1

Linear Systems (ME5401/EE5101) Lecturers:

A/Prof. C. J. Ong Prof. Q.G. Wang Rm: EA-05-27 Rm: E1-08-15 Tel : 6516-2217 Tel: 6516-2282 Email: mpeongcj@nus.edu.sg Email: elewqg@nus.edu.sg Meeting Time: Tuesday 6:00pm to 8:50pm Meeting Place: LT3

References :

1. Linear System Theory and Design by C.T.Chen. 3rd Edition, Published by Oxford University Press 1999.

2. Linear Systems by T. Kailath. Published by Prentice Hall 1980. 3. Modern Control Theory by W.Brogan, 3rd Edition, Prentice-Hall,

1991.

4. Fundamental of State Space Linear Systems, by John Bay, First Edition, McGraw Hill, 1999.

5. Modern Control Engineering by K.Ogata, 2nd Edition. Published

by Prentice Hall. 1990 6. Control System Design by B.Friedland. Published by McGraw Hill

1987.

2

Outline of course :

1. Introduction to Linear Control System 2. Review of Linear Algebra 3. State-space solutions and properties 4. Controllability and Observability 5. Canonical Forms and Realizations 6. Stability 7. Pole Placement 8. Quadratic Optimal Control 9. Decoupling Control 10. State Estimation 11. Servo Control

Additional Requirement: Project. (25%)

3

Journals on Control Systems

1. ASME Journal of Dynamic Systems, Measurement and Control. 2. IEEE Transactions on Automatic Control 3. AIAA Journal of Guidance, Control and Dynamics 4. IEEE Transactions on Control Systems Technology 5. Automatica 6. Control Theory and Practice 7. IEEE Control System Magazine 8. International Journal of Control 9. Control Theory and Advanced Technology 10. IEEE Transactions on Robotics and Automation 11. Journal of Robotic Systems 12. International Journal of Robotic Research 13. International Journal of Robotics and Automation 14. Robotica 15. Robotics and Autonomous Sytems 16. IEEE Robotics and Automation magazine. 17. Systems and Control Letters 18. SIAM Journal on Control and Optimization

4

Chapter 1. Introduction to Linear Control Systems

1.1 Review of Classical Control Systems Definition: A Control System is a system in which some physical quantities are controlled by regulating certain energy inputs. A system is a group of physical components assemblied to perform a specific function. It may be electrical, mechanical, thermal, biomedical, chemical, pneumatic or any combination of the the above. A physical quantity may be temperature, pressure, electrical voltage, mechanical position or velocity, liquid level, etc. Some applications of automatic control • Space vehicle systems • Missile Guidance Systems • Aircraft autopilot systems • Robotic systems • Weapon control systems • Power systems • Home appliances: aircons, refrigerators, dryers, microwave ovens etc. • Automatic assembly line • Chemical processes – controlling pressures, temperature,humidity, pH values,

etc. • Manufacturing systems – CNC machines, machine tool control, EDM

control, etc • Quality control and Inspection of manufactured parts

5

The following are some of the features of Classical Control Theory: 1. It uses extensively the Laplace operator in the description of a system.

2. It is based on input/output relationship called transfer function. In

computing the transfer function of a system, all initial conditions are set to zero.

3. It was primarily developed for single-input-single-output systems,

although extension to multi-input and multi-output is now possible. 4. It can only describe linear time-invariant differential equation systems. 5. Based on methods like frequency response, root locus etc. 6. Its formulation is not very well suited for digital computer simulation. The following are some of the features of Modern Control Theory 1. Uses the time domain representation of the system known as the state

space representation. 2. Can be used to describe multi-input-multi-output systems. Extensions to

linear time varying, nonlinear and distributed systems are easy and straight forward.

3. The formulation incorporates initial conditions into descriptions. 4. It is based on time domain approach and is therefore, very well suited for

computer simulations. 5. Provides a complete description of the system.

6

Controlled variable (output)

Sensed output

Open-loop control systems Typically represented in block diagram form

Figure 1.1

• Output has no effect on the control action.

• Examples: washing machine, traffic lights, microwave oven, dryers, toasters, etc.

Closed-loop control systems Typically represented as:

Figure 1.2

• Actuating signal, and hence output, depends on the input and the current state of the output.

