Introduction• Classical design methods (root locus and frequency response method)

requires that the physical systems be modeled in the form of transfer function .

• Limitations: a transfer function is only defined under zero initial condition. transfer function model is only applicable to linear time invariant

systems and their too it is generally restricted to single input single output systems.

it reveals only the system output for a given input and provide no information regarding the interval state of the system.

These methods are trial and error procedures so difficult to visualize and organize even in moderately complex systems and may not lead to a control system which yields an optimum performance in some defined sense

Why State Space Model ?• It provides information about the state of the system variables at some

predetermined points along the flow of signals along with the output.• It is a Direct time domain approach which provides basis for modern

control theory and system optimization.• It is a powerful technique for the analysis and design of linear and

nonlinear, time invariant or time varying MIMO systems• In transform domain analysis, Laplace transform is needed for continuous time system and z-transform is needed for discrete time systems but state variable techniques are same for both linear continuous time systems and linear discrete time systems.

Concepts Of State, State variables And State Model

• A mathematical abstraction to represent or model the dynamics of a system utilizes three types of variables called the input , the output and the state variables.

• Consider the mechanical system shown in below fig a .

We observe that the displacement x(t) (output variable) at any time t≥ be computed if we know the applied force F(t) from t=,provided v() the initial velocity and x() the initial displacement are known. At t= we conceive the initial velocity and initial displacement of the system . The state of the system of the fig at any time t is given variables x(t) and v(t) which are called state variables of the system

Definition of state and state variable

The State of a dynamic system is a minimal set of variables (known as state variables) such that knowledge of these variables at t = t0 together with the knowledge of the inputs for t >= t0 completely defines the behavior of the system for t > t0.

In state variable representation state variables are represented as x1(t),x2(t),…….., inputs by u1(t),u2(t)……, and output by y1(t),y2(t)………… .

Structure of general control system

For simple mechanical system the state variable representation is given by two first order differential equation 1 and 2,solution of these equation 3 and 4 gives the two state variable x(t) and u(t) of the system . for general system of above figure, the state representation can be arranged in the form of n first order differential equations.

Thus the n state variables and hence the state of the system can be determined uniquely at any t>to if each state variable is known at t=to and all the m control forces are known through out the interval t to to.

The ‘n’ differential equations may be written in vector notations as

STATE MODEL OF LINEAR SYSTEMS

• Where x(t) is n x 1 state vector , u(t) is m x 1 input vector,A is n x n system matrix define by

• A=And B is n x m input matrix defined

B=

Similarly , the output variable at time t are linear combination of the value of the input and state variables at time t, i.e

(t). .. .(t)Where the coefficients are constants. This set of equation may be put in the vector matrix form

Where y(t) is p x 1 Output vector , C is p x n output matrix define by

C=

And D is p x m transmission matrix define by

D=

v

Now = 2 = 2

= -2==

= -+

We immediately observe that 12 c and 12 d given an alternative state variable model of the systems previously represented by 12 a and 12 b.

We , therefore conclude that the state variables of a system are non unique.

Though the state model of a system is not unique , however ,

all such models have one characteristic is common for a given systems , namely the number of elements in the state vector is equal and minimal. This number n is referred to as the order of the system .

The state model for a linear time-varying systems is of the same form as define in eqs.11 except for the fact that the coefficient of the matrices A,B,C and D are no more constants but are functions of time.

STATE MODEL FOR SINGLE INPUT SINGLE OUTPUT LINEAR SYSTEMS

Where B and C are now respectively (n x 1)and (1 x n)

matrices, d is a constant and u is a scaler control variables. The block diagram representation of this state model is shown in fig 4.

STATE MODEL FOR LINEAR CONTINUOUS TIME SYSTEMS

State space representation using Physical Variables

• We shall consider state variable for a simple electrical system which is an RLC network shown in Fig .

• The network has three energy storage elements: a capacitor C and two inductors . If we have a knowledge of initial conditions and the input signal e(t) for tthen the behaviour of the network is completely specified for t.however ,if one of the initial conditions is not known, we are unable to determine the complete response of the network to a given input.

