APPLICATION OF DIGITAL CELLULAR RADIO FOR MOBILE LOCATION ESTIMATION


IIUM Engineering Journal, Vol. 22, No. 1, 2021 Abdullah et al. 
https://doi.org/10.31436/iiumej.v22i1.1538 

IMPROVED CONSTRAINT HANDLING APPROACH 

FOR PREDICTIVE FUNCTIONAL CONTROL USING 

AN IMPLIED CLOSED-LOOP PREDICTION

MUHAMMAD ABDULLAH1, JOHN ANTHONY ROSSITER2,

AND ALIA FARHANA ABDUL GHAFFAR1* 

1
Department of Mechanical Engineering, International Islamic University Malaysia, 

Jalan Gombak, 53100 Kuala Lumpur, Malaysia 
2
Department of Automatic Control and Systems Engineering, 

The University of Sheffield, S1 3JD, UK 

*
Corresponding author: aliafarhana@iium.edu.my 

     (Received: 6th July 2020; Accepted: 24th September 2020; Published on-line: 4th January 2021) 

ABSTRACT:   Predictive Functional Control is a simple alternative to the traditional PID 

controller which has the capability to handle process constraints more systematically. 

Nevertheless, the most basic form of PFC has suffered from ill-posed prediction due to its 

simplicity in formulation and assumption of constant future input dynamics. Although 

some constraints can be satisfied, nevertheless the performance may be very conservative 

due to this issue. The main objective of this paper is to improve the constrained 

performance of a PFC controller with a minimum modification of the existing formulation. 

Specifically, a novel constraint handling approach for PFC is proposed based on an implied 

closed-loop prediction. Instead of assuming a constant input as deployed in the 

conventional open-loop prediction, the implied closed-loop input dynamics are utilised to 

detect future constraint violations. In addition, a future perturbation is introduced into the 

prediction structure as an extra degree of freedom for satisfying the constraints. Two 

simulation results confirm that the proposed approach gives far less conservative 

constraint handling and thus better control performance compared to the nominal PFC. 

Furthermore, this novel implementation also alleviates the well-known tuning difficulties 

and prediction inconsistency issues that are associated with conventional PFC when 

handling constraints.  

ABSTRAK: Kawalan Kefungsian Ramalan adalah alternatif mudah kepada kawalan 

tradisional PID yang mempunyai kekangan keupayaan bagi mengawal proses secara lebih 

tersusun. Namun, keadaan paling asas pada kesan PFC adalah daripada ramalan tak teraju-

rapi yang disebabkan oleh formula ringkas dan anggapan dinamik input yang sama bagi 

masa depan. Walau kekangan ini dapat diatasi, namun prestasi akan berubah secara 

konservatif disebabkan oleh isu ini. Objektif utama kajian ini adalah bagi membaiki 

kekangan prestasi kawalan PFC dengan modifikasi minimum formula yang ada. Secara 

spesifik, pendekatan nobel kawalan PFC dicadangkan berdasarkan ramalan lingkaran-

tertutup. Selain anggapan input tetap seperti yang dilakukan pada ramalan lingkaran-

terbuka yang konservatif, dinamik input yang dibuat pada lingkaran-tertutup telah 

digunakan bagi mengesan kekangan masa depan yang bertentangan. Tambahan, gangguan 

yang bakal berlaku pada masa depan telah diperkenalkan ke dalam struktur ramalan 

sebagai tambahan darjah pada kebebasan bagi mengatasi kekangan. Dua dapatan simulasi 

kajian menyetujui pendekatan yang dicadangkan dan menyebabkan sangat kurang 

kekangan pengendalian pada sistem konservatif, oleh itu kawalan yang lebih bagus pada 

prestasi  berbanding pada PFC nominal. Selain itu, pendekatan nobel ini juga 

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menghilangkan kesukaran pelarasan yang dikenali ramai dan ramalan isu tidak konsisten 

yang terdapat pada PFC konvensional apabila mengendali kekangan. 

KEYWORDS: predictive functional control (PFC); constraints handling; implied closed-loop 

prediction; constrained predictive controller 

1. INTRODUCTION  

Advances in the industrial revolution triggered the need for advanced control methods 

that can work within a constrained environment. Generally, constraints can be classified into 

two categories: i) a hard constraint refers to a physical limitation of process hardware which 

typically is in terms of input and output rates and ii) a soft constraint refers to a state or 

output limit which may be violated to some extent when required or perhaps when 

unavoidable. Satisfying all of these constraints can provide several benefits such as lower 

maintenance costs, maximisation of profits, and a safer control environment [1]. 

Nevertheless, dealing with constrained systems poses several challenges. For example, one 

often needs to deploy a classical controller such as Proportional Integral Derivative (PID) 

and ensuring this control loop does not violate the limits may not be straightforward or 

systematic in general. The more systematic, but far more expensive alternative, is to 

implement Model Predictive Control (MPC); this approach utilises predictions explicitly 

and optimises the expected behaviour by minimising a quadratic cost function subjected to 

predictions satisfying process constraints [2, 3]. However, conventional MPC solutions [4] 

are expensive and computationally demanding and hence may not be viable for applications 

where the costs and complexity need to be similar to those of PID. 

An alternative prediction-based method which is computationally simple and relatively 

inexpensive is Predictive Functional Control (PFC) [5-7]. Whereas a conventional MPC 

algorithm uses optimisation of a quadratic cost function and the entire prediction trajectory, 

the most basic PFC utilises the prediction at just one point in the future, which is far simpler 

to compute, and avoids optimisation by forcing its future prediction to match the desired 

target trajectory (often first order dynamics) at a specific coincidence horizon. Again, for 

simplicity, a well-known assumption is that the predicted future input dynamics are taken 

to be constant, as this simplifies the control law definition into the solution of a single degree 

of freedom (d.o.f) equality expression. Thus, programming the algorithm will be simple and 

it can be used on a low cost processor including a Programmable Logic Controller (PLC) 

[5,8]. Of course, as a consequence, PFC performance is suboptimal in general (e.g. 

measured against typical MPC objective functions), but because it provides straightforward 

tuning, implementation, and constraint handling, it can be effective for a number of Single 

Input Single Output (SISO) processes when compared to both PID and MPC considering its 

low computation demand and simple coding requirement [6,9]. Indeed, these features 

explain the widespread acceptance and successful implementation of PFC in many real 

industrial applications ranging from aerospace, automotive, and chemical processes [5,10]. 

