Pole Placement Calculator

Ackermann's Formula for State-Feedback Pole Placement

K = [0 0 … 0 1] · Ctrb⁻¹ · φ_d(A), with φ_d(A) = Aⁿ + a₁·Aⁿ⁻¹ + … + aₙ₋₁·A + aₙ·I

Variables: K = state-feedback gain row vector, 1×n, applied as u = −K·x (entry kᵢ has units of plant input per unit of state xᵢ — e.g. N per m and N per m/s for a force-driven mass); A = state matrix, n×n (entries in s⁻¹ when the states share a common unit); B = input column, n×1; Ctrb = [B A·B … Aⁿ⁻¹·B] = controllability matrix, n×n; [0 … 0 1] = the n-th unit row vector (dimensionless); φ_d(A) = the DESIRED characteristic polynomial evaluated at the MATRIX A (an n×n matrix, not a scalar); a₁ … aₙ = coefficients of φ_d(s) = sⁿ + a₁sⁿ⁻¹ + … + aₙ; I = n×n identity; n = system order, 1 to 4 in this tool (dimensionless count). This calculator solves Ctrbᵀ·y = eₙ and then forms K = yᵀ·φ_d(A) rather than inverting Ctrb explicitly, which is better conditioned. Source: Ogata, Modern Control Engineering, 5th ed., §10-2 (Ackermann's formula, eq. 10-16); Franklin, Powell & Emami-Naeini, Feedback Control of Dynamic Systems, 7th ed., §7.5.

Controllability Matrix and the Rank Test (the gate on K)

Ctrb = [ B A·B A²·B … Aⁿ⁻¹·B ] controllable ⇔ rank(Ctrb) = n ⇔ det(Ctrb) ≠ 0

Variables: Ctrb = controllability (reachability) matrix, n×n for a single input, column j = Aʲ⁻¹·B; rank(Ctrb) = number of independent columns (dimensionless integer, 0 to n); det(Ctrb) = determinant (dimensionless in the state coordinates supplied); n = system order (dimensionless). Ackermann's formula requires Ctrb⁻¹, which exists only at full rank. When rank(Ctrb) < n the pair (A, B) has n − rank(Ctrb) modes that NO state feedback can move, and this calculator reports the rank and refuses to print a gain rather than returning a K that would appear to place unmovable poles. Numerical rank here is taken from Gaussian elimination with partial pivoting and a pivot tolerance of 1e-9 × ‖Ctrb‖_max × n, so a pair that is only marginally controllable is flagged as ill-conditioned rather than silently trusted. Source: Ogata, Modern Control Engineering, 5th ed., §9-6 (complete state controllability) and §10-2 (necessary and sufficient condition for arbitrary pole placement); Chen, Linear System Theory and Design, 4th ed., Ch. 6.

Desired Closed-Loop Characteristic Polynomial

φ_d(s) = Π_{i=1..n} (s − pᵢ) = sⁿ + a₁·sⁿ⁻¹ + … + aₙ₋₁·s + aₙ

Variables: φ_d(s) = desired closed-loop characteristic polynomial (monic); pᵢ = desired closed-loop pole i, in rad/s (s-plane, continuous time); s = Laplace variable (rad/s); a₁ … aₙ = real polynomial coefficients (aₖ has units of (rad/s)ᵏ). The coefficients aₖ are real ONLY if the pole set is closed under complex conjugation — every complex pole σ + jω_d must be entered together with σ − jω_d — otherwise no real gain K exists and this calculator rejects the request. A pole at Re(pᵢ) < 0 is stable; Re(pᵢ) > 0 is unstable; Re(pᵢ) = 0 is marginally stable. Source: Ogata, Modern Control Engineering, 5th ed., §10-2; Nise, Control Systems Engineering, 6th ed., §12.2.

Closed-Loop System and the Verification Identity

ẋ = (A − B·K)·x with u = −K·x det(sI − (A − B·K)) = φ_d(s) and eig(A − B·K) = { p₁, …, pₙ }

Variables: x = state vector, n×1 (units are whatever the user's states carry); ẋ = state derivative (state units per second); u = scalar control input, u = −K·x; A − B·K = closed-loop state matrix, n×n; det(sI − (A − B·K)) = achieved closed-loop characteristic polynomial (s in rad/s); eig(·) = eigenvalues, i.e. the achieved closed-loop poles (rad/s). This tool computes the achieved polynomial independently by Faddeev–LeVerrier on A − B·K, finds its roots, and reports the largest distance in rad/s between each requested pole and its nearest achieved eigenvalue. A well-conditioned placement returns an error of order 1e-12 rad/s or below; a visibly non-zero error means (A, B) is nearly uncontrollable and K should not be trusted. Source: Ogata, Modern Control Engineering, 5th ed., §10-2; Franklin, Powell & Emami-Naeini, 7th ed., §7.5 (verifying a placed design).

Controllable Canonical Form Shortcut (hand check for Ackermann)

If A is the companion matrix of sⁿ + a₁sⁿ⁻¹ + … + aₙ with B = [0 … 0 1]ᵀ, then K = [ aₙ′ − aₙ , aₙ₋₁′ − aₙ₋₁ , … , a₁′ − a₁ ]

Variables: aₖ = coefficient of sⁿ⁻ᵏ in the OPEN-loop characteristic polynomial det(sI − A) (units (rad/s)ᵏ); aₖ′ = the same coefficient of the DESIRED polynomial φ_d(s); K = gain row, ordered to match the state ordering x₁ = output, x₂ = ẋ₁, … of the phase-variable (controllable canonical) form. This is the fastest hand check on any Ackermann result: for the double integrator A = [[0,1],[0,0]], B = [0,1]ᵀ with poles at −2 ± 2j, φ_d = s² + 4s + 8 against an open loop s², so K = [8 − 0, 4 − 0] = [8, 4]. The shortcut applies ONLY in controllable canonical form; for any other state coordinates use the general Ackermann expression above, which this calculator evaluates directly. Source: Ogata, Modern Control Engineering, 5th ed., §10-2, eq. 10-13; Nise, Control Systems Engineering, 6th ed., §12.2 (phase-variable pole placement).

Choosing the Desired Poles from Transient Specifications

p = −ζ·ω_n ± j·ω_n·√(1 − ζ²) ζ = −ln(OS) / √(π² + ln²(OS)) T_s(2 %) ≈ 4 / (ζ·ω_n) T_p = π / (ω_n·√(1 − ζ²))

Variables: p = dominant closed-loop pole pair (rad/s); ζ = damping ratio (dimensionless, 0 to 1); ω_n = undamped natural frequency (rad/s); ω_d = ω_n·√(1 − ζ²) = damped natural frequency (rad/s); OS = fractional peak overshoot (dimensionless, e.g. 0.095 for 9.5 %); T_s = 2 % settling time (s); T_p = peak time (s). Pick ζ and ω_n from the specification, place the dominant pair at −ζω_n ± jω_d, and place any remaining poles at least 5 to 10 times further left so they decay quickly and the second-order approximation holds. Every angle in these expressions is in radians; the pole values entered into the calculator are in rad/s, not Hz (divide by 2π for Hz). Source: Nise, Control Systems Engineering, 6th ed., §4.6 and §12.2; Ogata, Modern Control Engineering, 5th ed., §5-3.