Design of Robust Policies for Uncertain Natural Resource Systems : Application to the Classic Gordon-Schaefer Fishery Model

Introduction. A critical challenge faced by sustainability science is to develop robust strategies to cope with highly uncertain social and ecological dynamics. The increasing intensity with which human societies utilize (limited) natural resources is fueling the global debate and urging the development of resource management methodologies/policies to effectively deal with very demanding socio-bio-economical issues. Unfortunately, despite concerted efforts by governments, many natural resources continue to be poorly managed. The collapse of many fisheries worldwide is the most notable example (Clark, 2006; Clark et al., 2006; Holland, Gudmundsson; Myers, Worm 2003; Sethi et al., 2005) but other examples include forests (Moran, Ostrom), groundwater basins (Shah, 2000), and soils (ISRIC, 1990). The suggested causes are varied but (Clark, 2006) highlights two: (1) lack of consideration of economic incentives actually faced by economic agents and (2) uncertainty associated with the dynamics of biological populations. In the case of fisheries, Clark notes that “complexity and uncertainty will always limit the extent to which the effects of fishing can be understood or predicted” (Clark, 2006, p. 98). This suggests that we need policies capable of effectively managing natural resource systems despite the fact that we understand them poorly at best. Real-World Management Issues. Real-world resource management must address three components: goal setting, practical (robust) implementation, and learning. Clark and others (Clark, 2007; 2006; Clark et al., 2006) have recently noted that practical implementation issues are frequently at the root of fishery management failures. For most fisheries, the necessary institutional contexts exist (Wilen, Homans) and we know what to do, yet management efforts fail. This suggests a need to focus on the actual process of resource management. For example, how can managers make decisions with incomplete information concerning how the resource and the resource users will respond to management actions? Design of Robust Policies for Uncertain Natural Resource Systems: Application to the Classic Gordon-Schaefer Fishery Model

cycle behavior (nonlinear oscillations) when parametric uncertainty is present.As such, it cannot be implemented in practice (because of prohibitively expensive switching costs).This motivates the need for robust policies that (1) do not exhibit limit cycle behavior and (2) offer performance (returns) as close to the optimal perfect information policy as model parameter (and derived fishery biomass target) uncertainty permits.Given the state of most world fisheries, our presentation focuses on a fishery that is nominally (i.e.believed to be) biologically over exploited (BOE); i.e. the optimal equilibrium biomass lies below the maximum sustainable yield biomass (Clark, 2006;Clark et al., 2006;Clark, 1990;Holland, Gudmundsson;Myers, Worm 2003;Sethi et al., 2005).By so doing, we directly address a globally critical renewable resource management problem.As in our prior work (Anderies et al., 2007), (Rodriguez et al., 2010), we do not seek "a best policy."Instead, we seek families of policies that are robust with respect to uncertainties that are likely to occur.Such families can, in principle, be used by a fishery manager to navigate the many tradeoffs (biological, ecological, social, economic, political) that must be confronted.More specifically, our effort to seek robust performance focuses on reducing the worst case downside performance; i.e. maximizing returns when we have the worst case combination of parameters.Such worst case (conservative) planning is critical to avoid/minimize the possibility of major regional/societal economical shortfalls; case in point, the recent "Great Recession."It is important to note that the simplicity of our model (vis-a-vis our performance objective of maximizing the net present value of returns) permits us to readily determine the worst case combination of model parameters (i.e.growth rate, carrying capacity, catchability, discount rate, price, cost of harvesting).Given this, we seek control laws that do not exhibit limit cycle behavior and whose returns are close (modulo limitations imposed by uncertainty) to that of the worst case perfect information optimal control policy -the best we could do in terms of return if we knew the parameters perfectly.Other design strategies are also examined; e.g.designing for the best case set of parameters."Blended strategies" that attempt to do well for the worst case downside perturbation (i.e.minimize the economic downside) as well as the best case upside perturbation (i.e.maximize the economic upside) are also discussed.Such strategies seek to flatten the return-uncertainty characteristics over a broad range of likely parameters.The above optimal control (derived) policies are used as performance benchmarks/targets for the development of robust control policies.While our focus is on bounded deterministic parametric uncertainty, we also show how probability distributions for uncertain model parameters can be exploited to help in the selection of benchmark (optimal) policies.After targeting a suitable optimal (benchmark) policy, we show how robust policies can be used to approximate the benchmark (as closely as the uncertainty will permit) in order to achieve desired performance-robustness-vulnerability tradeoffs; e.g. have a return that is robust to worst case parameter perturbations.While the presentation is intended to provide an introduction into how concepts from optimal and robust control may be used to address critical issues associated with renewable resource management, the presentation also attempts to shed light on challenges for the controls community.Although the presentation builds on the prior work presented in (Anderies et al., 2007), (Rodriguez et al., 2010), the focus here is more on defining the problem, describing the many issues, and sufficiently narrowing the scope to permit the presentation of a design methodology (framework) for robust control policies.Finally, it must be noted that the robust policies that we present are not intended to be viewed as final policies to be implemented.Rather, they should be viewed as policy targets -providing guidance to resource managers for the development of final implementable policies (based on taxes, quotas, etc. (Clark, 1990, Chapter 8)) that will (in some sense) approximate our robust policies.While our focus has been on parametric uncertainty, it must be noted that robustness to unmodeled dynamics (e.g.lags, time delays) is also important.While some discussion on this is provided, this will be examined in future work.Contributions of Work.The main contributions of this chapter are as follows: • Benefits of Robust Control in Renewable Resource Management.The chapter shows how robust control laws can be used to eliminate the limit cycle behavior of the optimal control law while increasing robustness to parametric uncertainty and achieving a return that is close (modulo limitations imposed by uncertainty) to the perfect information optimal control law.Special attention is paid to minimizing worst case economic downside.As such, the policies presented shed light on fundamental performance limitations in the presence of (parametric) uncertainty.The policies presented are intended to serve as targets/guidelines that fishery managers may try to approximate using available tools (e.g.taxes, quotas, etc. (Clark, 1990, Chapter 8).
• Tutorial/Introductory Value.The chapter serves as an introduction for the controls community to a very important resource management problem in the area of global sustainability.As such, the chapter offers a myriad of challenging problems for the controls community to address in future work.

