Advanced Control Engineering - Chapter 8 ppsx

8.1.2 The state vector differential equationThe state of a system is described by a set of first-order differential equations in terms of the state variables x1, x2,.. Determine a The st

State-space methods for control system design

8.1 The state-space-approachThe classical control system design techniques discussed in Chapters 5±7 are gener-ally only applicable to

(a) Single Input, Single Output (SISO) systems(b) Systems that are linear (or can be linearized) and are time invariant (haveparameters that do not vary with time)

The state-space approach is a generalized time-domain method for modelling, lysing and designing a wide range of control systems and is particularly well suited todigital computational techniques The approach can deal with

ana-(a) Multiple Input, Multiple Output (MIMO) systems, or multivariable systems(b) Non-linear and time-variant systems

(c) Alternative controller design approaches

8.1.1 The concept of stateThe state of a system may be defined as: `The set of variables (called the statevariables) which at some initial time t0, together with the input variables completelydetermine the behaviour of the system for time t  t0'

The state variables are the smallest number of states that are required to describethe dynamic nature of the system, and it is not a necessary constraint that they aremeasurable The manner in which the state variables change as a function of timemay be thought of as a trajectory in n dimensional space, called the state-space.Two-dimensional state-space is sometimes referred to as the phase-plane when onestate is the derivative of the other

8.1.2 The state vector differential equation

The state of a system is described by a set of first-order differential equations in terms

of the state variables (x1, x2, , xn) and input variables (u1, u2, , un) in the

The equations set (8.1) may be combined in matrix format This results in the state

vector differential equation

Equation (8.2) is generally called the state equation(s), where lower-case boldface

represents vectors and upper-case boldface represents matrices Thus

x is the n dimensional state vector

x1

x2

xn

266

37

u is the m dimensional input vector

u1u2

um

266

37

A is the n  n system matrix

a11 a12 a1na21 a22 a2n

an1 an2 ann

266

37

B is the n  m control matrix

b11 b1m

b21 b2m

bn1 bnm

266

37

In general, the outputs ( y1, y2, , yn) of a linear system can be related to the statevariables and the input variables

Equation (8.7) is called the output equation(s)

Example 8.1Write down the state equation and output equation for the spring±mass±dampersystem shown in Figure 8.1(a)

SolutionState variables

P(t) Ky C _y ˆ myor

From equations (8.9), (8.10) and (8.11) the set of first-order differential equations are

_x1ˆ x2

and the state equations become

_x1_x2

 

m

Cm

24

24

Example 8.2

For the RCL network shown in Figure 8.2, write down the state equations when

(a) the state variables are v2(t) and _v2

(b) the state variables are v2(t) and i(t)

and the state equations are

_x1_x2

 

LC

RL

24

3

5 x1x2

 

‡ 01LC

24

 

L

RL

24

24

3

Example 8.3For the 2 mass system shown in Figure 8.3, find the state and output equation whenthe state variables are the position and velocity of each mass

SolutionState variables

x1ˆ y1 x2ˆ _y1x3ˆ y2 x4ˆ _y2System outputs

y1, y2System inputs

X

Fyˆ m2y2

From (8.24), (8.25) and (8.26), the four first-order differential equations are

_x1ˆ x2_x2ˆ K1

m1

K2m1

x1 C1m1x2‡

K2m1x3‡

1m1u_x3ˆ x4

_x4ˆK2m2x1

K2m2x3

(8:27)

Hence the state equations are

_x1_x2_x3_x4

264

37

5 ˆ

K1‡ K2m1

Cm1

37775

37

5 ‡

01m100

266

37

and the output equations are

y1y2

 

ˆ 1 0 0 00 0 1 0

x2x3x4

264

37

8.1.3 State equations from transfer functionsConsider the general differential equation

dny

dtn‡ an 1ddtn 1n 1y‡


‡ a1dydt‡ a0y ˆ bn 1ddtn 1n 1u‡


‡ b1dudt‡ b0u (8:30)Equation (8.30) can be represented by the transfer function shown in Figure 8.4.Define a set of state variables such that

