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Wang, L., Gawthrop, P., Young, P.C. and Taylor, C.J. (2009) Non–minimal state space
model–based continuous–time model predictive control with constraints, International
Journal of Control, 82, 6, pp. 1122–1137.
Non-minimal State Space Model-based Continuous-Time
Model Predictive Control with Constraints
Liuping Wang †, Peter C. Young ††, Peter J. Gawthrop †††
and C. James Taylor ††††
†School of Electrical and Computer Engineering
RMIT University, Melbourne, Australia
†† Lancaster Environment Centre,
University of Lancaster, U.K.∗
††† Department of Mechanical Engineering, University of Glasgow, U.K.
†††† Department of Engineering, University of Lancaster, U.K.
e-mail:liuping.wang@rmit.edu.au
Abstract
This paper proposes a model predictive control scheme based on a non-minimal state-space
(NMSS) structure. Such a combination yields a continuous-time state-space model predictive
control system that permits hard constraints to be imposed on both plant input and output
variables, whilst using NMSS output-feedback without the need for an observer. A comparison
between the NMSS and observer-based approaches using Monte Carlo uncertainty analysis shows
that the former design is considerably less sensitive to plant-model mismatch than the latter.
Through simulation studies, the paper also investigates the role of the implementation filter
in noise attenuation, disturbance rejection and robustness of the closed-loop predictive control
system. The results show that the filter poles become a subset of the closed-loop poles and this
provides a straightforward method of tuning the closed-loop performance to achieve a reasonable
balance between speed of response, disturbance rejection, measurement noise attenuation and
robustness.
Key words Predictive control, continuous time systems, non-minimal state space realization, mul-
tivariable systems, Laguerre functions
1 INTRODUCTION
The conventional framework of model predictive control, designed using a state space model, consists
of an observer and a state feedback controller. An on-line optimization scheme is then used to
calculate the state feedback control law subject to plant operational constraints (see [18], [3] and
[16] for an overview). In the context of discrete-time MPC, it has been shown [24] that it is possible
to use a non-minimal state-space (NMSS) representation of the controlled system and thus avoid
the need for an observer, with its attendant sensitivity to errors in the system model. The present
∗Young also has Professorial appointments at the School of Environment and Society, Australian National Uni-
versity, Canberra; and the School of Electrical Engineering and Telecommunications, University of New South Wales,
Sydney, Australia.
1
paper shows that the NMSS approach can also be used in a continuous-time context, giving output
feedback without the use of an observer or output derivatives.
Continuous time MPC is more recent and less developed than the discrete-time formulation.
Following on the discrete-time design framework of Generalised Predictive Control [4], a continuous-
time version was presented [7, 6] and a non-linear version was derived later [9]. More recent work
on continuous time model predictive control includes several publications [2, 1, 10, 21]. There
are a number of reasons why design in the continuous-time domain might be preferable. First,
the continuous-time design is based on a continuous-time model and its implementation is less
sensitive to the choice of sampling interval. For example, the continuous-time implementation, or
its alternative based on the use of delta operator models [17, 27, 31, 28]), are preferable at high
sampling rates. Second, as shown in a recent publication [11], intermittent predictive control can be
employed. Finally, as we shall see later, the poles of the implementation filter become part of the
closed-loop poles, hence they can be used directly in the tuning of the closed-loop response.
The present paper uses the Laguerre function approach [21], combined with the continuous-time
NMSS formulation (for examples, see [19], [13]). The central idea is to model the derivative of the
future control signal using a set of orthonormal exponential basis functions, i.e. Laguerre functions
in the present paper. To help the reader understand the basic idea, a simple example is presented in
Section 2, where the derivative of the control signal is illustrated as a suitable candidate for modelling
by Laguerre functions. With this description of the future control signal, the continuous-time model
predictive control problem is converted into an optimisation problem where the design objective is
to find the set of coefficients that will minimise a given cost function subject to constraints. The
reason why we use orthonormal basis functions is that the control trajectory captured by the basis
functions converges to the underlying optimal control trajectory defined by the cost function.
In the remainder of the paper, Section 2 introduces the continuous-time NMSS model in the con-
text of model predictive control, where ‘state-variable filters’ are proposed for the implementation of
the control system. Section 3 begins with a simple motivational example to explain the methodology
used in the paper, followed by a discussion of predictive control both with and without constraints.
Section 4 presents simulation results illustrating the performance of the predictive control scheme.
These include the role of the implementation filter; comparison studies between an observer and
NMSS based designs; and simulation results for a multivariable predictive control system design
with constraints.
2 NMSS-BASED CONTINUOUS TIME MODEL FOR PRE-
DICTIVE CONTROL DESIGN
2.1 Design framework
Consider that a continuous time system is represented by the input-output, Laplace transform
relation:
A(s)Y(s) = B(s)U(s) + C(s)
sξ(s) (1)
where
A(s) = sn+a1sn−1+. . . +an,
B(s) = b1sn−1+b2sn−2+. . . +bn,
and the integrated noise model C(s)
swith C(s) being a stable, monic polynomial with order n;
while Y(s), U(s) and ξ(s) are the Laplace transforms of the output, input and disturbance signals,
2
respectively. The assumption of an integrated noise model introduces an embedded integrator into
the continuous time predictive control system, so ensuring the type one servo performance that is
essential for most practical applications.