• Examples are servo controlled motor, missile, robots etc.

Laplace transformation

Controller Controlled process

Reference (input) Actuating signal

Sensors

+

-

Error signal

Controller Controlled process

Reference (input)

Actuating signal Controlled variable (output)

7

Given a function f(t) where t ≥ 0. Let s be the complex variable defined as s = σ + j ω where σ , ω are variables. Then the Laplace transform of f(t), denoted as F(s), is F(s) = L[f(t)] = ∫

∞0

-)( dtetf st Existence: The Laplace transform of a function f(t) exists if the integral converges. Properties of Laplace transform

1. L [Af(t)] = A L [f(t)]; where A is a constant. 2. L [A1f1(t) +A2f2(t)] = A1 L [f1(t)] + A2 L [f2(t)]; where A1,A2 are constants. Transfer Function: Consider the system given by

where

)()()()(

)()()()(

11

1

10

11

1

10

trbtrdtdbtr

dtdbtr

dtdb

tcatcdtdatc

dtdatc

dtda

mmm

m

m

m

nnn

n

n

n

++++

=++++

−−

−

−−

−

LL

LL

with n ≥ m.

• The transfer function of a linear, time-invariant, differential equation

system is defined

][][

)()()(

inputLoutputL

sRsCsG == .

12

21

10

12

21

10

nnnnn

mmmmm

asasasasabsbsbsbsb

++++++++++

=−

−−−

−−

LL

LL

• In deriving a transfer function, all initial conditions are assumed zero.

Other definitions:

system

Input r(t) Output c(t) G(s)

R(s) C(s)

8

Proper transfer function: The transfer function G(s) is called proper if n ≥ m (i.e. G(s) evaluated at s = ∞ is a constant) Strictly proper transfer function: The transfer function G(s) is called strictly proper if n > m (i.e. G(s) evaluated at s = ∞ is zero) Improper transfer function: The transfer function G(s) is called improper if n < m (i.e. G(s) evaluated at s = ∞ is infinite) Order of a system: The highest power of “s” in the denominator of the transfer function G(s) is called the order of the system. Poles of a system: The roots of the denominator of the transfer function G(s) are called the poles of the system. Zeros of a system: The roots of the numerator of the transfer function G(s) are called the zeros of the system.

e.g. )23(

22)( 2

2

++++

=ssssssG

Here n = 3 and m = 2. So the transfer function G(s) is strictly proper. Order of the system is 3. Poles of the system are s = 0, s = −2 , s = −1. Zeros of the system are s = −1 + j1, s = −1 − j1. When is a SISO system stable? Ans: depends on the definition of stability. Simplest answer: all poles have negative real parts.

9

1.2 The problem with Classical Control Consider the following system

Figure 1.2.1

where the plant is unstable given by

11)(−

=s

sH f and the compensator is chosen

to be 11)(

+−

=sssH C . Then, the overall transfer function is given by

Figure 1.2.2

11)()(

)()(

+==

ssHsH

sVsY

cf , which is BIBO stable.

Will this work?? If we let )()(2 tytx = , then from Figure 1.2.1, we have

)()()( 22 tutxtx +=& . (1.2.1)

Similarly, from Figure 1.2.1, we have 11

)()(

+−

=ss

sVsU , or )(

12)()( sV

ssVsU

+−= .

If we let )(1

2)(1 sVs

sx+

−= , then

)(2)()( 11 tvtxtx −=+& . (1.2.2)

If we include the initial conditions of (1.2.1) and (1.2.2), it can be shown that

10

∫ −−= −−t

t dtveextx0

11 )(2)0()( τττ

and

τττ dtvexeexetxtyt

ttt ∫ −+−+== −−

012

122 )()0()()0()()(

Therefore, because of the term et, unless the initial conditions can be guaranteed to be zero, or )0(2)0( 21 xx −= , the output will grow without bounds! Some further insight into the problem can be obtained by studying the situation in which the two blocks are interchanged. Consider the situation where )(sH c follows )(sH f rather than precede it as shown below.

In this case, the overall system 1

1)()()()(

+==

ssHsH

sVsY

cf is again BIBO stable

and has the same overall transfer function as shown in Figure 1.2.2.

