Vehicle suspension control 3: PID + LQR + Resonance analysis

Learn how a PID controller uses proportional, integral, and derivative terms to track references, damp overshoot, and eliminate steady-state error, with tuning via pole placement.

Explore pid control in a rocket demonstration, and examine how lqr and resonance analysis enhance vehicle suspension performance.

Implement a pid controller for a vehicle using pole placement in the Laplace domain with a model; tune kp, kd, and k from the error between desired and actual displacement.

Compute the closed-loop transfer function for a PID-controlled plant by substituting the plant and controller expressions into the standard form and deriving it in terms of kp, kd, and Ki.

Derive pid constants in terms of p1, p2, p3 by equating polynomials, linking to closed-loop poles -p1, -p2, -p3 and to desired poles -15, -16, -17.

Explore how to implement a Python pid controller for a suspension system, define physical parameters, derive closed-loop poles, simulate with Euler integration, and plot amplitude ratios across frequency ratios.

Tuning pid for a mimo system extends siso methods to a two-input, two-output model with x and theta, using fu and mu and addressing coupling and pid constants.

Derive the closed loop transfer function matrix for a two-input, two-output system in the Laplace domain using the open loop matrix and a two by two controller.

Tune six pid constants for a fourth-order system. Equate open-loop and closed-loop polynomials and solve a nonlinear six-equation system using perplexity and matlab to place closed-loop poles near desired locations.

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Introduce the quarter car model with two masses—the sprung and unsprung masses—modeling tire dynamics and road interaction using springs and dampers, and derive a linear increment-based model from equilibrium.

The lecture derives a state-space model for the quarter car, detailing the upper and lower mass dynamics under spring, damper, and actuator forces, enforcing Newton's third law.

Put the quarter-car model into state-space form, derive the a, b, c matrices, and explain how four open-loop poles with three pid constants produce five closed-loop poles.

Explore dominant pole approximation by replacing a second-order transfer function with a matched first-order model, preserving dc gain and comparing unit step responses and the final value via Laplace.

Apply dominant pole approximation to a third-order system while preserving the dc gain, and observe how near-origin poles dominate the unit step response.

Dominant pole approximation reduces a third‑order vehicle suspension model to first‑ and second‑order forms that match the DC gain. It highlights step responses and function expansion using Python or Matlab for comparison.

Apply the dominant pole approximation to a four-pole quarter car model, reducing it to a second-order system for PID tuning via open-loop pole placement.

Apply dominant pole approximation to a quarter car transfer function, match the dc gain, and compare step responses for dominant versus insignificant poles.

Tune PID constants for a two-dof suspension by approximating the fourth-order model with a second-order reduced model using imaginary parameters, then apply the derived formulas to the real system.

Apply the linear quadratic regulator (LQR) to compute a gain matrix K that shapes the closed-loop poles from the open-loop A matrix, balancing state reduction and control versus pole placement.

Configure the LQR cost function with weight matrices to minimize states and inputs over time. Ensure the weight matrix is symmetric and positive definite for a single global minimum.

Set the lqr weights to prioritize states or control effort, so q drives aggressive state reduction while r conserves control energy, and ensure positive definite weight matrices.

Derive the LQR gain matrix K by reformulating the cost with a constant M and an unknown P, ensuring closed-loop stability with eigenvalues on the S plane.

Derive the optimal gain k from the Riccati framework by linking cost function terms to r, q, and p, and show symmetry of r and p yields identical k expressions.

Explain a quarter-car model implemented with PID or LQR, including custom C/D for x2 dot, symmetric positive-definite weights, Riccati analysis, and Euler-time simulations.

Show how the controller reduces sprung-mass oscillation while increasing wheel motion; tune LQR with identity Q and large R, and enable PID to avoid instability from insignificant poles.

Review linearization to obtain lti models. Apply modal analysis to decouple multi-degree-of-freedom systems and tune pid and lqr for quarter- and half-car models, including resonance and mimo considerations.

Model the half-car suspension with Newton's laws to derive bounce, pitch, and wheel dynamics, accounting for cosine theta effects, and simulate with Euler's method using absolute variables and spring-damper forces.

Linearize a nonlinear half-car model about chosen x1 and x3 equilibria using two actuators, deriving A, B matrices and x* via inverse relations to obtain a resting linear model.

Linearize the half-car model around an equilibrium, derive the A and B matrices from Jacobians for delta state and delta input, and design an LQR controller using those matrices.

Explore why dominant pole approximation struggles for a half-car model, highlighting the impact of zeros on transfer functions, and why LQR outperforms PID for large MIMO systems.

Compute undamped natural frequencies for the four-degree-of-freedom half-car using modal analysis. Recognize multiple omega_n values and resonance risks as forcing frequencies approach these values.

Explore how undamped natural frequencies are determined for multi-degree-of-freedom systems, comparing s-plane pole distances with modal analysis and noting when proportional damping enables direct calculations.

Explore the half-car model code with resonance analysis and LQR, comparing resonance mode on and off and how linearization about the desired x and theta suppresses oscillations.

Explore the half-car model code for resonance analysis with base excitation, matching forcing frequencies to natural frequencies, and linearizing to build A, B, C, D for LQR design.

Explore the full car model with seven degrees of freedom, 14 states, and four input actuators, and compare pid vs lqr control for robust suspension design.

Install python on Windows 11 with path, test with hello world, then install numpy, matplotlib, scipy solvers, cvxopt, sympy, and control for MPC, LQR, and suspension system analysis.