Biological pest control using a model‐based robust feedback

Abstract

Biological control is the artificial manipulation of natural enemies of a pest for its regulation to densities below a threshold for economic damage. The authors address the biological control of a class of pest population models using a model‐based robust feedback approach. The proposed control framework is based on a recursive cascade control scheme exploiting the chained form of pest population models and the use of virtual inputs. The robust feedback is formulated considering the non‐linear model uncertainties via a simple and intuitive control design. Numerical results on three pest biological control problems show that the proposed model‐based robust feedback can regulate the pest population at the desired reference via the manipulation of a biological control action despite model uncertainties.

1 Introduction

Pests are species that interfere with human activity or cause damage to another most valuable population. Pest control approaches are aimed to maintain the density of the pest population in an equilibrium level below economic damage [1]. Thus, the study of pest management methods is of great practical value in ecology, epidemiology and agricultural production [1–3].

The pest control problem has been addressed by many methods such as spraying pesticides (or chemical control) and releasing natural enemies (biological control) [3]. The spraying of pesticides is the principal method used to control pests. Nevertheless, the use of chemical pesticides might cause pollution to the environment, consumption of resources and reduction of natural enemies of the pests [3]. Given this, the biological control has attracted much attention [1, 2].

Biological control is based on the use of living organisms to reduce pest populations [1]. In this method, natural enemies of pests are released artificially when pests have caused damage. Natural enemies include the use parasites, pathogens and predators. Three main methods of biological control are [2] (i) conservation of existing natural enemies, (ii) the introduction of new natural enemies to establish a permanent population and (iii) mass rearing and periodic release.

It has been established elsewhere [1–3] that successful biological control can be achieved when the biological control agent has the following features: (i) It is host specific. (ii) It is synchronous with the pest. (iii) It can increase in density rapidly when the host does. Thus, the natural enemies’ population changes according to the pest population at a level that can control the pests. Furthermore, during the process of controlling the pests, as the pest population declines, the natural enemies’ population also decreases due to the lack of food. In long term, the populations coexist in a common environment.

Biological control is essentially a population phenomenon, resulting from the action of a natural enemy population interacting with a host population. Population models are useful in biological control problems to make predictions, to evaluate control programs, to suggest prevention strategies, and to understand qualitative aspects of natural enemy–host interactions. Indeed, several works have focused on mathematical properties such as stability, boundedness and convergence. However, it should be noted that the existence or not of a stable equilibrium is neither necessary nor sufficient for satisfactory control in practice. On the one hand, a stable equilibrium not guarantees that the system will be at or near the equilibrium under external perturbations. In the real world, an ecosystem is subjected to considerable perturbations of its initial state and continuous disturbances on its dynamics. On the other hand, an unstable equilibrium does not preclude a satisfactory control. Pest densities may oscillate without exceeding the economic threshold [4].

Population biologists have usually followed the state variable approach to derive simple models of pest/natural enemy interactions, in which a separate state variable equation is written for each species population. This approach has been extended to account for age and size structure within populations by dividing a population into classes, each represented by a density. Two good examples of this approach are: (i) models based on the well‐known Lotka–Volterra model, and (ii) models derived from the classical susceptible‐infected‐recovered (SIR) model of Kermack and McKendrick. It is important to note that, in general, biological control has been modelled as a two‐species interaction.

Control studies for biological pest control include impulsive control [5–12] and optimal approaches [13–17]. Tang and Cheke [6, 7] addressed the effect of periodic impulsive release and spraying of pesticides on continuous and discrete models. Impulsive biological and chemical controls for susceptible‐infected (SI)‐type models have been studied by Georgescu and Moroşanu [8], Tan and Chen [9] and Jiao et al. [10]. Impulsive control for predator‐pest models has been addressed by Lu et al. [5] and Wang and Huang [12]. Optimal impulsive harvesting policies and optimal timing of pesticide applications for the maximum stock level of the fish at the end of a fishing season were studied by Xue et al. [13]. Impulsive control uses impulsive differential equations that need a more elaborate analysis than the corresponding differential equations.

Optimal control approaches have also been applied to biological pest control problems [14–17]. For instance, optimal control approaches using the Pontryagin maximum principle were addressed by Apreutesei for a PDE food chain model [14] and by Ghosh and Bhattacharya [15] for an epidemic model. Optimal biological pest control based on a generalised Lotka–Volterra model and using linear feedback was considered by Rafikov et al. [16]. However, the use of optimisation methods and optimal control is limited due to the assumption of a perfect model for design purposes.

To the best of the authors’ knowledge, only a recent contribution by Sakthivel et al. [18] has addressed the design of a reliable controller with robustness properties. The control design is based on state feedback in terms of a linear matrix inequality approach. Robust model‐based feedback permits to deal with model uncertainties and incorporate the knowledge of the interactions captured by a simple mathematical model. Uncertainties in models of ecological populations include random fluctuations, parameter and state estimation, and structural uncertainties [19–22]. Thus, given the many uncertainties outlined above, it is necessary to provide some degree of robustness to any model‐based management strategies for biological control. Indeed, those strategies that perform well across all scenarios are preferred over more sensitive ones.

