9-12 MPC mode predictive control system.pdf

C21 Model Predictive Control
Mark Cannon
4 lectures
Hilary Term 2023
Department of Engineering Science
eng.ox.ac.uk/control
0 - 1

Organisation
B 4 lectures – Tuesday & Thursday weeks 5 & 6
in person at 14:00 in LR2
or pre-recorded on Canvas
B Examples class – Thursday week 7
in person at 12:00, 14:00, 15:00 in LR6
1 - 3

Course outline
1. Introduction to predictive control
2. Prediction and optimization
3. Closed loop properties
4. Disturbances and integral action
5. Robust tube MPC
1 - 4

Books
B J.M. Maciejowski, Predictive control with constraints. Prentice Hall, 2002
Recommended reading: Chapters 1–3, 6 & 8
B J.B. Rawlings and D.Q. Mayne, Model Predictive Control: Theory and Design. Nob Hill
Publishing, 2009
B B. Kouvaritakis and M. Cannon, Model Predictive Control: Classical, Robust and Stochastic,
Springer 2015
Recommended reading: Chapters 1, 2 & 3
1 - 5

Motivating example: switching control
How does a thermostat regulate room temperature?

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Closed loop control system:
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1 - 6

System model:
C
dT
dt
= q − qL
qL = βT
q = αu
u =
(
U if on
0 if off
Switching controller:
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dŚĞƌŵŽƐǁŝƚĐŚŝŶŐƉůŽƚƐ
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? Single controller parameter: hysteresis band δ
? Accurate models aren’t needed to regulate T to [T0
− δ, T0
+ δ]
1 - 7

System model:
C
dT
dt
= q − qL
qL = βT
q = αu
u =
(
U if on
0 if off
Closed loop response:
T(t) = Tss + (T(0) − Tss)e−t/τ
Tss =
(
αU/β if on
0 if off
τ =
C
β
− δ, T0
+ δ]
1 - 7

System model:
C
dT
dt
= q − qL
qL = βT
q = αu
u =
(
U if on
0 if off
− δ, T0
+ δ]
1 - 7

Motivating example: proportional control (P)
System model:
C
dT
dt
= q − qL
qL = βT
q = αu
u = K(T0
− T)
T(t) = Tss + (T(0) − Tss)e−t/τ
Tss =
αK
αK + β
T0
τ =
C
αK + β
? Controller parameter: gain K
? Tss → T0
and τ → 0 as K → ∞ independent of parameters C, α, β
1 - 8

Controller: u = K(T0
− T)
Effect of increasing gain (ideal case), K1 K2 K3:
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High gain K is often de-stabilizing because of:
? nonlinearity, e.g. actuator saturation: u = min
n
ū, max

K(T0
− T), 0
o
? additional dynamics, e.g. sensor and actuator time-delay or lag
1 - 9

Controller: u = K(T0
− T)
Actual effect of increasing gain:
High gain K is often de-stabilizing because of:
? nonlinearity, e.g. actuator saturation: u = min
n
ū, max

K(T0
− T), 0
o
? additional dynamics, e.g. sensor and actuator time-delay or lag
1 - 9

Motivating example: proportional + integral control (PI)
Control signal proportional to tracking error and integral of tracking error:
u = K(T0
− T) +
K
Ti
Z t
(T0
− T) dt
? If closed loop system is stable
then T0
− T(t) → 0 as t → ∞, i.e. no steady state error
(assuming T0
= constant)
? Controller has no knowledge of model parameters
but increasing gain (K/Ti) generally degrades transient performance
(overshoot and oscillations)
? Two controller parameters K, Ti to be tuned/optimized
1 - 10

Motivating example: PID control
Include the rate of change of tracking error:
u = K(T0
− T) +
K
Ti
Z t
(T0
− T) dt + KTd
d
dt
(T0
− T)
? The derivative term provides anticipation of future error (“feedforward”)
? Three PID gains K, Ti, Td need tuning, either using a system model or heuristic rules
(e.g. Ziegler-Nichols)
? PID tuning is difficult with nonlinear dynamics and constraints
? Not obvious how to configure feedback loops for MIMO problems
1 - 11

Controller optimization
Can we optimize controller parameters for a given performance criterion?
e.g. mean square error: min
K,Ti,Td
Z ∞
0
E{(T0
− T)2
+ ρu2
} dt
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? Optimization of linear controller gains (e.g. K, Ti, Td) is generally nonconvex
? It’s more common to optimize over control signals (e.g. LQG control)
min
u
Z ∞
0
E{(T0
− T)2
+ ρ u2
} dt
Unconstrained linear system =⇒ solution is linear state feedback
but no closed-form solution in almost all other cases
1 - 12

Model predictive control
MPC optimizes predicted performance numerically over future control and state trajectories
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? The optimization is generally easier than optimizing feedback gains
(e.g. convex for linear systems with linear state and input constraints)
? Single-shot solution is an open loop control signal
MPC updates it by repeating the optimization periodically online
? This results in a feedback controller,
providing robustness to model and measurement uncertainty
and compensating for using finite numbers of optimization variables
1 - 13

1 Prediction using a dynamic model constraints
2 Online optimization
3 Receding horizon implementation
1. Prediction
? Plant model: xk+1 = f(xk, uk)
? Simulate forward in time (over a prediction horizon of N steps)
predicted
input
sequence
uk =





u0|k
u1|k
.
.
.
uN−1|k





predicted
state
sequence
xk =





x0|k
x1|k
.
.
.
xN|k





Notation: (ui|k, xi|k) = predicted i steps ahead | evaluated at time k
x0|k = xk
1 - 14

Overview of MPC
2. Optimization
? Performance cost: J(xk, uk) =
N
X
i=0
`i(xi|k, ui|k)
`i(x, u): stage cost
? Optimize numerically to determine the optimal input sequence:
u∗
k = arg min
uk
J(xk, uk)
= u∗
0|k(xk), . . . , u∗
N−1|k(xk)

3. Implementation
? Use first element of u∗
k =⇒ MPC law: uk = u∗
0|k(xk)
? Repeat optimization at each sampling instant k = 0, 1, . . .
1 - 15

Overview of MPC
time
prediction horizon
time
past predicted
u
x
k k + N
1 - 16

Overview of MPC
time
prediction horizon at time
prediction horizon at time
time
u
x
k
k + 1
k k + N
k + 1 k + N + 1
1 - 16

Example
Plant model: xk+1 =

1.1 2
0 0.95

xk +

0
0.0787

uk
yk =

−1 1

xk
Cost:
N−1
X
i=0
(y2
i|k + u2
i|k) + y2
N|k
Prediction horizon: N = 3
Predicted input and state sequences: uk =


u0|k
u1|k
u2|k

, xk =




x0|k
x1|k
x2|k
x3|k




1 - 17

Example
0 1 2 3 4 5 6 7 8 9 10
input
-4
-2
0
2
4
6
sample
0 1 2 3 4 5 6 7 8 9 10
output
-1
-0.5
0
0.5
1
1 - 18

Example
0 1 2 3 4 5 6 7 8 9 10
input
-4
-2
0
2
4
6
predicted at k=0
closed loop
sample
0 1 2 3 4 5 6 7 8 9 10
output
-1
-0.5
0
0.5
1
1 - 18