In this case, let )()(1 sUsx = and )(1

2)( 12 sxs

sx+

−= . Then,

)()()( 21 txtxty += .

By incorporating the initial conditions, it can be shown that

∫ −+=t

t detvextx0

11 )()0()( ττ τ ,

∫∫ −−−+−+= −−−tt

ttt dtvedtveeexextx00

122 )()())(0()0()( ττττ ττ

and

11

∫ −++= −−t

t dtveexxty0

21 )())0()0(()( τττ

Now the system is stable as far as y(t) goes, even if the initial conditions are nonzero. However the realization is internally unstable because )(1 tx and )(2 tx have terms that will grow as te , thus after awhile the realization will saturate or burn out. Conclusion: The internal behaviour of a system is more complicated than is indicated by the external behaviour. It was the state equation analysis in the examples that first clarified these and related questions.

12

1.3 Mathematical Descriptions of Systems Class of systems The class of systems studied in this course has

• Some input terminals and output terminals. • If an excitation or input is applied, a unique response or output can be

measured at the output terminals.

Figure 1.3.1

• A system with only one input terminal and one output terminal is termed a

single-variable system or single-input-single-output (SISO) system. • A system with two or more input terminals/or two or more output

terminals is called a multivariable system (MIMO) system. Continuous time system

• A system is called a continuous-time system if it accepts continuous-time signals as its input and generates continuous-time signals as its output.

• If the input has one terminal, u(t) denotes a scalar quantity, otherwise

u(t)=[u1 u2 … up]T denotes a p×1 vector. Causality and Lumpedness

• A system is called memoryless system if its output y(t0) depends only on the input applied at t0; it is independent of the input applied before or after t0.

• Most systems, however, have memory. By this, we mean that y(t0)

depends on u(t) for t < t0, t = t0 and t > t0.

• A system is called causal or nonanticipatory system if its current output depends on past and current inputs but not on future input. If a system is non-causal, then its current output will depend on future input. No

13

physical system has such capability. ( This corresponds to the n < m condition in Classical transfer function representation.)

• Current output of a causal system is affected by its past inputs. How far

back in time will the past input affect the current output? In general, the time should go all the way back to -∞. However, tracking u(t) for -∞ ≤ t ≤ t0 is not convenient. The concept of state can deal with this problem.

Definitions: State: The state x(t0) of a system at time t0 is the information at t0 that, together with the input u(t) for t > t0 determines uniquely the output y(t) for all time t > t0. State Variable: The state variables of a dynamical system are the smallest set of variables that determine the state of the system. If at least n variables x1, …, xn are needed to completely describe the behavior of a dynamical system, then such n variables are a set of state variables. State Vector: The n state variables can be considered as the elements or components of the n-dimensional vector x(t). Such a vector x(t) is called a state vector. State Space: State space is defined as the n-dimensional space in which the components of the state vector represent its' coordinate axes. State Trajectory: The path produced in the state space by the state vector x(t) as it changes with the passage of time.

• With the concept of state, it is now easy to track the output of the system as only x(t0) and u(t) for t ≥ t0 are needed.

• A system is said to be lumped if its number of state variables is finite or

its state is a finite vector. A system is said to be distributed if its state has infinitely many state variables.

14

1.4 Linear Systems Based on the above definition of state, let us now denote the triplet {x(t0), u(t) for t ≥ t0, y(t) for t ≥ t0} as the output y(t) for t ≥ t0 using u(t) for t ≥ t0 with state x(t0) at time t0. Definition: A system is called a linear system if for every t0 and any pair of state-input-output triplets {x1(t0), u1(t) for t ≥ t0, y1(t) for t ≥ t0} and {x2(t0), u2(t) for t ≥ t0, y2(t) for t ≥ t0}, we have

{x1(t0)+ x2(t0), u1(t)+u2(t) for t ≥ t0, y1(t)+ y2(t) for t ≥ t0} (additivity) and

{αx1(t0), αu1(t) for t ≥ t0, αy1(t) for t ≥ t0} (homogeneity) The above two properties (additivity and homogeneity) can be combined into one as

{α1x1(t0)+ α2x2(t0), α1u1(t)+ α2u2(t) for t ≥ t0, α1y1(t)+ α2y2(t) for t ≥ t0} for any two real constants α1 and α2.

