Improving MPC performance by stabilizing feedbacks Roberto - PowerPoint PPT Presentation

Improving MPC performance by stabilizing feedbacks Roberto Guglielmi Gran Sasso Science Institute (GSSI) - L ’Aquila, IT based on a joint work with Eduardo Cerpa, CL, Lars Grüne, DE, and Dante Kalise, UK VI Latin American Workshop on Optimization and Control Escuela Politécnica Nacional, Quito, Ecuador September 3-7, 2018

Introduction Common tasks in an optimal control problem min u ∈ U J ( x 0 , u ) � T E J ( x 0 , u ) := ℓ ( x ( t ) , u ( t )) d t , T E ∈ ( 0 , ∞ ] 0 ˙ x ( t ) = f ( x ( t ) , u ( t )) , t ∈ ( 0 , T E ) , x ( 0 ) = x 0 , include ◮ Steering the state to a desired equilibrium and keep it there ◮ Following a reference trajectory

Introduction Common tasks in an optimal control problem min u ∈ U J ( x 0 , u ) � T E J ( x 0 , u ) := ℓ ( x ( t ) , u ( t )) d t , T E ∈ ( 0 , ∞ ] 0 ˙ x ( t ) = f ( x ( t ) , u ( t )) , t ∈ ( 0 , T E ) , x ( 0 ) = x 0 , include ◮ Steering the state to a desired equilibrium and keep it there ◮ Following a reference trajectory (Possibly) infinite horizon optimal control problem = ⇒ hard computational complexity

Introduction MPC scheme applied to the nonlinear infinite horizon OCP splits the problem into several iterative OCPs on a finite horizon u ∈ U T ( x 0 ) J T ( x 0 , u ) min � T J T ( x 0 , u ) := ℓ ( x ( t ) , u ( t )) d t , 0 ˙ x ( t ) = f ( x ( t ) , u ( t )) , t ∈ ( 0 , T ) , x ( 0 ) = x 0 , where T > 0 optimization horizon, x 0 , x ( t ) ∈ X , U T ( x 0 ) admissible controls for IC x 0 up to time T ℓ ( x , u ) running cost f ( x , u ) controlled nonlinear dynamics

In a discrete-time setting, MPC scheme constructs a feedback law µ according to the following steps: 1. Given and initial condition z µ ( 0 ) ∈ X and a horizon N ≥ 2, set n = 0. 2. Minimize N − 1 � J N ( z 0 , u ) := ℓ ( z u ( k ; z 0 ) , u ( k )) k = 0 subject to z ( k + 1 ) = f ( z ( k ) , u ( k )) , with initial value z 0 = z µ ( n ) . Apply the first value of the resulting optimal control sequence u ∗ ∈ U N , that is, set µ ( z µ ( n )) := u ∗ ( 0 ) . 3. Evaluate z µ ( n + 1 ) according to z µ ( k + 1 ) = f ( z µ ( k ) , µ ( z µ ( k ))) , set n := n + 1 and run again from step 2.

MPC by a trajectory point of view Past Future Reference trajectory t T E Consider an optimal control problem on [ 0 , T E ] .

MPC by a trajectory point of view Past Future Reference1trajectory Measured1state Past1control (Prediction)1Horizon t Sample1rate ... n n+1 n+2 n+N T E Choose a horizon N ∈ N and a sample rate T > 0. For each time t n := nT , n = 0 , 1 , 2 , ... :

MPC by a trajectory point of view Past Future Reference1trajectory Measured1state Past1control (Prediction)1Horizon t Sample1rate ... n n+1 n+2 n+N T E 1. Measure the current state z ( n ) . 2. Set z 0 := z ( n ) and solve the optimal control problem on the current time horizon [ t n , t n + N ] .

MPC by a trajectory point of view Past Future Reference1trajectory Measured1state Past1control Predicted1state Predicted1control (Prediction)1Horizon t Sample1rate ... n n+1 n+2 n+N T E 3. Denote the calculated optimal control sequence by u ∗ ( · ) and apply its first value u ∗ ( 0 ) on [ t n , t n + 1 ] .

MPC by a trajectory point of view Past Future Reference1trajectory Measured1state Past1control Predicted1state Predicted1control (Prediction)1Horizon t Sample1rate ... n n+1 n+2 n+N T E 4. If t n + 1 < T E , set n := n + 1 and go to 1. Otherwise end. Adaption of http://en.wikipedia.org/wiki/File:MPC_scheme_basic.svg (CC BY-SA 3.0)

Stability of the MPC closed loop Main theoretical question: How to guarantee stability of the MPC closed loop system z µ ( k + 1 ) = f ( z µ ( k ) , µ ( z µ ( k ))) ?

Stability of the MPC closed loop Main theoretical question: How to guarantee stability of the MPC closed loop system z µ ( k + 1 ) = f ( z µ ( k ) , µ ( z µ ( k ))) ? Two possible approaches: - Add stabilizing terminal conditions and/or constraints - Tune the horizon length N and/or the stage cost ℓ

Stability of the MPC closed loop Main theoretical question: How to guarantee stability of the MPC closed loop system z µ ( k + 1 ) = f ( z µ ( k ) , µ ( z µ ( k ))) ? Two possible approaches: - Add stabilizing terminal conditions and/or constraints - Tune the horizon length N and/or the stage cost ℓ

