State Space Analysis of Control System

State space analysis of control system represents internal dynamics with a vector of state variables. It complements transfer-function and frequency-response methods rather than replacing them.

1. Classical control often uses transfer functions, root locus and frequency response, but it also analyses time response.
2. State-space models naturally represent linear SISO and MIMO systems. A fixed A, B, C, D model is linear; nonlinear or time-varying dynamics require state equations or matrices that depend on state, input or time. The linked SISO term means single input and single output.
3. State-space methods support analytical and numerical work on stability, response, controllability, observability and feedback design.

State variables can describe linear or nonlinear models with SISO or MIMO signals in continuous or discrete time. The equations and solution methods must match the model class. The following terms define the representation.

1. State in State Space Analysis: The state at t = t₀, together with the model and input for t ≥ t₀, determines the system’s future evolution for t ≥ t₀.
2. State Variables in State Space Analysis: State variables are the components chosen to represent the state, written x₁(t), x₂(t), … , x_n(t). Different coordinate choices can represent the same dynamics.
3. State Vector: The state vector x(t) collects the n state variables into one n-dimensional column vector.
4. State Space: State space is the n-dimensional set of possible state vectors, with coordinates such as x₁, x₂, … , x_n.

State Space Equations

Consider a continuous linear time-invariant system.
Let r be the number of inputs and m the number of outputs.
The input vector contains u₁, u₂, u₃, … , u_r.
The output vector contains y₁, y₂, … , y_m.
Let n be the number of state variables x₁, x₂, … , x_n.
These counts determine the matrix dimensions.
The input column vector is:

Here T denotes transpose.

The output column vector is:

It has m components.
The state column vector is:

It has n components.
The standard equations are x-dot = Ax + Bu and y = Cx + Du:

Representation of State Model using Transfer Function

Decomposition: A transfer function can be realised as a state model in several forms. Three common construction methods are:

1. Direct decomposition,
2. Cascade or series decomposition,
3. Parallel decomposition.

A proper rational transfer function can be converted to differential equations and then to a state-space realisation. The inverse Laplace transform relates the algebraic transfer description to time-domain dynamics. Direct, cascade and parallel forms use different state coordinates but can represent the same input-output map. Improper, delayed or descriptor models need additional treatment.

The transfer matrix relates Y(s) to U(s) under zero initial conditions.
Taking the Laplace transform of the state and output equations gives:

Solving the state equation gives:

Here I is the n-by-n identity matrix.
Substitution into the output equation gives G(s) = C(sI-A)^-1B + D. The displayed special case assumes D = 0:

The adjugate-over-determinant expression is:

Setting det(sI-A)=0 gives the characteristic equation of A. A non-minimal realisation can have eigenvalues cancelled from the transfer matrix.

Concept of Eigen Values and Eigen Vectors

The roots of det(sI-A)=0 are the eigenvalues of A. They govern the natural modes of the LTI state equation.
The following properties apply with the stated conditions and with algebraic multiplicity counted:

1. A square matrix A and its transpose Aᵀ have the same eigenvalues.
2. The sum of the eigenvalues of A equals trace(A).
3. The product of the eigenvalues of A equals det(A).
4. Multiplying A by a scalar α multiplies every eigenvalue by α.
5. If A is nonsingular, the eigenvalues of A inverse are the reciprocals of the eigenvalues of A.
6. For a real matrix, non-real eigenvalues occur in complex-conjugate pairs.

For each eigenvalue eₖ, every nonzero vector Pₖ satisfying (eₖ × I – A)Pₖ = 0 is an eigenvector. An eigenspace can contain many eigenvectors. A repeated eigenvalue may have fewer independent eigenvectors than its algebraic multiplicity.

State Transition Matrix and Zero State Response

Take the Laplace transformation of x-dot = Ax + Bu with initial state x(0):

Solving for X(s) separates the initial-condition and input terms:

Let [sI-A]^-1 = θ(s). Its inverse Laplace transform is θ(t)=exp(At), giving:

The matrix θ(t) is the state transition matrix.
The θ(t)x(0) term is the zero-input response. The L^-1 transform of θ(s)BU(s), equivalently the convolution integral, is the zero-state response.
For an LTI system, the state transition matrix has these properties:

1. At t = 0, θ(0) equals the identity matrix I, not the scalar 1.
2. Time reversal gives θ(-t) = [θ(t)]^-1.
3. The composition rule is θ(t₁)θ(t₂)=θ(t₁+t₂). Therefore [θ(t)]ⁿ = θ(nt) for integer n.