Chapter - 4 Control System | Additional Study Material for Mechanical Engineering PDF Download

1. Basics of Control System

Definition of Control system (CS): It is a system by means of which any quantity of interest in a machine or mechanism is controlled (maintained or altered) in accordance with the desired manner. Control systems can be characterized mathematically by Transfer function' or 'State model'. Transfer function is defined as the ratio of Laplace Transform (LT) of output to that of input assuming that initial conditions are zero. Transfer function is also obtained as Laplace transform of the impulse response of the system.


For any arbitrary input r(t), output c(t) of control system can be obtained as below,
r(t) = L^-1 (R (s)) = L^-1 (T (s) . R (s))= L^-1 (T(s)) * r(t)
Where L and L^-1 are forward and inverse Laplace transform operators and * is convolution operator.

Classification of Control Systems
Open-Loop Control System

• The reference input controls the output through a control action process. Here output has no effect on the control action, as the output is not fed-back for comparison with the input.
• Due to the absence of feedback path, the systems are generally stable

Closed-Loop Control System (Feedback Control Systems): Closed-Loop control systems can be classified as positive and negative feedback (f/b) control systems. In a closed-loop control system, the output has an effect on control action through a feedback.

Let T(s) be the overall transfer function of the closed-loop control system, then

Here negative sign in denominator is considered for positive feedback and vice versa.
G(S)H(S) → Open loop transfer function
Error transfer function =

Effect of Feedback

• Effect of Feedback on Stability
  (i) Stability is a notion that describes whether the system will be able to follow the input command.
  (ii) A system is said to be unstable, if its output is out of control or increases without bound.
• Effect of Feedback on overall gain
  (i) Negative feedback decreases the gain of the system and positive feedback increase the gain of the system.
• Effect of Feedback on Sensitivity
  The sensitivity of the gain of the overall system T to the variation in G is defined as
  
  Similarly,
• Negative feedback makes the system less sensitive to the parameter variation.
• Negative feedback improves the dynamic response of the system
• Negative feedback reduces the effect of disturbance signal or noise.
• Negative feedback improves the bandwidth of the system.

Signal Flow Graphs (SFG): A signal flow graph is a graphical representation of portraying the input-output relationships between the variables of a set of linear algebraic equations. Also following are the basic properties of signal flow graphs.

• A signal flow graph applies to only linear systems.
• The equations based on which a signal flow graph is drawn must be algebraic equation in the form of effects as functions of causes.
• Signals travel along branches only in the direction described by the arrows of the branches.
• The branch directing from node y_k to y_j represents the dependence of the variable yt upon y_j but not the dependence of y_j upon y_k
• A signal y_k travelling along a branch between nodes y_k and y_j is multiplied by the gain of the branch, a_kj, so that signal a_kj y_k is delivered at node y_j.

Mason's Gain Formula: 
The general gain formula is, T =
Where T = Overall gain between y_in and y_out
y_out = output node variable
Y_in = input node variable
N = total number of forward paths
P_k = gain of the k^th forward path
Δ = determinant of the graph =
= 1 - (sum of all individual loop gains) + (sum of gain products of all possible combinations of two non- touching loops) - (sum of the gain products of all possible combinations of three non- touching loops) + ....
 = gain product of the m^th possible combination of V non-touching loops
Δ_k = that part of the signal flow graph which is non-touching with the k^th forward path

2. Time Domain Analysis

Introduction: Whenever an input signal or excitation is given to a system, the response or output of the system with respect to time is known as time response of the system. The time response of a control system is divided into two parts namely, transient and steady state response.
Total response of a system = transient response + steady state response (or C (t) — C_tr (t) + C_ss(t))
Where C(t) is overall response of the system,
C_tr(t) is transient response component of the system and
C_ss(t) is steady state response component of the system

Following are salient characteristics of transient response of a control system.

• This part of the time response which goes to zero after a large interval of time.
• It reveals the nature of response (e.g. oscillatory or over damped)
• It gives an indication about the speed of response.
• It does not depend on the input signal, rather depends on nature of the system.

Following are the salient properties of steady state response of a control system.