Organization of Chapter.
The remainder of the chapter is organized as follows.
• Section 2 describes the classic Gordon-Schaefer nonlinear fishery model (Clark, 1990) to be used.
• Section 3 describes the optimal control law and its properties.The latter motivates the need for robust control laws for fishery management -laws that try to achieve robust near optimal performance in some sense.
• Section 4 describes a class of robust control laws to be examined.
• Section 5 contains the main results of the work -comparing the properties of the optimal policy to those of the robust policies being considered.
• Finally, Section 6 summarizes the chapter and presents directions for future research.

Nonlinear bioeconomic model
In this section, we describe the nonlinear bioeconomic model to be used for control design.The model is then analyzed.

Nonlinear Gordon-Schaefer Bioeconomic Model.
The nonlinear model to be used is as follows:

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Challenges and Paradigms in Applied Robust Control www.intechopen.com where represents the natural regeneration rate of the resource and x, x o ,a n du p represent resource biomass, initial resource biomass, and harvesting effort, respectively.The parameters r, k,a n dq, retain their traditional definitions of intrinsic growth rate, carrying capacity, and catchability, respectively.
The feasibility condition can then be written in terms of (3), i.e.
where u is the control signal.When there is no risk of confusion, we will write u p = sat(u).Performance Measure.The fishery performance measure to be used, denoted J, is the net present value of future returns: where price p, cost per unit effort c, discount rate δ, and planning horizon T are assumed constant.(We will use T = ∞ to develop the optimal control law.)

Equilibrium analysis of bioeconomic model
One of the desired control objectives will be for the fishery to operate at specific equilibrium (set) points.Given this, the set of equilibria for the nonlinear model are as follows: x e = ou e = 0( 6 ) Observe that as the equilibrium effort increases, the equilibrium biomass decreases (as expected).

LTI small signal model
To further understand the local characteristics of the above nonlinear model, we can linearize it about equilibria.Doing so yields the following small signal linear time invariant (LTI)

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Design of Robust Policies for Uncertain Natural Resource Systems: Application to the Classic Gordon-Schaefer Fishery Model model: The associated transfer function from δu to δx is given by:

Control objectives
The control objectives for the fishery may be summarized (roughly) as follows: 1 A planning horizon of T = 50 years was selected because the nominal discount rate is δ = 0.1 and in roughly T = 5 δ = 50 years, the integrand within J is negligible.Focus of Work: Biologically Exploited (BOE) Fishery.The focus of our presentation will be on a fishery that biologically overly expoilted (BOE) as opposed to biologically under exploited (BUE).This is because most of the world's critical fisheries are overly exploited (Clark, 1990).
• BOE with the 'low cost' c = 13.24.BOE occurs when the cost is sufficiently small.For the parameters indicated, it can be shown that: x * e = 0.75 • x MSY = 37.5 < x MSY = k 2 = 50 i.e. the optimal equilibrium biomass is below the maximum sustainable yield biomass.

