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International Journal of Control, Automation, and Systems (2012) 10(5):1042-1048

DOI 10.1007/s12555-012-0522-2

ISSN:1598-6446

eISSN:2005-4092

http://www.springer.com/12555

A Probabilistic Approach for Model Following of Markovian Jump Linear

Systems Subject to Actuator Saturation

Linpeng Wang, Jin Zhu*, and Junhong Park

Abstract: This paper is concerned with the model following problem of Markovian jump linear sys-

tems (MJLSs), which suffer from stochastic uncertainties and actuator saturation. By applying a proba-

bilistic approach based on particles, a sequence of control inputs is designed to guarantee that the mod-

el following error remains within a desired region in a certain probability, as well as the control cost is

optimal. Motivated by this, the stochastic control problem is represented by chance constrained pro-

gramming, and approximated as a determinate optimization one, which is solved by mixed integer li-

near programming (MILP). Furthermore, an improved particle control approach is proposed to reduce

the computation complexity. The effectiveness of this improved approach is demonstrated by an exam-

ple along with complexity comparison.

Keywords: Actuator saturation, Markovian jump linear systems, model following, particle control

approach.

1. INTRODUCTION

The problems of model following with uncertainties

have received a great deal of attention over the past

decades. Up to date, much work has been done in this

field with lots of achievements. For instance, robust

tracking and model following problem is discussed in [1]

and a linear controller is designed to track dynamic

signals for uncertain linear systems; a linear memoryless

controller is developed in [2] to track reference model for

uncertain linear time-delay systems such that the tracking

error can be made arbitrarily small; [3] considers the

problem of robust tracking and model following for linear

systems with multiple delayed state perturbations, time-

varying uncertain parameters and disturbance; while [4]

proposes a discrete-time integral sliding mode control

scheme to achieve zero-tracking error for robust tracking

and model following of uncertain linear systems. All these

literature helps a deep understanding of model following

problems. However, the practical systems are getting

more and more complex recently and far from ideal

assumptions. Thus current research interests are focusing

on how to model the system dynamics accurately.

The practical systems of model following are affected

by many factors. Firstly, system dynamics, which is

usually driven by only continuous time, will vary with

some random discrete events occurring, resulting the

system states are now driven by both continuous time

and discrete events. Next, system dynamics usually

suffers from uncertainties arising from modeling error,

external disturbances or a combination of these. Such

uncertainties, which are best represented by a stochastic

model instead of a set-bounded one, can be of any

distributions rather than Gaussian. Thirdly, actuator

saturation or input constraint, is another important factor

for system dynamics if physical limitation is considered.

The performance of dynamic systems is degraded by

actuator saturation, even the systems are unstable due to

this phenomenon [5,6]. Considering above factors, the

development of theory along with technology is facing

big challenges for model following problems. Motivated

by this, Markovian jump linear system (MJLS) model

with stochastic uncertainties and actuator saturation is

used in this paper to describe such dynamics, where the

mentioned discrete events are governed by a Markov

chain [7]. Differing from the existed achievements about

model following of MJLS, the distribution of uncertainty

herein is of arbitrary rather than Gaussian such that jump

linear quadratic Gaussian (JLQG) method will be invalid

in this case [8]. All these practical considerations mean

that a new approach should be developed to deal with the

complex situations.

In this paper, a probabilistic approach for model

following of MJLS subject to stochastic uncertainties and

actuator saturation is proposed. This probabilistic ap-

proach, roughly speaking, is to approximate the stochas-

tic problem as a determinate one using particles, and the

possible sample paths can be represented by particles

which are generated according to the distributions of

________

Manuscript received May 22, 2011; revised March 9, 2012;

accepted June 4, 2012. Recommended by Editorial Board membe

Shengyuan Xu under the direction of Editor Myotaeg Lim.

This work is supported by the National Natural Science Foun-

dation of China (Grants no. 60904021) and the Fundamental Re-

search Funds for the Central Universities (Grants no. WK210006

0004).

Linpeng Wang and Jin Zhu are with the Department of Auto-

mation, University of Science and Technology of China, 230027,

Hefei, Anhui, China (e-mails: wlp816@mail.ustc.edu.cn, jinzhu@

ustc.edu.cn).

Junhong Park is with the Department of Mechanical Engineer-

ing, Hanyang University, Seoul, 133-791, Korea (e-mail: parkj@

hanyang.ac.kr).

* Corresponding author.