State space representation using Phase Variables

Phase variable formulations for transfer function with poles and zeros

From the result given in (eq 29),(eq 30) and (eq 36) we observe that the phase variable formulation can be obtained by inspection from the transfer function and vice versa.

A disadvantage of phase variable formation is that the phase variable , in general , are not physical variable of the system and therefore are not available for measurement and control purposes .In spite of the disadvantage phase variables provide a powerful method of state variable formation. A link between Transfer function and time domain design approach is established through phase variables.

State space Representation Using Canonical variables

Observation. Notice the dotted block in matrix A . Therefore, x1 and x2, are not in decoupled form (because of repeated roots at s=-1). This block is Known as JORDAN BLOCK. This is the simplest decoupled form as far as possible.

Derivation of Transfer Function from state model.For a general state model . x = Ax + Bu y = Cx + DuThe transfer function may be obtained as follows.Taking the Laplace transform of the above equations we have sX(s) – x(0) = Ax(s) + BU(s) Y(s) = CX(s) + dU(s)Solving for Y(s) we obtain Y(s) = C(sI – A) x(0) + C(sI - A) BU(s) + dU(s)Assuming zero initial conditions, we get the system transfer function as T(s) = Y(s)/U(s) = C(sI - A) B + d = C adj(sI - A)B + d ………….(*) det(sI - A)

An important observation that needs to be is that while the state model is non unique the transfer function is unique. Setting the denominator of eq(*) we get I sI – A I = 0 STATE VARIABLES AND LINEAR DISCREET TIME

SYSTEMSThe general form of state model for a multivariable discreet time system x[(k + 1)T] = f[x(kT), u(kT)] …………(*) y(kT) = g[x(kT), u(kT)] …………(**)Where x(kT) state vector u(kT) input vector y(kT) output vector.From these equations we see that given the initial state x(0) and the values of inputs u(0), u(T), u(2T)…………u(kT); we can uniquely deduce the evolution of state x(T), x(2T)………….x[(k+1)T] and evolution of outputs y(0)………y(kT) . Also note that the above equation is a set of first order difference equations.

For an nth order linear time invariant system the equations (*)and (**) reduce to

x(k + 1) = Ax(k) + Bu(k); x(kT) y(k) = Cx(k) + Du(k) x(k) = n x 1 state vector u(k) = m x 1 input vector y(k) = p x 1 output vector A = n x n system matrix B = n x m input matrix C = p x m output matrix D = p x m transmission matrix

State models from linear differential equations Z transfer functions

This is the canonical state model . When some of the poles of T(z) are repeated we get the Jordon canonical form. The method of obtaining this form is identical to that of continous case.

Derivation of Z transfer function from discreet time state model

We know the equation x = Ax + Bu y = Cx + DuTaking Z transform on both sides of the above equation. zX(z) –zx(0) = AX(z) + BU(z)Solving for X(z) we get X(z) = (zI – A)-1 zx(0) + (zI - A)-1 BU(z)Taking the Z transform of the above equation of the above

equation Y(z) = CX(z) + dU(z)

Solving for Y(z) we obtain Y(z) = C(zI - A)-1 zx(0) + C(zI - A)-1 BU(z) + dU(z)Assuming zero initial conditions, T(z)= C(zI-A)-1 B + d = C adj(zI-A)B + d I zI-A ISetting the denominator equal to zero I zI – A I = 0The roots of the characteristic equation are the eigen

values of Matrix A.

Diagonalization

• We observed in earlier sections that the state model of a system is not unique.

• From application point of view physical variables for system representation are most useful as resulting state variable are real physical variables which can be easily measured and used for control purposes .

• However the corresponding state model in this case generally not convenient for investigation of system properties and evaluation of time response.