The current implementation of PFC in handling input and rate constraints is by simply 

applying a clipping or saturation method. If the current manipulated input is violating the 

limits, then a maximum or minimum value will be sent to the plant. This simple framework 

can alleviate issues such as integral wind-up, due to its use of predictions, although it may 

still be suboptimal [5]. As for state and output constraints, the traditional PFC approach uses 

multiple controllers in parallel whereby the primary regulator is tuned to track the target 

trajectory while the other one is tuned to satisfy the limits [11], as shown in Fig. 1. A real-

time prediction is made based on the primary input by a supervisor and if it violates the 

limits, then the input will be switched. Although this simple concept often works and 

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provides a fast control solution, it can nevertheless be considered obsolete with modern 

computing facilities and moreover lacks sufficient rigour. Besides, it is easy to formulate 

scenarios where this approach fails or leads to significant performance degradation [12] and 

thus improvements are needed.  

 

Fig. 1: Schematics of traditional PFC when handling output constraint. 

A core aspect of efficient and accurate constraint handling is to ensure that the 

optimised predictions and the expected closed-loop behaviour are consistent [1,3, 12-13]. 

When there is inconsistency, ensuring that the model predictions meet constraints may be a 

poor reflection of whether the resulting closed-loop responses satisfy constraints, leading to 

either unpredicted constraint violations or the controlled response becomes very 

conservative in satisfying the constraints. In fact, the core weakness of the traditional PFC 

is its conventional assumption of its future input predictions being constant; in reality, this 

assumption is often inconsistent with the resulting closed-loop input to a significant degree 

[12]. Hence, even though PFC can check the expected constraint violations over a long 

prediction horizon, yet due to the prediction mismatch, the control law will produce a very 

conservative solution.  

In previous work [12], an alternative parameterisation of the d.o.f. had been introduced 

within the PFC framework in order to improve the predictions and closed-loop behaviour’s 

consistency, thus resulting in more accurate constraint handling and systematic tuning. 

Specifically, a Laguerre based parameterisation of the predicted future input replaced the 

conventional constant input assumption of PFC. This modification showed improvements 

in the prediction consistency with low order models and thus enabled less conservative 

constraint handling [14]. However, the use of a simple first order Laguerre function is still 

somewhat limited, and for higher-order systems, significant prediction inconsistency may 

still exist. While it is possible to increase the order of the Laguerre polynomial [15], this is 

not straightforward in general [16] and not in line with the simplicity concept which is an 

essential facet of PFC.  

In summary, the specific objective of this new work is to propose a novel constraint 

handling concept for PFC, which can be applied to higher order systems and which 

minimises the prediction inconsistency, thus ensuring that the inequalities used for 

constraint handling are as accurate as possible and reducing conservatism and risk. In real 

applications, this is very beneficial as the system can be pushed to work nearer to its high 

profit boundary region without causing unexpected violations and thus fatigue damage. A 

core research gap in PFC is that there are no attempts in deploying the well-known concept 

from dual-mode MPC [3] whereby the predictions are made up by assuming a known 

feedback control law is in place for the asymptotic part of the predictions. In the case of 

PFC, a nominal control law is available for the constraint-free case and thus a simple 

proposal is to exploit this control law within the prediction formulation used for constraint 

handling. Nevertheless, the nominal PFC control law also needs to be modified further by 

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| · · · 

introducing a new slack variable or degree of freedom (d.o.f.) in the formulation to be 

utilised for handling the constraints. Thus, this paper will make two main contributions: (i) 

first, it will demonstrate how an effective d.o.f. can be incorporated alongside a suitable 

dual-mode formulation and (ii) second, it will show how the constraint handling can be 

implemented.  

2.   BACKGROUND ON PREDICTIVE FUNCTIONAL CONTROL 

A brief review of the key assumptions, notations, and principles of conventional 

Predictive Functional Control laws together with a constraint handling method as proposed 

in [12] is described in this section. 

2.1  Target Trajectory and Control Law Definition 

A core principle of PFC is, at sample 𝑘, a desired target trajectory 𝑟(𝑘 +𝑖|𝑘) is defined, 
where  𝑖 = 1,2,… for future samples 𝑘 +𝑖 based on a desired steady-state target 𝑅. The 
target trajectory is defined as a first-order-response from the current process output 𝑦𝑝(𝑘) 
to 𝑅, hence: 

𝑟(𝑘 +𝑖|𝑘) = (1−𝜆𝑖)𝑅 +𝜆𝑛𝑦𝑝(𝑘),     𝑖 = 1,2,… (1) 

where, 𝜆 is the implied desirable pole position. For industrial users, one can use a desired 
settling time, Ts and determine 𝜆 from the relationship [5]:  

𝜆 = 𝑒
−3𝑇

𝑇𝑠  (2) 

with the sampling period, T. The second core principle in PFC is to force the system 

prediction, that is 𝑦𝑝(𝑘 +𝑛|𝑘), to match the target n samples into the future, where n is 

representing the coincidence horizon (second tuning parameter) [5]. Hence, the control law 

can be summarised as: 

𝑟(𝑘 +𝑛|𝑘) = (1−𝜆𝑛)𝑅 +𝜆𝑛𝑦𝑝(𝑘) = 𝑦𝑝(𝑘 +𝑛|𝑘) (3) 

This is shown in Fig. 2 for clarity. The two tuning parameters or user choices are the 
desired closed-loop pole, λ, and the coincidence horizon, n.  

