I denna avhandling studeras optimal återkopplad styrning av olinjära deskriptorsystem. Ett deskriptorsystem är en matematisk beskrivning som kan innehålla både differentialekvationer och algebraiska ekvationer. En av anledningarna till intresset för denna klass av system är att objekt-orienterade modelleringsverktyg ger systembeskrivningar på denna form. Här kommer det att antas att det, åtminstone lokalt, är möjligt att eliminera de algebraiska ekvationerna och få ett system på tillståndsform. Teoretiskt är detta inte så inskränkande för genom att använda någon indexreduktionsmetod kan ganska generella deskriptor\-system skrivas om så att de uppfyller detta antagande. För system på tillståndsform kan Hamilton-Jacobi-Bellman-ekvationen användas för att bestämma den optimala återkopplingen. Ett liknande resultat finns för deskriptor\-system där istället en Hamilton-Jacobi-Bellman-liknande ekvation ska lösas. Denna ekvation innehåller dock en extra term för att hantera de algebraiska ekvationerna. Eftersom antagandena i denna avhandling gör det möjligt att skriva om deskriptorsystemet som ett tillståndssystem, undersöks hur denna extra term måste väljas för att båda ekvationerna ska få samma lösning. Ett problem med att beräkna den optimala återkopplingen med hjälp av Hamilton-Jacobi-Bellman-ekvationen är att det leder till att en olinjär partiell differentialekvation ska lösas. Generellt har denna ekvation ingen explicit lösning. Ett lättare problem är att beräkna en lokal optimal återkoppling. För analytiska system på tillståndsform löstes detta problem på 1960-talet och den optimala lösningen beskrivs av serieutvecklingar. I denna avhandling generaliseras detta resultat så att även deskriptor-system kan hanteras. Metoden illustreras med ett exempel som beskriver en faslåsande krets. I många situationer vill man veta om ett område är möjligt att nå genom att styra på något sätt. För linjära tidsinvarianta system fås denna information från styrbarhetgramianen. För olinjära system används istället styrbarhetsfunktionen. Tre olika metoder för att beräkna styrbarhetsfunktionen har härletts i denna avhandling. De framtagna metoderna är också applicerade på några exempel för att visa beräkningsstegen. Dessutom har observerbarhetsfunktionen studerats. Observerbarhetsfunktionen visar hur mycket utsignalenergi ett visst initial tillstånd svarar mot. Ett par olika metoder för att beräkna observerbarhetsfunktionen för deskriptorsystem tagits fram. För att beskriva en av metoderna, studeras ett litet exempel bestående av en elektrisk krets. / In this thesis, optimal feedback control for nonlinear descriptor systems is studied. A descriptor system is a mathematical description that can include both differential and algebraic equations. One of the reasons for the interest in this class of systems is that several modern object-oriented modeling tools yield system descriptions in this form. Here, it is assumed that it is possible to rewrite the descriptor system as a state-space system, at least locally. In theory, this assumption is not very restrictive because index reduction techniques can be used to rewrite rather general descriptor systems to satisfy this assumption. The Hamilton-Jacobi-Bellman equation can be used to calculate the optimal feedback control for systems in state-space form. For descriptor systems, a similar result exists where a Hamilton-Jacobi-Bellman-like equation is solved. This equation includes an extra term in order to incorporate the algebraic equations. Since the assumptions made here make it possible to rewrite the descriptor system in state-space form, it is investigated how the extra term must be chosen in order to obtain the same solution from the different equations. A problem when computing the optimal feedback law using the Hamilton-Jacobi-Bellman equation is that it involves solving a nonlinear partial differential equation. Often, this equation cannot be solved explicitly. An easier problem is to compute a locally optimal feedback law. This problem was solved in the 1960's for analytical systems in state-space form and the optimal solution is described using power series. In this thesis, this result is extended to also incorporate descriptor systems and it is applied to a phase-locked loop circuit. In many situations, it is interesting to know if a certain region is reachable using some control signal. For linear time-invariant state-space systems, this information is given by the controllability gramian. For nonlinear state-space systems, the controllabilty function is used instead. Three methods for calculating the controllability function for descriptor systems are derived in this thesis. These methods are also applied to some examples in order to illustrate the computational steps. Furthermore, the observability function is studied. This function reflects the amount of output energy a certain initial state corresponds to. Two methods for calculating the observability function for descriptor systems are derived. To describe one of the methods, a small example consisting of an electrical circuit is studied. / Report code: LiU-TEK-LIC-2006:8

<p>Aggressive fiscal and monetary policies by governments of countries and central banks in developed markets could somehow push inflation to some very high level in the long run. Due to the decreasing of pension fund benefits and increasing inflation rate, pension companies are selling inflation-linked products to hedge against inflation risk. Such companies are seriously considering the possible effects of inflation volatility on their investment, and some of them tend to include inflationary allowances in the pension payment plan. In this dissertation we study the management of pension funds of the defined contribution type in the presence of inflation-recession. We study how the fund manager maximizes his fund&rsquo / s wealth when the salaries and stocks are affected by inflation. In this regard, we consider the case of a pension company which invests in a stock, inflation-linked bonds and a money market account, while basing its investment on the contribution of the plan member. We use a benchmarking approach and martingale methods to compute an optimal strategy which maximizes the fund wealth.</p>

The main contributions in this thesis are advances in parametric programming. The thesis is divided into three parts; theoretical advances, application areas and constrained control allocation. The first part deals with continuity properties and the structure of solutions to convex parametric quadratic and linear programs. The second part focuses on applications of parametric quadratic and linear programming in control theory. The third part deals with constrained control allocation and how parametric programming can be used to obtain explicit solutions to this problem.

