Transfer Function from State Space

While state space representation captures the complete internal dynamics of a system, engineers often need the transfer function for frequency-domain analysis, root locus design, and Bode plots. The transfer function from state space is derived using the formula H(s) = C(sI - A)^{-1}B + D, which is one of the most important relationships in modern control theory and appears regularly in GATE.

Core Concept Explanation

Starting from the state space equations dx/dt = Ax + Bu and y = Cx + Du, taking the Laplace transform with zero initial conditions gives sX(s) = AX(s) + BU(s) and Y(s) = CX(s) + DU(s). Rearranging the first equation: (sI - A)X(s) = BU(s), where sI is the scalar s multiplied by the identity matrix of size n. Solving for X(s) gives X(s) = (sI - A)^{-1} BU(s), provided the matrix (sI - A) is invertible.

Substituting X(s) into the output equation: Y(s) = C(sI - A)^{-1} BU(s) + DU(s). Factoring out U(s) gives the transfer function matrix H(s) = Y(s)/U(s) = C(sI - A)^{-1}B + D. For a SISO system, H(s) is a scalar rational function. For MIMO systems, H(s) is a matrix of transfer functions relating each input to each output.

The matrix inverse (sI - A)^{-1} is computed as the adjugate of (sI - A) divided by its determinant: (sI - A)^{-1} = adj(sI - A) / det(sI - A). The denominator det(sI - A) is the characteristic polynomial of A, whose roots are the eigenvalues of A and the poles of H(s). This is why eigenvalues of A and poles of the transfer function are always identical.

Mathematical Expression

For a 2x2 system matrix A = [[a, b],[c, d]], the matrix (sI - A) = [[s-a, -b],[-c, s-d]]. Its determinant is (s-a)(s-d) - bc, which equals s^2 - (a+d)s + (ad-bc). This is the characteristic polynomial. The inverse is (sI-A)^{-1} = [[s-d, b],[c, s-a]] / det(sI-A). Substituting into H(s) = C(sI-A)^{-1}B + D gives the final scalar transfer function after matrix multiplication.

Note that trace(A) = a + d = sum of eigenvalues, and det(A) = ad - bc = product of eigenvalues. These two identities allow quick pole calculation for 2x2 systems: poles are roots of s^2 - trace(A)s + det(A) = 0. This is frequently exploited in GATE numerical questions.

Practical Understanding

The formula H(s) = C(sI-A)^{-1}B + D is significant because it proves that the transfer function and state space model are equivalent representations of the same LTI system (for zero initial conditions). However, state space carries more information. Two different state space models with different A, B, C matrices but identical H(s) are related by a similarity transformation and represent the same external behavior with different internal coordinates.

When a state space model is obtained from a non-minimal realization (for example through an intermediate block diagram), the derived transfer function may have pole-zero cancellations. A pole of A that cancels with a zero in the numerator does not appear in the final H(s). Such cancelled poles are either uncontrollable or unobservable modes and are important for checking minimality of the representation.

Solved Numerical Example

Given:
A = [[0, 1], [-2, -3]]
B = [0, 1]^T
C = [1, 0]
D = 0

Why this formula applies:
H(s) = C(sI-A)^-1 B + D

Step 1: Compute (sI - A)
(sI - A) = [[s, -1], [2, s+3]]

Step 2: Compute det(sI - A)
det = s*(s+3) - (-1)*(2) = s^2 + 3s + 2

Step 3: Compute adj(sI - A)
adj = [[s+3, 1], [-2, s]]

Step 4: Compute (sI-A)^-1 = adj / det
(sI-A)^-1 = [[s+3, 1], [-2, s]] / (s^2 + 3s + 2)

Step 5: Compute C * (sI-A)^-1 * B
C * (sI-A)^-1 = [1, 0] * [[s+3, 1], [-2, s]] / (s^2+3s+2)
              = [(s+3), 1] / (s^2+3s+2)

[(s+3), 1] * [0, 1]^T = 0*(s+3) + 1*1 = 1

H(s) = 1 / (s^2 + 3s + 2) + 0

Final Answer:
H(s) = 1 / (s^2 + 3s + 2) = 1 / ((s+1)(s+2))
Poles at s = -1 and s = -2 (stable system)

Exam Tip: For a 2x2 system, the characteristic polynomial is s^2 - trace(A)*s + det(A). Use this to quickly find poles without computing the full matrix inverse. Also remember: poles of H(s) = eigenvalues of A always. If C(sI-A)^-1 B has any s terms in numerator, check if pole-zero cancellation occurs, indicating a non-minimal realization.

Mechanism Diagram

Key Observations

• Poles of H(s) are always the eigenvalues of matrix A, regardless of B, C, D values.
• Zeros of H(s) depend on all four matrices and appear in the numerator polynomial of C(sI-A)^{-1}B.
• If the same eigenvalue of A appears as both a pole and a zero of H(s), it is a pole-zero cancellation indicating an uncontrollable or unobservable mode.
• D matrix adds a constant to H(s), making the system improper if D is nonzero (numerator degree equals denominator degree).
• For MIMO systems, H(s) becomes an p x r matrix. Each element H_{ij}(s) relates the jth input to the ith output.

Quick Revision

• Formula: H(s) = C(sI-A)^{-1}B + D. Derived from Laplace transform of state equations with zero initial conditions.
• (sI-A)^{-1} = adj(sI-A) / det(sI-A). Denominator is the characteristic polynomial.
• Poles of H(s) = eigenvalues of A = roots of det(sI-A) = 0.
• For 2x2 A: characteristic poly = s^2 - trace(A)s + det(A). Use this shortcut in GATE.
• D = 0 means strictly proper system. D nonzero means degree of numerator equals degree of denominator.
• Pole-zero cancellation in H(s) signals non-minimal realization (loss of controllability or observability).
• Common trap: forgetting the minus sign in (sI - A). Writing (A - sI) gives wrong signs in the characteristic polynomial.