fundamentals of control theory - bme eetmizsei/mikrorejegy/the mems handbook(complete)/0077_p… ·...

12Fundamentals ofControl Theory

12.1 Introduction12.2 Classical Linear Control

Mathematical Preliminaries • Control System Analysis and Design • Other Topics

12.3 “Modern” ControlPole Placement • The Linear Quadratic Regulator • Basic Robust Control

12.4 Nonlinear ControlSISO Feedback Linearization • MIMO Full-State Feedback Linearization • Control Applications of Lyapunov Stability Theory • Hybrid Systems

12.5 Parting Remarks

12.1 Introduction

This chapter reviews the fundamentals of linear and nonlinear control. This topic is particularly importantin microelectromechanical systems (MEMS) applications for two reasons. First, as electromechanicalsystems, MEMS devices often must be controlled in order to be utilized in an effective manner. Second,important applications of MEMS technology are controls-related because of the utility of MEMS devicesin sensor and actuator technologies. Because the area of control is far too vast to be entirely presented inone self-contained chapter, the approach adopted for this chapter is to outline a variety of techniquesused for control system synthesis and analysis, provide at least a brief description of their mathematicalfoundation, discuss the advantages and disadvantages of each of the techniques and provide sufficientreferences so that the reader can find a starting point in the literature to fully implement any describedtechniques. The material varies from the extremely basic (e.g., root locus design) to relatively advancedmaterial (e.g., sliding mode control) to cutting-edge research (hybrid systems). Some examples areprovided; additionally, many references to the literature are provided to help the reader find further examplesof a particular analysis or synthesis technique.

This chapter is divided into three sections, each of which considers both the stability and performanceof a control system. The term performance includes both the qualitative nature of any transient response ofthe system, reference signal tracking properties of the system and the long-term or steady-state perfor-mance of the system. The first section considers “classical control,” which is the study of single-input,single-output (SISO) linear control systems, which relies heavily upon mathematical techniques fromcomplex variable theory. The material in this section outlines what is typically covered in an elementaryundergraduate controls course. The second section considers so-called “modern control” which is thestudy of multi-input, multi-output (MIMO) control systems in state space. Included in this section is

Bill GoodwineUniversity of Notre Dame

© 2002 by CRC Press LLC

what is sometimes called “post-modern control” (Zhou, 1996) which is a study of robust system perfor-mance and stability in the presence of unmodeled system dynamics. Finally, the third section considersnonlinear control techniques. Not considered in this chapter are model-free control techniques basedupon concepts from soft computing, which are outlined in Chapter 14. Also not considered in this chapterare nonlinear, open-loop control techniques (for recent advances in this area, the reader is referred toLafferriere and Sussmann [1993]; Bullo et al. [2000]; Goodwine and Burdick [2000]).

12.2 Classical Linear Control

Classical linear control relies heavily upon mathematical techniques from complex variable theory. Thisis apparently an historical consequence of the importance of frequency analyses of feedback amplifiers,which motivated much of the development of classical control theory, as well as a consequence of thefact that convolution in the time domain is simple multiplication in the frequency domain which greatlysimplifies the analysis of the natural input/output and “block diagram” structure of many control systems.This topic is typically thoroughly covered in undergraduate controls courses. Good references includeDorf (1992), Franklin et al. (1994), Gajec and Lelic (1996), Kuo (1995), Ogata (1997), Raven (1995) andShinners (1992).

12.2.1 Mathematical Preliminaries

The main tool is the Laplace transform, which transforms the linear ordinary differential equation (ODE)into an algebraic equation, thus reducing the task of solving an ODE into simple algebra. The Laplacetransform of a function, f(t), is defined as:

(12.1)

and the inverse Laplace transform of F(s) as:

(12.2)

A discussion of extremely important mathematical details concerning convergence and the proper lowerlimit of integration is found in Ogata (1997). As a practical matter, evaluating the integrals in thedefinition of the Laplace transform and the inverse Laplace transform is rarely necessary as extensivetables of Laplace transform pairs are readily available. A few Laplace transform pairs for typical functionsare listed in Table 12.1. More complete tables can be found in any undergraduate text on classical controltheory such as the references listed previously.

Important properties of the Laplace transform are as follows:

1. Real differentiation: L[ ] = sF(s) − f(0).2. Linearity: L[αf1(t) ± βf2(t)] = αF1(s) ± βF2(s).3. Convolution: 4. Final value theorem: If all the poles of sF(s) are in the left half of the complex plane, then

A basic result from the first three properties is that to solve a linear ordinary differential equation, onecan take the Laplace transform of each side of the equation, which converts the differential equation intoan algebraic equation in s, then algebraically solve the expression for the Laplace transform of the dependentvariable, and then take the inverse Laplace transform of the resulting function.

L[f(t)] F(s) e st– f(t) td0

∞

∫= =

L 1– [F(s)] f(t)1

2πj-------- F(s)est s, for t 0>d

c− jω

c+ jω

∫= =

ddt----- f(t)

L[ f1(t τ– )f2(τ) τ]d0

t∫ F1(s)F2(s).=

f(t)t→∞lim sF(s).

s→0lim=


Example

As a simple example, consider the differential equation:

(12.3)

Taking the Laplace transform of the equation yields:

(12.4)

Algebraic manipulation gives:

(12.5)

consequently, from the table of Laplace transform pairs:

(12.6)

For more examples, see Ogata (1997), Raven (1995), Kuo (1995), Franklin et al. (1994) etc.Due to the convolution property of Laplace transforms, a convenient representation of a linear control

system is the block diagram representation illustrated in Figure 12.1. In such a block diagram represen-tation, each block contains the Laplace transform of the differential equation representing that componentof the control system that relates the input to the block to its output. Arrows between blocks indicatethat the output from the preceding block is transferred to the input of the subsequent block. The outputof the preceding block multiplies the contents of the block to which it is an input. Simple algebra willyield the overall transfer function of a block diagram representation for a system.

TABLE 12.1 Laplace Transform Pairs for Basic Functions

F(t) F(s)

1 Unit impulse, δ(t) 1

2 Unit step, 1(t)

3 t

4 tn, n = 1, 2, 3, …

5 e−at

6 tne

−at

7 sinωt

8 cosωt

9 e−at

cosbt

8 e−at

sinbt

1s--

1

s2---

n!

sn+1--------

1s a+-----------

n!

(s a)+ n+1----------------------

ωs2 ω2+----------------

s

s2 ω2+----------------

s a+(s a)+ 2 b2+-----------------------------

b

(s a)+ 2 b2+-----------------------------

x x+ 0=

x(0) 0=

x(0) 1=

s2X(s) sx(0) x(0) X(s)+–– 0=

X(s)1

s2 1+-------------=

x(t) (t)sin=


Example

The transfer function for the system illustrated in Figure 12.2 can be computed by observing that:

(12.7)

and

(12.8)

which can be combined to yield

(12.9)

A more complete exposition on block diagram algebra can be found in any of the previously citedundergraduate texts. Note that the numerator and denominator of the transfer function will typically bepolynomials in s. The denominator is called the characteristic equation for the system. From the table ofLaplace transform pairs, it should be clear that if the characteristic polynomial has any roots with apositive real part, then the system will be unstable, as it will correspond to an exponentially increasingsolution (see entry 5 in Table 12.1). Given a reference input, R(s), determining the response of the systemis straightforward: Multiply the transfer function by the reference input, perform a partial fractionexpansion (i.e., expand):

(12.10)

where each term in the sum on the right-hand side of the equation is something in the form of one ofthe entries in Table 12.1. In this manner, the contribution to the response of each individual term canbe determined by referring to a Laplace transform table and can be superimposed to determine the overallsolution:

(12.11)

where each term in the sum is the inverse Laplace transform of the corresponding term in the partialfraction expansion.

FIGURE 12.1 Typical block diagram representation of a control system.

FIGURE 12.2 Generic block diagram including transfer functions.

InputController Process

Output

Sensor

Actuator

−+

R(s) E(s)C(s) P(s)

Y(s)

S(s)

A(s)

−+

E(s) R(s) Y(s)S(s)–=

Y(s) E(s)C(s)A(s)P(s)=

Y(s)R(s)----------- C(s)A(s)P(s)

1 C(s)A(s)P(s)S(s)+-----------------------------------------------------=

Y(s)R(s)C(s)A(s)P(s)

1 C(s)A(s)P(s)S(s)+-----------------------------------------------------

C1

s p1–------------

C2

s p2–------------ … Cn

s pn–-------------+ + += =

y(t) y1(t) y2(t) … yn(t)+ + +=


Example

For the block diagram in Figure 12.2 if C(s) = , A(s) = 1, P(s) = , S(s) = 1 and R(s) = (a unitstep input), then:

(12.12)

where ωd = ωn Referring to the table of Laplace transform pairs (Table 12.1), and assuming thatζ < 1,

(12.13)

12.2.2 Control System Analysis and Design

Control system analysis and design considers primarily stability and performance. One approach to theformer is briefly dispensed with first. The stability of a system with the closed-loop transfer function(note that in such a case a controller has already been specified):

(12.14)

is determined by the roots of the denominator, or characteristic equation. In fact, it is possible todetermine whether or not the system is stable without actually computing the roots of the characteristicequation. A necessary condition for stability is that each of the coefficients ai appearing in the characteristicequation be positive. Because this is only a necessary condition, if any of the ai are negative, then thesystem is unstable, but the converse is not necessarily true: Even if all the ai are positive, the system maystill be unstable. Routh (1975) devised a method to check necessary and sufficient conditions for stability.