Advantages
B Flexible plant model
– multivariable
– linear or nonlinear
– deterministic, stochastic or fuzzy
B Handles constraints on control inputs and states
– actuator limits
– safety, environmental and economic constraints
B Approximately optimal control
Disadvantages
B Requires online optimization
– quadratic programming (QP) problem for linear-quadratic problems
– high computational requirement for nonlinear systems
1 - 19

MPC development
Control strategy reinvented several times
LQG optimal control 1950’s
industrial process control 1980’s
constrained nonlinear MPC 1990’s
robust MPC 2000’s
stochastic MPC 2010’s
Current research challenges:
– high sample rates, long prediction horizons, uncertain nonlinear models
– embedded optimization sparse solvers
– adaptive and stochastic MPC
1 - 20

Prediction model
Linear plant model: xk+1 = Axk + Buk
B Predicted xk depends linearly on uk [details in Lecture 2]
B Therefore LQ cost is quadratic in uk u
k Huk + 2f
uk + g(xk)
and constraints are linear Acuk ≤ b(xk)
B Online optimization:
min
u
u
Hu + 2f
u s.t. Acu ≤ bc
This is a convex Quadratic Program (QP),
which is reliably and efficiently solvable
1 - 21

Prediction model
Nonlinear plant model: xk+1 = f(xk, uk)
B Predicted xk depends nonlinearly on uk
B In general the cost is nonconvex in uk: J(xk, uk)
and the constraints are nonconvex: gc(xk, uk) ≤ 0
B Online optimization:
min
u
J(xk, u) s.t. gc(xk, u) ≤ 0
– may be nonconvex
– may have local minima
– may not be solvable efficiently or reliably
1 - 22

Prediction model
Discrete time prediction model
B Predictions optimized periodically at t = 0, T, 2T, . . .
B Usually T = Ts = sampling interval of model
B But T = nTs for any integer n ≥ 1 is possible, (e.g. if Ts time needed
for online optimization)
1 - 23

Prediction model
Continuous time prediction model
B Predicted u(t) need not be piecewise constant,
e.g. continuous, piecewise linear u(t)
or u(t) = polynomial in t (piecewise quadratic, cubic etc)
B Continuous time prediction models can be solved online
B This course: discrete-time model and T = Ts assumed
1 - 24

Constraints
Classify state and input constraints as either hard or soft
B Hard constraints must be satisfied at all times,
if this is not possible, then the problem is infeasible
B Soft constraints can be violated to avoid infeasibility
B Strategies for handling soft constraints:
? impose (hard) constraints on the probability of violating each soft constraint
? or remove active constraints until the problem becomes feasible
1 - 25

Constraints
Typical methods for handling input constraints:
(a) Saturate the unconstrained control law
(ignore constraints in controller design)
(b) De-tune the unconstrained control law
by increasing the penalty on u in the performance objective
(c) Use an anti-windup strategy to limit the state of a dynamic controller
(typically the integral term of a PI or PID controller)
(d) Use MPC with inequality-constrained optimization
1 - 26

Example: input constraints
(a) Effects of controller saturation, u ≤ uk ≤ u
unconstrained LQ optimal control: u0
(x) = Klqx
saturated: u = max

min{u0
, u}, u
Input constraints:
u ≤ u ≤ u
u = −1, u = 1
Controller saturation causes
? poor performance
? possible instability
0 5 10 15 20 25 30 35 40
−2
0
2
4
6
8
u
0 5 10 15 20 25 30 35 40
−4
−2
0
2
4
6
y
sample
saturated lqr
unconstrained lqr
1 - 27

(b) Effects of de-tuning the unconstrained optimal control law:
Klq = optimal gain for LQ cost
∞
X
k=0
y2
k + ρ u2
k

Increase ρ until u = Klqx satisfies constraints (locally)
Example
10−2
≤ ρ ≤ 103
⇓
settling time increased
from 6 to 40
? yk → 0 slowly
? stability can be ensured
0 10 20 30 40 50 60
−2
0
2
4
6
y
sample
0 10 20 30 40 50 60
−2
0
2
4
6
8
u
lqr, R=1000
lqr, R=0.01
1 - 28

(c) Effects of Anti-windup:
Anti-windup attempts to avoid instability while control input saturated
Many possible approaches, e.g. anti-windup PI controller:
u = max

min{(Ke + v), u}, u
Tiv̇ + v = u
⇓
u ≤ u ≤ u =⇒ u = K

e +
1
Ti
Z t
e dt

u = u or u =⇒ v(t) → u or u exponentially
Heuristic strategy may not prevent instability
1 - 29

(d) Comparison with optimal constrained LQ control (MPC)
Example
MPC vs saturated LQ
(both using the same cost):
? settling time reduced to 20 by
MPC
? stability is guaranteed with
MPC
0 5 10 15 20 25 30 35 40
−1
0
1
2
3
u
0 5 10 15 20 25 30 35 40
−4
−2
0
2
4
6
y
sample
mpc, N=16
saturated lqr
1 - 30

Summary
B Predict performance using plant model
e.g. linear or nonlinear, discrete or continuous time
B Optimize future (open loop) control sequence
computationally much easier than optimizing over feedback laws
B Implement first sample, then repeat optimization
provides feedback to reduce effect of uncertainty
B Comparison of common methods of handling constraints:
saturation, de-tuning, anti-windup, MPC
1 - 31

Lecture 2
Prediction and optimization
2 - 1

Prediction and optimization
Input and state predictions
Unconstrained finite horizon optimal control
Infinite prediction horizons and connection with LQ optimal control
Incorporating constraints
Quadratic programming
2 - 2

Review of MPC strategy
At each sampling instant:
1 Use a model to predict system behaviour over a finite future horizon
2 Compute a control sequence by solving an online optimization problem
3 Apply the first element of optimal control sequence as control input
Advantages
? flexible plant model
? constraints taken into account
? optimal performance
Disadvantage
? online otimization required
2 - 3

Predicted cost:
Jk =
N−1
X
i=0
kxi|kk2
Q + kui|kk2
R

+ kxN|kk2
P
= x
k Q xk + u
k R uk

Q = diag{Q, . . . , Q, P}
R = diag{R, . . . , R, R}
⇓
Jk = u
k Huk + 2x
k F
uk + x
k Gxk
where
H = C
Q C + R ← u × u terms
F = C
Q M ← u × x terms
G = M
Q M ← x × x terms
time-invariant model =⇒ H, F, G can be computed offline
2 - 5

Prediction equations – example
Plant model: xk+1 = Axk + Buk, yk = Cxk
A =

1.1 2
0 0.95

, B =

0
0.079

, C =

−1 1

Prediction horizon N = 4: C =











0 0 0 0
0 0 0 0
0 0 0 0
0.079 0 0 0
0.157 0 0 0
0.075 0.079 0 0
0.323 0.157 0 0
0.071 0.075 0.079 0
0.497 0.323 0.157 0
0.068 0.071 0.075 0.079











Cost matrices Q = C
C, R = 0.01, and P = Q:
H =


0.271 0.122 0.016 −0.034
? 0.086 0.014 −0.020
? ? 0.023 −0.007
? ? ? 0.016

 F =


0.977 4.925
0.383 2.174
0.016 0.219
−0.115 −0.618


G =
h
7.589 22.78
? 103.7
i
2 - 6

Prediction equations: LTV model
Linear time-varying model: xk+1 = Akxk + Bkuk
Predictions: x0|k = xk
x1|k = Akxk + Bku0|k
x2|k = Ak+1Akxk + Ak+1Bku0|k + Bk+1u1|k
.
.
.
xi|k =
0
Y
j=i−1
Ak+jxk + Ci(k)uk, i = 0, . . . , N
Ci(k) =
1
Y
j=i−1
Ak+jBk
2
Y
j=i−1
Ak+jBk+1 · · · Bk+i−1 0 · · · 0
#
?
Q0
j=i−1 Ak+j = Ak+i−1 · · · Ak for i ≥ 1 and
Q0
j=i−1 Ak+j = 0 for i = 0
? H(k), F(k), G(k) depend on k and must be computed online
2 - 7

