Integrating the receding horizon LQG for nonlinear systems into intelligent control scheme

Những vấn dề chung

INTEGRATING THE RECEDING HORIZON LQG FOR
NONLINEAR SYSTEMS INTO INTELLIGENT CONTROL SCHEME
Nguyen Doan Phuoc
Abstract: In this paper the principle operation of a conventional output feedback
receding horizon controller based on LQR is presented quickly first and then some
opportunities to improve its performance by integrating into intelligent control
scheme are discussing. The paper is ending with a simulative application of the
intelligent receding horizon LQG for temperature tracking control inside of a thick
slab in a heating furnace. The simulation results have confirmed the effectiveness of
integrating intelligent components into a conventional receding horizon controller.
Key words: Receding horizon controller; Intelligent control; GA; PSO; Fuzzy estimator.

1. INTRODUCTION
The intelligent control is portrayed in [[1]] as the “control of tomorrow” with many
characteristics of human intelligence and biological systems in it such as planning under
large uncertainty, adaptation, learning.... Since then it is arised an actual question, whether
the intelligent control could replace completely the conventional control in tomorrow?
Based on a particular receding horizon controller the paper will show that the pure
intelligent control can not and never supersede the conventional control in the future. The
intelligent control is gainful only for being integrated additionally into a conventional
controller to improve its control performance.
The organization of this paper is as follows. The main results are presented first in
Section 2, including a short description of output feedback receding horizon controllers for
nonlinear systems, a brief introduction of intelligent control concept with its components,
then some approaches to integrating intelligent components into a conventional receding
horizon controller, and at last an example for the illustration of intelligent receding horizon
LQG performance. Finally some concluding remarks are given in Section 3.
2. MAIN CONTENT
2.1. Brief description of receding horizon LQG for nonlinear systems

It is already presented in [[2],[3]] the main content of a state feedback receding
horizon LQR for output tracking control of a time-variante, nonlinear continuous-time
systems:

x  f (x , u , t )  


y  g (x , t )  

(1)

where
 ,  , u and y
are state disturbances, measurement noise, vector of inputs and outputs respectively, as
well as the number of inputs and outputs are assumed to be equal. With this controller the
output y of closed-loop system will converge asymptotically to a desired reference r (t ) .
An extension of this controller to the compatible output feedback receding horizon
LQG based on separation principle is illustrated in Figure 1. At every time instant
, where tk  tk 1   and  is a sufficiently small positive value, and
tk , k  0,1,
together with measurable system outputs y k  y (tk ) , the nonlinear system (1) can be
replaced approximately during a short time interval tk  t  tk 1 by a LTI system:

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Kỹ thuật điều khiển – Tự động hóa


x  Ak x  Bk u   k
Hk : 

y  C k x   k , k  0,1,

(2)

where
Ak 

f
x

, Bk 
x k ,uk 1 ,tk

f
u

g

, Ck 

x

x k ,uk 1 ,tk

x k ,tk

, u k 1  u (tk 1 ), y k  y (tk )

(3)

 k  f (x k , u k 1 , tk )  Ak x k  Bk u k 1 and  k  g (x k , tk )  C k x k
and x k is an optimality estimated value of x (t ) at the current time instant tk in order to
eliminate the effects of both disturbances  and  .
In the case of particularity nonlinear structure of (1) as follow:

x  A(x , u , y , t )x  B (x , u , y , t )u  


y  C (x , u , t )x   ,

(4)

the determination of matrices Ak , Bk ,Ck given in (3) will be simplified with:
Ak  A(x k , u k 1, y k , tk ), Bk  B (x k , u k 1, y k , tk ), C k  C (x k ,u k 1,tk ),

(5)
 k  0 and  k  0.
After approximating the original nonlinear system (1) to LTI (2) for a short time
interval tk  t  tk 1 , the design of linear output feedback controller for (2) would be
carried out. Obviously the obtained controller is then valid for (1) only during the
corresponding time interval tk  t  tk 1 .


tk 1

tk

u (t )
tk  2


Getting

y (tk )

t


u (t )

u (t )

