Modern Control Engineering

Modern Control Engineering
Katsuhiko Ogata

Chapter 1 Introduction 1.1 Automatic control
21 century — information age, cybernetics(control theory), system approach and information theory , three science theory mainstay(supports) in 21 century. 1.1 Automatic control A machine(or system) work by machine-self, not by manual operation. 1.2 Automatic control systems 1.2.1 examples Figure 1.1 1) A water-level control system * Operating principle…… * Feedback control……

Chapter 1 Introduction 2) A temperature Control system
Another example of the water-level control is shown in figure 1.2. * Operating principle…… * Feedback control…… 2) A temperature Control system (shown in Fig.1.3) * Operating principle… * Feedback control(error)…

Chapter 1 Introduction 3) A DC-Motor control system * Principle…
* Feedback control(error)…

Chapter 1 Introduction 4) A servo (following) control system
Fig. 1.5 * principle…… * feedback(error)……

Chapter 1 Introduction 5) A feedback control system model of the family planning (similar to the social, economic, and political realm(sphere or field)) Fig. 1.6 * principle…… * feedback(error)……

Chapter 1 Introduction 1.2.2 block diagram of control systems Fig. 1.7
The block diagram description for a control system : Convenience Fig. 1.7 Example:

Chapter 1 Introduction Figure 1.1 For the Fig.1.1, The water level control system: resistance comparator Actuator Actual water level Output Desired water level Input amplifier Motor Gearing Valve Water container Error Process controller Float Feedback signal measurement (Sensor) Fig. 1.8

Chapter 1 Introduction For the Fig. 1.4, The DC-Motor control system

Chapter 1 Introduction 1.2.3 Fundamental structure of control systems
1) Open loop control systems Features: Only there is a forward action from the input to the output.

Chapter 1 Introduction 2) Closed loop (feedback) control systems
Features: 1) measuring the output (controlled variable) . 2) Feedback. not only there is a forward action , also a backward action between the output and the input (measuring the output and comparing it with the input).

Chapter 1 Introduction Notes: 1) Positive feedback; 2) Negative feedback—Feedback. 1.3 types of control systems 1) linear systems versus Nonlinear systems. 2) Time-invariant systems vs. Time-varying systems. 3) Continuous systems vs. Discrete (data) systems. 4) Constant input modulation vs. Servo control systems. 1.4 Basic performance requirements of control systems 1) Stability. 2) Accuracy (steady state performance). 3) Rapidness (instantaneous characteristic).

Chapter 1 Introduction 1.5 An outline of this text
1) Three parts: mathematical modeling; performance analysis ; compensation (design). 2) Three types of systems: linear continuous; nonlinear continuous; linear discrete. 3) three performances: stability, accuracy, rapidness. in all: to discuss the theoretical approaches of the control system analysis and design. 1.6 Control system design process shown in Fig.1.12

Chapter 1 Introduction 1. Establish control goals
2. Identify the variables to control 3. Write the specifications for the variables 4. Establish the system configuration Identify the actuator 5. Obtain a model of the process, the actuator and the sensor 6. Describe a controller and select key parameters to be adjusted 7. Optimize the parameters and analyze the performance Performance does not Meet the specifications Finalize the design Performance meet the specifications Fig.1.12

Chapter 1 Introduction 1.7 Sequential design example: disk drive read system A disk drive read system Shown in Fig.1.13 Actuator motor Arm Spindle Track a Track b Head slider Rotation of arm Disk Fig.1.13 A disk drive read system ◆ Configuration ◆ Principle　

Chapter 1 Introduction Sequential design:
here we are concerned with the design steps 1,2,3, and 4 of Fig.1.12. Identify the control goal: Position the reader head to read the date stored on a track on the disk. (2) Identify the variables to control: the position of the read head. (3) Write the initial specification for the variables: The disk rotates at a speed of between 1800 and 7200 rpm and the read head “flies” above the disk at a distance of less than 100 nm. The initial specification for the position accuracy to be controlled: ≤ 1 μm (leas than 1 μm ) and to be able to move the head from track a to track b within 50 ms, if possible.

