State space analysis Classical Vs modern control theory

The development of control system analysis and design can be divided into three eras. In the first era, we have classical control theory , which deals with techniques developed before 1950. classical control embodies such methods as , Bode, Nyquist and Routh- Hurwitz. These methods have in common the use of transfer functions in the complex frequency(s) domain, emphasis on the use of graphical techniques, the use of and the use of simplifying assumptions to approximate the time response . since computers were not available at that time , a great deal of emphasis was placed on developing methods that were amenable to manual computation and graphics. A major limitation of classical control methods was the use of single –input , single output (SISO) methods. Multivariable (i.e multiple – input multiple – output or MIMO) systems were analyzed and designed one loop at a time . Also the use of transfer functions and the frequency domain limited one to linear time – invariant systems.

In the second era , we have modern control ( which is not so modern any longer ) which referees to state – space methods developed in the late 1950,s and early 1960s. In modern control, system models are directly written in the time domain. Analysis and design are done in time domain. It should denoted that before Laplace transforms and transfer functions became popular in the 1920s Engineering were studying systems in the time domain. Therefore the resurgence of time domain analysis was not unusual, but it was triggered by the development of computers and advances in numerical analysis. Because computers were available, it was no longer necessary to development analysis and design methods that were strictly manual. An Engineer could use computers to numerically solve or simulate large system that were nonlinear and time varying. State space methods removed the previously mentioned limitations of classical control. The period of the 1960s was the heyday of modern control.

System representation in state variable form

This chapter introduce the concept of state variable and the various means of representing control systems in state variable form. Each method of state variable representation results in a system description interms of ‘n’ first –order differential equations, as opposed to the usual nth –order equation. A convenient tool for this new system representations is matrix notation.

System state and state variable

It is important to stress at out set that the concept of system state is first of all, a physical concept. However it is often convenient to interms of mathematical model. Here this mathematical model is assumed to consist of ordinary differential equations which have a unique solution for all inputs and initial conditions. It is in terms of this mathematical model that the ‘system state’ or simply ‘state’ is defined. Definition The state of system at any time ‘t0’ is the minimum set of numbers X1 (to) ,X2(to) ------Xn(to) which along with the input to the system for t ≥ to is sufficient to determine the behaviour of the system for all t ≥ to.

In other words, the state of a system represents the minimum amount of information that we need to know about a system at to such that its future behaviour can be determined without reference to the input before to.

The idea of state is familiar from a knowledge of the physical world and the means of solving the differential equations used to model the physical world . Consider a ball flying through the air. Intuitively we feel that if we know the ball’s position and velocity, we also know its future behaviour. It is on this basis that an outfielder positions himself to catch a ball. Exactly the same information is needed to solve a differential equation model of the problem. Consider for example the second order differential equation

· · · X +aX + bX = f(t)

The solution to this equation be found as the sum of the forced response, due to f(t) and the natural or unforced response ie the solution of homogeneous equation

· · · X +aX + bX = 0

If X1 (t), X2(t) ------Xn(t) are state variables of the system chosen then the initial conditions of the state variables plus the u(t)’s for t > 0 should be sufficient to decide the future behaviour i.e y(t)’s for t > 0. Note that the state variables need not be physically measurable or observable quantities. Practically however it is convenient to chose easily measurable quantities. The number of state variable is then equal to the order of the differential equation which is normally equal to the number of energy storage elements in the system.

State equations for linear systems in matrix form

The state of a linear time –invariant nth order system is represented by the following set of ‘n’ number of first order differential equations with constant coefficients in terms of n state variable X1,X2------Xn. · X1 = a11 X1+a12 X2 + ------+a1n Xn + b11U1 + ------+b1m U m · X2 = a21 X1+a22 X2 + ------+a21 Xn + b21U1 + ------+b2m U m

· · Xn = an1 X1+an2 X2 + ------+ann Xn + bn1U1 + ------+bnm U m In matrix form the above equations may be written as

X1 a11 a12 ------a1n x1 b11 b12 ------b1m u1

u2 X2 a21 a22 ------a2n x2 b21 b22 ------b2m

= +

a a ------a u Xn n1 n2 nn xn bn1 bn2 ------bnm m

X = A X + B U X is called derivative of state vector whose size is (nx1) · X is called state vector whose size is (n x 1)

A is called system matrix whose size is (n x n)

B is called input matrix whose size is ( n x m)

U is called input vector whose size is (m x 1)

Output equation The state variables X1(t) ------X n(t) represents the dynamic state of a system. The system output/ outputs may be used as some of the state variables themselves ordinarily, the output variables Y1 = C11 X1 + C12 X2 + ------C1n Xn Y2 = C21 X1 + C22 X2 + ------C2n Xn

Yp = Cp1 X1 + Cp2 X2 + ------CPn Xn In matrix form,

y1 c11c 12 ------c1n x1

y2 c21 c22 ------c2n x2

=

yp cp1 cp2 ------cpn xn

or Y = C X Y = output vector of size ( P x 1) C = Transmission matrix of size ( P x n) X = State vector of size ( n x 1)

Sometimes the output is a function of both state variables and inputs . for this general case Y = CX +DU or Y1 c11 c12 ------c1n x1 D11 D12 ------D1m u1

u2 Y2 c21 c22 ------c2n x2 D21 D22 ------D2m D1m D2m = +

D D ------D Yp cpn ------cpn xn P1 P2 Pm um D matrix is of size ( p x m) State Model

The state equation of a system determines its dynamic state and the output equation gives its output at any time t > 0 provided the state at t = 0 and the control forces for t ≥ 0 are known. These two equations together form the state model of the system. The state model of a linear system is therefore given by · X = AX +BU (1 ) Y = CX+ DU

State Model of SISO linear and time invariant system.