 
Fig. 2: PFC control law definition. 

Remark 2.1: For clarity of formulation, this work will not discuss the PFC framework that 

is related to a delay control problem and non-constant targets because these issues require 

more complex algebra and moreover have no effect on the core concepts proposed in this 

paper. Details are available in the references [2,5]. 

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2.2  Prediction Structure and Degrees of Freedom 

For solving the control law in Eq. (3), it is necessary to define the output prediction             

𝑦𝑝(𝑘 +𝑛|𝑘) and its dependence on the control signal 𝑢(𝑘|𝑘). Prediction is standard in the 

literature [1] so just a brief illustration is given here. It is noted that this formulation ensures 

unbiased prediction in the steady-state and therefore offset free tracking. 

Without loss of generality, the formulation in this paper utilises a state space model as 

it is easier to demonstrate the proposed control law in the next section. The one-step ahead 

output prediction for a state-space model with states, inputs and output 𝑥(𝑘),𝑢(𝑘),𝑦𝑚(𝑘) 
can be formed as: 

𝑥(𝑘 +1) = 𝐴𝑥(𝑘)+𝐵𝑢(𝑘) ;  𝑦𝑚(𝑘 +1) = 𝐶𝑥(𝑘 +1) (4) 

Note the use of subscript m to denote model. Typically, the real process output 𝑦𝑝 differs 

from the model 𝑦
𝑚

, so process predictions are estimated using the measured signal 

𝑑(𝑘) given in the structure illustrated in Fig. 3. Hence: 

𝑑(𝑘) = 𝑦𝑝(𝑘)−𝑦𝑚(𝑘)      →  𝑦𝑝(𝑘 +𝑖|𝑘) = 𝑦𝑚(𝑘 +𝑖|𝑘)+𝑑(𝑘) (5) 

Consequently, the predicted plant output with the assumed predicted constant future input 

(𝑢(𝑘) =  𝑢(𝑘 +1|𝑘) =  𝑢(𝑘 +2|𝑘),…) can be written as:  

𝑦𝑝(𝑘 +𝑖|𝑘) = 𝐶𝐴
𝑖𝑥𝑘 +[𝐶𝐵,𝐶𝐴𝐵, . .𝐶𝐴

𝑖−1𝐵][
1
⋮
1
]
⏟
𝐿

𝑢(𝑘)+𝑑(𝑘)  (6) 

Remark 2.2: For constraint handling it is convenient to stack the output predictions into 

matrix form over some prediction horizon and hence one can define: 

𝑦
𝑝
→ 
(𝑘+𝑛|𝑘) = 𝐹𝑥(𝑘) +  𝐻𝑢(𝑘) +  𝐿𝑑(𝑘)  (7) 

𝑦𝑝
→ 
(𝑘 +𝑛|𝑘) =

[
 
 
 
𝑦𝑝(𝑘 +1|𝑘)

𝑦𝑝(𝑘 +2|𝑘)

⋮
𝑦𝑝(𝑘 +𝑛|𝑘)]

 
 
 

 ;  𝐹 = [

𝐶𝐴
𝐶𝐴2

⋮
𝐶𝐴𝑛

] ;  𝐻 = [

𝐶𝐵 0 0 ⋯
𝐶𝐴𝐵 𝐶𝐵 0 ⋯
⋮ ⋮ ⋮ ⋱

𝐶𝐴𝑛−1𝐵 𝐶𝐴𝑛−2𝐵 𝐶𝐴𝑛−3𝐵 …

]  

 
Fig. 3: Structure of an Independent Model (IM). 

2.3  The PFC Control Law 

  The conventional PFC control law is defined by substituting the predictions in Eq. (6) 

into the control law definition Eq. (3). It is understood that one must first choose the desired 

closed-loop pole, λ (0 < λ< 1), and the coincidence horizon, 𝑛. In fact, the choice of 𝑛 is less 
simple in general [13], but that discussion is outside the focus of this paper. Defining the nth 

row of matrices 𝐹, 𝐻 to be 𝐹𝑛, 𝐻𝑛 respectively, the control law Eq. (3) is defined by solving 
the following for 𝑢𝑘 (this form would be used in real implementation): 

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HnLu(k)+Fnx(k)+d(k) = (1−λ
n)R+λnyp(k) (8) 

Equation (8) can be solved to determine 𝑢𝑘 but, in order to facilitate some loop analysis 

(nominal case), this can also be expressed by substituting 𝑦𝑝(𝑘) = 𝑦𝑚(𝑘)+𝑑(𝑘) = 𝐶𝑥(𝑘)+

𝑑(𝑘). Hence, the control law Eq. (3) is presented as: 

(1−𝜆𝑛)(𝑅 −𝑑(𝑘)) = 𝐻𝑛𝐿𝑢(𝑘)+𝐹𝑛𝑥(𝑘)−𝜆
𝑛𝐶𝑥(𝑘) (9) 

Thus, the required control input can be represented as an augmented state feedback: 

𝑢(𝑘) = [
−(𝐹𝑛−𝐶𝜆

𝑛)

𝐻𝑛𝐿

(1−𝜆𝑛)

𝐻𝑛𝐿
]

⏟            
𝑘

[
𝑥(𝑘)

(𝑅 −𝑑(𝑘))
]

⏟        
�̅�(𝑘)

 (10) 

where K is the state feedback gain and �̅�𝑘 is the augmented state. 

Remark 2.3: The main reason why PFC is attractive is due to its simplicity in computing 

the control law of Eq. (10). Since 𝐻𝑛𝐿 and 𝐹𝑛 are only needed for a single horizon, 𝑛, the 
required computation is relatively straightforward and indeed can be calculated off-line [2, 

5]. 