_x1 ˆ x2_x2 ˆ x3

_xn 1xn

2666

377

3775

x1x2

xn 1xn

2666

377

7‡

00

01

2664

377

xn

2666

377

State equation

_x1_x2_x3

24

3

5 ˆ 00 10 01

24

3

5 xx12

x3

24

3

5 ‡ 001

24

The state equation is the same as (8.35) The output equation is

y ˆ [ 4 7 5 ] xx12

x3

24

where the integral term in equation (8.41) is the convolution integral and  is a

dummy time variable Note that

eatˆ 1 ‡ at ‡a2!2t2‡


‡ak!ktk (8:42)

Consider now the state vector differential equation

8.2.1 Transient solution from a set of initial conditions

Example 8.6

For the spring±mass±damper system given in Example 8.1, Figure 8.1, the state

equations are shown in equation (8.13)

_x1_x2

 

m

Cm

24

3

5 x1x2

 

‡ 01m

24

3

Given: m ˆ 1 kg, C ˆ 3 Ns/m, K ˆ 2 N/m, u(t) ˆ 0 Evaluate,

(a) the characteristic equation, its roots, !nand 

(b) the transition matrices f(s) and f(t)

(c) the transient response of the state variables from the set of initial conditions

y(0) ˆ 1:0,_y(0) ˆ 0Solution

Since x1ˆ y and x2ˆ _y, then x1(0) ˆ 1:0, x2(0) ˆ 0

Inserting values of system parameters into equation (8.53) gives

_x1_x2

Using the standard matrix operations given in Appendix 2, equation (A2.12)

F(s) ˆ

(s ‡ 3)(s ‡ 1)(s ‡ 2)

1(s ‡ 1)(s ‡ 2)2

(s ‡ 1)(s ‡ 2)

s(s ‡ 1)(s ‡ 2)

264

37

37

Hence

x1x2

 

ˆ (2e2(ett ee2t2t)) ( e(e tt‡ 2ee 2t)2t)

10

 

(8:64)x1(t) ˆ (2e t e 2t)

The time response of the state variables (i.e position and velocity) together with thestate trajectory is given in Figure 8.5

Example 8.7For the spring±mass±damper system given in Example 8.6, evaluate the transientresponse of the state variables to a unit step input using

(a) The convolution integral(b) Inverse Laplace transformsAssume zero initial conditions

11(t ) 12(t )

21(t ) 22(t )

1m

24

x1x2

(b) An alternative method is to inverse transform from an s-domain expression

Equation (8.45) may be written

–1

1

Fig 8.5 State variable time response and state trajectory for Example 8.4.

Hence from equation (8.61)X(s) ˆ F(s) 00

37

7 01

 1

Simplifying

X(s) ˆ

1s(s ‡ 1)

12

2s(s ‡ 2)

1s(s ‡ 1)‡

2s(s ‡ 2)

264

37

Equation (8.74) is the same as equation (8.69)

The step response of the state variables, together with the state trajectory, is shown

The continuous-time solution of the state equation is given in equation (8.47) Ifthe time interval (t t0) in this equation is T, the sampling time of a discrete-time

system, then the discrete-time solution of the state equation can be written as

recursive discrete-time simulation of multivariable systems

The discrete-time state transition matrix A(T) may be computed by substituting

T ˆ t in equations (8.49) and (8.50), i.e

or

A(T) ˆ I ‡ AT ‡A22!T2‡


‡Akk!Tk (8:78)Usually sufficient accuracy is obtained with 5 < k < 50

The discrete-time control matrix B(T) from equations (8.75) and (8.76) is

BTPut T within the brackets

kˆ0

AkTk‡1(k ‡ 1)!

