Application of a merit function based interior point method to linear model predictive control

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 2, No.2, May 2014
DOI : 10.5121/ijitmc.2014.2205 37
APPLICATION OF A MERIT FUNCTION BASED INTERIOR
POINT METHOD TO LINEAR MODEL PREDICTIVE
CONTROL
Prashant Bansode1
, D. N. Sonawane2
and Prashant Basargi3
1,2,3
Department of Instrumentation and Control Engineering, College of Engineering,
Pune, Maharashtra
ABSTRACT
This paper presents robust linear model predictive control (MPC) technique for small scale linear MPC
problems. The quadratic programming (QP) problem arising in linear MPC is solved using primal dual
interior point method. We present a merit function based on a path following strategy to calculate the step
length α, which forces the convergence of feasible iterates. The algorithm globally converges to the optimal
solution of the QP problem while strictly following the inequality constraints. The linear system in the QP
problem is solved using LDLT
factorization based linear solver which reduces the computational cost of
linear system to a certain extent. We implement this method for a linear MPC problem of undamped
oscillator. With the help of a Kalman filter observer, we show that the MPC design is robust to the external
disturbances and integrated white noise.
KEYWORDS
Model Predictive Control, Quadratic Programming, Primal Dual Interior Point Methods, Merit Function
1. INTRODUCTION
MPC is an advanced control strategy. It predicts the effect of the input control signal on internal
states and the output of the plant. At each sampling interval of this strategy, the plant output is
measured, the current state of the plant is estimated and based on these calculations a new control
signal is delivered to the plant. The purpose of the new control input is to ensure that the output
signal tracks the reference signal while satisfying the objective function of the MPC problem
without violating the given constraints, see [1]-[3]. The objective function is defined in such a
way that the output signal tracks the reference signal while it eliminates the effect of known
disturbances and noise signals to achieve closed loop control of the plant. The constraints can be
given in terms of bounds on input and output signals. In reality, these constraints can be the
physical limitations on actuator movements, often called as hard constraints. MPC strategy
handles physical constraints effectively which makes it suitable for industrial applications.
MPC problem can be formulated as a quadratic programming (QP) optimal control problem, see
[2]-[4]. This QP problem is solved at a specific sampling interval to compute a sequence of
current and future optimal control inputs from the predictions made on the current state and the
plant output over a finite horizon known as prediction horizon; see [4]. Only the current optimal
input is implemented as the plant input and the plant is updated for internal states and the plant
output. Again, at next sampling interval the updated plant information is used to formulate a new