(superposition) • A system is called a nonlinear system if the superposition property does

not hold. If u(t) = 0 for all t ≥ t0, the output will be excited exclusively by the initial state x(t0) and is denoted as yzi(t). This output is called the zero-input response and the triplet is denoted as {x(t0), 0, yzi(t) for t ≥ t0}. If x(t0) = 0, the output will be excited exclusively by the input and is denoted as yzs(t). This output is called the zero-state response and the triplet is denoted as { 0, u(t) for t ≥ t0 , yzs(t) for t ≥ t0}. Using the additivity property, it can be shown that

y(t) = yzi(t) + yzs(t).

• Thus, the response of every linear system can be decomposed into the zero-state response and the zero-input response. These two responses can be studied separately and their sum yields the complete response.

15

Input-Output Description

We now develop a mathematical equation to describe the zero-state response of a linear system.

• Assume that the initial state is zero and consider a SISO system.

Let δΔ(t-t1) be the pulse function as shown in Figure below

Δ+1t1t

Δ

Δ

Figure 1.4.1

where the pulse is of width Δ and height 1/Δ located at time t1. Thus,

δΔ(t− t1) Δ =1 for t1≤ t ≤ t1+Δ, and δΔ(t− t1) Δ = 0 for all other t. and

Δ−≈ Δ∑ )()()( ii

i ttδtutu (1.4.1)

as shown in Figure 1.4.2.

Let gΔ(t, ti) be the output at time t excited by the pulse u(t)= δΔ(t− ti) applied at time ti. Then, by linearity we have Input Output δΔ ( t− ti) gΔ ( t, ti ) δΔ ( t− ti) u(ti) Δ gΔ ( t, ti ) u(ti) Δ (homogeneity) ∑i δΔ ( t− ti) u(ti) Δ ∑i gΔ ( t, ti ) u(ti) Δ (additivity)

16

Thus, the output y(t) excited by the input u(t) can be approximated by

∑ Δ= Δi

ii tuttgty )(),()( (1.4.2)

Let Δ → 0, and denote limΔ→0 δΔ(t− ti) as δ(t− ti), and

limΔ→0 gΔ ( t, ti ) as g( t, ti ). Then

• approximation (1.4.1) becomes an equality, • the summation of (1.4.2) becomes an integration, • the discrete ti becomes a continuum, denoted by τ and (1.4.2) becomes

∫∞

∞−

= τττ dutgty )(),()( (1.4.3)

• g( t, τ ) refers to the output observed at time t due to impulse applied at

time τ and hence, is referred to as the impulse response. By causality, g( t, τ ) = 0 for all t < τ . This implies that the upper limit of the integral of (1.4.3) becomes t. A system is said to be relaxed at t0 if its initial states at t0 is 0. In this case, the output y(t) for t ≥ t0 is excited exclusively by the input u(t) for t ≥ t0. Hence, for a relaxed causal system, (1.4.3) becomes

∫=t

tdutgty

0

)(),()( τττ (1.4.4)

• The condition of lumpedness is not used in the above derivation. Hence,

any lumped or distributed linear system has such an input-output description.

If a linear system has p input terminals and q output terminals, then (1.4.4) can be extended to

17

∫=t

tdutGty

0

)(),()( τττ

where

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

),(),(),(

),(),(),(),(),(),(

),(

21

22221

11211

τττ

ττττττ

τ

tgtgtg

tgtgtgtgtgtg

tG

qpqq

p

p

L

MMM

L

L

where ),( τtgij is the response at time t at the ith output due to an impulse applied at time τ on the jth input. • G(t, τ ) is known as the impulse response matrix of the system.