BHence

B(T) ˆ IT ‡AT2!2‡A23!T3‡


‡A(k ‡ 1)!kTk‡1

Example 8.8 (See also Appendix 1, examp88.m)

(a) Calculate the discrete-time transition and control matrices for the

spring-mass-damper system in Example 8.6 using a sampling time T ˆ 0:1 seconds

(b) Using the matrix vector difference equation method, determine the unit step

response assuming zero initial conditions

Solution(a) The exact value of A(T ) is found by substituting T ˆ t in equation (8.62)

kT ˆ 0

x1(0:1)x2(0:1)

ˆ 0:9910:172 0:7330:086

00

ˆ 0:9910:172 0:7330:086

0:004530:0861

ˆ 0:9910:172 0:7330:086

0:0330:192

u(t) ˆ t Determine

(a) The state and output equations

(b) The transition matrix F(s)

(c) Expressions for the time response of the state variables

 

‡ 01

 u

1(s ‡ 1)(s ‡ 1)1

(s ‡ 1)(s ‡ 1)

s(s ‡ 1)(s ‡ 1)

264

375

8.4 Control of multivariable systems 8.4.1 Controllability and observabilityThe concepts of controllability and observability were introduced by Kalman (1960)and play an important role in the control of multivariable systems.

A system is said to be controllable if a control vector u(t) exists that will transferthe system from any initial state x(t0) to some final state x(t) in a finite time interval

A system is said to be observable if at time t0, the system state x(t0) can be exactlydetermined from observation of the output y(t) over a finite time interval

If a system is described by equations (8.2) and (8.7)

The system described by equations (8.87) is completely observable if the n  nmatrix

Example 8.10 (See also Appendix 1, examp810.m)

Is the following system completely controllable and observable?

_x1_x2

 

0

 u

y ˆ [ 1 1 ] x1

x2

 

SolutionFrom equation (8.88) the controllability matrix is

Equation (8.90) is non-singular since it has a non-zero determinant Also the two row

and column vectors can be seen to be linearly independent, so it is of rank 2 and

therefore the system is controllable

From equation (8.89) the observability matrix is

N ˆ C T:ATCTwhere

ATCTˆ 02 35

11

are linearly dependent since the second column is 5 times the first column and

therefore the system is unobservable

8.4.2 State variable feedback design

Consider a system described by the state and output equations

_x ˆ Ax ‡ Bu

Select a control law of the form

In equation (8.93), r(t) is a vector of desired state variables and K is referred to as the

state feedback gain matrix Equations (8.92) and (8.93) are represented in state

variable block diagram form in Figure 8.7

Substituting equation (8.93) into equation (8.92) gives

_x ˆ Ax ‡ B(r Kx)or

In equation (8.94) the matrix (A ± BK) is the closed-loop system matrix

For the system described by equation (8.92), and using equation (8.52), thecharacteristic equation is given by

The roots of equation (8.95) are the open-loop poles or eigenvalues For the

closed-loop system described by equation (8.94), the characteristic equation is

The roots of equation (8.96) are the closed-loop poles or eigenvalues

Regulator design by pole placementThe pole placement control problem is to determine a value of K that will produce

a desired set of closed-loop poles With a regulator, r(t) ˆ 0 and therefore equation(8.93) becomes

u ˆ KxThus the control u(t) will drive the system from a set of initial conditions x(0) to a set

of zero states at time t1, i.e x(t1) ˆ 0

There are several methods that can be used for pole placement

(a) Direct comparison method: If the desired locations of the closed-loop poles(eigenvalues) are

then, from equation (8.96)

ˆ sn‡ n 1sn 1‡


‡ 1s ‡ 0 (8:99)Solving equation (8.99) will give the elements of the state feedback matrix

(b) Controllable canonical form method: The value of K can be calculated directly using

k ˆ [0 a0:1 a2: :n 2 an 2:n 1 an 1]T 1 (8:100)where T is a transformation matrix that transforms the system state equation intothe controllable canonical form (see equation (8.33))

Fig 8.7 Control using state variable feedback.

where M is the controllability matrix, equation (8.88)

377

Note that T ˆ I when the system state equation is already in the controllable

canonical form

(c) Ackermann's formula: As with Method 2, Ackermann's formula (1972) is a direct

evaluation method It is only applicable to SISO systems and therefore u(t) andy(t) in equation (8.87) are scalar quantities Let

where M is the controllability matrix and

where A is the system matrix and iare the coefficients of the desired closed-loopcharacteristic equation

