State Space Representation

The state space representation is a unified mathematical framework that describes a dynamic system using a set of first-order differential equations. Unlike the transfer function approach, it captures internal system dynamics, handles multiple inputs and outputs, and enables modern controller and observer design. It is one of the most tested topics in GATE control systems.

Core Concept Explanation

The standard state space model consists of two vector equations. The state equation dx/dt = Ax(t) + Bu(t) describes how the internal state evolves over time. The state vector x(t) is an n-dimensional column vector. Matrix A captures the natural dynamics of the system (how states interact with each other), and matrix B describes how the input drives the states. The system order n is the dimension of A.

The output equation y(t) = Cx(t) + Du(t) maps the internal states and inputs to the measurable outputs. Matrix C selects which combination of states appears in the output. Matrix D represents the direct feedthrough from input to output, and is zero for most physical systems because real systems cannot respond instantaneously to input changes. When D is nonzero, the system is said to have feedforward.

The eigenvalues of matrix A are the poles of the system. This is one of the most important results in state space theory. The characteristic equation is det(sI - A) = 0, which produces the same poles as the denominator of the transfer function. If all eigenvalues have negative real parts, the system is asymptotically stable. This direct link between matrix algebra and system stability is central to modern control design.

Mathematical Expression

For a second-order SISO system, the state space model in matrix form is written explicitly. Let the state vector be x = [x1, x2]^T. Then the state equation becomes [x1-dot, x2-dot]^T = [[a11, a12],[a21, a22]] [x1, x2]^T + [b1, b2]^T u. The output equation becomes y = [c1, c2] [x1, x2]^T + d u. The solution to the state equation is given by the state transition matrix phi(t) = e^(At), where e^(At) = L^(-1)[(sI - A)^(-1)]. The complete solution is x(t) = e^(At) x(0) + integral from 0 to t of e^(A(t-tau)) B u(tau) d(tau).

Practical Understanding

The transfer function can be derived directly from the state space matrices using the relation H(s) = C(sI - A)^(-1) B + D. This confirms that state space and transfer function descriptions are equivalent for LTI systems, though state space carries more information for non-zero initial conditions. Multiple state space representations can produce the same transfer function, differing only in basis choice.

For a series RLC circuit with input voltage Vs and output capacitor voltage Vc: choosing x1 = Vc and x2 = iL (inductor current), the A matrix entries contain 1/LC and R/L terms directly from Kirchhoff's equations. This physical derivation approach is more reliable in exams than memorizing canonical forms.

Solved Numerical Example

Given:
Transfer function G(s) = (s + 3) / (s^2 + 3s + 2)
Convert to phase variable state space form.

Why this formula applies:
Denominator degree n=2, so 2 state variables needed.
Phase variable (controllable canonical) form is directly readable from coefficients.

Denominator: s^2 + 3s + 2 → a1=3, a0=2
Numerator: s + 3 → b1=1, b0=3

State matrix A (companion form):
A = [[0, 1], [-2, -3]]
(last row = negative of denominator coefficients in order a0, a1)

Input matrix B:
B = [0, 1]^T

Output matrix C:
C = [b0, b1] = [3, 1]
(numerator coefficients)

Feedthrough D:
D = 0 (degree numerator < degree denominator)

Verification: H(s) = C(sI-A)^-1 B
det(sI - A) = s^2 + 3s + 2 (matches denominator)

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

Exam Tip: The eigenvalues of A are the system poles. To find poles quickly in GATE, compute det(sI - A) = 0. For the controllable canonical form, the last row of A is always the negatives of the denominator coefficients. D = 0 whenever the numerator degree is strictly less than denominator degree.

Quick Revision

• State equation: dx/dt = Ax + Bu. Output equation: y = Cx + Du.
• A is n x n, B is n x r, C is p x n, D is p x r where n=order, r=inputs, p=outputs.
• Eigenvalues of A = poles of the system = roots of det(sI-A) = 0.
• Transfer function from state space: H(s) = C(sI-A)^(-1)B + D.
• D = 0 for proper systems (numerator degree strictly less than denominator degree).
• State transition matrix: phi(t) = e^(At) = L^(-1)[(sI-A)^(-1)].
• Common trap: confusing matrix dimensions. A must be square n x n. B has n rows, not n columns.