An Open-loop Model Predictive Control Based on the Heuristic
Algorithm
Yuanqing Yang
1
, Baocang Ding
1
1. Department of Automation, School of Electronic and Information Engineering, Xian Jiaotong University, Xian, 710049, P.R. China.
E-mail: daveyoung@163.com, baocangding@126.com
Abstract: This paper proposes an open-loop model predictive control (MPC) scheme for the constrained uncertain linear sys-
tem. Accounting for model uncertainty, the regulator uses a tree trajectory to forecast the time-varying model uncertainty. The
open-loop MPC is parameter-dependent, which optimizes the vertex control moves for all corners of the uncertainty evolution.
Applying the proposed scheme, offset-free control is achieved. The simulation results and analysis demonstrate the effectiveness
of the approach.
Key Words: Model predictive control, process control, heuristic algorithm
1 Introduction
Model predictive control (MPC), is one of the impor-
tant advanced control algorithms in both theory and prac-
tice. With the features of handling physical constraints and
the multi-variable problems, MPC has been widely applied
in the industrial circle since 1978 [1], meanwhile, was ex-
panded to theoretical field to improve and advance its theo-
retical bases. One can refer to [2–5] for details about various
MPC algorithms. Usually, MPC optimizes, at each sampling
time, a cost function associated with the future state/output
evolutions and constraints, so as to obtain a sequence of con-
trol moves. However, only the first control move is imple-
mented.
Linear parameter varying (LPV) system is a typical
methodology to approximate nonlinear dynamics, which
embeds the nonlinear system trajectories into a family of lin-
ear models [6]. Although some extent of the conservatism is
unavoidable in this approximation, using the LPV approach
can yield the convex optimization and allows for the appli-
cation of powerful linear design tools. MPC for LPV system
with guaranteed stability and recursive feasibility is referred
as a synthesis approach which is a hot topic. Free pertur-
bation items have been added to the feedback control laws
in order to enlarge regions of attraction [7]. The class of lin-
early parameter-dependent Lyapunov functions are proposed
for uncertain polytopic discrete-time plants in [8], which
gives rise to less conservative stability conditions than those
arising from classical quadratic Lyapunov functions in [9].
In [10], the authors presented a class of nonlinearly parame-
terized Lyapunov functions instrumental to the achievement
of more efficient relaxed stability conditions. In [11], an ef-
ficient algorithm, which constructs the maximal admissible
set for linear systems with polytopic model uncertainty, is
designed. Considering high-speed control for constrained
LPV systems, the explicit MPC is developed [12, 13]. The
future model prediction and MPC design for LPV systems
with bounded parameter variations are investigated in [14–
16]. In [17], the output feedback MPC is proposed for LPV
systems based on the quasi-min-max algorithm. In [18], the
This work is supported by National Key R&D Program of China
(No. 2017YFA0700300), by Natural Science Foundation of China (No.
61573269), by NSFC-Zhejiang Joint Fund for the Integration of Industrial-
ization and Informatization (No. U1509209)
authors considered robust MPC schemes for LPV systems
where the parameter is assumed to be measured online and
exploited for feedback.
In practical applications, one of major drawbacks of the
synthesis MPC is the high calculation effort compared with
heuristic methods. This paper aims at the LPV system, with
consideration of state and input constraints. we adopt an
open-loop MPC scheme based on the heuristic algorithm.
Considering the model uncertainty, the regulator uses a tree
trajectory to forecast the time-varying model uncertainty,
which is inspired by [19, 20]. The optimization problem
is properly formulated as a classic quadratic form where a
new cost function involves vertices of steady-state targets.
The open-loop MPC is parameter-dependent, which calcu-
lates the vertex control moves for all corners of the uncer-
tainty evolution. By applying the scheme, the computational
burden is less than the synthesis approach of MPC and the
offset-free control is achieved.
Notation: R
n
is the n-dimensional Euclidean space, and
R
m×n
is the m × n-dimensional real matrix set. For any
matrix A, A
T
denotes the transpose of A. For the column
vectors x, x(i|k) denotes the value of x at the future time k+
i, predicted at the kth instant. The symbol ⋆ implies that the
element can be deduced from the symmetry of the matrix and
a variable with ∗ as superscript indicates that it is the optimal
solution of the optimization problem. For the column vectors
x and y, [x; y] = [x
T
, y
T
]
T
. The time-dependence of the
MPC variables is often omitted for simplicity.
2 Problem Statement
Consider the following discrete-time LPV system:
x(k + 1) = A(k)x(k) + B(k)u(k),
y(k) = Cx(k), (1)
where x(k) ∈ R
n
, u(k) ∈ R
m
are measurable state and
input, respectively. We assume that [A(k)|B(k)] ∈ Ω, for
all k ≥ 0, where
Ω = Co {[A
1
|B
1
], [A
2
|B
2
], · · · , [A
L
|B
L
]}
i.e., there exist l time-varying nonnegative combining pa-
rameter ω
l
(k), l ∈ {1, . . . , L} such that
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