H0K03a : Advanced Process Control Model-based Predictive Control 3 : Stability 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 3 : Stability bert.pluymers@esat.kuleuven.be
Overview • Introduction • Example • Stability Theory • Set Invariance • Implementations Lecture 3 : Stability • Introduction • Example • Stability Theory • Set Invariance • Implementations S ignal processing I dentification S ystem T heory 1 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
MPC Paradigm • Introduction • Example • At every discrete time instant , given information about • Stability Theory • Set Invariance the current system state , calculate an ‘optimal’ input • Implementations sequence over a finite time horizon : N N • Apply the first input to the real system • Repeat at the next time instant , using new state S ignal processing I dentification measurements / estimates. S ystem T heory 2 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Optimality of input sequence • Introduction • Example • Stability Theory to be applied at • Set Invariance • Implementations computed at S ignal processing I dentification Optimal input sequences Input sequence applied to the system S ystem T heory 3 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Stability Analysis • Introduction • Example • Stability Theory • Set Invariance • Classical Way : • Implementations x u r + - linear controller plant Analyse poles/zeros of and associated transfer functions. • Modelbased Predictive Control : x u r + - MPC controller plant S ignal processing I dentification Lyapunov theory for stability. S ystem T heory 4 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Inverted Pendulum • Introduction • Example • Stability Theory • Set Invariance • Implementations • 1 input : • 4 states : • open loop unstable system S ignal processing I dentification S ystem T heory 5 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Inverted Pendulum • Introduction • Example • Stability Theory • Set Invariance • Implementations Non-minimum phase behaviour S ignal processing • 4 different horizon lengths I dentification • 3 different MPC variants (to be defined later) S ystem T heory 6 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Stability Theory • Introduction • Example • Stability Theory • Set Invariance • Explicit vs. Optimization-based controller • Implementations • Transfer functions → Lyapunov theory Stability is obtained / proven in 2 steps : 1. Recursive feasibility i.e. controller well-defined for all k 2. Lyapunov function construction i.e. trajectories converge to equilibrium S ignal processing I dentification S ystem T heory 7 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Stability Theory • Introduction • Example • Stability Theory • Set Invariance Limited validity of MPC stability framework : • Implementations • only for ‘stabilization’ problems : • initial state • system steered towards • no disturbances allowed (but extension possible) • no general stability framework for ‘tracking’ problems S ignal processing I dentification S ystem T heory 8 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Stability Theory • Introduction • Example • Stability Theory • Set Invariance Recursive Feasibility • Implementations If the optimization problem is feasible for time , then it is also feasible for time . ( and hence for all ) Feasible Region The region in state space, defined by all states for which the MPC optimization problem is feasible. → Recursive feasibility proven : all states within feasible region lead to trajectories for which the MPC-controller is feasible and hence well-defined. S ignal processing I dentification S ystem T heory 9 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Stability Theory • Introduction • Example • Stability Theory → Recursive feasibility proven : all states within feasible • Set Invariance region lead to trajectories for which the MPC-controller is • Implementations feasible and hence well-defined. feasible region S ignal processing I dentification S ystem T heory 10 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
MPC Stability Measures • Introduction • Example • Stability Theory • Set Invariance • Implementations S ignal processing I dentification S ystem T heory 11 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
MPC Stability Measures • Introduction • Example • Stability Theory • Set Invariance • Implementations recursive feasibility ( terminal constraint ) Lyapunov stability ( terminal cost ) S ignal processing I dentification S ystem T heory 12 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Step 1 : Recursive Feasibility • Introduction • Example • Stability Theory Problem : given the optimal (and hence feasible) solution • Set Invariance to the optimization at time , construct a feasible solution • Implementations for the optimization at time . S ignal processing I dentification S ystem T heory 13 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Step 1 : Recursive Feasibility • Introduction • Example • Stability Theory • Set Invariance Given : • Implementations To be found : Observe / Choose : S ignal processing ? I dentification S ystem T heory 14 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Step 1 : Recursive Feasibility • Introduction • Example • Stability Theory Plant state at time predicted at time : • Set Invariance • Implementations Real plant state at time : Assumption : No plant model mismatch, i.e. Hence, reusing the overlapping part of the input sequence S ignal processing I dentification will also result in an identical state sequence S ystem T heory 15 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Step 1 : Recursive Feasibility • Introduction • Example • Stability Theory • Set Invariance • Implementations OK OK ??? ??? S ignal processing I dentification OK OK OK ??? S ystem T heory 16 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Step 1 : Recursive Feasibility • Introduction • Example • Stability Theory Condition 1 : • Set Invariance • Implementations Satisfied if Condition 2 : Condition 3 : How to choose and ? S ignal processing I dentification S ystem T heory 17 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Step 1 : Recursive Feasibility • Introduction • Example • Stability Theory How to choose and ? • Set Invariance • Implementations Assume we know a locally stabilizing controller : i.e. such that is locally stable. Then choose S ignal processing I dentification S ystem T heory 18 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Step 1 : Recursive Feasibility • Introduction • Example • Stability Theory • Set Invariance OK OK OK ??? • Implementations OK OK OK ??? Condition 2 : Condition 3 : S ignal processing I dentification S ystem T heory 19 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
Step 1 : Recursive Feasibility • Introduction • Example • Stability Theory • Set Invariance Condition 2 : • Implementations Condition 3 : Since we know that … Condition 2 is satisfied if Condition 3 is satisfied if S ignal processing I dentification S ystem T heory 20 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be
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