Unconstrained optimization
Minimize cost: u∗
= arg min
u
J, J = u
Hu + 2x
F
u + x
Gx
differentiate w.r.t. u: ∇uJ = 2Hu + 2Fx = 0
⇓
u = −H−1
Fx
= u∗
if H is positive definite i.e. if H 0
Here H = C
Q C + R 0 if:

R 0 Q, P 0 or
R 0 Q, P 0 C is full-rank
m
(A, B) controllable
Receding horizon controller is linear state feedback:
uk = −

I 0 · · · 0

H−1
Fxk
is the closed loop response optimal? is it even stable?
2 - 8

Example
Model: A, B, C as before, cost: Jk =
N−1
X
i=0
y2
i|k + 0.01u2
i|k

+ y2
N|k
I For N = 4: u∗
k = −H−1
Fxk =



−4.36 −18.7
1.64 1.24
1.41 3.00
0.59 1.83


xk
uk =

−4.36 −18.7

xk
I For general N: uk = L(N)xk
N = 4 N = 3 N = 2 N = 1
L(N)

−4.36 −18.69

−3.80 −16.98

1.22 −3.95

5.35 5.10

λ A + BL(N)

0.29 ± 0.17j 0.36 ± 0.22j 1.36, 0.38 2.15, 0.30
stable stable unstable unstable
2 - 9

Example
Horizon: N = 4, x0 = (0.5, −0.5)
0 2 4 6 8 10
−2
0
2
4
6
8
u
0 2 4 6 8 10
−1
−0.5
0
0.5
1
sample, k
y
closed−loop
predicted
2 - 10

Example
Horizon: N = 3, x0 = (0.5, −0.5)
0 2 4 6 8 10
−5
0
5
10
u
0 2 4 6 8 10
−1
−0.5
0
0.5
1
sample, k
y
closed−loop
predicted
2 - 10

Example
Horizon: N = 2, x0 = (0.5, −0.5)
0 2 4 6 8 10
−10
−5
0
5
u
0 2 4 6 8 10
−5
0
5
10
15
sample, k
y
closed−loop
predicted
2 - 10

Example
Horizon: N = 2, x0 = (0.5, −0.5)
0 2 4 6 8 10
−80
−60
−40
−20
0
20
u
0 2 4 6 8 10
−5
0
5
10
15
sample, k
y
closed−loop
predicted
Observation: predicted and closed loop responses are different for small N
2 - 10

Receding horizon control
Why is this example unstable for N ≤ 2?
System is non-minimum phase
⇓
impulse response changes sign
⇓
hence short horizon causes instability
N 3
C21 Model Predictive Control 2 - 12
Unconstrained optimization: example
• Unstable for
short horizon closed-loop
instability
impulse response changes sign
non-minimum phase
System is
• Solution:
use infinite cost horizon
but retain finite number of d.o.f. in predictions
N =
Solution:
? use an infinite horizon cost
? but keep a finite number of optimization variables in predictions
2 - 11

Dual mode predictions
An infinite prediction horizon is possible with dual mode predictions:
ui|k =

optimization variables i = 0, . . . , N − 1, mode 1
Kxi|k i = N, N + 1, . . . mode 2
mode 1
optimized explicitly
mode 2
feedback law: u = Kx
Feedback gain K: stabilizing and determined offline
e.g. unconstrained LQ optimal for
P∞
i=0(kxik2
Q + kuik2
R)
2 - 12

Infinite horizon cost
If the predicted input sequence is
{u0|k, . . . , uN−1|k, KxN|k, KΦxN|k, . . .}
then
∞
X
i=0
kxi|kk2
Q + kui|kk2
R) =
N−1
X
i=0
kxi|kk2
Q + kui|kk2
R) + kxN|kk2
P
where
P − (A + BK)
P(A + BK) = Q + K
RK
Lyapunov matrix equation (discrete time)
? If Q + K
RK 0, then the solution P is unique and P 0
? Matlab: P = dlyap(Phi’,RHS);
Phi = A+B*K; RHS = Q+K’*R*K;
? P is the steady state Riccati equation solution if K is LQ optimal
2 - 13

Proof that the predicted cost over the mode 2 horizon is kxN|kk2
P :
Let J∞
(x) =
∞
X
i=0
kxik2
Q + kuik2
R

, with ui = Kxi, xi+1 = Φxi ∀i
x0 = x
– then J∞
(x) =
∞
X
i=0
x
Φi
QΦi
x + x
K
Φi
RKΦi
x

= x
∞
X
i=0
(Φi
)
(Q + K
RK)Φi
| {z }
=P

x = kxk2
P
– but Φ
PΦ =
∞
X
i=1
(Φi
)
(Q + K
RK)Φi
= P − (Q + K
RK)
so P − Φ
PΦ = Q + K
RK
2 - 14

Connection with LQ optimal control
Let J(xk, uk) =
N−1
X
i=0
kxi|kk2
Q + kui|kk2
R

+ kxN|kk2
P
P − (A + BK)
P(A + BK) = Q + K
RK, K = LQ optimal
Then the solution of the unconstrained optimization satisfies
u∗
0|k = Kxk where u∗
k = arg min
u
J(xk, u) = (u∗
0|k, . . . , u∗
N−1|k)
since
{u0|k, u1,k, . . .} is optimal iff

uk = {u0|k, . . . , uN−1|k} is optimal
and {uN|k, uN+1|k, . . .} is optimal
2 - 15

Connection with LQ optimal control – example
I Model parameters (A, B, C) as before
LQ optimal gain for Q = C
C, R = 0.01: K =

−4.36 −18.74

Lyapunov equation solution: P =

3.92 4.83
13.86

I Cost matrices for N = 4:
H =




1.44 0.98 0.59 0.26
? 0.72 0.44 0.20
? ? 0.30 0.14
? ? ? 0.096



 F =




3.67 23.9
2.37 16.2
1.36 9.50
0.556 4.18



 G =

13.8 66.7
? 413

I Predictive control law: uk = −

1 0 0 0

H−1
Fxk
=

−4.35 −18.74

xk
2 - 16

Connection with LQ optimal control – example
I Response for N = 4, x0 = (0.5, −0.5)
0 2 4 6 8 10
u
-2
0
2
4
6
8
sample, k
0 2 4 6 8 10
y
-1
-0.5
0
0.5
closed loop
predicted
no constraints

=⇒ identical predicted and closed loop responses
2 - 17

Pre-stabilize predictions to provide better numerical stability:
B Vectorized form: xk = Mxk + Cck
xk :=



x0|k
.
.
.
xN|k


 , ck :=



c0|k
.
.
.
cN−1|k



M =







I
Φ
Φ2
.
.
.
ΦN







, C =







0 0 · · · 0
B
ΦB B
.
.
.
.
.
.
...
ΦN−1
B ΦN−2
B · · · B







B Cost: J xk, (u0|k, . . . , uN−1|k)