Model uncertainty and scope of presentation
Within this presentation, we focus on uncertainty associated with the nominal model parameters: r, k, q, p, c, δ.The following uncertainty will not be addressed in this presentation but it is duly noted: 2. All plant parameters are uncertain.They may even change with time.Moreover, the plant contains additional dynamics; e.g. it takes time for the fishery workers to mobilize.This can contribute additional lags, time delays, and rate limiters within the plant.One can use a decentralized or distributed model in order to capture the decision making made by individual fisher people (Clark, 1990, Ch. 8 & 9).
3. Input and output disturbances are uncertain.
5. The biomass is not known; it must be estimated 6.The output (biomass) is sampled at some rate; if this rate is not sufficiently high, it could cause aliasing (Ogata, 1995); the sampling rate should be (as a rule-of-thumb)greater than ten times the control system bandwidth.
In contrast to many control applications where the "controller" is implemented with great fidelity, fishery controllers are implemented by an organization.As such, there can be considerable implementation issues/uncertainty.This will be discussed further below.

Optimal control law and properties
Within this section, we present the optimal control problem, the associated solution (optimal control law), and the properties of the optimal control law.

Optimal control law
We begin with a brief derivation of the classical optimal control policy stated in a way that will facilitate comparison to the class of LTI policies described later in this section.
The solution of the traditional optimal control problem: maximize is obtained by forming the Hamiltonian: where G(x, t) def = e −δt (pqx − c) − λqx and λ is the co-state variable.Pontryagin's Maximum Principle then implies that an optimal control policy will satisfy: Because the objective functional is linear, the Maximum Principle says nothing about the case when G(x, t)=0.However, using the co-state variable relationship λ = − ∂H ∂x ,t h e well-known implicit formula for the singular control path can be determined (Clark, 1990): 422 Challenges and Paradigms in Applied Robust Control www.intechopen.com Optimal Steady State Equilibrium Biomass.When F(x)=rx(1 − x/k), the above equation can be used to determine the optimal (steady state) equilibrium biomass x * e : x where is the maximum sustainable yield biomass and is the optimal equilibrium when δ = ∞; i.e. open-access equilibrium (Clark, 1990).The above shows that the optimal biomass x * e depends on the three independent parameters x ∞ , x MSY , and δ r .I tc a nb esh ow nt h a t for all δ ∈ [0, ∞] where the quantity x MSY + x ∞ 2 is the optimal x * e for δ = 0.The associated optimal (steady state) equilibrium control is given by: Optimal Control Policy.Define the tracking error as the difference between the desired (reference) state and the actual state, i.e.
Setting x re f = x * e and combining ( 19) with (25) yields following expression for the control law: The saturation function is then applied to this control signal to capture the physical constraints on the system, i.e. u p (t)=sat(u(t)).This control law implies the following: . Below, we show that this policy (in general) exhibits limit cycle behavior in the presence of parameter uncertainty (see Figure 6).

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Design of Robust Policies for Uncertain Natural Resource Systems: Application to the Classic Gordon-Schaefer Fishery Model www.intechopen.com

Nominal optimal control policy numerics
The numerics for the nominal optimal perfect information control law are summarized in Table 2.