If we let m =1 and p=1 in the state model of a multiple input multiple output linear time invariant system we obtain the following state model for SISO linear system.

· X = AX +bu (2 ) Y = CX+ du

Where b and C are ( nx1) vectors

State Model using phase –variables ( BUSH FORM)

Let us now consider how the state model defined by equation (2) may be obtained for an nth order SISO system whose describing differential equation relating output y with input u is given by

n-1 n-2 dny d y d y dy + an-1 + an-2 + ------a1 + a0 y = b0 u (3) dtn dtn-1 dtn-2 dt

where an-1 , an - 2 ------a1, a0 are constant coefficients

dy dn-1y y(0), (0), ------(0), are initial conditions dt + dt +

To arrive at the state model of equation (3) it is rewritten in shorthand form as

n n-1 n-2 · (4) y +an-1 y + an-2 y +------a1y + a0y = b0u We first define the state variables

X1, X2, ------Xn which can be done in many possible ways. A convenient way is define

X1 = y · X2 = y

· n-1 Xn = y

With the above definition of state variables equation (4) is reduced to a set of ‘n’ first order differential equations given below; · · X1 = y = X2 · · · X2 = y = X3

· n-1 Xn-1 = y = Xn

· n X = y = -a0X1 – a1X2 – a2X3 -----a n-1Xn +b0u · the above equations result in the following state equations

·

It is to be noted that the matrix A has a special form. It has all 1’s in the upper off – diagonal, its last row is comprised of the negative of the coefficients of the original differential equation and all other elements are zero. This form of matrix A is known as the ‘BUSH FORM’ . The set of state variables which yield ‘BUSH FORM’ for the matrix A are called ‘Phase variables’ When ‘A’ is in ‘BUSH FORM’, the vector b has the specialty that all its elements except the last are zero. In fact A and B and therefore the state equation can be written directly by inspection of the linear differential equation.

The output being y = X1 , the output equation is given by

Where C = [ 1 0 ------0]

Note: There is one more state model called canonical state model . we shall consider this model after going through .

Derivation of transfer function from a given state model

Having obtained the state model we next consider the problem of determining transfer function from a given state model of SISO / MIMO systems.

1) SISO SYSTEM

u(s) y(s) G(s)

G(s) is called transfer function defined as

y(s) G(s) = ...... y(s) = u(s) G(s) or y (s) = G(s) u(s) ------(1) u(s)

State Model is given by

X (t) = AX (t) + Bu ------(2) Y(t) = CX (t) + Du ------(3)

Taking laplace transformation on both sides of equations ( 2) and (3) and neglecting initial conditions We get sX(s) = AX (s) + Bu(s) -----(4) Y(s) = CX (s) + Du(s) ---- (5)

From (4) (sI –A ) X(s) = Bu(s)

Or X(s) = ------(sI –A )-1Bu(s) ------(6)

Substituting ( 6) in (5)

Y(s) = C (sI –A )-1Bu(s) + Du(s)

Y(s) = ( C (sI –A )-1B+D) u(s) ------(7)

Comparing ( 7 ) with (1)

G(s) = C( sI –A ) -1B + D ------( 8)

An important observation that needs to be made here is that while the state model is not unique, the transfer function is unique. i.e. the transfer function of equation (8) must work out to be the same irrespective of which particular state model is used to describe the system.

(ii) MIMO SYSTEM

y (s) u1(s) 1 G(s) u2(s) y2(s) m inputs P out puts

um(s) yn(s)

G(s) = C( sI –A ) -1B + D

Where y(s) =G(s) U(s) G(s) matrix is called transfer matrix of size (p x m) y(s) matrix is of size (px1) u(s) matrix is of size (mx1) y1(s) = G 11(s) u1(s) + G 12(s) u2(s) + ------+ G 1m(s)um(s)

y1(s)

Transfer function G 11(s) = u1(s)

u2(s) = u3(s) = ----- um(s) = 0

Similarly G 12(s), ------are defined

Derivative of state models from transfer function

More often the system model is known in the transfer function form. It therefore becomes necessary to have methods available for converting the transfer function model to a state model. The process of going from the transfer function to the state equations is called decomposition of the transfer function. In general there are three basic ways of decomposing a transfer function in direct decomposition, parallel decomposition, and cascaded decomposition has its own advantage and is best suited for a particular situation.