Example 8.11 (See also Appendix 1, examp811.m)

A control system has an open-loop transfer function

Y

U(s) ˆ

1s(s ‡ 4)When x1ˆ y and x2ˆ _x1, express the state equation in the controllable canonical form

Evaluate the coefficients of the state feedback gain matrix using:

(a) The direct comparison method

(b) The controllable canonical form method

(c) Ackermann's formula

such that the closed-loop poles have the values

s ˆ 2, s ˆ 2Solution

From the open-loop transfer function

Equation (8.106) provides the output equation and (8.107) the state equation

_x1_x2

(s ‡ 2)(s ‡ 2) ˆ 0or

k1ˆ 4

(b) Controllable canonical form method: From equation (8.100)

M ˆ [B:AB]

AB ˆ 00 14

01

Thus proving that equation (8.108) is already in the controllable canonical form

Since T 1 is also I, substitute (8.118) into (8.116)

(A) ˆ A2‡ 1A ‡ 0I

Equations (8.124) requires that all state variables must be measured In practice thismay not happen for a number of reasons including cost, or that the state may notphysically be measurable Under these conditions it becomes necessary, if full statefeedback is required, to observe, or estimate the state variables.

A full-order state observer estimates all of the system state variables If, however,some of the state variables are measured, it may only be necessary to estimate a few

of them This is referred to as a reduced-order state observer All observers use someform of mathematical model to produce an estimate ^x of the actual state vector x.Figure 8.8 shows a simple arrangement of a full-order state observer

In Figure 8.8, since the observer dynamics will never exactly equal the systemdynamics, this open-loop arrangement means that x and ^x will gradually diverge Ifhowever, an output vector ^y is estimated and subtracted from the actual outputvector y, the difference can be used, in a closed-loop sense, to modify the dynamics ofthe observer so that the output error (y ^y) is minimized This arrangement, some-times called a Luenberger observer (1964), is shown in Figure 8.9

Let the system in Figure 8.9 be defined by

Assume that the estimate ^x of the state vector is

where Keis the observer gain matrix

If equation (8.127) is subtracted from (8.125), and (x ^x) is the error vector e, then

and, from equation (8.127), the equation for the full-order state observer is

Thus from equation (8.128) the dynamic behaviour of the error vector depends upon

the eigenvalues of (A KeC) As with any measurement system, these eigenvalues

should allow the observer transient response to be more rapid than the system itself(typically a factor of 5), unless a filtering effect is required.

The problem of observer design is essentially the same as the regulator poleplacement problem, and similar techniques may be used

(a) Direct comparison method: If the desired locations of the closed-loop poles(eigenvalues) of the observer are

s ˆ 1, s ˆ 2, , s ˆ nthen

∫

0 y

Fig 8.9 The Luenberger full-order state observer.

(b) Observable canonical form method: For the generalized transfer function shown

in Figure 8.4, the observable form of the state equation may be written

_x1_x2

_xn

266

37

377

x1

x2

xn

266

37

7‡

b0

b1

bn 1

266

377u

y ˆ [ 0 0 0 1 ]

x1x2

xn

266

377

(8:131)

Note that the system matrix of the observable canonical form is the transpose of the

controllable canonical form given in equation (8.33)

The value of the observer gain matrix Kecan be calculated directly using

Keˆ Q

0 a0

1 a1

n 1 an 1

266

37

Q is a transformation matrix that transforms the system state equation into the

observable canonical form

where W is defined in equation (8.102) and N is the observability matrix given in

equation (8.89) If the equation is in the observable canonical form then Q ˆ I

(c) Ackermann's formula: As with regulator design, this is only applicable to systems

where u(t) and y(t) are scalar quantities It may be used to calculate the observergain matrix as follows

Keˆ (A)N 1[ 0 0 0 1 ]T

or alternatively

Keˆ (A)

CCA

CAn 1

264

375

1 00

1

264

37

where (A) is defined in equation (8.104)

Example 8.12 (See also Appendix 1, examp812.m)

A system is described by

_x1_x2

y ˆ [ 1 0 ] x1

x2