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 Grne, DE, and Dante Kalise, UK VI Latin American Workshop on
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
Common tasks in an optimal control problem min
u∈U J(x0, u)
J(x0, u) := TE ℓ(x(t), u(t)) dt , TE ∈ (0, ∞] ˙ x(t) = f(x(t), u(t)) , t ∈ (0, TE) , x(0) = x0 , include ◮ Steering the state to a desired equilibrium and keep it there ◮ Following a reference trajectory
Common tasks in an optimal control problem min
u∈U J(x0, u)
J(x0, u) := TE ℓ(x(t), u(t)) dt , TE ∈ (0, ∞] ˙ x(t) = f(x(t), u(t)) , t ∈ (0, TE) , x(0) = x0 , 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
MPC scheme applied to the nonlinear infinite horizon OCP splits the problem into several iterative OCPs on a finite horizon min
u∈UT (x0) JT(x0, u)
JT(x0, u) := T ℓ(x(t), u(t)) dt , ˙ x(t) = f(x(t), u(t)) , t ∈ (0, T) , x(0) = x0 , where T > 0 optimization horizon, x0, x(t) ∈ X, UT(x0) admissible controls for IC x0 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:
set n = 0.
JN(z0, u) :=
N−1
ℓ(zu(k; z0), u(k)) subject to z(k + 1) = f(z(k), u(k)) , with initial value z0 = zµ(n). Apply the first value of the resulting optimal control sequence u∗ ∈ UN, that is, set µ(zµ(n)) := u∗(0).
zµ(k + 1) = f(zµ(k), µ(zµ(k))) , set n := n + 1 and run again from step 2.
Past Future t TE Reference trajectory
Consider an optimal control problem on [0, TE].
Past Future t TE Reference1trajectory Measured1state Past1control
...
Sample1rate (Prediction)1Horizon n n+1 n+2 n+N
Choose a horizon N ∈ N and a sample rate T > 0. For each time tn := nT, n = 0, 1, 2, ... :
Past Future t TE Reference1trajectory Measured1state Past1control
...
Sample1rate (Prediction)1Horizon n n+1 n+2 n+N
the current time horizon [tn, tn+N].
Past Future t TE Reference1trajectory Measured1state Past1control Predicted1state Predicted1control
...
Sample1rate (Prediction)1Horizon n n+1 n+2 n+N
and apply its first value u∗(0) on [tn, tn+1].
Past Future t TE Reference1trajectory Measured1state Past1control Predicted1state Predicted1control
...
Sample1rate (Prediction)1Horizon n n+1 n+2 n+N
Adaption of http://en.wikipedia.org/wiki/File:MPC_scheme_basic.svg (CC BY-SA 3.0)
Main theoretical question: How to guarantee stability of the MPC closed loop system zµ(k + 1) = f(zµ(k), µ(zµ(k))) ?
Main theoretical question: How to guarantee stability of the MPC closed loop system zµ(k + 1) = f(zµ(k), µ(zµ(k))) ? Two possible approaches:
Main theoretical question: How to guarantee stability of the MPC closed loop system zµ(k + 1) = f(zµ(k), µ(zµ(k))) ? Two possible approaches:
Main theoretical question: How to guarantee stability of the MPC closed loop system zµ(k + 1) = f(zµ(k), µ(zµ(k))) ? Two possible approaches:
Main motivation: even for small optimization horizons N we can
values for which the finite horizon problem is well defined
Describe a general framework that enable stability of the MPC closed loop by an instantaneous control
Describe a general framework that enable stability of the MPC closed loop by an instantaneous control
z
z
feedback towards ¯ z
Exponential Controllability Condition Improving MPC performance by stabilizing feedback Linear Heat Equation The Schlögl system The Burgers’ Equation Conclusions
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.
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
ℓ(zu¯
z(k, ¯
z), u¯
z(k)) ≤ Cσk min u∈U ℓ(¯
z, u) for all k ∈ N.
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 2z(k) − ¯ z2 + λ 2u(k) − ¯ u2 , λ > 0 .
