lqr-control

name	lqr-control
description	Linear Quadratic Regulator (LQR) control design — state space formulation, Riccati equation, Q and R weighting matrices, closed-loop pole placement, robustness margins, integral augmentation, LQI (LQR with integral), observer design (LQE/Kalman), output feedback, MATLAB implementation, mechanical system applications.
metadata	{"priority":7,"promptSignals":{"phrases":["LQR control","linear quadratic regulator","optimal control LQR","Riccati equation","state feedback LQR","LQR Q R matrix"],"minScore":3}}

Linear Quadratic Regulator (LQR) — Complete Skill

Problem Formulation

System Description

Continuous-time state space: ẋ = A x + B u [state equation; A ∈ ℝⁿˣⁿ; B ∈ ℝⁿˣᵐ; x = state; u = control input] y = C x + D u [output equation; y = measured output]

Control objective: regulate state x to zero (or to reference) while minimizing cost

LQR Cost Function

Infinite-horizon quadratic cost: J = ∫₀^∞ (xᵀ Q x + uᵀ R u) dt [J = total cost; Q ≥ 0 (positive semi-definite); R > 0 (positive definite)]

Q matrix: penalizes state deviation (large Q_ii → state x_i tightly regulated) R matrix: penalizes control effort (large R → less aggressive control; saves actuator energy/wear)

Interpretation: Large Q/R ratio → aggressive control, tight tracking, potentially large control effort Small Q/R ratio → sluggish control, large steady-state error, minimal control effort Design freedom: choose Q and R to achieve desired closed-loop performance

Optimal Solution — Algebraic Riccati Equation (ARE)

Continuous-Time ARE

Solution procedure:

1. Solve the Algebraic Riccati Equation (ARE) for P > 0: Aᵀ P + P A - P B R⁻¹ Bᵀ P + Q = 0 [P ∈ ℝⁿˣⁿ, symmetric positive definite]

2. Optimal gain matrix: K = R⁻¹ Bᵀ P [K ∈ ℝᵐˣⁿ]

3. Optimal control law: u*(t) = -K x(t) [state feedback; requires full state measurement or observer]

Closed-loop system: ẋ = (A - B K) x [stable if (A, B) controllable and Q, R chosen properly] Eigenvalues of (A - BK): closed-loop poles; placed optimally by LQR

Discrete-Time LQR

Discrete-time ARE: P = Q + Aᵀ P A - Aᵀ P B (R + Bᵀ P B)⁻¹ Bᵀ P A [solve iteratively or with MATLAB] K = (R + Bᵀ P B)⁻¹ Bᵀ P A

Control law: u_k = -K x_k [applied at each time step k]

Q and R Matrix Design

Physical Design Approach (Bryson's Rule)

Bryson's Method (practical starting point): Q = diag(1/x_max_i²) [Q_ii = 1/(max acceptable deviation of state i)²] R = diag(1/u_max_j²) [R_jj = 1/(max acceptable control effort j)²]

Example — active suspension: States: x = [z_s, ż_s, z_u, ż_u]ᵀ (sprung/unsprung mass displacement and velocity) Q = diag([1/(0.05)², 1/(1)², 1/(0.02)², 1/(2)²]) [max 5 cm body displacement, 1 m/s body velocity, etc.] R = diag([1/(5000)²]) [max 5 kN actuator force]

Iterative refinement: Start with Bryson → simulate → observe performance → increase Q_ii to tighten regulation of x_i → decrease R_jj to allow more aggressive control

Performance vs. Robustness Trade-off

Aggressive gains (large Q/R): Faster response, smaller tracking error Larger bandwidth → amplifies sensor noise Reduced robustness margins → susceptible to model uncertainty

Conservative gains (small Q/R): Slower response, possible steady-state error Lower bandwidth → better noise rejection Higher robustness margins

LQR Properties

Guaranteed Stability Margins

Classical LQR gain and phase margins (single input, full state feedback): Gain margin: [0.5, ∞) — infinite upward gain margin; can increase gain without instability Phase margin: ≥ 60° — robust to phase lag in loop

Return Difference Inequality: |(I + K(sI - A)⁻¹ B)| ≥ 1 → singular values of loop gain ≥ 1 → guaranteed return difference This implies: gain margin [0.5, ∞); phase margin ≥ 60°

Practical caveat: these margins apply to SISO LQR with full state feedback; output feedback (with observer) has reduced guaranteed margins (Doyle's counterexample — observer-based LQR can have zero gain margin)

Controllability Requirement

LQR solvable if: (A, B) is stabilizable (all unstable modes are controllable) LQR optimal if: (A, B) is controllable AND (A, Q^(1/2)) is observable Verify: rank([B AB A²B...Aⁿ⁻¹B]) = n (Kalman controllability matrix)

LQR with Integral Action (LQI — Tracking)

Integral Augmentation

Problem: standard LQR regulates to zero; for setpoint tracking with disturbance rejection, add integrator states

Augmented state: x_aug = [x; x_I] where ẋ_I = r - y = r - C x [x_I = integral of tracking error; r = reference]

Augmented system: ẋ_aug = A_aug × x_aug + B_aug × u + B_r × r A_aug = [A 0; -C 0]; B_aug = [B; 0]; B_r = [0; I]