Calculation
unit

Figure 1. Repeatedly operational performance of the receding horizon LQG [[2],[3]].
Based on the variation technique, the tracking controller u (t ) , which guarantees
definitely the asymptotic convergence of the output y of LTI system (2) to the longing
constant reference:
w k  r (tk )  r (tk 1 )  y k 1 ,

(6)

is determined as follows [[1],[3]]:
u  Kk x  xs [k ] Tk z  u s [k ] with Pk Bk Rk1BkT Pk  AkT Pk  Pk Ak  Qk
  k   Ak



 w k   k  C k

t
Bk   x s [k ] 
,
z



 y (t )  w k dt and
0   u s [k ] 
0





1 T

Kk  Rk Bk Pk

T

Tk  Tk  0

(7)

where Qk  QkT  0, Rk  RkT  0 and Tk  TkT  0 are chosen arbitrarily. The constant

reference (6) for LTI in (2) is established from the desired value r (tk ) of the original

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Những vấn dề chung

nonlinear system (1) at the current time instant tk and the tracking error r (tk 1 )  y k 1
remaining from the previous horizon k  1 , which will be compensated now during the
current control horizon k , i.e. during the time interval tk  t  tk 1 .
2.2. Meaning of intelligent control and its components
As it was remarked in [[1],[4]-[16]], there are involved in intelligent control concept
all unconventional control approaches, in which the control performance imitates the
acting behaviour of biological system or human knowledge to solving a control problem.
The unconventionality of intelligent control here means that to design an intelligent
controller the mathematical models of controlled plants or processes could not be required
as usually by conventional control.
In a precise view, the intelligent control is a class of control techniques, where various
control approaches of artificial intelligence are applied either separately or simultaneously,
such as neural networks, fuzzy logic, evolutional computing and machine learning [[4][9]].
1. Artificial neural network (ANN) is a computerized system of enormously connected
artificial neurons, which attempt to copy the behaviour of biological neurals in the
nature and the message exchange between them. Every artificial neuron i in an ANN is
essentially a sub-system with multi inputs ui1 ,
, uin and single output yi , which is
modelled by an associated activation function:
yi  fi (ui1 
 uin ) .
In an ANN, all neurals are arranged in different layers, including input-, hidden-, and
output layer. Any artificial neuron i of each layer is connected permanently with an

neural j of the other layer through a connection bond. Every connection bond i  j is
always assigned with a weight wij to perform the information flow u jk  wij yi from
single output signal yi of neural i to an input u jk of another neural j . The set of
weighted factors w  (wij ) plus a suitably updating strategy for them (most optimimal),
together with the accordingly chosen connection topology (feedforeward or backward)
and an activation functions f j () in each neural j , decide the essentially operational
performance of ANN, under which the weighted factors w  (wij ) cover most useful
informations for ANN to solve a particular control problem [[6],[7]].
w1j
wnj

u1

i

wij



z

f j (z )

y1

j

u2

y2

un

ym

Input
layer

Hidden layers

Output
layer

Figure 2. Multi-layers model with two hidden layers.
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Figure 2 illustrates the typically feedforward topology of an ANN with one input layer,
one output layer and two hidden layers. Although this connection topology is a quite
simple one, it has been nevertheless shown in [[4],[6],[7]] that this feedforward ANN
with suitably chosen nonlinear, continuous and differentiable activation functions f j ()
for all neurals j are adequately for universal approximation and control through
adjusting optimality the weighted factors w  (wij ) in it.
2. Fuzzy controller (FC) is a computerized system, in which the human knowledge about
how to control a plant is reproduced and implemented [[6],[9]].
To design an output feedback fuzzy controller, the human knowledge for control a

plant with n input u1 ,
, un and m outputs y1 ,  , ym will be first reproduced in a
so-called rule-base as follows:
Ri : If y1  Yi1 and y 2  Yi 2 and  and ym Yim
then u1  U i1 and u2  U i 2 and  and un  U in , i  1, 2,  , p ,