Chapter 1 Introduction (4) Establish an initial system configuration:
It is obvious : we should propose a closed loop system , not a open loop system. An initial system configuration can be shown as in Fig.1.13. Control device Actuator motor Read arm sensor Desired head position error Actual Fig.1.13 system configuration for disk drive We will consider the design of the disk drive further in the after-mentioned chapters.

Chapter 1 Introduction Exercise: E1.6, P1.3, P1.13

Chapter 2 mathematical models of systems
2.1 Introduction Why？ 1) Easy to discuss the full possible types of the control systems—in terms of the system’s “mathematical characteristics”. 2) The basis — analyzing or designing the control systems. For example, we design a temperature Control system : Controller Actuator Process Disturbance Input r(t) desired output temperature Output T(t) actual output Control signal Actuating u k a c Fig. 2.1 measurement Feedback signal b(t) + - (－) e(t)= r(t)-b(t) The key — designing the controller → how produce uk.

Different characteristic of the process — different uk: For T1 T(t) Ⅰ T2 Ⅱ uk21 T1 uk11 uk12 uk For T1 What is ？ Mathematical models of the control systems—— the mathematical relationships between the system’s variables. How get？ 1) theoretical approaches 2) experimental approaches 3) discrimination learning

types 1) Differential equations 2) Transfer function 3) Block diagram、signal flow graph 4) State variables(modern control theory) 2.2 Input-output description of the physical systems — differential equations The input-output description—description of the mathematical relationship between the output variable and the input variable of the physical systems. 2.2.1 Examples

Example 2.1 : A passive circuit define: input → ur output → uc。 we have：

Example 2.2 : A mechanism Define: input → F ，output → y. We have: Compare with example 2.1: uc→y; ur→F ─ analogous systems

Example 2.3 : An operational amplifier (Op-amp) circuit Input →ur output →uc (2)→(3); (2)→(1); (3)→(1)：

Example 2.4 : A DC motor Input → ua， output → ω1 (4)→(2)→(1) and (3)→(1):

Make:

The differential equation description of the DC motor is: Assume the motor idle: Mf = 0, and neglect the friction: f = 0, we have:

Example 2.5 : A DC-Motor control system Input → ur，Output → ω; neglect the friction:

（2）→（1）→（3）→（4），we have： steps to obtain the input-output description (differential equation) of control systems 1) Determine the output and input variables of the control systems. 2) Write the differential equations of each system’s components in terms of the physical laws of the components. * necessary assumption and neglect. * proper approximation.

3) dispel the intermediate(across) variables to get the input-output description which only contains the output and input variables. 4) Formalize the input-output equation to be the “standard” form: Input variable —— on the right of the input-output equation . Output variable —— on the left of the input-output equation. Writing polynomial—— according to the falling-power order. General form of the input-output equation of the linear control systems—A nth-order differential equation: Suppose: input → r ，output → y

2.3 Linearization of the nonlinear components what is nonlinearity？ The output is not linearly vary with the linear variation of the system’s (or component’s) input → nonlinear systems (or components). How do the linearization？ Suppose: y = f(r) The Taylor series expansion about the operating point r0 is:

Examples: Example 2.6 : Elasticity equation Example 2.7 : Fluxograph equation Q —— Flux; p —— pressure difference

2.4 Transfer function Another form of the input-output(external) description of control systems, different from the differential equations. definition Transfer function: The ratio of the Laplace transform of the output variable to the Laplace transform of the input variable,with all initial condition assumed to be zero and for the linear systems, that is:

C(s) —— Laplace transform of the output variable R(s) —— Laplace transform of the input variable G(s) —— transfer function Notes: * Only for the linear and stationary(constant parameter) systems. * Zero initial conditions. * Dependent on the configuration and the coefficients of the systems, independent on the input and output variables. How to obtain the transfer function of a system 1) If the impulse response g(t) is known

We have: Because: Then: Example 2.8 : 2) If the output response c(t) and the input r(t) are known We have:

Example 2.9: Then: 3) If the input-output differential equation is known Assume: zero initial conditions; Make: Laplace transform of the differential equation; Deduce: G(s)=C(s)/R(s).