2.4  Conventional Constraint Handling with Open-loop Predictions 

A good constraint handling approach for PFC should be systematic and also simple 

enough to avoid the usage of common optimisers deployed in more expensive MPC 

algorithms and reference governor strategies [11, 17]. Let the rate, input and output 

constraints at every sample be defined respectively as: 

Δ𝑢 ≤ Δ𝑢(𝑘) ≤ Δ𝑢  ;  𝑢 ≤ 𝑢(𝑘) ≤ 𝑢  ;  𝑦𝑝 ≤ 𝑦𝑝(𝑘) ≤ 𝑦𝑝̅̅̅ (11) 

As discussed before, the classical PFC algorithm deploys a saturation method to deal 

with input and rate constraints [1,5], but that can be severely suboptimal in practice due to 

the difference between the input prediction and the implied closed-loop evolution. This 

inaccuracy is even more notable when output/state constraints are introduced in the 

optimised open-loop predictions, [14], as detailed next. The constraint handling procedure 

of PFC, as proposed by [12], is outlined as: 

(1) Given a suitably long validation horizon, 𝑛𝑖 (horizon used to check a future limit 
violation), the whole set of future predictions as formulated in Eq. (7) is computed. 

(2) Group all the constraints (input, rate, and output) and the system predictions into a 
single set of linear inequalities as: 

𝑀𝑢(𝑘) ≤ 𝒇(𝐾) (12) 

𝑀 =

[
 
 
 
 
 
1
−1
1
−1
𝐻𝐿
−𝐻𝐿]

 
 
 
 
 

 ;  𝒇(𝑘) =

[
 
 
 
 
 
 
𝑢
−𝑢

Δ𝑢
−Δ𝑢

𝐿𝑦𝑝
−𝐿𝑦𝑃]

 
 
 
 
 
 

−

[
 
 
 
 
 

0
0

𝑢(𝑘 −1)

−𝑢(𝑘 −1)

𝐹𝑥(𝑘)+𝐿𝑑(𝑘)

−𝐹𝑥(𝑘)−𝐿𝑑(𝑘)]
 
 
 
 
 

 

where 𝒇(𝑘) depends on the states and the limits. It is advisable that the output 
predictions horizon, 𝑦

𝑝
→ 
(𝑘+𝑛), and the row dimension of H, need to be long enough 

in order to capture all the important dynamics [17].  

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(3) It is noted that the predictions will satisfy constraints if the selected input u(k) 
satisfies Eq. (12), that is, inequalities 𝑀𝑢(𝑘) ≤  𝒇(𝑘) can be used explicitly to 

determine this condition. 

Next, a modern PFC constraint handling algorithm is given. This concept utilises a simple 

“for” loop to choose the 𝑢(𝑘), that is nearest to the unconstrained solution shown in Eq. (10) 
but does not violate the limits in Eq. (12). 

Algorithm 2.1. At each time step or sample: 

(1) Compute the unconstrained input 𝑢(𝑘) as defined in Eq. (9). 

(2) Define the vector 𝒇(𝑘), from Eq. (13) where vector M is fixed. 

(3) A simple for loop is utilised to inspect each row of M, here: 

(a) The ith constraint; represented by the ith row of 𝑀𝑢(𝑘) ≤  𝒇(𝑘), is 

checked using 𝑎𝑖 = 𝑀𝑖𝑢(𝑘)−𝒇𝑖(𝑘). 

(b) If 𝑎𝑖 > 0, then 𝑢(𝑘) = 𝒇𝑖(𝑘)/𝑀𝑖; else the original value of 𝑢(𝑘) is 
retained. 

It has been proven in [14] that for the nominal case where 𝑑(𝑘) is assumed constant and also 
for stable open-loop systems, a recursive feasibility property for Algorithm 2.1 is guaranteed 

where 𝑢(𝑘) converges to a feasible value nearest to the unconstrained choice. 

Remark 2.4: Since Algorithm 2.1 utilises a straightforward for-loop, the programming, 

coding, and computation becomes far simpler and faster compared to more demanding 

MPC approaches based on optimisation of a quadratic program. In addition, the algorithm 

is more systematic and faster than most of the conventional PID methods. However, it 

should be noted that the usage of PFC is often limited to a single-input-single-output (SISO) 

process and is rarely used for multi-input, multi-output (MIMO) systems [5]. 

2.5  Limitations of Conventional PFC Constraint Handling 

The main reason behind the popularity of PFC in industrial processes is due to its 

transparent tuning parameters. Here, one only needs to select the desired closed loop time 

constant, which is equivalent to 𝜆. Then, the different choices of coincidence horizon, 𝑛, can 
be quickly explored using a computer while displaying the associated responses. Thus, a 

user can then select a suitable coincidence horizon, 𝑛, for their system. Nevertheless, there 
are two major trade-offs that need to be considered [8, 13, 15]: 

(1) Smaller values of coincidence horizon, 𝑛, drive the output response closer to the 
target trajectory where the effect of tuning parameter, 𝜆, becomes more significant. 
However, smaller, 𝑛, typically leads to poorer consistency between the predicted and 

actual responses, and thus inaccurate constraint handling and potentially less 

desirable behaviour. 

(2) Conversely, larger, 𝑛, improves the prediction consistency, yet the closed-loop 
behaviour then approaches open-loop dynamics which decreases the significance 

and efficacy of 𝜆 as a tuning parameter. 

Prediction mismatch is potentially catastrophic for reliable constraint handling as 

predictions satisfying constraints need not imply that the resulting closed-loop responses 

will satisfy constraints, and vice versa. Consequently, the constraint handling algorithm may 

be either highly conservative [12, 14] and/or overly aggressive. 