(8)

where Uij ,Yik , 1  i  p, 1  j  n , 1  k  m are the linguistic values of plant inputs and
outputs repectively. These linguistic values are obtained from numeric values of plant
via a so-called fuzzification process. Each item Ri of the rule-base above is called the
inference clause.
To implement the rule-base is needed an inference mechanism, which is established
based on axiomatic implication of fuzzy logic [[9]]. Since many inference mechanisms
are available for an implementation, there are also many compatible fuzzy controller
available to represent a fixed human control knowledge (8).
Finally, since the inference mechanism returns a result of rule-base (8) as a linguistic
value in output, for being useable to control plants it has now to be converted
correspondingly into a numeric value, and which will be realized by using an
appropriate so-called defuzzifiacation process [[9]].
Hence, a fuzzy controller consists of three main components:
 Fuzzification for transforming the numeric values of plant inputs and output into
linguistic values to be useable for the inference mechanism.
 The rule-base and the inference mechanism for realizing the human control
experience. Whereas the rule-based is a set of rules for reproducing the human
knowledge about how to control, the inference mechanism is a tool to implement it.
 Defuzzification for converting the linguistic results of rule-base into a numeric
value to be useable for the plant control.
Figure 3 below illustrates the main optional structure of a fuzzy controller.
Inference mechanism

y  (y1 ,

, ym )T

U

Y
Fuzzification

Rule-base

Defuzzification

u  (u1 ,

, un )T

Figure 3. Fuzzy controller.
3. Evolutional computing is a concept of all unconventional approaches or algorithms,
inspired by biological evolution such as natural inheritance, mutation, selection and
crossover, or by social behaviour of biological systems such as bird flocking, fish
schooling, to determine the global solution of an optimization problem.

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Những vấn dề chung

In evolutional computing, an initial set of candidate solutions is generated first and then

updated or moved iteratively towards the optimal solution. Each new set of candidate
solutions is produced by using artificial selection, mutation principle or flocking
behaviour of biological systems for removing less desired solutions and supplementing
other solutions, which increase the fitness (objective function). The optimal solution is
picked finally then from the last set of candidate solutions, which is obtained when
either a maximum number of iterations has been exceeded or a satisfactory fitness level
has been reached. The main advantage of evolutionary computation techniques is that
for applying them no any particular assumtions about problems being optimized are
needed, such as derivate or convexness, which are required commonly by using
conventional techniques [[7],[11],[12],[13]].
Genetic algorithm (GA) and Particle swarm optimization (PSO) are two representative
examples of evolutionary computation techniques. Nowadays these two evolutionary
techniques become very popular, because they are not only suited for solving a wide
range of optimization problems, but also have been applied successfully to selfadaptive and self-organized control differently nonlinear systems [[6],[7],[10]].
 Genetic algorithm (GA) is a computational search technique for finding the optimal
parameters x * of:
x  arg min f (x ) , subjected to x  X  Rn ,
*

x X

(9)

where x is a vector (or a point) of n variables x1 , x 2 ,
, xn to be optimized.
In the language of evolutionary computation generally, the objective function value
f (x ) at a definite poin x and the constraint X  Rn are often referred to as the
fitness of particle x and the search space for optimal solution respectively.
GA is different from conventional optimization methods in the senses, that firstly it
is working initialy with a set of parameter codes (i.e. bit strings of particles) instead

of parameters themselves, secondly GA updates iteratively a set of particles (i.e. of
population) not a single particle, and finally GA uses probability updating rule
instead of deterministic one [[6],[10]].
GA begins with a selected population of N particles denoted with x i , i  1,  , N
(typically random). Then each particle x i will be assigned a bit string of length L .
It means that each string will consist of 2L discrete values.
After that, GA will update iteratively this initial population of strings step by step to
regenerate a new one (new population) in the evolutionary sense, that based on the
invididual fitness of each string the worse strings will be discarded and some betters
will be supplemented. Keys for the reproduction of new population of strings are
selection, crossover and mutation rule.
Denote all strings in the iteration k , i.e. in the k  th population, with:
G (k )  x 1 (k ), x 2 (k ),
, x N (k ) ,
then each string in G (k ) will be assigned a selection probability:
pi  f x i (k )  f , where f 

1
N

 f x j (k )  .
N

(10)

j 1

Correspondingly, the string x i (k ) will have a chance of pi to be copied into a socalled intermediate population I (k ) . With this copying approach more fit strings
will tend to I (k ) more often.

