State Space Representation of Control System | Control Systems - Electrical Engineering (EE) PDF Download

State Space Analysis

The state-space description provides the dynamics of a system as a set of coupled first-order ordinary differential equations in a set of internal variables called state variables, together with algebraic equations that combine the state variables to produce the physical output variables.

State

The state of a dynamic system is a minimum set of variables that, together with knowledge of future inputs, completely determines the future behaviour of the system. Formally, a set of variables x_i(t), i = 1,...,n, is a state if knowledge of x(t0) at some initial time t₀ and the inputs u(t) for t ≥ t₀ suffice to predict x(t) and all outputs y(t) for t > t₀.

• The dynamic behaviour of a state-determined system is fully characterised by the responses of the n state variables x_i(t). The integer n is called the order of the system.
• If a system has r inputs u₁(t),...,u_r(t) and m outputs y₁(t),...,y_m(t), then the state variables form an internal description from which any output can be computed.

The State Equations

In general the dynamics are expressed as n coupled first-order ordinary differential equations, the state equations, in which the time derivative of each state variable is a (possibly nonlinear, time varying) function of the state variables, the system inputs and time:

ẋ₁(t) = f₁(x(t), u(t), t)

ẋ₂(t) = f₂(x(t), u(t), t)

...

ẋ_n(t) = f_n(x(t), u(t), t)

• Here ẋ_i = d x_i/dt and the functions f_i(·) may be nonlinear and explicitly time dependent.
• For compactness we use vector notation. Define the state vector x(t) = [x₁(t), x₂(t), ..., x_n(t)]^T and the input vector u(t) = [u₁(t), ..., u_r(t)]^T.

In vector form the general nonlinear state equations become

ẋ(t) = f(x(t), u(t), t)

Linear Time-Invariant (LTI) State Equations

For linear time-invariant systems the functions are linear and time independent. The state equations take the matrix form

ẋ(t) = A x(t) + B u(t)

where A is an n × n matrix of constant coefficients a_ij, and B is an n × r matrix of coefficients b_ij that weight the inputs.

Compactly:

Output Equations

An output is any variable of interest produced by the system. For linear systems an arbitrary output y(t) (scalar) may be written as a linear combination of state variables and inputs:

y(t) = c₁x₁(t) + c₂x₂(t) + ... + c_nx_n(t) + d₁u₁(t) + ... + d_ru_r(t)

In matrix form, when m outputs are defined, the output equation is

y(t) = C x(t) + D u(t)

• Here y(t) is an m × 1 column vector of outputs, C is an m × n matrix of coefficients that weight the state variables, and D is an m × r feedthrough matrix that weights the inputs.
• For many physical systems the direct feedthrough is zero, D = 0, so y = C x.

Block Diagram Representation of Linear Systems Described by State Equations

A system of order n is represented in block diagram form by n integrators. Each integrator's output is a state variable and the integrator input is the derivative of that state. The block diagram is constructed by following these rules:

• Draw n integrator blocks and assign a state variable to the output of each block (integrator output = x_i(t)).
• At the input to each integrator place a summing junction that forms ẋ_i from weighted sums of state variables and inputs.
• From the state equations connect state outputs and inputs to the summing junctions through scaling blocks (gains equal to elements of A and B).
• Form the outputs by summing scaled state variables and inputs according to the C and D matrices.

Example: Second-order SISO System

Consider a general second-order single-input single-output system described by state equations (canonical form). A block diagram for this system is drawn by following the four steps above.

Transformation from Classical (Transfer Function) Form to State-Space Representation

Consider a scalar transfer function represented by polynomial operators of order n:

(a_nsⁿ + a_n-1s^n-1 + ··· + a₀)Y(s) = (b_nsⁿ + b_n-1s^n-1 + ··· + b₀)U(s)

• Multiplying both sides by s^-n (or dividing by sⁿ) converts the operator polynomials to powers of s^-1, which helps in defining a set of first-order differential (or difference) relations corresponding to state equations.

One standard procedure is to select state variables as successive derivatives (or integrals) of the output (or of an intermediate variable). This yields equivalent state equations and output equations in the time domain. Different choices of state variables produce different state-space realizations (for example, controllable canonical form, observable canonical form, or diagonal/ modal forms), all of which are algebraically equivalent.