A Probabilistic Approach for Model Following of Markovian Jump Linear Systems Subject to Actuator Saturation

1043

stochastic uncertainties. Some particles are called failure

ones if they fail to achieve the control objective. The

proportion of failure particles, i.e., failure probability or

chance constraint, should be ensured to remain in a given

bound with appropriate controller adopted. Approach

based on particles has been studied for filter problems

[9,10], and has shown advantages over traditional

Kalman filter for disturbance in the form of non-

Gaussian disturbance [10]. Meanwhile, control method

using particles is also investigated in [11-13]. The

particle control method discussed in these literature is

applied for unmanned aircrafts and ground vehicles. The

approximated determinate problem, which consists of

linear control cost function, linear system state and linear

input constraint, can be solved using mixed integer linear

programming (MILP) [14,15]. However, there is a

drawback for this method: the computation time is

exponential with the size of MILP such as the number of

particles or time steps [13].

The purpose of this paper is to design a sequence of

optimal constrained control inputs such that the model

following error is guaranteed to be in a small region with

a certain failure probability. In consideration of the

stochastic uncertainties in form of arbitrary distributions

and the existence of input constraints, particle control

approach is applied to solve this chance constrained

programming problem. Meanwhile, computation com-

plexity is also a consideration since model following

problem may be a long-time process. Motivated by this,

an improved algorithm is developed to obtain the solu-

tion of MILP for the reduction of computation time. By

dividing the whole time interval into some subintervals,

optimal controllers can be obtained for each subinterval.

All these controllers combine together and constitute a

new control law over the whole time interval to

guarantee the performance of model following. In this

sense, the desired optimal programming is replaced by

several sub-optimal programming problems, and the

complexity decreases greatly with an acceptable control

cost ensured.

This paper is organized as follows. Section 2 intro-

duces the model following problem including controlled

system and reference system. Next, an optimal controller

is designed for model following via particle control

approach in Section 3. Section 4 proposes an improved

algorithm to reduce the computation complexity. Section

5 presents an example demonstrating this probabilistic

approach to verify its effectiveness with comparison. A

brief conclusion is drawn in Section 6.

2. PROBLEM STATEMENT

Consider the controlled discrete-time MJLS,

() ()( ) ,

() .

k kk kkk

kkk

xArxBruw

yCrx

=+ +

(1)

where

x ∈ R is system state vector at time step k,

y ∈ R is system output vector,

u ∈ R is control

input vector, and

():

uσ →RR is saturation func-

tion as follows,

()

sgn( ) , ,

uuU

uUu U

θθ



≤





⋅>





(2)

where u

kθ

denotes the θ th element of vector u

, and U is

the limitation of control input. w

∈ R

is stochastic

uncertainty, which may be of arbitrary distribution rather

than Gaussian and is independent of the system state.

A(r

), B(r

), C(r

) are matrices with appropriate

dimensions. r

denotes the system mode, and {r

} is a

Markov chain on the complete probability space (Ω,

,),FP takes value in a finite set {1, 2, , }.

= I For

the sake of convenience, A(r

) is denoted by A

when r

i. The mode transition probability matrix is P={P

}

I×I

and the transition probability is

(

)

kk ij

pr j r i p

=== (3)

where

p ≥

∈

∑

For the model following problem, consider that the

mode of reference model is of the same as the controlled

MJLS,

(1)

() ,

mk m k mk

xArx

yCrx

(4)

where

x ∈ R is reference model state vector.

∈ R is reference model output vector. For notation

simplification, A

) can be denoted by A

when r

= i.

The model following error is defined as,

kkmk

eyy=− (5)

The objective is to design a sequence of optimal

constrained control inputs such that e

is guaranteed to be

in a small region with probability 1 – δ at least, where δ

is a desired failure probability.

Similar to [1-4], the requirement for system (1) to

follow (4) is that the solution G

, H

of following matrix

equation should exist.

ii i imi

iimi

ABG CA

CHC



   



   



   

(6)

If a solution cannot be found to satisfy (6), a different

reference model or output matrix C

should be chosen.

Considering that the uncertainty is represented by a

stochastic model rather than a set-bounded one, the

control method proposed in [2-4] is not suitable.

Meanwhile, the JLQG approach has disadvantage to deal

with this model following problem due to the arbitrary

distribution of uncertainty. Then a probabilistic approach

is developed in Section 3.

3. COTROLLER DESIGN

Considering the reference model (4) and controlled

system (1) with uncertain initial state x

, the model

following error e

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