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After completing the copy of G (k ) into I (k ) , the crossover can occur over I (k ) in
the way that a pair of strings in I (k ) is picked randomly first and then recombined
with a specified probability pc (usually very big) by swapping their bit fragments as
illustrated in Figure 4. Two new obtained strings will replace the original pair in
intermediate population I (k ) .
Finally, the new population G (k  1) for the next iteration k  1 is created from I (k )
with a randomly chosen mutation probability pm (often very small) so, that all
strings of I (k ) satisfied pi  pm will be copied directly into G (k  1) . The others
have to be flipped in a bit via mutation rule before copying into G (k  1) .
x i (k )

before

swapping area of strings

after

x j (k )

Figure 4. Crossover occures through swapping two fragments of strings.
In summary, the procedure of GA is as follows: (a) Use a random generator to
create an initial population of strings  (b) Use the fitness function to evaluate
each string  (c) Use a suitable selection method to select best strings  (d)

Use crossover and mutation rules to create the new population of strings 
Back to the step (b) or exit if a stop condition is encountered.
 Particle swarm optimization (PSO) is an optimization technique to solving the
problem (9), in which a random chosen population of particles, also called the
swarm, will be moved iteratively around search space, instead of updating it
iteratively what happened in GA. The moving rule of PSO was inspired by
collective behaviour of bird flocking [[7],[11]-[13]].
Motivated by social behaviour of bird flocking, PSO will solve the optimization
problem given in (9) above by iteratively moving m randomly chosen particles
within bounds (i.e. swarm or population):
S  x i , i  1,
, m
in the search space X according to formulas (11) over their own position x i and
velocity v i . The movement of each particle is influenced by its local best position
p i and the global best position g among all particles in the swarm, which are also
updated iteratively together with the moverment. At the end of iterations it is
expected, that the whole swarm will flock to the optimal solution x * .
Denote the position, velocity, local best position and global best position of each
particle x i  S during the iteration k with x ki , v ki , pki , g k respectively, then the
basic fomulas for moving them involve [[12]]:
v ki 1  wv ki  c1r1k ( pk  x ki )  c2r2k (g k  x ki )
i

x ki 1  x ki  v ki 1  X

 If f (x ki 1 )  f ( pk ) then set pk 1  x ki 1, otherwise pk 1  pk
i
i
i
i


g k 1  arg min f (x ik 1 )
i


(11)

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Những vấn dề chung

In formulas (11) above 0  w  1.2, 0  c1  2, 0  c2  2 are user-definited constants
and 0  r1  1, 0  r2  1 are random values regenerated for each iterative update.
Following steps exhibit the iterative algorithm of PSO: (a) Create an initial swarm
of particles within bounds and assign them initial velocities.  (b) Evaluate the
fitness of each particle  (c) Update velocity and position of each particle 
(d) Update local and global best fitnesses and positions  Go back to the step
(b) or exit if a stop condition is encountered.
4. Machine learning, as mentioned in [[14]-[18]], is a field of computer science, which
provides computer systems the learning ability with data, without being explicitly
programmed. Machine learning is related closely to many different domains of
computer science such as pattern-recognition, decision-making, prediction-making,
optimizing from past data, data analysis, machine vision and etc..
From the data analysis and optimization characteristic points of view, the machine
learning is then for control engineering very helpful, if there the controlled systems are
too complex to modelling mathematically, or the obtained model may be too
complicated for a conventional synthesis of controllers, as well as when the systems are
effected additionally by large environmental disturbances, which are difficult to predict
or eliminate. A subfield of machine learning faciliates to overcome these

circumstances, and additionally to find some meaningful anomalies of systems, is the
learning control.
Because controllers provided from learning control framework are established based on
data analysis and intelligent optimization, they would possess properties of human
knowledge and therefore they could be considered as intelligent controllers. These
controllers are also called learning controllers. Figure 5 exhibits a control configuation
using learning controllers, which is nowadays often met in high-tech manufacture,
robot manipulator and flying machine control [[17],[18]].
Learning
controller

ul
r

e

Conventional
controller

uc

u

Plant or
process

y

Figure 5. Control a complex system with an additional learning controller.
At a certaint time instant tk the learning controller uses past data of process inputs,