18

State-space description Every linear lumped system can be described by a set of equation of the form

)()()()()( tutBtxtAtx +=& (1.4.5) )()()()()( tutDtxtCty += (1.4.6)

where the overdot denotes differentiation with respect to time t, and pqnqpnnn RtDRtCRtBRtA ×××× ∈∈∈∈ )(,)(,)(,)( and 11, ×× ∈∈ pn RuRx . Here, the system consists of two sets of equations, collectively known as the state-space equation. The first (1.4.5) is a set of n first-order differential equations with p inputs. The second set of equation (1.4.6) consists of q algebraic equations relating the state x(t) and input u(t) to the output y(t). Linear Time-Invariant (LTI) systems A system is said to be time invariant if for every triplet {x(t0), u(t) for t ≥ t0, y(t) for t ≥ t0} and any scalar T, y(t) = ŷ( t− T ) where ŷ is the solution of the same system with initial state x(t0) at time t0+T and input u(t−T) for t ≥ t0+T. Hence, for LTI system, the output is the same no matter when the input is applied as long as the initial state and the input are the same.

• Without loss of generality, t0 is chosen to be 0. If the system is time-invariant, then the impulse function g(t,τ) is

),(),( TTtgtg ++= ττ and for any T. In particular, if T= −τ ,

)()0,(),( τττ −=−=++ tgtgTTtg The equation (1.4.4) then becomes

19

∫∫ −=−=tt

dtugdutgty00

)()()()()( ττττττ (1.4.7)

• Unlike the time-varying case where g is a function of two variables, g is

now a function of a single variable. • The integration above is known as the convolution integral, used

extensively in Classical control system analysis.

• The result of (1.4.7) can be easily extended to the MIMO case in which the Impulse matrix is G(t−τ).

State-space equation From (1.4.5) and (1.4.6), if )(),(),(),( tDtCtBtA are not function of time, then the state-space system describes a linear time-invariant system given by

)()()( tButAxtx +=& (1.4.8) )()()( tDutCxty += (1.4.9)

where again pqnqpnnn RDRCRBRA ×××× ∈∈∈∈ ,,, and 11, ×× ∈∈ pn RuRx . Nonlinear System Most physical systems are nonlinear and time varying and some of them can be described by nonlinear differential equation of the general form

),,(),,(

),,(),,(

1

tuxhytuxf

tuxftuxfx

n

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡== M&

(1.4.10)

where f and h are nonlinear functions. The behavior of system described by such equations can be very complicated. Some nonlinear systems can be approximated by linear equations under certain conditions.

20

Suppose for some input function u0(t) and some initial states, x0(t) is the solution of (1.4.10). Then

)),(),(()( 000 ttutxftx =&

Suppose the input is perturbed slightly to become u0(t)+δu(t) and the initial state

is also perturbed only slightly in the form of x0(t)+δx(t). Then we can linearize

the above equation about (u0(t), x0(t) ) using Taylor’s series expansion.

L

&&

+++=

++=+

uufx

xfttutxf

ttututxtxfxx

tutxtutx

δ∂∂δ

∂∂

δδδ

)(),()(),(00

000

0000

)),(),((

)),()(),()((

If we neglect higher order terms, we have

uufx

xfx

tutxtutx

δ∂∂δ

∂∂δ

)(),()(),( 0000

+=&

This is of the form

utBxtAx δδδ )()( +=& with A(t), B(t) given by

A(t) =

)(,)(21

2

1

2

1

2

1

1

1

)(,)(

00

00

tutxn

nnn

n

n

tutx

xf

xf

xf

xf

xf

xf

xf

xf

xf

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

L

OMM

L

L

and

21

B(t) =

)(,)(21

2

1

2

1

2

1

1

1

)(,)(

00

00

tutxp

nnn

p

p

tutx

uf

uf

uf

uf

uf

uf

uf

uf

uf

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

L

MMMM

MM

L

The non-linear output equation

)),(),(()( 000 ttutxhty = can be similarly linearized to give the approximate output

utDxtCty δδδ )()()( += with

)()()( 0 tytyty −=δ

C(t) =

)(,)(21

2

1

2

1

2

1

1

1

)(,)(

00

00

tutxn

qqq

n

n

tutx

xh

xh

xh

xh

xh

xh

xh

xh

xh

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

L

OMM

L

L

and

D(t) =

)(,)(21

2

1

2

1

2

1

1

1

)(,)(

00

00

tutxp

qqq

p

p

tutx

uh

uh

uh

uh

uh

uh

uh

uh

uh

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

L

MMMM

MM

L

22

Examples of LTI system: Example 1. Mass-Spring-Damper System

From Newton’s laws, we have

)()()()(2

2

tkzdt

tdzbtfdt

tzdm −−=

Choose

zxzx&=

=

2

1

then

))(( 121

2

21

kxbxtfmzx

xzx

−−==

==−&&&

&&

i.e.