• The part of the time response that remains even after the transients have died out is said to be steady state response.
• The steady state part of time response reveals the accuracy of a control system.
• Steady state error is observed if the actual output does not exactly match with the input.
• It depends on the input signal applied.

Time Response of a First Order Control System: A first order control system is one for which the highest power of 's' in the denominator of its transfer function is equal to 1. Thus a first order control system is expressed by a transfer function,

Time Response of a First Order Control System Subjected to Unit Step Input Function: 
As the input is a unit step function, r(t) = u(t) and R(s) = 1/s
Output is given by c(t) = (1 - e^-t/T) u(t)
The error is given by e(t) = r(t) - c(t) = e^-t/T. u(t)
The steady state error

Time Response of a First Order Control System Subjected to Unit Ramp Input Function: 
As the input is a unit ramp function, r(t) = t.u(t) and R(s) = 1 / s²
Output is given as c (t) = (t - T + Te ^-t/T) u(t)
The error is given by e(t) = r(t) - c(t) =  T - Te^-t/T) u(t)
The steady state error is
From above we see that the output velocity matches with the input velocity but lags behind the input by time T and a positional error of T units exists in the system.

Time Response of A First Order Control System Subjected to Unit Impulse Input Function:
As
The error is given by e(t) = r(t) - c(t)
The steady state error is

Time Response of Second Order Control System:
A second order control system is one for which the highest power 's' in the denominator of its transfer function is equal to 2. A general expression for the T.F. of a second order control system is given by,

Characteristic Equation: The characteristic equation of a second order control system is given by

The roots are s =
Here ω_n is called natural frequency of oscillations,
 is called damped frequency of oscillations,
 is called damping ratio and affects damping and
 is called damping factor or damping coefficient.
Based on roots of characteristic equation, following can be highlighted.

• The real part of the roots denotes the damping
• Imaginary part denotes the damped frequency of oscillation
• Sustained oscillations are observed if the roots are lying on imaginary axis (j ω axis).
• As  increases, system becomes less oscillatory and more sluggish.

Time Response of a Second Order Control System Subjected to Unit Step Input Function:
 where
The steady state error is
Here the time constant T = and Speed of the system

Transient Response Specification of Second Order Under-Damped Control System:
The time response of an under damped control system exhibits damped oscillations prior to reaching steady state.

• Delay Time (t_d): It is the time required for the response to rise to 50% of the final value from zero, in first attempt.
• Rise Time (t_r): The time needed for the response to reach from 10% to 90% (for overdamped system) or 0 to 100% (for underdamped systems) of the desired value of the output at the very first instant.
  
• Peak Time (t_p): It is the time required for the response to reach the peak value at the first instant
  
  Also the response exhibits overshoot and undershoot at the instants,The local overshoots occur for n = 1, 3.... and local undershoots occur for n =2, 4 , .... Hence the first undershoot occurs at the instant,
• Maximum Overshoot (M_p): The maximum positive deviation of the output with respect to its desired value/ steady state value is called Maximum overshoot.
  Percentage overshoot
• Settling Time (t_s):
  For 2% tolerance band, the settling time is given by,
  For 5% tolerance band, the settling time is given by,

Time Response of The Higher Order System And Error Constants:
Steady state error,
Using Final value theorem, e_ss = Lim_s→0 s E(s)
But C(s) = E(s) G(s) ⇒ E(s) =

Type and Order of System: For the open-loop transfer function,

• The type indicates the number of poles at the origin and the order indicates the total number of poles.
• The type of the system determines steady state response and the order of the system determines transient response.

Let k_p = Lim_s→0 G(s) = Position error constant
Let k_v = Lim_s→0 s G(s) = Velocity error constant
Let k_a = Lim_s→0 s² G(s) = Acceleration error constant

3. Stability & Routh Hurwitz Criterion

Introduction: Any system is said to be a stable system, if the output of the system is bounded for a bounded input (stability in BIBO sense) and also in the absence of the input, output should tend to zero (asymptotic stability).
Based on above discussion, systems are classified as below,

• Absolutely stable systems
• Unstable systems
• Marginally stable or limitedly stable systems
• Conditionally stable systems

Depending on the location of poles for a control system, stability of the system can be characterized in following ways.