= J (xk, ck)
2 - 18

Input and state constraints
Infinite horizon unconstrained MPC = LQ optimal control
but MPC can also handle constraints
Consider constraints applied to mode 1 predictions:
? input constraints: u ≤ ui|k ≤ u, i = 0, . . . , N − 1
⇐⇒

I
−I

uk ≤

u
−u

where
u =

u
· · · u

u =

u
· · · u

? state constraints: x ≤ xi|k ≤ x, i = 1, . . . , N
⇐⇒

Ci
−Ci

uk ≤

x
−x

+

−Ai
Ai

xk, i = 1, . . . , N
2 - 19

Input and state constraints
Constraints on mode 1 predictions can be expressed
Acuk ≤ bc + Bcxk
where Ac, Bc, bc can be computed offline since model is time-invariant
The online optimization is a quadratic program (QP):
minimize
u
u
Hu + 2x
k F
u
subject to Acu ≤ bc + Bcxk
which is a convex optimization problem with a unique solution if
H = C
QC + R is positive definite
2 - 20

QP solvers: (a) Active set
Consider the QP: u∗
= arg min
u
u
Hu + 2f
u
subject to Au ≤ b
and let (Ai, bi) = ith row/element of (A, b)
B Individual constraints are active or inactive
active inactive
Aiu∗
= bi, ∀i ∈ I Aiu∗
≤ bi, ∀i 6∈ I
bi affects solution bi does not affect solution
B Equality constraint problem: u∗
= arg min
u
u
Hu + 2f
u
subject to Aiu = bi, ∀i ∈ I
B Solve QP by searching for I
? one equality constraint problem solved at each iteration
? optimality conditions (KKT conditions) identify solution
2 - 21

Active constraints – example
−3 −2 −1 0 1 2
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
u1
u
2
u∗
i =
4
i = 5
i
=
3
i = 2
i = 1
A QP problem with 5 inequality constraints
active set at solution: I = {2}
2 - 22

Active constraints – example
−3 −2 −1 0 1 2
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
u1
u
2
u∗
i = 2
An equivalent equality constraint problem
2 - 22

QP solvers: (a) Active set
B Computation:
O(N3
n3
u) additions/multiplications per iteration (conservative estimate)
upper bound on number of iterations is exponential in problem size
B At each iteration choose trial active set using: cost gradient
constraint sensitivities
⇓
number of iterations needed is often small in practice
B In MPC u∗
k = u∗
(xk) and Ik = I(xk)
hence initialize solver at time k using the solution computed at k − 1
2 - 23

QP solvers: (b) Interior point
B Solve an unconstrained problem at each iteration:
u(µ) = min
u
µ u
Hu + 2f
u

+ φ(u)
where
φ(u) = barrier function (φ → ∞ at constraints)
u → u∗
as µ → ∞
Increase µ until φ(u∗
) 1/ ( = user-defined tolerance)
B # arithmetic operations per iteration is constant, e.g. O(N3
n3
u)
# iterations for given is polynomial in problem size
⇓
Computational advantages for large-scale problems
e.g. # variables 102
, # constraints 103
B No general method for initializing at solution estimate
2 - 24

Interior point method – example
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
0
1
2
3
4
5
6
7
8
9
10
u
u(0.5)
u(1)
u(2)
φ(u)
A
A
K
u
Hu + 2f
u
6
u(µ) → u∗
= 1 as µ → ∞
but minu µ u
Hu + 2f
u

+ φ(u) becomes ill-conditioned as µ → ∞
2 - 25

QP solvers: (c) Multiparametric
Let u∗
(x) = arg min
u
u
Hu + 2x
F
u
subject to Au ≤ b + Bx
then:
? u∗
is a continous function of x
? u∗
(x) = Kjx + kj for all x in a polytopic set Xj
B In principle each Kj, kj and Xj can be determined offline
B Large number of sets Xj (combinatorial in problem size)
so online determination of j such that xk ∈ Xj is difficult
2 - 26

Multiparametric QP – example
−8 −6 −4 −2 0 2 4 6 8
−1.5
−1
−0.5
0
0.5
1
1.5
x1
x
2
Model: (A, B, C) as before,
cost: Q = C
C, R = 1, horizon: N = 10
constraints: −1 ≤ u ≤ 1, −1 ≤ x/8 ≤ 1
2 - 27

Summary
B Predicted control inputs: uk =



u0|k
.
.
.
uN−1|k



and states: xk =



x1|k
.
.
.
xN|k


 = Mxk + Cuk
B Predicted cost: J(xk, uk) =
N−1
X
i=0
kxi|kk2
Q + kui|kk2
R

+ kxN|kk2
P
= u
k Huk + 2x
k F
uk + x
k Gxk
B Online optimization subject to linear state and input constraints is a QP:
minimize
u
u
Hu + 2x
k F
u
subject to Acu ≤ bc + Bcxk
2 - 28

Lecture 3
Closed loop properties of MPC
3 - 1

Closed loop properties of MPC
Review: infinite horizon cost
Infinite horizon predictive control with constraints
Closed loop stability
Constraint-checking horizon
Connection with constrained optimal control
3 - 2

Short prediction horizons cause poor performance and instability, so
? use an infinite horizon cost: J(xk, uk) =
∞
X
i=0
kxi|kk2
Q + kui|kk2
R

? keep optimization finite-dimensional by using dual mode predictions:
ui|k =

optimization variables i = 0, . . . , N − 1, mode 1
Kxi|k i = N, N + 1, . . . mode 2
mode 1: uk =



u0|k
.
.
.
uN−1|k


 uk optimized online
mode 2: ui|k = Kxi|k K chosen offline
3 - 3

Review: MPC online optimization
B Unconstrained optimization: ∇uJ(x, u∗
) = 2Hu∗
+ 2Fx = 0, so
u∗
(x) = −H−1
Fx
=⇒ linear controller: uk = KMPCxk
KMPC = LQ-optimal if K = LQ-optimal (in mode 2)
B Constrained optimization:
u∗
(x) = arg min
u
u
Hu + 2x
F
u
subject to Acu ≤ bc + Bcx
=⇒ nonlinear controller: uk = KMPC(xk)
3 - 5

Constrained MPC – example
B Plant model: xk+1 = Axk + Buk, yk = Cxk
A =

1.1 2
0 0.95

, B =

0
0.0787

, C =

−1 1

Constraints: −1 ≤ uk ≤ 1
B MPC optimization (constraints applied only to mode 1 predictions):
minimize
u
N−1
X
i=0
kxi|kk2
Q + kui|kk2
R

+ kxN|kk2
P
subject to − 1 ≤ ui|k ≤ 1, i = 0, . . . , N − 1
Q = C
C, R = 0.01, N = 2
. . . performance? stability?
3 - 6

Closed loop response for x0 = (0.8, −0.8)
0 20 40 60 80 100
−1
−0.5
0
0.5
1
u
0 20 40 60 80 100
−200
−150
−100
−50
0
50
sample, k
y
unstable
3 - 7

Closed loop response for x0 = (0.5, −0.5)
0 20 40 60 80 100
−1
−0.5
0
0.5
1
u
0 20 40 60 80 100
−2
0
2
4
6
sample, k
y
stable, but . . .
3 - 8

Optimal predicted cost x0 = (0.5, −0.5)
0 20 40 60 80 100
0
10
20
30
40
50
60
70
80
90
sample, k
Predicted
cost
J(k)
. . . increasing Jk =⇒ closed loop response does not follow predicted trajectory
3 - 9