fm

xmbx

mkx

xx

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−=

=

1212

21

&

&

or, in matrix form

fmx

x

mb

mk

xx

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−=⎥

⎦

⎤⎢⎣

⎡ 1010

2

1

2

1

&

&

If we are interested in finding the position of the mass, then the output y(t) is

[ ] ⎥⎦

⎤⎢⎣

⎡=

2

101)(xx

ty

23

More compactly we can write,

)()()()()()(tDutCxtytButAxtx

+=+=

where

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−=

mb

mkA

10;

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

mB 1

0; [ ] .0;01 == DC

If we are interested in the force acting on the spring, then the output y(t) will become

[ ] fxx

kkxty .00)(2

11 +⎥

⎦

⎤⎢⎣

⎡==

If we are interested in both the force acting on the dashpot as well as the force acting on the spring, then the output becomes

fxx

kb

tyty

t .00

0)()(

)(y2

1

2

1 +⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

or if we want to know about the restoring forces acting on the mass, then the output is

[ ] fxx

bkty .0)(2

1 +⎥⎦

⎤⎢⎣

⎡=

24

Example 2 : Electrical Circuit

Let x1(t) be the voltage across the capacitor x2(t) be the current in the circuit. By Kirchoff’s law, we have

)()()()( 12

2 txdt

tdxLtxRtv +⋅+⋅=

)(1)( 21 txc

tx =&

Rearranging, we have

vLx

x

LR

L

cx

x

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡10

1

10

2

1

2

1

&

&

Suppose we are interested in the voltage of the inductor, then let y(t) denote the

output variable, we have

vL

LxLRx

LLxL

dttdxLty 1)1()()( 212

2 +−−=== &

vRxx +−−= 21

or

[ ] vx

xRty +

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−=

2

11)(

25

If we denote the voltage across the resistor as x3, then x3=Rx2. The state

equations can also be written in terms of x1 and x3 as

)(1)(1)( 321 txcR

txc

tx ==&

vL

xLRx

Ltx

Rtx 11)(1)( 2132 +−−== &&

or vLRx

RLRx

LRtx +⋅−−= 3

2

131)(& .

Expressed in matrix form, we have

vLR

x

x

LR

LR

cRx

x

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ 010

3

1

3

1

&

&

Suppose we are still interested in the voltage across the inductor, we see that

vxLRx

LR

RLx

RL

dttdxLty +−−=== )()()( 313

2 &

vxx +−−= 31

[ ] vx

xty +

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−=

3

111)(

26

Example 3: DC Motor

On the electrical side, by KVL,

e(t) – R i(t)= vb (KVL)

τ(t) = ki i(t) (Torque-current relation)

vb(t) = k2 ω(t) (back emf)

On the mechanical side, using Newton’s law

dtdJ ωτ =

Hence, we have,

)(

)(1

2ωω

ω

keRk

dtdJ

veR

kikdtdJ

i

bii

−=

−==

eJRk

JRkk

eRk

RkkJ

ii

ii

+−=

+−=

ωω

ωω

2

2

&

&

Since dtdθω = , we have

ωθ =&

eJRk

JRkk ii +−= ωω 2&

27

or

eJRk

JRkk ii ⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ 0

0

102

ω

θ

ω

θ

&

&

Example 4: Quarter Car Model

Let v1(t) = velocity of the mass

u(t) = velocity of the road profile

x1(t) = displacement of spring

Then,

uvx −= 11&

)( 11111 uvbkxxbkxvm −−−=−−= &&

Hence, we have

uvx −= 11&

umbv

mbx

mkv −−−= 111&

or

umb

v

x

mb

mk

v

x

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ 110

1

1

1

1

&

&

Suppose we are interested in the forces in the spring, then

28

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡=

1

10

v

xky

Suppose we are interested in the velocity of the car, then we can choose the

output equation as

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡=

1

110

v

xy

The above examples illustrate that the choice of the state variables may not be

obvious and some particular choice maybe more insightful than others. In fact,

state-space models are not unique! If there is one state-space model of a system,

there is an infinite number of representations. More on this later.