• Stability of any system depends only on the location of poles but not on the location of zeros.
• If the poles are located in left side of s-plane, then the system is stable.
• If any of the poles is located in right half of s-plane, then the system is unstable.
• if the repeated roots are located on imaginary axis including the origin, the system is unstable.
• When non-repeated roots are located on imaginary axis, then the system is marginally stable.
• As a pole approaches origin, stability decreases.
• The pole which is closest to the origin is called dominant pole.
• If the variable parameter is varied from 0 to ∞ and the poles are always located on left side of s-plane, then the system is absolutely stable.
• When variable parameter is varied from 0 to ∞, if some point onwards, there is a pole in right half of S-plane. Then system is called conditionally stable and typically stability is conditioned on variable parameter.

Absolute Stability Analysis: Absolute stability analysis is by the qualitative analysis of stability and is determined by location of roots of characteristic equation in s-plane.
Relative Stability Analysis: The relative stability can be specified by requiring that all the roots of the characteristic equation be more negative than a certain value, i.e. all the roots must lie to the left of the line; s = - σ₁, (σ₁ > 0). The characteristic equation of the system under study is modified by shifting the origin of the s - plane to s = - σ₁ i.e. by substitution s = z - σ_1. If the new characteristic equation in z satisfies the Routh criterion, it implies that all the roots of the original characteristic equation are more negative than - σ₁.
Also if it is required to find out number of roots of characteristic equation between the lines S = σ₂ and S = -σ₂ perform Routh analysis by putting S = z - σ₁, and find out number of roots to right of S = -σ₁. Similarly find out number of roots to the right of S = σ₂. The difference between above two numbers gives the number of roots of characteristic equation between S = -σ₁ and 5 = -σ₂.
Routh-Hurwitz Criterion: The Routh-Hurwitz criterion represents a method of determining the location of poles of polynomial with constant real coefficient with respect to the left half and the right half of the s- plane. Routh-Hurwitz criterion mainly gives a flexibility to determine the stability of the closed loop control system without actually solving for poles.
F(s) = a₀sⁿ + a₁s^n-1 + a₂s^n-2 ............ + a_n-1S + a_n = 0

If any power of s is missing in the characteristic equation, it indicates that there is at least one root with positive real part, hence the system is unstable. If the characteristic equation contains only odd or even powers of s, then roots are purely imaginary. Thus, the system will have sustained oscillations in output response. Also when Routh - Hurwitz criterion is applied, following difficulties can be faced.

Difficulty 1: When the first term in any row of the Routh array is zero while rest of the row has at least one non-zero term.

• The difficulty is solved if zero of the first column is replaced by a small positive number 'ε' and Routh array is formed as usual. Then as ε → 0 from positive side, elements in the first column of Routh array are found out and stability analysis is done as usual.

Difficulty 2: When all the elements in any one row of the Routh array are zero.

• This situation is overcome by replacing the row of zeros in the Routh array by a row of coefficients of the polynomial generated by taking the first derivative of the auxiliary polynomial and Routh's test is performed as usual.

4. Root Locus Technique

Introduction: Root locus is a locus of poles of transfer function of a closed loop control system when the variable parameter is varied from 0 to ∞. Depending on nature of variable parameter and range of variation, root locus can be classified as below.

• Root Locus (RL) - (K is varied from 0 to ∞)
• Complementary RL - (K is varied from 0 to -∞)
• Complete RL - (K is varied from -∞ to ∞)
• Root counter - (Multiple parameter variation)

Characteristic equation of above system is 1 + G(S)•H(S) = 0. Usually while plotting root locus, a forward path gain, K which is inherently present in G(S) is considered as independent variable and roots of characteristic equation are considered as dependent variables. Any root of 1 + G(S)H(S) = 0 satisfies following two conditions,

• |G(S)H(S)| = 1
• LG(S)H(S) = (2 K + 1)180° where K = 0, 1 ,2 .....

