On Performance for Tracking MPC

On Performance for Tracking MPC

Johannes P. Maree^∗, James B. Rawlings^∗∗, Lars S. Imsland^∗

∗Norwegian University of Science and Technology, Norway

∗∗University of Wisconsin-Madison, USA

17^th Nordic Process Control Workshop, January 26, 2012

Outline

1 Optimize for plant-wide performance

2 Control performance for MPC

3 Stabilizing MPC formulations

4 Plant-wide performance for infeasible MPC formulations

5 Closed-loop asymptotic performance: Future work

6 Conclusive remarks

Optimize for plant-wide performance

Multi-level Hierarchical Control Structure

©Skogestad

Multi-layer control and optimization structure for plant-wide performance.

Functional segregation

Executed at different time periods

Economical optimal operation addressed by a two-layered structure.

Optimize for plant-wide performance

Multi-level Hierarchical Control Structure

Multi-layer control and optimization structure for plant-wide performance.

Functional segregation

Executed at different time periods

Economical optimal operation addressed by a two-layered structure.

Optimize for plant-wide performance

Multi-level Hierarchical Control Structure

Production optimization (RTO).

(x_sp,u_sp) = arg min

x,u E(x,u) s.t.

f (x,u) = 0 g(x,u)60

Dynamic regulation and disturbance rejection (MPC).

minx,u

1 2

N

X

k=0

kx(k)−xspk²_Q+ku(k)−uspk²_R s.t.

x(k+ 1) =Ax(k) +Bu(k), ∀k ∈I[0,N]

x(k)∈X,u(k)∈U,∀k ∈I[0,N]

Optimize for plant-wide performance

Multi-level Hierarchical Control Structure

Production optimization (RTO). (xsp,usp) = arg min

x,u E(x,u) s.t.

f (x,u) = 0 g(x,u)60

Dynamic regulation and disturbance rejection (MPC).

minx,u

1 2

N

X

k=0

kx(k)−xspk²_Q+ku(k)−uspk²_R s.t.

x(k+ 1) =Ax(k) +Bu(k), ∀k ∈I[0,N]

x(k)∈X,u(k)∈U,∀k ∈I[0,N]

Control performance for MPC

How does control performance in MPC improve economy?

Usetextbook approach ofsqueeze-and-shift method

We reduce output variance, and steady-state tracking offset

Control performance for MPC

How does control performance in MPC improve economy?

Usetextbook approach ofsqueeze-and-shift method

We reduce output variance, and steady-state tracking offset

Control performance for MPC

How does control performance in MPC improve economy?

Usetextbook approach ofsqueeze-and-shift method

Active constraint Backoff

(loss)

Time

We reduce output variance, and steady-state tracking offset

Control performance for MPC

How does control performance in MPC improve economy?

Usetextbook approach ofsqueeze-and-shift method

Active constraint Backoff

(loss)

Time

Active constraint

Time Squeeze

We reduce output variance, and steady-state tracking offset

Control performance for MPC

How does control performance in MPC improve economy?

Usetextbook approach ofsqueeze-and-shift method

Active constraint Backoff

(loss)

Time

Active constraint

Time Squeeze

Active constraint

Time Shift

We reduce output variance, and steady-state tracking offset

Control performance for MPC

How does control performance in MPC improve economy?

Usetextbook approach ofsqueeze-and-shift method

Active constraint Backoff

(loss)

Time

Active constraint

Time Squeeze

Active constraint

Time Shift

We reduce output variance, and steady-state tracking offset

Stabilizing MPC formulations

Stable systems promote performance

For a dynamic system x⁺=f (x), x∈X, f :X→Rⁿ

Lyapunov function V :X →R≥ such that α₁(|x|)6V(x)6α₂(|x|) V(f (x))−V(x)6−α₃(|x|)

Theorem: Lyapunov stability [J.B. Rawlings, 2009 [2]]