In this paper, we derive a simple and robust control approach for biological control in pest management, aimed to provide to an ecologist with a policy for pest regulation. To this end, we exploit the chained form of a class of models of biological pest control problems to introduce a framework for robust feedback control design. Chained models describe hierarchically vertically structured population systems, where the i th state is affected by the following state until the control input appears. A large class of models proposed in the literature for biological pest control problems can be written as non‐linear dynamical systems in a chained form. For instance, prey–predator, host–parasitoid, and SIR type epidemic models [23–26].

Various robust control techniques, linear and non‐linear, are available in the literature to include model uncertainties and state constraints in the control design. For instance, $H_{\mathrm{\infty }}$ optimisation approach [27], internal model control [28], and linear model predictive control [29], which are based on linear models. However, the high non‐linearity present in some ecological models limits the performance of linear approaches. Non‐linear approaches include non‐linear adaptive control [30], active disturbance adaptive control [31], Lyapunov‐based control [32], and sliding mode control techniques [33]. However, the mathematics behind the theory of that control design is quite involved. Moreover, their practical application is limited because of its structure and high computing cost as well as involved control designs. Thus, though various approaches have been studied to deal with the non‐linearity and model uncertainty, the existing control methods have some drawbacks to be applied in ecological problems. Hence, in this paper, a simple and practical robust control approach is proposed for a class of biological control problems with model uncertainties and additive disturbances. For control design, the adaptation is based on the estimation of all uncertain terms lumped in a modelling error function.

The control approach is formulated using modelling error compensation (MEC) techniques [34, 35]. The proposed control approach has three nice features for biological control applications: (i) robustness against model uncertainties (structural and parametric), (ii) simplicity in the control design, and (iii) the use of the minimum information from the system. Numerical results on three examples of biological control show that the proposed control framework can regulate the desired pest population via the release of natural enemies.

The main contributions of this work can be summarised in three aspects: (i) Based on the model structure of the population dynamics of a class of biological pest control problems, we introduce a control approach to compute the required biological action to achieve a desired equilibrium of the pest population. (ii) Our control approach is based on the direct chained non‐linear model, avoiding the change of coordinates as in backstepping approaches or more involved control designs for chained systems. (iii) The resulting control approach has a simple structure and provides good robustness capabilities against parameters with random uncertainties, commonly found in the modelling of biological and ecological systems.

This work is organised as follows: In Section 2, we introduce three case studies, as well as the generic model that we are using to develop our robust feedback approach. In Section 3, we address the biological pest control problem as a robust feedback problem. In Section 4, numerical studies show the implementation and performance of the proposed robust feedback approach. Finally, we close this paper with some concluding remarks.

2 Class of biological control models

A variety of models of biological control have been reported in [5, 7–9, 12, 15, 23–26]. In this section, we first describe three case studies: (i) A two populations host–parasitoid model describing interactions between soybean caterpillars and a parasitoid population [16]. (ii) A three populations host–parasitoid model applied to sugarcane borer, its egg parasitoid, and the larvae density of the sugarcane borer [11]. (iii) A three populations SI epidemic model describing interactions between a susceptible pest, infected pest and a prey [9]. Then we introduce a simple and general chained model for biological control problems.

2.1 Host–parasitoid model of soybean caterpillars

As a first case study, it is considered a host–parasitoid model of two soybean caterpillars, Rachiplusia nu, and Pseudoplusia includes, lumped in a single population (taking into account that the main characteristics of the biological cycle of these pests are practically identical) and a parasitoid [16]. Host–parasitoid interactions based on Lotka–Volterra model have been used extensively to describe the process of biological control by parasitoids. Indeed, host–parasitoid models have been favoured for the modelling of biological control problems due to several realistic simplifying assumptions, including that host attack is confined to the parasitoid adult female stage only, this allows age structure to be ignored or handled in a more simplified form [23].

The assumptions to derive the model are (i) In the absence of any predators, the prey populations grow unboundedly in a Malthusian way. (ii) The effect of the predation is to reduce the prey's per capita growth rate by a term proportional to the prey and predator populations. (iii) In the absence of any prey for sustenance, the predator's death rate results in exponential decay.

The model is given as

$$\frac{\mathrm{d}x(t)}{\mathrm{d}t}=r{x}_1(t)-\frac{r}{K}{x}_1(t){}^2-\beta x_1(t)x_2(t)$$ (1)

$$\frac{\mathrm{d}{x}_1(t)}{\mathrm{d}t}=\beta x_1(t)x_2(t)-m{x}_2(t)+u(t)$$

where x ₁ (t) and x ₂ (t) are caterpillar and parasitoid densities, respectively, r is the net reproduction rate, β is the rate of parasitism, K is the carrying capacity of the environment, m is the mortality rate of the parasitoid, and u (t) represents the introduction of natural enemies.

Pest control strategy is aimed to maintain the pest population at x _1,ref  = x_d by using the control u (t), where x_d is a pest population density below the economic injury level (EIL).

