Vehicle suspension control 2: Modal Analysis+ Pole Placement

Learn that eigenvectors must be nonzero to preserve direction, while eigenvalues are scalars that can be zero and map vectors to zero, scale them in diagonal or upper triangular cases.

Examine how span and basis affect linear systems, distinguishing no, unique, and infinite solutions, and how determinant and invertibility of A determine reachability of b.

Identify the second eigenvector by solving A tilde v2 = 0, yielding eigen-directions for lambda=5 and lambda=3. Eigenspaces are the null spaces along these lines, with nonzero magnitudes preserving direction.

Learn how diagonalization decouples linear odes by converting A to a diagonal matrix of eigenvalues. Complex eigenvalues yield complex eigenvectors, increasing computation for large systems; noninvertible P makes diagonalization fail.

Show that a noninvertible P makes the full eigenvector set linearly dependent, while the smaller subset stays independent, so the equation V c = 0 has only the trivial solution.

Project vector b onto u1 and u2 to extract components parallel to each. Express projections in magnitude and cartesian form, normalize u1, and solve for coefficients via a linear system.

Discover how orthogonal bases simplify vector projections, with coefficients k1 and k2 becoming one when u1 and u2 are orthogonal, and extend to orthogonal vector sets.

Learn the Gram-Schmidt process to build an orthogonal, then orthonormal basis from non-orthogonal vectors, using projections to remove components and enable simple matrix diagonalization.

Apply Gram-Schmidt to form orthogonal eigenvectors, normalize to create an orthonormal P, and diagonalize a symmetric matrix A as A = P D P^T, illustrating eigenspaces, multiplicities, and orthogonality.

Apply modal analysis to a double mass-spring system by transforming to mass-normalized coordinates, diagonalizing the stiffness matrix with an orthonormal eigenvector basis, and deriving decoupled modal equations.

Decouple the truck with modal analysis by making the damping matrix a linear combination of mass and stiffness, mass-normalize to modal coordinates, and solve impulse responses in that frame.

Derive the truck’s state-space model, define the states, and form the A and B matrices, distinguishing real and equivalent inputs and a disturbance term for pole placement.

Apply a pole placement controller to the truck, using place to obtain a real, negative closed-loop pole set and a gain matrix k, with Matlab or Python code.

Explore step-by-step code for a vehicle suspension control system, detailing open-loop and closed-loop matrices, pole placement gains, equilibrium inputs, state increments, and plant input conversion.

Explore how the final value theorem yields steady-state values from Laplace and time-domain limits. Note its applicability to stable LTI systems only, with examples of unstable or marginal cases.

Apply the final value theorem to derive a universal CR matrix that, with a fixed delta r vector, drives delta x and delta theta to specified reference values.

Review modal analysis and pole placement for linear time invariant systems, contrasting analytical and numerical odes. Explain how controllable and disturbance inputs shape the system response and equilibrium values.

Explore solving an LTI state-space system with the state transition matrix, using Laplace transforms to obtain x(t) from x0, with inputs or disturbances, and verify via modal analysis.

Explore solving an LTI state-space system with the state transition matrix, separating the response from initial conditions and external inputs. Learn how outputs are formed with c and d matrices.

Solve an LTI state-space system via modal analysis, transforming initial-condition and input solutions to modal coordinates with p matrix; symmetry in K, C, M yields real eigenvalues and simpler analysis.

Apply the Cayley-Hamilton theorem to rewrite A^k in terms of lower powers and the identity for any n by n matrix, yielding a finite n-term state-transition series with time-dependent alphas.

Explore linear algebra, modal analysis, pole placement, and differential equations for MIMO systems, and learn how LQR automatically selects pole locations by prioritizing error minimization and input usage.