The so-called state-space description provides 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.

- A mathematical description of the system in terms of a minimum set of variables,
**x**, together with knowledge of those variables at an initial time t_{i}(t), i = 1,...,n_{0}, and the system inputs for time**t ≥ t**, is sufficient to predict the future system state and outputs for all time t>t_{0}_{0}.

- The dynamic behavior of a state-determined system is completely characterized by the response of the set of n variables xi(t), where the number n is defined to be the order of the system.
- The system shown in the above figure has two
**inputs u**, and_{1}(t) and u_{2}(t)**four output**variables**y**. If the system is state-determined, knowledge of its state variables_{1}(t),...,y_{4}(t)**(x**at some initial time_{1}(t_{0}), x_{2}(t_{0}),...,x_{n}(t_{0}))**t**and the inputs_{0},**u**and_{1}(t)**u**for_{2}(t)**t ≥ t**is sufficient to determine all future behavior of the system._{0} - The state variables are an internal description of the system which completely characterize the system state at any time t, and from which any output variables y
_{i}(t) may be computed.

**The State Equations**

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 x_{1}(t),...,x_{n}(t) and the system inputs u_{1}(t),...,u_{r}(t). In the general case the form of the n state equations is:

- where =
**dx**and each of the functions f_{i}/dt_{i}(x, u, t), (i = 1,...,n) may be a general nonlinear, time-varying function of the state variables, the system inputs, and time. - It is common to express the state equations in a vector form, in which the set of n state variables is written as a state vector
**x(t)=[x**, and the set of r inputs is written as an input vector_{1}(t), x_{2}(t),...,x_{n}(t)]^{T}**u(t)=[u**. Each state variable is a time-varying component of the column vector x(t)._{1}(t), u_{2}(t),...,u_{r}(t)]^{T} - In vector notation the set of n equations in Eqs. (1) maybe written as

** ** where f (x, u, t) is a vector function with n components fi (x, u, t).

where the coefficients** a _{ij}** and

**b**are constants that describe the system.

_{ij}The last equation can be written in matrix form as below

which may be summarized as:

- Where state vector
**x**is a column vector of length**n,**the input vector**u**is a column vector of length**r, A**is an**n × n**square matrix of the constant coefficients**a**, and_{ij}**B**is an**n × r**matrix of the coefficients**b**that weight the inputs._{ij}

**Output Equations**

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) = c _{1}x_{1} + c_{2}x_{2} + ... + c_{n}x_{n} + d_{1}u_{1} + ... + d_{r}u_{r}**

where the** c _{i}** and

**d**are constants. If a total of

_{i}**m system variables**are defined as outputs, then the output equation can also be obtained as State Equation in compact form

**y = Cx + Du**

- where y is a column vector of the output variables y
_{i}(t), C is an m×n matrix of the constant coefficients c_{ij}that weight the state variables, and D is an m × r matrix of the constant coefficients d_{ij}that weight the system inputs. - For many physical systems, the matrix D is the null matrix, and the output equation reduces to a simple weighted combination of the state variable

**if D= 0 ; then Y = Cx**

**Block Diagram Representation of Linear Systems Described by State Equations**

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:

- Draw n integrator (S
^{−1}) blocks, and assign a state variable to the output of each block. - At the input to each block (which represents the derivative of its state variable) draw a summing element.
- Use the state equations to connect the state variables and inputs to the summing elements through scaling operator blocks.
- Expand the output equations and sum the state variables and inputs through a set of scaling operators to form the components of the output.

**Example:** Draw a block diagram for the general second-order, single-input single-output system

the block diagram shown below is drawn using the four steps described above

**Transformation from Classical Form to State-Space Representation**

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.

**(a _{n}s^{n} + a_{n−1}s^{n−1}+ ··· + a_{0})Y(s) =(b_{n}s^{n} + b_{n−1}s^{n−1} + ··· + b_{0})U(s)**

- We may multiply both sides of the equation by
**s**to ensure that all differential operators have been eliminated^{−n}

**a _{n}+ a_{n−1}s^{−1} + ··· + a_{1}s^{−(n−1)} + a_{0}s−nY(s) =b_{n}+b_{n−1}s^{−1} + ··· + b_{1}s^{−(n−1)}+ ··· + b_{0}s^{−n}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) = (a _{n} + a_{n−1}s^{−1} + ··· + a_{1}s^{−(n−1)} + a_{0}s^{−n}X(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) ^{-1}BU(s)+ DU(s) **

** Y(s)/U(s) = G(s) = C(sI-A) ^{-1}B+D**

**State-Transition Matrix**

The state-transition matrix is defined as a matrix that satisfies the linear homogeneous state

equation:

Let ϕ(t) be the n × n matrix that represents the state-transition matrix; then it must satisfy the

equation:

Furthermore, let x(0) denote the initial state at t = 0; then ϕ(t) is also defined by the matrix

equation:

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

Solving for X(s) from Eq. (v). we get

where it is assumed that the matrix (s1 – A) is non-singular. Taking the inverse Laplace

transform on both sides of Eq. (v) yields

By comparing Eq.(iv) with Eq. (v), the state-transition matrix is identified to be

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

for t ≥ 0, where eAt represents the following power series of the matrix At, and

**Properties of State-Transition Matrices.**

**Controllability & Observability**

**Controllability: **Controllability can be defined 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 t_{0} 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 C _{B} = [ B : AB :**

**..... A**

^{n-1}B is equal to n ], then the system is controllable.**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 t_{0} if, with the system in state **X(t _{0})**, 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 O _{B} = [ C^{T} : A^{T}C^{T} : ..... A^{T})^{n-1}C^{T} is equal to n], then the system is sail to be Observable.**

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