Properties of the optimal control law
In this section, we describe the properties of the optimal control law assuming perfect information (i.e.model parameters are known with no error) and imperfect information (i.e.model parameters are not perfectly known).Understanding the properties of the optimal policy is very important for several reasons.We wish to understand (1) the fundamental robustness properties (e.g. economic inefficiency) of an optimal policy (e.g. one based on nominal, worst case, or best case parameters); (2) implementation issues associated with the optimal control policy; (3) how the robustness properties for our robust policies compare to those of a particular optimal control policy; (4) how x * e depends on parameter perturbations.The latter is important because we will using x * e as the reference command x re f for our robust control law policies.This is an issue because the optimal x * e (in general) is uncertain; i.e. x * e is only known for specific value selections (e.g.nominal, worst case, best case).As such, we will have to address this uncertainty to clearly understand what our robust control policies (with built-in command following) will be driving the state of the fishery to.In short, we show below that: (1) Since x * e is, in general, uncertain, if x * e is the desired (reference) state, then we have a major issue in that we will be driving the fishery to the incorrect state.This can have severe economic as well as biological implications (e.g.driving x below the critical depensation parameter k c , will destroy the fishery).(2) The optimal policy exhibits limit cycle behavior when x * e is uncertain.Moreover, it is very sensitive to any discrete sampling.As such, the (imperfect information) optimal policy is prohibitively expensive to implement (see Figure 6).Optimal Perfect Information Control Law Sensitivity: Single Parameter Results (x o = x * e ).The following shows how the the performance of the optimal perfect information control law changes with parameter perturbations.Results for our BOE fishery when x o = x * e are as follows: 1. (J Robustness with Respect to Parametric Uncertainty: Imperfect Versus Perfect Information.
Figure 1 shows how the optimal control law performs in the presence of parametric uncertainty.
x o = x * e for the perfect information policy.
x o is at the unperturbed/nominal x * e for the imperfect information policy.The plots compare the performance of the optimal control law with imperfect parameter knowledge to that with perfect parameter knowledge.The perfect information optimal control law (by definition) results in the maximum achievable return.While it represents a suitable benchmark to compare with, it must be emphasized that x * e is always uncertain.This is particularly crucial when x * e is being used as the target biomass (reference command) for a robust control law (see Sections 4, 5) because an incorrect reference command x re f will fundamentally limit the achievable performance.Moreover, no (inner loop) robust policy can address this.To properly address this, one needs some combination of parameter estimation, system identification, and learning coupled with some adaptive outer loop policy that adjusts the target based on collected information.While this is challenging and exciting to pursue, it is beyond the scope of our presentation.Figure 1 specifically shows the maximum theoretical (perfect information) return on the left in blue.The return associated with the imperfect information optimal policy (designed for nominal parameter values) is shown on the left in red.On the right in blue, we see how much the imperfect information optimal control law under performs the perfect information optimal control law.When k is perturbed by −30%, the imperfect law under performs the perfect information law by nearly 10%. Figure 1 shows that for a similar perturbation in r, the imperfect policy under performs by nearly 2%.It can be shown (figures not provided) that for a similar perturbation in δ, the imperfect policy under performs by less than 1%.It can be shown (figures not provided) that for similar perturbations in p, c,o rq the imperfect policy under performs by a very small percentage.Why is it that the biological parameters k and r matter more in closing the perfect-imperfect information performance gap than δ, p, c,o rq?T h i si s b e c a u s ex * e is more sensitive to uncertainty in k and r for the BOE case under consideration.In short, the plots show that we should be concerned primarily with uncertainty in k.More generally, we seek (robust) policies that perform closer to the perfect information optimal policy for likely parametric modeling errors.Imperfect information obviously limits how close we can get.This and associated issues will be addressed below.

Impact of Extremal Parameter Uncertainty on Perfect Information (J *
e , x * e , u * e )-AtOptimal Equilibrium.In what follows, x * e will be used as a reference command x re f to a robust control law with good command following properties.Since x * e is uncertain, it is important to understand how commanding an incorrect target will limit achievable performance.Given this, suppose that x o = x * e .We now ask, what is the worst case combination of perturbations for the model parameters (r, k, q, p, c, δ)?While an analytical proof is difficult, it can be shown (numerically) that J * e , in general, decreases when (k, r, p, q)aredecreasedand/or(c, δ)areincreased.
This result is independent of the initial condition x o for the BOE case under consideration.
Given uncertainty bounds for each of the model parameters, this observation permits us to readily determine the worst case set of parameter perturbation -something that, in general, is very difficult to do.x * e and perfect information is assumed.The figures show the dependence of the perfect information optimal control law on worst case and best case (extremal) parameter perturbations as defined below.
• Worst Case Extremal Parameter Perturbations.Within figures 2-3, negative (worst case extremal) parameter perturbations correspond to i.e. equal parametric perturbations that result in a smaller return.Here, Δθ def = θ − θ o represents a perturbation in the parameter θ with respect to the nominal parameter θ o .
• Best Case Extremal Parameter Perturbations.Within figures 2-3, positive (best case extremal) parameter perturbations correspond to i.e. equal parametric perturbations that result in a larger return.

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Challenges and Paradigms in Applied Robust Control www.intechopen.com The green curves within figures 2-3 represent actual optimal perfect information values.The blue curves give the percent deviation with respect to the nominal value.Assuming ±20% uncertainty for each nominal parameter value, figure 2 shows that the worst case perfect information optimal return is $215.6 M (65.25% below the nominal of J * e = 620.4M).In contrast, the best case perfect information optimal return is $1482M (138.95%above the nominal of J * e = 620.4M)-a 687% improvement with respect to the worst case perfect information optimal return.Also note from figure 3 that the worst case parameter combination results in a 20% reduction in x * e with respect to the nominal.From figure 3, we see that u * e is increased by less than 1%.