Example 2.10: 4) For a circuit * Transform a circuit into a operator circuit. * Deduce the C(s)/R(s) in terms of the circuits theory.

Example 2.11: For a electric circuit:

Example 2.12: For a op-amp circuit

5) For a control system Write the differential equations of the control system, and Assume zero initial conditions; Make Laplace transformation, transform the differential equations into the relevant algebraic equations; Deduce: G(s)=C(s)/R(s). Example 2.13 the DC-Motor control system in Example 2.5

In Example 2.5, we have written down the differential equations as: Make Laplace transformation, we have:

(2)→(1)→(3)→(4), we have:

2.5 Transfer function of the typical elements of linear systems A linear system can be regarded as the composing of several typical elements, which are: Proportioning element Relationship between the input and output variables: Transfer function: Block diagram representation and unit step response: Examples: amplifier, gear train, tachometer…

Integrating element Relationship between the input and output variables: Transfer function: Block diagram representation and unit step response: Examples: Integrating circuit, integrating motor, integrating wheel…

Differentiating element Relationship between the input and output variables: Transfer function: Block diagram representation and unit step response: Examples: differentiating amplifier, differential valve, differential condenser…

Inertial element Relationship between the input and output variables: Transfer function: Block diagram representation and unit step response: Examples: inertia wheel, inertial load (such as temperature system)…

Oscillating element Relationship between the input and output variables: Transfer function: Block diagram representation and unit step response: Examples: oscillator, oscillating table, oscillating circuit…

Delay element Relationship between the input and output variables: Transfer function: Block diagram representation and unit step response: Examples: gap effect of gear mechanism, threshold voltage of transistors…

block diagram models (dynamic) Portray the control systems by the block diagram models more intuitively than the transfer function or differential equation models. Block diagram representation of the control systems Examples:

Example 2.14 For the DC motor in Example 2.4 In Example 2.4, we have written down the differential equations as: Make Laplace transformation, we have:

Draw block diagram in terms of the equations (5)～(8): Consider the Motor as a whole:

Example 2.15 The water level control system in Fig 1.8:

The block diagram model is:

2.1 Introduction Why？ 1) Easy to discuss the full possible types of the control systems —only in terms of the system’s “mathematical characteristics”. 2) The basis of analyzing or designing the control systems. What is ？ Mathematical models of systems — the mathematical relationships between the system’s variables. How get？ 1) theoretical approaches 2) experimental approaches 3) discrimination learning

types 1) Differential equations 2) Transfer function 3) Block diagram、signal flow graph 4) State variables 2.2 The input-output description of the physical systems — differential equations The input-output description—description of the mathematical relationship between the output variable and the input variable of physical systems. 2.2.1 Examples

Example 2.1 : A passive circuit define: input → ur output → uc。 we have：

Example 2.2 : A mechanism Define: input → F ，output → y. We have: Compare with example 2.1: uc→y, ur→F---analogous systems

Example 2.3 : An operational amplifier (Op-amp) circuit Input →ur output →uc (2)→(3); (2)→(1); (3)→(1)：

Example 2.4 : A DC motor Input → ua， output → ω1 (4)→(2)→(1) and (3)→(1):

Make:

the differential equation description of the DC motor is: Assume the motor idle: Mf = 0, and neglect the friction: f = 0, we have: Compare with example 2.1 and example 2.2: ----Analogous systems

Example 2.5 : A DC-Motor control system Input → ur, Output →ω; neglect the friction:

（2）→（1）→（3）→（4），we have： steps to obtain the input-output description (differential equation) of control systems 1) Identify the output and input variables of the control systems. 2) Write the differential equations of each system’s component in terms of the physical laws of the components. * necessary assumption and neglect. * proper approximation. 3) dispel the intermediate(across) variables to get the input- output description which only contains the output and input variables.