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Remark 2.5: It is well noted that one of the weaknesses of conventional PFC algorithm is 

its prediction mismatch [1] between its optimised predictions and the closed-loop behaviour 

that results. This inconsistency arises mainly due to the assumed constant future input 

prediction. Consistency between predictions and actual behaviour is needed to ensure good 

behaviour, especially for cases dealing with higher order systems. 

3.   PFC CONSTRAINT HANDLING BASED ON AN IMPLIED 

CLOSED-LOOP PREDICTION 

The main weakness of conventional PFC, as indeed with many conventional MPC 

algorithms, is its assumption of a constant future input. This assumption is typically 

inconsistent with the actual closed-loop behaviour, and thus incorrect constraint handling 

decisions are made [12]. To some extent, dual-mode approaches [18] can overcome this 

issue by ensuring the predictions are as close as possible to the actual closed-loop behaviour, 

but of course these are computationally demanding. The main contribution here is to exploit 

a similar concept, that is, to use implied closed-loop predictions for constraint handling, but 

in a simplified manner to maintain the core selling benefits of PFC. The proposal is novel 

in that dual-mode approaches still use open-loop predictions in transients, but with a long-

term input trajectory that emulates the implied closed-loop after transients. Conversely, we 

will predict the expected closed loop behaviour explicitly for the entire horizon. 

In summary, the expected benefits of this proposal are: 

• More accurate predictions when checking output/state constraints implies less 

conservative decision making. 

• A smaller coincidence horizon (n) can be selected without worrying about the 

prediction inconsistency which will retain more tuning capability. 

• The proposed approach only requires a simple modification in the offline 

computations required for constraint handling. 

3.1  Closed-loop Prediction PFC (CL-PFC) with Constant Input Perturbation 

In order to use closed-loop predictions for constraint handling, an extra d.o.f. is required 

in the predictions. The proposal here is to add a constant perturbation to the implied closed-

loop input [1] as this structure has been found to be effective elsewhere. Hence, based on 

the control law of Eq. (10), the closed-loop prediction model combines the open-loop model 

and control law, with a constant input perturbation, 𝑐(𝑘), as the new d.o.f. Thus: 

{𝑥(𝑘 +1|𝑘) = 𝐴𝑥(𝑘)+𝐵𝑢(𝑘);𝑢(𝑘) = 𝐾𝑥(𝑘)+𝑐(𝑘);𝑐(𝑘 +𝑖|𝑘) = 𝑐(𝑘), 𝑖 > 0}    (13) 

𝐴,𝐵 are the augmented model with respect to �̅�𝑘.The implied closed-loop predictions take 

the following form (for clarity of presentation, we define ϕ = [𝐴+𝐵𝐾] ): 

          

[
 
 
 
𝑦𝑝(𝑘 +1|𝑘)

𝑦𝑝(𝑘 +2|𝑘)

⋮
𝑦𝑝(𝑘 +𝑛|𝑘)]

 
 
 

= [

𝐶ϕ

𝐶ϕ2

⋮
𝐶ϕ𝑛

]

⏟  
𝑃𝑐

𝑥(𝑘)+

[
 
 
 
𝐶𝐵 0 0 ⋯

𝐶ϕ�̅� 𝐶𝐵 0 ⋯
⋮ ⋮ ⋮ ⋱

𝐶ϕ𝑛−1𝐵 𝐶ϕ𝑛−2𝐵 𝐶ϕ𝑛−3𝐵 …]
 
 
 

⏟                      
𝐻𝑐

𝐿𝑐(𝑘)       (14) 

         [

𝑢(𝑘|𝑘)

𝑢(𝑘 +1|𝑘)
⋮

𝑢(𝑘 +𝑛|𝑘)

] = [

−𝐾
−𝐾𝜙
⋮

−𝐾𝜙𝑛−1

]

⏟      
𝑃𝑐𝑢

𝑥(𝑘)+[

𝐼 0 0 ⋯
−𝐾�̅� 𝐶�̅� 0 ⋯
⋮ ⋮ ⋮ ⋱

−𝐾ϕ𝑛−2�̅� 𝐶ϕ𝑛−3�̅� 𝐶ϕ𝑛−4�̅� …

]

⏟                        
𝐻𝑐𝑢

𝐿𝑐(𝑘)          (15) 

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         [

Δ𝑢(𝑘|𝑘)

Δ 𝑢(𝑘 +1|𝑘)
⋮

Δ𝑢(𝑘 +𝑛|𝑘)

] = 𝑃𝑐𝐷𝑢𝑥(𝑘)+𝐻𝑐𝐷𝑢𝑐(𝑘)+𝑄𝑐𝐷𝑢𝑢(𝑘)                                                 (16 

Remark 3.1: For the unconstrained case, the value of perturbation as in Eq. (13) will be 

𝑐(𝑘) =  0 since the term 𝑢(𝑘) =  𝐾�̅�𝑘 is derived from the original control law. Thus, a non-
zero 𝑐(𝑘) is required solely for constraint handling and the underlying loop tuning is based 
on the original PFC law. 

3.2  Constraint Handling 

For constraint handling, a similar procedure as discussed in Section 2.4, will be 

employed. The prime conceptual improvement here is the d.o.f., which is now expressed in 

terms of perturbation 𝑐(𝑘) instead of the input 𝑢(𝑘) and one needs to consider the entire 
input prediction and not just the first value. Thus: 

𝑀𝑐𝑘 ≤ 𝒇𝑘, (17) 

𝑀 =

[
 
 
 
 
 
𝐻𝑐𝑢𝐿
−𝐻𝑐𝑢𝐿
𝐻𝑐𝐷𝑢𝐿
−𝐻𝑐𝐷𝑢𝐿
𝐻𝑐𝐿
−𝐻𝑐𝐿 ]

 
 
 
 
 

 ; 𝒇(𝑘) =

[
 
 
 
 
 
 
 