Dealing with Uncertain x *
e and J * e .Let x re f denote the reference biomass at which the fishery manager wishes to operate the fishery2 .How does a manager choose the target fishery biomass x re f ?A biologically conservative manager may wish to keep the fishery at the maximum sustainable yield x re f = x MSY = k 2 .A financially aggressive manager may chose to operate the fishery at the infinite discount (δ = ∞) ) optimal value x re f = x ∞ = c pq .M o r e generally, a manager could use the optimal value x re f = x * e as the point at which to operate.Given that x * e is known to within a percentage To select x re f ,weofferthefollowingapproaches.
1. Best-Worst Case Approach.Assume that we have good bounds on parametric uncertainty (not necessarily tight, but encompassing) for the 6 model parameters under consideration: k, r, q, p, c, δ.Suppose that we design for the best worst case scenario; i.e. try to approach the return of the perfect information optimal policy when the worst case parameter perturbations occur; i.e.
We assume worst case maximal parameter perturbations.(For simplicity, we assume that all parameters are perturbed by their maximum worst case percentage and that this percentage is the same for all of the parameters.)One could, for example, pick a worst case percentage which bounds all of the parameters.Doing so can be conservative.Parameter estimation can be used to narrow tighten this worst case percentage.If we have fixed percentage bounds for each of the parameters, our approach remains the same.(Recall: Determining the worst case perturbation in our problem is easy.This is not true in most practical scenarios.)Now choose x re f equal to the associated worst case x * e ;i.e.thex * e that results from choosing the worst case parameters.By so doing, the actual x * e will be greater than x re f .A ss u c h , only cases 1-3 can occur; i.e. cases 4-6 cannot occur.The only way, cases 4-6 can occur is if our uncertainty bounds were not truly encompassing.
2. Best-Best Case Approach.Assume that we have good bounds on parametric uncertainty (not necessarily tight, but encompassing)for the 6 model parameters under consideration: k, r, q, p, c, δ.Suppose that we design for the best best case scenario; i.e. try to approach the return of the perfect information optimal policy when the best case parameter perturbations occur; i.e.
We assume best case maximal parameter perturbations.(For simplicity, we assume that all parameters are perturbed by their maximum best case percentage and that this percentage is the same for all of the parameters.)Now choose x re f equal to the associated best case x * e ;i.e.thex * e that results from choosing the best case parameters.By so doing, the actual x * e will be smaller than x re f .A ssuc h ,on ly cases 4-6 can occur; i.e. cases 1-3 cannot occur.The only way, cases 1-3 can occur is if our uncertainty bounds were not truly encompassing.
3. Blended Best-Worst-Best-Best Approach.One can also try to offer a blended approach that attempts to offer decent returns when either worse case or best case parameter perturbations occur.We shall illustrate this below.
4. Probabilistic Approach.If a probability density function for the parameter percentage θ is available, it can be used to determine where to operate.Let f θ denote a density function for θ.This can be used to derive the density function f J for J.Given this, the expected value for J is given by E The density function for θ can be used to reflect what parameter perturbations are most likely to occur.The above expectation can then be used to choose x re f to maximize the expectation.
To illustrate the above ideas, consider figures 4-5 for small and large initial conditions, respectively under extremal parameter perturbations.The figures show results for the perfect information designs (black), best worst case design (blue), best best case design (red), and the nominal design (green).
To summarize, the following specific optimal control laws were implemented: 1. Perfect Information Optimal Designs: A Best-Worst Case Design: x re f = 29.7,u re f = 0.629

