( ) Known possibly nonlinear regression function Unknown parameters

27 Feedback Linearization Controller A dynamic controller r e e u v y e d Feedforward terms y y d d e e e I r(t)? v controller u plant y y Tracking Loop The equations give the FB controller structure

28 Adaptive Control Error Dynamics r f( x) v Control input r(t)= control error Unknown nonlinearities Assume: f(x) is known to be of the structure f ( x) W T ( x) Known basis set= regression vector DEPENDS ON THE SYSTEM Unknown parameter vector LINEAR-IN-THE-PARAMETERS (LIP) Error Dynamics T r W ( x) v

29 Adaptive Control Controller ˆ( ) ˆ T v f x K r W ( t) ( x) K r v v Pos. def. control gain ESTIMATE of unknown parameters closed-loop system becomes T T ( ) ( ) ˆ T r W x vw x W ( x) Kvr T r W ( x) K r v Est. error drives the control error Parameter estimation error W () t W Wˆ () t Parameter estimate is updated (tuned) using the adaptive tuning law dw ˆ ˆ W F ( x ) r T dt

30 Feedback Linearization Adaptive Controller A dynamic controller r e e u v y e d ˆ( ) ˆ T v f x K r W ( t) ( x) K r v v ˆ T W () t ( x) Tunable inner loop ˆ W F( x) r T y y d d e e e I r(t) K v ˆ( ) f x Feedback terms v u plant y y Tracking Loop The equations give the FB controller structure

31 Neural Networks for Control

32 F.L. Lewis, S. Jagannathan, and A. Yesildirek, Neural Network Control of Robot Manipulators and Nonlinear Systems, Taylor and Francis, London, NN control in Chapter 4

33 Control System Design Approach Robot dynamics Tracking Error definition Siding variable Mqq ( ) V ( qqq, ) Fq ( ) Gq ( ) m et () q() t qt () d r e e d Error dynamics Mr M( e e ) M( q ( t) q( t) e ) d Where the unknown function is M r V r f ( x) m f ( x) M( q)( q e ) V ( q, q )( q e) F( q ) G( q) d m d d

34 Tracking error e( t) qd ( t) q( t) Robot dynamics M ( q) q V ( q, q ) q G( q) F( q ) m d q d Sliding variable r e e e r [I]? controller Robot System q PD Tracking Loop The equations give the FB controller structure

35 Kung Tz 500 BC Confucius Tian xia da tong Harmony under heaven 孔子 Man s relations to Family Friends Society Nation Emperor Ancestors Archery Chariot driving Music Rites and Rituals Poetry Mathematics 124 BC - Han Imperial University in Chang-an

36 Handling High-Frequency Dynamics

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38 Actuator Dynamics

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