𝐿𝑢
−𝐿𝑢

𝐿Δ𝑢
−𝐿Δ𝑢

𝐿(𝑦𝑝 −𝑑(𝑘))

−𝐿(𝑦𝑃 −𝑑(𝑘))]
 
 
 
 
 
 
 

−

[
 
 
 
 
 
 

𝑃𝑐𝑢𝑥𝑘
−𝑃𝑐𝑢𝑥𝑘

𝑃𝑐𝑢𝑥𝑘 +𝑄𝑐𝐷𝑢𝑢(𝑘 −1)

−𝑃𝑐𝑢𝑥𝑘 +𝑄𝑐𝐷𝑢𝑢(𝑘 −1)

𝑃𝑐𝑥𝑘
−𝑃𝑐𝑥𝑘 ]

 
 
 
 
 
 

 

In a similar manner to Algorithm 2.1, the goal is to select the d.o.f. c(k) that is closest to its 

unconstrained value while satisfying the constraints in Eq. (11) (equivalently inequalities 

(17)). 

Algorithm 3.1. At each sample: 

(1) Set 𝑐(𝑘) = 0. 

(2) Define vector �̅�𝑘 as in Eq. (17) with the assumption that �̅�
 does not change. 

(3) A simple loop is utilised to inspect each row of �̅�, where: 

(a) The ith constraint represented by the ith row of  �̅�𝑐(𝑘) ≤ �̅�(𝒌) is checked 

with 𝑎𝑖 = �̅�𝑖𝑐(𝑘)− �̅�𝑖(𝑘). 

(b) If 𝑎𝑖 > 0, then 𝑐(𝑘) = �̅�𝑖(𝑘)/�̅�𝑖, else use the existing 𝑐(𝑘). 
(4) Define the input from 𝑢(𝑘) = −𝐾�̅�𝑘  +  𝑐(𝑘). 

One can argue that better prediction consistency and more accurate constraint handling 

is obvious (and will be demonstrated in the examples section). Nevertheless, it is also 

important to establish that the proposed algorithm does indeed retain feasibility, that is the 

recursive feasibility property in the nominal case and moreover subject to changes in the 

target R. 

Theorem 3.1: When there is no change in 𝑑(𝑘) and also in the nominal case, Algorithm 3.1 
can guarantee recursive feasibility with any desired target, R. 

Proof: Assume at sample k−1, the solution is feasible, then with no change in R, the choice 
𝑐(𝑘) =  𝑐(𝑘 −1)  must retain feasibility as the implied predictions at the previous and current 

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sample would then be identical. Hence, it remains only to show that 𝑐(𝑘) can be chosen to 

retain feasibility, irrespective of any changes in R. Consider then the nominal control law 

given in Eq. (10) and augment this with the proposed perturbation term 𝑐(𝑘) as in Eq. (13), 
hence: 

𝑢(𝑘) = [
−(𝐹𝑛−𝐶𝜆

𝑛)

𝐻𝑛𝐿

(1−𝜆𝑛)

𝐻𝑛𝐿
1][

𝑥(𝑘)

𝑅 −𝑑(𝑘)

𝑐(𝑘)
] (18) 

It is clear that the terms 𝑅 and 𝑐(𝑘) both act as scaled additive terms to 𝑢(𝑘), and thus any 
change in R can be countered by a suitable scaled change in 𝑐(𝑘) so that the predictions at 
sample 𝑘 match those from sample k − 1. In other words, recursive feasibility is guaranteed!  

Corollary 3.2: In practice, Algorithm 2.1 uses the smallest perturbation 𝑐(𝑘) possible so 
that performance is as close as possible to that desired. This also means that, if it is feasible, 

then 𝑐(𝑘) is driven to zero and offset free tracking will be ensured. 

The reader will note that we have not attempted to guarantee recursive feasibility 

subject to parameter uncertainty and disturbance changes as the literature on that area is far 

more complicated than would fit with the PFC philosophy: that is, simple coding and 

implementation. Of course, in practice, all MPC algorithms have some inherent degree of 

robustness and thus this lack of a guarantee is rarely a problem except for exceptional and 

challenging cases where one is unlikely to use PFC anyway. 

4.   METHODOLOGY FOR RESULT ANALYSIS 

In this paper, the developed formulation will be used to set up the control algorithm for 

an arbitrary SISO system with constraints. Instead of quantitative analysis, this paper adopts 

a qualitative analysis since the main objective is to show the improvement between the 

proposed PFC with a conventional PFC. The control performance can be evaluated by 

observing its convergence speed to the desired setpoint within the constrained environment. 

Besides, for clarity of presentation, the obtained simulation results will not be compared 

with other types of controllers as that is not the main contribution of this paper and 

moreover, those comparisons are already well known in the existing literature. Indeed PFC 

performance cannot be contrasted with more comprehensive MPC alternatives as those 

deploy more complicated algorithms and are far more costly to implement and maintain. 

Interested readers can refer to these references on the comparison between MPC, PID and 

PFC [19, 20].  

5.   NUMERICAL EXAMPLES 

This section provides two numerical examples with the characteristics of typical 

industrial processes to highlight the effectiveness of the proposed Algorithm 3.1 compared 

to the traditional PFC algorithm such as Algorithm 2.1. For clarity, the following 

comparisons are made: 

(1) Investigation of the prediction behaviour of the two algorithms and their associated 
consistency with the resulting closed-loop dynamics. 

(2) Comparison of the closed-loop controller performance when dealing with output 
constraints. 

(3) Demonstration of the ability of the proposed controller to handle multiple constraints 
and uncertainties. 