Rules for the Construction of Root Locus (RL): Let P be the number of open-loop poles and Z be the number of open - loop zeroes of a control system. Then the following are the salient features for construction of root locus plot.

• The root locus is always symmetrical about the real axis.
• The root locus always starts from open-loop poles for K=0 and ends at either finite open - loop zeroes or infinity for K → ∞.
• The number of branches of root locus terminating at infinity is equal to (P - Z).
• The number of separate branches of the root locus equals either the number of open - loop poles or the number of open-loop zeroes, whichever is greater.
  N = max(P, Z)
• A section of root locus lies on the real axis, if the total number of open-loop poles and zeroes to the right of the section is odd and is helpful in determining presence of root locus at any point on real axis.
• If P > Z, (P-Z) branches will terminate at '∞' along straight line asymptotes whose angles are ∝_q as given below,
  
  If Z > P, (Z-P) branches will start at 'oo1 along straight line asymptotes whose angles are β_q as given below,
  
• The asymptotes meet the real axis at centroid s = -σ_A as given below
  
• Break - away point is calculated when root locus lies between two poles and break - in point is calculated when root locus lies between two zeros. Break - away / Break - in points are determined from the roots of the equation  Also r branches of the root locus which meet at a point, break away at an angle of
• Angle of departure is calculated when there are complex poles. Also, angle of departure from an open loop pole is given as below
  ∅_P = ±180° (2q + 1) + ∅; q = 0, 1, 2, 3 .....
  Where ∅ is the net contribution at the pole of all other open loop poles and zeros. Also, angle of departure is tangent to root locus at complex pole.
• Angle of arrival is calculated when there are complex zeroes. Also, angle of arrival at the open loop zero is given as below
  ∅_Z = ±180° (2q + 1) - ∅, q = 0, 1, 2, 3 ......
  Also, angle of arrival is tangent to root locus of complex zero.
• The value of X and the point at which root locus branch crosses the imaginary axis is determined by applying Routh criterion to the characteristic equation. The roots at the intersection point are imaginary. Also the points of intersection are conjugate, if all the coefficients of Sⁿ are real in the characteristic equation.
• The value of open loop gain X at any point s₀ on the root locus can be calculated by using the magnitude criteria,
  

5. Frequency Response Analysis Using Nyquist plot

Frequency Domain Specifications: Consider a dosed - loop control system of open - loop transfer function G(S) and feed - back transfer function, H(S). If the system has negative feedback, the overall transfer function is given by M(S) =

The plot of |M(Jω)| with respect to ω is shown in figure below.

The response falls by 3 dB at frequency ω_c, from its low frequency value, called cut-off frequency and the frequency range 0 to ω_c is called the bandwidth of the system. The resonant peak, M_r occurs at resonance frequency ω_r. The bandwidth is defined as the frequency at which the magnitude gain of frequency response plot reduces to 1/√2 = 0.707 (i.e. 3 db) of its low frequency value.
for a second order control system.

Polar Plot: Consider a control system of transfer function G(s). The sinusoidal transfer function G(jω) is a complex function which is given as,
G(jω) = Re [G(jω)] +j lm [G(jω)] (or) G(jω) = |G(jω)| ∠G(jω) = M ∠∅ Where M = |G(Jω)| and ∅ = ∠G(Jω))
G(jω) may be represented as a phasor of magnitude M and phase angle ∅. As the input frequency ω is varied from 0 to ∞, the magnitude M and phase angle ∅ change and hence the tip of the phasor G(jω) traces a locus in the complex plane, which is known as polar plot.
Consider a transfer function which consists of P poles and Z zeroes.

• If transfer function doesn't contain poles at origin, then the polar plot starts from 0° with non-zero magnitude and terminates at -90° x (P - Z) with zero magnitude.
• If the transfer function consists of poles at origin, then the polar plot starts from - 90° with '∞' magnitude and ends at -90° x (P - Z) with zero magnitude.

Special Cases of LTI Control Systems

Minimum Phase System
If G(S) has no poles and zeroes in the R.H.S of S-plane, then the system is called "minimum phase system'. As zeroes are also on left half on s-plane, inverse system of a minimum phase system is also stable.