system references and system errors, i.e. of uc (t ), r (t ),e (t ), t  tk , to determine the
correction signals ul (tk ) for controlled systems in oder to minimize the prediction
tracking error by using a so-called learning mechanism.
The most essential component of every learning controller is a function approximator
f (, x ) in it, where x are parameters to be determined frequently (during the control) by
learning from the past data so that the predictive errors of tracking behaviour will be
minimized. In this sense, the learning controller could be considered as a model
predictive controller (MPC) or an adaptive controller, but it is applicalbe for more
larger class of systems, espectially for such complex systems, which are difficult or
even unable to modelling mathematically [[18]].
The design of a learning controller will occur through following steps:

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 Select a suitable function approximator f (, x ) . The suitableness here means that the
application of this selected function approximator must guarantee the long term
stability of systems despite the undesired changing of plants. Furthermore, the
convergence behaviour of tracking error must be indicated.
At the present time there are many function approximators are available for the
selection, such as the Multi Layer Perceptron network (MLP), B-Spline Network
(BSP) or Cerebellar Model Articulation Controller network (CMAC)... [[18]].
 Choose an appropriate learning mechanism. The learning mechanism denotes a
computational algorithm for updating x frequently from past data so that during
control the tracking error of system will be constantly minimized.
Learning mechanisms are distinguished mainly to each other through their

frequency for the update of parameters x . Some gainful mechanisms are the
iterative learning, the repetitive learning and the non-repetitive learning.
The frequency of applied learning algorithm and the charactertistics itself effect
obviously the stability of the system. Therefore, according to a certain learning
mechanism being used the stability of closed-loop system has to be investigated
individually and it is usually carried out with theoretical approaches [[18]].
2.3. Integrating receding horizon LQG into intelligent control scheme
It is simply recognisable from Subsession 2.1 above that there are for the presented
receding horizon LQG some opportunities to improve its control performance through
changing intelligently parameters in it. Precisely there are:
1. Since all positive definite matrices Qk , Rk ,Tk in each control horizon k  0,1, are
chosen arbitrarily, it arises here an oppportunity to select them so that the obtained
controller could satisfy additionally some more required constraints, such as the input
contraint u k U , the state constraint x k  X , the constraint of transient time and etc..
However, an analytically unified objective function by combining all these constraints
would be too compicated for the utilization of any conventional optimization method,
and fathermore it is even impossible in some theoretical situations. Hence, in these
circumstances, the usage of an intelligent optimization method is a good remedy.
Intelligent
Intelligent
outputfeedback estimator
controller

a

r

Receding
horizon LQR

u

Plant or
process

y

Q, R

x

Intelligent
optimizing
State
observer

Figure 6. Intelligent receding horizon LQG.
The utilization of GA, PSO, DE (differential evolution) and ICA (imperialist
competitive algorithm) to determine optimale weighted matrices Q , R for multiconstrained LQR design on a flexible robot manipulator LTI model had been presented
and compared in detail in [[19]], which could be here employed gainfully to determine

12 Nguyen Doan Phuoc, “Intergrating the receding horizon LQG … intelligent control scheme.”


Những vấn dề chung

optimality weighted matrices Qk , Rk ,Tk for the receding horizon LQG (7) subjected to
several constraints of each horizon moving k  0,1, .
2. To be converted correlatively a state feedback controller for nonlinear systems into
output feedback one based on separation principle is needed a suitable state observer.

There are many intelligent state observers for nonlinear systems (1) available, under
which the Kalman filter and the particle filter are known as representative examples. In
spite of the fact that these observers often require system models or measurement
models for their implementation and hence they are related closely to theoretical
approaches, these observers would be still considered as intelligent components of
Bayesian
control
as
remarked
in
[ />3. Despite the receding horizon controller (7) was designed originally for certain systems
(1), it could be also outspreaded to employ for an analytically uncertain one:

x  f (x , u ,a , t )  
(12)


y  g (x ,a , t )  
where a is a vector of all system parameters, which depend on system inputs and
outputs u , y but can not be determined analytically.
However, if the dependance of a on system inputs u and outputs y could be captured
by human experiences, then together with a fuzzy estimator to reproduce the human
knowledge of how the vector a depends on u , y , the receding horizon controller (7) is
applicable again for this analytically uncertain nonlinear system (12).
Figure 6 exhibits an example of intelligent receding horizon LQG, which is obtained
by integrating intelligent control components into the original receding horizon LQG (7).
2.4. An illustrated example
To illustrate the integration of intelligent components into a conventional receding
horizon LQG for improving intelligently its control performance it will be considered here
an intelligent controller based on the receding horizon LQR for the temperature tracking

control at a particular position z inside of a thick slab in a heating furnace, which has
been basically presented in [[20]].
Slab temperature at the position z