It has been established [16] that densities of big soybean caterpillars longer than 1.5 cm above the value of x_d  = 20 pests/m² cause economic damage to the soybean crop. Fig. 1 shows the density variations of the caterpillar and parasitoid populations using the nominal parameters r  = 0.17, r/K  = 0.0003825, β  = 0.000935, m  = 0.119. It is clear from Fig. 1 that system (1) has a stable fix point equilibrium at x ₁ (t) = 127.27 and x ₂ (t) = 129.75. Despite the existence of stable equilibrium, pest densities exceed the economic threshold. Thus, it is necessary to apply pest control on the model (1) for the parameters used.

Fig. 1.

Dynamics of caterpillar x₁ (t) (continuous blue line) and parasitoid x₂ (t) (dotted green line) populations for the biological control of soybean caterpillar pest

2.2 Host–parasitoid system for sugarcane borer

The second case study consists of a simple model of the sugarcane borer (Diatraea saccharalis) [11]. A host–parasitoid model is considered for the transformations of pest eggs into egg parasitoid and a larvae stage.

The model consists of following three differential equations [11]:

$$\frac{\mathrm{d}{x}_1(t)}{\mathrm{d}t}=n_1{x}_2(t)-m_1{x}_1(t)-n_2{x}_1(t)$$ (2)

$$\begin{array}{rl}\frac{\mathrm{d}{x}_2(t)}{\mathrm{d}t}& =r\left(1-\frac{x_2(t)}{K}\right)x_2(t)-m_2{x}_2(t)-n_1{x}_2(t)\\ & -\beta x_2(t)x_3(t)\end{array}$$

$$\frac{\mathrm{d}{x}_3(t)}{\mathrm{d}t}=\beta x_2(t)x_3(t)-m_3{x}_3(t)-n_3{x}_3(t)+u(t)$$

where x ₁ (t) is the larvae density of the sugarcane borer, x ₂ (t) is the egg density of the sugarcane borer and x ₃ (t) is the density of eggs parasitised by Trichogramma galloi, n ₁ is the fraction of the eggs from which the larvae emerge at time t, n ₂ is the fraction of the larvae population which moults into pupal stage at time t, n ₃ is the fraction of the parasitised eggs from which the adult parasitoids emerge at time t, m ₁ is the mortality rate of the larvae population, m ₂ is the mortality rate of the egg, m ₃ is the mortality rate of parasitised egg, r is the intrinsic oviposition rate of female sugarcane borer, β is the rate of parasitism, and K is the potential maximum of oviposition rate of female sugarcane borer.

For parameter values n ₁  = n ₃  = 0.1, n ₂  = 0.02439, m ₁  = 0.00256, m ₂  = m ₃  = 0.03566, β  = 0.0001723, r  = 0.1908, and K  = 25,000, the operating steady state of the open‐loop system is stable, where all populations move to an asymptotically stable steady state with damping oscillations. This set of parameters results from field experiments [25]. One can see from Fig. 2 that the sugarcane borer larvae density x ₁ (t) takes on values more than the EIL = 2500 numbers/ha³ for this pest. Thus, also in this case, it is necessary to apply biological control via the release of parasitised eggs.

Fig. 2.

Dynamics of the larvae density x₁ (t) (continuous blue line), egg density x₂ (t) (dashed green line) and eggs parasitised by Trichogramma galloi x₃ (t) (dotted red line) for the sugarcane borer pest control problem

2.3 SI epidemic model of a generic pest problem

An epidemic model is considered as the third case study. In simple SI models, the host population is assumed to consist of only two disease classes: susceptible and infective individuals.

The model consists of susceptible and infected pests and a crop population as the food of the susceptible pest [9]. The model is written as

$$\frac{\mathrm{d}{x}_1(t)}{\mathrm{d}t}=k\theta x_1(t)x_3(t)-\alpha x_1(t)x_2(t)-m_1{x}_1(t)$$ (3)

$$\frac{\mathrm{d}{x}_2(t)}{\mathrm{d}t}=\alpha x_1(t)x_2(t)-m_2{x}_2(t)+u(t)$$

$$\frac{\mathrm{d}{x}_3(t)}{\mathrm{d}t}=x_3(t)\left(r-\omega x_3(t)\right)-\theta x_1(t)x_3(t)$$

where x ₁ (t) is the susceptible pests, x ₂ (t) is the infected pests, x ₃ (t) denotes the food population, m ₁ is the natural death rate of susceptible population, m ₂ is the death rate of the infected pest population, α denotes the transmission coefficient, r is the intrinsic growth rate of the food, and the control variable u (t) denotes the release rate of infected pests. The contribution of food to the susceptible pest growth is given by kθx ₁ (t) x ₃ (t).

The assumptions to derive the model are (i) The pest population is divided into two classes which are the susceptible class and the infective class. (ii) The susceptible pest is infected via a bilinear incidence. (iii) The food in the absence of any predator grows logistically. (iv) When the pests are infected, they will die soon and do not consume food.