38
optimal control problem and the process is repeated. MPC problem may require a long
sampling time depending upon computational complexities associated with the QP problem
solving algorithm. Therefore, the application of MPC is restricted to systems with slow dynamic
performance; such applications are found in chemical industries. Interior point methods are
widely accepted as the QP problem solving techniques in MPC applications. In last two decades,
many research literatures have discussed the application of interior point methods to MPC
problems with relatively faster dynamics by exploiting the structure of QP problems arising in
MPC, see [1], [5], [6], [11]. The general discussion on interior point methods is seen in [1], [5],
[6], [7], [8], [9], [10], [11]. For discussion on MPC as an optimal control strategy, see [2], [3], [4],
[12], [13]. For small scale MPC problems (with state dimension not more than five) we need not
exploit the structure of the problem, rather the effort should be in the direction to render faster
execution of QP solving techniques by introducing new step length strategies and improving
linear solvers while retaining the stability of the system, see [10], [11].
If we consider the barrier method (earlier form of interior point method) for MPC problems, it has
computational complexities associated with calculating the inverse of the Hessian matrix [9]. The
computational cost of inverting the Hessian is O(n3
), Secondly, the barrier method requires two
distinct iterations to update primal and dual variables, see [9]. Moreover, this method works only
for strictly feasible problems. If barrier methods are considered for MPC problems, the above
issues will dramatically affect the computational time of a numerical QP algorithm, as a result,
also the sampling time of the MPC. The primal dual interior point method has several advantages
over barrier methods such as updates of primal and dual variables are computed in a single
iteration, efficiency in terms of accuracy and ability to work even when problem is not strictly
feasible, and inverting the Hessian matrix is not required. Hence, it is more cost effective to select
the primal dual interior point methods for QP problems than selecting the barrier methods.
Further, faster convergence of iterates can be achieved by considering new step length strategies
in the primal dual algorithm. One of such strategies is to measure progress to the solution by
monitoring a merit function. We can measure progress to the solution between two successive
iterates using a proper merit function, see [15]-[17]. We consider a logarithmic merit function that
contains all the possible information required to minimize a convex quadratic objective function.
This paper discusses the primal dual interior point method to solve linear model predictive control
problems with convex quadratic objective function and linear inequality constraints on the control
input. The proposed method utilizes a log barrier penalty function as a merit function which
estimates the progress to the solution and forces convergence of the primal dual feasible iterates,
hence making the algorithm to execute faster while strictly maintaining the feasibility. To solve a
linear system for computation of Newton steps, we use LDLT
factorization linear solver which
reduces the computational cost of the linear solver from O(n3
) to its O((1/3)n3
), see [9]. The step
length selection is based on the mathematical condition derived using the merit function, and only
that step length value is selected for which a sufficient decrease in the derived condition is
observed. Finally, we implement this algorithm to a problem of undamped oscillator described in
[3] using MATLAB platform. In results, we show that the proposed method solves a MPC
problem within the specified sampling interval. Secondly, using Kalman filter observer we also
prove that our MPC design is robust in terms of disturbance and noise rejection.
We organize the paper as follows: Section 2 describes linear MPC plant model and its QP
formulation. In section 3, we discuss the proposed QP solving algorithm; section 4 includes MPC
implementation on the undamped oscillator problem with simulation results. Section 5 concludes
our paper.

39
2. MPC PROBLEM FORMULATION
The linear MPC comprises a linear plant model, convex quadratic objective function and linear
inequality constraints. The plant model is described in the section below.
2.1. Plant Model
We assume a state space model of the plant as given below:
.
,
1
t
t
t
t
t
t
t
t
v
Du
Cx
y
Gw
Bu
Ax
x
+
+
=
+
+
=
+
(1)
Where .
,
, y
u
x
n
t
n
t
n
t R
y
R
u
R
x ∈
∈
∈
Further, we assume that the plant is subjected to white noise disturbances i.e. unbiased process
noise wt and measurement noise vt which are Gaussian distributed with zero mean. We design a
Kalman filter observer to calculate the estimates of the current state and the plant output. If Np is
a prediction horizon, MPC computes these estimates over the entire prediction horizon from time
t+1 until time t+Np based on the information about previous plant measurements available from t-
1 up to current time t. Let us represent the optimal estimates of the state space equations as given
below:
.
,
1
i
t
i
t
i
t
i
t
i
t
i
t
i
t
i
t
v
Du
x
C
y
Gw
Bu
x
A
x
+
+
+
+
+
+
+
+
+
+
+
=
+
+
=
(2)
Where, i =1,…, Np.
2.2. Control Objective
The main objective of MPC is to force yt to track the reference signal, denoted by rt while
rejecting the disturbance signal, denoted by dt. A control objective can be represented
mathematically using an objective function which obeys certain inequality constraints. By
including a penalty term such as ||yt-rt|| in the objective function, we penalize deviations of the
output from the reference. Secondly by adding a term like ||ut-ut-1|| to the objective functions we
penalize the control signal ut not to exceed a given limit, say lb ≤ ut ≤ ub, where lb and ub are
lower and upper bounds on the input respectively. The objective function for the MPC problem is
defined below as:
.
||
||
||
||
2
1 2
1
2
1
S
k
t
k
t
Q
k
t
k
t
P
k
u
u
r
y
J −
+
+
+
+
=
−
+
−
= ∑ (3)
In the above, matrices Q and S act as weight factors on yt and ut respectively, and they are
assumed to be symmetric and positive definite. Further, we can illustrate that:
).
(
)
(
||
|| 2
k
t
k
t
T
k
t
k
t
Q
k
t
k
t r
y
Q
r
y
r
y +
+
+
+
+
+ −
−
=
− (4)