The continuous-time model presented in (1) involves the derivatives of the plant input and output
signal. Nominally, therefore, the state variables of the NMSS model are defined in terms of these
derivatives. If there is measurement noise in the system, then the direct differentiation of the input
and output signals is impossible because of the infinite gain of the time-derivative operation at high
frequencies. In order to avoid this difficulty, the following stable, all-pole ‘state variable filter’ is
introduced (see [25, 8]) in conjunction with the noise model C(s)
s:
F(s) = tn
sn+t1sn−1+t2sn−2+. . . +tn
=tn
T(s)(2)
where T(s) = C(s)E(s), and degree of E(s) = n−m, with E(s) a stable polynomial. Note that the
role of the F(s) filter and how it can be selected by the designer is discussed later in Section 4.1.
Prefiltering the input and output signal with F(s), equation (1) can be re-written as
A(s)F(s)(sY (s)) = B(s)F(s)(sU(s)) + tnξ(s)
E(s)(3)
The selection of F(0) = 1 maintains the same steady state values of the filtered input and output
signals.
If the filtered derivatives y(n)
f(t) and u(n)
f(t) of the input and output signals are now defined as,
y(n)
f(t) = L−1{F(s)snY(s)};y(n−1)
f(t) = L−1{F(s)sn−1Y(s)};. . . yf(t) = L−1{F(s)Y(s)}
and
u(n)
f(t) = L−1{F(s)snU(s)};u(n−1)
f(t) = L−1{F(s)sn−1U(s)};. . . uf(t) = L−1{F(s)U(s)}
ξf(t) = L−1{tnξ(s)
E(s)}
where L−1{K}denotes inverse Laplace transform of function K, then equation (3) can be recaptured
in the time domain by the filtered input and output relationship:
y(n+1)
f(t) + a1y(n)
f(t) + . . . +an˙yf(t) = b1u(n)
f(t) + b2u(n−1)
f(t) + . . . +bn˙uf(t) + ξf(t) (4)
The filtered input signal is related to the derivative of the actual control signal ˙u(t), through
u(n+1)
f(t) = −t1u(n)
f(t)−t2u(n−1)
f(t)−. . . −tn˙uf(t) + tn˙u(t) (5)
while the derivative of the actual output signal ˙y(t) is related to the filtered output signal yf(t)
through
tn˙y(t) = (t1−a1)y(n)
f(t)+(t2−a2)y(n−1)
f(t)+. . .+(tn−an) ˙yf(t)b1u(n)
f(t)+b2u(n−1)
f(t)+. . .+bn˙uf(t)+ξf(t)
(6)
The predictive control design relies on the use of Equations (4)-(6) to generate the predicted output
responses. In the discrete time situation, it is trivial to solve the equivalent equations in an iterative
3
manner, which is precisely how this is achieved in Generalized Predictive Control [4]. However, in
the present continuous-time situation, this is not possible and so an alternative strategy is required.
One straightforward solution is to compute the output prediction by utilizing a NMSS formulation
of the model. This was first formulated in a discrete-time context [26] but has been used subsequently
for both continuous-time [19] and closely related delta operator control systems [27, 31, 28]. In a
similar manner to these references, the NMSS state vector x(t)Tis chosen as:
hy(n)
f(t)y(n−1)
f(t). . . ˙yf(t)u(n)
f(t)u(n−1)
f(t). . . ˙uf(t)y(t)i
and the NMSS model is defined as follows,
˙x(t) = Ax(t) + B˙u(t)+Ωmξf(t)
y(t) = Cx(t) (7)
where
A=
−a1−a2. . . −an−1−anb1. . . bn−1bn0
1 0 . . . 0 0 0 . . . 0 0 0
0 1 . . . 0 0 0 . . . 0 0 0
.
.
..
.
.. . . .
.
..
.
.. . . .
.
..
.
..
.
. 0
0 0 . . . 1 0 0 . . . 0 0 0
0 0 . . . 0 0 −t1. . . −tn−1−tn0
0 0 . . . 0 0 1 . . . 0 0 0
.
.
..
.
.. . . .
.
..
.
.. . . .
.
..
.
..
.
. 0
0 0 . . . 0 0 0 . . . 1 0 0
t1−a1
tn
t2−a2
tn
t3−a3
tn. . . tn−an
tn
b1
tn
b2
tn. . . bn
tn0
BT=000. . . 0tn000
C=000. . . 000. . . 0 1
Ωm=100. . . 000. . . 01
tn
Note that this non-minimal state space realization extends the dimensionality of the model from
nto 2n+ 1 and that, while the pair (A, B ) is controllable, the pair (A, C) is not observable from
the output [20]. Consequently, when the predictive control scheme is chosen to optimise the output
response, the unobservable modes in the NMSS are not changed by output feedback control. In this
case, the poles of the polynomial F(s) form part of the closed- loop poles.
The filtered plant input and output signals can be generated quite easily using the well known
state variable filter framework [25, 8]. For instance, in order to obtain the derivatives of the filtered
output responses, we define a state variable vector
Xy(t) = hy(n)
f(t)y(n−1)
f(t). . . ˙yf(t)iT
4
with the associated state space model formulation in the following control canonical form:
dy(n)
f(t)
dt
dy(n−1)
f(t)
dt
···
dy(1)
f(t)
dt
=
−t1−t2. . . −tn
1 0 . . . 0
.........