4) Formalize the input-output equation to be the “standard” form: Input variable —— on the right of the input-output equation . Output variable —— on the left of the input-output equation. Writing the polynomial—according to the falling-power order. General form of the input-output equation of the linear control systems ——A nth-order differential equation: Suppose: input → r ，output → y

2.3 Linearization of the nonlinear components what is nonlinearity？ The output of system is not linearly vary with the linear variation of the system’s (or component’s) input → nonlinear systems (or components). How do the linearization？ Suppose: y = f(r) The Taylor series expansion about the operating point r0 is:

Examples: Example 2.6 : Elasticity equation Example 2.7 : Fluxograph equation Q —— Flux; p —— pressure difference

2.4 Transfer function Another form of the input-output(external) description of control systems, different from the differential equations. definition Transfer function: The ratio of the Laplace transform of the output variable to the Laplace transform of the input variable with all initial condition assumed to be zero and for the linear systems, that is:

C(s) —— Laplace transform of the output variable R(s) —— Laplace transform of the input variable G(s) —— transfer function Notes: * Only for the linear and stationary(constant parameter) systems. * Zero initial conditions. * Dependent on the configuration and coefficients of the systems, independent on the input and output variables. How to obtain the transfer function of a system 1) If the impulse response g(t) is known

Because: We have: Then: Example 2.8 : 2) If the output response c(t) and the input r(t) are known We have:

Example 2.9: Then: 3) If the input-output differential equation is known Assume: zero initial conditions; Make: Laplace transform of the differential equation; Deduce: G(s)=C(s)/R(s).

Example 2.10: 4) For a circuit * Transform a circuit into a operator circuit. * Deduce the C(s)/R(s) in terms of the circuits theory.

Example 2.11: For a electric circuit:

Example 2.12: For a op-amp circuit

5) For a control system Write the differential equations of the control system; Make Laplace transformation, assume zero initial conditions, transform the differential equations into the relevant algebraic equations; Deduce: G(s)=C(s)/R(s). Example 2.13 the DC-Motor control system in Example 2.5

In Example 2.5, we have written down the differential equations as: Make Laplace transformation, we have: (2)→(1)→(3)→(4), we have:

2.5 Transfer function of the typical elements of linear systems A linear system can be regarded as the composing of several typical elements, which are: Proportioning element Relationship between the input and output variables: Transfer function: Block diagram representation and unit step response: Examples: amplifier, gear train, tachometer…

Integrating element Relationship between the input and output variables: Transfer function: Block diagram representation and unit step response: Examples: Integrating circuit, integrating motor, integrating wheel…

Differentiating element Relationship between the input and output variables: Transfer function: Block diagram representation and unit step response: Examples: differentiating amplifier, differential valve, differential condenser…

Inertial element Relationship between the input and output variables: Transfer function: Block diagram representation and unit step response: Examples: inertia wheel, inertial load (such as temperature system)…

Oscillating element Relationship between the input and output variables: Transfer function: Block diagram representation and unit step response: Examples: oscillator, oscillating table, oscillating circuit…

Delay element Relationship between the input and output variables: Transfer function: Block diagram representation and unit step response: Examples: gap effect of gear mechanism, threshold voltage of transistors…

block diagram models (dynamic) Portray the control systems by the block diagram models more intuitively than the transfer function or differential equation models Block diagram representation of the control systems Examples:

Example 2.14 For the DC motor in Example 2.4 In Example 2.4, we have written down the differential equations as: Make Laplace transformation, we have:

Draw block diagram in terms of the equations (5)～(8): 1 ) ( 2 + f e m T s C J U a (s) W M - Consider the Motor as a whole:

Example 2.15 The water level control system in Fig 1.8:

Example 2.16 The DC motor control system in Fig 1.9

Block diagram reduction purpose: reduce a complicated block diagram to a simple one. Basic forms of the block diagrams of control systems Chapter 2-2.ppt

Three basic forms G1 G2 G2 G1 G1 G2 G1 G2 1+ G1 G2 G1 G2 cascade
parallel feedback G1 G2 G2 G1 G1 G2 G1 G2 1+ G1 G2 G1 G2

2.6 block diagram models (dynamic)
block diagram transformations 1. Moving a summing point to be: behind a block x1 y G x2 x1 x2 y G Ahead a block x1 x2 y G x1 y G x2 1/G