40
And
).
(
)
(
||
|| 1
1
2
1 −
+
+
−
+
+
−
+
+ −
−
=
− k
t
k
t
T
k
t
k
t
S
k
t
k
t u
u
S
u
u
u
u (5)
We derive the objective function in a compact form as given below:
.
2
1
t
T
t
T
t U
h
HU
U
J +
= (6)
Where Ut is the optimal control input as a solution to the above QP problem. The state space
matrices and weighting factors do not change unless modified by the user. Hence, the matrix H
can be computed prior to the plant simulation i.e. it is computed offline. The basic formulation of
a MPC problem as a QP optimal control problem is cited in [2]-[4].
2.3. Inequality Constraints
Inequality constraints on the control input signal prevent it from exceeding a specific limit. We
describe the inequality constraints on the control input as given below:
max
min
U
U
U <
< or





−
=





−
max
min
U
U
U
I
I
. (7)
With the control objective and the inequality constraints formulated as shown in (6) and (7)
respectively, we compute the optimal control input Ut* as a solution to QP optimal control
problem shown below:
.
s.t
min
*
in
t
in
t
q
U
P
J
U
≤
=
(8)
3. QP SOLVER
In this section, we discuss the algorithm for primal dual interior point method. At first, we define
Lagrangian function and derive its K-K-T optimality conditions. In later part, we discuss the
merit function and finally the algorithm.
Consider an inequality constrained QP problem in general form as given below:
.
subject to
,
2
1
min
q
Pu
u
h
Hu
u
J T
T
≤
+
=
(9)
For the sake of simplicity of the algorithm, we use notations u, P and q for Ut, Pin and qin
respectively.

41
3.1. K-K-T Optimality Conditions
The Lagrangian of the above QP problem is given below:
).
(
)
,
( q
Pu
u
h
Hu
u
u
L T
T
T
−
+
+
= 
 (10)
Where, λ is the Lagrange multiplier for the inequality constraints and s is the slack variable
associated with it. The optimality conditions for the general QP in (9) with u as a primal variable
and λ as a dual variable are given below:
.
0
,
,
0
,
,
>
=
=
+
−
=
+
s
s
q
s
Pu
h
P
Hu
T
T



(11)
Since we consider MPC-QP problems, they are assumed to be convex in nature and H is assumed
to be positive symmetric semidefinite matrix. Hence, the Optimality conditions derived above are
both necessary and sufficient.
The augmented linear system for above optimality conditions is given by:
.
1 





−
=






∆
∆





 −
−
p
d
T
r
r
u
s
P
P
H


(12)
Where d
r and p
r are the residues as given below:
h
P
Hu
r T
d +
+
=  and .
q
s
Pu
rp −
+
= (13)
3.2. Merit Function
We consider a force field interpretation theory [3] for selection of a proper merit function. We
consider force field acting on a particle in a feasible region as given below:
.
)
(
)
(
)))
(
log(
(
)
(
u
f
u
f
u
f
u
F
i
i
i
i
∇
=
−
−
−∇
= (14)
The force )
(u
Fi is associated with each constraint acting on a particle when it is at position u.
The potential associated with the total force field generated by constraints is summation of all
such force fields which is given as the logarithmic barrier function . As the particle moves
toward the boundary of a feasible set, the bound on the particle grows strong repelling it away
from the boundary.

42
The merit function we considered depends essentially upon a log barrier function given by:
)),
(
log(
1
∑
=
−
=
m
i
i
T
i
T
q
u
p
f 
 with }.
,...,
1
,
0
)
(
|
{
dom m
i
u
f
R
u i
m
=
<
∈
=

Where m=number of inequality constraints.
Now, if we penalise the Lagrangian L with the log barrier function f, we get:
)).
(
log(
)
,
(
)
(
1
∑
=
−
−
=
m
i
i
T
i
T
q
u
p
u
L
w 


 (15)
Substituting for )
,
( 
u
L in (15), we get:
)).
(
log(
)
(
)
(
1
∑
=
−
−
−
+
=
m
i
i
T
i
T
T
q
u
p
q
Pu
J
w 


 (16)
Where, )
,
( 
u
w = which satisfies .
0
)
,
( >

u
The merit function )
(w

 can be thought of as another QP minimization problem because f is
convex and strictly satisfy the constraints, it is given below as:
.
),
(
min
q
Pu
w
≤