Slab in
heating
furnace

Measurement noise

Actuator disturbance
Intelligent receding
horizon LQG

e (t )
Fuzzy
estimator

PSO

r

Qk , Rk

z

c (T ),  (T )

Receding
horizon LQR

u

Galerkin
model

x

Figure 7. Simulation configuration.

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The principle integration for this LQG controller to becoming intelligent is depicted in
Fig.7 and which will be used here as the rightful simulation configuaration. This controller
would be called intelligent, because it consists of two intelligent control components in it.
The first intelligent component is the fuzzy estimator for determining heat transfer and
conductive parameters c (T ),  (T ) of the temperature transfer inside the slab respectively.
The second component is the evolutional optimization algorithm PSO, which is applied for
finding out all weighted matrices Qk , Rk , k  0,1, appropriately to oblige constraints of
the control problem. Furthermore, in this simulation configuaration the Galerkin model
will be used for the state observation, instead of any state observer as usual such as
Kalman or particle filter. Hence it would be called the internal model controller.
The simulation according to the configuration given above will be carried out as close
as possible to real envionments. Concretely, it will be realized in presence of both
impulsive disturbance in inputs and high-frequency measurement noise at system outputs.

The controlled object, i.e. the thick slab in a heating furnace, is modelled with Simscape
libraries and the desired reference r (t ) is a continuous-time function as follows:
r (t )  289  711  1  exp(0.005t )  .

(13)

Figure 8. Simulation results.
As shown in obtained simulation results given in Figure 8, due to using additionally a
fuzzy estimator and an intelligent optimization algorithm PSO, the receding horizon LQG
has tracked the slab temperature in the middle asymptotically to the desired value (13) and
the required constraints u  2000, T5%  2500s are satisfied as expected, without having
to determine analytically c (T ),  (T ) as well as the unified objective function.
3. CONCLUSIONS
It was shown in this paper, the intelligent control alone with all its components, such
as ANN, Fuzzy controller, evolutionary computing, machine learning..., can not supersede
entirely the conventional output-feedback receding horizon controller for nonlinear
systems. However, the intelligent components can be integrated additionally into a
conventional controller for improving its control performance, when the systems are too
complex or are affected by large environmental disturbances.
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Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san FEE, 08 - 2018

15


Kỹ thuật điều khiển – Tự động hóa

TÓM TẮT
THÔNG MINH HÓA BỘ ĐIỀU KHIỂN LQG DỊCH CHUYỂN
THEO THỜI GIAN CHO HỆ PHI TUYẾN
Trong bài báo này, trước tiên nguyên lý làm việc của bộ điều khiển tối ưu phản
hồi đầu ra không dừng truyền thống LQG được trình bày ngắn gọn, sau đó là bàn
đến những khả năng tích hợp thêm các thành phần của điều khiển thông minh nhằm
nâng cao chất lượng điều khiển cho nó. Bài báo kết thúc với một ví dụ ứng dụng của
bộ điều khiển LQG thông minh này vào điều khiển trường nhiệt độ trong vật nung
dày. Kết quả mô phỏng đã xác nhận tính hiệu quả của việc tích hợp thêm các thành
phần thông minh vào một bộ điều khiển truyền thống.
Từ khóa: Bộ điều khiển LQG không dừng; Điều khiển thông minh; GA; PSO; Ước lượng mờ.

Received date, 01st July 2018

Revised manuscript, 10th September 2018
Published on 20th September 2018

Author affiliations:
Department of Automatic Control, Hanoi University of Science and Technology.
Email:

16 Nguyen Doan Phuoc, “Intergrating the receding horizon LQG … intelligent control scheme.”