The method is to construct the Routh array, defined as follows:

Row n sn: 1 a2 a4 …

Row n−1 sn−1

: a1 a3 a5 …

Row n−2 sn−2

: b1 b2 b3 …

Row n−3 sn−3

: c1 c2 c3 …

Row 2 s2: ∗ ∗

Row 1 s1: ∗

Row 0 s0: ∗

1s--

ωn2

s 2ζωn+--------------------- 1

s--

Y(s)ωn

2

s(s2 2ζωns ωn2)+ +

---------------------------------------------=

1s--

s 2ζωn+s2 2ζωns ωn

2+ +-------------------------------------–=

1s--

s ζωn+(s ζωn)+ 2 ωd+-------------------------------------

ζωn

(s ζωn)+ 2 ωd+-------------------------------------––=

1s--

s ζωn+(s ζωn)+ 2 ωd+-------------------------------------

ζωn

ωd

---------ωd

(s ζωn)+ 2 ωd+-------------------------------------––=

1 ζ2.–

y(t) 1 eζωnt–

(ωdt)cosζ

1 ζ2–------------------ (ωdt)sin+

–=

T(s)b0sm b1sm−1 … bm+ + +

sn a1sn−1 … an+ + +-------------------------------------------------------=

… … … … … …


in which the ai are from the denominator of Eq. (12.14); bi and ci are defined as:

The basic result is that the number of poles in the right-half plane (i.e., unstable solutions) is equal tothe number of sign changes among the elements in the first column of the Routh array. If they are allpositive, then the system is stable. When a zero is encountered, it should be replaced with a small positiveconstant, ε, which will then be propagated to lower rows in the array. Then, the result can be obtainedby taking the limit as ε → 0.

Example

Construct the Routh array and determine the stability of the system described by the transfer function:

(12.15)

The Routh array is

(12.16)

so the system is stable, as there are no sign changes in the elements in the first column of the array.One aspect of performance concerns the steady-state error exhibited by the system. For example, from

the time-domain solution from the example above, it is clear that as t → ∞, y(t) → 1. However, the finalvalue theorem can be utilized to determine this without actually solving for the time domain solution.

Example

Determine the steady-state value for the time-domian function, y(t), if its Laplace tranfrorm is given byY(s) = . Because all the solutions of have a negative real part, all

the poles of sY(s) lie in the left half of the complex plane. Therefore, the final value theorem can beapplied to yield:

(12.17)

which clearly is identical to the limit of the time domain solution as t → ∞.

b1

det1 a2

a1 a3

a1

----------------------------– b2

det1 a4

a1 a5

a1

----------------------------– b3

det1 a6

a1 a7

a1

----------------------------–= = =

c1

deta1 a3

b1 b2

b1

----------------------------– c2

deta1 a5

b1 b3

b1

----------------------------– c3

deta1 a7

b1 b4

b1

---------------------------- .–= = =

Y(s)R(s)----------- 1

s4 4s3 9s2 10s 8+ + + +------------------------------------------------------=

s4: 1 9 8

s3: 4 10 0

s2:(10 36)––

1------------------------- 26= (0 32)––

1----------------------- 32= 0

s1:(128 260)––

4------------------------------- 33= 0 0

s0:(0 1056)––

26---------------------------- 40.6= 0 0

ωn2

s(s2 2ζωns ωn2 )+ +-------------------------------------------- s2 2ζωns ωn

2+ + 0=

y(t)t→∞lim sY(s)

s→0lim s

s→0lim

ωn2

s(s2 2ζωns ωn2)+ +

--------------------------------------------- 1= = =


12.2.2.1 Proportional–Integral–Derivative (PID) Control

Perhaps the most common control implementation is so-called PID control, where the commandedcontrol input (the output of the “controller” box in Figures 12.1 and 12.2) is equal to the sum of threeterms: one term proportional to the error signal (the input to the “controller” box in Figures 12.1 and12.2), the next term proportional to the derivative of the error signal and the third term proportional tothe time integral of the error signal. In this case, from Figure 12.2, D(s) = KP + + Kds, where KP is theproportional gain, KI is the integral gain and Kd is the derivative gain. A simple analysis of a second-ordersystem shows that increasing KP and KI generally increases the speed of the response at the cost of generallyreducing stability; whereas, increasing Kd generally increases damping and stability of the response. WithKI = 0, there may be a nonzero steady-state error, but when KI is nonzero the effect of the integral controleffort is to typically eliminate steady-state error.

Example—PID Control of a Robot Arm

Consider a robot arm illustrated in Figure 12.3. Linearizing the equations of motion about θ = 0 (theconfiguration in Figure 12.3) gives:

(12.18)

where I is the moment of inertia of the arm, m is the mass of the arm, θ is the angle of the arm and uis a torque applied to the arm. For PID control,

(12.19)

If we set I = 1 and m = 1/g, the block diagram representation for the system is illustrated in Figure 12.4.Thus, the closed-loop transfer function is

(12.20)

Figure 12.5 illustrates the step response of the system for proportional control (KP = 1, KI = 0, Kd = 0),PD control (KP = 1, KI = 0, Kd = 1) and PID control (KP = 1, KI = 1, Kd = 1). Note that for proportional

FIGURE 12.3 Robot arm model.

KI

s-----

I θ mgθ+ u=

u Kp(θdesired θactual)–= Kd(θdesired θactual)– KI (θdesired θactual– ) td0

t

∫+ +

T s( )Kds2 Kps KI+ +

s3 Kds2 (Kp 1+ )s KI+ + +-------------------------------------------------------------=


control and PD controls, there is a final steady-state error, which is eliminated with PI control (also notethat both of these facts could be verified analytically using the final value theorem). Finally, note thatthe system response for pure proportional control is oscillatory, whereas with derivative control theresponse is much more damped.

The subjects contained in the subsequent sections consider controller synthesis issues. For PID controllers,tuning methods exist and the interested reader is directed to the undergraduate texts cited previously orthe papers by Ziegler and Nichols (1942; 1943).

12.2.2.2 The Root Locus Design Method

Although, as mentioned above in the discussion of PID control, various rules of thumb can be determinedto relate system performance to changes in gains, a systematic approach is clearly more desirable. Becausepole locations determine the characteristics of the response of the system (recall the partial fractionexpansion), one natural design technique is to plot how pole locations change as a system parameter orcontrol gain is varied [Evans, 1948; 1950]. Because the real part of the pole corresponds to exponentialsolutions, if all the poles are in the left-half plane, the poles closest to the jω-axis will dominate the systemresponse. In particular, if we focus a second-order system of the form:

(12.21)

FIGURE 12.4 Robot arm block diagram.

FIGURE 12.5 PID control response.

K + K s + K s

1

s +12 p d I

θ actualθdesired

+−

Time (sec.)

Am

plitu

de

Step Response

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

1.2

1.4

Proportional ControlPD Control PID Control

H(s)ωn

2

s2 2ζωns ωn2+ +

-------------------------------------=


the poles of the system are as illustrated in Figure 12.6. The terms ωn, ωd and ζ are the natural frequency,the damped natural frequency and the damping ratio, respectively. Multiplying H(s) by (unit step) andperforming a partial fractions expansion give:

(12.22)

so the time response for the system is

(12.23)

where ωd = ωn and 0 ≤ ζ < 1. Plots of the response for various values of ζ are illustrated inFigure 12.7. Clearly, referring to the previous equation and Figure 12.7, if the damping ratio is increased,the oscillatory nature of the response is increasingly damped.

Because the natural frequency and damping are directly related to the location of the poles, one effectiveapproach to designing controllers is to pick control gains based upon desired pole locations. A root locus

FIGURE 12.6 Complex conjugate poles, natural frequency, damped natural frequency and damping ratio.

FIGURE 12.7 Step response for various damping factors.

Im(s)

Re(s)

x

x

ωd ωn

ζωn

Time (sec.)

Am

plitu

de

Step Response

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1s--

Y(s)1s--

s ζωn+(s ζωn)+ 2 ωn

2(1 ζ2)–+---------------------------------------------------------

ζωn

(s ζωn)+ 2 ωn2(1 ζ2)–+

---------------------------------------------------------––=

y(t) 1 eζωnt–

ωdtcosζ

1 ζ2–------------------ ωdtsin+

–=

1 ζ2–


plot is a plot of pole locations as a system parameter or controller gain is varied. Once the root locushas been plotted, it is straightforward to pick the location on the root locus with the desired pole locationsto give the desired system response. There is a systematic procedure to plot the root locus by hand (referto the cited undergraduate texts), and computer packages such as Matlab (using the rlocus() andrlocfind() functions) make it even easier. Figure 12.9 illustrates a root locus plot for the robot armabove with the block diagram as the single gain K is varied from 0 to ∞ as illustrated in Figure 12.8. Notethat for the usual root locus plot, only one gain can be varied at a time. In the above example, the ratioof the proportional, integral and derivative gains was fixed, and a multiplicative scaling factor is what isvaried in the root locus plot.

Because the roots of the characteristic equation start at each pole when K = 0 and approach each 0of the characteristic equation as K → ∞, the desired K can be determined from the root locus plot byfinding the part of the locus that most closely matches the desired natural frequency ωn and dampingratio ζ (recall Figure 12.7).

Typically, control system performance is specified in terms of time-domain specifications, such as risetime, maximum overshoot, peak time and settling time, each of which is illustrated in Figure 12.10. Roughestimates of the relationship between the time domain specifications and the natural frequency anddamping ratio are given in Table 12.2 [Franklin et al., 1994].