State-Space and Transfer Function

Given the LTI state equations

ẋ(t) = A x(t) + B u(t)

y(t) = C x(t) + D u(t)

take Laplace transforms and neglect initial conditions to obtain the input-output transfer function G(s) = Y(s)/U(s).

s X(s) = A X(s) + B U(s)

(sI - A) X(s) = B U(s)

X(s) = (sI - A)^-1 B U(s)

Substitute into the output equation:

Y(s) = C X(s) + D U(s) = C (sI - A)^-1 B U(s) + D U(s)

Therefore the transfer function is

G(s) = C (sI - A)^-1 B + D

State-Transition Matrix

For the homogeneous linear system

ẋ(t) = A x(t)

the solution with initial condition x(0) is expressed using the state-transition matrix ϕ(t):

x(t) = ϕ(t) x(0)

ϕ(t) must satisfy the matrix differential equation

ϕ̇(t) = A ϕ(t)

and the initial condition ϕ(0) = I.

The state-transition matrix is related to the matrix exponential:

ϕ(t) = e^At

It can also be obtained by the inverse Laplace transform:

ϕ(t) = L^-1[(sI - A)^-1].

Computation of e^At

• Power series definition: e^At = I + At + (At)²/2! + (At)³/3! + ···.
• If A is diagonalizable, A = V Λ V^-1, then e^At = V e^Λt V^-1, where e^Λt is a diagonal matrix of exponentials of the eigenvalues.
• When (sI - A) has a known partial fraction expansion, ϕ(t) is obtained by inverse Laplace transform term by term.

Properties of the State-Transition Matrix

1. ϕ(0) = I (identity matrix).
2. ϕ^-1(t) = ϕ(-t) (when the matrix exponential exists for the required t range).
3. ϕ(t₂ - t₁) ϕ(t₁ - t₀) = ϕ(t₂ - t₀) for any t₀, t₁, t₂.
4. ϕ(kt) = [ϕ(t)]^k for positive integer k when the exponential series interpretation is used (matrix properties must be respected).

Controllability and Observability

Controllability

Controllability expresses whether the input can move the system between arbitrary states. A continuous-time LTI system ẋ = A x + B u is said to be controllable on [t₀, t₁] if, for any initial state x(t₀) and any final state x(t₁), there exists an input u(t) defined on [t₀, t₁] that transfers the state from x(t₀) to x(t₁).

For finite-dimensional LTI systems the following algebraic test is standard:

The controllability matrix is

W_c = [ B   A B   A² B   ...   A^n-1 B ]

If rank(W_c) = n, the system is controllable (reachable).

Observability

Observability expresses whether the internal state can be determined from output measurements. The system ẋ = A x + B u, y = C x + D u is observable if the initial state x(t₀) can be determined from knowledge of the output y(t) over a finite time interval t ∈ [t₀, t₁].

The observability matrix is

W_o = [ C^T   A^T C^T   ...   (A^T)^n-1 C^T ]^T

Equivalently, written row-wise,

W_o = [ C   C A   C A²   ...   C A^n-1 ]^T.

If rank(W_o) = n, the system is observable.

Remarks on Controllability and Observability

• Controllability and observability are dual concepts: tests for one can be derived from tests for the other by transposition.
• If a system is not controllable, it is not possible to place all closed-loop poles arbitrarily by state feedback; only the controllable subsystem poles can be assigned.
• If a system is not observable, some internal modes cannot be seen from outputs and cannot be estimated by any observer that uses only the measured outputs.
• Canonical state representations such as controllable canonical form and observable canonical form make these properties transparent and are useful for controller and observer design.

Worked Example: Controllability of a Second-Order System

Consider a second-order system with state matrices

A = [ [0, 1], [-k/m, -c/m] ] and B = [ 0, 1/m ]^T (typical mass-spring-damper form).

Form the controllability matrix W_c = [ B   A B ].

Compute A B and then check rank(W_c). If rank = 2 the system is controllable; if rank < 2="" it="" is="">

Summary

The state-space method provides a general and compact framework to represent linear and nonlinear dynamic systems using state variables, state equations and output equations. It is especially suited to multivariable systems and modern control design (state feedback, observers). Key algebraic relations include the LTI state equations ẋ = A x + B u and y = C x + D u, the transfer function relation G(s) = C(sI - A)^-1B + D, the state-transition matrix ϕ(t) = e^At, and the algebraic tests for controllability and observability based on the matrices [B AB ... A^n-1B] and [C; CA; ...; C A^n-1].