European Rabbits is a biological control problem that can be represented by this model, where x ₁ (t) denotes the European Rabbits (Oryctolague cuniculus), x ₂ (t) denotes the infected ones, and x ₃ (t) denotes the grass in the pasture.

Model (3) can display a stable steady state and complex oscillations, including chaos and quasi‐periodicity for different parameter values. The set of parameters values corresponding to the steady state are m ₁  = 0.2, m ₂  = 0.5, α  = 0.8, r  = 0.2, ω  = 0.2, k  = 0.9, θ  = 0.5. The corresponding dynamics is illustrated in Fig. 3.

Fig. 3.

Dynamics of susceptible (continuous blue line), infected pests (dashed green line) and crop populations (dotted red line) for a biological epidemic pest control problem

Although the susceptible pest converges to a steady state below the given threshold (EIL = 1), the susceptible pest has some transitory dynamics whose maximum value exceeds the EIL. From the practical point of view, it is desirable to establish a control policy of the release of infected pests such that the number of the susceptible pests is always less than the given EIL.

2.4 Class of biological pest control chained models

In reviewing the examples, it can be noted similar structural features of the models. Thus, we consider the following general class of mathematical models of biological control:

$$\frac{\mathrm{d}{x}_i(t)}{\mathrm{d}t}=f_i(x(t))+g_i(x(t))x_{i+1}(t),1\le i\le j-1$$ (4)

$$\frac{\mathrm{d}{x}_j(t)}{\mathrm{d}t}=f_j(x(t))+g_j(x(t))u(t),j\le n$$

$$\frac{\mathrm{d}{x}_{j+l}(t)}{\mathrm{d}t}=f_{j+1}(x(t)),j+l=n,l>j$$

$$y(t)=x_1(t)$$

where x (t) is the state (population densities) of the system, y (t) is the controlled variable (pest population) to be regulated, and u (t) is the control input (biological action). Functions f (x (t)) and g (x (t)) represent either linear or non‐linear functions depending on the state of the system. The first j equations in the model (4) are given in a chained form.

The following comments are in order:

1. General model structure: The motivation behind the model structure is that in several biological control problems the manipulated variable (e.g. parasitoid release) affects the desired controlled variable indirectly (e.g. pest population). Indeed, it is not hard to recast several models of biological control in the form given by model (1) [5, 7–9, 12, 15, 23–26].

2. Economic injury level: The point where these financial losses caused by the pest equal the cost to control the pest is known as EIL and is usually expressed as a number of pests per unit area [1, 2]. Thus, for a biological control situation to be successful, the natural enemy should be able to eradicate the pest or regulate it to densities below the threshold for economic damage.

3. Pest eradication: Although pest eradication is appealing due to perpetual freedom from the pest and its harmful effects, it is unlikely to occur [36, 37]. Two main barriers exist to pest eradication: (i) high costs relative to measurable benefits, and (ii) lack of techniques to locate and remove pests at low densities. Furthermore, the pest will persist either because absolute spatial or temporal refuges exist, because there are invulnerable stages, or because the enemy cannot wipe out the pest everywhere simultaneously.

4. Simple versus complex models: Biological control models attempt to describe fundamental population processes (birth, death, growth and migration) and the way that these are affected by population size and environmental factors. Real host–pathogen systems have a variety of additional features. For instance: (i) host species exhibit age and size‐structured predation, (ii) host–pathogen dynamics often play out in spatially explicit settings, (iii) host and pathogen species coexist with other species in communities. More realistic is more complex and parameter rich. Moreover, is likely to apply to a class of interactions rather than all parasitoid–host interactions. While on the one hand, they provide greater opportunities for field tests, on the other hand, such tests require more information from the real system. Nevertheless, for control design purposes, a model with small dimension and less complexity is more useful.

5. Environmental impact of the biological control agent: Biological control agents can affect the environment in a variety of ways [38, 39]. For instance, enhanced the targeted pest, synergistically interacted with other organisms to enhance pest problems, affected public health, and attacked non‐target organisms. Thus, natural enemies must be well chosen and carefully tested on native hosts.

6. Eradication of the biological agent: The eradication of the biological control agent after establishment is extremely difficult at best [38–40]. Possible elimination of the biological control agent can be achieved in two ways: (i) When a small pathogen is introduced in a pest population with the expectation that it will generate an epidemic which will persist at an endemic level. (ii) When a pathogen is applied, and there is no expectation that the pathogen persists in the environment for a long time.

7. Continuous biological action: Control actions (e.g. introduction and release of natural enemies) in biological control can be applied continuously or impulsively in a constant amount in periodic form [3, 7, 9]. Model (4) considers that the control input is acting continuously.

8. Uncertainties: The uncertainty is pervasive in ecology, where the variation in the system itself exacerbates the difficulties of dealing with it. Uncertainties in biological control models arise in two main ways, structural and parametric. Structural uncertainty refers to different choices of population densities in a model and relationships between them. The uncertainty that arises from the approximation of complex models to simpler ones also fits into the category of model uncertainty. Parametric uncertainty refers to variation in the numerical base values of different parameters of the model. These parameters may include birth rate, carrying capacity, transmission coefficient and the growth rate. Two critical factors of parameter variations are environmental fluctuations and weather events.