0. . . 1 0
y(n)
f(t)
y(n−1)
f(t)
···
y(1)
f(t)
+
tn
0
. . .
0
˙y(t)
=AF
y(n)
f(t)
y(n−1)
f(t)
.
.
.
y(1)
f(t)
+BF˙y(t) (8)
Assuming zero initial conditions, the solution of the state space equation (8) yields the derivatives of
the filtered output responses. In a similar operation on the input signal, the state vector is defined
as
Xu(t) = hu(n)
f(t)u(n−1)
f(t). . . u(1)
f(t)iT
These state variables satisfy the following differential equation, again assuming zero initial conditions
˙
Xu(t) = AFXu(t) + BF˙u(t) (9)
Up to this stage, the plant model is assumed to be a single-input and single-output system. In the
case of a multi-input and multi-output system with an equal number of inputs and outputs, a left
matrix fraction (LMF) representation of the input-output relationship leads to similar expressions
for the NMSS structure in Equation (7), as shown in [29, 30]. Note that, in this multi-input,
multi-output situation, the coefficient matrices ti,i= 1,2, . . . , n are chosen to be diagonal matrices.
3 CONTINUOUS TIME MODEL PREDICTIVE CONTROL
Based on the Non-minimal state space model presented in the previous section, the algorithm pro-
posed in [21] is used to design a continuous-time model predictive control with or without con-
straints1. In order to facilitate further the reader’s understanding of the design method, the next
Subsection 3.1 illustrates the method of approximating the control trajectory by means of a very
simple example, while Subsection 3.2 then describes the NMSS design procedures.
3.1 Modelling of the control trajectory
Let us consider the design example of a state-feedback control system using the pole-assignment
technique [14]. Suppose that the system is described by the state space model:
˙x1
˙x2=0 1
−ω2
00 x1
x2+0
1u(t) (10)
y(t) = 1 0 x1
x2(11)
1Of course, standard NMSS control, which is equivalent to model predictive control [20], provides a simpler
implementation in the case where there are no constraints.
5
Note that, in order to simplify the explanation, x1and x2are assumed to be directly measurable, so
eliminating the need to consider the higher order NMSS form, since this does not affect the problem
of approximating the control trajectory. A slightly extended form of the above model is given below,
where the input to the model is the derivative of the control signal:
˙x1(t)
˙x2(t)
˙x3(t)
=
0 1 0
−ω2
00 0
1 0 0
x1(t)
x2(t)
x3(t)
+
0
1
0
˙u(t) (12)
y(t) = 001
x1(t)
x2(t)
x3(t)
(13)
The eigenvalues for this system are λ1=jω0,λ2=−j ω0and λ3= 0. The model defined by
equations (12) and (13) is both controllable and observable, which can be checked through the
computation of the controllability and observability matrices. Using a pole assignment design based
on a desired closed-loop polynomial (s+ 3ω0)3, the state feedback control law takes the form,
˙u(t) = −k1k2k3
x1(t)
x2(t)
x3(t)
(14)
where the coefficients of the controller are k1= 26ω2
0,k2= 9ω0and k3= 27ω3
0and the closed-loop
system is
˙x1(t)
˙x2(t)
˙x3(t)
=
0 1 0
−27ω2
0−9ω2
0−27ω2
0
1 0 0
x1(t)
x2(t)
x3(t)
(15)
with its three eigenvalues at −3ω0. Hence, for a given set of initial state values
x(0) = x1(0) x2(0) x3(0) T,
the response of the closed-loop state variable vector is
x(t) = eAcltx(0) (16)
leading to
˙u(t) = −K eAcl tx(0) (17)
where K= [ k1k2k3]. The expression for the derivative of the control signal ˙u(t) given by (17) is
a continuous function and satisfies the condition that
Z∞
0
˙u(t)2dt < ∞(18)
This can be seen in Figure 1 (for ω0= 1), which shows that the closed-loop control system is stable
and the area under the plot ˙u(t)2is bounded, i.e. R∞
0˙u(t)2dt < ∞.
The above example leads us to a link that connects the receding horizon control principle, which
is used in the design of predictive control, and the state-variable-feedback control computed using
techniques such as pole-assignment and linear quadratic regulator (LQG) design: see, for example,
[14] and also [20], which examines the relationship between LQ/LQG and predictive control in the
6
0 0.5 1 1.5 2 2.5 3
−20
−10
0
10
du/dt
Time (sec)
0 0.5 1 1.5 2 2.5 3
0
100
200
300
(du/dt)2
Time (sec)
Figure 1: Results for example 1: ˙u(t) and ˙u(t)2
-√2p
s+p
δ(t) -s−p
s+p··· -s−p
s+p
?
lN(t)l2(t)l1(t)
? ?
Figure 2: Laguerre network
7
context of NMSS design. In the design of model predictive control, for a given optimization window
and initial states, the derivative of the control would satisfy (18). Therefore, ˙u(t) can be described
accurately in an optimization window using a set of orthonormal basis functions [15], which was the
rationale behind the original work [21]. In particular, Laguerre functions form a set of orthonormal
basis functions and their Laplace transforms can be expressed in a simple network formulation, as
shown in Figure 2. With this network formulation, a set of Laguerre functions are defined explicitly
by the differential equation (19) below, with initial condition L(0) = √2p[1 1 . . . 1]T
| {z }
N
.
