The so-called state-space description provide the dynamics as a set of coupled first-order differential equations in a set of internal variables known as state variables, together with a set of algebraic equations that combine the state variables into physical output variables.
State
The state of a dynamic system refers to a minimum set of variables, known as state variables, that fully describe the system.
The State Equations
In the standard mathematical form of the system is expressed as a set of n coupled first-order ordinary differential equations, known as the state equations, in which the time derivative of each state variable is expressed in terms of the state variables x1(t) ,..., xn(t) and the system inputs u1(t) ,..., ur(t). In the general case the form of the n state equations is:
x1 = f1(x, u, t)
x2 = f2(x, u, t)
xn = fn(x, u, t)
Output Equations
A system output is defined to be any system variable of interest. An arbitrary output variable in a system of order n with r inputs may be written as
y(t) = c1x1 + c2x2 + ... + cnxn + d1u1 + ... + drur
where the ci and di are constants. If a total of m system variables are defined as outputs, then the output equation can also be obtained as State Equation in compact form
y = Cx + Du
A system of order n has n integrators in its block diagram. The derivatives of the state variables are the inputs to the integrator blocks, and each state equation expresses a derivative as a sum of weighted state variables and inputs. A detailed block diagram representing a system of order n may be constructed directly from the state and output equations as follows:
Example: Draw a block diagram for the general second-order, single-input single-output system
block diagram shown below is drawn using the four steps described above
Let the differential equation representing the system be of order n, and without loss of generality assume that the order of the polynomial operators on both sides is the same.
(ansn + an−1sn−1+ ··· + a0)Y(s) =(bnsn + bn−1sn−1 + ··· + b0)U(s)
from which the output may be specified in terms of a transfer function. If we define a dummy variable Z(s), and split into two parts
Eq. of Z(s) can ne be solved for U(s)
U(s) = (an + an − 1s−1 + ··· + a1s−(n − 1) + a0s−nX(s)
State-Space and Transfer Function
The state equation form can be transformed into transfer function.
Tanking the Laplace transform and neglect initial condition then
sX(s) - X(0) = AX(s) + BU(s)
y(s) = CX(s) + DU(s)
then sX(s) - AX(s)= X(0) + BU(s)
By Neglecting Initial Conditions
(sI - A)X(s) = BU(s)
X(s) = (sI - A)-1 BU(s)
Then Put the value of X(s) for Y(s)...
then Y(s) = C(sI - A)-1BU(s) + DU(s)
Y(s) / U(s) = G(s) = C(sI - A)-1B + D
State-Transition Matrix
The state-transition matrix is defined as a matrix that satisfies the linear homogeneous state equation:
dx(t) / dt = Ax(t) ...(i)
Let ϕ(t) be the n × n matrix that represents the state-transition matrix; then it must satisfy the equation:
dϕ(t) / dt =Aϕ(t) ...(ii)
Furthermore, let x(0) denote the initial state at t = 0; then ϕ(t) is also defined by the matrix equation:
x(t) = ϕ(t) x (0) ...(iii)
which is the solution of the homogeneous state equation for t ≥ 0. One way of determining ϕ(t) is by taking the Laplace transform on both sides of Eq. (i); we have
s(X)(s) - x(0) = AX(s) ...(iv)
Solving for X(s) from Eq. (v). we get
X(s) = (SI - A)-1X(s) ...(v)
where it is assumed that the matrix (s1 – A) is non-singular. Taking the inverse Laplace transform on both sides of Eq. (v) yields
x(t) = L-1[(S1 - A)-1 x (0) t ≥ 0 ...(vi)
By comparing Eq.(iv) with Eq. (v), the state-transition matrix is identified to be
ϕ(t) = L-1[(sI - A)-1] ...(vii)
An alternative way of solving the homogeneous state equation is to assume a solution, as in the classical method of solving linear differential equations. We let the solution to be
x(t) = eATX(0)
for t ≥ 0, where eAt represents the following power series of the matrix At, and
Properties of State-Transition Matrices.
The state-transition matrix OM possesses the following properties
1. Controllability: Controllability can be define in order to be able to do whatever we want with the given dynamic system under control input, the system must be controllable. A system is said to be controllable at time t0 if it is possible by means of an unconstrained control vector to transfer the system from any initial state to any other state in a finite interval of time.
Condition for Controllability;
If the rank of CB = [ B : AB : ..... An - 1B is equal to n ], then the system is controllable.
2. Observability: In order to see what is going on inside the system under observation, the system must be observable. A system is said to be observable at time t0 if, with the system in state X(t0),it is possible to determine this state from the observation of the output over a finite interval of time.
Condition for Observability: If the rank of OB = [ CT : ATCT : ..... AT)n - 1CT is equal to n], then the system is sail to be Observable.
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