Table of contents for Analysis and control of linear systems analysis and control of linear systems / edited by Philippe de Larminat.

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TABLE OF CONTENTS
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CHAPTER 2	57
STATE REPRESENTATION	57
2.1. REMINDER NOTES ON THE SYSTEMS	58
2.1.1. INTERNAL REPRESENTATION OF 
DETERMINIST SYSTEMS: THE CONCEPT OF 
STATE	58
2.1.2. EQUATIONS OF STATE AND EQUATIONS OF 
MEASUREMENT FOR CONTINUOUS SYSTEMS
	60
2.1.3. CASE OF LINEAR SYSTEMS	61
2.1.4. CASE OF CONTINUOUS AND INVARIANT 
LINEAR SYSTEMS	61
2.2. RESOLVING THE EQUATION OF STATE	61
2.2.1. FREE STATE	62
2.2.2. FORCED STATE	62
2.2.3. PARTICULAR CASE OF LINEAR AND 
INVARIANT SYSTEMS	63
2.3.4. CALCULATION METHOD OF THE TRANSITION 
MATRIX 	64
USE OF SERIAL DEVELOPMENT	64
?	SYLVESTER FORMULA (DISTINCT 
EIGENVALUES OF A)	65
?	SYLVESTER INTERPOLATION METHOD	66
?	METHOD OF MODES	67
2.2.5. APPLICATION FOR THE MODELING OF 
LINEAR DISCRETE SYSTEMS	68
2.3. SCALAR REPRESENTATION OF LINEAR AND 
INVARIANT SYSTEMS	70
2.3.1. STATE ? TRANSFER PASSAGE	70
2.3.2. CHANGE OF BASIS IN THE STATE SPACE	72
2.3.3. TRANSFER ? STATE PASSAGE	73
?	COMPANION FORM FOR THE CONTROL	74
2.3.4. SCALAR REPRESENTATION OF INVARIANT 
AND LINEAR DISCRETE SYSTEMS	77
2.4. CONTROLABILITY OF SYSTEMS	78
2.4.1. GENERAL DEFINITIONS	78
?	EXAMPLE 2.6.- LET US CONSIDER THE SIZE 
4MONO-VARIABLE SYSTEM DEFINED BY THE 
EQUATION OF STATE:	78
2.4.2. CONTROLABILITY OF LINEAR AND 
INVARIANT SYSTEMS	78
2.4.3. CANONIC REPRESENTATION OF PARTIALLY 
CONTROLABLE SYSTEMS	81
2.4.4. SCALAR REPRESENTATION OF PARTIALLY 
CONTROLABLE SYSTEMS	85
2.5. OBSERVABILITY OF SYSTEMS	86
2.5.1. GENERAL DEFINITIONS	86
2.5.2. OBSERVABILITY OF LINEAR AND INVARIANT 
SYSTEMS	86
2.5.3. CASE OF PARTIALLY OBSERVABLE SYSTEMS
	89
2.5.4. CASE OF PARTIALLY CONTROLABLE AND 
PARTIALLY OBSERVABLE SYSTEMS	90
2.6. BIBLIOGRAPHY	90
CHAPTER 3	93
DISCRETE-TIME SYSTEMS	93
3.1. INTRODUCTION	93
3.2. DISCRETE SIGNALS: ANALYSIS AND 
MANIPULATION	94
3.2.1. REPRESENTATION OF A DISCRETE SIGNAL	94
3.2.2. DELAY AND LEAD OPERATORS	95
3.2.3. Z TRANSFORM	96
3.2.3.1. DEFINITION	96
?	3.2.3.2. INVERSE TRANSFORM	97
?	3.2.3.3. PROPERTIES OF THE Z TRANSFORM	98
?	3.2.3.4. RELATIONS BETWEEN THE FOURIER-
LAPLACE TRANSFORMS AND THE Z 
