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MBW:Robustness Analysis of an Observer-Based Controller in a Food Web

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by Jay Brasch and Kirk Nichols

Executive Summary

We took the observer model of an ecosystem addressed in Observability and observers in a food web, by López, Gámez and Mólnar, and extended the model into an observer-based controller. We then studied robustness properties of this controller by considering perturbations resulting from inaccuracies in approximating the interactions between species in the ecosystem. In this article we will primarily use the example of wetland ecosystems as described by a system of first order ordinary differential equations. We will use control theory to both model and manipulate our model for life in a wetlands area and finally to draw meaningful real world conclusions from our model.


Motivating Example

A situation where decoupled source and consumer populations are connected in a food web occurs in wetlands, where physical separation stymies water-bound populations. For example, adjacent ponds might be dominated by duckweed and algae, and respectively consumed by carp and minnows. A top predator like an otter, though, could traverse the terrain between habitats and consume both types of fish. Additionally, duckweed and algae are difficult to distinguish from a distance, and thus are resistant to aerial surveys.

FoodWeb.jpg

Background

The original paper presents a method for obtaining complete state-space data using only local observation of a single species. They present requirements for convergence of the system using the community matrix Γ where Γ is invertible and the equilibrium point

Background1.jpg

Given the additional requirement that γ12 γ21x1*x2* = γ45γ54 x4*x5*, the authors claim that the system is locally observable at x*. Now an observer can be constructed using the matrices C=[0 0 1 0 0] and K=[k1 k2 k3 k4 k5], where

   k1 = k2 = k4 = k5 = 0 and k3 > 0.

Since, by the Routh-Hurwitz criteria, the matrix [A-KC] - a “local exponential observer” - has been defined, i.e. the state-space of the system can be determined by observing only the top predator in the food web.

For more on a 3 species food chain model see APPM4390:Three Species Food Chain Modeling

Basic Control Theory

The basic vocabulary terms used in control theory are plant, compensator, and feedback. The plant models how the system behaves with respect to itself and an external actuation. A typical model for a plant is.

BCT1.jpg Here x is called the “state” and in terms of ecology, is a vector with entries corresponding to the individual populations. u represents an actuation of the system. Matrix A looks like a Leslie matrix, describing how the system will behave without external influence. The B matrix maps the actuation vector to the same size as the state of our system. The second half of the plant is modeled by the output equation

BCT2.jpg

y is our output vector, or how we see the system. C is an observer matrix, mapping the elements of the state-space to our output observations. Matrix D is a feed-forward term, and for the remainder of this paper is assumed to be equal to zero. A realization of a plant is accomplished with the set of matrices {A,B,C,D}.

BCT3.jpg

Fig. 2: A basic feedback loop Feedback is the mechanism by which the compensator typically operates. The compensator uses the current state of the system and feeds back this value with proper adjustments necessary to actuate the input of the system in such a way such that the state vector of the system converges to a reference signal “r”.

Control Design In Observability and Observers in a Food Web

Here the system has the following system diagram.

Robust1.jpg

Fig. 3: Diagram of an observer model

We see here that the feedback term is simply the observer matrix C multiplied on the left by the observer gain matrix L, which appears as matrix K in the paper. The dynamics of the equation are governed by the following equations.

Robust2.jpg

which reduces to


Robust3.jpg


Here we see that when we reduce the system the eigenvalues of A-LC govern the dynamics of the system. The further these eigenvalues are in the left half plane, the faster the estimated state of the solutions converges to the true state of the system. This paper focused on the result that if the system is said to be observable, that is, the matrix File:Robust4.jpg where n is the number of states in the system has full normal-rank, the initial states of the system can be calculated by looking at the output throughout time.

Observer Based Controller

We decided to extend this work by considering the question, “What if we not only wanted to observe the system, but control it as well?” A general full-state feedback system is seen in Fig. 4.

Observer1.jpg Fig. 4: A full-state feedback loop Here we see that the true state of the system, vector x, is being fed back through the gain matrix K and injected into the system. The equations modeling these dynamics are:

Observer2.jpg which implies


Observer3.jpg

Here we can see that the eigenvalues of (A-BK) govern the overall behavior. That is, the further the eigenvalues of (A-BK) are in the left half plane, the faster the state of the system converges to a reference value. Considering the paper created and analyzed an observer to the plant, we were led to create an observer-based controller. An observer-based controller uses its best-guess realization of the plant and uses the guessed state vector to build a controller for governing the actual plant. A picture of this system is seen in Fig. 5.