Example

Returning to the robot arm example, assume that the desired system performance is to have the systemrise time be less than 1.4 sec, the maximum overshoot less than 30%, and the 1% settling time less than10 sec. From the first row in Table 12.2, the natural frequency must be greater than 1.29, and from thethird and fourth rows the damping ratio should be greater than approximately 0.4. Figure 12.11 illustratesthe root locus plot along with the pole locations and corresponding gain, and K (rlocfind() is the

FIGURE 12.8 Robot arm block diagram.

FIGURE 12.9 Root locus for robot arm PID controller.

1+ s + 1s

1

s +12

θactualθdesired

+−

K

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

Real Axis

Imag

inar

y A

xis


Matlab command for retrieving the gain value for a particular location on the root locus), which providea damping ratio of approximately .45 and a natural frequency of approximately 1.38. Figure 12.12illustrates the step response of the system to a unit step input verifying these system parameters.

12.2.2.3 Frequency Response Design Methods

An alternative approach to controller design and analysis is the so-called frequency response methods.Frequency response controller design techniques have two main advantages: They provide good controllerdesign even with uncertainty with respect to high-frequency plant characteristics and using experimentaldata for controller design purposes is straightforward. The two main tools are Bode and Nyquist plots(see Bode [1945] and Nyquist [1932] for first-source references) and stability analyses are considered first.

A Bode plot is a plot of two curves. The first curve is the logarithm of the magnitude of the responseof the open-loop transfer function with respect to unit sinusoidal inputs of frequency ω. The second curveis the phase of the open-loop transfer function response as a function of input frequency ω. The Bode plotfor the transfer function:

(12.24)

TABLE 12.2 Time-Domain Specifications as a Function of Natural Frequency, Damped Natural Frequency and Damping Ratio

Rise time: tr ≅

Peak time: tp ≅

Overshoot: Mp =

Settling time (1%): ts =

Note: Results are from Franklin et al., 1994.

FIGURE 12.10 Time domain control specifications.

1.8ωn

-------

πωd

------

e πζ 1 ζ2–⁄–

4.6ζωn

---------

Time (sec.)

Am

plitu

de

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

Maximum Overshoot

Rise Time

Peak Time

Settling Time

Steady-State Error

G(s)1

s3 25s2 s+ +----------------------------=


is illustrated in Figure 12.13. Clearly, as the frequency of the sinusoidal input in increased, the magnitudeof the system response decreases. Additionally, the phase difference between the sinusoidal input andsystem response starts near −90° and approaches −270° as the input frequency becomes large.

An advantage of Bode plots is that they are easy to sketch by hand. Because the magnitude of thesystem response is plotted on a logarithmic scale, the contributions to the magnitude of the responsedue to individual factors in the transfer function simply add together. Similarly, due to basic facts relatedto the polar representation of complex numbers, the phase contributions of each factor simply add as well.

FIGURE 12.11 Selecting pole locations for a desired system response.

FIGURE 12.12 Robot arm step response.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

Real Axis

Ima

gin

ary

Axi

sK = 0

K Large

K = 2.5

Natural Frequency = 1.38

Damping Ratio = 0.44

Time (sec.)

Am

plitu

de (

radi

ans)

0 1.4 2.8 4.2 5.6 70

0.2

0.4

0.6

0.8

1

1.2

1.4


Recipes for sketching Bode plots by hand can be found in any undergraduate controls text, such as Franklinet al. (1994), Raven (1995), Ogata (1997) and Kuo (1995).

For systems where the magnitude of the response passes through the value of 1 only one time and forsystems where increasing the transfer function gain leads to instability (the most common, but notexclusive scenario), the gain margin and phase margin can be determined directly from the Bode plot toprovide a measure of system stability under unity feedback. Figure 12.13 also illustrates the definition ofgain and phase margin. Positive gain and phase margins indicate stability under unity feedback; con-versely, negative gain and phase margins indicate instability under unity feedback. The class of systemsfor which Bode plots can be used to determine stability are called minimum phase systems. A system isminimum phase if all of its open-loop poles and zeros are in the left-half plane.

Bode plots also provide a means to determine the steady-state error under unity feedback for varioustypes of reference inputs (steps, ramps etc.). In particular, if the low-frequency asymptote of the mag-nitude plot has a slope of zero and if the value of this asymptote is denoted by K, then the steady-stateerror of the system under unity feedback to a step input is

(12.25)

If the slope of the magnitude plot at low frequencies is –20 dB/decade and if the value where the asymptoteintersects the vertical line ω = 1 is denoted by K, then the steady-state error to a ramp input is

(12.26)

Example

Consider the system illustrated in Figure 12.2 where C(s) = A(s) = S(s) = 1 and P(s) = The Bodeplot for the open-loop transfer function P(s) = is illustrated in Figure 12.14. The low-frequencyasymptote is approximately at −6, so 20logK = −6 ⇒ K ≈ 0.5012 ⇒ yss ≈ 0.6661, where .Figure 12.15 illustrates the unity feedback closed-loop step response of the system, verifying that thesteady-state value for y(t) is as computed from the Bode plot.

FIGURE 12.13 Bode plot.

Frequency (rad/sec)

Pha

se (

deg)

; Mag

nitu

de (

dB)

−150

−100

−50

0

50

100

Gm = 27.959 dB (at 1 rad/sec), Pm = 10.975 deg. (at 0.19816 rad/sec)

10−3

10−2

10−1

100

101

102−300

−250

−200

−150

−100

−50

Positive Gain Margin

Positive Phase Margin

et→∞lim

11 K+-------------=

et→∞lim

1K---=

1 2⁄s 1+----------- .

1 2⁄s 1+-----------

yss p(t)t ∞→lim=


A Nyquist plot is a more sophisticated means to determine stability and is not limited to cases whereonly increasing gain leads to system instability. It is based on the well-known result from complex variabletheory called the principle of the argument. Consider the (factored) transfer function:

(12.27)

FIGURE 12.14 Bode plot for example problem.

FIGURE 12.15 Step response for example problem.

Frequency (rad/sec)

Pha

se (

deg)

; Mag

nitu

de (

dB)

−30

−25

−20

−15

−10

−5

10−1

100

101

−100

−80

−60

−40

−20

0

To: Y

(1)

Time (sec.)

Am

plitu

de

0 0.5 1 1.5 2 2.5 3 3.5 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

G s( )(s zi)+

i

∏(s pj)+

j

∏-------------------------=


By basic complex variable theory, ∠G(s) = Σiθi − Σj ϕj, where θi are the angles between s and the zeroszi, and the φj are the angles between s and the poles pj . Thus, a plot of G(s) as s follows a closed contour(in the clockwise direction) in the complex plane will encircle the origin in the clockwise direction thesame number of times that there are zeros of G(s) within the contour minus the number of times thatthere are poles of G(s) within the contour. Thus, an easy check for stability is to plot the open loop G(s)on a contour that encircles the entire left-half complex plane. Assuming that G(s) has no right-half planepoles (poles of G(s) itself, in contrast to poles of the closed-loop transfer function), an encirclement of–1 by the plot will indicate a right-half plane zero of 1 + G(s), which is an unstable right-half plane poleof the unity feedback closed-loop transfer function:

(12.28)

The Nyquist plot for a unity feedback system with open-loop transfer function given by:

(12.29)

which is stable under unity feedback is illustrated in Figure 12.16, and a Nyquist plot for a system thatis unstable under unity feedback is illustrated in Figure 12.17.

12.2.2.4 Lead-Lag Compensation

Lead-lag controller design is another extremely popular compensation technique. In this case, the com-pensator (the C(s) block in Figure 12.2) is of the form:

(12.30)

FIGURE 12.16 Nyquist plot for a stable system.

Real Axis

Imag

inar

y A

xis

Nyquist Diagrams

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8From: U(1)

To:

Y(1

)

G(s)1 G(s)+--------------------

G(s)1

(s 1+ )(s 1)+--------------------------------=

C(s) Kβ As 1+αAs 1+------------------- Bs 1+

βBs 1+------------------=


where α < 1 and β > 1. The first fraction is the lead portion of the compensator and can provideincreased stability with an appropriate choice for A. The second term is the lag compensator andprovides decreased steady-state error. To understand the first of these assertions, Figure 12.18 plotsthe Bode plot for a lead compensator for various values of the parameter A. Because the lead com-pensator shifts the phase plot up, by an appropriate choice of the parameter A, the crossover pointwhere the magnitude plot crosses through the value of 0 dB can be shifted to the right, increasing thegain margin.

FIGURE 12.17 Nyquist plot of an unstable system.

FIGURE 12.18 Bode plots of various lead compensators.

Real Axis

Imag

inar

y A

xis

Nyquist Diagrams

−25 −20 −15 −10 −5 0−15

−10

−5

0

5

10

15From: U(1)

To:

Y(1

)

Frequency (rad/sec)

Pha

se (

deg)

; Mag

nitu

de (

dB)

0

1

2

3

4

5

6

7

10−2

10−1

100

101

0

5

10

15

20

A

A


Example

Figure 12.19 plots the bode plot for the compensated system:

(12.31)

where A = 0, 1 and α = 0.5. As can be seen, the magnitude crossover point has been shifted to the left,increasing the gain margin. In a similar manner, unstable systems (which would originally have negativegain and phase margins) can possibly be stabilized.

Lag compensation works in a very similar manner to increase the magnitude plot for low frequencies,which decreases the steady-state error for the system. Lead and lag controllers can be used in series toboth increase stability as well as decrease steady-state error. Systematic approaches for determining theparameters, α, β, A and B can be found in the references, particularly Franklin et al. (1994).