Stability analysis
How can we guarantee the closed loop stability of MPC?
(a). Show that a Lyapunov function exists demonstrating stability
(b). Ensure that optimization feasible is at each time k = 0, 1, . . .
B For Lyapunov stability analysis:
? consider first the unconstrained problem
? use predicted cost as a trial Lyapunov function
B Guarantee feasibility of the MPC optimization recursively
by ensuring that feasibility at time k implies feasibility at k + 1
3 - 10

Discrete time Lyapunov stability
Consider the system xk+1 = f(xk), with f(0) = 0
B Definition: x = 0 is a stable equilibrium point if
maxk kxkk can be made arbitrarily small
by making x0 sufficiently small
B If continuously differentiable V (x) exists with
(i). V (x) is positive definite and
(ii). V (xk+1) − V (xk) ≤ 0
then x = 0 is a stable equilibrium point
3 - 11

B Definition: x = 0 is a stable equilibrium point if
for all R 0 there exists r such that
kx0k r =⇒ kxkk R for all k
(ii). V (xk+1) − V (xk) ≤ 0
then x = 0 is a stable equilibrium point
3 - 11

B Definition: x = 0 is an asymptotically stable equilibrium point if
(i). x = 0 is stable and
(ii). r exists such that kx0k r =⇒ lim
k→∞
xk = 0
(ii). V (xk+1) − V (xk) 0 whenever xk 6= 0
then x = 0 is an asymptotically stable equilibrium point
3 - 12

Lyapunov stability
Trial Lyapunov function:
J∗
(xk) = J(xk, u∗
k)
where J(xk, uk) =
∞
X
i=0
kxi|kk2
Q + kui|kk2
R

? J∗
(x) is positive definite if:
(a). R 0 and Q 0, or
(b). R 0 and Q 0 and (A, Q) is observable
since then J∗
(xk) ≥ 0 and J∗
(xk) = 0 if and only if xk = 0
? J∗
(x) is continuously differentiable
. . . from analysis of MPC optimization as a multiparametric QP
3 - 13

Lyapunov stability
Construct a bound on J∗
(xk+1) − J∗
(xk) using the “tail” of the
optimal prediction at time k
u
i
optimal at k
tail at k + 1
0 1 N −1 N
Optimal predicted sequences at time k:
u∗
k =











u∗
0|k
u∗
1|k
.
.
.
u∗
N−1|k
Kx∗
N|k
.
.
.











x∗
k =











x∗
0|k
x∗
1|k
.
.
.
x∗
N|k
Φx∗
N|k
.
.
.











(Φ = A + BK)
optimal at k : J∗
(xk) = J(xk, u∗
k) =
∞
X
i=0
kx∗
i|kk2
Q + ku∗
i|kk2
R

tail at k + 1 : ˜
J(xk+1) = J(xk+1, ũk+1) =
∞
X
i=1
kx∗
i|kk2
Q + ku∗
i|kk2
R

3 - 14

Lyapunov stability
(xk+1) − J∗
Predicted cost for the tail:
˜
J(xk+1) = J∗
(xk) − kxkk2
Q − kukk2
R
but ũk+1 is suboptimal at time k + 1, so
J∗
(xk+1) ≤ ˜
J(xk+1)
Therefore
J∗
(xk+1) ≤ J∗
(xk) − kxkk2
Q − kukk2
R
3 - 15

Lyapunov stability
The bound J∗
(xk+1) − J∗
(xk) ≤ −kxkk2
Q − kukk2
R implies:
(i). the closed loop cost cannot exceed the initial predicted cost,
since summing both sides over all k ≥ 0 gives
∞
X
k=0
kxkk2
Q + kukk2
R

≤ J∗
(x0)
(ii). x = 0 is asymptotically stable
? if R 0 and Q 0, this follows from Lyapunov’s direct method
? if R 0, Q 0 and (A, Q) observable, this follows from:
(a). stability of x = 0 ⇐ Lyapunov’s direct method
(b). lim
k→∞
(kxkk2
Q + kukk2
R) = 0 ⇐
P∞
k=0 kxkk2
Q + kukk2
R

∞
3 - 16

Stability analysis
How can we guarantee the closed loop stability of MPC?
(a). Show that a Lyapunov function exists demonstrating stability
(b). Ensure that optimization feasible is at each time k = 0, 1, . . .
B For Lyapunov stability analysis:
? consider first the unconstrained problem
? use predicted cost as a trial Lyapunov function
B Guarantee feasibility of the MPC optimization recursively
by ensuring that feasibility at time k =⇒ feasibility at k + 1
3 - 17

Terminal constraint
The basic idea
terminal set
origin
stabilizing linear controller satisfies constraints

more sophisticated controller needed to satisfy constraints

?
@
@
@
@
R
3 - 18

Terminal constraint
Terminal constraint: xN|k ∈ Ω, where Ω = terminal set
xk
x1|k
xN−1|k
xN|k
xN+1|k
safe region
for mode 2
control law
Choose Ω so that:
(a). x ∈ Ω =⇒

u ≤ Kx ≤ u
x ≤ x ≤ x
(b). x ∈ Ω =⇒ (A + BK)x ∈ Ω
then Ω is invariant for the mode 2 dynamics and constraints, so
xN|k ∈ Ω =⇒

u ≤ ui|k ≤ u
x ≤ xi|k ≤ x
for i = N, N + 1, . . .
i.e. constraints are satisfied over
the infinite mode 2 prediction horizon
3 - 19

Stability of constrained MPC
Prototype MPC algorithm
At each time k = 0, 1, . . .
(i). solve u∗
k = arg min
uk
J(xk, uk)
s.t. u ≤ ui|k ≤ u, i = 0, . . . , N − 1
x ≤ xi|k ≤ x, i = 1, . . . , N
xN|k ∈ Ω
(ii). apply uk = u∗
0|k to the system
Asymptotically stabilizes x = 0 with region of attraction FN ,
FN =





x0 : ∃ {u0, . . . , uN−1} such that
u ≤ ui ≤ u, i = 0, . . . , N − 1
x ≤ xi ≤ x, i = 1, . . . , N
xN ∈ Ω





= the set of all feasible initial conditions for N-step horizon
and terminal set Ω
3 - 20

Terminal constraints
Make Ω as large as possible so that the feasible set FN is maximized, i.e.
Ω = X∞ = lim
j→∞
Xj
where
? Xj = initial conditions for which constraints are satisfied for j steps
with u = Kx
=

x :
u ≤ K(A + BK)i
x ≤ u
x ≤ (A + BK)i
x ≤ x
i = 0, . . . , j

? X∞ = Xν for some finite ν if |eig(A + BK)| 1
⇓
x ∈ X∞ if constraints are satisfied on a finite constraint checking horizon
3 - 21

Terminal constraints – Example
Plant model: xk+1 = Axk + Buk, yk = Cxk
A =

1.1 2
0 0.95

B =

0
0.0787

C =

−1 1

input constraints: −1 ≤ uk ≤ 1
mode 2 feedback law: K =

−1.19 −7.88

= KLQ for Q = C
C, R = 1
3 - 22

Terminal constraints – example
Constraints: −1 ≤ u ≤ 1
−5 0 5
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
x1
x
2
X0
Kx = −1

Kx = 1 -
3 - 23

−5 0 5
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
x1
x
2
X1
KΦx = −1

KΦx = 1 -
Kx = −1

Kx = 1 -
3 - 23

−5 0 5
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
x1
x
2
X2
KΦ2
x = −1

