State Feedback Control

State feedback control is a powerful design methodology where the control input is computed as a linear function of all system state variables rather than just the output error. This full-state information allows the designer to place the closed-loop poles at any desired locations in the s-plane, enabling precise specification of system transient behavior. It is one of the most important topics in modern control theory and appears regularly in GATE examinations.

Core Concept Explanation

In classical control, feedback is based only on the output y(t), which is a limited projection of the full state. State feedback control uses the complete state vector x(t) to compute the control input. The control law is u = -Kx + r, where K is the 1×n (for single-input systems) feedback gain vector and r is the reference input. This law feeds back every state variable, weighting each by the corresponding element of K.

Substituting u = -Kx + r into the state equation dx/dt = Ax + Bu gives:

dx/dt = Ax + B(-Kx + r) = (A - BK)x + Br

The closed-loop system matrix is (A - BK). The eigenvalues of this matrix are the closed-loop poles. Since K has n free parameters (for single-input systems), the designer has n degrees of freedom to place n closed-loop poles anywhere in the complex plane, provided the system is completely controllable.

Mathematical Expression

The pole placement problem asks: given desired pole locations p₁, p₂, ..., pₙ, find K such that the characteristic polynomial of (A - BK) equals (s - p₁)(s - p₂)...(s - pₙ). The standard method is Ackermann's formula:

K = eₙᵀ · Mc⁻¹ · φ(A)

where eₙᵀ = [0, 0, ..., 0, 1] is the last standard basis row vector, Mc is the controllability matrix, and φ(A) = (A - p₁I)(A - p₂I)...(A - pₙI) is the desired characteristic polynomial evaluated at A. This formula is elegant but requires computing matrix polynomial products and inverting Mc.

For simple 2nd order systems, direct comparison of characteristic polynomials is faster: expand det(sI - A + BK), set it equal to the desired polynomial s² + 2ζωₙs + ωₙ², and solve for K elements by matching coefficients.

Practical Understanding

Placing poles deeper in the left half plane (more negative real parts) gives faster response but requires larger control effort and higher gain K. This leads to a fundamental design trade-off: speed of response versus control effort and actuator saturation.

An important practical limitation is that state feedback requires all state variables to be known. In practice, states are often not directly measurable, so a state observer (Luenberger observer) is designed alongside the state feedback controller. The combined structure is the observer-based output feedback controller, and the separation principle guarantees that the observer and controller can be designed independently when the system is both controllable and observable.

Numerical Example

Consider a second-order system with A = [[0, 1], [0, -1]], B = [[0], [1]]. Design state feedback K = [k₁, k₂] to place closed-loop poles at s = -2 and s = -3. The desired characteristic polynomial is (s+2)(s+3) = s² + 5s + 6.

Given:
A = [[0,1],[0,-1]],  B = [[0],[1]],  n = 2
Desired poles: s = -2, s = -3

Why this formula applies:
System is controllable (verify Mc). Closed-loop char. poly. = det(sI - A + BK).
Match with desired polynomial to find K.

Formula:
det(sI - (A - BK)) = s² + 5s + 6

Substitution:
A - BK = [[0,1],[0,-1]] - [[0],[1]]·[k₁,k₂]
       = [[0,1],[0,-1]] - [[0,0],[k₁,k₂]]
       = [[0,     1  ],
          [-k₁, -1-k₂]]

Calculation:
det(sI - (A-BK)) = det([[s, -1],[k₁, s+1+k₂]])
= s(s+1+k₂) + k₁
= s² + (1+k₂)s + k₁

Matching: s² + (1+k₂)s + k₁ = s² + 5s + 6
1 + k₂ = 5  →  k₂ = 4
k₁ = 6

Final Answer:
K = [6, 4]
Closed-loop matrix A-BK = [[0, 1],[-6, -5]]
Eigenvalues: s = -2, -3 (verified)

Exam Tip: Always verify controllability before attempting pole placement. If rank(Mc) < n, the desired poles cannot be achieved by any state feedback gain K. This is one of the most common GATE traps.

Mechanism Points

• The control law u = -Kx + r modifies the system matrix from A to (A - BK). Every element of K shifts the closed-loop poles from their open-loop positions.
• For single-input systems, n free parameters in K allow arbitrary placement of n poles. Multiple-input systems offer more degrees of freedom.
• Ackermann's formula K = eₙᵀ · Mc⁻¹ · φ(A) directly computes K from the desired pole polynomial φ(A) and the controllability matrix Mc.
• Complex poles must appear as conjugate pairs to ensure real-valued K. Purely real desired poles are simpler to design for.
• When states are not directly measurable, the observer-based output feedback controller combines the Luenberger observer with state feedback. The separation principle guarantees independent design of both components.

Quick Revision

• State feedback law: u = -Kx + r, where K is the 1×n gain vector.
• Closed-loop system: dx/dt = (A - BK)x + Br. Closed-loop poles are eigenvalues of (A - BK).
• Pole placement is possible if and only if rank(Mc) = n (complete controllability).
• Design method: expand det(sI - A + BK), match coefficients with desired characteristic polynomial to find K.
• Ackermann formula: K = eₙᵀ · Mc⁻¹ · φ(A), where φ is desired characteristic polynomial evaluated at A.
• Exam trap: Do not confuse closed-loop poles (eigenvalues of A - BK) with open-loop poles (eigenvalues of A). State feedback changes the closed-loop but not open-loop poles.
• Separation principle: observer and state feedback can be designed independently when system is controllable and observable.