12.2.3 Other Topics

Various other topics are typically considered in classical control but will not be outlined here due to spacelimitations. Such topics include, but are not limited to, systematic methods for tuning PID regulators,lead-lag compensation and techniques for considering and modeling time delay. Interested readers shouldconsult the references, particularly Franklin et al. (1994), Ogata (1997), Kuo (1995) and Raven (1995).

12.3 “Modern” Control

In contrast to classical control, which is essentially a complex-variable, frequency-based approach forSISO systems, modern control is a time-domain approach that is amenable to MIMO systems. The basictools are from the theory of ordinary differential equations and matrix algebra. The topics outlined inthis section are the pole placement and linear quadratic regulator (LQR) problems. Additionally, the basicsof robust control are outlined.

FIGURE 12.19 Lead compensated example system.

Frequency (rad/sec)

Pha

se (

deg)

; Mag

nitu

de (

dB)

−150

−100

−50

0

50

100

10−3

10−2

10−1

100

101

102

−300

−250

−200

−150

−100

−50

G(s)As 1+

αAs 1+------------------- 1

s3 25s2 s+ +----------------------------=


12.3.1 Pole Placement

First, a multistate but single-input control system will be examined. Consider a control system writtenin state space:

(12.32)

where x is the 1 × n state vector, u is the scalar input, A is an n × n constant matrix and B is an n × 1constant matrix. If we assume that the control input u can be expressed as a combination of the currentstate variables (called full state feedback), we can write:

(12.33)

where K is a row vector comprised of each of the gains ki . Then, the state-space description of the systembecomes:

(12.34)

so that the solution of this equation is

(12.35)

where is the matrix exponential of the matrix A − BK defined by:

(12.36)

Basic theory from linear algebra and ordinary differential equations [Hirsch and Smale, 1974] indicatesthat the stability and characteristics of the transient response will be determined by the eigenvalues ofthe matrix A − BK . In fact, if

(12.37)

then it can be shown that the eigenvalues of A − BK can be placed arbitrarily as a function of theelements of K. Techniques to solve the problem by hand by way of a similarity transformation exist(see the standard undergraduate controls books), and Matlab has functions for the computations aswell.

Example—Pole Placement for Inverted Pendulum System

Consider the cart/pendulum system illustrated in Figure 12.20. In state-space form, the equations ofmotion are

(12.38)

x Ax Bu+=

u −k1x1 k2x2… knxn––– Kx–= =

x (A BK– )x=

x(t) e(A BK)– tx(0)=

e A BK–( )t

e(A BK– )t I (A BK– )t(A BK)– 2t2

2!----------------------------- (A BK)– 3t3

3!----------------------------- …+ + + +=

rank B|AB|A2B|A3B|…|An 1– B[ ] n=

ddt-----

x

x

θ

θ

0 1 0 0

0 0gm–M

---------- 0

0 0 0 1

0 0(m M+ )g–

lM--------------------------- 0

x

x

θ

θ

0

1M-----

0

1–lM-------

u+=


Setting M = 10, m = 1, g = 9.81 and l = 1 and letting u = if the desired polelocations for the system are at

(12.39)

the Matlab function place() can be used to compute the values for the corresponding ki. For thisproblem, the gain values are

(12.40)

With initial conditions x(0) = 0.25, and the response of the systemis illustrated in Figure 12.21. Note that the cart position, x, initially moves in the “wrong” direction inorder to compensate for the pendulum position.

12.3.2 The Linear Quadratic Regulator

The LQR problem is not limited to scalar input problems and seeks to find a control input,

u = −Kx(t) (12.41)

for the system:

(12.42)

that minimizes the performance index:

(12.43)

FIGURE 12.20 Cart and pendulum system.

xM

l

θ

u

mg

−k1x k2x k3θ k4θ+ + +

λ1 −1 i–=λ2 −1 i+=λ3 8–=λ4 9–=

k1 122.32=k2 151.21=k3 849.77–=k4 38.79–=

x(0) 0 θ(0), 0.25= = θ(0) 0,=

x Ax Bu+=

J (xTQx uTRu+ ) td0

∞

∫=


where Q and R are positive definite, real symmetric matrices. By the second method of Lyapunov [Khalil,1996; Sastry, 2000], the control input that minimizes the performance index is

(12.44)

where R and B are from the performance index and equations of motion, respectively, and P satisfiesthe reduced matrix Riccati equation:

ATP + PA − PBR

−1B

TP + Q = 0 (12.45)

Example—LQR for Inverted Pendulum System

For the same cart and pole system as in the previous example with

(12.46)

(which weights all the states equally), and R = 0.001, the optimal gains (computed via the Matlab lqr()function) are

(12.47)

FIGURE 12.21 Cart and pendulum system pole placement response.

0 1 2 3 4 5 6 7 8 9 10−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (sec.)

Ca

rt P

osi

tion

(m

) a

nd

An

gle

(ra

d.)

Cart Position

Pendulum Angle

u R 1– BTPx(t)–=

Q

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

=

k1 31.62=k2 145.75=k3 95.53–=k4 21.65–=


and the response of the system with initial conditions x(0) = 0.25, and is illustrated in Figure 12.22. If the Q matrix is modified to provide a heavy weighting for the

θ state,

(12.48)

the system response is illustrated in Figure 12.23. Note that the pendulum angle goes to zero very rapidly,but at the “expense” of a slower response and greater deviation for the cart position.

12.3.3 Basic Robust Control

The main idea motivating modern robust control techniques is to incorporate explicitly plant uncertaintyrepresentations into system modeling and control synthesis methods. The basic material here outlinesthe presentation in Doyle et al. (1992), and the more advanced material is from Zhou (1996). Modernrobust control is a very involved subject and only the briefest outline can be provided here.

Consider the unity feedback SISO system illustrated in Figure 12.24, where P and C are the plant andcontroller transfer functions; R(s) is the reference signal; Y(s) is the output; D(s) and N(s) are externaldisturbances and sensor noise, respectively; E(s) is the error signal; and U(s) is the control input.

Now, define the loop transfer function, L = CP, and the sensitivity function:

(12.49)

which is the transfer function from the reference input R(s) to the error E(s) which provides a measureof the sensitivity of the closed loop (or complementary sensitivity) transfer function:

(12.50)

FIGURE 12.22 Cart and pendulum LQR response.

0 1 2 3 4 5 6 7 8 9 10−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time (sec.)

Ca

rt P

osi

tion

(m

) a

nd

Pe

nd

ulu

m A

ng

le (

rad

.)

Cart Position

Pendulum Angle

x(0) 0 θ(0), 0.25= =θ(0) 0=

Q

1 0 0 0

0 1 0 0

0 0 100 0

0 0 0 1

=

S1

1 L+------------=

TPC

1 PC+----------------=


to infinitesimal variations in the plant P. Now, given a (frequency-dependent) weighting function W1(s),a natural performance specification (relating tracking error to classes of reference signals) is

(12.51)

where denotes the infinity norm. An easy graphical test for the performance specification is that theNyquist plot of L must always lie outside a disk of radius centered at –1.

To incorporate plant uncertainty into the model, consider a nominal plant P and perturbed plant where P and differ by some multiplicative or other type of uncertainty. Let W2 be a stable transferfunction and ∆ be a variable stable transfer function satisfying Then, common uncertaintymodels can be constructed by appropriate combinations of P, ∆ and W. It can be shown that the systemis internally stable (this is a stronger definition than simple I/O stability; see Doyle [1992]) for the conditionsshown in Table 12.3.

Recall that the nominal performance condition was Not surprising, then, the robustperformance condition is a combination of the two (for the (1 + ∆W2)P perturbation):

(12.52)

Other robust performance measures for various types of uncertainty can be found in Doyle et al. (1992)and Zhou (1996).

Recall that W1 is the performance specification weighting function and W2 is the plant uncertaintytransfer function. Consider the following facts:

FIGURE 12.23 Cart and pendulum LQR response with large pendulum angle weighting.

FIGURE 12.24 Robust control feedback block diagram.

0 1 2 3 4 5 6 7 8 9 10−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (sec.)

Ca

rt P

osi

tion

(m

) a

nd

Pe

nd

ulu

m A

ng

le (

rad

.)

Cart Position

Pendulum Angle

R(s) E(s)C(s)

D(s)P(s)

N(s)

Y(s)U(s)

−+

W1S ∞ 1<

⋅ ∞W1

PP

∆ ∞ 1.≤

W1S ∞ 1.<

W1S W2T+ ∞ 1<


1. Plant uncertainty is greatest for high frequencies.2. It is only reasonable to demand high performance for low frequencies.

Typically, then,

(12.53)

for low frequencies, and

(12.54)

for high frequencies (it can be shown that the magnitude of either W1 or W2 must be less than 1). Now,by considering the relationship between L, S and T, the following can be derived:

(12.55)

and

(12.56)

Loopshaping [Bower and Schultheiss, 1961; Horowitz, 1963] controller design is the task of determiningan L (and hence C) that satisfies the low-frequency performance criterion as well as the high-frequencyrobustness criterion. Hence, the task is to design C so that the magnitude versus frequency plot of Lappears as in Figure 12.25. In the figure, the indicated low-frequency performance bound is a plot of:

(12.57)

for low frequencies, and the high-frequency stability bound is a plot of:

(12.58)

for high frequencies.Two more aspects of this problem have been developed in recent years. The first concerns optimality

and the second concerns multivariable systems. For both aspects of these recent developments, interestedreaders are referred to the comprehensive book by Zhou (1996).

