Control System Engineering Module I

MODULE-I


State space analysis.
State space analysis is an excellent method for the design and analysis of control systems.
The conventional and old method for the design and analysis of control systems is the
transfer function method. The transfer function method for design and analysis had many
drawbacks.
Advantages of state variable analysis.
▪ It can be applied to non linear system.
▪ It can be applied to tile invariant systems.
▪ It can be applied to multiple input multiple output systems.
▪ Its gives idea about the internal state of the system.


State Variable Analysis and Design

State: The state of a dynamic system is the smallest set of variables called state variables such that
the knowledge of these variables at time t=to (Initial condition), together with the knowledge of input
for ≥ 𝑡0 , completely determines the behaviour of the system for any time 𝑡 ≥ 𝑡0 .

State vector: If n state variables are needed to completely describe the behaviour of a given system,
then these n state variables can be considered the n components of a vector X. Such a vector is called
a state vector.

State space: The n-dimensional space whose co-ordinate axes consists of the x1 axis, x2 axis,. .... xn
axis, where x1 , x2 ,. .... xn are state variables: is called a state space.

State Model

Lets consider a multi input & multi output system is having

r inputs 𝑢1 𝑡 , 𝑢2 𝑡 , ........... 𝑢𝑟 (𝑡)

m no of outputs 𝑦1 𝑡 , 𝑦2 𝑡 , ...........𝑦𝑚 (𝑡)

n no of state variables 𝑥1 𝑡 , 𝑥2 𝑡 ,........... 𝑥𝑛 (𝑡)

Then the state model is given by state & output equation

X t = AX t + BU t................... state equation

Y t = CX t + DU t ............... output equation

A is state matrix of size (n×n)

B is the input matrix of size (n×r)

C is the output matrix of size (m×n)


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,D is the direct transmission matrix of size (m×r)

X(t) is the state vector of size (n×1)

Y(t) is the output vector of size (m×1)

U(t) is the input vector of size (r×1)




(Block diagram of the linear, continuous time control system represented in state space)

𝐗 𝐭 = 𝐀𝐗 𝐭 + 𝐁𝐮 𝐭

𝐘 𝐭 = 𝐂𝐗 𝐭 + 𝐃𝐮 𝐭



STATE SPACE REPRESENTATION OF NTH ORDER SYSTEMS OF LINEAR
DIFFERENTIAL EQUATION IN WHICH FORCING FUNCTION DOES NOT INVOLVE
DERIVATIVE TERM

Consider following nth order LTI system relating the output y(t) to the input u(t).

𝑦𝑛 + 𝑎1𝑦𝑛−1 + 𝑎2𝑦𝑛−2 + ⋯ + 𝑎𝑛−1𝑦1 + 𝑎𝑛 𝑦 = 𝑢

Phase variables: The phase variables are defined as those particular state variables which are
obtained from one of the system variables & its (n-1) derivatives. Often the variables used is
the system output & the remaining state variables are then derivatives of the output.

Let us define the state variables as

𝑥1 = 𝑦

𝑑𝑦 𝑑𝑥
𝑥 = =
2 𝑑𝑡 𝑑𝑡
𝑑𝑦 𝑑𝑥2
𝑥 = =
3 𝑑𝑡 𝑑𝑡
⋮ ⋮ ⋮

𝑑𝑥𝑛−1
𝑥𝑛 = 𝑦𝑛−1 =
𝑑𝑡

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, From the above equations we can write

𝑥1 = 𝑥2

𝑥2 = 𝑥3

⋮ ⋮



𝑥 𝑛 − 1 = 𝑥𝑛

𝑥𝑛 = −𝑎𝑛 𝑥1 − 𝑎𝑛−1𝑥2 − ⋯ … … … − 𝑎1𝑥𝑛 + 𝑢

Writing the above state equation in vector matrix form

X t = AX t + Bu t

𝑥1 0 1 0…… 0
𝑥2 0 0 1…… 0
Where 𝑋 = ⋮ , 𝐴= ⋮ ⋮ ⋮ ⋮
⋮ 0 0 0…… 1
𝑥𝑛
𝑛×1 −𝑎𝑛 −𝑎𝑛−1 −𝑎𝑛−2 … … . −𝑎1 𝑛×𝑛

Output equation can be written as

Y t = CX t

𝐶= 1 0……. 0 1×𝑛




Example: Direct Derivation of State Space Model (Mechanical Translating)
Derive a state space model for the system shown. The input is fa and the output is y.




We can write free body equations for the system at x and at y.


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