˙
L(t) = ApL(t) (19)
where
Ap=
−p0. . . 0
−2p−p . . . 0
.
.
.
−2p . . . −2p−p
Here the parameter pis referred to as the scaling factor and Nthe number of terms used in the
system description. Note that the matrix Apis a lower triangular matrix and this formulation
leads to a closed-form solution in the computation of the predicted output, as shown in the next
Sub-section 3.2.
3.2 Prediction and Optimization
Suppose that, in a general description of dynamic systems, we have mcontrol signals. For a given
prediction horizon Tpand 0 ≤τ≤Tp, let the derivative of the control signal be expressed as
˙u(τ) = [ ˙u1(τ) ˙u2(τ). . . ˙um(τ)]T
and the input matrix be partitioned as
B= [B1B2. . . Bm]
where Biis the ith column of the Bmatrix. Then the ith control signal ˙ui(τ) (i= 1,2. . . m) is
described by using the following set of Laguerre functions:
˙ui(τ) = L(τ)Tηi
where L(t)T=l1(t)l2(t). . . lNi(t)and ηi=ξi
1ξi
2. . . ξi
NiT.
In the case of the NMSS model, at the current time ti, the state variable x(ti) is available through
the measurement of input and output signals. Then at the future time τ,τ > 0, the predicted state
variable x(τ|ti) is described by the following equation
x(τ|ti) = eAτ x(ti) + Zτ
0
eA(τ−γ)B˙u(γ)dγ (20)
and the predicted future state at time τis parameterised by ηas
x(τ|ti) = eAτ x(ti) + Zτ
0
eA(τ−γ)B1L1(γ)TB2L2(γ)T... BmLm(γ)Tdγη
=eAτ x(ti) + Iint(τ)1Iint (τ)2. . . Iint(τ)m
η1
η2
.
.
.
ηm
(21)
8
where Iint(τ)iis the analytical solution of the ith integral equation [21] given by the algebraic
equation
AIint(τ)i−Iint (τ)iAT
p=−BiL(τ)T+eAτ BiL(0)T(22)
where L(τ)T,L(0)Tand Apare the Laguerre function vector, initial vector and the state matrix,
respectively. Since the state matrix of the Laguerre functions Apis a lower triangular matrix,
Equation (22) is solved in a closed-form through a set of linear equations (see [21],[22]). So that,
finally, the plant output prediction can be represented by
y(τ|ti) = Cx(τ|ti) = C eAτ x(ti) + φ(τ)Tη(23)
where φ(τ) = (CIint(τ)1Iint (τ)2. . . Iint(τ)r)T. Note that the predicted plant output is
expressed in terms of the coefficient vector η.
Suppose that, at time ti, the future setpoint signals are given as
r(ti) = r1(ti)r2(ti). . . rq(ti).
Such trajectories can be as simple as a set of constants or responses from a target dynamical system.
The common objective of model predictive control is to find the control law that will drive the
predicted plant output y(τ|ti) as close as possible, in a least squares sense, to the future trajectory
of the setpoint r(ti). In continuous time, the equivalent of the well known discrete time cost function
(for example, [4], [18]) is
J=ZTp
0
[r(ti)−y(τ|ti)]TQ[r(ti)−y(τ|ti)]dτ +ZTp
0
˙u(τ)TR˙u(τ)dτ (24)
where Qand Rare symmetric matrices with Q > 0 and R≥0. By taking advantage of the
orthonormal property of the Laguerre functions, the cost function Jis then given equivalently by
J=ZTp
0
[r(ti)−y(τ|ti)]TQ[r(ti)−y(τ|ti)]dτ +ηT¯
Rη (25)
where ¯
R=diag{Ri}and Ri=λiINi×Ni(a unit matrix with dimension Ni).
Now, if we define
w(τ|ti) = r(ti)−CeAτ x(ti) (26)
then the quadratic cost function (25) can be considered in the following standard form:
J=ηT{ZTp
0
φ(τ)Qφ(τ)Tdτ +¯
R}η−2ηTZTp
0
φ(τ)Qw(τ|ti)dτ +ZTp
0
wT(τ|ti)Qw(τ|ti)dτ (27)
which can be written explicitly in terms of setpoint signal r(ti) and the state variable x(ti):
J=ηTΠη−2ηT{Ψ1r(ti)−Ψ2x(ti)}+ZTp
0
wT(τ|ti)Qw(τ|ti)dτ (28)
where
Π = ZTp
0
φ(τ)Qφ(τ)Tdτ +¯
R
Ψ1=ZTp
0
φ(τ)Qdτ; Ψ2=ZTp
0
φ(τ)QCeAτ dτ
9
The minimum of (28), without hard constraints on the variables, is then given by the simple least
squares solution:
η= Π−1{Ψ1r(ti)−Ψ2x(ti)}(29)
By applying the principle of receding horizon control (i.e. the control action will use only the
derivative of the future control signal at τ= 0), the derivative of the optimal control input for the
unconstrained problem with finite horizon prediction is
˙u(ti) =
L1(0)T0. . . 0
0L2(0)T... 0
.
.
.