If there exists a Lyapunov function V :X →R_>0 for systemx⁺=f (x) in a positive invariant admissible set XN, then the system isasymptotically stable with a region of attractionXN

StabilizingMPC formulations are guaranteed with the addition of,

a terminal penalty cost functionVf(x),and an invariant terminal constraint regionXf

Resulting MPC value function aLyapunov candidate

Stabilizing MPC formulations

Stable systems promote performance

For a dynamic system x⁺=f (x), x∈X, f :X→Rⁿ Lyapunov function V :X →R≥ such that

α₁(|x|)6V(x)6α₂(|x|) V(f (x))−V(x)6−α₃(|x|)

Theorem: Lyapunov stability [J.B. Rawlings, 2009 [2]]

If there exists a Lyapunov function V :X →R_>0 for systemx⁺ =f (x) in a positive invariant admissible set XN, then the system isasymptotically stable with a region of attractionXN

StabilizingMPC formulations are guaranteed with the addition of,

a terminal penalty cost functionVf(x),and an invariant terminal constraint regionXf

Resulting MPC value function aLyapunov candidate

Stabilizing MPC formulations

Stable systems promote performance

For a dynamic system x⁺=f (x), x∈X, f :X→Rⁿ Lyapunov function V :X →R≥ such that

α₁(|x|)6V(x)6α₂(|x|) V(f (x))−V(x)6−α₃(|x|)

Theorem: Lyapunov stability [J.B. Rawlings, 2009 [2]]

If there exists a Lyapunov function V :X →R_>0 for systemx⁺ =f (x) in a positive invariant admissible set XN, then the system isasymptotically stablewith a region of attraction XN

StabilizingMPC formulations are guaranteed with the addition of,

a terminal penalty cost functionVf(x),and an invariant terminal constraint regionXf

Resulting MPC value function aLyapunov candidate

Stabilizing MPC formulations

Stable systems promote performance

For a dynamic system x⁺=f (x), x∈X, f :X→Rⁿ Lyapunov function V :X →R≥ such that

α₁(|x|)6V(x)6α₂(|x|) V(f (x))−V(x)6−α₃(|x|)

Theorem: Lyapunov stability [J.B. Rawlings, 2009 [2]]

If there exists a Lyapunov function V :X →R_>0 for systemx⁺ =f (x) in a positive invariant admissible set XN, then the system isasymptotically stablewith a region of attraction XN

StabilizingMPC formulations are guaranteed with the addition of,

a terminal penalty cost functionVf(x),and

an invariant terminal constraint regionXf

Resulting MPC value function aLyapunov candidate

Stabilizing MPC formulations

Stable systems promote performance

For a dynamic system x⁺=f (x), x∈X, f :X→Rⁿ Lyapunov function V :X →R≥ such that

α₁(|x|)6V(x)6α₂(|x|) V(f (x))−V(x)6−α₃(|x|)

Theorem: Lyapunov stability [J.B. Rawlings, 2009 [2]]

If there exists a Lyapunov function V :X →R_>0 for systemx⁺ =f (x) in a positive invariant admissible set XN, then the system isasymptotically stablewith a region of attraction XN

StabilizingMPC formulations are guaranteed with the addition of, a terminal penalty cost functionV_f(x),and

an invariant terminal constraint regionX^f

Resulting MPC value function aLyapunov candidate

Stabilizing MPC formulations

Stable systems promote performance

For a dynamic system x⁺=f (x), x∈X, f :X→Rⁿ Lyapunov function V :X →R≥ such that

α₁(|x|)6V(x)6α₂(|x|) V(f (x))−V(x)6−α₃(|x|)

Theorem: Lyapunov stability [J.B. Rawlings, 2009 [2]]

If there exists a Lyapunov function V :X →R_>0 for systemx⁺ =f (x) in a positive invariant admissible set XN, then the system isasymptotically stablewith a region of attraction XN