A Nominal Design Based on the Nominal Parameters:
x re f = 37.5, u re f = 0.625 4. A Best-Best Case Design: The performance of the perfect information designs are always best (by definition).The performance of the best-worst case design (blue) duplicates that of the perfect information design for 20% worst case perturbations since it is based on the worst case parameter model and x * e .The performance of the best-best case design (blue) duplicates that of the perfect information design for 20% best case perturbations since it is based on the best case parameter model and x * e .The following key observations are in order within figure 4 (small IC case): 1.The best-worst case design does better than the best-best case design when its parameter assumptions are maximally incorrect; falling by less than 20% (with respect to perfect information optimal return) while the best best falls by more than 40% (with respect to perfect information optimal return) when its parameter assumptions are maximally incorrect.
2. The nominal design can be viewed as a nice compromise or blend between the two prior policies.It is based on the nominal parameter model and x * e .Its returns deteriorates by a little more than 10% for worst case parameter uncertainty and by a little more than 5% for best case parameter uncertainty.In short, the returns associated with this nominal (blended) policy offers flatter returns over a wider range of extremal parameter perturbations.Each of the above three approaches offer a specific design model (to base the control design upon) and a specific x * e to use as a target.Control laws are always evaluated with the true (nonlinear) plant.In what follows, we will use the above as benchmarks whose performance we shall target via robust control laws.Similar patterns are observed for the large IC case in figure 5. Sensitivity Analysis: Extremal Perturbations, Small Initial The expected value for each of the design cases considered are as follows:  A uniform distribution has been assumed for the parameter uncertainty.The optimal perfect information control law is included for comparison purposes.Its performance can only be approximated over a range of parameter perturbations.This is because the (1) design plant parameters differ from those of the true plant and the desired target x re f differs from the perfect information target x * e .The following additional points are in order: • Although the best-best case design appears worse in terms of percentages at off design conditions, it has a higher expected return across all cases versus the best-worst case design.
• A manager could readily design a policy that limited the worst case downside return to a certain percentage of the maximum possible.For example, if the manager wanted a worst case downside return no worse than 5% of the maximum possible, a policy should be designed around roughly a −2 % parameter perturbation.
• A manager may also be interested in implementing the following policy: Sensitivity Analysis: Extremal Perturbation, Large Initial Condition.The expected value for each of the design cases considered are as follows: Finally, it should be noted that in contrast to the low initial condition study conducted, the Best-Best Case Design performs worse both in terms of the percentage possible and the expected return when compared to the Best-Worst Case Design.Limit Cycles In the Presence of Uncertainty.Finally, consider figure 6.The optimal control law is based on the nominal BOE parameters.The initial condition is above the uncertain x * e .The simulation is conducted with a truth plant possessing a 10% reduction in k -h e n c et h e limit cycle behavior.The figure shows that: (1) The optimal control policy (in general) will exhibit limit cycle behavior when we have imperfect information; i.e. model parameters are not known exactly.(2) The optimal control policy (in general) will be very sensitive to finer 431 Design of Robust Policies for Uncertain Natural Resource Systems: Application to the Classic Gordon-Schaefer Fishery Model sampling (ΔT smaller) under imperfect information; i.e. more oscillations (switching) will be exhibited as our x time samples are spaced closer together.The figure also shows that low pass filtering the optimal with a lag can be used to smooth oscillations a bit.To significantly reduce the oscillations, however, there is no easy fix.We either need a penalized up term within J to penalize switching or we need policies that are inherently more robust (like the ones we will describe subsequently).As such, this implies that, in practice, the optimal control policy is prohibitively expensive to implement in the presence of parametric uncertainty because of the inherent limit cycle behavior and the associated switching costs.Motivation for Robust Control Laws.The above motivates the need for more robust control laws; i.e. control laws that (1) exhibit an acceptable return (i.e. return robustness) in the presence of anticipated (likely) parametric uncertainty; (2) do not exhibit limit cycle behavior in the presence of anticipated (likely) parametric uncertainty.As such, the above motivates the robust control laws to be considered in our presentation.Control Law Implementation Issues.Unlike many control applications where controllers are implemented with great fidelity (within state-of-the-art digital computing units), controllers within a resource management system are implemented by an organization by setting rules for the fishery worker community (e.g.quotas, taxes (Clark, 1990, Chapter 8).As such, many types of uncertainties can be introduced by the organization.These could include any of the following: (1) parameter uncertainty, (2) additional uncertain actuation/sensing dynamics (e.g.lags, time delays, rate limiters, etc.), (3) nonlinearities (e.g.rate limiters, saturations, quantization, dead zones), (4) actuation/incentive errors (e.g.quota/tax miscalculations), (5) sensing, measurement, and estimation errors (e.g.sensor dynamics, biomass sampling/aliasing/quntization errors, noise, disturbances).

Robust control laws
The model under consideration is very simple.Many tools from the controls literature may be applied (e.g.classical control (Rodriguez, 2003), H-infinity (Rodriguez, 2004), feedback linearization, SDRE's, etc.).Given the introductory/tutorial nature of the paper, the simplicity of the model being used, as well as the fact that this text covers advanced control methodologies, we shall focus on simple control strategies from classical control theory.We will show that such control laws can be used to avoid limit cycles, increase robustness with 432 Challenges and Paradigms in Applied Robust Control www.intechopen.comrespect to parametric uncertainty, and achieve returns that are close to those of the perfect information optimal control law.Control System Structure.The structure of the control system may be visualized as shown in Figure 7. 2. Reference State or Command.x re f is the desired reference biomass state.Ideally, we would like to use x re f = x * e .Parameter uncertainty prevents us from commanding the desired state.As such, we are forced to choose x re f more judiciously.Given this, we will give special attention to maximizing our return under the worst case parameter uncertainty.
3. PI Controller.K is a proportional-plus-integral (PI) controller possessing the structure: where g > 0, z > 0, and p ro > 0. The integrator within the controller will ensure that step biomass commands are followed in the steady state while step input/output disturbances are rejected in the steady state.The (s + z) term will ensure that the LTI plant-integrator pair will be stabilized.The term p ro s+p ro provides high frequency roll-off to ensure that high frequency sensor noise n is suitably attenuated.4. Command Pre-Filter.W is a reference command pre-filter possessing the structure: This pre-filter can be used to ensure that the overshoot to step reference commands is suitably bounded.