0 0 ... Lr(0)T
Π−1{Ψ1r(ti)−Ψ2x(ti)}(30)
The predictive controller gain matrix K, which is associated with state variable x(ti), is calculated
using
K=
L1(0)T0. . . 0
0L2(0)T... 0
.
.
.
0 0 ... Lr(0)T
Π−1Ψ2(31)
and the closed-loop system matrix is
Acl =A−BK (32)
from which we can assess the closed-loop performance of the predictive control system.
The predictive control system contains integral action and we only need to integrate in order to
reveal the control law and the action of this integral control,
u(t) = Zt
0
˙u(τ)dτ (33)
The key strength of this particular formulation of model predictive control lies in its ability to handle
hard constraints. The essence of such constrained control is to minimize the quadratic cost function
given by (28), subject to linear inequality constraints. Because the future derivative control signal
is parameterized by the Laguerre function, we can express the constraints on the control derivative,
for 0 ≤τ≤τM, as
˙umin ≤
L1(τ)T0. . . 0
0L2(τ)T... 0
.
.
.
0 0 ... Lr(τ)T
η1
η2
.
.
.
ηr
≤˙umax
where ˙umin and ˙umax are the data vectors that contain the lower and upper limits of the derivative
of the control signals. Similarly, the constraints for the ith control signal are formulated as
ui
min ≤Ap(L(τ)T−L(0)T)ηi≤ui
max
This constrained control problem is then solved by using a quadratic programming algorithm.
10
4 SIMULATION STUDIES
The continuous-time model predictive control presented previously [21] was designed and imple-
mented using a state observer. As an alternative, the current paper focuses on the NMSS-based
alternative, where the implementation filter F(s) = tn
T(s)plays an important role in the analysis and
design since it can be used to modify the closed loop performance.
4.1 The role of filter F(s)
The filter F(s) could be selected on the basis of stochastic or other design considerations [19,
12]. Here, however, we will consider it simply from the standpoint of closed loop performance
enhancement. As pointed out previously, the poles of F(s) are a subset of the closed-loop poles,
hence their locations can be selected in conjunction with the consideration of dynamic response speed,
disturbance rejection, and robustness of the closed-loop MPC system. The example considered in
this section illustrates the effect of the filter parameters on these important properties of the control
system.
Consider an unstable system with the continuous-time transfer function
G(s) = 1
(s2−2s+ 2)(s+ 5) (34)
and a sampling interval for control of 0.0004 sec. This system has a pair of complex poles at 1 ±j. In
the design of predictive control, the weights in the cost function are selected as Q= 1 and R= 0.01:
in pole assignment terms, this selection of Qand Rensures that the dominant closed-loop poles
of the predictive control are close to −1±j[14]. Based on this observation, the scaling factor in
the Laguerre function is selected as p= 0.8, and the number of terms Nis selected to be 4. The
prediction horizon is selected as 10
p= 12.5, approximately 10 times the dominant time constant of
the Laguerre functions, so ensuring that the Laguerre functions approximate well the underlying
optimal control trajectory. The four closed-loop poles that result from this predictive control design
are −4.9999 −1.1008 + 0.6805i−1.1008 −0.6805i−0.9328
It is seen that the first three closed-loop poles are an approximation to the optimal closed-loop poles
of a linear quadratic regulator in this high gain situation, which is a consequence of the relatively
small input weighting (R= 0.01). Therefore, when constraints are not imposed, there is little scope
for further improvement in the speed of response by employing different design parameters in the
predictive control component. However, we are able to vary the locations of the filter poles and
obtain different types of closed-loop behaviour. In order to compare the performance with different
filter settings, four cases are examined with increasing response speed induced by the selection of
the three filter poles: Case A: poles at −1,−1.2,−1.3; Case B: poles at −5,−5.2,−5.3; Case C: poles
at −10, −10.2, −10.3; and Case D: poles at −50, −50.2, −50.3. Table (1) gives the details of the
closed-loop poles for these four cases: as pointed out previously, it shows that the closed-loop poles
consist of predictive control poles and the filter poles.
Table (2) shows that the feedback controller gains all change with the location of the filter poles,
except the one related to the output signal y(t). The closed-loop responses for cases A, B and C
are compared in Figure 3, where a unit setpoint signal is applied at time t= 0 and a constant, unit
input disturbance enters the system at t= 18 sec. This demonstrates that, although the nominal
setpoint response is independent of the filter pole locations, the disturbance rejection speed is largely
dependent on these locations. As expected, the closed-loop MPC system has a faster disturbance
11
Closed-loop poles
Case A −1,−1.2,−1.3, −4.99, −1.1008 ±0.6805i,−0.9328
Case B −5,−5.2,−5.3, −4.99, −1.1008 ±0.6805i,−0.9328
Case C −10, −10.2, −10.3, −4.99, −1.1008 ±0.6805i,−0.9328
Case D −50, −50.2, −50.3, −4.99, −1.1008 ±0.6805i,−0.9328
Table 1: Closed-loop Poles for Cases A-D.
State Feedback Control Gain
Case A 69.8698, 262.4908, −490.5861, 3.2913, 19.2099, 50.0790, 7.8113
Case B 13.8153, 62.2304, −34.2297 0.0373, 0.6646, 4.4434, 7.8113
Case C 9.1177, 43.8322, −8.5769 0.0049, 0.1605, 1.8781, 7.8113
Case D 5.8700, 30.5201, 6.9898 0.0000, 0.0062, 0.3215, 7.8113
Table 2: State feedback controller gains for Cases A-D obtained from equation (31).
rejection speed when the filter poles are of higher magnitude and so these can be used as very
effective closed-loop tuning parameters.