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 2z(k) − ¯ z2 + λ 2u(k) − ¯ u2 , λ > 0 .
In particular, ℓ(¯ z, ¯ u) = 0 and ℓ(z, u) > 0 for (z, u) = (¯ z, ¯ u) .
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 2z(k) − ¯ z2 + λ 2u(k) − ¯ u2 , λ > 0 .
Then there exists N0 ≥ 2 such that the equilibrium (¯ z, ¯ u) is globally asymptotically stable for the MPC closed loop for any optimization horizon N ≥ N0 .
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 2z(k) − ¯ z2 + λ 2u(k) − ¯ u2 , λ > 0 .
Then there exists N0 ≥ 2 such that the equilibrium (¯ z, ¯ u) is globally asymptotically stable for the MPC closed loop for any optimization horizon N ≥ N0 .
with C = 1 , then N0 = 2 (instantaneous control).
Exponential controllability condition ℓ(zu¯
z(k, ¯
z), u¯
z(k)) ≤ Cσk min u∈U ℓ(¯
z, u) , ∀k ∈ N
Exponential controllability condition ℓ(zu¯
z(k, ¯
z), u¯
z(k)) ≤ Cσk min u∈U ℓ(¯
z, u) , ∀k ∈ N Dependence of C and σ on N
(Figure: Harald Voit)
Exponential Controllability Condition Improving MPC performance by stabilizing feedback Linear Heat Equation The Schlögl system The Burgers’ Equation Conclusions
Let ¯ z be an unstable equilibrium of z(k + 1) = f(z(k), u(k))
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
Let ¯ z be an unstable equilibrium of z(k + 1) = f(z(k), u(k)) Assume that ◮ the stage cost ℓ is of tracking type towards ¯ z , i.e., ℓ(z, u) = 1 2z − ¯ z2 + λ 2u2 , λ > 0 . ◮ there exists us(z0) a given feedback stabilizing the system towards ¯ z , at an exponential rate
Let ¯ z be an unstable equilibrium of z(k + 1) = f(z(k), u(k)) Assume that ◮ the stage cost ℓ is of tracking type towards ¯ z , i.e., ℓ(z, u) = 1 2z − ¯ z2 + λ 2u2 , λ > 0 . ◮ there exists us(z0) a given feedback stabilizing the system towards ¯ z , at an exponential rate Then, suitably tuning the stage cost ℓ (or the dynamics f), we can achieve stability of the MPC closed loop with N = 2 (instantaneous control)
The equilibrium ¯ z is globally asimptotically stable for the MPC closed loop with dynamics z(k + 1) = f(z(k), u(k)) , z(0) = z0 and stage cost ℓs(z, u) = 1 2z − ¯ z2 + λ 2u−us(z0)2 , λ > 0 , for any optimization horizon N ≥ 2 .
The equilibrium ¯ z is globally asimptotically stable for the MPC closed loop with dynamics z(k + 1) = f(z(k), u(k)) , z(0) = z0 and stage cost ℓs(z, u) = 1 2z − ¯ z2 + λ 2u−us(z0)2 , λ > 0 ,
z(k + 1) = fs(z(k), u(k)) = f(z(k), us(z0) + u(k)) , z(0) = z0 ℓ(z, u) = 1 2z − ¯ z2 + λ 2u2 , λ > 0 . for any optimization horizon N ≥ 2 .
Exponential Controllability Condition Improving MPC performance by stabilizing feedback Linear Heat Equation The Schlögl system The Burgers’ Equation Conclusions
Consider yt − yxx − µy = χωu , in (0, 1) × (0, +∞) , y(0, t) = 0 , t ∈ (0, +∞) , y(1, t) = 0 , t ∈ (0, +∞) , y(x, 0) = y0(x) , x ∈ (0, 1) .
