Optimal Control: LQR and Performance Index Quiz Quiz

Explore the principles of Linear Quadratic Regulator (LQR) design and performance index formulation in optimal control. Assess your understanding of cost functions, weighting matrices, and solution properties within this foundational area of control engineering.

1. Purpose of the Performance Index

   In the context of the Linear Quadratic Regulator (LQR), which statement best describes the role of the performance index J for a given linear system?

   1. The performance index J represents the set of all possible state trajectories.
   2. The performance index J quantifies how well the control law minimizes a quadratic cost over time.
   3. The performance index J is used to determine the time delay in system response.
   4. The performance index J acts as a stability margin indicator.

   Explanation: The performance index J in LQR is specifically designed to assign a cost to state and input trajectories, allowing the optimization of control laws that minimize this value over time. It does not represent the set of all possible state trajectories, which are determined by system dynamics. J is not a direct measure of stability margin, nor is it related to calculating system time delay. The distractors are related to system analysis but not to the purpose of the performance index.

2. Weighting Matrices Selection

   In the standard LQR problem, what is the main effect of increasing the values in the input weighting matrix R relative to the state weighting matrix Q?

   1. It leads to less aggressive control actions with smaller input magnitudes.
   2. It ensures the closed-loop system always becomes unstable.
   3. It only affects the system's time delay, not the control effort.
   4. It causes the system to become uncontrollable.

   Explanation: Increasing the matrix R in the performance index penalizes control effort more heavily, resulting in more conservative control actions. The system does not become uncontrollable simply due to R being larger, and R does not inherently cause instability. Additionally, while R may influence system transients, its main impact is on the aggressiveness and size of control inputs rather than time delay alone.

3. LQR Riccati Equation

   Which equation must be solved to obtain the optimal state feedback gain K in the finite-horizon LQR problem for a continuous-time system?

   1. The Lyapunov stability equation
   2. The state transition matrix equation
   3. The continuous-time algebraic Riccati equation (CARE)
   4. The Hill equation

   Explanation: The continuous-time algebraic Riccati equation (CARE) is required to find the optimal LQR gain by providing the matrix P used in the feedback law. The Lyapunov stability equation assesses stability, not optimality; the state transition matrix determines state evolution but not optimal control gains; and the Hill equation is unrelated to LQR or optimal control. Only CARE directly relates to solving for K in LQR.

4. Positive Definiteness Requirement

   Why must the input weighting matrix R in the LQR performance index be positive definite?

   1. To make the closed-loop system uncontrollable.
   2. To guarantee the plant matrix A is invertible.
   3. To ensure the cost function has a unique minimum and the optimization problem is well-posed.
   4. To minimize only the state variables without considering the inputs.

   Explanation: A positive definite R ensures that the cost of large control inputs grows quickly, providing uniqueness and well-posedness in the minimization problem. Making R positive definite does not make the system uncontrollable, nor does it relate to the invertibility of A. Ignoring inputs by making R zero would undermine one of the LQR's core functions.

5. Feature of the Optimal LQR Controller

   Given a discrete-time linear system regulated with LQR, what is a primary feature of the optimal control law obtained?

   1. The optimal control law ignores the state and only depends on the input.
   2. The optimal control law is a linear state feedback of the form u = -Kx.
   3. The optimal control law is always nonlinear and time-varying.
   4. The optimal control law is always independent of Q and R matrices.

   Explanation: The hallmark of LQR design is the optimal feedback law being linear in the state, represented by u = -Kx. While time-varying solutions exist for certain conditions, the standard LQR formulation produces a linear, often constant gain. Ignoring the state makes the control ineffective, and the optimal control law heavily depends on Q and R, which define the trade-off in the cost function.