StabilizingMPC formulations are guaranteed with the addition of, a terminal penalty cost functionV_f(x),and

an invariant terminal constraint regionX^f

Resulting MPC value function aLyapunov candidate

Stabilizing MPC formulations

Stable systems promote performance

For a dynamic system x⁺=f (x), x∈X, f :X→Rⁿ Lyapunov function V :X →R≥ such that

α₁(|x|)6V(x)6α₂(|x|) V(f (x))−V(x)6−α₃(|x|)

Theorem: Lyapunov stability [J.B. Rawlings, 2009 [2]]

If there exists a Lyapunov function V :X →R_>0 for systemx⁺ =f (x) in a positive invariant admissible set XN, then the system isasymptotically stablewith a region of attraction XN

StabilizingMPC formulations are guaranteed with the addition of, a terminal penalty cost functionV_f(x),and

an invariant terminal constraint regionX^f

Resulting MPC value function aLyapunov candidate

Plant-wide performance for infeasible MPC formulations

V_N(x,u) = min

x,u N−1

X

k=0

l(x(k),u(k)) s.t.

x⁺=f (x,u) (x,u)∈X×U x(N) =x_sp

.

.

Plant-wide performance for infeasible MPC formulations

V_N(x,u) = min

x,u N−1

X

k=0

l(x(k),u(k)) +V_f (x(N)) s.t.

x⁺=f (x,u) (x,u)∈X×U x(N)∈Xf

.

.

Xf :=O∞(xsp) x(N)

Vf(x(N))

Plant-wide performance for infeasible MPC formulations

What ifx_sp is not feasible?

Solutions exists, calledsafe-park procedures Relax terminal constraint region by choosing intermediate feasible set-pointx_s

Choose cost function objectiveg(x,u) as cost objective to intermediate steady-statex_s Add offset costV_o(x_s) and penalize distance cost fromxsp

VN(x,u) = min

x,u N−1

X

k=0

g(x(k),u(k)) +V0(xs) s.t.

x⁺=f (x,u) (x,u)∈X×U x(N) =x_s

Plant-wide performance for infeasible MPC formulations

What ifx_sp is not feasible?

Solutions exists, calledsafe-park procedures

Relax terminal constraint region by choosing intermediate feasible set-pointx_s

Choose cost function objectiveg(x,u) as cost objective to intermediate steady-statex_s Add offset costV_o(x_s) and penalize distance cost fromxsp

VN(x,u) = min

x,u N−1

X

k=0

g(x(k),u(k)) +V0(xs) s.t.

x⁺=f (x,u) (x,u)∈X×U x(N) =x_s

Plant-wide performance for infeasible MPC formulations

What ifx_sp is not feasible?

Solutions exists, calledsafe-park procedures Relax terminal constraint region by choosing intermediate feasible set-pointx_s

Choose cost function objectiveg(x,u) as cost objective to intermediate steady-statex_s Add offset costV_o(x_s) and penalize distance cost fromxsp

VN(x,u) = min

x,u N−1

X

k=0

g(x(k),u(k)) +V0(xs) s.t.

x⁺=f (x,u) (x,u)∈X×U x(N) =x_s

Plant-wide performance for infeasible MPC formulations

What ifx_sp is not feasible?

Solutions exists, calledsafe-park procedures Relax terminal constraint region by choosing intermediate feasible set-pointx_s

Choose cost function objectiveg(x,u) as cost objective to intermediate steady-statex_s

Add offset costV_o(x_s) and penalize distance cost fromxsp

VN(x,u) = min

x,u N−1

X

k=0

g(x(k),u(k)) +V0(xs) s.t.

x⁺=f (x,u) (x,u)∈X×U x(N) =x_s

Plant-wide performance for infeasible MPC formulations

What ifx_sp is not feasible?