Consider yt − yxx − µy = χωu , in (0, 1) × (0, +∞) , y(0, t) = 0 , t ∈ (0, +∞) , y(1, t) = 0 , t ∈ (0, +∞) , y(x, 0) = y0(x) , x ∈ (0, 1) . Set λ1 the smallest eigenvalue of the operator (−∂xx) in H1
0(0, 1) , for µ ≥ λ1 the origin ¯
y = 0 is not asymptotically stable for the uncontrolled equation (with u = 0).
Consider yt − yxx − µy = χωu , in (0, 1) × (0, +∞) , y(0, t) = 0 , t ∈ (0, +∞) , y(1, t) = 0 , t ∈ (0, +∞) , y(x, 0) = y0(x) , x ∈ (0, 1) . Set λ1 the smallest eigenvalue of the operator (−∂xx) in H1
0(0, 1) , for µ ≥ λ1 the origin ¯
y = 0 is not asymptotically stable for the uncontrolled equation (with u = 0).
Discrete-time setting: y(k) := y(·, kT) , u(k) := u(·, kT) for some T > 0 (sampled data system with sampling time T)
Discrete-time setting: y(k) := y(·, kT) , u(k) := u(·, kT) for some T > 0 (sampled data system with sampling time T) Compare the cost functionals ℓy(z(k), u(k)) := 1 2y(k)2
2,Ω + λ
2u(k)2 , and ℓyx(z(k), u(k)) := 1 2yx(k)2
2,Ω + λ
2u(k)2 ,
Discrete-time setting: y(k) := y(·, kT) , u(k) := u(·, kT) for some T > 0 (sampled data system with sampling time T) Compare the cost functionals ℓy(z(k), u(k)) := 1 2y(k)2
2,Ω + λ
2u(k)2 , and ℓyx(z(k), u(k)) := 1 2yx(k)2
2,Ω + λ
2u(k)2 ,
Discrete-time setting: y(k) := y(·, kT) , u(k) := u(·, kT) for some T > 0 (sampled data system with sampling time T) Compare the cost functionals ℓy(z(k), u(k)) := 1 2y(k)2
2,Ω + λ
2u(k)2 , and ℓyx(z(k), u(k)) := 1 2yx(k)2
2,Ω + λ
2u(k)2 , In particular, the minimal horizon for ℓy is N∗ = 6
On the other hand, it is well understood that a Riccati feedback
u = Py stabilizes the problem to the origin ¯ y = 0
On the other hand, it is well understood that a Riccati feedback
u = Py stabilizes the problem to the origin ¯ y = 0
We thus plug the stabilizing feedback ¯ u in the equation. Let ys the solution of the controlled (stabilized) equation yt − yxx − µy = χω¯ u + χωu , in (0, 1) × (0, +∞) =: Q , y(0, t) = 0 , t ∈ (0, +∞) , y(1, t) = 0 , t ∈ (0, +∞) , y(x, 0) = y0(x) , x ∈ (0, 1) .
On the other hand, it is well understood that a Riccati feedback
u = Py stabilizes the problem to the origin ¯ y = 0
We thus plug the stabilizing feedback ¯ u in the equation. Let ys the solution of the controlled (stabilized) equation yt − yxx − µy = χω¯ u + χωu , in (0, 1) × (0, +∞) =: Q , y(0, t) = 0 , t ∈ (0, +∞) , y(1, t) = 0 , t ∈ (0, +∞) , y(x, 0) = y0(x) , x ∈ (0, 1) . The MPC closed loop with cost functional ℓ(ys, u) := 1 2ys(t)2
2,Ω + λ
2u(t)2 is asymptotically stable towards ¯ y = 0 for any N ≥ 2 .
The origin ¯ y = 0 is unstable for the free equation ( u ≡ 0 ) Applying MPC with N = 2 to the heat equation fails to stabi- lize ys towards ¯ y = 0
Applying MPC with N = 9 to the heat equation without Ric- cati feedback stabilizes ys to- wards ¯ y = 0 Applying MPC with N = 2 to the heat equation with Riccati feedback stabilizes ys towards ¯ y = 0
Exponential Controllability Condition Improving MPC performance by stabilizing feedback Linear Heat Equation The Schlögl system The Burgers’ Equation Conclusions
Given y1 ≤ y2 ≤ y3 real numbers, consider the polynomial ϕ(y) = (y − y1)(y − y2)(y − y3) , ∀y ∈ R .
