
MBW:Robustness Analysis of an ObserverBased Controller in a Food WebFrom MathBioby Jay Brasch and Kirk Nichols ContentsExecutive SummaryWe 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 observerbased 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 ExampleA situation where decoupled source and consumer populations are connected in a food web occurs in wetlands, where physical separation stymies waterbound 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. BackgroundThe original paper presents a method for obtaining complete statespace 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 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 RouthHurwitz criteria, the matrix [AKC]  a “local exponential observer”  has been defined, i.e. the statespace 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 TheoryThe 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. 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 y is our output vector, or how we see the system. C is an observer matrix, mapping the elements of the statespace to our output observations. Matrix D is a feedforward 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}. 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 WebHere the system has the following system diagram. 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. which reduces to
Observer Based ControllerWe 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 fullstate feedback system is seen in Fig. 4. Fig. 4: A fullstate 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: Here we can see that the eigenvalues of (ABK) govern the overall behavior. That is, the further the eigenvalues of (ABK) 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 observerbased controller. An observerbased controller uses its bestguess 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. 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: Subtracting the estimated state from the actual state gives us Defining , we finally see that 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 (ABK) with (ALC). 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 AnalysisWe 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,
ResultsGiven 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 smallgain criterion, we were able to establish elementwise maximal perturbations in A, such that systemwide control was retained. The smallgain thresholds are presented below. 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 CarpOtter interaction, and observe the result. The graphs below illustrate the results for a perturbation of magnitude .02, well below the stability threshold: The values presented represent the transfer functions between the two inputs (duckweed and algae concentrations) and the two outputs (otter and algae+duckweed). The zeroconvergence indicates stability. To illustrate the sufficiency but not necessity of the stability criterion, we plot the values for the carpotter interaction at a magnitude of .071, just outside the threshold. As can be seen below, the system is still stable. Finally, we will illustrate an unstable system by setting our carpotter perturbation to 1: As can be seen from the diverging transfer functions, we have lost the ability to control our system from the algae and duckweed inputs. ConclusionGiven 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 multielement 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 realworld analogue. Finally, the current code is limited to perturbations in one parameter of the system, and not multiple parameters at once. Other ApplicationsAnother 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). References1. López, I., Gamaz M. , and Molnár, S. Applied Mathematics Letters 20:8, 951957 (2007). "Observability and observers in a food web." 2. López, I., Gamaz M. , and Molnár, S. Biosystems Volume 93, Issue 3, September 2008, Pages 211217 "Monitoring environmental change in an ecosystem" 