Non Minimum Phase System:
If G(S) has at least one pole or zero in the R.H.S of S plane, then system is called non-minimum phase system. Also the inverse system of a non-minimum phase system is unstable.

All Pass System:
If G(S) has symmetric poles and zeroes about the about the imaginary axis, then system is called "All pass system".
Also |G(jw)l = K; ∀ω where K is a constant.

Linear Phase System:
A system is called linear phase if plot of ∠G(Jω) with respect to to is linear.
∴  where K is a constant.

Nyquist Plot & Nyquist Stability Criteria:
Nyquist criterion is helpful to identify the presence of roots in a specified region based on polar plot of G(S).H(S). Thus, Nyquist stability analysis is more generalized than Routh criterion. By inspection of polar plot of G(S).H(S) more information is obtained than the stability of the control system.
If N is the number of encirclements of G(s).H(s) around (-1 + j0) in counter-clockwise direction, then
N = P⁺ - Z⁺

Where P⁺ is number of open loop poles with +ve real part and Z⁺ is number of close-loop poles with +ve real part

Tips for Getting Nyquist Plot:

• Nyquist plot is symmetric with respect to real axis. So the plot from ω = 0⁺ to ∞⁺ is M(ω), the plot from ω = ∞^- to 0^- is M*(ω)
• If the system is type N system, the angle subtended by the plot at origin as co varies from 0^- to 0⁺ is - Nπ in clockwise direction.

Gain Margin: The gain margin is a factor by which the gain of a stable system can be increased to bring the system on the verge of instability. If the phase cross-over frequency is denoted by ω_c, and the magnitude of G(jω) H(jω) at
ω = ω_c is |G(jω_c)H(jω_c)|. The gain margin is given by

Phase Margin: The phase margin of a stable system is the amount of additional phase lag required to bring the system to the point of instability. Phase margin is given as,
PM = 180° + ∠G(Jω_g)H(Jω_g) where ω_g is gain crossover frequency.

6. Frequency Response Analysis Using Bode Plot

Bode Plots: Given open - loop transfer function of a dosed - loop control system as G(S) H(S), the stability of the control system can also be determined based on its sinusoidal frequency response (obtained by substituting S = Jω). The quantities, M = 20 log₁₀ |G(Jω)H(Jω)| (in dB) and phase, ∅ = ∠G(Jω)H(Jω) (in degrees) are plotted with respect to frequency on logarithmic scale (log₁₀ω) in rectangular axes. The plot obtained above is called "Bode plot". G.M and PM can be found out from Bode plots, thus relative stability of closed loop control system can be assessed.
Bode Plots of K: 

Bode Plots of :

Bode Magnitude Plot of :

Bode Plot of (1 + ST)

Bode Magnitude Plot of Second Order Control System:

Bode magnitude plot of any open-loop transfer function G(s) H(s) can be found out by superimposing individual magnitude plots of basic pole and zero terms. However phase response can be found out as usual by substituting S = Jω.
M & N Circles
Constant Magnitude Loci: M-Cirdes:
The constant magnitude contours are known as M-circles. M-circles are used to determine the magnitude response of a close-loop system using open-loop transfer function. It is applicable only for unity feedback systems.

The above Eq. represents a family of circles with center at  and radius as  On a particular circle the value of M (magnitude of close-loop transfer function) is constant, therefore these circles are called M-circles.
Constant Phase Angles Lod: N-Circles: The constant phase angle contours are known as N-circles. N-circles are used to determine the phase response of a close-loop system using open-loop transfer function.

For different values of N, above equation represents a family of circles with center at x = -1/2, y = 1/2N and radius as  On a particular circle, the value of N or the value of phase angle of the closed-loop transfer function is constant; therefore, these circles are called N-circles.
Nichol's Chart: The transformation of constant - M and constant - N circles to log-magnitude and phase angle coordinates is known as the Nichols chart.

7. Compensators & Controllers

Introduction: Many a times, performance of a control system may not be upto the expectation, in which case the performance of the same can be improved by controllers or compensating networks.