9. Model properties: For control design purposes, the following model properties are considered: (i) non‐linear functions f_i (x) and f_j (x) are smooth functions. (ii) Non‐linear functions g_i (x) and g_j (x) are bounded away from zero, globally bounded, and their sign (the directionality of the control action) is known. These assumptions are realistic. Indeed, the primary source of non‐linearity in biological control models is the functional response. For instance, all possible Holling type I, II or III structures of functional responses meet these assumptions.

3 Robust control design for biological pest control

In this section, the control problem is stated, and the control design based on MEC ideas is presented. By exploiting the chained form of the class of biological pest control problems described by the model (1), a general feedback framework based on a recursive cascade control configuration is introduced.

3.1 Biological control using a robust feedback

The control objective is stated as the regulation at or below the EIL of the pest population via the manipulation of a biological control action. The following assumptions complete the control problem description:

A1 : States x ₁ (t) to x_n (t) are measured and available for control design purposes.

A2 : The non‐linear function f_j (x (t)) includes uncertain terms and uncertain parameters, and without loss of generality, for control design purposes it is assumed entirely unknown.

A3 : There exist rough estimates of non‐linear function g (x (t)) given by $g\left(\widehat{x(t)}\right)$.

The following comments are in order:

1. The population measurement is not an easy task in ecology. Indeed, available methods for population estimation in ecology include statistically based measurements, image and video recordings, which are subject to significant uncertainties [1–3, 21, 22, 41]. However, even in the absence of such measurements, state estimators can be designed, if the model of the pest control problem is observable [42].

2. A2 and A3 consider model uncertainties, or in the worst case that the whole terms are unknown. The rough estimate of $g\left(\widehat{x(t)}\right)$ can be obtained using a nominal constant value of this term. These are realistic assumptions. Indeed, the same ecosystem, which consists of different pests and its natural enemies can be modelled in different forms. Moreover, both interactions between species and model parameters (e.g. reproduction and mortality rates, carrying capacity of the environment etc.) have some degree of uncertainties [21–23, 41].

3.2 Recursive cascade control configuration

It can be seen from (4) that the biological action (the control input) does not affect the pest density (the output) directly. The relative degree of the system (4) is the number of times we have to differentiate the output before the input u (t) appears explicitly. Thus, for the control design, we have a high‐order relative degree order. Based on the chained form of the model (4), it is introduced a recursive cascade control configuration reducing the control design to an affine control system (i.e. relative one‐degree control problems where the control input appears linearly).

Fig. 4 shows the recursive cascade control configuration for the class of biological control model given in (4). The controller design is recursive because the computation of u_i+1 (t) requires the computation of u_i (t). The cascade control configuration consists of a master controller driven by the regulation error, e (t) = y (t) − y _ref, which provides reference values to a chain of slave controllers, where the last one is driven by the real control input u (t).

Fig. 4.

Recursive cascade control configuration for the class of biological pest control problems

3.3 Robust feedback approach

The control approach is based on MEC ideas that lead to controllers with a simple linear structure, good closed‐loop performance and robustness properties [34, 35].

The control design consists of the following steps:

1. Lump the uncertain model terms, including terms containing uncertain parameters, in a single new state η (t). From model (1) and assumptions A2 and A3, the modelling error η (t) and the equivalent model are written as

   $$\eta (t)=f(x(t))$$ (5)

   $$\frac{\mathrm{d}x(t)}{\mathrm{d}t}=\eta (t)+\widehat{g(x(t))}u(t)$$ (6)
2. Estimate the uncertain term η (t) via a reduced order observer

   $$\frac{\mathrm{d}\eta (t)}{\mathrm{d}t}={\tau }_{\mathrm{e}}^{-1}\left(\eta (t)-\widehat{\eta (t)}\right)$$ (7)

   where the real uncertain term is obtained from model (6) and introduced in the reduced order observer (7), which lead to

   $$\frac{\mathrm{d}\eta (t)}{\mathrm{d}t}={\tau }_{\mathrm{e}}^{-1}\left(\frac{\mathrm{d}x(t)}{\mathrm{d}t}-\widehat{g\left(x(t)\right)}u(t)-\widehat{\eta (t)}\right)$$

   using $\omega (t)={\tau }_{\mathrm{e}}\widehat{\eta (t)}-y(t),$ the reduced order observer can be written as follows:

   $$\frac{\mathrm{d}\omega (t)}{\mathrm{d}t}=\widehat{g\left(x(t)\right)}u(t)-\widehat{\eta (t)}$$ (8)

   where τ _e is the only observer design parameter.
3. Design a feedback action to assign a desired closed‐loop behaviour. We consider a desired first‐order linear asymptotic behaviour, which is obtained using the following feedback:

   $$u(t)=-{\widehat{g\left(x(t)\right)}}^{-1}\left\{{\tau }_{\mathrm{c}}^{-1}e(t)+\widehat{\eta (t)}-\frac{\mathrm{d}{y}_{\text{ref}}(t)}{\mathrm{d}t}\right\}$$ (9)

where τ _c is the only controller design parameter.