TRANSFORM	99
3.3. DISCRETE SYSTEMS (LTID)	100
3.3.1. EXTERNAL REPRESENTATION	100
?	3.3.1.1. SYSTEMS DEFINED BY A DIFFERENCE 
EQUATION	100
?	3.3.1.2. REPRESENTATION USING THE IMPULSE 
RESPONSE	100
3.3.2. INTERNAL RESPONSE	101
3.3.3. REPRESENTATION IN TERMS OF OPERATOR
	103
?	USE OF OPERATOR 	105
?	COMBINED USE OF OPERATORS AND 	106
3.3.4. TRANSFER FUNCTION AND FREQUENCY 
RESPONSE	107
3.3.5. TIME RESPONSE OF BASIC SYSTEMS	110
?	3.3.5.1. FIRST ORDER SYSTEM	110
?	3.3.5.2. SECOND ORDER SYSTEM	110
3.4. DISCRETIZATION OF CONTINUOUS-TIME 
SYSTEMS	111
3.4.1. DISCRETIZATION OF ANALOG SIGNALS	112
3.4.2. TRANSFER FUNCTION OF THE DISCRETIZED 
SYSTEM	113
3.4.3. STATE REPRESENTATION OF THE 
DISCRETIZED SYSTEM	114
3.4.4. FREQUENCY RESPONSES OF THE 
CONTINUOUS AND DISCRETE SYSTEM	115
3.4.5. THE PROBLEM OF SUB-SAMPLING	116
3.4.6. THE PROBLEM OF OVER-SAMPLING	117
3.5. CONCLUSION	118
3.6. BIBLIOGRAPHY	119
CHAPTER 4	121
STRUCTURAL PROPERTIES OF LINEAR SYSTEMS
	121
4.1. INTRODUCTION: BASIC TOOLS FOR A 
STRUCTURAL ANALYSIS OF SYSTEMS	122
4.1.1. VECTOR SPACES, LINEAR APPLICATIONS	123
4.1.2. INVARIANT SUB-SPACES	123
4.1.3. POLYNOMIALS, POLYNOMIAL MATRICES	125
4.1.4. SMITH FORM, COMPANION FORM, JORDAN 
FORM	126
4.1.5. NOTES AND REFERENCES	128
4.2. BEAMS, CANONICAL FORMS, AND INVARIANTS
	128
4.2.1. MATRIX BEAMS AND GEOMETRY	130
4.2.2. KRONECKER'S CANONICAL FORM	130
4.2.3. CONTROLLABLE, OBSERVABLE CANONICAL 
FORM (BRUNOVSKY)	133
4.2.4. MORSE'S CANONICAL FORM	138
4.2.5. NOTES AND REFERENCES	140
4.3. INVARIANT STRUCTURES UNDER 
TRANSFORMATION GROUPS	140
4.3.1. CONTROLLABILITY INDICES	140
4.3.2. OBSERVABILITY INDICES	141
4.3.3. INFINITE ZEROS	141
4.3.4. INVARIANT, TRANSMISSION FINITE ZEROS	143
4.3.5. NOTES AND REFERENCES	144
4.4. AN INTRODUCTION TO A STRUCTURAL 
APPROACH OF THE CONTROL	144
4.4.1. DISTURBANCE REJECTION AND 
DECOUPLING: EXISTENCE OF SOLUTIONS	145
4.4.2. DISTURBANCE REJECTION AND 
DECOUPLING: EXISTENCE OF STABLE 
SOLUTIONS	146
4.4.3. DISTURBANCE REJECTION AND 
DECOUPLING: FLEXIBILITY IN THE 
LOCATION OF POLES/FIXED POLES	147
4.4.4. NOTES AND REFERENCES	148
4.5. CONCLUSION	148
4.4.5. OPTIMAL ATTENUATION OF DISTURBANCE
	149
4.6. BIBLIOGRAPHY	149
CHAPTER 5. SIGNALS: DETERMINISTIC AND 
STATIC MODELS	
CHAPTER 6	171
KALMAN'S FORMALISM FOR STATE 