5.
Observer-Based Integrator Anti-Windup Logic.Anti-windup logic is included so that the integrator in the PI controller does not windup.That is, the integrator is turned off so that it does not integrate constant errors which occur when the input to the plant is saturated (Aström, Hägglund).The structure of the anti-windup logic is as follows ẋk = A k x k + B k e + L(sat(u) − u) where L is an observer gain matrix.The PI controller with the anti-windup logic may be described by the following equations: where G AW is the anti-windup gain.
Nominal Design Methodology.The nominal design methodology can be described as follows (Rodriguez, 2003): 1. Plant Approximant.The following small signal LTI model P ≈ P d def = b s−a will be used to approximate our nonlinear plant.Here, P d is referred to as the design plant;i .e .t h e plant upon which we will base our control law design.While any design we obtain can be evaluated using plant approximants such as P d , control designs must be evaluated with the actual nonlinear plant model.√ bgz is the damping factor and ω n = √ gz is the undamped natural frequency.
5. Closed Loop Poles.Determine the nominal closed loop poles (assumed complex for rapid transient response): Given this, we will have nominal (local) closed loop exponential stability with an associated time constant τ = 1 ζω n .The associated (approximate 1%) settling time is t s = 5τ.6.Standard Second Order Closed Loop Transfer Function and Percent Overshoot.With the command pre-filter W, the associated closed loop transfer function takes the standard second order form: As such, the associated percent overshoot to a step reference command is given by so that the added high frequency roll-off does not significantly degrade the nominal phase margin within the loop.It could also be selected in order to satisfy a specific sinusoidal steady state noise attenuation specification.
11. Anti-Windup Gain.Choose the anti-windup gain G AW > 0 to be sufficiently large so that the integrator suitably shuts down in order to "maximally recapture" the dominant second order response characteristics described above.A family of gains is examined below.

Control law comparisons
In this section, we compare the properties of the nominal optimal control law with those for the robust policies based upon the nominal LTI plant model P d = b s−a .

Sample control law time responses
Within this section, sample time responses are provided for families of robust control laws (based upon the nominal LTI plant model) -families that approximate the performance of the nominal optimal control law.(Note: There will be an approximation gap when uncertainty is considered.)Reference Biomass Tracking: Anti-Windup Gain Study. Figure 8 shows closed loop biomass tracking time responses for a family of robust control law designs where ζ = 1, t s = 1.The anti-windup gain G AW is varied to control how well the responses approximate that of the optimal with no limit cycle behavior.As the anti-windup gain G AW is increased, the responses come closer to the (nominal) optimal control law (with no limit cycle behavior).The limit cycle behavior of the (nominal) optimal has been cleaned up in order to improve the readability of the figure (see Figure 6).A Note On Robustness with Respect to High Frequency Unmodeled Dynamics.It should be noted that as the speed of a policy is increased, the significance of unmodeled high frequency dynamics within the fishery or within the policy implementing organization/evironment (e.g.lags, time delays, rate limiters) becomes an issue to consider in final policy evaluation.It is well known from fundamental robustness theory (Rodriguez, 2004;2003) that fast control laws can result in closed loop oscillatory responses or instability when high frequency unmodeled dynamics are "significantly excited."This issue will be examined in future work.Reference Biomass Tracking: Damping Factor Study.Here, the reference command is large (x re f = 18.75), the controls saturate, windup is exhibited in the linear w/Sat case, and we observe relatively good agreement between the linear (particularly linear w/AW) and nonlinear responses.Biomass Tracking Robustness In Presence Of Capacity Uncertainty: Anti-Windup Gain Study. Figure 13 shows how our robust control laws can be adjusted to achieve the "flatter" economic inefficiency of the nominal optimal control law (see Figures 4-5).We observe the following: • With an anti-windup implementation, a PI control law can come arbitrarily close to matching the performance of the nominal optimal control law with imperfect information.
• Improving upon the nominal optimal control law with imperfect information requires some outer loop control logic as well as system identification to more appropriately select the reference/target biomass.
The observed performance gap (or inefficiency) is fundamentally because the target x re f differs from the perfect information x * e ; not because the nominal design plant differs from the truth plant.Closing the observed performance gap further requires an outer loop controller and/or parameter estimation techniques in order to get a more accurate target x re f that is closer to the perfect information target x * e .