(17)
The problem in (17) decreases along the direction w
∆ forcing the convergence of w towards the
solution of (12) which is unique, say w*. We can say that:
)
(
)
( 1 i
i
w
w 


 
 ≤
+
. (18)
We derive the conditions for a stationary point w* by computing 0
)
( =
∇ w

 using its directional
gradient )
(w


∇ as given below:
)
(
)
(
)
( w
w
w u 


 

 ∇
+
∇
=
∇ , (19)
∑
= −
−
+
∇
=
∇
m
i i
T
i
i
u
u
q
u
p
p
P
J
w
1 )
(
)
( 

 , (20)
∑
=
−
−
=
∇
m
i
q
Pu
w
1
1
)
(



 . (21)

43
Further, we can modify (20) and (21) as:
1
1
)
(
)
(
)
(
)
( −
=
−
−
∇
−
−
∇
+
∇
=
∇ ∑ i
T
i
m
i
i
i
u
u
u q
u
p
q
u
P
q
Pu
J
w 

 . (22)
∑
=
−
Λ
−
−
=
∇
m
i
e
q
Pu
w
1
1
)
( 

 . (23)
In equations (22) and (23), )
(w
u 

∇ and )
(w


∇ act as force fields on particles at position u
and λ respectively, forcing them away from the boundaries of the convex region to follow the
central path defined by the set of points (u*, λ*), for u>0 and λ>0.
Where )
,...,
(
Diagonal m
i 

=
Λ and e is the unit vector associated with .
Λ
The directional gradient )
(w


∇ at point w is nonpositive, to assure that )
(w

 is only
decreasing as w moves along the central path toward the optimal solution .
*
w Now, the
progress to the QP solution can be monitored using the Newton step w
∆ , associated step length
 and a point .
w
w
w ∆
+
=  The choice of sufficient decrease condition is in spirits with the
sufficient decrease condition mentioned in [9], [12]. We consider the Taylor’s series expansion of
)
( w
w ∆
+
 at a point w which is given as:
.
)
(
)
(
)
( w
w
w
w
w T
∆
∇
+
≤
∆
+ 

 



 (24)
The above expansion term satisfies the condition given in (18); it also shows that )
(w

 reduces
as w moves towards its optimal solution. The backtracking search algorithm is used to compute
 such that 0
≥
∆
+ w
w  . We stop the search algorithm if sufficient decrease in the condition
given in (24) is satisfied, if sufficient decrease in (24) is not observed then the value of  is
decreased and set to a value using 

 *
= where )
1
,
0
(
∈
 and the process is repeated until the
sufficient decrease condition is satisfied. Hence, only those values of  are chosen for which
algorithm generates feasible iterates and values of  for which algorithm deviates from the
central path are rejected.
3.2. Algorithm
Let )
,
( 0
0
0 
u
w = be an initial point satisfying 0
)
,
( 0
0 >

u and assume that 0
H is available for k=0,
1 , . . . , .
0
,
0
,
1 >
>
> 

 feas
Do
1. Choose )
1
,
0
(
∈
k
 and set gap.
duality
*
k
k 
 =
Where duality gap = m
sk
T
k /
 and m = number of inequality constraints.
2. Compute Newton steps k
k
u 
∆
∆ & by solving following augmented linear system. As
proposed, we solve the linear system using T
LDL factorization method.
,
1








−
=






∆
∆





 −
−
k
p
k
d
k
k
k
k
T
k
k
k
r
r
u
s
P
P
H



44
and ).
(
1

 ∆
−
=
∇ −
S
r
s s
k
Where h
P
Hu
r T
d +
+
=  , Se
q
s
Pu
rp 
−
−
+
= and e
S
rs Λ
=
And )
,...,
(
Diagonal m
i s
s
S = .
3. Compute a step size  by using a backtracking line search.
Set ,
1
max =
= 

and test the sufficient decrease condition: ,
)
(
)
(
)
( w
w
w
w
w T
∆
∇
+
≤
∆
+ 

 