KΦ2
x = 1 -
KΦx = −1

KΦx = 1 -
Kx = −1

Kx = 1 -
3 - 23

−5 0 5
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
x1
x
2
X∞
X4 = X5 = · · · = Xj for all j 4 so X4 = X∞
3 - 23

In this example X∞ is determined in a finite number of steps because
A (A + BK) is strictly stable, and
B (A + BK), K

is observable
A ⇒

shortest distance of hyperplane
K(A + BK)i
x ≤ 1 from origin

=
1
kK(A + BK)ik2
→ ∞ as i → ∞
B ⇒ X∞ is bounded because x0 /
∈ X∞ if x0 is sufficiently large
Here {x : −1 ≤ K(A + BK)i
x ≤ 1} contains X4 for i 4
⇓
X∞ = X4
constraint checking horizon: ν = 4
3 - 24

Terminal constraints
General case
Let Xj = {x : FΦi
x ≤ 1, i = 0, . . . j} with

Φ strictly stable
(Φ, F) observable
then:
(i). X∞ = Xν for finite ν
(ii). Xν = X∞ iff x ∈ Xν+1 whenever x ∈ Xν
Proof of (ii)
(a). for any j, Xj+1 = Xj ∩

x : FΦj+1
x ≤ 1
so Xj ⊇ Xj+1 ⊇ limj→∞ Xj = X∞
(b). if x ∈ Xν+1 whenever x ∈ Xν , then Φx ∈ Xν whenever x ∈ Xν
but Xν ⊆

x : Fx ≤ 1 and it follows that Xν ⊆ X∞
(a) (b) ⇒ Xν = X∞
3 - 25

Terminal constraints – constraint checking horizon
Algorithm for computing constraint checking horizon Nc
for input constraints u ≤ u ≤ u:
C4A Model Predictive Control 3 - 28
Terminal constraint set
Algorithm (computation of ) :
umax := max
x
K(A+ BK)N +1
x s.t. u ! K(A+ BK)i
x ! u, i = 0,…N
umin := min
x
K(A+ BK)N +1
x s.t. u ! K(A+ BK)i
x ! u, i = 0,…N
Nc := N
N := 0
umax ! u ?
and
umin u?
N := N +1
no
yes
2 linear programs
solved at each step
3 - 26

Constrained MPC
Define the terminal set Ω as XNc
MPC algorithm
At each time k = 0, 1, . . .
(i). solve u∗
k = arg min
uk
J(xk, uk)
s.t. u ≤ ui|k ≤ u, i = 0, . . . , N + Nc
x ≤ xi|k ≤ x, i = 1, . . . , N + Nc
(ii). apply uk = u∗
0|k to the system
Note
? predictions for i = N, . . . N + Nc:

xi|k = (A + BK)i−N
xN|k
ui|k = K(A + BK)i−N
xN|k
? xN|k ∈ XNc
implies linear constraints so online optimization is a QP
3 - 27

Closed loop performance
Longer horizon N ensures improved predicted cost J∗
(x0)
and is likely (but not certain) to give better closed-loop performance
Example: Cost vs N for x0 = (−7.5, 0.5)
N 6 7 8 11 11
J∗
(x0) 364.2 357.0 356.3 356.0 356.0
Jcl(x0) 356.0 356.0 356.0 356.0 356.0
Closed loop cost: Jcl(x0) :=
P∞
k=0 kxkk2
Q + kukk2
R

For this initial condition:
MPC with N = 11 is identical to constrained LQ optimal control (N = ∞)!
3 - 28

Closed loop performance – example
Predicted and closed loop inputs for N = 6
0 5 10 15 20 25 30
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
time step, k
input,
u
predicted
closed−loop
3 - 29

Predicted and closed loop states for N = 6
−8 −7 −6 −5 −4 −3 −2 −1 0 1
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x
1
x
2
predicted
closed−loop
Ω
F6
x6|0 -
3 - 29

Predicted and closed loop states for N = 11
x
1
-8 -7 -6 -5 -4 -3 -2 -1 0 1
x
2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
predicted
closed-loop
Ω
F6
x11|0
3 - 29

Choice of mode 1 horizon – performance
B For this x0: N = 11 ⇒ xN|0 lies in the interior of Ω
m
terminal constraint is inactive
⇓
no reduction in cost for N 11
B Constrained LQ optimal performance is always obtained with N ≥ N∞
for some finite N∞ dependent on x0
B N∞ may be large, implying high computational load
but closed loop performance is often close to optimal for N N∞
(due to receding horizon)
in this example Jcl(x0) ≈ optimal for N ≥ 6
3 - 30

Choice of mode 1 horizon – region of attraction
Increasing N increases the feasible set FN
−10 −8 −6 −4 −2 0 2 4 6 8 10
−1.5
−1
−0.5
0
0.5
1
1.5
x
1
x
2
N = 26
N = 24
N = 20
N = 16
N = 12
N = 8
N = 4
N = 0
Ω
F4

F8

F12

3 - 31

Summary
B Linear MPC ingredients:
? Infinite cost horizon (via terminal cost)
? Terminal constraints (via constraint-checking horizon)
B Constraints are satisfied over an infinite prediction horizon
B Closed-loop system is asymptotically stable
with region of attraction equal to the set of feasible initial conditions
B Ideal optimal performance if mode 1 horizon N is large enough
3 - 32

Lecture 4
Robustness to disturbances
4 - 1

Robustness to disturbances
Review of nominal model predictive control
Setpoint tracking and integral action
Robustness to unknown disturbances
Handling time-varying disturbances
4 - 2

Review
MPC with guaranteed stability – the basic idea
terminal set
origin
stabilizing linear controller satisfies constraints

more sophisticated controller needed to satisfy constraints

?
@
@
@
@
R
4 - 3

Review
MPC optimization for linear model xk+1 = Axk + Buk
minimize
uk
N−1
X
i=0
kxi|kk2
Q + kui|kk2
R

+ kxN|kk2
P
subject to u ≤ ui|k ≤ u, i = 0, . . . , N + Nc
x ≤ xi|k ≤ x, i = 1, . . . , N + Nc
where
? ui|k = Kxi|k for i ≥ N, with K = unconstrained LQ optimal
? terminal cost: kxN|kk2
P =
∞
X
i=N
(kxi|kk2
Q + kui|kk2
R), with
P − ΦT
PΦ = Q + KT
RK, Φ = A + BK
? terminal constraints are defined by the constraint checking horizon Nc:
u ≤ KΦi
x ≤ u
x ≤ Φi
x ≤ x

i = 0, . . . , Nc =⇒

u ≤ KΦNc+1
x ≤ u
x ≤ ΦNc+1
x ≤ x
4 - 4

Comparison
B Linear MPC
terminal cost ←− exact cost over the mode 2 horizon
terminal constraint set ←− contains all feasible initial conditions
for mode 2
B Nonlinear MPC
terminal cost ←− upper bound on cost over
mode 2 horizon
terminal constraint set ←− invariant set (usually not the largest)
for mode 2 dynamics and constraints
4 - 6

Model uncertainty
Uncertain
model,
robust
constraints
Stochastic
model,
probabilistic
constraints
Nominal,
unconstrained
Nominal,
constrained
4 - 7

Model uncertainty
Uncertain
model,
robust
constraints
Stochastic
model,
probabilistic
constraints
Nominal,
unconstrained
LQ-optimal
control
Classical
MPC
Robust
MPC
Stochastic
MPC
4 - 7