LQI gain: K_aug = [K_x | K_I] from LQR on augmented system u = -K_x × x - K_I × x_I [state feedback + integral]

Effect: integral action eliminates steady-state error to step references and constant disturbances

Observer Design (LQG)

Linear Quadratic Estimator (Kalman Filter Observer)

When full state not measurable: combine LQR with Kalman filter (LQE) → LQG (Linear Quadratic Gaussian) control

Observer (Luenberger / Kalman): x̂̇ = A x̂ + B u + L(y - C x̂) [L = observer gain] u = -K x̂ [control law uses estimated state]

Kalman filter gain for LQE: Minimize: E[xᵀ Q_e x] (state estimation error) subject to process noise W and measurement noise V L = P_e Cᵀ V⁻¹ [P_e from dual Riccati equation: A P_e + P_e Aᵀ - P_e Cᵀ V⁻¹ C P_e + W = 0] W = process noise covariance; V = measurement noise covariance (analogous to Q, R for controller)

Separation principle: LQR controller and LQE observer can be designed independently Combined LQG: closed-loop poles = LQR poles ∪ LQE poles (exact separation in linear case)

LQG robustness: generally LESS robust than pure LQR (as noted by Doyle 1978) LQG/LTR (Loop Transfer Recovery): design procedure to recover LQR robustness margins in LQG

Mechanical Engineering Applications

Active Suspension (Quarter-Car Model)

States: x = [z_s, ż_s, z_u, ż_u]ᵀ (body and wheel displacements and velocities) Inputs: u = F_actuator (active force) Road disturbance: w = ż_road (disturbance input)

Equations of motion: m_s ẑ̈_s = -k_s(z_s - z_u) - c_s(ż_s - ż_u) + F_act m_u ẑ̈_u = k_s(z_s - z_u) + c_s(ż_s - ż_u) - k_t(z_u - z_road) - F_act

LQR objectives: Minimize body acceleration (ride comfort): high Q for ż_s Maintain tire contact: high Q for (z_u - z_road) Limit actuator stroke: high Q for (z_s - z_u) Limit force: high R

Typical LQR performance vs. passive: LQR: 30–50% reduction in RMS body acceleration at 5 Hz road input

Magnetic Bearing Control

Unstable plant (single DOF magnetic bearing): A = [0 1; k_x/m 0] [k_x = position stiffness (positive = destabilizing); m = rotor mass] System has unstable eigenvalue at √(k_x/m)

LQR requirement: (A, B) controllable (B = [0; 1/m]) → rank condition satisfied LQR stabilizes the open-loop unstable mode naturally

Gains: large Q₁₁ (penalize position error) → stiff levitation; large Q₂₂ → damp velocity; R → limit current

Flexible Structure Control (Vibration Suppression)

Modal model: ẍ_i + 2ζ_i ω_i ẋ_i + ω_i² x_i = φ_i^T B u [decoupled modal equations; i = 1 to N modes] State: [x_i, ẋ_i] for each mode; truncated to first M modes (residual modes = spillover)

LQR for vibration suppression: Penalize modal displacements and velocities: Q = block-diag of Q_mode_i Result: active damping added to all controllable modes

Spillover warning: control of truncated model may excite unmodeled high-frequency modes → instability Prevention: penalize control bandwidth (high R) + notch filters for unmodeled modes

MATLAB Implementation

% System matrices
A = [0 1; -omega_n^2 -2*zeta*omega_n];  % 2nd order oscillator
B = [0; 1/m];
C = [1 0];  % position output

% LQR weighting
Q = diag([1/(x_max)^2, 1/(v_max)^2]);  % Bryson's rule
R = 1/(u_max)^2;

% Solve Riccati equation and find gain
[K, P, e] = lqr(A, B, Q, R);
% e = closed-loop eigenvalues (poles of A - B*K)

% Verify stability
eig(A - B*K)  % all must have negative real part

% Observer design (Kalman)
W = 1e-4 * eye(2);   % process noise covariance
V = 1e-2;            % measurement noise variance
[L, ~, ~] = lqe(A, eye(2), C, W, V);  % L = observer gain

% Simulate LQG (LQR + Kalman filter)
sys_cl = ss(A - B*K - L*C, L, C, 0);  % closed-loop TF with observer

% Closed-loop response
t = 0:0.001:5;  % time vector
[y, t, x] = lsim(sys_cl, u_input, t, x0);

Output

Provide: system type (suspension/bearing/robot/vibration), state vector x (physical meaning; n states), input vector u (m inputs; actuator type), A and B matrices (from physics equations), controllability check (rank = n), Q matrix (Bryson's rule; diagonal entries with justification [units²]), R matrix (diagonal entries [units²]), P solution from ARE [Riccati], optimal gain K = R⁻¹Bᵀ P [gain vector/matrix], closed-loop poles (eigenvalues of A-BK; check negative real parts → stable), LQI augmentation (integral state for tracking; yes/no), observer gain L (Kalman; process noise W, measurement noise V), LQG combined performance (guaranteed gain margin > 0.5, phase margin > 60° for state feedback), MATLAB code snippet, and applicable reference (CCPS LQR, Bryson-Ho "Applied Optimal Control", MATLAB Control Toolbox documentation).