Overview

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### Summary of Systems and Controllers Mentioned in the Document

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#### 1. Linear Time-Invariant (LTI) Systems

- Description: Systems where the dynamics and outputs are governed by linear equations that remain constant over time.

- State-Space Representation:

$\dot{x}(t) = Ax(t) + Bu(t), \quad y(t) = Cx(t) + Du(t),$

where:

- $x(t)$: State vector,

- $u(t)$: Input vector,

- $y(t)$: Output vector.

- Key Use Case: Foundation for designing controllers and observers.

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#### 2. Mass-Spring-Damper System

- Example of an LTI System: Describes the motion of a mass subjected to forces, damping, and a spring.

- Dynamics:

$m\ddot{x}(t) + c\dot{x}(t) + kx(t) = w(t),$

where $w(t)$ is a disturbance force.

- State-Space Model:

$\dot{x}(t) = \begin{bmatrix} 0 & 1 \\ -k/m & -c/m \end{bmatrix}x(t) + \begin{bmatrix} 0 \\ 1/m \end{bmatrix}w(t),$

$y(t) = \begin{bmatrix} 1 & 0 \end{bmatrix}x(t) + v(t).$

- Key Use Case: Demonstrates the effects of process and measurement noise on state estimation.

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#### 3. Luenberger Observers

- Purpose: Estimates the state of an LTI system based on measurements.

- Structure:

$\dot{\hat{x}}(t) = A\hat{x}(t) + Bu(t) + L\left(y(t) - \hat{y}(t)\right),$

$\hat{y}(t) = C\hat{x}(t) + Du(t).$

- Gain Matrix ($L$): Chosen to ensure the error dynamics:

$\dot{e}(t) = (A - LC)e(t),$

are asymptotically stable.

- Limitations:

1. Does not address noise explicitly.

2. Optimal placement of eigenvalues for $A - LC$ is unclear.

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#### 4. Kalman Filter

- Purpose: An optimal observer that minimizes estimation error covariance in the presence of process and measurement noise.

- Noise-Driven State-Space Model:

$\dot{x}(t) = Ax(t) + Bu(t) + w(t), \quad y(t) = Cx(t) + Du(t) + v(t),$

where:

- $w(t)$: Process noise (zero-mean, Gaussian, white),

- $v(t)$: Measurement noise (zero-mean, Gaussian, white).

- Optimal Gain:

$L(t) = P(t)C^\top R^{-1},$

where $P(t)$ is the error covariance matrix, governed by the Riccati equation.

- Applications:

- Provides accurate state estimates even when the initial state is unknown.

- Handles stochastic disturbances.

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#### 5. Linear Quadratic Regulator (LQR)

- Purpose: A state-feedback controller that minimizes a quadratic cost function of the states and inputs.

- Control Law:

$u(t) = -Kx(t),$

where $K$ is the feedback gain matrix derived from solving the Riccati equation.

- Key Connection: The LQR and Kalman filter are duals, as both solve Riccati equations to optimize system performance under constraints (control vs. estimation).

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#### 6. Infinite Horizon Kalman Filter

- Purpose: The steady-state version of the Kalman filter.

- Key Requirements:

- Observability of the pair $(A, C)$,

- Controllability of $(A, \sqrt{Q})$, where $Q$ is the process noise covariance.

- Features:

- Constant Kalman gain:

$L = PC^\top R^{-1},$

where $P$ satisfies the algebraic Riccati equation:

$0 = AP + PA^\top + Q - PC^\top R^{-1}CP.$

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#### 7. Infinite Horizon Linear Quadratic Regulator (LQR)

- Purpose: A steady-state state-feedback controller designed to optimize system performance over an infinite time horizon.

- Objective: Minimize the quadratic cost function:

$J = \int_0^\infty \left(x^\top Q x + u^\top R u\right) dt,$

where $Q$ and $R$ are positive semi-definite and positive definite matrices, respectively.

- Control Law:

$u(t) = -Kx(t),$

where $K = R^{-1}B^\top P$, and $P$ solves the algebraic Riccati equation:

$A^\top P + PA - PBR^{-1}B^\top P + Q = 0.$

- Features:

- Feedback gain $K$ is constant.

- Assumes steady-state behavior of the system.

- Applications: Used in control systems requiring long-term optimization, such as robotics and aerospace applications.

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### Key Concepts and Relationships

1. State Observers (Luenberger Observer vs. Kalman Filter):

- Luenberger observers rely on deterministic dynamics, while the Kalman filter explicitly models noise to achieve optimal estimation.

2. Duality (Kalman Filter vs. LQR):

- The Kalman filter minimizes estimation errors, while the LQR minimizes control costs.

- Both are rooted in solving Riccati equations, emphasizing their mathematical duality.

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This updated list now includes the Infinite Horizon Linear Quadratic Regulator (LQR) and its connections to the other systems and controllers. Let me know if you'd like further clarifications or refinements!