2. Moving a pickoff point to be: behind a block G x1 x2 y G x1 x2 y 1/G ahead a block G x1 x2 y G x1 x2 y

3. Interchanging the neighboring— Summing points x3 x1 x2 y ＋－ x1 x3 y ＋－ x2 Pickoff points y x1 x2 y x1 x2

4. Combining the blocks according to three basic forms. Notes: 1. Neighboring summing point and pickoff point can not be interchanged！ 2. The summing point or pickoff point should be moved to the same kind！ 3. Reduce the blocks according to three basic forms！ Examples:

G1 G2 G3 G4 H3 H2 H1 a b Moving pickoff point G1 G2 G3 G4 H3 H2 H1
Example 2.17 G4 1 G1 G2 G3 G4 H3 H2 H1 a b

G3 G1 G2 H1 G3 G1 G2 G1 H1 Moving summing point Move to the same kind
Example 2.18 G2 H1 G1 G3 G1

Disassembling the actions
Example 2.19 H3 H1

2.7 Signal-Flow Graph Models Block diagram reduction ——is not convenient to a complicated system. Signal-Flow graph —is a very available approach to determine the relationship between the input and output variables of a sys-tem, only needing a Mason’s formula without the complex reduc-tion procedures. Signal-Flow Graph only utilize two graphical symbols for describing the relation-ship between system variables。 Nodes, representing the signals or variables. G Branches, representing the relationship and gain Between two variables.

2.7 Signal-Flow Graph Models
Example 2.20: f c x0 x1 x2 g x3 x4 a d h b e some terms of Signal-Flow Graph Path — a branch or a continuous sequence of branches traversing from one node to another node. Path gain — the product of all branch gains along the path.

Loop —— a closed path that originates and terminates on the same node, and along the path no node is met twice. Loop gain —— the product of all branch gains along the loop. Touching loops —— more than one loops sharing one or more common nodes. Non-touching loops — more than one loops they do not have a common node. Mason’s gain formula

Example 2.21 x4 x3 x2 x1 x0 h f g e d c b a

Portray Signal-Flow Graph based on Block Diagram Graphical symbol comparison between the signal-flow graph and block diagram: Block diagram Signal-flow graph and G(s) G(s)

Example 2.22 － C(s) R(s) G1 G2 H2 H1 G4 G3 H3 E(s) X1 X2 X3 －H1 R(s) 1 E(s) G1 X1 G2 X2 G3 X3 G4 C(s) －H3 －H2

R(s) －H2 1 G4 G3 G2 G1 C(s) －H1 －H3 X1 X2 X3 E(s)

Example 2.23 G1 G2 + － C(s) R(s) E(s) Y2 Y1 X1 X2 -1 X1 Y1 G1 -1 1 -1 R(s) E(s) 1 C(s) 1 1 1 -1 X2 Y2 G2 -1

1 -1 G1 G2 R(s) E(s) C(s) X1 X2 Y2 Y1 7 loops: 3 ‘2 non-touching loops’ :

1 -1 G1 G2 R(s) E(s) C(s) X1 X2 Y2 Y1 Then: 4 forward paths:

We have

Chapter 5 Frequency Response Method
Concept Graphics mode Analysis Introduction Frequency Response of the typical elements of the linear systems Bode diagram of the open loop system Nyquist-criterion System analysis based on the frequency response Frequency response of the closed loop systems

5.1 Introduction Three advantages:
* Frequency response(mathematical modeling) can be obtained directly by experimental approaches. * easy to analyze the effects of the system with sinusoidal voices. * easy to analyze the stability of the systems with a delay element ur uc R C 5.1.1 frequency response For a RC circuit: We have the steady-state response:

5.1 Introduction Make: then: We have: Here: We call:
Frequency Response(or frequency characteristic) of the electric circuit.

