#### Transcript Nonlinear Predictive Control for Fast Constrained Systems

Nonlinear Predictive Control for Fast Constrained Systems By Ahmed Youssef Introduction What’s MBPC N=N2-N1 CV t N1 N2 MV Nu CV: controlled variable MV: manipulated variable Introduction Shortcomings of current industrial nonlinear MBPC •Computing the MBPC control law demands significant on-line computation effort •Inability to deal explicitly with the plant model uncertainty. Objective of research work Reducing the computational complexity of nonlinear MBPC & adding the robustness property whilst preserving its good attributes to make it more effective practical tool for controlling systems of fastconstrained dynamics. State Dependent State-Space Models Given the nonlinear dynamic model xk 1 f ( xk , uk ) yt g ( xt ) Reformulate into nonlinear state-dependent form xt 1 A ( xt ) xt B ( xt ) ut yt C ( xt ) xt This is not a linearisation Trivial example xk 1 sin(xk ) xk cos(xk ) u k sin(xk ) A( xk ) ; B( xk ) xk cos(xk ) xk Hamilton-Jacobi-Bellman x f ( x) g ( x) u V ( x) min x T Q( x) x u T R( x) u dt u (t ) T 0 T V V V 1 V 1 T f ( x) g ( x) R g ( x) x T Q x 0 t 4 x x x NLQGPC Control Law The NLQGPC quadratic infinite horizon cost function: 1 T J lim Jt T T 1 t 0 N Jt j 0 y t j 1 rt j 1 T T Qe yt j 1 rt j 1 ut j Qu ut j The optimal control vector in terms of the states of the system and reference model: Ut,N (t ,t N ) A(t ) T H 1 A(t ) 1 T S (t ,t N ) Qe T xˆt T H 2 N S(t ,t N ) Qe Rt , N T Qu S(t ,t N ) Qe S(t ,t N ) (t ) H (t ) T 1 The Coupled Algebraic Riccati Equations S Q H 1j AT N Qe N H 1j 1 A AT N Qe S N H 1j 1 T T Qu S N Qe S N T T H 1j 1 1 T N e T 1 H j 1 A N H 2j AT N Qe AT H 2j 1 N AT T Qe S N H 1j 1 T Qu S N Qe S N T T H 1j 1 1 T H 2j 1N S N T Qe Dealing with Stability Issue Control Lyapunov Function A C1 function V(x): n is said to be a discrete CLF for the system: if V(x) is positive definite, unbounded, and if for all x 0 Stability via Satisficing Satisficing is based on a point-wise cost / benefit comparison of an action. The benefits are given by the “Selectability” function Ps(u,x), while the costs are given by the “Rejectability” function Pr(u,x). The “satisficing” set is those options for which selectability exceeds rejectability: CLF-Based Satisficing Technique Start f, B, P, uNLQGPC Calculate implies uS No implies uaug = uNLQGPC - ( )(BTPB)-1BTPf uS u NLQGPC B T P f T f T P B ( B T P B) 1 B T P f f T P f xT P x T f P B ( B T P B) 1 B T P f Satisficing generates the state dependent set of controls that render the closed-loop system stable with respect to a known CLF. Dealing with Input Constraints Examples of Actuator Constraints • Magnitude Saturation u max , u out sat( u in ) u in , u , min • Rate-Limited Actuators max , u out sat( u in ) u in , min , • Actuator Dead-Zone u in u max , u min u in u max , u in u min , u in max , min u in max , u in min , Dd (u) u sat d (u) Therefore the common term is the Saturation function Approximation of Magnitude Saturation the actuator range of operation is limited 0 u Limiting functions that map the interval (-,) onto (0, 1) u Limiting functions that map the interval (-,) onto (-1, 1) Approximation of Magnitude Saturation Sigmoid function S 01 (Black) Error function (Blue) Tanh function (Green) Sigmoid function S 11 (Red) Case Studies F-8 fighter aircraft F-16 fighter aircraft Caltech Ducted Fan Controlling of F-8 Fighter 0.5 Angle of Attack [rad] 0.4 Unconstrained NLQGPC Constrained NLQGPC CNLQGPC-Satisficing (Nominal) CNLQGPC-Satisficing (Wind Gust) 0.3 0.2 0.1 0 -0.1 0 2 4 6 8 Time [s] 10 12 14 0.1 0 Pitch Rate [rad/s] -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 0 2 4 6 8 Time [s] 10 12 14 Elevator Deflection 0.01 0 -0.01 u [rad] -0.02 -0.03 u 0.05236 rad -0.04 -0.05 -0.06 -0.07 -0.08 0 2 4 6 8 Time [s] 10 12 14 Trajectories of Controlled System 0.8 0.6 0.4 x3 0.2 0 -0.2 -0.4 -1 -0.6 0 -0.8 0.5 1 0 -0.5 x1 x2 CONCLUSIONS Properties of NLQGPC controller: 1. High performance 2. Less computational burden 3. Dealing with input constraints 4. Guaranteeing asymptotic stability to the closed-loop system. 5. Possesses both performance robustness & stability robustness