Model uncertainty
Common causes of model error and uncertainty:
I Unknown or time-varying model parameters
B unknown loads inertias, static friction
B unknown d.c. gain
I Random (stochastic) model parameters
B random process noise or sensor noise
I Incomplete measurement of states
B state estimation error
4 - 7

Setpoint tracking
I Output setpoint: y0
y → y0
⇒

x → x0
u → u0
where
x0
= Ax0
+ Bu0
y0
= Cx0
⇓
y0
= C(I − A)−1
Bu0
I Setpoint for (u0
, x0
) is unique iff C(I − A)−1
B is invertible
e.g. if dim(u) = dim(y), then

u0
= C(I − A)−1
B
−1
y0
x0
= (I − A)−1
Bu0
I Tracking problem: yk → y0
subject to

u ≤ uk ≤ u
x ≤ xk ≤ x
is only feasible if u ≤ u0
≤ u and x ≤ x0
≤ x
4 - 8

Setpoint tracking
I Unconstrained tracking problem:
minimize
uδ
k
∞
X
i=0
kxδ
i|kk2
Q + kuδ
i|kk2
R

where xδ
= x − x0
uδ
= u − u0
has optimal solution: uk = Kxδ
k + u0
, K = KLQ
I Constrained tracking problem:
minimize
uδ
k
∞
X
i=0
kxδ
i|kk2
Q + kuδ
i|kk2
R

subject to u ≤ uδ
i|k + u0
≤ u, i = 0, 1, . . .
x ≤ xδ
i|k + x0
≤ x, i = 1, 2, . . .
has optimal solution: uk = uδ∗
0|k + u0
4 - 9

Setpoint tracking
If û0
is used instead of u0
(e.g. if d.c. gain C(I − A)−1
B unknown)
then uk = uδ∗
0|k + û0
implies
uδ
k = uδ∗
0|k + (û0
− u0
)
xδ
k+1 = Axδ
k + Buδ∗
0|k + B (û0
− u0
)
| {z }
constant disturbance
and if uδ∗
0|k → Kxδ
k as k → ∞, then
lim
k→∞
xδ
k = (I − A − BK)−1
B(û0
− u0
) 6= 0
lim
k→∞
yk − y0
= C(I − A − BK)−1
B(û0
− u0
)
| {z }
steady state tracking error
6= 0
4 - 10

Additive disturbances
Convert (constant) setpoint tracking problem into a regulation problem:
x ← xδ
, y ← yδ
, u ← uδ
Consider the effect of additive disturbance w:
xk+1 = Axk + Buk + Dwk,
yk = Cxk
Assume that wk is unknown at time k, but is known to be:
? constant (wk = w for all k) or time-varying
? within a known polytopic set: wk ∈ W for all k
where W = conv{w(1)
, . . . , w(r)
}
or W = {w : Hw ≤ 1}
4 - 11

Additive disturbances
Convert (constant) setpoint tracking problem into a regulation problem:
x ← xδ
, y ← yδ
, u ← uδ
Consider the effect of additive disturbance w:
xk+1 = Axk + Buk + Dwk,
yk = Cxk
Assume that wk is unknown at time k, but is known to be:
? constant (wk = w for all k) or time-varying
? within a known polytopic set: wk ∈ W for all k
where W = conv{w(1)
, . . . , w(r)
}
or W = {w : Hw ≤ 1}
W
w(1)
w(2)
w(3)
w(4)
w(5)
w(6)
4 - 11

Integral action (no constraints)
Introduce integral action to remove steady state error in y
by considering the augmented system:
zk =

xk
vk

, zk+1 =

A 0
C I

zk +

B
0

uk +

D
0

wk
vk = integrator state
vk+1 = vk + yk
? Linear feedback uk = Kxk + KIvk
is stabilizing if eig

A + BK BKI
C I

1
? If the closed-loop system is (strictly) stable and wk → w = constant
then uk → uss
=⇒ vk → vss
=⇒ yk → 0 even if w 6= 0
but arbitrary KI may destabilize the closed loop system
4 - 12

Integral action (no constraints)
Ensure stability by using a modified cost:
minimize
uk
∞
X
i=0
kzi|kk2
Qz
+ kui|kk2
R

Qz =

Q 0
0 QI

0
with predictions generated by an augmented model
zi+1|k =

A 0
C I

zi|k +

B
0

ui|k, z0|k =

xk
vk

? this is a “nominal” prediction model since wk = 0 is assumed
? unconstrained solution: uk = Kzzk = Kxk + KIvk
? if R 0,

A 0
C I

,

Q 0
0 QI

is observable and wk → w = constant
then uk → uss
=⇒ vk → vss
=⇒ yk → 0
4 - 13

Integral action – example
Plant model:
xk+1 = Axk + Buk + Dw yk = Cxk
A =

1.1 2
0 0.95

B =

0
0.0787

D =

1
0

C =

−1 1

Constraints: none
Cost weighting matrices: Qz =

CT
C 0
0 0.01

, R = 1
Unconstrained LQ optimal feedback gain:
Kz =

−1.625 −9.033 0.069

4 - 14

+ integrator
no integrator
0 10 20 30 40 50
u
-1
0
1
2
3
4
+ integrator
no integrator
sample k
0 10 20 30 40 50
y
-1
-0.5
0
0.5
1
1.5
Closed loop response for initial condition: x0 = (0.5, −0.5)
no disturbance: w = 0
4 - 15

+ integrator
no integrator
0 10 20 30 40 50
u
-2
-1
0
1
2
3
4
+ integrator
no integrator
sample k
0 10 20 30 40 50
y
-5
-4
-3
-2
-1
0
Closed loop response for initial condition: x0 = (0.5, −0.5)
constant disturbance: w = 0.75
4 - 16

Constrained MPC
Naive constrained MPC strategy: w = 0 assumed in predictions
minimize
uk
N−1
X
i=0
kzi|kk2
Qz
+ kui|kk2
R

+ kzN|kk2
P
subject to u ≤ ui|k ≤ u, i = 0, . . . , N + Nc
x ≤ xi|k ≤ x, i = 1, . . . , N + Nc
with: P and Nc determined for mode 2 control law ui|k = Kzzi|k
and initial prediction state: z0|k =

xk
vk

where vk+1 = vk + yk
? If closed loop system is stable
then uk → uss
=⇒ vk → vss
=⇒ yk → 0
? but disturbance wk is ignored in predictions, so

J∗
(zk+1) − J∗
(zk) 6≤ 0
feasibility at time k 6⇒ feasibility at k + 1
therefore no guarantee of stability
4 - 17

constraints
violated

*
Closed loop response with initial condition: x0 = (0.5, −0.5)
constraints: −1 ≤ u ≤ 1 disturbance: w = 0.75
4 - 18

Pre-stabilized predictions – example
Scalar system: xk+1 = 2xk + uk + wk, constraint: |xk| ≤ 2
uncertainty: ei|k =
i−1
X
j=0
2j
w = (2i
− 1)w, disturbance: wk = w
|w| ≤ 1
4 - 20

Scalar system: xk+1 = 2xk + uk + wk, constraint: |xk| ≤ 2
uncertainty: ei|k =
i−1
X
j=0
2j
w = (2i
− 1)w, disturbance: wk = w
|w| ≤ 1
Robust constraints:
|si|k + ei|k| ≤ 2 for all |w| ≤ 1
m
|si|k| ≤ 2 − max|w|≤1|ei|k|
⇓
|si|k| ≤ 2 − (2i
− 1)
⇓
infeasible for all i 1 sample
0 1 2 3 4 5
-10
-8
-6
-4
-2
0
2
4
6
8
10
upper bound on ei|k
@
@
R
lower bound on ei|k

