ELEC4632 Chap.1 Linear Algebra for State-Space Control
Linear Algebra for State-Space Control
The opening tutorials refresh the matrix structure needed for computer control: rank, full-rank status, range, null space, determinants, inverses, eigenvalues and characteristic equations. This chapter interprets those calculations as questions about independent state directions. It also connects the tutorial basis b, Ab and A squared b to controllability, giving a direct bridge from linear algebra to state-feedback design.
What this chapter covers
- 01
Rank as the number of independent directions preserved by a matrix
- 02
Range bases chosen from pivot columns of the original matrix
- 03
Null-space directions that disappear under the matrix map
- 04
Determinant, invertibility and identity-multiplication checks
- 05
Eigenvalues as modes of a state update
- 06
Characteristic polynomials and companion-form coefficients
- 07
Controllability columns b, Ab and A squared b
Worked example · free
Find range and null-space bases from a tutorial matrix
- +1Row reduction leaves two pivot rows, so the rank is two.
- +1The first two original columns are independent and form a convenient range basis.
- +1The third column equals negative four times the first plus three times the second.
- +1The resulting null-space basis vector is (4,-3,1), and direct multiplication returns zero.
Key terms
- Rank
- Rank is the number of independent row or column directions represented by a matrix.
- Range
- The range is the set of output vectors produced by multiplying the matrix by possible inputs.
- Null Space
- The null space contains input directions that the matrix maps to the zero vector.
- Characteristic Polynomial
- The characteristic polynomial is obtained from the determinant of lambda times identity minus the matrix.
- Companion Form
- Companion form stores characteristic coefficients in a structured state matrix useful for pole calculations.
Linear Algebra for State-Space Control FAQ
Why must range bases use the original columns?
Row operations change the columns while preserving row-equivalent equations. Pivot locations reveal which original columns are independent, but the reduced columns belong to a different matrix. Selecting the original pivot columns keeps the basis tied to the original range.
How can an inverse calculation be checked quickly?
Confirm that the determinant is nonzero, then multiply the proposed inverse by the original matrix. At least one diagonal and several off-diagonal entries should be checked explicitly so a sign error cannot hide behind a partial verification.
How do characteristic coefficients connect to pole placement?
Desired poles define a target characteristic polynomial. Expanding its factors supplies the coefficients that a controllable closed-loop matrix must match. The companion-form tutorial makes that coefficient relationship visible before the full feedback-gain calculation is introduced.
What should follow a rank calculation in a control answer?
Translate the number into structure. State whether the columns span the required space, whether an inverse or coordinate basis is available, or whether a controllability test reaches every state direction. This one sentence connects elimination to the controller question instead of leaving rank as isolated arithmetic.
Why verify a characteristic polynomial by substitution?
Coefficient expansion is vulnerable to sign errors, especially when roots become companion-matrix coefficients. Substituting one desired root checks the scalar polynomial, while evaluating the completed polynomial at the matrix supports the Cayley-Hamilton identity used by later state-space methods.
How much working should a matrix answer show?
Show enough elimination, expansion or multiplication to make the result auditable, then label the selected basis or coefficient set. A bare rank or inverse hides sign and pivot decisions, while every elementary step can waste time without strengthening the structural conclusion.
Exam move
Rework every matrix in the first tutorial without copying the answer path. For rank questions, mark pivot columns and write one sentence about full-rank status. For null-space questions, multiply the final basis vector by the original matrix. For determinant and inverse questions, use the determinant to decide whether an inverse can exist before row reduction.
Finish by expanding the desired-pole polynomial and checking one root through substitution. Build a three-column error log headed operation, verification and control meaning. Under operation, record the row reduction, determinant, inverse or polynomial step that failed.
Under verification, write a check that does not merely repeat the same arithmetic: multiply the proposed inverse, substitute a root, or apply the matrix to a null vector. Under control meaning, state whether the result changes reachability, coordinate validity or mode placement. Repeat the tutorial matrices after a delay and compare the error log rather than only the final answer.
Time one complete matrix question and reserve the closing minute for the structural sentence. When moving into later chapters, return to this sheet whenever a controllability matrix, observability matrix or desired polynomial appears. The notation changes, but the habits remain the same: preserve dimensions, expose signs and verify with the original object.
A concise exam response should show enough elimination or expansion to be auditable, label the basis it selects and end with the implication for the state-space design. Use one final mixed drill to connect the topics: decide rank, extract a valid basis, test invertibility and form a target polynomial without resetting your notation.
Check the dimensions after every multiplication and circle any sign introduced by a negative coefficient. On review, explain aloud why each result matters to a controller rather than rereading the arithmetic silently. This forces the transition from computational fluency to state-space interpretation.
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