ELEC4631 Chap.7 State Feedback and Pole Placement: Ackermann and Canonical Form
State Feedback and Pole Placement: Ackermann and Canonical Form
With u=-Kx, the nominal closed-loop matrix is A-BK. Exact pole placement begins with the permission condition rank(𝒞)=n. Desired roots are expanded into a monic target polynomial; controllable canonical form exposes how feedback changes its coefficients, while Cayley–Hamilton and Ackermann package the same logic into K=e_n^T 𝒞^-1 R(A) under the convention used here. Similarity offers a transparent alternative: design in controllable coordinates and transform the gain back to the stated state variable. A complete solution does not end with K. It substitutes the gain into A-BK, recomputes the characteristic polynomial or eigenvalues and states the stability or modal conclusion. Pole placement assigns nominal eigenvalues; it does not by itself optimise effort, guarantee robustness, preserve zeros or control eigenvector conditioning.
What this chapter covers
- 01Feedback sign convention u=-Kx
- 02Controllability permission
- 03Desired characteristic polynomial
- 04Controllable canonical form
- 05Cayley–Hamilton reduction
- 06Ackermann formula
- 07Closed-loop polynomial verification and method limits
Place two real poles and verify by coefficients
- permission𝒞=[[0,1],[1,-3]] has determinant -1, so rank two and the pair is controllable.
- targetThe target polynomial is (s+4)(s+5)=s^2+9s+20.
- closureA-BK=[[0,1],[-2-k1,-3-k2]].
- polynomialIts characteristic polynomial is s^2+(3+k2)s+(2+k1).
- gainCoefficient matching gives k2=6 and k1=18, so K=[18,6].
- verificationThe verified polynomial is s^2+9s+20 with roots -4,-5; the closed loop is asymptotically stable.
Key terms
- State feedback
- A control law using the state, here u=-Kx, which changes the system matrix from A to A-BK.
- Pole assignment
- Choosing K so the eigenvalues of A-BK equal a prescribed set, possible for all poles when (A,B) is controllable.
- Desired polynomial
- The monic polynomial formed from target roots; it is the coefficient object matched by canonical form or Ackermann.
- Controllable canonical form
- A coordinate form whose characteristic coefficients appear directly in a row altered by feedback.
- Cayley–Hamilton theorem
- A satisfies its own characteristic polynomial, reducing higher powers in R(A) to a finite controllability-basis combination.
- Ackermann formula
- A compact single-input gain expression using 𝒞^-1 and the desired polynomial evaluated at A; it assumes controllability and a declared convention.
State Feedback and Pole Placement: Ackermann and Canonical Form FAQ
Why must controllability be shown first?
An uncontrollable mode cannot be moved by any state-feedback gain. Ackermann’s inverse also assumes a full-rank square single-input controllability matrix. Stating rank n is the mathematical permission, not ceremonial setup.
How do I catch the feedback sign error?
Write the supplied law before forming the matrix. For u=-Kx, x-dot=(A-BK)x. For u=Kx it is A+BK. After computing K, the trace and determinant of a second-order A-BK should match the desired root sum and product.
Does pole placement determine the transient completely?
No. Eigenvectors, nonnormality, gain magnitude, actuator limits and robustness also affect behaviour. Exact poles are a nominal spectral promise. LQR and LMIs appear later because they encode different objectives or certificates.
When is canonical form preferable?
It is useful when the question asks for derivation, when coefficient logic should remain visible or when Ackermann conventions are ambiguous. Declare z=Tx, design the transformed gain and convert it back so the final K multiplies the original x.
What is a theorem-complete pole-placement answer?
State u=-Kx and form 𝒞. Show rank n before any gain formula. Expand target roots into a real monic polynomial, including conjugate pairs when necessary. Use coefficient matching, controllable canonical form or Ackermann with declared conventions. Report K with dimensions, substitute it into A-BK and recompute the characteristic polynomial. Compare every coefficient or root with the target and then state stability from root location. If a similarity transform is used, show how the transformed gain returns to the original state. This chain distinguishes permission, computation and verification. It also gives a marker recoverable steps if a single inverse or sign is wrong, whereas a gain copied from software has no visible route and conflicts with the analytical-working expectation shown by the current quiz.
How should I compare pole placement with later design methods?
Use pole placement when the requirement specifies exact nominal roots and the pair is controllable. Use LQR when the requirement specifies an infinite- or finite-horizon quadratic state/effort trade-off; the poles then emerge from Q and R. Use an LMI when the requirement is a convex feasibility certificate, a fixed decay rate or another supported affine matrix constraint. All three may produce a stabilising K, but their promises differ. Pole placement does not minimise a cost. LQR does not impose arbitrary roots. An LMI feasibility solve is not optimal without an objective. On an exam, write the requested objective in words before selecting the method; this prevents technically correct algebra from answering a different design question.
What practice exposes convention errors early?
Solve the same controllable second-order plant by coefficient matching and Ackermann, then compare K and the final polynomial. Next define z=Tx and repeat in canonical coordinates, writing the gain transformation explicitly. Change the control law from u=-Kx to u=Kx and predict the sign changes before calculating. For every version, check trace and determinant of the closed-loop matrix against the target roots. Ask an AI tutor for a fresh controllable matrix with integer arithmetic, but provide your convention and require only a polynomial residual check after your derivation. The goal is to make rank, target and verification automatic, not to memorise one gain.
Exam move
Use a five-line routine: rank 𝒞, expand target roots, calculate K, substitute into A-BK, verify its polynomial. Practise both coefficient matching and Ackermann on the same second-order example so the formula is connected to the canonical logic. State u=-Kx in the first line and keep target coefficients in descending powers. For transformed-coordinate solutions, verify T is invertible and write the relationship between gains before returning to physical coordinates. Add one deficient-rank example and explain which pole cannot move. Finally, compare method language: exact roots point to pole placement; a quadratic cost points to LQR; a feasibility or rate certificate points to an LMI.
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