Stability of the MPC closed loop Main theoretical question: How to guarantee stability of the MPC closed loop system z µ ( k + 1 ) = f ( z µ ( k ) , µ ( z µ ( k ))) ? Two possible approaches: - Add stabilizing terminal conditions and/or constraints - Tune the horizon length N and/or the stage cost ℓ Main motivation: even for small optimization horizons N we can - in principle - obtain large feasible sets, i.e., sets of initial values for which the finite horizon problem is well defined

Objective of the talk Describe a general framework that enable stability of the MPC closed loop by an instantaneous control

Objective of the talk Describe a general framework that enable stability of the MPC closed loop by an instantaneous control 1. The method shall be suitable for - Unstable equilibrium ¯ z - Stage cost of tracking type towards ¯ z 2. The method requires the knowledge of a given stabilizing feedback towards ¯ z

Outline Exponential Controllability Condition Improving MPC performance by stabilizing feedback Linear Heat Equation The Schlögl system The Burgers’ Equation Conclusions

Exponential Controllability condition Without terminal constraints, stability is known to hold for "sufficiently large optimization horizon N " [Alamir/Bornard ’95, Jadbabaie/Hauser ’05, Grimm/Messina/Tuna/Teel ’05] In order to get an estimate of how large shall N be, we need quantitative information.

Exponential Controllability condition Without terminal constraints, stability is known to hold for "sufficiently large optimization horizon N " [Alamir/Bornard ’95, Jadbabaie/Hauser ’05, Grimm/Messina/Tuna/Teel ’05] In order to get an estimate of how large shall N be, we need quantitative information. Exponential Controllability w.r.t. stage cost ℓ The system z ( k + 1 ) = f ( z ( k ) , u ( k )) is called exponentially controllable w.r.t. stage cost ℓ iff there exist an overshoot bound C ≥ 1 and a decay rate σ ∈ ( 0 , 1 ) such that for each state ¯ z = z ( 0 ) ∈ X there is a control u ¯ z ∈ U satisfying z ( k )) ≤ C σ k min z ( k , ¯ u ∈ U ℓ (¯ ℓ ( z u ¯ z ) , u ¯ z , u ) for all k ∈ N .

Stability of MPC without stabilizing terminal constraints Theorem (Grüne, Pannek, 2011) Let (¯ z , ¯ u ) be an equilibrium, i.e., f (¯ z , ¯ u ) = ¯ z . Consider the MPC scheme with stage cost ℓ ( z ( k ) , u ( k )) = 1 z � 2 + λ u � 2 , 2 � z ( k ) − ¯ 2 � u ( k ) − ¯ λ > 0 .

Stability of MPC without stabilizing terminal constraints Theorem (Grüne, Pannek, 2011) Let (¯ z , ¯ u ) be an equilibrium, i.e., f (¯ z , ¯ u ) = ¯ z . Consider the MPC scheme with stage cost ℓ ( z ( k ) , u ( k )) = 1 z � 2 + λ u � 2 , 2 � z ( k ) − ¯ 2 � u ( k ) − ¯ λ > 0 . In particular, ℓ (¯ z , ¯ u ) = 0 and ℓ ( z , u ) > 0 for ( z , u ) � = (¯ z , ¯ u ) .

Stability of MPC without stabilizing terminal constraints Theorem (Grüne, Pannek, 2011) Let (¯ z , ¯ u ) be an equilibrium, i.e., f (¯ z , ¯ u ) = ¯ z . Consider the MPC scheme with stage cost ℓ ( z ( k ) , u ( k )) = 1 z � 2 + λ u � 2 , 2 � z ( k ) − ¯ 2 � u ( k ) − ¯ λ > 0 . 1. Assume the exponential controllability w.r.t. ℓ hold. Then there exists N 0 ≥ 2 such that the equilibrium (¯ z , ¯ u ) is globally asymptotically stable for the MPC closed loop for any optimization horizon N ≥ N 0 .

Stability of MPC without stabilizing terminal constraints Theorem (Grüne, Pannek, 2011) Let (¯ z , ¯ u ) be an equilibrium, i.e., f (¯ z , ¯ u ) = ¯ z . Consider the MPC scheme with stage cost ℓ ( z ( k ) , u ( k )) = 1 z � 2 + λ u � 2 , 2 � z ( k ) − ¯ 2 � u ( k ) − ¯ λ > 0 . 1. Assume the exponential controllability w.r.t. ℓ hold. Then there exists N 0 ≥ 2 such that the equilibrium (¯ z , ¯ u ) is globally asymptotically stable for the MPC closed loop for any optimization horizon N ≥ N 0 . 2. If, in addition, the exponential controllability property holds with C = 1 , then N 0 = 2 (instantaneous control).

Stability chart for C and σ Exponential controllability condition z ( k )) ≤ C σ k min z ( k , ¯ u ∈ U ℓ (¯ ℓ ( z u ¯ z ) , u ¯ z , u ) , ∀ k ∈ N

Stability chart for C and σ Exponential controllability condition z ( k )) ≤ C σ k min z ( k , ¯ u ∈ U ℓ (¯ ℓ ( z u ¯ z ) , u ¯ z , u ) , ∀ k ∈ N Dependence of C and σ on N (Figure: Harald Voit)

Outline Exponential Controllability Condition Improving MPC performance by stabilizing feedback Linear Heat Equation The Schlögl system The Burgers’ Equation Conclusions

Setting of the problem Let ¯ z be an unstable equilibrium of z ( k + 1 ) = f ( z ( k ) , u ( k ))

Setting of the problem Let ¯ z be an unstable equilibrium of z ( k + 1 ) = f ( z ( k ) , u ( k )) ¯ z unstable = ⇒ usually a long horizon N to ensure stability of the MPC