Solution of state equationsWe shall develop methods for solution of the state equationFrom which the system transient response can then be obtained.First consider the classical method considering first orderdifferential equation. dx = ax ; x(0) = dtThis equation has the solution

The comparison of vector coefficients of equal powers of t a1 = Aa0

a2 = (½)Aa1 = (½!)A2 a0

ai = (1/i!) Ai a0

In the assumed solution equating x(t=0)= x0 , we find that

a0 = x0

The solution x(t) is thus found to be x(t) = (I + At + (½!)A2 t 2+…………..(1/i!)Ai t i+.. )x0

Each of the terms inside brackets is an nxn matrix. Because of thesimilarity of the entity inside the brackets with the scalar exponent ofeq(**) we call it a matrix exponential which can be written as. eAt = I + At + (½!)A2t2 + ……………………(1/i!)Aiti +…….

The solution X(t) can now be written as x(t) = eAt x0 …………………………..(***)

From eq (***) it is observed that the initial state x0 at t=0 is driven to a

State x(t) at time t. this transition in state is carried out by the matrixexponent eAt . Because of this property , eAt is known as state transitionmatrix and is denoted by Ø(t).

Let us now determine the solution of the non homogenous state eq ẋ(t) = Ax(t) + Bu(t) ; x(0) = x0

Rewrite this eq in this form ẋ(t) – Ax(t) = Bu(t)Multiplying both sides by e-At we can write e-At(ẋ(t) – Ax(t)) = (d/dt)[e-Atx(t)] = e-AtBu(t)Integrating both sides with respect to t between limits 0 to t , we get.

e-Atx(t) = e-AtBu(Ʈ)dƮ e-Atx(t) – x(0) = e-AtBu(Ʈ)dƮ

Now pre multiplying both side by eAt, we have

x(t) = eAtx(0) + eA(t-Ʈ)Bu(Ʈ)dƮ ………..(****)

homogenous solution forced solution

If the initial state is known at t = t0 rather than at t = 0,

eq (****) becomes. x(t) = eA(t-t0)x(t) + eA(t-t0)Bu(Ʈ)dƮ

Properties of the State Transition Matrix :

Computation of state transition matrixComputation by Laplace transform.Let us consider an unforced system. ẋ(t) = Ax where A is a constant matrixTaking the laplace transform of this equation sX(s) – x(0) = AX(s)Where X(s) is the laplace transform of the unforced response and x(0) isthe initial condition vector . The above equation may be rearranged as [sI - A]X(s) = x(0) X(s) = [sI - A]-1 x(0)Taking the inverse laplace transform we get x(t) = L-1 [(sI - A) -1 ]x(0)Where x(t) is the unforced response of the system this solution must beObviously identical with the one obtained earlier . The computationyields a different approach to determine the state transition matrix givenbelow.

ɸ(t) = eAt = L-1[(sI -A)-1] = L-1ɸ(s) ɸ(s) = (sI - A)-1 is called the resolvent matrix.Let us consider now the response when the controlForce vector u is applied. The state equation for thiscase is ẋ = Ax +BuPerforming the laplace transform gives. sX(s) – x = AX(s) +BU(s) ; x0=x(0)

(sI - A)X(s) = x0 + BU(s)

Therefore X(s) = [(sI - A)-1 ]x0 + [(sI - A)-1 BU(s)]

By inverse laplace transform X(t) = L-1[(sI - A)-1 ]x0 + L-1[(sI -A)-1BU(s)]

= ɸ(t)x0 + L-1[ɸ(s)BU(s)]

Concepts of controllability and observability

Controllability:

A system is said to be completely state controllable if it impossible to transfer the system state from any initial state x() to any desired state x(t) in specified finite time by a control vector u(t).

A general nth order multi-input linear time-invariant system (with an m-dimensional control vector)

= Ax +Bu

is completely controllable if and only if the rank of the composite matrix

= [B : AB : : B]

is n. Since only matrices A and B are involved in above eqn, it can be said that pair (A, B) is controlled if rank of is n.

The composite matrix defined by

= [B : AB : B]

=

It is easily seen that det 0 i.e., its rank is r = n = 3. The system is therefore completely controllable.