Solutions exists, calledsafe-park procedures Relax terminal constraint region by choosing intermediate feasible set-pointx_s

Choose cost function objectiveg(x,u) as cost objective to intermediate steady-statex_s Add offset costV_o(x_s) and penalize distance cost fromxsp

VN(x,u) = min

x,u N−1

X

k=0

g(x(k),u(k)) +V0(xs) s.t.

x⁺=f (x,u) (x,u)∈X×U x(N) =x_s

Plant-wide performance for infeasible MPC formulations

What ifx_sp is not feasible?

Solutions exists, calledsafe-park procedures Relax terminal constraint region by choosing intermediate feasible set-pointx_s

Choose cost function objectiveg(x,u) as cost objective to intermediate steady-statex_s Add offset costV_o(x_s) and penalize distance cost fromxsp

VN(x,u) = min

x,u N−1

X

k=0

g(x(k),u(k)) +V0(xs) s.t.

x⁺=f (x,u) (x,u)∈X×U x(N) =x_s

.

x=Ax+Bu

.

.

Plant-wide performance for infeasible MPC formulations

Or, we can terminate in some control invariant regionXf :=O∞(xs)

V_N(x,u) = min

x,u N−1

X

k=0

g(x(k),u(k))

+Vf (x(N)) +V0(xs) s.t.

x⁺=f (x,u) (x,u)∈X×U x(N)∈Xf

Xf :=O∞(xs) x(N)

.

f(x(N)-xs)

x=Ax+Bu

.

.

Plant-wide performance for infeasible MPC formulations

Or, we can terminate in some control invariant regionXf :=O∞(xs) V_N(x,u) = min

x,u N−1

X

k=0

g(x(k),u(k))

+V_f (x(N)) +V0(xs) s.t.

x⁺=f (x,u) (x,u)∈X×U x(N)∈Xf

Xf :=O∞(xs) x(N)

.

f(x(N)-xs)

x=Ax+Bu

.

.

Plant-wide performance for infeasible MPC formulations

There exists room for improvement!

We solve the wrong problem, penalizing withxs!

We loose our Lyapunov candidate using offset penalty V_o(x_s) What is desirable?

Use set-points received from RTO layer Want stabilizing MPC formulations

Plant-wide performance for infeasible MPC formulations

There exists room for improvement!

We solve the wrong problem, penalizing withxs!

We loose our Lyapunov candidate using offset penalty V_o(x_s)

What is desirable?

Use set-points received from RTO layer Want stabilizing MPC formulations

Plant-wide performance for infeasible MPC formulations

There exists room for improvement!

We solve the wrong problem, penalizing withxs!

We loose our Lyapunov candidate using offset penalty V_o(x_s) What is desirable?

Use set-points received from RTO layer

Want stabilizing MPC formulations

Plant-wide performance for infeasible MPC formulations

There exists room for improvement!

We solve the wrong problem, penalizing withxs!

We loose our Lyapunov candidate using offset penalty V_o(x_s) What is desirable?

Use set-points received from RTO layer Want stabilizing MPC formulations

Plant-wide performance for infeasible MPC formulations

Definitions of steady state manifold

X_s ={x|x =Ax+Bu,x∈X,u ∈U}

We propose

Redefine terminal state constraint set,Xf := S

∀x∈Xs

O∞(x) Use original stage costl(x,u) with RTO set-points Assume there exists terminal control law κ_f(x) such that V_f (f (x, κ_f(x)))−V_f(x) +l(x, κ_f(x))60

Implement standard stabilizing MPC formulation VN(x,u) := min

x,u N−1

X

k=0

l(x(k),u(k)) +Vf (x(N)) s.t. x⁺ =f (x,u)

(x,u)∈X×U,x(N)∈Xf

Plant-wide performance for infeasible MPC formulations

Definitions of steady state manifold

X_s ={x|x =Ax+Bu,x∈X,u ∈U} We propose

Redefine terminal state constraint set,Xf := S

∀x∈Xs

O∞(x)