TABLE 12.3 Internal Stability Conditions

Perturbation Condition

(1 + ∆W2)P

P + ∆W2

W2T ∞ 1<

W2CS ∞ 1<P

1 ∆W2P+------------------------ W2PS ∞ 1<

P1 ∆W2+-------------------- W2S ∞ 1<

W1 1 W2> >

W1 1 W2<<

W1 >>1 W2 LW1

1 W2–------------------->⇒>

W1 1<< W2 L1 W1–

W2

-------------------<⇒<

W1

1 W2–-------------------

1 W1–W2

-------------------


12.4 Nonlinear Control

Aside from the developments of robust optimal control briefly outlined in the previous section, the areaof most recent development in control theory has been nonlinear control, where, in contrast to ignoringnonlinear effects via linearization, the nonlinearities in the control system are either expressly recognizedor are even exploited for control purposes. Much, but not all, development in nonlinear control has utilizedtools from differential geometry. While the control techniques will be outlined here, the basics of differ-ential geometry will not, and the interested reader is referred to Abraham et al. (1988), Boothby (1986),Isidori (1996) and Nijmeijer and van der Schaft (1990) for details.

The general nonlinear model considered here is of the form:

(12.59)

where x is a 1 × n vector, the f(x) and g i(x) are smooth vector fields, and the ui are scalar control inputs.Note that this is not the most general form for nonlinear systems, as the ui are assumed to enter theequations in an affine manner (i.e., they simply multiply the g i(x) vector fields). For some aerodynamicproblems, for example, this assumption may not be true.

12.4.1 SISO Feedback Linearization

In contrast to the standard Jacobian linearization of a nonlinear control system, feedback linearizationis a technique to construct a nonlinear change of coordinates which converts a nonlinear system in theoriginal coordinates to a linear system in the new coordinates. Thus, whereas the Jacobian linearizationis an approximation of the original system, a feedback linearized system is still exactly the original system.For clarity of presentation, SISO systems will be considered first, followed by MIMO systems. Excellentreferences for feedback linearization are Isidori (1996), Nijmeijer and van der Schaft (1990), Krener(1987), Khalil (1996) and Sastry (2000). Developmental papers or current research in this area are considered

FIGURE 12.25 Loopshaping concepts.

Frequency (rad/sec)

Pha

se (

deg)

; Mag

nitu

de (

dB)

−20

−10

0

10

20

30

10−1

100

101−200

−150

−100

−50

0

Low Frequency Performance Requirements

High FrequencyStability Requirements

x f(x) gi(x)ui

i=1

n

∑+=


in Slotine and Hedrick (1993), Brockett (1978), Dayawansa et al. (1985), Isidori et al. (1981a; 1981b)and Krener (1987).

Consider the nonlinear system:

(12.60)

where the function h(x) is called the output function. Let Lfh denote the Lie derivative of the function hwith respect to the vector field f, which is defined in coordinates as:

(12.61)

where

(12.62)

so it is simply the directional derivative of h along f. Now, because the system evolves according to thestate equations, the time derivative of the output function is simply the directional derivative of theoutput function along the control system:

(12.63)

The relative degree of a system is defined as follows: A SISO nonlinear system is said to have strict relativedegree γ at the point x if:

1. (12.64)

2. (12.65)

In the case where γ = n, the system is full state feedback linearizable, and it is possible to construct thefollowing change of coordinates where the original coordinates xi are mapped to a new set of coordinatesξi as follows:

(12.66)

x f(x) g(x)u+=y h(x)=

Lfh(x)∂h∂xi

-------(x)fi(x)i=1

n

∑=

x

x1

x2

xn−1

xn

and f

f1(x)

f2(x)

fn−1(x)

fn(x)

= =

y

y h∂h∂x------ x

∂h∂x------(f(x) g(x)u)+ Lf+guh Lfh Lghu+= = = = =

LgLfih(x) 0 i≡ 0, 1, 2,…, γ 2–=

LgLfγ −1h(x) 0≠

ξ1 h(x)=

ξ2 ξ1 h Lfh= = =

ξ3 ξ2 h Lf2h= = =

ξn ξn−1 Lfγ −1h= =


Computing derivatives, the control system becomes:

(12.67)

or, setting

(12.68)

the system is

(12.69)

which is both linear and in controllable canonical form. At this point, it is simple to determine anappropriate v to stabilize the system or track desired values of h(x). One approach is pole placement;that is, v = −Kξ. Note that the overall approach was to determine an output function h that could bedifferentiated n times before the control input appeared. This essentially constructs a system known asa chain of integrators, as the derivative of the ith state in the ξ variables is equal to the (i + 1)th state variable.

There are two main limitations to feedback linearization approaches. The first is that not all systemsare feedback linearizable, although analytical tests exist to determine whether a particular system islinearizable. Second, determining the output function, h(x) involves solving a system of partial differentialequations (generally, not easy to do).

Example—SISO Full State Feedback Linearization

Consider, as a mathematical example of the computations involved in feedback linearization, the system:

(12.70)

with output function, y = h(x) = x1. The system has a relative degree equal to 4, so the system is full statefeedback linearizable and the coordinate transformation is given by:

(12.71)

ξ1 ξ2=

ξ2 ξ3=

ξn−1 ξn=

ξn Lfγ h LgLf

γ −1hu+=

u1

LgLfγ −1h

------------------( Lfγ h– v)+=

ξ1 ξ2=

ξ2 ξ3=

ξn−1 ξn=

ξn v=

x1

x2

x3

x4

x3

x4

x1 x2 x3+ +

x1 x3–

0

0

0

1

u+ f(x) g(x)u+= =

ξ1 h(x) x1= =ξ2 Lfh(x) x3= =

ξ3 Lf2h(x) x1 x2 x3+ += =

ξ4 Lf3h(x) x1 x2 2x3 x4+ + += =


COLOR FIGURE 5.6 Close-up of the vorticity contours for Re = 30 simulation at the left valve (meshes shown onright side). Top: τω = 0.28, corresponding to the beginning of the suction stage; start-up vortices due to the motionof the inlet valve can be identified. Middle: τω = 0.72, corresponding to the end of the suction stage; a vortex jet pairis visible in the pump cavity. Bottom: τω = 0.84, corresponding to early ejection stage; further evolution of the vortex jet and the start-up vortex of the exit valve can be identified. (Courtesy of A. Beskok.)

a

b

c

d

e

f


COLOR FIGURE 6.1 Micro heat exchanger constructed from rectangular channels machined in metal. (Courtesy ofK. Schubert and D. Cacuci, Forschungszentrum, Karlsruhe.)


COLOR FIGURE 13.8 Localized controller gains relating the state estimate x inside the domain to the control forc-ing u at the point x = 0, y = -1, z = 0 on the wall. Visualized are a positive and negative isosurface of the convolu-tion kernels for (left) the wall-normal component of velocity and (right) the wall-normal component of vorticity.(From Högberg, M., and Bewley, T.R., Automatica (submitted). With permission from Elsevier Science.)

COLOR FIGURE 13.9 Localized estimator gains relating the measurement error (y – y) at the point x = 0, y = -1,z = 0 on the wall to the estimator forcing terms v inside the domain. Visualized are a positive and negative isosur-face of the convolution kernels for (left) the wall-normal component of velocity and (right) the wall-normal com-ponent of vorticity. (From Högberg, M., and Bewley, T.R., Automatica (submitted). With permission from ElsevierScience.)

x xz

z

yy

x

z

y

xz

y


COLOR FIGURE 13.11 Visualization of the coherent structures of uncontrolled near-wall turbulence at Reτ = 180.Despite the geometric simplicity of this flow (see Figure 13.1), it is phenomenologically rich and is characterized bya large range of length scales and time scales over which energy transport and scalar mixing occur. The relevant spec-tra characterizing these complex nonlinear phenomena are continuous over this large range of scales, thus such flowshave largely eluded accurate description via dynamic models of low state dimension. The nonlinearity, the distrib-uted nature and the inherent complexity of their dynamics make turbulent flow systems particularly challenging forsuccessful application of control theory. (Simulation by Bewley et al. (2001). Reprinted with permission ofCambridge University Press.)


COLOR FIGURE 13.12 Example of the spectacular failure of linear control theory to stabilize a simple nonlinearchaotic convection system governed by the Lorenz equation. Plotted are the regions of attraction to the desired sta-tionary point (blue) and to an undesired stationary point (red) in the linearly controlled nonlinear system, and typ-ical trajectories in each region (black and green, respectively). The cubical domain illustrated is Ω = (–25, 25)3 in allsubfigures; for clarity, different viewpoints are used in each subfigure. (Reprinted from Bewley, T.R. (1999) Phys.Fluids 11, 1169–1186. Copyright 1999, American Institute of Physics. With permission.)

COLOR FIGURE 13.14 Performance of optimized blowing/suction controls for formulations based on minimiz-ing T0(φ), case c (see Section 13.9.1.2), as a function of the optimization horizon T+. The direct numerical simulationsof turbulent channel flow reported here were conducted at Reτ = 100. For small optimization horizons (T+ = O(1),sometimes called the “suboptimal approximation”), approximately 20% drag reduction is obtained, a result that canbe obtained with a variety of other approaches. For sufficiently large optimization horizons (T+ ≥ 25), the flow isreturned to the region of stability of the laminar flow, and the flow relaminarizes with no further control effortrequired. No other control algorithm tested in this flow to date has achieved this result with this type of flow actua-tion. (From Bewley, T.R., Moin, P., and Temam, R., J. Fluid Mech., to appear. With permission of CambridgeUniversity Press.)