STABILIZATION AND ESTIMATION	171
6.1. THE ACADEMIC PROBLEM OF STABILIZATION 
THROUGH STATE FEEDBACK	171
6.2. STABILIZATION THROUGH POLE PLACEMENT
	173
6.2.1. RESULTS	173
6.2.2. EXAMPLE	175
6.3. REBUILD OF STATE AND OBSERVERS	176
6.3.1. GENERAL PRINCIPLES	176
6.3.2. CONTINUOUS-TIME OBSERVER	176
6.3.3. DISCRETE-TIME OBSERVER	178
6.3.4. CALCULATION OF OBSERVER THROUGH 
POLE PLACEMENT	178
6.3.5. BEHAVIOR OF THE OBSERVER OUTSIDE THE 
IDEAL CASE	179
6.3.6. EXAMPLE	180
6.4. STABILIZATION THROUGH QUADRATIC 
OPTIMIZATION	182
6.4.1. GENERAL RESULTS FOR CONTINUOUS-TIME
	182
6.4.2. GENERAL RESULTS FOR DISCRETE-TIME	184
6.4.3. INTERPRETATION OF THE RESULTS	185
6.4.4. EXAMPLE	186
6.5. RESOLUTION OF THE STATE REBUILD 
PROBLEM BY DUALITY OF THE QUADRATIC 
OPTIMIZATION	187
6.5.1. CALCULATION OF A CONTINUOUS-TIME 
OBSERVER	187
6.5.2. CALCULATION OF A DISCRETE-TIME 
OBSERVER	189
6.5.3. INTERPRETATION IN A STOCHASTIC 
CONTEXT	189
6.5.4. EXAMPLE	191
6.6. STATE FEEDBACK CONTROL AND OBSERVER
	192
6.6.1. IMPLEMENTATION OF THE CONTROL	192
6.6.2. DYNAMICS OF THE LOOPED SYSTEM	193
6.6.3. INTEREST AND LIMITATIONS OF THIS 
RESULT	194
6.6.4. INTERPRETATION IN THE FORM OF 
EQUIVALENT CORRECTOR	195
6.6.5. EXAMPLE	196
6.7. A FEW WORDS ON THE RESOLUTION OF 
RICCATI'S EQUATIONS	198
6.8. CONCLUSION	201
6.9. BIBLIOGRAPHY	201
CHAPTER 9	21
ANALYSIS BY CLASSIC SCALAR APPROACH	21
9.1. CONFIGURATION OF FEEDBACK LOOPS	21
9.1.1. OPEN LOOP - CLOSED LOOPS	21
9.1.2. CLOSED LOOP HARMONIC ANALYSIS	23
BAND WIDTH	23
?	APPROXIMATE TRACE	23
?	9.1.2.1. BLACK-NICHOLS ABACUS	24
?	9.1.2.2. ESTIMATION OF CLOSED LOOP TIME 
PERFORMANCES FROM THE HARMONIC 
ANALYSIS	25
9.2. STABILITY	26
9.2.1. NYQUIST'S CRITERION	27
?	CASE 1. OPEN LOOP STABLE SYSTEM- NYQUIST'S 
TRACES	28
?	CASE 2. OPEN LOOP INTEGRATOR SYSTEM	30
?	9.1.2.3. STATEMENT OF NYQUIST CRITERION.	31
?	CASE 3. UNSTABLE SYSTEM IN OPEN LOOP	31
9.2.2. ROUTH'S ALGEBRAIC CRITERION	33
?	9.1.2.5. STATEMENT OF ROUTH'S CRITERION	34
9.2.3. STABILITY MARGINS	35
?	A) PHASE MARGIN - GAIN MARGIN	35
?	B) DELAY MARGIN - MODULE MARGIN	36
?	C) DEGREE OF STABILITY OF A SECOND ORDER 
SYSTEM	37
?	D) DEGREE OF STABILITY OF ANY ORDER 
SYSTEM	38
9.3. PRECISION	38
9.3.1. PERMANENT ERROR	40
?	9.3.1.2. UNIT STEP INTERFERENCE - NULL 
INPUT	40
9.3.1.3. RAMP INPUT - NULL INTERFERENCE	41