Use original stage costl(x,u) with RTO set-points Assume there exists terminal control law κ_f(x) such that V_f (f (x, κ_f(x)))−V_f(x) +l(x, κ_f(x))60

Implement standard stabilizing MPC formulation VN(x,u) := min

x,u N−1

X

k=0

l(x(k),u(k)) +Vf (x(N)) s.t. x⁺ =f (x,u)

(x,u)∈X×U,x(N)∈Xf

Plant-wide performance for infeasible MPC formulations

Definitions of steady state manifold

X_s ={x|x =Ax+Bu,x∈X,u ∈U} We propose

Redefine terminal state constraint set,Xf := S

∀x∈Xs

O∞(x) Use original stage costl(x,u)with RTO set-points

Assume there exists terminal control law κ_f(x) such that V_f (f (x, κ_f(x)))−V_f(x) +l(x, κ_f(x))60

Implement standard stabilizing MPC formulation VN(x,u) := min

x,u N−1

X

k=0

l(x(k),u(k)) +Vf (x(N)) s.t. x⁺ =f (x,u)

(x,u)∈X×U,x(N)∈Xf

Plant-wide performance for infeasible MPC formulations

Definitions of steady state manifold

X_s ={x|x =Ax+Bu,x∈X,u ∈U} We propose

Redefine terminal state constraint set,Xf := S

∀x∈Xs

O∞(x) Use original stage costl(x,u)with RTO set-points

Assume there exists terminal control law κf(x) such that V_f (f (x, κ_f(x)))−V_f(x) +l(x, κ_f(x))60

Implement standard stabilizing MPC formulation VN(x,u) := min

x,u N−1

X

k=0

l(x(k),u(k)) +Vf (x(N)) s.t. x⁺ =f (x,u)

(x,u)∈X×U,x(N)∈Xf

Plant-wide performance for infeasible MPC formulations

Definitions of steady state manifold

X_s ={x|x =Ax+Bu,x∈X,u ∈U} We propose

Redefine terminal state constraint set,Xf := S

∀x∈Xs

O∞(x) Use original stage costl(x,u)with RTO set-points

Assume there exists terminal control law κf(x) such that V_f (f (x, κ_f(x)))−V_f(x) +l(x, κ_f(x))60

Implement standard stabilizing MPC formulation VN(x,u) := min

x,u N−1

X

k=0

l(x(k),u(k)) +Vf (x(N)) s.t. x⁺=f (x,u)

(x,u)∈X×U,x(N)∈Xf

Plant-wide performance for infeasible MPC formulations

What is the big picture?

VN(x,u) := min

x,u N−1

X

k=0

l(x(k),u(k)) +Vf (x(N)) s.t. x⁺=f (x,u)

(x,u)∈X×U,x(N)∈Xf ≡ [

∀x∈Xs

O∞(x)

.

x=Ax+Bu

.

.

O∞(xs,i-1) O∞(xs,0)

O∞(xs,i) O∞(xsp) Xs

Plant-wide performance for infeasible MPC formulations

Construction of κ

(x )

Terminal control law construction

κ_f (x(k)) :=K x(k)−x¯_i⁰(x(k))

+¯u_i⁰(x(k)), ji−16k6ji, j0 = 0,

i =i(k), i,j,k ∈I>0

Bx⁰1

Bx⁰2

^Bxs

 ^ 0

x k j

 _ 

0 1

x k j1 ₀ _ 

2

x k j2





sp i

x k j

Xs

j0 j₁ j₂ ji*

Plant-wide performance for infeasible MPC formulations

Terminal cost V

(x) from κ

(x )

Terminal cost

V_f (x(k =j₀)) :=kx(k =j₀)−x_spk²_P +

i^∗−1

X

i=1

¯x_i⁰(k =j_i)−x_sp

2 P,

ji−16k 6j_i, j₀ = 0, i =i(k), i,j,k ∈I>0

Bx⁰1

Bx⁰2

Bxs

  ₀

x k j

Xs

j0 j₁ j₂ ji*





sp i

x k j

 _ 

0 1

x k j1

 _ 

0 2

x k j2

Plant-wide performance for infeasible MPC formulations

Results: Closed-loop trajectories

−8 −6 −4 −2 0 2 4 6 8

−5

−4

−3

−2

−1 0 1 2 3 4 5

Tracking MPC: Closed−loop trajectories

x2

(b.) (a.) (d.) (c.)