(a) History of drag. (b) History of turbulent kinetic energy.

(a) = 10 (b) = 0.5 (c) = 0.025


COLOR FIGURE 13.20 A MEMS tile integrating sensors, actuators and control logic for distributed flow controlapplications. (Developed by Profs. Chih-Ming Ho, UCLA, and Yu-Chong Tai, Caltech.)

Actuator electronics

Control logic

Microflap actuator

Shear-stress sensor

Sensor electronics

COLOR FIGURE 13.21 Simulation of a proposed driven-cavity actuator design (Prof. Rajat Mittal, University ofFlorida). The fluid-filled cavity is driven by vertical motions of the membrane along its lower wall. Numerical simu-lation and reduced-order modeling of the the influence of such flow-control actuators on the system of interest willbe essential for the development of feedback control algorithms to coordinate arrays of realistic sensor/actuator con-figurations.


COLOR FIGURE 13.22 Future interdisciplinary problems in flow control amenable to adjoint-based analysis:(a) Minimization of sound radiating from a turbulent jet (simulation by Prof. Jon Freund, UCLA)(b) Maximization of mixing in interacting cross-flow jets (simulation by Dr. Peter Blossey, UCSD) [Schematic of jet

engine combustor is shown at left. Simulation of interacting cross-flow dilution jets, designed to keep the turbineinlet vanes cool, is visualized at right.]

(c) Optimization of surface compliance properties to minimize turbulent skin friction(d) Accurate forecasting of inclement weather system

(a)

(b)

(c) (d)


COLOR FIGURE 14.11 “Old” fitness.

COLOR FIGURE 14.13 “New” fitness.


COLOR FIGURE 14.29 Response surface for pendulum fuzzy controller.


COLOR FIGURE 18.1 DXRL-based (direct LIGA) microfabrication process.


COLOR FIGURE 21.11 Packaged SiC pressure-sensor chip. Through the semitransparent SiC can be seen the edgeof the well that has been etched in the backside of the wafer to form the circular diaphragm. The metal-covered, n-type SiC that connects the strain gauges is highly visible, while the n-type SiC strain gauges are more faintly visible.There is a U-shaped strain gauge over the edge of the diaphragm at top and bottom, and there are two vertically ori-ented linear gauges in the center of the diaphragm.


COLOR FIGURE 23.8 Thermal stress and strain distribution in SiC die and Au thick-film layer. (a) Von-Mises stressdistribution in SiC die; (b) principal stress distribution in SiC die; (c) equivalent plastic strain in Au thick-film layer.

(a)

(b)

(c)


COLOR FIGURE 24.1 Examples of two high-volume accelerometer products. On the left is Analog Devices, Inc.ADXL250 two-axis lateral monolithically integrated accelerometer. On the right is a Motorola, Inc. wafer-scale pack-aged accelerometer and control chips mounted on a lead frame prior to plastic injection molding. (Photographscourtesy of Analog Devices, Inc. and Motorola, Inc.)

COLOR FIGURE 27.31 The blackened trace near the “+” sign wascaused by a dimple on a grounded gear touching a high potentialsignal line. (Photograph courtesy of S. Barnes, Sandia NationalLaboratories.)


COLOR FIGURE 28.17 Photograph of the microrobot platform used for walking during a load test. The load of2500 mg is equivalent to maximum 625 mg/leg (or more than 30 times the weight of the robot itself). The powersupply is maintained through three 30-µm-thin and 5- to 10-cm-long bonding wires of gold. The robot walks usingthe asynchronous ciliary motion principle described in Figure 28.6c. The legs are actuated using the polyimide V-groove joint technology which was described in Figure 28.10. (Photo by P. Westergård; published with permission.)(Note: Videos on different experiments that have been performed using the microrobot shown above are availableat: http://www.s3.kth.se/mst/research/gallery/microrobot_video.html.)

COLOR FIGURE 35.35 SEM view of University of Florida MEMS wall-shear sensor.


http://www.s3.kth.se/mst/research/gallery/microrobot_video.html

COLOR FIGURE 35.37 Streamlines of the flow over the step without (a) and with (b) actuation.

(a)

(b)


COLOR FIGURE 35.38 Phase-averaged streamline plots at different phases of the forcing cycle.

(a) 5.4% of input waveform, flap deflection = 14.1 µm

(a) 25.4% of input waveform, flap deflection = 22.3 µm

(a) 45.4% of input waveform, flap deflection = –0.3 µm

(a) 65.4% of input waveform, flap deflection = –22.5 µm


The above equations and the fact that the system has a relative degree of 4 is verified by the followingdetailed calculations:

(12.72)

Therefore, a controller of the form:

(12.73)

with the gains ki picked via pole placement, for example, will allow the system to track trajectories of theoutput function h(x) = x1.

So far, this section has considered full state feedback linearization where the relative degree of a systemis equal to the dimension of its state space. Partial feedback linearization is also possible where the relative

Lgh(x) Lgx1 1 0 0 0

0

0

0

1

0,= = =

LgLfh(x) Lglf x1 Lg 1 0 0 0

x3

x4

x1 x2 x3+ +x1 x3–

Lgx3 0 0 1 0

0

0

0

1

0,= = = = =

LgLf2h(x) LgLf

2x1 LgLf x3 Lg 0 0 1 0

x3

x4

x1 x2 x3+ +x1 x3–

= = =

Lg(x1 x2 x3)+ + 1 1 1 0

0

0

0

1

0,= = =

LgLf3h(x) LgLf

3x1 LgLf (x1 x2 x3)+ + Lg 1 1 1 0

x3

x4

x1 x2 x3+ +x1 x3–

= = =

Lg(x1 x2 2x3 x4)+ + + 1 1 2 1

0

0

0

1

1= = =

u1

LgLf3h

--------------( Lf4h v)+–=

( (3x1 2x2 2x3 x4)+ + +– k1x1 k2x3 k3(x1 x2 x3)+ + k4(x1 x2 2x3 xer4))+ + ++ + + +=


degree is less than the dimension of the state space; however, for such systems an analysis of the stabilityof the zero dynamics is necessary. In particular, if the relative degree γ < n, then the change of coordinatesis typically expressed in the form:

(12.74)

where the ηi are chosen so that the matrix:

(12.75)

is full rank. The dynamics of the system in the new coordinates will be of the form:

(12.76)

The zero dynamics are the dynamics expressed by the η equations, the stability of which must beconsidered independently of the linearized ξ equations. The interested reader is referrred to texts byIsidori (1996), Khalil (1996), Nijmeijer and van der Schaft (1990) and Sastry (2000) for the relevantdetails.

ξ1 h(x)=

ξ2 ξ1 h Lfh= = =

ξ3 ξ2 h Lf2h= = =

ξγ ξγ −1 Lfγ −1h= =

η1 η1(x)=η2 η2(x)=

ηn−γ ηn−γ (x)=

dh(x)

dLfh(x)

dLfγ −1h(x)

dη1(x)

dηn−γ (x)

ξ1 ξ2=

ξ2 ξ3=

ξγ b(ξ, η) a(ξ, η)u+=η1 q1(ξ, η)=

ηn−γ qn−γ (ξ, η)=


12.4.2 MIMO Full-State Feedback Linearization

The MIMO feedback linearization is a slight extension of the SISO feedback linearization by which theSISO linearization construction is repeated for m output functions for a system with m control inputs:

(12.77)

Now, the vector relative degree is defined as a combination of relative degrees for each of the outputfunctions. Considering the jth output yj ,

(12.78)

If for each i, then the inputs do not appear in the derivative. Now let γj be the smallest integersuch that for at least one i. Define the matrix:

(12.79)

Now, the system has vector relative degree γ1, γ2,…, γm at x if for i = 1,…,mand the matrix A(x) is nonsingular.

12.4.3 Control Applications of Lyapunov Stability Theory

Lyapunov theory for autonomous differential equations states that if x = 0 is an equilibrium point for adifferential equation and there exists a continuously differentiable function, V(x) > 0 exceptfor V(0) = 0 and and in some domain containing zero where:

(12.80)

then the point x = 0 is an asymptotically stable equilibrium point for the differential equation. The utilityof Lyapunov theory in control is that controller synthesis techniques can be designed to ensure thenegative definiteness of a Lyapunov function to ensure stability or boundedness of the system trajectories.

As fully described in Khalil (1996), the main applications of Lyapunov stability theory to control systemdesign are Lyapunov redesign, backstepping, sliding mode control and adaptive control. Due to spacelimitations, the basic concepts of each will only be briefly outlined here.

Lyapunov redesign is an instance of nonlinear robust control design; however, there is a severerestriction upon how the uncertainties are expressed in the equations of motion with a correspondingrestriction on the types of systems that are amenable to this technique. In particular, consider the system:

(12.81)

where f and G are known, δ is unknown but is bounded by a known, but not necessarily small, function.The main restriction here is that the uncertainty enters the system in exactly the same manner as the

x f(x) g1(x)u1 g2(x)u2… g(x)mum+ + + +=

y1 h1(x)=y2 h2(x)=

ym hm(x)=

yj Lf hj Lg1hju1 Lg2

hju2… Lgm

hjum+ + + +=

Lgihj 0≡

LgLf

γ j−1hj 0≠

A(x)

Lg1Lf

γ 1−1h1

… LgmLf

γ 1−1h1

Lg1Lf

γ m−1hm

… LgmLf

γ m−1hm

=

LgiLf

kh1 0 0 k γ i 2–≤ ≤≡

x f(x),=V(x) 0<

V(x)∂V∂xi

------- xi

i=1

n

∑ ∂V∂xi

-------fi(x)i=1

n

∑= =

x f(t, x) G(t, x)u G(t, x)δ(t, x, u)+ +=


control input. In order to use Lyapunov redesign, a stabilizing control law exists for the nominal system(ignoring δ) and a Lyapunov function for the nominal system must be known. (Note that one nice aspectof the feedback linearization discussed previously is that if a controller is designed using that technique,a Lyapunov function is straightforward to determine due to the simple form of the equations of motionafter the nonlinear coordinate transformation.) Due to the way that the uncertainty enters the system, itis easy to modify the nominal control law to compensate for the uncertainty to ensure that References concerning Lyapunov redesign include Corless (1993), Corless and Heitmann (1981), Barmishet al. (1983) and Spong and Vidyasager (1989).