?	9.3.1.4. SINUSOIDAL INPUT - NULL 
INTERFERENCE	43
9.3.2. TRANSITIONAL ERROR	44
?	9.3.2.1. CASE OF A UNITARY FEEDBACK - NULL 
INTERFERENCE	44
9.4. PARAMETRIC SENSITIVITY	45
9.4.1. OPEN LOOP SENSITIVITY	45
9.4.2. CLOSED LOOP SENSITIVITY	47
9.5. BIBLIOGRAPHY	49
CHAPTER 10	49
SYNTHESIS OF CLOSED-LOOP CONTROL SYSTEMS
	49
10.1. ROLE OF CORRECTORS: PRECISION-
STABILITY DILEMMA	49
10.1.1. ANALYSIS OF SYSTEMS' BEHAVIOR	50
10.1.1.1. Static errors	50
?	13.1.1.2. STABILITY	50
?	10.1.1.3. RAPIDITY	51
?	CLOSED LOOP FREQUENCY BEHAVIOR, BAND 
WIDTH	51
?	TIME BEHAVIOR IN UNIT-STEP RESPONSE	52
?	DYNAMICAL PRECISION	52
10.1.2. SERIAL CORRECTION	54
10.1.3. PARALLEL CORRECTION	55
10.1.4. CORRECTION BY ANTICIPATION	56
?	10.1.4.1. COMPENSATION OF INTERFERENCES 
(NULL INPUT)	56
?	10.1.4.2. COMPENSATION OF THE INPUT (NULL 
INTERFERENCE)	57
10.1.5. CONCLUSIONS	58
10.2. SERIAL CORRECTION	58
10.2.1. Correction by phase lead	58
?	10.2.1.1. TRANSFER FUNCTION	58
?	10.2.1.2. ACTION MECHANISM OF A PHASE LEAD 
CORRECTOR	60
10.2.2. CORRECTION BY PHASE LAG	64
10.2.2.1. Transfer function	64
10.2.2.2.	ACTION MECHANISM OF THESE 
CORRECTORS	65
10.3. CORRECTION BY COMBINED ACTIONS	68
10.3.1. Transfer function	68
?	10.3.1.1. ACTION MECHANISM OF THESE 
CORRECTORS	69
10.4. DERIVATIVE PROPORTIONAL (DP) 
CORRECTION	71
10.4.1. TRANSFER FUNCTION	71
10.5. INTEGRAL PROPORTIONAL (IP) CORRECTION
	72
10.5.1. Transfer function	72
10.6. DERIVATIVE INTEGRAL PROPORTIONAL (DIP) 
CORRECTION	74
10.6.1. Transfer function	74
10.6.2. METHOD OF EXPERIMENTAL ADJUSTMENT
	77
10.7. PARALLEL CORRECTION	79
10.7.1. GENERAL PRINCIPLE	79
10.7.2. SIMPLE TACHYMETRIC CORRECTION (C(P) = 
?P)	81
10.7.3. FILTERED TACHYMETRIC CORRECTION	83
10.7.4. CORRECTION OF LAG SYSTEMS: SMITH'S 
PREDICTOR	87
10.8. BIBLIOGRAPHY	88
CHAPTER 11	87
RELIABLE SINGLE-VARIABLE CONTROL 
THROUGH POLE PLACEMENT	87
11.1. INTRODUCTION	87
11.1.2. REMINDER NOTES ON POLYNOMIAL 
ALGEBRA	89
11.2. THE OBVIOUS OBJECTIVES OF THE 
CORRECTION	92
11.2.1. INTERNAL STABILITY	92
11.2.2. STATIONARY BEHAVIOR	93
11.2.3. GENERAL FORMULATION	94
11.3. RESOLUTION	95
11.3.1 RESOLUTION OF A PARTICULAR CASE	96
?	11.3.1.1. CONDITIONS ON THE DEGREES	97
?	11.3.1.2. STANDARD SOLUTION	98
?	11.3.1.3. EXAMPLE	99
?	B1 = S+1 AND B2 = 1	99
?	B1 = 1	100