(e.) x(0)

Xf

XN

Xs

Plant-wide performance for infeasible MPC formulations

Results: Cost and performance

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 200 400 600 800 1000

time (normalized) PN,1: V

N 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 100 200 300

time (normalized) PN,2: V

N 0

(a.) (b.) (c.) (d.) (e.)

(a.) (b.) (c.) (d.) (e.)

Closed-loop asymptotic performance: Future work

Closed-loop asymptotic performance: Future work

Question?

What can we say abouteconomic closed-loop performance as we approach(x_sp,u_sp) asymptotically for these approaches?

Intuitively our proposed method, using correct RTO set-point should have better performance?

We denote our economic objective to bel(x,u), defined as a l(x,u) := 1

2kx−xspk²_Q +1

2ku−uspk²_R Closed-loop asymptotic performance measure

V_T =

T

X

k=0

l(φ(k;x, κ_N,i(x)), κ_N,i(φ(k;x, κ_N,i(x)))),i ∈I[1,2]

Closed-loop asymptotic performance: Future work

Closed-loop asymptotic performance: Future work

Question?

What can we say abouteconomic closed-loop performance as we approach(x_sp,u_sp) asymptotically for these approaches?

Intuitively our proposed method, using correct RTO set-point should have better performance?

We denote our economic objective to bel(x,u), defined as a l(x,u) := 1

2kx−xspk²_Q +1

2ku−uspk²_R Closed-loop asymptotic performance measure

V_T =

T

X

k=0

l(φ(k;x, κ_N,i(x)), κ_N,i(φ(k;x, κ_N,i(x)))),i ∈I[1,2]

Closed-loop asymptotic performance: Future work

Closed-loop asymptotic performance: Future work

Question?

What can we say abouteconomic closed-loop performance as we approach(x_sp,u_sp) asymptotically for these approaches?

Intuitively our proposed method, using correct RTO set-point should have better performance?

We denote our economic objective to bel(x,u), defined as a l(x,u) := 1

2kx−xspk²_Q +1

2ku−uspk²_R

Closed-loop asymptotic performance measure V_T =

T

X

k=0

l(φ(k;x, κ_N,i(x)), κ_N,i(φ(k;x, κ_N,i(x)))),i ∈I[1,2]

Closed-loop asymptotic performance: Future work

Closed-loop asymptotic performance: Future work

Question?

What can we say abouteconomic closed-loop performance as we approach(x_sp,u_sp) asymptotically for these approaches?

Intuitively our proposed method, using correct RTO set-point should have better performance?

We denote our economic objective to bel(x,u), defined as a l(x,u) := 1

2kx−xspk²_Q +1

2ku−uspk²_R Closed-loop asymptotic performance measure

V_T =

T

X

k=0

l(φ(k;x, κ_N,i(x)), κ_N,i(φ(k;x, κ_N,i(x)))),i ∈I[1,2]

Closed-loop asymptotic performance: Future work

The MPC value function of proposed approach can be expressed as VN,1(x,u) =

N−1

X

k=0

l(x(k),u(k)) +Vf (x(N))

We denote g(x,u)stage cost the penalty to any admissible set-point (¯x,¯u)

The MPC value function of thesafe-park strategy is expressed as VN,2(x,u) =

N−1

X

k=0

g(x(k),u(k)) +kx(N)−xk¯ ²_P +k¯x−xspk²_T,T P

Consider the respective control laws of the two MPC formulations to be κ_N,1(x) andκ_N,2(x)