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In order to emphasise a variety of characteristics, the examples use two second order 

dynamics systems as follows: 

(1) A second order over-damped system model (pole at 0.4 and another pole at 0.8), 
with constraints of 𝑢 = 0.12, ∆𝑢 = 0.05, and 𝑦 = 1.5: 

𝐺1 =
𝑧−1 +0.3𝑧−2

1−1.2𝑧−1 +0.32𝑧−2
;  𝑛 = 2 ;λ = 0.7 (19) 

and to include the model uncertainty in the analysis, the real process representation is 

given by: 

𝐺1,𝑝 =
𝑧−1 +0.28𝑧−2

1−1.21𝑧−1 +0.3𝑧−2
 (20) 

(2) A second order non-minimum phase model (zero at 1.6 and poles at 0.6 and 0.85, 

respectively), with constraints of 𝑢 = −3, ∆𝑢 = −1, and 𝑦 = 1.5: 

𝐺2 =
0.1𝑧−1 −0.16𝑧−2

1−1.45𝑧−1 +0.51𝑧−2
;  𝑛 = 5 ;λ = 0.7 (21) 

     and its real process representation is given by: 

𝐺2,𝑝 =
0.11𝑧−1 −0.15𝑧−2

1−1.44𝑧−1 +0.5𝑧−2
 (22) 

The reader is reminded that λ is selected based on the user preference, which is 

associated to the desired closed-loop time response (CLTR). The selection range is between 

0 < λ < 1. In principle, a smaller λ gives faster convergence and vice versa. Normally, the 

coincidence horizon, n, is taken to be at the low end of the values which gives good 
performance as this ensures that λ is also an effective tuning parameter. 

5.1  Effect of Prediction Consistency when Identifying Future Constraint Violations 

In this subsection, only the output constraint will be considered to highlight the 

performance improvement where the output limit for both examples is set to 𝑦 =  0.8. In 
addition, the plant transfer function is set to be equal as the model 𝐺𝑝 = 𝐺. Notably, and as 

expected, the unconstrained control law for both PFC and CL-PFC for examples 1 and 2 are 

the same as shown in Figs. 3 and 4, respectively. Since PFC is a discrete controller, the 

results are plotted in terms of sampling instant instead of time.   

However, in the presence of output constraints, their performances may differ since the 

adjustment is linked to different inequalities shown in Eq. (12) and Eq. (17), respectively. 

The first example is based on open-loop predictions, while the second example is based on 

implied closed-loop predictions. In this section, we present the optimised predictions for 

each algorithm, with and without constraint handling, and demonstrate the differences in the 

resulting decision making. The core observations are as follows: 

• The unconstrained predictions at the first sample (same for both approaches) violate 

output constraints for large prediction horizons (refer to Figs. 5 and 6). 

• Based on Fig. 7 and Fig. 8, Algorithm 2.1 (PFC, red dotted line) reduces the choice 

of constant 𝑢(𝑘) substantially to ensure that the maximum value of 𝑦𝑝(𝑘 +𝑖|𝑘) 
satisfies the constraints, but the consequence is a small initial value of 𝑢(𝑘)  and a 
much slower responding output prediction for both examples 1 and 2 respectively. 

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• Conversely, Algorithm 3.1 (CL-PFC, blue dashed line) recognises that future inputs 

will change, thus the predictions used for constraint handling can retain the fast 

transients while not exceeding the constraints. 

 

  

Fig. 3: Unconstrained closed-loop input and output of PFC 

and CP-PFC for example G1. 
 

 
 

 
Fig. 4: Unconstrained closed-loop input and output of PFC  

and CP-PFC for example G2. 
 

 
 

 
Fig. 5: Input and output predictions at the first sample of PFC  

and CP-PFC for example G1. 

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Fig. 6: Input and output predictions at the first sample of PFC  

and CP-PFC for example G2. 
 

 
Fig. 7: Constrained input and output behaviour of PFC and CL-PFC for example 

G1. 
 

 
Fig. 8: Constrained input and output behaviour of PFC and CL-PFC for example 

G2. 

In summary, due to significant prediction mismatch in the open-loop predictions and 

the corresponding closed-loop behaviour in Algorithm 2.1, the constraint handling is highly 

conservative/suboptimal. Conversely, the proposed Algorithm 3.1 removes this mismatch 

thus ensuring effective and accurate constraint handling, therefore, no more loss of 

performance than necessary.  

5.2  Handling Multiple Constraints and Uncertainties 

In this section, the CL-PFC algorithm is tested with multiple constraints that include 

input, input rate, and output constraints to demonstrate that Algorithm 3.1 deals effectively 

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with all of these simultaneously. Moreover, some parameter uncertainties are introduced to 

demonstrate inherent robustness as discussed before in the introduction part of Section 4. In 

addition, to demonstrate the recursive feasibility properties more strongly, the examples 

include a switch from a feasible to an infeasible R. Figures 9 and 10 show the constrained 
closed-loop performance of CL-PFC for both example 1 and 2, respectively. It is noted that 

all the constraints are satisfied without a conflict and good performance is retained even in 

the presence of uncertainty while the system converges to the closest feasible output. 

 
Fig. 9: Constrained and unconstrained performance for example G1. 

 

 

Fig. 10: Constrained and unconstrained performance for example G2. 

 

5.3   Possible Application and Practical Significance 

 As discussed in Section 1, PFC is a simple and practical controller that can be 

implemented in any processor with a few lines of coding. The main attraction of PFC is that 

it can provide a satisfactory control performance when handling constraints with minimum 

computation and transparent tuning parameter of the desired closed-loop time constant. It 

should be noted that this work does not claim that PFC is better than MPC and in fact, it can 

never be, due to the simplification of the algorithm. For low order processes and SISO 

systems, PFC can work well whereas the implementation of MPC will be very expensive 

for a simple application.  

With the proposed framework, the user can get a less conservative control performance 

compared to a traditional PFC. For example, if this controller is implemented in a car, the 

car will slow down at the right time before cornering. Conversely, with a traditional PFC, 

the car will slow down long before the corner which will make it unnecessarily 

conservative/slow. Given the nominal performance is the same as conventional PFC, the 

authors expect that application of CL-PFC to real systems will demonstrate similar benefits 

[3, 17] in the constrained case; this constitutes future work.    