Given y1 ≤ y2 ≤ y3 real numbers, consider the polynomial ϕ(y) = (y − y1)(y − y2)(y − y3) , ∀y ∈ R . Then ϕ has the following properties: i) mϕ := inf ϕ′(y) > −∞ , ii) the infimum mϕ < 0 is attained at the point (y1 + y2 + y3)/3.
Given y1 ≤ y2 ≤ y3 real numbers, consider the polynomial ϕ(y) = (y − y1)(y − y2)(y − y3) , ∀y ∈ R . Then ϕ has the following properties: i) mϕ := inf ϕ′(y) > −∞ , ii) the infimum mϕ < 0 is attained at the point (y1 + y2 + y3)/3. We consider the semilinear parabolic PDE yt = yxx − Kϕ(y) , (x, t) ∈ (0, L) × (0, ∞) , (SchlogEQ) with positive constants K, L , and appropriate initial and boundary conditions.
Global exponential stability result from Gugat, Tröltzsch, 2015:
Global exponential stability result from Gugat, Tröltzsch, 2015: Assume that L2K <
1 2|mϕ| , and let yd be a desired trajectory,
that is, yd ∈ H2(Q) satisfies (SchlogEQ).
Global exponential stability result from Gugat, Tröltzsch, 2015: Assume that L2K <
1 2|mϕ| , and let yd be a desired trajectory,
that is, yd ∈ H2(Q) satisfies (SchlogEQ). Consider the solution y to (SchlogEQ) with initial state y(. , 0) = y0 ∈ L∞(0, L) and with boundary conditions yx(0, t) = C(y(0, t) − yd(0, t)) + yd
x (0, t) ,
yx(L, t) = −C(y(L, t) − yd(L, t)) + yd
x (L, t) ,
(SchlogBC) where C ≥
1 2L .
Global exponential stability result from Gugat, Tröltzsch, 2015: Assume that L2K <
1 2|mϕ| , and let yd be a desired trajectory,
that is, yd ∈ H2(Q) satisfies (SchlogEQ). Consider the solution y to (SchlogEQ) with initial state y(. , 0) = y0 ∈ L∞(0, L) and with boundary conditions yx(0, t) = C(y(0, t) − yd(0, t)) + yd
x (0, t) ,
yx(L, t) = −C(y(L, t) − yd(L, t)) + yd
x (L, t) ,
(SchlogBC) where C ≥
1 2L .
Then, setting µ = 1 L2 − 2K|mϕ| , y converges exponentially fast in the L2−sense to yd, i.e., L
2 dx ≤ L
2 dx e−µt holds for all t ≥ 0 .
We now introduce the cost functional J(ys, u) = 1 2ys − yd2
L2(Q) + λ1
2 u12
L2(0,∞) + λ2
2 u22
L2(0,∞) ,
constrained to the control system yt = yxx − Kϕ(y) , (x, t) ∈ Q , y(x, 0) = y0(x) , x ∈ (0, L) , yx(0, t) − C(y(0, t) − yd(0, t)) − yd
x (0, t) = u1(t) ,
t ∈ (0, ∞) , yx(L, t) + C(y(L, t) − yd(L, t)) − yd
x (L, t) = u2(t) ,
t ∈ (0, ∞) .
We now introduce the cost functional J(ys, u) = 1 2ys − yd2
L2(Q) + λ1
2 u12
L2(0,∞) + λ2
2 u22
L2(0,∞) ,
constrained to the control system yt = yxx − Kϕ(y) , (x, t) ∈ Q , y(x, 0) = y0(x) , x ∈ (0, L) , yx(0, t) − C(y(0, t) − yd(0, t)) − yd
x (0, t) = u1(t) ,
t ∈ (0, ∞) , yx(L, t) + C(y(L, t) − yd(L, t)) − yd
x (L, t) = u2(t) ,
t ∈ (0, ∞) . From Gugat, Tröltzsch: the closed loop system with ui = 0 is exponentially stable towards yd .