The resulting feedback controlled depends only on the measure of y (t), the estimated value of the lumped uncertain terms $\widehat{\eta (t)}$, and the rough estimate of $\widehat{g(x(t))}$. It is also noted that the proposed controller has only two parameters, one for the observer and other for the feedback control law. Moreover, the time derivative of the desired reference is equal to zero for regulation (constant reference) control problems and for tracking (time‐dependent reference) control problems it can be included in the modelling error function.

The tuning of both parameters follows the simple rule [34, 35]: τ _p  > 0.5τ _c  > 0.5τ _e, where τ _p is a characteristic time constant of the controlled systems. τ _c can be seen as a closed‐loop time constant and, determines the desired closed‐loop convergence, and τ _e determines the smoothness of the modelling error estimation. It is noted that smaller values of τ _c lead to a faster closed‐loop system. However, a major effort of the control action will be commonly required.

3.4 Robustness and stability properties

To obtain successful biological control strategies, they should be robust in response to both model uncertainties and external perturbations. Furthermore, it is essential to establish that the closed‐loop system will be stable to finite perturbations of its initial state and its dynamics. The robustness properties against model uncertainties of the closed‐loop system are related to the compensation of the estimated lumped uncertain terms. Thus, the closed‐loop systems provide good robustness properties via the feedback function (9).

The stability analysis of closed‐loop systems is based on singular perturbation arguments. For the sake of completeness in presentation, a sketch of main ideas of the stability results for the MEC approach is provided as follows [34].

Given the regulation error e (t) and defining the estimation error as, ϕ (t) = η (t) −  $\widehat{\eta (t)}$, then the closed‐loop system becomes

$$\frac{\mathrm{d}e(t)}{\mathrm{d}t}=-{\tau }_{\mathrm{c}}^{-1}e(t)+\varphi (t)$$ (10)

$$\frac{\mathrm{d}\varphi (t)}{\mathrm{d}t}=-{\tau }_{\mathrm{e}}^{-1}\frac{g\left(x(t)\right)}{\widehat{g\left(x(t)\right)}}\varphi (t)+\mathrm{\Gamma }\left(e(t),\varphi (t)\right)$$

where Γ(e (t), ϕ (t)) stands for the time derivative of lumped uncertain terms, which does not depend on τ _e. By using model properties described in Section 2, it can be shown that such time derivative is a continuous function of its arguments. Thus, there exist two positive constants ν ₁ and ν ₂ both independent of τ _e, such that

$$\left|\mathrm{\Gamma }\left(e(t),\varphi (t)\right)\right|\le v_1\left|e(t)\right|+v_2\left|\varphi (t)\right|$$ (11)

The system (10) can be seen as a non‐linear singularly perturbed system with τ _e as the perturbation parameter, and e (t) and ϕ (t) as the slow and fast variables, respectively. The reduced system (obtained by taking τ _e  = 0) and the boundary‐layer system (obtained by taking the time‐scaling t ′ = t/τ _e and τ _e  = 0) are linear asymptotically stable. Hence, there exists a maximum estimation time constant τ _e *, such that for all τ _e  < τ _e * the regulation error e (t) goes asymptotically to zero. The maximum estimation time constant can be taken as a measure of the robustness of the controlled plant. Larger values of τ _e * lead to better robustness capabilities of the closed‐loop system. Smaller values of τ _e lead to a faster estimation of the modelling error. However, excessively small values of τ _e may induce high sensitivity of the controller to high‐frequency measurement noise. Stability results imply that the environmental stochasticity, dynamical noise and the stochasticity of human actions do not affect the stability of the pest‐eradication periodic solution if these exogenous forces are sufficiently small.

4 Numerical studies

To illustrate our methodology, we have performed several numerical simulations on case studies. For numerical simulations, the resulting ordinary differential equation (ODE) systems were integrated via a fourth‐order Runge–Kutta routine programmed in Matlab v.7 with model parameters values given in Section 2 and control parameters provided in this section for each case study. To show the controller robustness against model uncertainties, the control design is based on nominal parameters with a parameter mismatch of the parameters used in the simulations. That is, while the system is simulated with random fluctuations of about 10% in typical uncertain parameters (e.g. birth and growth rates, carrying capacity, transmission coefficient), the control design is based on nominal parameter values given in Section 2.