0 5 10 15 20 25 30 35 40
0
0.5
1
1.5
2
2.5
Output
Fast
Med
Slow
0 5 10 15 20 25 30 35 40
−5
0
5
10
15
20
25
Control signal
Time (sec)
Fast
Med
Slow
Figure 3: Step command input and disturbance response: plant output y(t) (top figure) for Case A,
B and C; and control signal u(t) (bottom figure) for Case A,B and C
Since the filter pole locations affect the gain of the closed-loop control system, they directly
12
0 5 10 15 20 25 30 35 40
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Output
Fast
Med
Slow
0 5 10 15 20 25 30 35 40
−5
0
5
10
15
20
Control signal
Time (sec)
Fast
Med
slow
Figure 4: Responses with model mismatch: plant output y(t) (top figure) for Case B,C and D; and
control signal u(t) (bottom figure) for Case B,C and D
affect the robustness of the MPC system. In order to examine this, we reduce the plant gain by
20 percent (i.e. multiply by a factor of 0.8) and simulate the closed-loop response for all the above
Cases. Figure 4 shows the closed-loop response for Cases B, C, and D. In Case A, the closed-loop
system actually becomes unstable, so this is not shown in the Figure. As expected, therefore, the
closed- loop system has a better tolerance to the model-plant mismatch for the higher feedback gains
that are induced by selection of faster responding filter dynamics. This result is consistent with the
previous results reported in connection with the linear robust control of unstable systems [23].
Although we see that faster filter dynamics give rise to a higher rate of response to disturbance
inputs (see Figure 3), the fast filter dynamics will tend to amplify the effects of measurement noise
in the system. In order to illustrate this, Case C is compared with Case D in a simulation study
where a nonstationary stochastic signal, in the form of integrated white noise, is added to the input,
while discrete white noise is added to the output. Here, the control objective is to maintain the
output signal at a steady value (unity). Figure 5 shows the noise and disturbance in the upper panels
and, in the lower panels, compares the output and control signal responses in the presence of these
stochastic disturbances. As expected, this demonstrates that, although the disturbance rejection
response is slower when the filter poles are at the vicinity of −10, the control signal is much less
affected by the measurement noise, and so more likely to be acceptable in most practical situations,
than when they are selected at around −50.
13
10 15 20 25 30 35
−5
0
5
Measurement Noise
10 15 20 25 30 35
−1
−0.5
0
Integrated White Noise
Time (sec)
(a) Top Figure Measurement noise; Bottom Figure Non-stationary disturbance
10 15 20 25 30 35
0.8
0.9
1
1.1
1.2
Output
10 15 20 25 30 35
−10
0
10
20
30
40
Control
Time (sec)
(b) Slower Filter Dynamics (case D)
10 15 20 25 30 35
0.8
0.9
1
1.1
1.2
Output
10 15 20 25 30 35
−10
0
10
20
30
40
Control
Time (sec)
(c) Faster Filter Dynamics (Case C)
Figure 5: The effect of the filter dynamics on the NMSS controlled system in the case of a non-
stationary stochastic disturbance
14
Closed-loop poles
N= 2 −10.3000 −10.2000 −10.0000 −5.0000 −0.8000 ±0.0001i0.0002
N= 3 −10.300 −10.2−10.0000 −5.0000 −0.9015 −0.7902 ±0.2122i
N= 4 −10, −10.2, −10.3, −4.99, −1.1008 ±0.6805i,−0.9328
N= 5 −10.30 −10.2−10.0−4.9997 −1.1393 ±1.0634i−0.9332
N= 6 −10.300 −10.2000 −10.0030 −4.9996 −1.0058 ±1.1992i−0.9333
LQR −10.3000 −10.2000 −10.0000 −4.9994 −0.9249 ±1.1348i−0.9332
Table 3: Comparison of Closed-loop Poles
4.2 Choice of Design Parameters
Another important design issue connected with the proposed MPC system is the choice of the
Laguerre function parameters pand N. In principle, the scaling parameter pis selected corresponding
to the dominant closed-loop pole of the underlying optimal control system. By judicious selection of
this parameter, the Laguerre functions can be used to capture well the underlying optimal control
trajectory with a relatively small number of terms, N. However, for an arbitrary choice of p > 0,
the accuracy of the approximation increases as the number of term Nincreases, which is guaranteed
by the convergence of orthonormal basis function model, as pointed out in Subsection 3.1. The
optimal number of terms Nis reached when the closed-loop poles of the predictive control system
do not change significantly with respect to the variation of N. This is demonstrated in Table 3,
which compares the closed-loop poles of the underlying linear quadratic regulation system with those
obtained by the predictive control system for N= 2,N= 3,N= 4, N= 5 and N= 6, with the filter
poles maintained at −10,−10.2 and −10.3. When N= 2, the closed-loop predictive control system
is unstable but it stabilizes when Nis increased to 3. And after 4, the closed-loop poles remain
relatively unchanged. Note that the closed-loop poles from the predictive control system are very
similar to the closed-loop poles of the underlying linear quadratic regulation system for N= 5 and
above. However, because the open-loop system is unstable, there is a minimum number of terms N
required for stabilization of the unstable plant (in this example, N= 3 for the choice of p= 0.8).