We now introduce the cost functional J(ys, u) = 1 2ys − yd2
L2(Q) + λ1
2 u12
L2(0,∞) + λ2
2 u22
L2(0,∞) ,
constrained to the control system yt = yxx − Kϕ(y) , (x, t) ∈ Q , y(x, 0) = y0(x) , x ∈ (0, L) , yx(0, t) − C(y(0, t) − yd(0, t)) − yd
x (0, t) = u1(t) ,
t ∈ (0, ∞) , yx(L, t) + C(y(L, t) − yd(L, t)) − yd
x (L, t) = u2(t) ,
t ∈ (0, ∞) . From Gugat, Tröltzsch: the closed loop system with ui = 0 is exponentially stable towards yd . Thus, we look for controls u1, u2 which improve the tracking performance of the feedback (SchlogBC) with respect to the cost functional J(ys, u) .
In a discrete-time setting, the cost functional is recast as J∞(ys, u) =
∞
ℓ(ys(k), u(k)) , where the running cost ℓ is given by ℓ(ys(k), u(k)) = 1 2ys(k)−yd(k)2
L2(0,L)+λ1
2 |u1(k)|2+λ2 2 |u2(k)|2 , for a given sampling rate T > 0.
In a discrete-time setting, the cost functional is recast as J∞(ys, u) =
∞
ℓ(ys(k), u(k)) , where the running cost ℓ is given by ℓ(ys(k), u(k)) = 1 2ys(k)−yd(k)2
L2(0,L)+λ1
2 |u1(k)|2+λ2 2 |u2(k)|2 , for a given sampling rate T > 0. The associated cost functional on a finite horizon N is JN(ys, u) =
N
ℓ(ys(k), u(k)) .
Moreover, the stage cost ℓ satisfies ℓ(ys(k; 0), 0) = 1 2y(k) − yd(k)2
L2(0,L)
≤ 1 2y0 − yd(0)2
L2(0,L) e−µkT = e−µkT min u∈U ℓ(y0, u)
Moreover, the stage cost ℓ satisfies ℓ(ys(k; 0), 0) = 1 2y(k) − yd(k)2
L2(0,L)
≤ 1 2y0 − yd(0)2
L2(0,L) e−µkT = e−µkT min u∈U ℓ(y0, u)
= ⇒ the exponential controllability condition is satisfied with σ = e−µT and C = 1
Moreover, the stage cost ℓ satisfies ℓ(ys(k; 0), 0) = 1 2y(k) − yd(k)2
L2(0,L)
≤ 1 2y0 − yd(0)2
L2(0,L) e−µkT = e−µkT min u∈U ℓ(y0, u)
= ⇒ the exponential controllability condition is satisfied with σ = e−µT and C = 1 = ⇒ the MPC closed loop is stable for any N ≥ 2 .
For ϕ = y3 − y , K = 15 , the
Applying MPC with N = 2 to (SchlogEQ) without feed- back (SchlogBC) fails to stabi- lize it towards yd = 0
Applying MPC with N = 2 to (SchlogEQ) with feed- back (SchlogBC) stabilizes ys towards yd = 0
Exponential Controllability Condition Improving MPC performance by stabilizing feedback Linear Heat Equation The Schlögl system The Burgers’ Equation Conclusions
Given a positive constant d, we consider the Burgers equation yt − dyxx + yyx = 0 , in (0, 1) × (0, +∞) =: Q , yx(0, t) = u0(t) , t ∈ (0, +∞) , yx(1, t) = u1(t) , t ∈ (0, +∞) , y(x, 0) = y0(x) , x ∈ (0, 1) . (BurgEQ)
Given a positive constant d, we consider the Burgers equation yt − dyxx + yyx = 0 , in (0, 1) × (0, +∞) =: Q , yx(0, t) = u0(t) , t ∈ (0, +∞) , yx(1, t) = u1(t) , t ∈ (0, +∞) , y(x, 0) = y0(x) , x ∈ (0, 1) . (BurgEQ) The controls ui act as Neumann boundary conditions and aim to stabilize the state y towards the equilibrium yd ≡ 0.