• Insertion of compensating network is nothing but addition of poles and zeros.
• We can reduce the steady state error by increasing the forward path gain, but it makes the system unstable and oscillatory.
• Addition of a pole to the open loop transfer function will lead the system towards instability. The speed of the response slows down, but the accuracy of the system increases.
• Addition of a zero to the open loop transfer function will lead the system towards stability. The speed of the response becomes faster. But the accuracy of the system is reduced.

Compensating Network

• Cascade Compensation: The compensating network is introduced in forward path in this case. Phase lag/ lead compensators fall into this category.
• Feedback Compensation: The compensating network is introduced in feedback path in this case.

Phase Lag Compensator: A compensator having the characteristic of a lag network is called a lag compensator. Hence, the poles of this network should be closer to origin than zeroes.

• Results in a large improvement in steady sate response (i.e. steady state error is reduced).
• Results in a sluggish response due to reduced bandwidth.
• It is low pass filter and so high frequency noise signals are attenuated.
• Acts as an Integrator.
• Settling time increases.
• Gain of the system decreases.

General form of lag compensator,

Frequency of maximum phase lag,
∴ Maximum lag angle,

Phase Lead compensator: A compensator having the characteristics of a lead network is called a lead compensator. Lead compensator has zero placed more closer to origin than a pole.

• Lead compensation appreciably improves the transient response.
• The lead compensation increases the bandwidth, which improves the speed of the response and also reduces the amount of overshoot.
• A lead compensator is basically a high pass filter and so it amplifies high frequency noise signals.
• Acts as a differentiator.
• Settling time decreases.
• Gain of the system increases.
• There is no improvement in steady state response.

General form of transfer function of lead compensator, G_c(s) =

Frequency of maximum phase lead,
Also,

Comparison of Phase Lag And Phase Lead Compensators:

Phase Lag - Lead Compensator: A compensator having the characteristics of lag -lead network is called a lag - lead compensator.

• A lag - lead compensator improves both transient and steady state response.
• Bandwidth of the system is increased.

The transfer function of lag - lead compensator,  where β> 1, 0 < α < 1 and αβ = l

Feedback Compensation: In this method, the compensating element is introduced in feedback path of a control system as shown.

After compensation, overall open-loop transfer function = G(S) • (1 + SK_t)
Depending on nature of G(S) and K_t, damping of response can be controlled.
Controllers: A closed loop control system tries to achieve the target output because of the feedback signal.. Many a times, the output response achieved is not smooth and also may have steady state error. Thus, the transient and steady state response can be improved by using a control action of transfer function G_c(s) as shown in figure below.

Proportional Controller: Transfer function of a proportional controller is given as, G_c(s) = K_p. Proportional controller is usually an amplifier with gain K_p. It is used to vary the transient response of the control system. One cannot determine the steady state response by changing K_p. Steady state response depends on the type of the system. However, maximum overshoot is increased in this case.

Integral Controller: Transfer function of a Integral controller is given as, G_c(s) = K₁/s. It is used to decrease the steady state error by increasing the type of the system. However, stability decreases in this case.

Derivative Controller: Transfer function of a derivative controller is given as, G_c(s) = K_D. s. It is used to increase the stability of the system by adding zeros, steady state error increases, as type of the system decreases in this case.

Proportional + Integral (PI) Controller: Transfer function of PI controller is given as, G_c(s) = (K_p + K₁/s). It is used to decrease the steady state error without effecting stability,, as a pole at origin and a zero are added. In P+1 controller, order of a system increases, i.e. it converts a second order system to third order.

Proportional + Derivative (PD) Controller: Transfer function of a PD controller is given as, G_c(s) = (K_D.s + K_p). It is used to increase the stability without effecting the steady state error. Here type of the system is not changed and a zero is added.

Proportional + Integral + Derivative (PID) Controller: Transfer function of a PID controller is given as, G_c(s) = (K_p + K₁/s + K_D. s) =  It is used to decrease the steady state error and to increase the stability as one pole at origin and two zeros are added. One zero compensates the pole and other zero will increase the stability. Hence response is faster and highly accurate.