This thesis concerns modeling and control of bilinear systems (BLS). BLS are linear but not jointly linear in state and control. In the first part of the thesis, a background to BLS and their applications to modeling and control is given. The second part, and likewise the principal theme of this thesis, is dedicated to theoretical aspects of identification, modeling and control of mainly BLS, but also linear systems. In the last part of the thesis, applications of bilinear and linear modeling and control to the activated sludge process (ASP) are given.

While the market for credit instruments grew continuously in the decade before 2008, its liquidity has dried up significantly in the current crisis, and investors have become aware of the possible consequences of being exposed to credit risk. In this thesis we address these issues by pricing credit instruments using utility indifference pricing, a method that takes into account the investor's personal risk aversion and which is not affected by the lack of liquidity. Through stochastic optimal control methods, we use indifference pricing with exponential utility to determine corporate bond prices and CDS spreads. In the first part we examine how these quantities are affected by risk aversion under different models of default. The emphasis lies on a hybrid model, in which a regime switch of the reference entity is triggered by a creditworthiness index correlated to its stock price. The second part generalizes this setup by introducing uncertainty in the model parameters. Robust optimal control has been used independently in the literature to address model uncertainty for portfolio selection problems. Here, we incorporate this approach with utility indifference and derive some analytical and numerical results on how model uncertainty affects credit spreads.

The design of optimal strictly positive real (SPR) compensators using numerical optimization is considered. The plants to be controlled are linear and nonlinear flexible manipulators. For the design of SISO and MIMO linear SPR controllers, the optimization objective function is defined by reformulating the H2-optimal control problem subject to the constraint that the controllers must be SPR. Various controller parameterizations using transfer functions/matrices and state-space equations are considered. Depending on the controller form, constraints are enforced (i) using simple inequalities guaranteeing SPRness, (ii) in the frequency domain, or (iii) by implementing the Kalman-Yakubovich- Popov lemma. The design of a gain-scheduled SPR controller using numerical optimization is also considered. Using a family of linear SPR controllers, the time dependent scheduling signals are parameterized, and the objective function of the optimizer seeks to find the form of the scheduling signals which minimizes the manipulator tip tracking error while minimizing the control effort.

This thesis contains three essays on topics in agricultural economics. Essays one and two share a focus on international trade and economic development, and essays two and three apply dynamic tools to agricultural economic policy issues.<p>Essay one analyses trade-related implications of a developing country's decision to adopt genetically-modified crop technology. A fixed-proportions model is constructed that evaluates the welfare implications of a range of adoption policies and export market responses. The model in this essay illustrates the importance of the prospective adopter formulating a projection of probable export market effects before making an adoption decision and of the role that high transaction costs may play in a developing country's adoption decision. The model also considers the effects of a new policy tool; a check-off style levy on genetically-modified technology in place of a technology-use agreement. A levy could be useful tool in developing countries, which are characterised by high transaction costs. <p>Essay two models the effects of emergency food aid on a recipient country's agricultural industry. This essay formulates a definition of needed aid in the context of a food emergency and constructs an optimal control model that solves a path of aid shipments that best meets that need. The effects of a range of food aid paths on recipient-country agricultural production are illustrated through numerical simulations. There are two key results. First, a non-optimal amount of aid can hinder a recipient-country's recovery from an exogenous food shock. Second, an exogenous shock can affect farmer revenue and therefore impact planting decisions. This effect must be considered in aid allocation policies. <p>Essay three uses time-series econometric techniques to develop a demand model that assesses the effectiveness of commodity advertising. This essay describes the importance of considering long-run and dynamic effects in demand systems, especially in the case of closely substitutable commodities. A demand system that tests for and accommodates dynamic and time-series properties is developed and applied to US meat data. The results of this model are compared to a traditional static demand system. The dynamic model produces econometrically and theoretically sound results and generates some more intuitively appealing estimates.

The design of optimal strictly positive real (SPR) compensators using numerical optimization is considered. The plants to be controlled are linear and nonlinear flexible manipulators. For the design of SISO and MIMO linear SPR controllers, the optimization objective function is defined by reformulating the H2-optimal control problem subject to the constraint that the controllers must be SPR. Various controller parameterizations using transfer functions/matrices and state-space equations are considered. Depending on the controller form, constraints are enforced (i) using simple inequalities guaranteeing SPRness, (ii) in the frequency domain, or (iii) by implementing the Kalman-Yakubovich- Popov lemma. The design of a gain-scheduled SPR controller using numerical optimization is also considered. Using a family of linear SPR controllers, the time dependent scheduling signals are parameterized, and the objective function of the optimizer seeks to find the form of the scheduling signals which minimizes the manipulator tip tracking error while minimizing the control effort.

While the market for credit instruments grew continuously in the decade before 2008, its liquidity has dried up significantly in the current crisis, and investors have become aware of the possible consequences of being exposed to credit risk. In this thesis we address these issues by pricing credit instruments using utility indifference pricing, a method that takes into account the investor's personal risk aversion and which is not affected by the lack of liquidity. Through stochastic optimal control methods, we use indifference pricing with exponential utility to determine corporate bond prices and CDS spreads. In the first part we examine how these quantities are affected by risk aversion under different models of default. The emphasis lies on a hybrid model, in which a regime switch of the reference entity is triggered by a creditworthiness index correlated to its stock price. The second part generalizes this setup by introducing uncertainty in the model parameters. Robust optimal control has been used independently in the literature to address model uncertainty for portfolio selection problems. Here, we incorporate this approach with utility indifference and derive some analytical and numerical results on how model uncertainty affects credit spreads.

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