Closed-loop asymptotic performance: Future work

The MPC value function of proposed approach can be expressed as VN,1(x,u) =

N−1

X

k=0

l(x(k),u(k)) +Vf (x(N))

We denote g(x,u)stage cost the penalty to any admissible set-point (¯x,¯u)

The MPC value function of thesafe-park strategy is expressed as VN,2(x,u) =

N−1

X

k=0

g(x(k),u(k)) +kx(N)−xk¯ ²_P +k¯x−xspk²_T,T P

Consider the respective control laws of the two MPC formulations to be κ_N,1(x) andκ_N,2(x)

Closed-loop asymptotic performance: Future work

The MPC value function of proposed approach can be expressed as VN,1(x,u) =

N−1

X

k=0

l(x(k),u(k)) +Vf (x(N))

We denote g(x,u)stage cost the penalty to any admissible set-point (¯x,¯u)

The MPC value function of thesafe-park strategy is expressed as VN,2(x,u) =

N−1

X

k=0

g(x(k),u(k)) +kx(N)−xk¯ ²_P +k¯x−xspk²_T,T P

Consider the respective control laws of the two MPC formulations to be κ_N,1(x) andκ_N,2(x)

Closed-loop asymptotic performance: Future work

The MPC value function of proposed approach can be expressed as VN,1(x,u) =

N−1

X

k=0

l(x(k),u(k)) +Vf (x(N))

We denote g(x,u)stage cost the penalty to any admissible set-point (¯x,¯u)

The MPC value function of thesafe-park strategy is expressed as VN,2(x,u) =

N−1

X

k=0

g(x(k),u(k)) +kx(N)−xk¯ ²_P +k¯x−xspk²_T,T P

Consider the respective control laws of the two MPC formulations to be κ_N,1(x) andκ_N,2(x)

Closed-loop asymptotic performance: Future work

Question

For initial state x ∈ X_N using economic asymptotic performance measure V_T =

T

P

k=0

l(φ(k;x, κ_N,i(x)), κ_N,i(φ(k;x, κ_N,i(x)))),i ∈I[1,2] how does the control laws κ_N,1(x) andκ_N,2(x) perform with respect to each other?

We know

VT 6V_N,1^o (x) =VN,1(x, κN,1(x))6VN,1(x, κN,2(x)) What is the relation? - future work

T

X

k=0

l(φ(k;x, κ_N,1(x)), κ_N,1(φ(k;x, κ_N,1(x))))

6=

T

X

k=0

l(φ(k;x, κ_N,2(x)), κ_N,2(φ(k;x, κ_N,2(x))))

Closed-loop asymptotic performance: Future work

Question

For initial state x ∈ X_N using economic asymptotic performance measure V_T =

T

P

k=0

l(φ(k;x, κ_N,i(x)), κ_N,i(φ(k;x, κ_N,i(x)))),i ∈I[1,2] how does the control laws κ_N,1(x) andκ_N,2(x) perform with respect to each other?

We know

VT 6V_N,1^o (x) =VN,1(x, κN,1(x))6VN,1(x, κN,2(x))

What is the relation? - future work

T

X

k=0

l(φ(k;x, κ_N,1(x)), κ_N,1(φ(k;x, κ_N,1(x))))

6=

T

X

k=0

l(φ(k;x, κ_N,2(x)), κ_N,2(φ(k;x, κ_N,2(x))))

Closed-loop asymptotic performance: Future work

Question

For initial state x ∈ X_N using economic asymptotic performance measure V_T =

T

P

k=0

l(φ(k;x, κ_N,i(x)), κ_N,i(φ(k;x, κ_N,i(x)))),i ∈I[1,2] how does the control laws κ_N,1(x) andκ_N,2(x) perform with respect to each other?