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6.   CONCLUSIONS 

This work presents a novel constraint handling approach for PFC that employs some of 

the recent ideas from a more conventional MPC. Specifically, by utilising the concept of an 

implied closed-loop prediction one can make the constraint handling decisions more 

accurate. With this modification, the constraint handling performance becomes more 

consistent and reliable when compared with the nominal PFC, as demonstrated in the 

numerical example simulations. With this finding, a simple system can work nearer to its 

constraints limits, but without constraint violation or sacrificing the desired dynamic 

performance. It is also worth highlighting that the required modifications for deploying this 

method are simple, thus in-line with the key requirements for PFC. In a real application, this 

benefit may be translated in terms of more production profit and a safer working 

environment as well as simple coding and maintenance.  

Nevertheless, it should be noted that the constrained control performance is not claimed 

to be optimal. The underlying algorithm is still based on a simplified and far cheaper version 

of conventional MPC so the performance is not expected to be comparable. A few aspects 

which can be addressed next may include the rigorous extension of this approach to handle 

more challenging open-loop dynamics such as unstable, marginally stable, and oscillating 

systems. In addition, a more systematic and robust tuning procedure in selecting a suitable 

coincidence horizon in the presence of parameter uncertainty, noise and disturbances also 

needs to be investigated. A systematic sensitivity analysis that describes the relationship 

between the PFC tuning parameter and those uncertainties may provide useful information 

to a user before deploying this controller.  

ACKNOWLEDGEMENTS 

The author would like to acknowledge the International Islamic University Malaysia 

Research Acculturation Grant Scheme 2018 (IRAGS18-013-0014) and Ministry of Higher 

Education Malaysia for funding this work. 

REFERENCES 

[1] Rossiter, J. A. (2018). A first course in predictive control. CRC press.  
[2] Haber, R., Bars, R., & Schmitz, U. (2012). Predictive control in process engineering: From 

the basics to the applications. John Wiley & Sons. 

[3] Mayne, D. Q., Rawlings, J. B., Rao, C. V., & Scokaert, P. O. (2000). Constrained model 
predictive control: Stability and optimality. Automatica, 36(6), pp 789-814.  

[4] Clarke, D. W., & Mohtadi, C. (1989). Properties of generalized predictive control. 
Automatica, 25(6), pp 859-875.  

[5] Richalet, J., & O'Donovan, D. (2009). Predictive functional control: principles and 
industrial applications. Springer Science & Business Media. 

[6] Richalet, J., & O'Donovan, D. (2011). Elementary predictive functional control: A tutorial. 
In 2011 International Symposium on Advanced Control of Industrial Processes 

(ADCONIP): May 2011; pp 306-313. 

[7] Richalet, J., Rault, A., Testud, J. L., & Papon, J. (1978). Model predictive heuristic control. 
Automatica (Journal of IFAC), 14(5), pp 413-428. 

[8] Khadir, M., & Ringwood, J. (2008). Extension of first order predictive functional 
controllers to handle higher order internal models. International Journal of Applied 

Mathematics and Computer Science, 18(2), pp 229-239. 

337



IIUM Engineering Journal, Vol. 22, No. 1, 2021 Abdullah et al. 
https://doi.org/10.31436/iiumej.v22i1.1538 

 

[9] Haber, R., Rossiter, J. A., & Zabet, K. (2016). An alternative for PID control: predictive 
functional control-a tutorial. In 2016 American Control Conference (ACC): July 2016; pp. 

6935-6940 

[10] Richalet, J., Lavielle, G., Mallet, J., & Dreyfus, G. (2005). La commande prédictive. 
Eyrolles. 

[11] Fiani, P., & Richalet, J. (1991). Handling input and state constraints in predictive functional 
control. In Proceedings of the 30th IEEE Conference on Decision and Control: December 

1991; pp. 985-990. 

[12] Abdullah, M., & Rossiter, J. A. (2019). Using Laguerre functions to improve the tuning and 
performance of predictive functional control. International Journal of Control, pp 1-13. 

[13] Rossiter, J. A., & Haber, R. (2015). The effect of coincidence horizon on predictive 
functional control. Processes, 3(1), pp 25-45. 

[14] Abdullah, M., Rossiter, J. A., & Haber, R. (2017). Development of constrained predictive 
functional control using Laguerre function based prediction. IFAC-PapersOnLine, 50(1), pp 

10705-10710.  

[15] Khan, B., & Rossiter, J. A. (2013). Alternative parameterisation within predictive control: a 
systematic selection. International Journal of Control, 86(8), pp 1397-1409.  

[16] Muehlebach, M., & D'Andrea, R. (2017). Basis functions design for the approximation of 
constrained linear quadratic regulator problems encountered in model predictive control. In 

2017 IEEE 56th Annual Conference on Decision and Control (CDC): December 2017; pp. 

6189-6196. 

[17] Gilbert, E. G., & Tan, K. T. (1991). Linear systems with state and control constraints: The 
theory and application of maximal output admissible sets. IEEE Transactions on Automatic 

control, 36(9), pp 1008-1020. 

[18] Scokaert, P. O., & Rawlings, J. B. (1998). Constrained linear quadratic regulation. IEEE 
Transactions on automatic control, 43(8), pp 1163-1169.  

[19] Valencia-Palomo, G., & Rossiter, J. A. (2012). Comparison between an auto-tuned PI 
controller, a predictive controller and a predictive functional controller in elementary 

dynamic systems. online publication, research gate. 

[20] Haber, R., Rossiter, J. A., & Zabet, K. (2016). An alternative for PID control: predictive 
functional control-a tutorial. In 2016 American Control Conference (ACC): July 2016; pp. 

6935-6940. 

 

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