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

Investigating an important overlooked phase of interorganizational relationship evolution, which is currently hypothesized to progress through five stages of awareness, exploration, expansion, commitment, and dissolution, this dissertation proposes that in the long road between commitment and dissolution, the quintessential interfirm relationship is likely to be characterized by aprolonged period of relationship alienation, which then becomes the immediate precursor to the dissolution stage. The dissertation utilizes social learning theory, behavior constraint theory, and alienation theory to explain apathetic behaviors of franchisees. The principal proposition is that certain characteristics of the franchise system’s operating environment inadvertently condition franchisee estrangement and failure, and the maladaptive behaviors persist even after environmental changes make success possible again. The dissertation proposes and empirically tests a conceptual model of franchisee alienation. Data from dyadic franchisee-franchisor relationships (N=185) across a wide variety of industries were obtained through a survey of franchisee organizations that were members of the Franchise Council of Australia (FCA). The results render support to the central hypothesis that franchisee alienation occurs as a result of the franchisee organization disconnecting its own actions from the outcomes of its interactions with the franchisor. Franchisee alienation is shown as a phenomenon that is extremely toxic for the franchise system as a whole, as the alienated franchisee is likely to engage in opportunistic behaviors, exhibits reduced productivity, and is inclined to litigate against the franchisor and to dissolve its relationship with the franchisor. On the basis of the findings, the dissertation offers a prescription in terms of the different strategies that can be used by the franchisor to prevent and combat franchisee alienation.

A quadratic form on \(\mathbb{R}^n\) is a map of the form \(x \mapsto x^T M x\), where M is a symmetric \(n \times n\) matrix. A quadratic map from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) is a map, all m of whose components are quadratic forms. One of the two central questions in this thesis is this: when is the image of a quadratic map \(Q: \mathbb{R}^n \rightarrow \mathbb{R}^m\) a convex subset of \(\mathbb{R}^m\)? This question has intrinsic interest; despite being only a degree removed from linear maps, quadratic maps are not well understood. However, the convexity properties of quadratic maps have practical consequences as well: underlying every semidefinite program is a quadratic map, and the convexity of the image of that map determines the nature of the solutions to the semidefinite program. Quadratic maps that map into \(\mathbb{R}^2\) and \(\mathbb{R}^3\) have been studied before (in (Dines, 1940) and (Calabi, 1964) respectively). The Roundness Theorem, the first of the two principal results in this thesis, is a sufficient and (almost) necessary condition for a quadratic map \(Q: \mathbb{R}^n \rightarrow \mathbb{R}^m\) to have a convex image when \(m \geq 4\), \(n \geq m\) and \(n \not= m + 1\). Concomitant with the Roundness Theorem is an important lemma: when \(n < m\), quadratic maps from \(\mathbb{R}^n\) to \(\mathbb{R}^m\)seldom have convex images. This second result in this thesis is a controllability condition for bilinear systems defined on direct products of the form \(\mathcal{G} \times\mathcal{G}\), where \(\mathcal{G}\) is a simple Lie group. The condition is this: a bilinear system defined on \(\mathcal{G} \times\mathcal{G}\) is not controllable if and only if the Lie algebra generated by the system’s vector fields is the graph of some automorphism of \(\mathcal{g}\), the Lie algebra of \(\mathcal{G}\). / Engineering and Applied Sciences

This study explores whether anti-fat bias is reduced when obesity in women is perceived to be the result of an uncontrollable versus controllable factor. An experiment was designed to test whether manipulating the causal attribution in favor of the target (i.e., offering an uncontrollable attribution that removes all responsibility from the target for her obesity) would have an impact on perceivers' anti-fat attitudes, beliefs about weight controllability, and tendency to blame a woman for her weight. The exploration of this topic was conducted within a greater context of obesity-related issues including body image, social stigma, responsibility, and a culture of blame. Equally important in this endeavor was gaining insight into the reasons why women are blamed more for their obesity than their male counterparts. Attribution theory (Heider, 1958) serves as the primary theoretical framework. One hundred and eighty undergraduates from a large, southwestern university participated in the study. Participants were randomly assigned to one of the following six conditions, five of which contained a version of the treatment embedded in an essay: Uncontrollable attribution, controllable attribution, potentially controllable attribution, manifest mention of weight, no mention of weight or control. The potentially controllable attribution was ultimately determined to be insufficiently manipulated and responses generated in response to it were not included in the data analysis. All participants completed a survey comprised of various measures designed to gauge anti-fat attitudes, beliefs about weight controllability, and blame tendency. A mediated model examining the relationships among the above three variables was also tested. Additionally, participants completed the Obesity-Induced Blame Scale (OIBS); a measure designed for the present study to test the degree to which a woman will be blamed and held responsible for her weight. While some results failed to statistically support the usefulness of causal attributions that remove controllability from the woman for her weight, other results demonstrated the embedded anti-fat bias some individuals hold for obese women. Findings that emerged from the qualitative data were especially revealing. The social implications of anti-fatism, beliefs about weight controllability, and blame tendency, as well as all key findings, are discussed.

In this thesis, we study uniform controllability properties of semi-discrete approximations for parabolic systems. In a first part, we address the minimization of the Lq-norm (q > 2) of semidiscrete controls for parabolic equation. Our goal is to overcome the limitation of [LT06] about the order 1/2 of unboundedness of the control operator. Namely, we show that the uniform observability property also holds in Lq (q > 2) even in the case of a degree of unboundedness greater than 1/2. Moreover, a minimization procedure to compute the approximation controls is provided. The study of Lq optimality in the first part is in a general context. However, the discrete observability inequalities that are obtained are not so precise than the ones derived then with Carleman estimates. In a second part, in the discrete setting of one-dimensional finite-differences we prove a Carleman estimate for a semi discrete version of the parabolic operator @t − @x(c@x) which allows one to derive observability inequalities that are far more precise. Here we consider in case that the diffusion coefficient has a jump which yields a transmission problem formulation. Consequence of this Carleman estimate, we deduce consistent null-controllability results for classes of linear and semi-linear parabolic equations.

Research Doctorate - Doctor of Philosophy (PhD) / This dissertation presents some new approaches to addressing the main issues encountered by practitioners in the implementation of linear model predictive control(MPC), namely, stability, feasibility, complexity and the size of the region of attraction. When stability guaranteeing techniques are applied nominal feasibility is also guaranteed. The most common technique for guaranteeing stability is to apply a special weighting to the terminal state of the MPC formulation and to constrain the state to a terminal region where certain properties hold. However, the combination of terminal state constraints and the complexity of the MPC algorithm result in regions of attraction that are relatively small. Small regions of attraction are a major problem for practitioners. The main approaches used to address this issue are either via the reduction of complexity or the enlargement of the terminal region. Although the ultimate goal is to enlarge the region of attraction, none of these techniques explicitly consider the upper bound of this region. Ideally the goal is to achieve the largest possible region of attraction which for constrained systems is the null controllable set. For the case of systems with a single unstable pole or a single non-minimum phase zero their null controllable sets are defined by simple bounds which can be thought of as implicit constraints. We show in this thesis that adding implicit constraints to MPC can produce maximally controllable systems, that is, systems whose region of attraction is the null controllable set. For higher dimensional open-loop unstable systems with more than one real unstable mode, the null controllable sets belong to a class of polytopes called zonotopes. In this thesis, the properties of these highly structured polytopes are used to implement a new variant of MPC, which we term reduced parameterisation MPC (RP MPC). The proposed new strategy dynamically determines a set of contractive positively invariant sets that require only a small number of parameters for the optimisation problem posed by MPC. The worst case complexity of the RP MPC strategy is polylogarithmic with respect to the prediction horizon. This outperforms the most efficient on-line implementations of MPC which have a worst case complexity that is linear in the horizon. Hence, the reduced complexity allows the resulting closed-loop system to have a region of attraction approaching the null controllable set and thus the goal of maximal controllability.

Dans cette thèse, nous étudions les propriétés de contrôlabilité uniforme des semidiscrets approximations de systèmes paraboliques. Dans une première partie, nous nous intéressons à la minimisation de Lq-norme (q > 2) des contrôles semidiscrete pour l'équation parabolique. Notre objectif est de dépasser la limitation de [LT06] à propos de l'ordre ½ de l'absence de limites d'opérateur de contrôle. Plus précisément, nous montrons que la propriété d'observabilité uniforme est également titulaire dans Lq (q > 2), même dans le cas d'un degré d'absence de limites supérieure à 1/2. En outre, une procédure de minimisation pour calculer les commandes d'approximation est fournie. L'étude de l'optimalité Lq dans lapremière partie est dans un contexte général. Cependant, les inégalités d'observabilité discrets qui sont obtenus ne sont pas aussi précises que celles dérivées puis avec des estimations de Carleman. Dans une seconde partie, dans le contexte particulier de unidimensionnels-finis différences nous démontrons une inégalité de Carleman pour une version semi-discret de l'opérateur parabole @t − @x(c@x) qui permet pour dériver les inégalités d'observabilité qui sont beaucoup plus précis. On considère ici que dans le cas où le coefficient de diffusion a un saut qui donne une formulation du problème de transmission. Conséquence de cette inégalité de Carleman, on en déduit cohérentes nul contrôlabilité des résultats pour les classes de linéaires et semi-linéaire des équations paraboliques. / In this thesis, we study uniform controllability properties of semi-discrete approximations for parabolic systems. In a first part, we address the minimization of the Lq-norm (q > 2) of semidiscrete controls for parabolic equation. Our goal is to overcome the limitation of [LT06] about the order 1/2 of unboundedness of the control operator. Namely, we show that the uniform observability property also holds in Lq (q > 2) even in the case of a degree of unboundedness greater than 1/2. Moreover, a minimization procedure to compute the approximation controls is provided. The study of Lq optimality in the first part is in a general context. However, the discrete observability inequalities that are obtained are not so precise than the ones derived then with Carleman estimates. In a second part, in the discrete setting of one-dimensional finite-differences we prove a Carleman estimate for a semi discrete version of the parabolic operator @t − @x(c@x) which allows one to derive observability inequalities that are far more precise. Here we consider in case that the diffusion coefficient has a jump which yields a transmission problem formulation. Consequence of this Carleman estimate, we deduce consistent null-controllability results for classes of linear and semi-linear parabolic equations.

De type parabolique et soumis à l’action d’un couple de contrôles (h, k) où h et k jouent des rôles différents ; le contrôle k étant de type "contrôlabilité" et h de type "contrôle optimal".Il est alors naturel de considérer un problème d’optimisation multi-critères. Il existe plusieurs façons d’étudier de tels problèmes. Nous proposons, dans cette thèse, le contrôle de Stackelberg. Il s’agit d’une notion d’optimisation hiérarchique avec, ici, h qui est le "Leader" et k le "Follower". / In this thesis, we are interested in evolution problems governed by parabolic equations subjected to the action of a pair of controls (h, k) where h and k play different roles : the control k being of "controllability" type and h of "optimal control" type.It is then natural to consider a multi-criteria optimization problem. There are several ways to study such problems. We propose in this thesis, the Stackelberg control which is a notion of hierarchical optimization with here, h which is the "Leader" and k the "Follower".

Made available in DSpace on 2015-05-15T11:46:02Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 425597 bytes, checksum: 2cb5c42e376ba77764139fbc3fcdaf15 (MD5) Previous issue date: 2011-07-02 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / We consider the semilinear heat equation involving gradient terms in a bounded domain of Rn. It is assumed the non-linearity is globally Lipschitz. We prove that the system is approximately controllable when the control acts on a bounded subset of the domain. The proof uses a variant of a classical fixed point method and is a simpler alternative to the earlier proof existing in the literature by means of the penalization of an optimal control problem. We also prove that the control may be built so that, in addition to the approximate controllability requirement, it ensures that the state reaches exactly a finite number of constraints. / Consideremos a equação do calor semilinear envolvendo termos do gradiente em um domínio limitado do Rn. Assumimos que a não-linearidade é globalmente Lipschtz. Usando o método do ponto fixo, provamos que o sistema é finito-aproximadamente controlável e nulamente controlável, quando o controle age em um subconjunto não vazio do domínio.

Made available in DSpace on 2015-05-15T11:46:03Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 876230 bytes, checksum: 3a204615891ef1a7232794e0c75afdc8 (MD5) Previous issue date: 2010-09-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This dissertation is devoted to prove the local exact controllability to the trajectories for a coupled system, of the Boussinesq kind. In the state system, the unknowns are the velocity field and pressure of the uid (y; p), the temperature (-) and an additional variable c that can be viewed as the concentration of a contaminant solute. We prove several results, that essentially show that it is sufficient to act locally in space on the equations satisfied by (-) and c. The controllability property of this system will be obtained by means of a Carleman inequality for apropriate system and of a inverse function theorem. / Esta dissertação é dedicada a provar a controlabilidade exata local ás trajetórias para um sistema acoplado do tipo Boussinesq. No sistema estado, as variáveis desconhecidas são o campo velocidade e pressão do fluido (y; p), a temperatura - e uma variável adicional c que pode ser vista como uma concentração de um soluto contaminante. A propriedade de controlabilidade nula desse sistema será obtida por meio de uma desigualdade de Carleman para um sistema apropriado e de um teorema de função inversa.