Where ).
1
,
0
(
∈

If the test function is not decreased sufficiently, decrease and compute  as given below:
,
0
min
,
1
min
max






<
∆
∆
−
= 





Where ).
1
,
0
(
∈

4. Update k
k
k
k w
w
w ∆
+
=
+ 
1 .
Until feas
dual
feas
pri r
r 
 ≤
≤ 2
2 ||
||
,
||
|| , where level
tolerance
=
feas
 .
3.3. LDLT
Factorization
The linear system given in (12) is of the form AX=B, where A is a Hermitian positive definite
matrix, the algorithm uniquely factors A as:
T
LDL
A = (Cost O((1/3)n3
))
L is a lower triangular square matrix with unity elements at diagonal positions, D is the diagonal
matrix, and LT
is the Hermitian transpose of L.
The equation B
X
LDLT
= is solved for X by the following steps.
1. Substitute
X
DL
Y T
=
2. Substitute
X
L
Z T
=
3. Solve
B
LY = , using forward substitution (Cost O(n2
))
Y
DZ = , solving diagonally (Cost O(n))
Z
X
LT
= , using backward substitution (Cost O(n2
))
The overall cost of the above linear solver is dominated by cost of T
LDL
A = which is O((1/3)n3
).

45
4. MPC IMPLEMENTATION
4.1. Plant Model
We consider a following problem of undamped oscillator for implementing linear MPC strategy;
this problem is cited from [3]. Its state space representation is given below:
[ ] .
)
(
)
(
1
0
)
(
),
(
0
1
)
(
)
(
0
4
1
0
)
(
)
(
2
1
2
1
2
1






=






+












−
=






t
x
t
x
t
y
t
u
t
x
t
x
t
x
t
x


(25)
The state space model given in (25) is both controllable and observable. The discrete time state
space representation of the system in (25) with sampling time of 0.1 seconds is given below
[ ] .
)
(
)
(
1
0
)
(
),
(
0199
.
0
0993
.
0
)
(
)
(
9801
.
0
3973
.
0
0993
.
0
9801
.
0
)
1
(
)
1
(
2
1
2
1
2
1






=






−
+












−
=






+
+
k
x
k
x
k
y
k
u
k
x
k
x
k
x
k
x
(26)
The control signal u(k) has inequality constraints given as -25 ≤ u(k) ≤ 25. We set the prediction
horizon to Np=10 and the control horizon to Nc=5 with the MPC sampling interval of Ts=1unit.
In our case, we introduce an integrated white noise signal in the plant which is assumed to be
Gaussian distributed with zero mean and covariance matrices Qo and Ro respectively.
The new plant model is given as:
[ ] .
)
(
)
(
)
(
),
(
0
)
(
)
(
0
0
)
1
(
)
1
(






=






+












=






+
+
k
d
k
x
I
C
k
y
k
u
B
k
d
k
x
I
A
k
d
k
x
(27)
4.2. Observer Design
The observer equations for the state space representation in (27) are given as:
[ ] )
(
)
(
0
)
1
(
)
1
(
0
0
)
1
(
)
1
(
k
y
L
L
k
u
B
k
d
k
x
I
C
L
L
I
A
k
d
k
x
d
x
d
x






+






+






+
+














−






=






+
+
. (28)
For the plant subjected to white noise disturbances, we design Kalman filter observer with the
gain matrix L such that mean value of the sum of estimation errors is minimum. We obtain L
recursively using MATLAB command dlqe, see [3], [4] which is given as:
)
,
,
,
,
(
dlqe o
o
o
o
o R
Q
C
G
A
L = . (29)

46
With A
Ao = , [ ] 2
2x
o I
G = , [ ]
1
1
=
o
C , [ ] 2
2x
o I
Q = and 1
=
o
R , we obtain 




−
=
52
.
0
43
.
0
L .
For the state space representation in (26), we obtain matrices Q, R, H and h as given below:






=
=
1
.
0
2
.
0
,
]
[ 2
2 R
I
Q x ,
















=
0387
.
0
0462
.
0
0557
.
0
0551
.
0
0563
.
0
0666
.
0
0763
.
0
1008
.
0
1117
.
0
1180
.
0
0763
.
0
1008
.
0
1233
.
0
1391
.
0
1498
.
0
0828
.
0
1117
.
0
1391
.
0
1629
.
0
1785
.
0
0860
.
0
1180
.
0
1498
.
0
1785
.
0
2022
.
0
H and
