Given a positive constant d, we consider the Burgers equation yt − dyxx + yyx = 0 , in (0, 1) × (0, +∞) =: Q , yx(0, t) = u0(t) , t ∈ (0, +∞) , yx(1, t) = u1(t) , t ∈ (0, +∞) , y(x, 0) = y0(x) , x ∈ (0, 1) . (BurgEQ) The controls ui act as Neumann boundary conditions and aim to stabilize the state y towards the equilibrium yd ≡ 0. However, the origin is not asymptotically stable for the free equation (with ui = 0), since in this case any nonzero constant is also an equilibrium of the system.
Global exponential stability result from Krstic 1999:
Global exponential stability result from Krstic 1999: Let y0 ∈ H1(0, 1) and c0, c1 positive parameters.
Global exponential stability result from Krstic 1999: Let y0 ∈ H1(0, 1) and c0, c1 positive parameters. Then the solution y to equation (BurgEQ) with boundary conditions ¯ u0 = 1
d
3y2(0, t)
¯ u1 = − 1
d
3y2(1, t)
(BurgBC) satisfies 1. sup
(x,t)∈¯ Q
|y(x, t)| < ∞;
t→∞ max x∈[0,1] |y(x, t)| = 0;
3. 1 |y(x, t)|2 dx decays to zero at an exponential rate given by c = min(d/2, c0, c1) > 0.
Global exponential stability result from Krstic 1999: Let y0 ∈ H1(0, 1) and c0, c1 positive parameters. Then the solution y to equation (BurgEQ) with boundary conditions ¯ u0 = 1
d
3y2(0, t)
¯ u1 = − 1
d
3y2(1, t)
(BurgBC) satisfies 1. sup
(x,t)∈¯ Q
|y(x, t)| < ∞;
t→∞ max x∈[0,1] |y(x, t)| = 0;
3. 1 |y(x, t)|2 dx decays to zero at an exponential rate given by c = min(d/2, c0, c1) > 0.
In particular, the equilibrium yd ≡ 0 is globally exponentially stable in L2(0, 1) for the closed loop system (BurgEQ)-(BurgBC)
We introduce the cost functional J(ys, u) = 1 2ys2
L2(Q) + λ
2
L2(0,∞) + u12 L2(0,∞)
We introduce the cost functional J(ys, u) = 1 2ys2
L2(Q) + λ
2
L2(0,∞) + u12 L2(0,∞)
where λ > 0 and ys is the solution to yt − dyxx + yyx = 0 , in (0, 1) × (0, +∞) =: Q , yx(0, t) − ¯ u0 = u0 , t ∈ (0, +∞) , yx(1, t) − ¯ u1 = u1 , t ∈ (0, +∞) , y(x, 0) = y0(x) , x ∈ (0, 1) , and ¯ u0, ¯ u1 are given by (BurgBC).
We introduce the cost functional J(ys, u) = 1 2ys2
L2(Q) + λ
2
L2(0,∞) + u12 L2(0,∞)
where λ > 0 and ys is the solution to yt − dyxx + yyx = 0 , in (0, 1) × (0, +∞) =: Q , yx(0, t) − ¯ u0 = u0 , t ∈ (0, +∞) , yx(1, t) − ¯ u1 = u1 , t ∈ (0, +∞) , y(x, 0) = y0(x) , x ∈ (0, 1) , and ¯ u0, ¯ u1 are given by (BurgBC). Rmk: we expect the optimal controls u0, u1 to approach asymptotically (as t → +∞) the control references ¯ u0, ¯ u1.
In a discrete-time setting, given a sampling rate T > 0, the cost functional is recast as J∞(ys, u) =
∞
ℓ(ys(k), u(k)) , with ℓ(ys(k), u(k)) = 1 2ys(k)2
L2(0,1) + λ
2
1
|ui(kT)|2 , where ys(k) = ys(. , kT) ∈ L2(0, 1) and u(k) = (u0(kT), u1(kT)).