4.3 Comparison of NMSS based design with Observer based MPC design
with constraints
An important reason for considering NMSS-based predictive control is its observer-free structure.
In the case of discrete-time control, it has been shown [24] that there are definite advantages in the
NMSS-based alternative, particularly in the situation when the constraints become activated. In
order to investigate these differences in performance in the present continuous-time context, let us
consider the MPC design for the following continuous-time system:
G(s) = b
s2(s+a)(35)
where a= 5 and b= 1, while the filter poles for the continuous time NMSS are chosen to be at
−2,−2.2,−2.3. In the case of the observer-based design, the observer is designed using a pole-
assignment approach with the desired poles allocated at −2,−2.2,−2.3,−25, where the fourth pole
is much faster than the three dominant poles. The design parameters for both predictive control sys-
tems are chosen to be identical with p= 0.8, N= 4, Q=Iand R= 0.01. The resulting closed-loop
15
poles for the continuous time NMSS-MPC are −2,−2.2,−2.3,−5,−0.8533 ±j1.0895,−1.2651, while
those for the observer based MPC are very similar at −2,−2.2,−2.3,−5,−0.6658±j1.2682,−1.1632.
In the situation when there is no uncertainty about the system, so that the model used in the
control system design is identical to the system model (35), the two predictive control systems
perform quite similarly. The main difference in performance is that the control signal of the NMSS
system leaves the constraints earlier than the observer-based system, leading to improved closed-
loop performance. However, the differences are not very great and are certainly smaller, in this
continuous-time case, than in those encountered in the discrete-time case considered previously [24].
When there is uncertainty in the system, so that there is model-plant mismatch, the difference
between the performance of the two control system designs is rather more pronounced, as we see in
Figures 6 and 7. These show the results obtained using uncertainty analysis based on 100 Monte
Carlo realizations. Here, for each random realization, the parameter vector a= [a b] in the system
model (35) is selected from a normally distributed random vector with mean [5 1], and an associated
covariance matrix P(a), where
P(a) = 1.745 0.347
0.347 0.070 (36)
This covariance matrix was obtained by estimating the parameters in the stable part of the system
model (35), under very high noise conditions, using the RIVC algorithm in the CAPTAIN Toolbox
(see e.g. [32, 33])2. This analysis provides a demanding test for the control system designs, since
each Monte Carlo realization is based on positively correlated aand bvalues centred about the true
values with large standard deviations of 1.32 and 0.26, respectively.
0 10 20 30
−0.2
0
0.2
0.4
0.6
0.8
1
Output Response
Time
NMSS−based Control
99% bounds
Mean
0 10 20 30
−0.2
0
0.2
0.4
0.6
0.8
1
Output Response
Time
Observer−based Control
99% bounds
Mean
Figure 6: Monte Carlo uncertainty analysis: comparison of controlled output response ensembles for
NMSS-based control (left hand panel) and observer-based control (right-hand panel).
Figure 6 compares the ensemble of output responses obtained for each of the two controlled sys-
tems, while Figure 7 compares the ensembles of associated control inputs, with the ensembles shown
via the grey area within their 99 percentile bounds. It is clear that the sensitivity of the observer-
based system is noticeably higher than that of the NMSS-based system: the variance of its output
2This is a toolbox for use in MatlabTM and it can be downloaded from http://www.es.lancs.ac.uk/cres/captain/.
16
0 10 20 30
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Control Input Response
Time
NMSS−based Control
99% bounds
Mean
0 10 20 30
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Control Input Response
Time
Observer−based Control
99% bounds
Mean
Figure 7: Monte Carlo uncertainty analysis: comparison of control input response ensembles for
NMSS-based control (left hand panel) and observer-based control (right-hand panel).
response ensemble is higher and, while both implementations exhibit oscillatory characteristics for
the lowest values of the system parameters (i.e. for the most negative model-plant mismatch), this
is particularly noticeable in the control input responses of Figure 7, where the observer-based sys-
tem has larger amplitude, higher variance and less damped oscillations. It should be noted that
the results obtained here are similar to those obtained in the case NMSS-LQ/LQG control of lin-
ear systems [20], which provides a simpler alternative to NMSS-MPC in the unconstrained input
situation.
4.4 Multi-input and multi-output system
Although, for simplicity, we have so far only considered single input, single output systems in the
paper, the theory and methodology presented here also apply to multivariable systems. In this case,
however, a multivariable NMSS model form has to be utilized. The most obvious form is the one
used for the closely related NMSS control of multivariable delta operator systems [31], to which the
reader is directed for further information in this regard.