Observer4.jpg

Fig. 5: Estimated state feedback loop Take note that the boxes in yellow represent the true plant, and that there are no signals (lines) connecting any part of this plant to the compensator that we don't have access to. That is, the only signals that we extract from the plant are the input and output. The green boxes represent the compensator. Note that the compensator has its own realization {A,B,C,0} of the plant, and also encompasses and observer gain matrix L, shown here in red. The estimated state of the system is fed back through a control gain matrix K and injected into the system.

The equations representing the plant and controller are:

Observer5.jpg Subtracting the estimated state from the actual state gives us

Observer6.jpg

Defining Observer7.jpg , we finally see that

Observer8.jpg

By the separation principle, we have that the overall system dynamics of an observer based controller are governed by the union of the eigenvalues of (A-BK) with (A-LC). In this fashion, we can place the poles of the compensator independent of the poles of the observer. For our system, we were assuming that we could only inject members into the population by controlling the individual populations of the algae and duckweed independently, and we could observe the number of otters in the system, and the sum total of the duckweed and algae. Our extension of the project placed the observer poles much further in the left half plane than the compensator poles (utilizing solutions to the Sylvester equation), and designed the compensator gain and observer gain appropriately.

Robustness Analysis

We next focused on robustness analysis of this controller. We supposed that our model of the A matrix in the plant, the “Leslie matrix”, was incorrect in one entry, meaning that there was one interaction between the species that was inaccurately modeled. We then studied how much this perturbation in this entry in our model of matrix A from the true plant's matrix A can be before the controller is no longer stabilizing. For this we created an interconnection structure of the system, separating the dynamics of the system (plant and compensator) from the perturbation and found the transfer function of the output of the perturbation to its input. By the small gain theorem, we have guaranteed stability for a controller given a perturbation if,


RobustAnal.jpg Our algorithm investigated the upper bound for the magnitude of the perturbation to any position in the Leslie matrix and stored this value in the same position it was derived from in the Leslie matrix in what we called a perturbation matrix. This matrix's entries corresponded to the maximum perturbation allowed in this parameter before the controller designed in the previous section would become unstable.

Results

Given the desire for control over the food web given control using two species inputs, we attempted to quantify our degree of control over the system. Using the small-gain criterion, we were able to establish element-wise maximal perturbations in A, such that system-wide control was retained. The small-gain thresholds are presented below.

Table 1.jpg

Note that since the small gain theorem is a sufficient condition, values outside our matrix will not necessarily create instability. To illustrate, we perturb a single value in the A matrix for the Carp-Otter interaction, and observe the result. The graphs below illustrate the results for a perturbation of magnitude .02, well below the stability threshold:

Picture 19.jpg

The values presented represent the transfer functions between the two inputs (duckweed and algae concentrations) and the two outputs (otter and algae+duckweed). The zero-convergence indicates stability. To illustrate the sufficiency but not necessity of the stability criterion, we plot the values for the carp-otter interaction at a magnitude of .071, just outside the threshold. As can be seen below, the system is still stable.

Picture 20.jpg

Finally, we will illustrate an unstable system by setting our carp-otter perturbation to 1:

Picture 21.jpg

As can be seen from the diverging transfer functions, we have lost the ability to control our system from the algae and duckweed inputs.

Conclusion

Given the promising preliminary results, this approach appears to be worth pursuing. Possible future steps would be data sampling on real systems to establish control matrices, extension of the modeling beyond two inputs and two outputs, and further research into multi-element perturbations. Additionally, there are several hurtles to jump before this theory is concrete. For example, we use continuous control, which isn't necessarily the best way to control an ecosystem, as fractional biological organisms have no real-world analogue. Finally, the current code is limited to perturbations in one parameter of the system, and not multiple parameters at once.

Other Applications

Another application of this type of ecosystem modeling is that of adding human effects into the system. This allows researchers to see how human interactions with the system could cause potential harm via pollution, climate change, or application of pesticides and fertilizer in nearby areas. An example of this can be found in [2] where López, Gamaz , and Molnár study a three species Lotka–Volterra model, but introduce a factor of environmental change into the system and then they studied the long term behavior of the system and compared it to models without the pollution introduced. In their paper various examples are given to demonstrate how these changes can affect an ecosystem. (For more information on how to model three species food chain models see: APPM4390:Three Species Food Chain Modeling).

References

1. López, I., Gamaz M. , and Molnár, S. Applied Mathematics Letters 20:8, 951-957 (2007). "Observability and observers in a food web."

2. López, I., Gamaz M. , and Molnár, S. Biosystems Volume 93, Issue 3, September 2008, Pages 211-217 "Monitoring environmental change in an ecosystem"