We know

VT 6V_N,1^o (x) =VN,1(x, κN,1(x))6VN,1(x, κN,2(x)) What is the relation? - future work

T

X

k=0

l(φ(k;x, κ_N,1(x)), κ_N,1(φ(k;x, κ_N,1(x))))

6=

T

X

k=0

l(φ(k;x, κ_N,2(x)), κ_N,2(φ(k;x, κ_N,2(x))))

Conclusive remarks

Conclusion

Introduction to plant-wide performance control strategies

Control performance when set-points are unreachable

Investigate previous safe-park solution to unreachable set-points Proposed new method that solves correcteconomic objective Proposed method admits Lyapunov candidate value function

A terminal control law was proposed, but further refinement/analysis is needed

Different control laws give favourable control performance - acceptable

With economic MPC, however, we want to promote economics!

Conclusive remarks

Conclusion

Introduction to plant-wide performance control strategies Control performance when set-points are unreachable

Investigate previous safe-park solution to unreachable set-points Proposed new method that solves correcteconomic objective Proposed method admits Lyapunov candidate value function

A terminal control law was proposed, but further refinement/analysis is needed

Different control laws give favourable control performance - acceptable

With economic MPC, however, we want to promote economics!

Conclusive remarks

Conclusion

Introduction to plant-wide performance control strategies Control performance when set-points are unreachable

Investigate previous safe-park solution to unreachable set-points

Proposed new method that solves correcteconomic objective Proposed method admits Lyapunov candidate value function

A terminal control law was proposed, but further refinement/analysis is needed

Different control laws give favourable control performance - acceptable

With economic MPC, however, we want to promote economics!

Conclusive remarks

Conclusion

Introduction to plant-wide performance control strategies Control performance when set-points are unreachable

Investigate previous safe-park solution to unreachable set-points Proposed new method that solves correcteconomic objective

Proposed method admits Lyapunov candidate value function

A terminal control law was proposed, but further refinement/analysis is needed

Different control laws give favourable control performance - acceptable

With economic MPC, however, we want to promote economics!

Conclusive remarks

Conclusion

Introduction to plant-wide performance control strategies Control performance when set-points are unreachable

Investigate previous safe-park solution to unreachable set-points Proposed new method that solves correcteconomic objective Proposed method admits Lyapunov candidate value function

A terminal control law was proposed, but further refinement/analysis is needed

Different control laws give favourable control performance - acceptable

With economic MPC, however, we want to promote economics!

Conclusive remarks

Conclusion

Introduction to plant-wide performance control strategies Control performance when set-points are unreachable

Investigate previous safe-park solution to unreachable set-points Proposed new method that solves correcteconomic objective Proposed method admits Lyapunov candidate value function

A terminal control law was proposed, but further refinement/analysis is needed

Different control laws give favourable control performance - acceptable

With economic MPC, however, we want to promote economics!

Conclusive remarks

Conclusion

Introduction to plant-wide performance control strategies Control performance when set-points are unreachable

Investigate previous safe-park solution to unreachable set-points Proposed new method that solves correcteconomic objective Proposed method admits Lyapunov candidate value function

A terminal control law was proposed, but further refinement/analysis is needed

Different control laws give favourable control performance - acceptable

With economic MPC, however, we want to promote economics!

Conclusive remarks

Conclusion

Introduction to plant-wide performance control strategies Control performance when set-points are unreachable

Investigate previous safe-park solution to unreachable set-points Proposed new method that solves correcteconomic objective Proposed method admits Lyapunov candidate value function

A terminal control law was proposed, but further refinement/analysis is needed

Different control laws give favourable control performance - acceptable

With economic MPC, however, we want to promote economics!

Bibliography

[D. Limon, 2008] D. Limon and I. Alvarado and T. Alamo and E.F.

Camacho.

MPC for tracking piecewise constant references for constrained linear systems,

Automatica 44, 2008.

[J.B. Rawlings, 2009] J.B. Rawlings and D.Q Mayne Model Predictive Control: Theory and Design Nob Hill Publishing, 2009.