Backstepping is a recursive controller design procedure where the entire control system is decomposedinto smaller, simpler subsystems for which it may be easier to design a stabilizing controller. By consideringthe appropriate way to modify a Lyapunov function after each smaller subsystem is designed, a stabilizingcontroller for the full system may be obtained. The main restriction for this technique is a limitation onthe structure of the equations of motion (a type of hierarchical structure is required). Extensions of thisprocedure to account for certain system uncertainties have also been developed. For references, see Krsticet al. (1995), Qu (1993) and Slotine and Hedrick (1993).

The basic idea in sliding mode control is to drive the system in finite time to a certain submanifoldof the configuration space, called the sliding manifold, upon which the system should indefinitely evolve.Because the sliding manifold has a lower dimension than the full state space for the system, a lower ordermodel can describe the evolution of the system on the sliding manifold. If a stabilizing controller can bedesigned for the sliding manifold, the problem reduces to designing a controller to drive the system tothe sliding manifold. The advantage of sliding mode control is that it is very robust with respect to systemuncertainties. One disadvantage is that it is a bit mathematically quirky as there are discontinuities in thecontrol law when switching from the full state of the system to the sliding manifold. Additionally,“chattering,” wherein the system constantly alternates between the two sides of the submanifold, is acommon problem. See Utkin (1992) and DeCarlo et al. (1988) for overviews of the approach.

Finally, there is a vast literature in the area of adaptive control. In adaptive control, some systemperformance index is measured and the adaptive controller modifies adjustable parameters in the con-troller in order to maintain the performance index of the control system close to a desired value (or setof desired values). This is clearly desirable in cases where system parameters are unknown or changewith time. Representative references concerning adaptive control include Anderson et al. (1986), Ioannouand Sun (1995), Krstic et al. (1995), Landau et al. (1998), Narendra and Annaswamy (1989) and Sastryand Bodson (1989).

12.4.4 Hybrid Systems

Hybrid systems are systems characterized by both continuous as well as discrete dynamics. Examples ofhybrid systems include, but certainly are not limited to, digital computer-controlled systems, distributedcontrol systems governed by a hierarchical logical interaction structure, multi-agent systems (such as theair traffic management system [Tomlin, 1998]) and systems characterized by intermittent physical contact[Goodwine, 2000]. Recent papers considering modeling and control synthesis methods for such compli-cated systems include Alur and Henzinger (1996), Antsaklis et al. (1995; 1997), Branicky et al. (1998),Henzinger and Sastry (1998) and Lygeros et al. (1999).

12.5 Parting Remarks

This chapter provided a brief overview of the fundamental concepts in analysis and design of controlsystems. Included was an outline of “classical linear control,” including stability concepts (the Routharray) and controller design techniques (root locus and lead-lag synthesis). Additionally, more recentadvances in control including pole placement, the linear quadratic regulator and the basic concepts fromrobust control were outlined and examples were provided. Finally, recent developments in nonlinear

V(x) 0.<


control, including feedback linearization (for both single-input, single-output and multi-input, multi-output systems), were outlined along with basic approaches utilizing Lyapunov stability theory.

Defining Terms

Adaptive control: A controller design technique wherein typically some parameter in the controller isvaried or changed in response to variations in the controlled system.

Backstepping: A nonlinear controller design technique based upon Lyapunov stability theory.Block diagram: A graphical representation of a differential equation describing a linear control system.Bode plot: A plot of the magnitude of the response of a system to sinusoidal inputs vs. the frequency of

the inputs and a plot of the phase difference between the sinusoidal input and response of the system.Characteristic equation: The equation obtained by equating the denominator of a transfer function with

zero. Analysis of the characteristic equation yields insight into the stability of a transfer function.Feedback linearization: A nonlinear controller design technique based upon determining a nonlinear

change of coordinates which transform the differential equations describing the system into asimple, canonical and controllable form.

Gain margin: A measure of stability or instability of a control system which can be determined from aBode plot by considering the magnitude of the system response when the phase difference betweenthe input and system response is –180°.

Laplace transform: A mathematical transformation useful to transform ordinary differential equationsinto algebraic equations.

Loopshaping: A controller design method based upon obtaining a desired “shape” for the Bode plot ofa system.

LQR control: An optimal control design technique based upon minimizing a cost function defined as acombination of the magnitude of the system response and the control effort. LQR is an acronymfor linear quadratic regulator.

Lyapunov redesign: A nonlinear robust controller design technique based upon Lyapunov stabilitytheory.

Lyapunov stability theory: The analysis of the stability of (nonlinear) differential equations based uponthe time derivative of a Lyapunov function.

Nonlinear control: The design and analysis of control systems described by nonlinear differential equa-tions.

Nyquist plot: A contour plot in the complex plane of a transfer function as the dependent variableencircles the entire right-half complex plane.

Phase margin: A measure of stability or instability of a control system which can be determined from aBode plot by considering the magnitude of the phase difference between the input and systemresponse when the magnitude of the response is equal to 0 dB.

PID control: A very common control law wherein the input to the system is proportional to the error,the derivative of the error and the time integral of the error. PID is an acronym for proportional–integral–derivative control.

Poles: Roots of the denominator of a transfer function.Pole placement: A state-space controller design technique based upon specifying certain eigenvalues for

the system.Robust control: The design and analysis of control systems which explicitly account for unmodeled

system dynamics and disturbances.Root locus: A controller synthesis technique wherein the poles of the transfer function of a control system

are plotted as a parameter (usually a controller gain) is varied.Routh array: A technique to determine the number of roots of a polynomial that have a positive real

part without having to factor the polynomial. This is used to determine the stability of a transferfunction based upon its characteristic equation.


Sliding mode control: A nonlinear controller design technique based upon Lyapunov stability theorywith the goal of driving the system to a center manifold upon which the dynamics of the systemare simpler and easily controlled.

Transfer function: The algebraic expression relating the Laplace transform of the input to the output ofa control system.

Zeros: Roots of the numerator of a transfer function.

References

Abraham, R., Marsden, J.E., and Ratiu, T.T. (1988) Manifolds, Tensor Analysis, and Applications, Springer-Verlag, New York.

Aeyels, D. (1985) “Stabilization of a Class of Nonlinear Systems by a Smooth Feedback Control,” Syst.Control Lett. 5, pp. 289–294.

Alur, R., and Henzinger, T., eds. (1996) Hybrid Systems III: Verification and Control, Springer-Verlag,New York.

Anderson, B.D.O., Bitmead, R.R., Johnson, C.R., Kokotovic, P.V., Kosut, R.L., Mareels, I.M.Y., Praly, L.,and Riedle, B.D. (1986) Stability of Adaptive Systems, MIT Press, Cambridge, MA.

Antsaklis, P., Kohn, W., Nerode, A., and Sastry, S., eds. (1995) Hybrid Systems II, Springer-Verlag, New York. Antsaklis, P., Kohn, W., Nerode, A., and Sastry, S., eds. (1997) Hybrid Systems IV, Springer-Verlag,

New York.Barmish, B.R., Corless, M., and Leitmann, G. (1983) “A New Class of Stabilizing Controllers for Uncertain

Dynamical Systems,” SIAM J. Control Optimization 21, pp. 246–355. Bestle, D., and Zeitz, M. (1983) “Canonical Form Observer Design for Nonlinear Time-Variable Systems,”

Int. J. Control 38, pp. 419–431. Bode, H.W. (1945) Network Analysis and Feedback Amplifier Design, D Van Nostrand, Princeton, NJ.Boothby, W.M. (1986) An Introduction to Differentiable Manifolds and Reimannian Geometry, Academic

Press, Boston.Bower, J.L., and Schultheiss, P. (1961) Introduction to the Design of Servomechanisms, Wiley, New York.Branicky, M., Borkar, V., and Mitter, S.K. (1998) “A Unified Framework for Hybrid Control: Model and

Optimal Control Theory,” IEEE Trans. Autom. Control AC-43, pp. 31–45. Brockett, R.W. (1978) “Feedback Invariants for Nonlinear Systems,” in Proceedings of the 1978 IFAC

Congress, Helsinki, Finland, Pergamon Press, Oxford.Bullo, F., Leonard, N.E., and Lewis, A. (2000) “Controllability and Motion Algorithms for Underactuated

Lagrangian Systems on Lie Groups,” IEEE Trans. Autom. Control 45, pp. 1437–1454.Byrnes, C.I., and Isidori, A. (1988) “Local Stabilization of Minimum-Phase Nonlinear Systems,” Syst.

