Notes in Week 6 - Observers

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Published	11/26/2024	A key requirement for MPC is the system model, defined by state dynamics equations like {{c1::\(x(t+1) = Ax(t) + Bu(t)\)}}.
Published	11/26/2024	MPC can be formulated as a {{c1::Quadratic Programming (QP)}} problem, given that most cost functions are quadratic.
Published	11/26/2024	Observers are used in control when some states {{c1::cannot be directly measured}}.
Published	11/26/2024	The estimation error is defined as {{c2::\( e(t) = x(t) - \hat{x}(t) \)}}, which ideally should approach {{c1::zero}} over time.
Published	11/26/2024	A simple observer design might set the error dynamics as {{c3::\( e(t+1) = A e(t) \)}} if {{c2::\( A \)}} is stable, meaning all eigenvalues…
Published	11/26/2024	Observer stability depends on the eigenvalues of the matrix \( A - {{c1::LC}} \), where \( L \) is the observer gain.
Published	11/26/2024	The observer gain \( L \) is selected to ensure that the {{c1::estimation error}} converges to zero.
Published	11/26/2024	In Observer Design, the error dynamics are modified by introducing \( L \) as a {{c1::tunable parameter}}.
Published	11/26/2024	A {{c1::reduced-order observer}} can be used when some states are measurable, allowing the observer to only estimate the unmeasurable states.
Published	11/26/2024	The {{c1::reduced-order observer}} is efficient because it only focuses on {{c2::unmeasured states}}, avoiding redundant calculations.
Published	11/26/2024	A system is considered {{c1::observable}} if a finite set of measurements can uniquely determine the {{c2::initial state}}.
Published	11/26/2024	Observability is verified if the {{c1::observability matrix}} has full rank.
Published	11/26/2024	In control theory, the concept of {{c1::controllability}} is analogous to observability, defined by the ability to reach any state from an initial con…
Published	11/26/2024	The {{c1::duality principle}} states that the pair \( (A, B) \) is controllable if and only if the pair \( (A^{\top}, B^{\top}) \) is {{c2::observable…
Published	11/26/2024	{{c1::Pole placement}} is a technique in which controllability and observability are used to assign eigenvalues to specific, desired locations.
Published	11/26/2024	Observer Design allows tuning the {{c1::convergence speed}} of the estimation error by adjusting the matrix \( L \), affecting the error dynamics' eig…
Published	11/26/2024	In a full-state observer, the estimated output \( \hat{y}(t) \) is represented as {{c1::\( C\hat{x}(t) \).}}
Published	11/26/2024	In Observer Design, adding \( L(y(t) - \hat{y}(t)) \) enables the observer to adjust its estimate based on the {{c1::error between measured and estima…
Published	11/26/2024	A {{c1::disturbance observer}} is designed to estimate the effect of unmeasurable disturbances on the system.
Published	11/26/2024	In systems with disturbances, the disturbance can be modeled as an {{c1::additional state with constant dynamics}}, such as \( x_d(t+1) = x_d(t) \).
Published	11/26/2024	In designing an observer for a system with disturbances, the system is augmented to include both the original and {{c1::disturbance states}}.
Published	11/26/2024	In MPC, the observer's estimates are used to adjust inputs to ensure the system remains within {{c1::constraints}}.
Published	11/26/2024	If some states are measurable, the output matrix \( C \) can be structured as \( \begin{bmatrix} I & 0 \end{bmatrix} \) to represent these {{c1::k…
Published	11/26/2024	A reduced-order observer leverages measurements to estimate only the {{c1::unmeasurable states}} in the system.
Published	11/26/2024	The {{c1::reduced-order observer}} is computationally efficient since it does not re-estimate already measurable states.
Published	11/26/2024	If an initial guess is required for the observer, the state estimation can start from an {{c1::initial approximation}}, such as \( \hat{x}(0) \).
Published	11/26/2024	{{c1::Pole placement}} is used to select the observer gain \( L \), ensuring the desired behavior in error dynamics.
Published	11/26/2024	Reduced-order observers are advantageous when some states are too {{c1::expensive}} or difficult to measure directly.
Published	11/26/2024	The estimation dynamics in an observer are influenced by the {{c1::system matrix}} \( A \) and the {{c2::gain matrix}} \( L \).
Published	11/26/2024	To estimate disturbances, the observer treats disturbances as part of an {{c1::augmented state}} in the system.
Published	11/26/2024	In MPC, the observer's predicted state is used to adjust future inputs, allowing the system to account for {{c1::future states}} rather than solely re…
Published	11/26/2024	Observers allow MPC to operate effectively even when some system variables are {{c1::unobservable}}.
Published	11/26/2024	The performance of an observer is judged by its ability to minimize the {{c1::estimation error}} over time.
Published	11/26/2024	The system's observability can be confirmed by verifying that the {{c1::observability matrix}} has full rank.
Published	11/26/2024	For stable estimation error dynamics, the matrix \( A - LC \) must have eigenvalues within the {{c1::unit circle}}.
Published	11/26/2024	Observer Design incorporates a correction term based on the {{c1::output error}} between the measured and estimated outputs.
Published	11/26/2024	Disturbance estimation through an observer allows the system to account for external influences, such as {{c1::wind or load torque}}.
Published	11/26/2024	The {{c1::reduced-order observer}} estimates only the unmeasurable states in a subsystem, enhancing computational efficiency.
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