=
4450
.
0
5758
.
0
6922
.
0
7845
.
0
8461
.
0
h .
5. SIMULATION RESULTS
For the problem mentioned in (26), the simulations are run for 150 sampling intervals with Ts=1,
output disturbance is introduced for samples with t>50. The results shown in figure 1 are the
plant states x1 and x2 and their state estimations. Figure 2 shows the output response yt to the plant
in the presence of white noise and the output disturbances. In figure 3, we show the plant output
response yt without the presence of the white noise and output disturbances. We testify the
proposed method by comparing it with Matlab’s standard QP solver ‘Quadprog’ for the accuracy
of computed values of ut. In figure 4, we plot the graph for computational accuracy as well as
time comparison between the proposed method and Quadprog tool. In the graphs below, we use
PD-IP as a legend for proposed interior point method.
Figure 1. Plant states and their estimation using Kalman filter

47
Figure 2. Plant output with disturbances and integrated white noise
Figure 3. Plant output without disturbances and integrated white noise

48
Figure 4. Plot of computational time comparison for u(t)
6. CONCLUSION
In this paper, we presented a linear model predictive control of an undamped oscillator. The
primal dual interior point method has been used to solve the QP optimal control problem arising
in MPC. The logarithmic barrier merit function is used in such a way that it enables faster
convergence of iterates while they remain strictly feasible. From the simulations; it is found that
the proposed method is robust in the sense that it considerably rejects the output disturbances. For
a given example, the proposed method was able to compute the optimal solution of the MPC
problem approximately 3 times faster than that of using Quadprog without affecting the accuracy
of ut computations.
ACKNOWLEDGEMENTS
We would like to thank our senior colleagues Vihang Naik and Deepak Ingole for their helpful
technical discussions with us on the concept of MPC problem formulation.
REFERENCES
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survey. Automatica, 25-335-348,1989.
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Verlag London Limited, 2009.
[4] J. M. Maciejowski, Predictive control with constraints. Eaglewood Cliffs, NJ: Prentice Hall, 2002.

49
[5] Yang Wang and Stephen Boyd, “Fast Model Predictive Control Using Online Optimization”, IEEE
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Authors
Prashant Bansode received the B.E degree in Instrumentation Engineering from The University of
Mumbai, in 2010 and the M.Tech degree in Instrumentation and Control Engineering from College of
Engineering Pune, in 2013. His current research interests include convex optimization with applications to
control, embedded system design, and implementation of numerical optimization algorithms using FPGA
platforms.
Dr. D. N. Sonawane received the B.E degree in Instrumentation and Control Engineering from S.G.G.S
College of Engineering, Nanded, in 1997. He received the M.E in Electronics and Telecommunication
Engineering and the Ph.D degree in Engineering from College of Engineering Pune, in 2000 and 2012
respectively. He joined College of Engineering Pune as a lecturer in 1998, currently he is an Associate
Professor with the Instrumentation and Control Department, College of Engineering Pune. He is the author
of the book Introduction to Embedded System Design (ISTE WPLP, May, 2010). His research interests
include Model Predictive Control, Quadratic Programming solver and their hardware implementation and
acceleration using FPGA platforms. He is also involved in various research projects funded by different
agencies of Govt. of India. He is a recipient of Uniken Innovation Award for the project “Sub Cutaneous
Vein Detection System for Drug Delivery Assistance: an Embedded Open Source Approach”, August,
2012, Followed by numerous National level awards. His paper titled “Design and Implementation of
Interior-point Method based Linear Model Predictive Control” received best paper award at International
Conference on Control, Communication and Power engineering, Bangalore, India, 2012.
Prashant Basargi received the B.E degree in Instrumentation Engineering from Shivaji University, in 2011
and the M.Tech degree in Instrumentation and Control Engineering from College of Engineering Pune, in
2013. His current research interests include Model Predictive Control, implementation of numerical
optimization algorithms and Quadratic programming solvers using FPGA platforms.

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