Control Lett. 11, pp. 9–17. Chen, M.J., and Desoer, C.A. (1982) “Necessary and Sufficient Condition for Robust Stability of Linear,

Distributed Feedback Systems,” Int. J. Control 35, pp. 255–267. Corless, M. (1993) “Control of Uncertain Nonlinear Systems,” J. Dynamical Syst. Meas. Control 115,

pp. 362–372. Corless, M., and Leitmann, G. (1981) “Continuous State Feedback Guaranteeing Uniform Ultimate

Boundedness for Uncertain Dynamic Systems,” IEEE Trans. Autom. Control AC-26, pp. 1139–1144. Dayawansa, W.P., Boothby, W.M., and Elliott, D. (1985) “Global State and Feedback Equivalence of

Nonlinear Systems,” Syst. Control Lett. 6, pp. 517–535. DeCarlo, R.A., Zak, S.H., and Matthews, G.P. (1988) “Variable Structure Control of Nonlinear Multi-

variable Systems: A Tutorial,” Proc. IEEE 76, pp. 212–232. Dorf, R.C. (1992) Modern Control Systems, Addison-Wesley, Reading, MA. Doyle, J.C., Francis, B. A., and Tannenbaum, A.R. (1992) Feedback Control Theory, Macmillan, New York. Doyle, J.C., and Stein, G. (1981) “Multivariable Feedback Design: Concepts for a Classical Modern

Synthesis,” IEEE Trans. Autom. Control AC-26, pp. 4–16. Evans, W.R. (1948) “Graphical Analysis of Control Systems,” AIEE Trans. Part II, pp. 547–551.


Evans, W.R. (1950) “Control System Synthesis by Root Locus Method,” AIEE Trans. Part II, pp. 66–69. Franklin, G.F., Powell, D.J., and Emami-Naeini, A. (1994) Feedback Control of Dynamic Systems, Addison-

Wesley, Reading, MA. Friedland, B. (1986) Control System Design, McGraw-Hill, New York.Gajec, Z., and Lelic, M.M. (1996) Modern Control Systems Engineering, Prentice-Hall, London. Gibson, J.E. (1963) Nonlinear Automatic Control, McGraw-Hill, New York.Goodwine, B., and Burdick, J. (2000) “Motion Planning for Kinematic Stratified Systems with Application

to Quasistatic Legged Locomotion and Finger Gaiting,” IEEE J. Robotics Autom., accepted forpublication.

Hahn, W. (1963) Theory and Application of Liapunov’s Direct Method, Prentice-Hall, Upper Saddle River,NJ.

Henzinger, T., and Sastry, S., eds. (1998) Hybrid Systems: Computation and Control, HSCC 2000,Pittsburgh, PA.

Hirsch, M., and Smale, S. (1974) Differential Equations, Dynamical Systems, and Linear Algebra, AcademicPress, Boston.

Horowitz, I.M. (1963) Synthesis of Feedback Mechanisms, Academic Press, New York.Ioannou, P.A., and Sun, J. (1995) Robust Adaptive Control, Prentice-Hall, Englewood Cliffs, NJ.Isidori, A. (1996) Nonlinear Control Systems, Springer-Verlag, Berlin.Isidori, A., Krener, A.J., Gori-Giorgi, C., and Monaco, S. (1981a) “Locally (f, g)-Invariant Distributions,”

Syst. Control Lett. 1, pp. 12–15.Isidori, A., Krener, A.J., Gori-Giorgi, C., and Monaco, S. (1981b) “Nonlinear Decoupling Via Feedback:

A Differential Geometric Approach,” IEEE Trans. Autom. Control AC-26, pp. 331–345. Jakubczyk, B., and Respondek, W. (1980) “On Linearization of Control Systems,” Bull. Acad. Polonaise

Sci. Ser. Sci. Math 28, pp. 517–522. Junt, L.R., Su, R., and Meyer, G. (1983) “Design for Multi-Input Nonlinear Systems,” in Differential

Geometric Control Theory, eds. R.W. Brockett, R.S. Millman, and H. Sussmann, Birkhäuser, Basel. Khalil, H.K. (1996) Nonlinear Systems, Prentice-Hall, Englewood Cliffs, NJ.Krener, A.J. (1987) “Normal Forms for Linear and Nonlinear Systems,” Contempor. Math 68, pp. 157–189. Krener, A.J. (1998) “Feedback Linearization,” in Mathematical Control Theory, eds. J. Baillieul and

J. Willems, Springer-Verlag, New York.Krstic, M., Kanellakopoulos, I., and Kokotovic, P. (1995) Nonlinear and Adaptive Control Systems Design,

John Wiley & Sons, New York. Kuo, B.C. (1995) Automatic Control Systems, Prentice-Hall, Englewood Cliffs, NJ.Lafferriere, G., and Sussmann, H. (1993) “A Differential Geometric Approach to Motion Planning,” in

Nonholonomic Motion Planning, eds. Z. Li and J. F. Canny, Academic Press, New York, pp. 235–270.Landau, Y.D., Lozano, R., and M’Saad, M. (1998) Adaptive Control, Springer-Verlag, Berlin.LaSalle, J.P., and Lefschetz, S. (1961) Stability by Liapunov’s Direct Method with Applications, Academic

Press, New York.Lygeros, J., Godbole, D., and Sastry, S. (1999) “Controllers for Reachability Specifications for Hybrid

Systems,” Automatica 35, pp. 349–370. Marino, R., and Tomei, P. (1995) Nonlinear Control Design, Prentice-Hall, New York.Marino, R. (1988) “Feedback Stabilization of Single-Input Nonlinear Systems,” Syst. Control Lett. 10,

pp. 201–206. Narendra, K.S., and Annaswamy, A.M. (1989) Stable Adaptive Systems, Prentice-Hall, Englewood Cliffs,

NJ.Nijmeijer, H., and van der Schaft, A.J. (1990) Nonlinear Dynamical Control Systems, Springer-Verlag,

New York.Nyquist, H. (1932) “Regeneration Theory,” Bell System Technol. J. 11, pp. 126–47. Ogata, K. (1997) Modern Control Engineering, Prentice-Hall, Englewood Cliffs, NJ.Qu, Z. (1993) “Robust Control of Nonlinear Uncertain Systems under Generalized Matching Conditions,”

Automatica 29, pp. 985–998.


Raven, F.H. (1995) Automatic Control Engineering, McGraw-Hill, New York.Routh, E.J. (1975) Stability of Motion, Taylor & Francis, London.Sastry, S. (2000) Nonlinear Systems: Analysis, Stability and Control, Springer-Verlag, New York.Sastry, S., and Bodson, M. (1989) Adaptive Systems: Stability, Convergence and Robustness, Prentice-Hall,

Englewood Cliffs, NJ.Shinners, S.M. (1992) Modern Control System Theory and Design, John Wiley & Sons, New York.Slotine, J.-J.E., and Hedrick, J.K. (1993) “Robust Input–Output Feedback Linearization,” Int. J. Control

57, pp. 1133–1139. Sontag, E.D. (1998) Mathematical Control Theory: Deterministic Finite Dimensional Systems, Second

Edition, Springer, New York, 1990.Spong, M.W., and Vidyasager, M. (1989) Robot Dynamics and Control, Wiley, New York.Su, R. (1982) “On the Linear Equivalents of Nonlinear Systems,” Syst. Control Lett. 2, pp. 48–52. Tomlin, C., Pappas, G., and Sastry, S. (1998) “Conflict Resolution in Air Traffic Management: A Study

in Multi-Agent Hybrid Systems,” IEEE Trans. Autom. Control 43(4), pp. 509–521. Truxal, J.G. (1955) Automatic Feedback Systems Synthesis, McGraw-Hill, New York.Utkin, V.I. (1992) Sliding Modes in Optimization and Control, Springer-Verlag, New York.Zames, G., and Francis, B.A. (1983) “Feedback, Minimax Sensitivity, and Optimal Robustness,” IEEE

Trans. Autom. Control AC-28, pp. 585–601. Zames, G. (1981) “Feedback and Optimal Sensitivity: Model Reference Transformations, Multiplicative

Seminorms, and Approximate Inverses,” IEEE Trans. Autom. Control AC-26, pp. 301–320. Zeitz, M. (1983) “Controllability Canonical (Phase-Variable) Forms for Nonlinear Time-Variable

Systems,” Int. J. Control 37, pp. 1449–1457. Zhou, K. (1996) Robust and Optimal Control, Prentice-Hall, Englewood Cliffs, NJ.Ziegler, J.G., and Nichols, N.B. (1942) “Optimum Settings for Automatic Controllers,” ASME Trans. 64,

pp. 759–768. Ziegler, J.G., and Nichols, N.B. (1943) “Process Lags in Automatic Control Circuits,” ASME Trans. 65,

pp. 433–444.

For Further Information

Most of the material covered in the chapter is thoroughly covered in textbooks. For the section on classicalcontrol, the undergraduate texts by Dorf (1992), Franklin et al. (1994), Kuo (1995), Ogata (1997) andRaven (1995) provide a complete mathematical treatment of root locus design, PID control, lead-lagcompensation and basic state-space methods. Textbooks for linear, robust control include Doyle et al.(1992) and Zhou (1996).

The standard textbooks for geometric nonlinear control are Isidori (1996) and Nijmeijer and van derSchaft (1990). Additional material concerning the mathematical basis for differential geometry is foundin Abraham et al. (1988) and Boothby (1986). Sastry (2000) provides an overview of differential geometrictechniques but also considers Lyapunov-based methods and nonlinear dynamical systems in general.Khalil (1996) focuses primarily on Lyapunov methods and includes a chapter on differential geometrictechniques.


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