4.1 Biological control of the soybean caterpillar pest

As was discussed below, the control objective of this case study consists in the regulation of the pest population below the EIL via the release of parasitoids. Proposed controllers based on MEC approach using the recursive cascade configuration are given as follows:

$$u_{v1}(t)=-{\hat{\beta }}^{-1}{x}_1(t){}^{-1}\left\{{\tau }_{c1}^{-1}(y_{1,\text{ref}}-y_1(t))+\widehat{{\eta }_1(t)}\right\}$$ (12)

$$\frac{\mathrm{d}{\omega }_1(t)}{\mathrm{d}t}=\hat{\beta }{x}_1(t)u_{v1}(t)-\widehat{{\eta }_1(t)}$$

$$\widehat{{\eta }_1(t)}={\tau }_{e1}^{-1}({\omega }_1(t)+y_1(t))$$

$$u(t)=-\left\{{\tau }_{c2}^{-1}(u_{v1}(t)-y_2(t))+\widehat{{\eta }_2(t)}\right\}$$ (13)

$$\frac{\mathrm{d}{\omega }_2(t)}{\mathrm{d}t}=u(t)-\widehat{{\eta }_2(t)}$$

$$\widehat{{\eta }_2(t)}={\tau }_{e2}^{-1}({\omega }_2(t)+y_2(t))$$

For control design purposes, x ₂ (t) = u_v1 (t) is chosen as a virtual control input for the regulation of the pest population y ₁ (t) = x ₁ (t). Thus, the first control loop is driven by the regulation error e ₁ (t) = y _1,ref  − y ₁ (t). The resulting virtual control input is the reference driving the real control input u, i.e. e ₂ (t) = u_v1 (t) − y ₂ (t). It can be noted that the proposed controllers depend only on a rough estimate of parameter $\hat{\beta }$, measurements of states y ₁ (t) = x ₁ (t) and y ₂ (t) = x ₂ (t), and four control design parameters [τ_c1 , τ_e1 , τ_c2 , τ_e2 ].

Fig. 5 shows the performance of proposed controller for three different sets of controller design parameters [τ_c1 , τ_e1 , τ_c2 , τ_e2 ]: (a) nominal control parameters [7.5, 4, 4, 2] (continuous black line), (b) higher control parameters [15, 8, 8, 4] (dashed blue line), and (c) smaller control parameters [2, 0.4, 1, 0.2] (dotted red line). From Fig. 5, it can be observed that to regulate the pest population, the control input consists of an initial high release of parasitoids which is stabilised after 100 time units. The control action achieves minimum and maximum constraints with decreasing time duration at these constraints. It is noted that for the base and small controller parameters, the desired reference is completely achieved after 100 time units of the activation of the control action. After that time, the release of natural enemies remains in a constant value. It can also be observed from Fig. 5 that small closed‐loop and estimation constants lead to small distortions of both the controlled and manipulated variables. On the other hand, higher values of these time constants lead to better robustness properties with a slow response.

Fig. 5.

Closed‐loop performance for the soybean pest control problem

(a) Controlled pest population, (b) Natural enemy population, (c) Release of natural enemies

To illustrate the performance of the proposed robust control design model uncertainties, we have assumed that all model parameters have a 10% uniformly distributed random fluctuations. For control design, we have set the rough estimate of the parameter $\hat{\beta }$ as the base value. Numerical results are shown in Fig. 5 with the continuous green line. It is noted despite the parameter mismatch between the simulated system and the rough estimate of the parameter $\hat{\beta }$ used in the control design, the controlled variable is regulated to the desired value without serious distortions.

4.2 Biological control of the sugarcane borer pest

In this case, the biological control problem is the regulation of the larvae density of the sugarcane borer via the release of parasitised eggs. The recursive cascade control configuration is given with the use of two virtual control inputs, x ₂ (t) = u_v1 (t) and x ₃ (t) = u_v2 (t), and three regulation errors e ₁ (t) = y _1,ref  − y ₁ (t), e ₂ (t) = u_v1 (t) − y ₂ (t), and e ₃ (t) = u_v2 (t) − y ₃ (t). The proposed control framework is given as

$$u_{v1}(t)=-{\widehat{n_1}}^{-1}\left\{{\tau }_{c1}^{-1}(y_{1,\text{ref}}-y_1(t))+\widehat{{\eta }_1(t)}\right\}$$

$$\frac{\mathrm{d}{\omega }_1(t)}{\mathrm{d}t}=-\widehat{n_1}{u}_{v1}(t)-\widehat{{\eta }_1(t)}$$

$$\widehat{{\eta }_1(t)}={\tau }_{e1}^{-1}({\omega }_1(t)+y_1(t))$$

$$u_{v2}(t)={\hat{\beta }}^{-1}{x}_2(t){}^{-1}\left\{{\tau }_{c2}^{-1}\left(u_{v1}(t)-y_2(t)\right)+\widehat{{\eta }_2(t)}\right\}$$ (15)

$$\frac{\mathrm{d}{\omega }_2(t)}{\mathrm{d}t}=\hat{\beta }{x}_2(t)u_{v2}(t)-\widehat{{\eta }_2(t)}$$

$$\widehat{{\eta }_2(t)}={\tau }_{e2}^{-1}({\omega }_2(t)+y_2(t))$$

$$u(t)=-\left\{{\tau }_{c3}^{-1}(u_{v2}(t)-y_3(t))+\widehat{{\eta }_3(t)}\right\}$$ (16)

$$\frac{\mathrm{d}{\omega }_3(t)}{\mathrm{d}t}=u(t)-\widehat{{\eta }_3(t)}$$

$$\widehat{{\eta }_3(t)}={\tau }_{e3}^{-1}({\omega }_3(t)+y_3(t))$$

Based on the EIL for this case study, we have set the desired reference at x _1,ref  = 2000 numbers/ha³. The control configuration is turned on at t  = 1500 time units. Model uncertainties were considered in the transmission coefficient, carrying capacity and transmission coefficient via 10% random fluctuations.