8. State Variable Analysis

Introduction: The analysis of control system, carried out till now using transfer function approach etc, assumes that system is initially at rest and system is single input single output (SISO) type. Hence the state-space approach is used to overcome above disadvantages and this approach is performed by writing differential equation in time domain and by suitably choosing state variables.

Advantages of State Variable Analysis:

• This method along with the output gives the information about the state of the system at some predetermined point along the flow of the state in state space.
• Used for linear as well as non-linear, time invariant or time varying systems.
• Analysis of multi-input-multi-output systems is less complex.
• Analysis is done by considering initial conditions.
• More accurate than transfer function.
• Organization of the state variables is easily amendable to the solution through digital computers.
• Can be used both for continuous time systems as well as discrete time systems with the same formulation.

Here u₁(t) and u₂(t) are inputs, x₁(t) and x₂(t) are state variables, y₁(t) and y₂(t) are outputs.

State Space Representation: Consider a differential equation,

State-space representation of above is obtained as below,
Let x(t) = x₁ and
∴
In matrix form,

Here x₁(t) and x₂(t) are called state variables. The n dimensional state variables are elements of n dimensional space called state space.

State Variable: The smallest set of variables, which determine the state of the dynamic system, are called the state variables.

State: It is the smallest set of state variables, the knowledge of these variable at t = t₀ together with the input completely determines the behavior of the system for any time t > t₀.

State Equation: Consider a system described as below,

State - space model can be described as below based on above state equation.

Where X(t) = State vector,
dX/dt = Rate of change of state vector,
U(t) = input vector,
A = System matrix or Evolution matrix
B = Control matrix

Output Equation: For the system described above, let the output equations be

By representing these in matrix form,

Where
y(t) is output vector,

C is observation matrix
D is transmission matrix

Different State - Space Representations
Direct Decomposition: In direct decomposition, the matrix A is of Bash's phase variable form as below,

If λ1,λ2 ,....λ_n are eigen values of A, eigen vector matrix, P can be represented as.

Cascade Decomposition: Here given system is converted into multiple systems in cascade and direct decomposition is performed to each of these sub-systems.

Parallel Decomposition: Here the given system transfer function is split into partial fractions first and by considering direct decomposition of each of the sub-system (partial fraction terms) in parallel, parallel decomposition can be performed.

State Transition Matrix: The transition matrix is defined as a matrix that satisfies the linear homogeneous state equation.

For t ≥ 0,

Here   is called state transition matrix.

Properties of the State Transition Matrix: The state transition matrix ∅(t) possesses the following properties

Time Response: Given a state space representation of a control system, the time response for any generic input and initial conditions contains the following.

Zero Input Response: Only initial conditions are considered and input is considered to be zero.
X(t) = ∅(t) X (0)

Zero State Response: Only input functions are considered and initial conditions are zero.

Total Response: Total response can be described as,

Transfer Matrix of System: Consider a MIMO described by,
X' = AX + BU
Y = CX + DU
Transfer matrix of system is given as G (s) = C(SI - A)^-1 B + D

Controllability of Linear Systems: A system is said to be controllable, if there exists an input to transfer the state of system from any given initial state X(t_i) to any final state X(t_f) in a finite time (t_f - t_i) ≥ 0. The condition of controllability depends on the coefficient matrices A and 13 of the system.

Kalman Test for Controllability: For the system to be completely state controllable, it is necessary and sufficient that the following matrix Q_c has a rank n, where n is order of A.
Q_c = [ B : A B : A²B : ........... A^n-1B]
For the system to be controllable, rank of Q_c should be n or |Q_c|≠0.

Observability of Linear Systems: A system is said to be observable if it's possible to get information about state variables from the measurements of the output and input.

Kalman Test for Observability: For the system to be completely observable, it is necessary and sufficient that the following composite matrix Q_o has a rank of n, where n is order of A.
Q_o = [C^T : A^TC^T : (A^T)²C^T : ........ (A^T)^n-1C^T].
For the system to be observable, rank of Q_o should be n or |Q₀| ≠ 0.