11.3.2. GENERAL CASE	101
11.3.2.1. CHOICE OF DEGREES	101
?	11.3.2.2. EXAMPLE	103
11.4. IMPLEMENTATION	103
11.4.1. FIRST POSSIBILITY	103
11.4.2. MINIMAL REPRESENTATION	104
?	11.4.2.1. EXAMPLE	105
?	11.4.2.2 GENERALIZATION	106
11.4.3. MANAGEMENT OF SATURATIONS	107
11.4.3.1. METHOD	107
?	11.4.3.2. JUSTIFICATION OF THE METHOD [KAI 
80]	109
11.5. METHODOLOGY	111
11.5.1. INTUITIVE APPROACH	112
11.5.2. REDUCTION OF THE NOISE ON THE 
CONTROL BY CHOICE OF DEGREES	114
11.5.3. CHOICE OF THE DYNAMICS OF AM AND AO
	115
?	11.5.3.1. OPTIMAL CHOICE OF AM	115
11.5.3.3. STUDY OF THE SENSITIVITY FUNCTION	117
?	11.5.3.4. SOME IMPROVEMENTS	119
?	11.5.3.5. METHOD OF ADJUSTMENT	120
11.5.4. EXAMPLES	121
11.6. CONCLUSION	127
11.7. BIBLIOGRAPHY	127
CHAPTER 12	127
FEEDFORWARD CONTROL	127
12.1. GENERAL PRINCIPLES OF FEEDFORWARD 
CONTROL	127
12.1.4. PRINCIPLE OF THE SLIDING HORIZON	128
12.2. GENERALIZED FEEDFORWARD CONTROL 
(GPC)	129
?	12.2.1.1. DEFINITION OF NUMERICAL MODEL	129
?	OPTIMAL PREDICTOR	130
?	12.2.1.2 DEFINITION AND MINIMIZATION OF 
THE QUADRATIC CRITERION	131
?	12.2.1.3. SYNTHESIS OF THE EQUIVALENT 
POLYNOMIAL RST REGULATOR	131
12.2.2 AUTOMATIC SYNTHESIS OF ADJUSTMENT 
PARAMETERS	133
12.2.2.1. CRITERION OF STABILITY AND 
RELIABILITY	134
12.2.3. EXTENSION OF THE BASIC VERSION	135
?	12.2.2.3. STRUCTURE OF MULTIPLE REFERENCE 
MODELS	135
?	12.2.2.4. CASCADE STRUCTURE	138
?	12.2.2.5. RECOGNITION OF EQUALITY TYPE 
TERMINAL CONSTRAINTS (CRHPC)	140
12.3. FUNCTIONAL FEEDFORWARD CONTROL (FPC)
	143
12.3.1. DEFINITION OF NUMERICAL MODEL	143
12.3.3. OBJECT-MODEL DIFFERENCE	144
12.3.4. STRUCTURE OF THE FUTURE CONTROL	145
12.3.5. STRUCTURE OF THE OPTIMAL PREDICTOR
	145
12.3.6. DEFINITION OF QUADRATIC CRITERION, 
CONCEPT OF MATCH POINTS	146
12.3.7. ADJUSTMENT PARAMETERS	147
12.4. CONCLUSION	148
12.5. BIBLIOGRAPHY	149
CHAPTER 13 METHODOLOGY OF THE STATE 
APPROACH CONTROL	151
13.2.1. STANDARDS	155
13.2.1.1. SIGNAL STANDARD	155
13.2.1.2. STANDARD INDUCED ON THE SYSTEMS	155
THE GRAMMIANS' ROLE IN THE CALCULATION OF 
THE STANDARD	156
13.2.2. H2 OPTIMIZATION	159
13.2.1.3. DEFINITION OF THE STANDARD H2 
PROBLEM [DOY 89]	159
13.2.1.4. RESOLUTION OF THE H2 STANDARD 
OPTIMIZATION PROBLEM	160
13.2.3. H2 - LQG	163
13.2.4. H2 - LTR	165
13.2.6. GENERALIZED H2 PROBLEM AND RELIABLE 
RPIS	172