In order to illustrate how the NMSS-MPC approach works in the case of multivariable systems,
consider a simple continuous-time plant described by the transfer function:
G(s) = (s+ 5)2(s+ 2) 0
0 (s+ 5)2(s+ 2) −120(s−2) −20(s+ 1)
20(s−2) −100 (37)
where the time units are in minutes. Following the specification of the multivariable NMSS model
form, as mentioned above, the filter F(s) is selected as
F(s) = 544.48
(s+ 8)(s+ 8.2)(s+ 8.3) (38)
with the weighting matrices in the predictive control scheme selected as Q=Iand R= 0.01 ×I,
where Iis the 2 ×2 identity matrix. The parameters for Laguerre functions are selected as p1= 1.1,
17
0 1 2 3 4 5 6
−1.5
−1
−0.5
0
0.5
1
1.5
0 1 2 3 4 5 6
−1.5
−1
−0.5
0
0.5
1
1.5
Time (min)
Figure 8: Derivative of control signal ˙u1(t) (top figure) and derivative of control signal ˙u2(t) (bottom
figure). −1.03 ≤˙u1(t)≤1.35, −1.03 ≤˙u2(t)≤1.15.
p2= 2.6, N1=N2= 3, and Tp= 2. With this selection of design parameters, the closed-loop poles
are located at −1.6211, −4.1752 ±j3.86461, −3.8101 ±j2.4482, −4.0461, −6.3519 and −6.9123, in
addition to the closed-loop poles derived from the filter, i.e. (−8,−8.2,−8.3). In the simulation, the
closed-loop control results are obtained with a unity set-point change for the output y2(t), while the
setpoint for plant output y1is set to zero. The following constraints are applied to the input variables:
Rates of change of the input signals :−1.03 ≤˙u1(t)≤1.35
−1.03 ≤˙u2(t)≤1.15
Amplitude of the input signals :−0.38 ≤u1(t)≤0.38
−0.7≤u2(t)≤0.8
Figures 8-10 show the the derivatives of these input signal ˙u1(t) and ˙u2(t), the control input signals
u1(t) and u2(t) and the plant output responses y1(t), y2(t). It is clear that the closed loop responses
are satisfactory and all the constraints are satisfied. Note that now attempt was made to obtain
better decoupling on the system in this example, although this is often possible by the adjustment
of the off-diagonal elements of the weighting matrices using multi-objective optimization methods
[].
4.5 Summary of the Performance Tuning Parameters
The following list summarizes the tuning parameters in the continuous time model predictive control.
18
0 1 2 3 4 5 6
−0.4
−0.2
0
0.2
0.4
0.6
0 1 2 3 4 5 6
−1
−0.5
0
0.5
1
Time (min)
Figure 9: Control signal u1(t) (top figure) and control signal u2(t) (bottom figure). −0.38 ≤u1(t)≤
0.38,−0.7≤u2(t)≤0.8.
0 1 2 3 4 5 6
−0.6
−0.4
−0.2
0
0.2
0.4
0 1 2 3 4 5 6
−1.5
−1
−0.5
0
0.5
1
1.5
Time (min)
Figure 10: Plant output y1(t) (top figure) and plant output y2(t) (bottom figure)
19
1. Closed-loop poles of the predictive control system. The closed-loop poles comprise two
sets of poles: implementation filter poles and predictive control-loop poles.
2. Implementation filter F(s).Since the poles of the implementation filter are part of closed-
loop poles, the choice of the filter poles is directly related to the noise attenuation and the
response speed of disturbance rejection. They also affects robustness of the predictive control
system. However, they does not affect the response speed of setpoint change.
3. The weighting matrices Qand R.As usual, the Qand Rmatrices have their most
important effect on the poles contributed by the predictive controller. In general, the smaller
the elements in R, the faster the closed-loop response speed.
4. Parameters in the Laguerre functions, pand N.The Laguerre scaling factor pplays a less
significant role in the closed-loop performance, since the predictive control trajectory converges
to the underlying optimal control dictated by the choice of Qand Rmatrices, provided that
Nis chosen large enough to ensure convergence. In effect, therefore, the pair of Laguerre
parameters define a closed-loop performance that is an approximation to the underlying LQR
system. Consequently, they can be used as the additional performance tuning parameters.
Further discussions on this topic can be found in the predictive control text [22].
5 CONCLUSIONS
This paper has proposed and evaluated an approach to the design of continuous-time model pre-
dictive control using a Non-Minimal State Space (NMSS) realization of the system model and a
Laguerre function-based design. This NMSS formulation has the advantage that it leads to an im-
plementation without an observer and so its performance is not dependent on uncertainty in the
parameters of the observer design. In particular, Monte Carlo simulation results demonstrate that
the NMSS-based design is considerably less vulnerable to plant-model mismatch than the observer-
based deign. Continuous-time filters are an inherent part the NMSS implementation and their
poles become a subset of the closed-loop poles. The simulation results in the paper show how this
provides a straightforward method for tuning the closed-loop performance in order to achieve a rea-
sonable balance between speed of response, disturbance rejection, measurement noise attenuation
and robustness. The paper also shows that the difference in behaviour between the NMSS and
observer-based designs is partly due to the fact that NMSS formulation allows for instantaneous
response of the control signal to changes in the output y(t), whereas the observer-based design does
not. This suggests that the use of a reduced-order observer [14] may improve the response of the
observer-based design and this is the subject of current research.
6 Acknowledgements
Part of the work reported here was accomplished whilst Young and Gawthrop were Visiting Profes-
sors at RMIT University, Melbourne, supported by the RMIT Professorial Fund. Young completed
part of the work while visiting the School of Electrical Engineering and Telecommunications, Uni-
versity of New South Wales, Sydney. The research was also partially funded by the Royal Academy
of Engineering through international travel grants to Young and Gawthrop.
20
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