4 - 20

Avoid infeasibility by using pre-stabilized predictions:
ui|k = Kxi|k + ci|k, K = −1.9, ci|k =
(
free i = 0, . . . , N − 1
0 i ≥ N
stable predictions: ei|k =
i−1
X
j=0
0.1j
w = (1 − 0.1i
)w/0.9, |w| ≤ 1
Robust constraints:
|si|k + ei|k| ≤ 2 for all |w| ≤ 1
m
|si|k| ≤ 2 − max|w|≤1|ei|k|
⇓
|si|k| ≤ 2 − (1 − 0.1i
)/0.9
| {z }
0 for all i
sample
0 1 2 3 4 5
-5
-4
-3
-2
-1
0
1
2
3
4
5
upper bound on ei|k
?
lower bound on ei|k
6
4 - 21

Pre-stabilized predictions
B Feedback structure of MPC with open loop predictions:
Plant
MPC
Optimization
- -
- x
u
?
w
B Feedback structure of MPC with pre-stabilized predictions:
Stablizing
feedback
Plant
MPC
Optimization
6
g
- - -

- x
u
c
Kx
+ +
?
w
4 - 22

Tube interpretation
The uncertainty in predictions: ei+1|k = Φei|k + Dwi, wi ∈ W
evolves inside a tube (a sequence of sets): ei|k ∈ Ei|k, where
Ei|k = DW ⊕ ΦDW ⊕ · · · ⊕ Φi−1
DW, i = 1, 2, . . .
Hence we can define:
? a state tube xi|k = si|k + ei|k ∈ Xi|k
Xi|k = {si|k} ⊕ Ei|k, i = 0, 1, . . .
? a control input tube ui|k = Kxi|k + ci|k = Ksi|k + ci|k + Kei|k ∈ Ui|k
Ui|k = {Ksi|k + ci|k} ⊕ KEi|k, i = 0, 1, . . .
and impose constraints robustly for the state and input tubes
(where ⊕ is Minkowski set addition)
4 - 24

Tube interpretation
The uncertainty in predictions: ei+1|k = Φei|k + Dwi, wi ∈ W
evolves inside a tube (a sequence of sets): ei|k ∈ Ei|k, where
Ei|k = DW ⊕ ΦDW ⊕ · · · ⊕ Φi−1
DW, i = 1, 2, . . .
e.g. for constraints Fx ≤ 1 (G = 0)
s0|k
X1|k

s1|k
s2|k
X2|k

s3|k
X3|k

s4|k
X4|k

sample
Fx = 1
?
Xi|k = {si|k} ⊕ Ei|k
4 - 24

Robust MPC
Prototype robust MPC algorithm
Offline: compute Nc and h1, . . . , hNc
. Online at k = 0, 1, . . .:
(i). solve c∗
k = arg min
ck
J(xk, ck)
s.t. (F + GK)si|k + Gci|k ≤ 1 − hi, i = 0, . . . , N + Nc
(ii). apply uk = Kxk + c∗
0|k to the system
? tightened linear constraints are applied to nominal predictions
? Nc is the constraint-checking horizon defined by:
(F + GK)ΦNc+1
s ≤ 1 − hNc+1
for all s satisfying (F + GK)Φi
s ≤ 1 − hi, i = 0, . . . , Nc
? the online optimization is robustly recursively feasible
4 - 25

Robust MPC
Prototype robust MPC algorithm
Offline: compute Nc and h1, . . . , hNc
. Online at k = 0, 1, . . .:
(i). solve c∗
k = arg min
ck
J(xk, ck)
s.t. (F + GK)si|k + Gci|k ≤ 1 − hi, i = 0, . . . , N + Nc
(ii). apply uk = Kxk + c∗
0|k to the system
nominal cost, evaluated assuming wi = 0 for all i:
J(xk, ck) =
∞
X
i=0
ksi|kk2
Q + kKsi|k + ci|kk2
R

= kxkk2
P + kckk2
Wc
(one possible choice)
4 - 25

Convergence of robust MPC with nominal cost
If ui|k = Kxi|k + ci|k for K = KLQ, then:
? the unconstrained solution is ck = 0, so the nominal cost is
J(xk, ck) = kxkk2
P + kckk2
Wc
and Wc is block-diagonal: Wc = diag{Pc, . . . , Pc}
? recursive feasibility ⇒ c̃k+1 = (c∗
1|k, . . . , c∗
N−1|k, 0) feasible at k + 1
? hence kc∗
k+1k2
Wc
≤ kc∗
kk2
Wc
− kc∗
0|kk2
Pc
⇒
∞
X
k=0
kc0|kk2
Pc
≤ kc∗
0k2
Wc
∞
⇒ limk→∞ c0|k = 0
? therefore uk → Kxk as k → ∞
xk → the (minimal) robustly invariant set
under unconstrained LQ optimal feedback
4 - 26

Robust MPC with constant disturbance
Assume wk = w = constant for all k
combine: pre-stabilized predictions
augmented state space model
? robust state constraints:
x ≤ xi|k ≤ x ⇐⇒ x + hi ≤ si|k ≤ x − hi
hi = max
w∈W

I 0
i−1
X
j=0
Φj

D
0

w
? robust input constraints:
u ≤ ui|k ≤ u ⇐⇒ u + h0
i ≤ Kzsi|k + ci|k ≤ u − h0
i
h0
i = max
w∈W
Kz
i−1
X
j=0
Φj

D
0

w
? Nc and hi, h0
i for i = 1, . . . , Nc can be computed offline
4 - 27

Robust MPC with constant disturbance – example
+ integrator
no integrator
0 10 20 30 40 50
u
-1
-0.5
0
0.5
1 + integrator
no integrator
sample k
0 10 20 30 40 50
y
-6
-5
-4
-3
-2
-1
0
1
Closed loop response with initial condition: x0 = (0.5, −0.5)
constraints: −1 ≤ u ≤ 1 disturbance: w = 0.75
4 - 28

Summary
B Integral action: augment model with integrated output error
include integrated output error in cost
then
(i). closed loop system is stable if w = 0
(ii). steady state error must be zero if response is stable for w 6= 0
B Robust MPC: use pre-stabilized predictions
apply constraints for all possible future uncertainty
then
(i). constraint feasibility is guaranteed at all times if initially feasible
(ii). closed loop system inherits the stability and convergence properties
of unconstrained LQ optimal control (assuming nominal cost)
4 - 29

Overview of the course
1 Introduction and Motivation
Basic MPC strategy; prediction models; input and state constraints; constraint handling:
saturation, anti-windup, predictive control
2 Prediction and optimization
Input/state prediction equations; unconstrained optimization. Infinite horizon cost; dual mode
predictions. Incorporating constraints; quadratic programming.
3 Closed loop properties
Lyapunov analysis based on predicted cost. Recursive feasibility; terminal constraints; the
constraint checking horizon. Constrained LQ-optimal control.
4 Robustness to disturbances
Setpoint tracking; MPC with integral action. Robustness to constant disturbances:
prestabilized predictions and robust feasibility. Handling time-varying disturbances.
4 - 30

9-12 MPC mode predictive control system.pdf

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