13.2.7. DISCRETIZATION OF THE H2 PROBLEM	174
13.3.1. MODEL OF THE PROCESS	177
13.3.3. ADDITIONAL DATA	181
13.4.2. DEFINITION OF THE H2 OPTIMIZATION 
PROBLEM	184
13.4.3. THE INTEREST IN STANDARDIZATION	186
13.5. CONCLUSION	188
13.6. APPENDICES	188
13.6.1. RESOLUTION OF LYAPUNOV'S EQUATIONS
	188
13.6.2. DUALITY PRINCIPLE	189
13.6.5. STANDARDIZATION OF A SYSTEM	192
13.7. BIBLIOGRAPHY	192
REFERENCES COMPLEMENTAIRES RELATIVES A 
LA COMMANDE LQG/LTR	194
REFERENCES COMPLEMENTAIRES RELATIVES A 
LA COMMANDE LQGPF/H2	194
REFERENCES RELATIVES AU PROBLEME DE 
REGULATION SOUS CONTRAINTE DE 
STABILITE INTERNE	195
CHAPTER 14
CHAPTER 15	233
THE ROBUST CONTROL IN H?/LMI	233
15.1. THE H? APPROACH	234
15.1.2. EXAMPLE	236
15.1.3. RESOLUTION METHODS	242
15.1.4. RESOLUTION OF PROBLEM STANDARD 
THROUGH RICCATI'S EQUATIONS	243
15.1.5. RESOLUTION OF THE PROBLEM 
STANDARD THROUGH LMI	245
15.1.6. RESTRICTED SYNTHESIS ON THE 
EQUALIZER ORDER	247
15.2. THE ?-ANALYSIS	248
15.2.1. ANALYSIS DIAGRAM AND STRUCTURED 
SINGLE VALUE	248
15.2.2. MAIN RESULTS OF ROBUSTNESS	249
15.2.3. EXAMPLE	250
15.2.4. EVALUATION OF STRUCTURED SINGLE 
VALUE	254
15.3. THE ?-SYNTHESIS	256
15.3.1. A H? ROBUST SYNTHESIS	256
15.3.2. APPROACH THROUGH D-K ITERATIONS	257
15.3.3. EXAMPLE	260
15.4. SYNTHESIS OF AN EQUALIZER DEPENDING 
ON VARYING PARAMETERS	261
15.4.1. PROBLEM CONSIDERED AND L2 GAIN	261
15.4.2. POLY-TOPIC APPROACH	263
15.4.3. A MORE GENERAL APPROACH	266
15.4.4. EXAMPLE	268
15.5. CONCLUSION	271
15.6. BIBLIOGRAPHY	271
CHAPTER 16	271
TIME-VARIANT LINEAR SYSTEMS	271
16.1. RING OF NON-COMMUTATIVE POLYNOMIALS
	272
16.1.1. DIVISION AND THE RIGHT HIGHEST DIVISOR 
(RHD)	273
16.1.2. RIGHT LEAST COMMON MULTIPLE (RLCM)
	273
16.1.3. EXPLICIT FORMULATION OF RLCM	274
16.1.4. FACTORING, ROOTS, RELATIONS WITH THE 
COEFFICIENTS	274
16.2. BODY OF RATIONAL FRACTIONS	275
16.3. TRANSFER FUNCTION	276
16.3.1. PROPERTIES OF TRANSFER FUNCTIONS	277
16.3.2. NORMAL MODES	277
16.3.3. STABILITY	277
16.4. ALGEBRA OF NON-STATIONARY LINEAR 
SYSTEMS	278
16.4.1. SERIAL SYSTEMS	278
16.4.2. PARALLEL SYSTEMS	279
16.5.1. MODELING	280
16.5.2. POLE PLACEMENT	281
16.6. CONCLUSION	283
16.7. BIBLIOGRAPHY	284
list of authors	
index

Library of Congress Subject Headings for this publication:

Linear control systems.
Automatic control.