Fig. 6 shows the closed‐loop performance of the proposed robust feedback control framework for three set of control parameters and parameter uncertainties. The following sets of control parameters are selected: [τ_c1 , τ_e1 , τ_c2 , τ_e2 , τ_c3 , τ_e3 ]: (a) nominal controller parameters [25, 1, 15, 0.5, 8, 0.2] without (continuous black line) and with parameter uncertainties (continuous green line), (b) higher controller parameters [50, 5, 25, 2.5, 15, 1] (dashed blue line) and (c) smaller controller parameters [15, 0.5, 10, 0.25, 5, 0.1] (dotted red line).

Fig. 6.

Closed‐loop performance for the sugarcane borer pest control problem

(a) Controlled pest population, (b) Natural enemy population, (c) Crop population, (d) Release of natural enemies

From Fig. 6, it is noted that to achieve the regulation of the larvae density of the sugarcane borer to the desired reference a high release of parasitised eggs is necessary. It is also noted that using the nominal controller parameters, the control action is fast and achieves a maximum peak and after some slight transitory dynamics converge to a constant value. Smaller controller parameters lead to slight oscillations in both the controlled and manipulated variables. Random fluctuations of the three uncertain parameters only induce small distortions around the controlled variable but with a good closed‐loop behaviour.

4.3 Biological control in an epidemic model

Consider the regulation of susceptible pests x ₁ (t) via the release rate of pest u (t) with the virtual control input x ₂ (t) = u_v1 (t) and the regulation errors e ₁ (t) = y _1,ref  − y ₁ (t) and e ₂ (t) = u_v1 (t) − y ₂ (t). Proposed controllers of the robust feedback control framework are given as

$$u_{v1}(t)={\hat{\alpha }}^{-1}{x}_1(t){}^{-1}\left\{{\tau }_{c1}^{-1}(y_{1,\text{ref}}-y_1(t))+\widehat{{\eta }_1(t)}\right\}$$ (17)

$$\frac{\mathrm{d}{\omega }_1(t)}{\mathrm{d}t}=\hat{\alpha }{x}_1(t)u_{v1}(t)-\widehat{{\eta }_1(t)}$$

$$\widehat{{\eta }_1(t)}={\tau }_{e1}^{-1}({\omega }_1(t)+y_1(t))$$

$$u(t)=-\left\{{\tau }_{c2}^{-1}(u_{v1}(t)-y_2(t))+\widehat{{\eta }_2(t)}\right\}$$ (18)

$$\frac{\mathrm{d}{\omega }_2(t)}{\mathrm{d}t}=u(t)-\widehat{{\eta }_2(t)}$$

$$\widehat{{\eta }_2(t)}={\tau }_{e2}^{-1}({\omega }_2(t)+y_2(t))$$

The control configuration is turned on at t  = 60 time units. We have considered parameter uncertainties in all model parameters via random 10% random fluctuations. Based on previous results for this problem using both continuous constant and periodic impulsive approaches [9], we have set the reference of susceptible pests at y _ref  = 0.5, which is below the ET for this case study.

Fig. 7 shows that the control approach is able to regulate the susceptible pest population via the release of infected pests. The set of control parameters are [τ_c1 , τ_e1 , τ_c2 , τ_e2 ]: (a) nominal control parameters [1.25, 0.7, 0.6, 0.4] without (continuous black line) and with random parameter fluctuations (continuous green line), (b) higher control parameters [5, 2.5, 2.5, 1] (dashed blue line), and (c) smaller control parameters [0.1, 0.05, 0.05, 0.02] (dotted red line).

Fig. 7.

Closed‐loop performance for the SI‐type pest control problem

(a) Controlled pest population, (b) Natural enemy population, (c) Release of natural enemies

It can be observed from Fig. 7 that the release of infected pests is continuous and smooth. Thus, based on our results, a continuously infected pest release can be implemented to control susceptible pests below the economic threshold (ET) level. Notice that controller takes the output to reference without excessive control effort despite uncertainty in parameter values. As in the previous cases, the lower closed‐loop and estimation‐time constants lead to a slight oscillation of the controlled and manipulated variable. On the other hand, higher controller parameters lead to a slow achievement of the desired reference.

5 Conclusions

In this paper, we have presented a model‐based strategy for biological control based on a robust feedback control approach. First, we have introduced a general class of pest population models under biological control actions given in a chained form, including prey–predator and SIR‐type models. Next, based on the model structure, a recursive cascade control configuration is proposed. Then, we introduce a model‐based robust control approach with a simple structure and good robustness and performance capabilities. Robustness properties of the proposed control approach are obtained via the lumping of model uncertainties, and its compensation using an uncertainty estimator. The control strategy offers an alternative for the release of natural enemies in biological control that can be feasible to implement in concrete circumstances.