In a discrete-time setting, given a sampling rate T > 0, the cost functional is recast as J∞(ys, u) =
∞
ℓ(ys(k), u(k)) , with ℓ(ys(k), u(k)) = 1 2ys(k)2
L2(0,1) + λ
2
1
|ui(kT)|2 , where ys(k) = ys(. , kT) ∈ L2(0, 1) and u(k) = (u0(kT), u1(kT)). The associated cost functional on a finite horizon N is JN(ys, u) =
N
ℓ(ys(k), u(k)) .
Since min
u∈U ℓ(ys(k), u(k)) = 1
2ys(k; 0)2
L2(0,1) ,
from the exponential stability result we deduce that ℓ(ys(k; 0), 0) = 1 2y(k; ¯ u(k))2
L2(0,1) ≤ 1
2e−ckTy(0)2
L2(0,1)
= σk min
u∈U ℓ(y(0), u) .
Since min
u∈U ℓ(ys(k), u(k)) = 1
2ys(k; 0)2
L2(0,1) ,
from the exponential stability result we deduce that ℓ(ys(k; 0), 0) = 1 2y(k; ¯ u(k))2
L2(0,1) ≤ 1
2e−ckTy(0)2
L2(0,1)
= σk min
u∈U ℓ(y(0), u) .
= ⇒ the exponential controllability condition is satisfied with σ = e−cT and C = 1
Since min
u∈U ℓ(ys(k), u(k)) = 1
2ys(k; 0)2
L2(0,1) ,
from the exponential stability result we deduce that ℓ(ys(k; 0), 0) = 1 2y(k; ¯ u(k))2
L2(0,1) ≤ 1
2e−ckTy(0)2
L2(0,1)
= σk min
u∈U ℓ(y(0), u) .
= ⇒ the exponential controllability condition is satisfied with σ = e−cT and C = 1 = ⇒ the MPC closed loop is stable for any N ≥ 2 .
Exponential Controllability Condition Improving MPC performance by stabilizing feedback Linear Heat Equation The Schlögl system The Burgers’ Equation Conclusions
The knowledge of an explicit stabilizing feedback may lead to stability of the MPC with the shortest possible reciding horizon.
The knowledge of an explicit stabilizing feedback may lead to stability of the MPC with the shortest possible reciding horizon. We compare the effectiveness of two different approaches: A) Apply MPC to an unstable equation with a sufficiently long horizon; B) Apply MPC to a stabilized equation with the shortest horizon possible.
The knowledge of an explicit stabilizing feedback may lead to stability of the MPC with the shortest possible reciding horizon. We compare the effectiveness of two different approaches: A) Apply MPC to an unstable equation with a sufficiently long horizon; B) Apply MPC to a stabilized equation with the shortest horizon possible. Cons of A): ◮ Need to compute the best horizon N∗ ensuring stability of the MPC scheme in connection with the specific cost functional ℓ ; ◮ Increased computational time and complexity at each iteration of the scheme.
Pros of B): ◮ Stability of the MPC with instantaneous control; ◮ Easier to adapt to different cost functionals targeting the same equilibrium state; ◮ Simplified computational complexity at each iteration of the scheme.
Pros of B): ◮ Stability of the MPC with instantaneous control; ◮ Easier to adapt to different cost functionals targeting the same equilibrium state; ◮ Simplified computational complexity at each iteration of the scheme. Cons of B): ◮ It requires the knowledge of an explicit stabilizing feedback; ◮ Not suitable in case of state target different from the equilibrium.
Pros of B): ◮ Stability of the MPC with instantaneous control; ◮ Easier to adapt to different cost functionals targeting the same equilibrium state; ◮ Simplified computational complexity at each iteration of the scheme. Cons of B): ◮ It requires the knowledge of an explicit stabilizing feedback; ◮ Not suitable in case of state target different from the equilibrium.