Summary and future directions
Summary.This chapter has shown how ideas from robust control may be applied to a fishery.It has been specifically shown how some small amount of income may be sacrificed for increased robustness with respect to uncertain fishery parameters.Directions for Future Research.Future work will examine more complex models (e.g.decentralized, distributed), pros/cons associated with parameter estimation schemes, more complex robust control laws (e.g.use of receding horizon control for long-term management), robustness with respect to plant and controller uncertainty (parametric and dynamic).

Fig. 1 .
Fig. 1.Economic Inefficiency for Imperfect Information (Nominal) Optimal Control Law: Capacity & Growth Rate Uncertainty Consider figures 2-3 for (J * e , x * e , u * e ), respectively.Within these figures, x o =x * e and perfect information is assumed.The figures show the dependence of the perfect information optimal control law on worst case and best case (extremal) parameter perturbations as defined below.

Fig. 2 .
Fig. 2. Perfect Information Optimal Control Law Returns: Extremal Percent Parameter Perturbations, x o = x * e

Fig. 4 .
Fig. 4. Economic Inefficiencies for Various Optimal Control Laws: Extremal Perturbations (Small IC) • E[J]=867.8for the Perfect Information Optimal Designs • E[J]=772.0for the Best-Worst Case Design • E[J]=838.0for the Nominal Design • E[J]=800.4for the Best-Best Case Design

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Challenges and Paradigms in Applied Robust Control www.intechopen.com

Fig. 5 .
Fig. 5. Economic Inefficiencies for Various Optimal Control Laws: Extremal Perturbations (Large IC) • E[J]=1368.2for the Perfect Information Optimal Designs • E[J]=1284.5 for the Best-Worst Case Design • E[J]=1336.8for the Nominal Design • E[J]=1259.3for the Best-Best Case Design Fig. 6.Optimal Control Law Robustness: Limit Cycles In Presence of (-10% Capacity) Uncertainty

Fig. 7 .
Fig. 7. Renewable Resource Management Problem Represented as a Standard Negative Feedback System with a Pre-Filter and Anti-Windup Logic 1. Plant.Here, P represents the plant under control.We shall use an LTI small signal model to approximate our nonlinear plant.

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Design of Robust Policies for Uncertain Natural Resource Systems: Application to the Classic Gordon-Schaefer Fishery Model www.intechopen.com Table1in Section 2.5 summarizes model parameter definitions, units, nominal values, and ranges.Model uncertainty will be addressed in Section 2.6.Saturating Nonlinearity.Typically, effort is bounded above by some maximum and below by zero, i.e. u p ∈ [0, u max ].Typically, this physical constraint is implicitly taken into account when the optimal control problem is solved.However, a more general family of controls may generate control signals outside the allowable range, and it is important to be explicit about how these signals are "clipped" by physical constraints.We thus define the saturation function sat(x;x min , x max ) def it follows that the equilibrium point (x e , u e ) is asymptotically stable with the rate of convergence (pole) being proportional to the equilibrium biomass x e ,t h e fishery growth rate r, and inversely proportional to the fishery's carrying capacity k.≈ x when u ≈ u e (i.e.δu(t) ≈ 0) and x o ≈ x e (i.e.δx(0)=x o − x e ≈ 0).

Table 3 .
In general, x o and x * e are uncertain.How does one choose the target x re f .We'd ideally likex re f = x * e ,b utx * e is uncertain.What can a manager do?The table below contains the six possible relations that can exist amongst the three scalars (x o , x * e , x re f ) -from smallest to biggest.In general, we would (ideally) like the state to move from x o toward x re f = x * .As such, it follows that we may issue reference commands x re f that move the state x in an incorrect direction.Since the state moves from x o toward x re f , it follows from the table below that in two cases the state moves in the incorrect direction.In the four other cases, the state moves in the correct direction.Six Possible Inequality Relations for (x o , x * e , x re f ): 6 Cases.e .S in c ex * e is uncertain, it follows that x re f (in general) will differ from x * e e , x re f ) 2. Controller Approximant.Use the controller approximant K ≈ Controller Roll-Of Parameter.Choose the roll-off parameter p ro as follows: