The category of dynamical systems most established and studied for the longest time is that of linear systems; however, most of the real systems are non-linear, the State Dependent Coefficient (SDC) approach proposes a method to represent them with structures similar to those of the linear world retaining their non-linear nature. Linear time invariant systems have been extensively studied, it is possible to clearly identify characteristics such as reachability, observability and stability. The control techniques can achieve the overall optimum in a mathematically certain way if the system meets the requirements. The most frequent strategy is to linearize the system around the working point, obtaining satisfactory performances valid in the local neighborhood, this approach in many cases is too restrictive. For example, in the aerospace field, where the devices work in every part of the atmosphere under significantly different conditions, or more simply in applications where the working point moves away from that of linearization. Unlike non-linear control techniques, such as sliding mode or feedback linearization, the State Dependent Riccati Equation (SDRE) control offers a hybrid alternative. Using the theory of linear optimal control, it is possible to extend the effectiveness of the Linear Quadratic Regulator (LQR) control to non-linear systems through the use of SDC matrices and SDRE control. This allows to control the system in an area of the state space that is larger than the neighborhood working point where the system has been linearized. The SDRE control technique has been applied to various devices, such as missiles, aircraft, UAVs, autonomous systems, biomedical systems and robots. The robustness, flexibility of this technique and today's ability to implement control in real time have attracted the attention of the community. Despite the growing documentation on the SDRE technique, there is still a significant lack of theoretical justifications on the choice of SDC matrices and their effectiveness in representing the system. This thesis illustrates how to represent a system in state space with terms dependent on the state itself. The most common methods for constructing SDC matrices have been collected, improving some aspects and proposing alternatives. The SDRE control with SDC structure is used and studied in efficiency, efficacy and ductility with methods known in the literature and new modalities. An innovative solution is proposed for this control in the choice of the Q matrix, which can also be extended to the well-known LQR. In conclusion, all the techniques on different case studies commonly used in literature and some real application cases are illustrated.
La categoria di sistemi dinamici più assodata e studiata da maggior tempo è quella dei sistemi lineari; tuttavia, la maggior parte dei sistemi reali è non lineare, l’approccio State Dependent Coefficient (SDC) propone un metodo per rappresentarli con strutture simili a quelle del mondo lineare ma conservando la loro natura non lineare. I sistemi lineari tempo invarianti sono stati ampiamente studiati, è possibile identificare chiaramente le caratteristiche come aggiungibilità, osservabilità e stabilità. Le tecniche di controllo possono raggiungere l’ottimo globale in modo matematicamente certo se il sistema soddisfa i requisiti richiesti. La strategia più frequente è quella di linearizzare il sistema attorno al punto di lavoro, ottenendo delle prestazioni soddisfacenti valide nell’intorno locale, questo approccio in molti casi è troppo restrittivo. Ad esempio, nel campo aerospaziale, dove i dispositivi lavorano in ogni parte dell’atmosfera in condizioni significativamente diverse, oppure più semplicemente in applicazioni dove il punto di lavoro si allontana da quello di linearizzazione. A differenza di tecniche di controllo non lineari, come sliding mode o feedback linearization, il controllo State Dependent Riccati Equation (SDRE) propone un’alternativa ibrida. Utilizzando la teoria del controllo ottimo lineare, è possibile estendere l’efficacia del controllo Linear Quadratic Regulator (LQR) a sistemi non lineari passando attraverso l’uso di matrici SDC ed il controllo SDRE. Questo permette di controllare il sistema in un’area dello spazio di stato più estesa dell’intorno del punto di lavoro dove si sarebbe linearizzato il sistema. La tecnica di controllo SDRE è stata applicata su diversi dispositivi, come missili, aereomobili, UAV, sistemi autonomi, sistemi biomedicali e robot. La robustezza, flessibilità di questa tecnica e la possibilità odierna di implementare il controllo in tempo reale hanno attirato l’attenzione della comunità. Nonostante la crescente documentazione sulla tecnica SDRE è presente tutt’oggi una significativa mancanza di giustificazioni teoriche sulla scelta delle matrici SDC e la loro efficacia nel rappresentare il sistema. In questa tesi viene illustrato come rappresentare un sistema in spazio di stato con termini dipendenti dallo stato stesso. Sono stati raccolti i metodi più comuni per costruire le matrici SDC migliorandone alcuni aspetti e proponendoalternative. Il controllo SDRE con struttura SDC viene utilizzato e studiato in efficienza, efficacia e duttilità con metodi noti in letteratura e nuove modalità. Viene proposta una soluzione alternativa per tale controllo nella scelta della funzione di costo, estendibile anche al noto LQR. In conclusione, sono illustrate tutte le tecniche su diversi casi di studio utilizzati comunemente in letteratura ed alcuni casi di applicazione reale.
Problemi di controllo ottimo per sistemi non lineari con struttura State-Dependent Coefficient, implicazioni ed aspetti computazionali / Scala, GIUSEPPE ANTONIO. - (2022 Mar 04).
Problemi di controllo ottimo per sistemi non lineari con struttura State-Dependent Coefficient, implicazioni ed aspetti computazionali
SCALA, GIUSEPPE ANTONIO
2022-03-04
Abstract
The category of dynamical systems most established and studied for the longest time is that of linear systems; however, most of the real systems are non-linear, the State Dependent Coefficient (SDC) approach proposes a method to represent them with structures similar to those of the linear world retaining their non-linear nature. Linear time invariant systems have been extensively studied, it is possible to clearly identify characteristics such as reachability, observability and stability. The control techniques can achieve the overall optimum in a mathematically certain way if the system meets the requirements. The most frequent strategy is to linearize the system around the working point, obtaining satisfactory performances valid in the local neighborhood, this approach in many cases is too restrictive. For example, in the aerospace field, where the devices work in every part of the atmosphere under significantly different conditions, or more simply in applications where the working point moves away from that of linearization. Unlike non-linear control techniques, such as sliding mode or feedback linearization, the State Dependent Riccati Equation (SDRE) control offers a hybrid alternative. Using the theory of linear optimal control, it is possible to extend the effectiveness of the Linear Quadratic Regulator (LQR) control to non-linear systems through the use of SDC matrices and SDRE control. This allows to control the system in an area of the state space that is larger than the neighborhood working point where the system has been linearized. The SDRE control technique has been applied to various devices, such as missiles, aircraft, UAVs, autonomous systems, biomedical systems and robots. The robustness, flexibility of this technique and today's ability to implement control in real time have attracted the attention of the community. Despite the growing documentation on the SDRE technique, there is still a significant lack of theoretical justifications on the choice of SDC matrices and their effectiveness in representing the system. This thesis illustrates how to represent a system in state space with terms dependent on the state itself. The most common methods for constructing SDC matrices have been collected, improving some aspects and proposing alternatives. The SDRE control with SDC structure is used and studied in efficiency, efficacy and ductility with methods known in the literature and new modalities. An innovative solution is proposed for this control in the choice of the Q matrix, which can also be extended to the well-known LQR. In conclusion, all the techniques on different case studies commonly used in literature and some real application cases are illustrated.File | Dimensione | Formato | |
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