H0K03a : Advanced Process Control Model-based Predictive Control 4 : Robustness Bert Pluymers Prof. Bart De Moor Katholieke Universiteit Leuven, Belgium Faculty of Engineering Sciences Department of Electrical Engineering (ESAT) Research Group SCD-SISTA H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Overview • Example • Robustness • Robust MPC • Conclusion Lecture 4 : Robustness • Example • Robustness • Robust MPC • Conclusion S ignal processing I dentification S ystem T heory 1 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Example • Example • Robustness • Robust MPC • Conclusion Linear state-space system of the form with bounded parametric uncertainty Aim : steer this system towards the origin from initial state without violating the constraint S ignal processing I dentification S ystem T heory 2 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Example • Example • Robustness • Robust MPC Results for 4 different parameter settings : • Conclusion S ignal processing • Recursive feasibility ? I dentification • Monotonicity of the cost ? S ystem T heory 3 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Robustness • Example • Robustness • Robust MPC Robust with respect to what ? • Conclusion • Disturbances Cause predictions of ‘nominal’ MPC to be inaccurate • Model uncertainty S ignal processing I dentification S ystem T heory 4 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Robustness • Example • Robustness • Robust MPC • Conclusion Main aims : • Keep recursive feasibility properties, despite model errors, disturbances • Keep asymptotic stability (in the case without disturbances) We need to have an idea about … • the size of the model uncertainty • the size of the disturbances S ignal processing I dentification S ystem T heory 5 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Uncertain Models • Example • Robustness • Robust MPC • Conclusion Linear Parameter-Varying state space models with polytopic uncertainty description S ignal processing I dentification S ystem T heory 6 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Uncertain Models • Example • Robustness • Robust MPC Linear Parameter-Varying state space models with • Conclusion norm-bounded uncertainty description S ignal processing I dentification S ystem T heory 7 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Bounded Disturbances • Example • Robustness • Robust MPC • Typically bounded by a polytope : • Conclusion • Can be described in two ways • • • Trivial condition for well-posedness : S ignal processing I dentification S ystem T heory 8 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Robust MPC • Example • Robustness • Robust MPC • Conclusion Main aims : • Keep recursive feasibility properties, despite model errors, disturbances • Keep asymptotic stability (in the case without disturbances) Necessary modifications : • Uncertain predictions (e.g predictions with all models within uncertainty region) • worst-case constraint satisfaction over all predictions • worst-case cost over all predictions • Terminal cost has to satisfy multiple Lyap. Ineq. • Terminal constraint has to be a robust invariant set S ignal processing I dentification S ystem T heory 9 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Robust MPC • Example • Robustness • Robust MPC • Conclusion Uncertain predictions : model uncertainty disturbances N N S ignal processing I dentification S ystem T heory 10 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Uncertain Predictions • Example • Robustness • Robust MPC • Conclusion Step 1) Robust Constraint Satisfaction Observations : • depends linearly on • is a convex polytopic set • is a convex set Result : Sufficient to impose constraint only for vert. of : S ignal processing I dentification S ystem T heory 11 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Uncertain Predictions • Example • Robustness LTI LPV • Robust MPC • Conclusion (L=1) (L>1, e.g. 2) S ignal processing I dentification S ystem T heory 12 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Uncertain Predictions • Example • Robustness • Robust MPC • Conclusion Impose state constraints on all nodes of state prediction tree → number of constraints increases expon. with incr. !!! S ignal processing I dentification S ystem T heory 13 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Worst-Case Cost Objective • Example • Robustness • Robust MPC • Conclusion Step 2) Worst-Case cost minimization Observations : • depends linearly on • is a convex polytopic set • cost function typically convex function of → Also for objective function sufficient to make S ignal processing I dentification predictions only with vertices of uncertainty polytope S ystem T heory 14 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Worst-Case Cost Objective • Example • Robustness • Robust MPC • Conclusion states inputs S ignal processing I dentification S ystem T heory 15 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Worst-Case Cost Objective • Example (1-norm) • Robustness • Robust MPC • Conclusion LP S ignal processing I dentification S ystem T heory 16 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Worst-Case Cost Objective • Example (2-norm) • Robustness • Robust MPC • Conclusion CVX ? S ignal processing I dentification S ystem T heory 17 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Worst-Case Cost Objective • Example (2-norm) • Robustness • Robust MPC • Conclusion Constraints of the form : CVX ? SOC S ignal processing I dentification S ystem T heory 18 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Worst-Case Cost Objective • Example (2-norm) • Robustness • Robust MPC • Conclusion SOCP S ignal processing I dentification S ystem T heory 19 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Robust MPC • Example (2-norm) • Robustness • Robust MPC By rewriting we now get • Conclusion SOCP S ignal processing Terminal constraint I dentification Terminal cost S ystem T heory 20 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Robust Terminal Cost • Example • Robustness • Robust MPC • Conclusion “non - robust” stability condition for terminal cost: In case of… • LPV system with polytopic uncertainty • linear feedback controller • quadratic cost criterion • quadratic terminal cost … this becomes : or equivalent : S ignal processing I dentification S ystem T heory 21 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Robust Terminal Cost • Example • Robustness • Robust MPC • Conclusion Robust stability condition for terminal cost: Observations : • inequality is convex and linear in and (i.e. LMI in ) • is a convex polytopic set Hence, inequality satisfied iff S ignal processing I dentification S ystem T heory 22 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
Robust Terminal Cost : Design • Example • Robustness • Robust MPC • Conclusion 1. Find a robustly stabilizing controller 2. Find a terminal cost satisfying by solving the following optimization problem : SDP Minimization of eigenvalues of S ignal processing I dentification optimization variables S ystem T heory 23 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be
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