5.1 Introduction Generalize above discussion, we have: Definition : frequency response (or characteristic) —the ratio of the complex vector of the steady-state output versus sinusoid input for a linear system, that is: Here: And we name: (amplitude ratio of the steady-state output versus sinusoid input) (phase difference between steady-state output and sinusoid input )

5.1.2 approaches to get the frequency characteristics
1. Experimental discrimination Measure the amplitude and phase of the steady-state output Input a sinusoid signal to the control system Get the amplitude ratio of the output versus input Get the phase difference between the output and input Are the measured data enough？ Data processing Change frequency y N

2. Deductive approach Theorem: If the transfer function is G(s), we have: Proof : Where — pi is assumed to be distinct pole (i=1,2,3…n).

In partial fraction form:
Here:

Taking the inverse Laplace transform: For the stable system all poles (-pi) have a negative real parts, we have the steady-state output signal:

the steady-state output: Compare with the sinusoid input , we have: The amplitude ratio of the steady-state output cs(t) versus sinusoid input r(t): The phase difference between the steady-state output and sinusoid input: Then we have :

5.1 Introduction Examples 5.1.1
a unity feedback control system, the open-loop transfer function: 1) Determine the steady-state response c(t) of the system. 2) Determine the steady-state error e(t) of the system. Solution: 1) Determine the steady-state response c(t) of the system. The closed-loop transfer function is:

5.1 Introduction The frequency characteristic :
The magnitude and phase response : The output response: So we have the steady-state response c(t) :

5.1 Introduction 2) Determine the steady-state error e(t) of the system. The error transfer function is : The error frequency response: The steady state error e(t) is:

5.1 Introduction 5.1.3 Graphic expression of the frequency response
Graphic expression —— for intuition 1. Rectangular coordinates plot Example 5.1.2

5.1.3 Graphic expression of the frequency response
2. Polar plot The polar plot is easily useful for investigating system stability. Example 5.1.3 The magnitude and phase response: Re Im Calculate A(ω) and 　　 for different ω: -117o -135o

5.1.3 Graphic expression of the frequency response
The shortage of the polar plot and the rectangular coordinates plot: to synchronously investigate the cases of the lower and higher frequency band is difficult. Idea: How to enlarge the lower frequency band and shrink (shorten) the higher frequency band？ 3. Bode diagram(logarithmic plots) Plot the frequency characteristic in a semilog coordinate: Magnitude response — Y-coordinate in decibels: X-coordinate in logarithm of ω: logω Phase response — Y-coordinate in radian: X-coordinate in logarithm of ω: logω First we discuss the Bode diagram in detail with the frequency response of the typical elements.

5.2 Frequency Response of The Typical Elements
The typical elements of the linear control systems — refer to Chapter 2. 1. Proportional element Transfer function: Frequency response: Re Im K 0dB, 0o 100 10 1 0.1 Polar plot Bode diagram

5.2 Frequency response of the typical elements
2. Integrating element Transfer function: Frequency response: 0dB, 0o 100 10 1 0.1 Re Im Polar plot Bode diagram

3. Inertial element Transfer function: 1/T: break frequency 0dB, 0o 100 10 1 0.1 Re Im 1 Polar plot Bode diagram

4. Oscillating element Transfer function: maximum value of ： Make:

The polar plot and the Bode diagram: Re Im 0dB, 0o 100 10 1 0.1 1 Polar plot Bode diagram

5. Differentiating element Transfer function: Re Im Re Im Re Im 1 1 differential 1th-order differential 2th-order differential Polar plot

Because of the transfer functions of the differentiating elements are the reciprocal of the transfer functions of Integrating element, Inertial element and Oscillating element respectively, that is: the Bode curves of the differentiating elements are symmetrical to the logω-axis with the Bode curves of the Integrating element, Inertial element and Oscillating element respectively. Then we have the Bode diagram of the differentiating elements:

0dB, 0o 100 10 1 0.1 0dB, 0o 100 10 1 0.1 differential 0dB, 0o 100 10 1 0.1 2th-order differential 1th-order differential

6. Delay element Transfer function: Re Im 0dB, 0o 100 10 1 0.1 R=1 Polar plot Bode diagram

5.3 Bode diagram of the open loop systems
Plotting methods of the Bode diagram of the open loop systems Assume: We have: That is, Bode diagram of a open loop system is the superposition of the Bode diagrams of the typical elements. Example 5.3.1

5.3 Bode diagram of the open loop systems
G(s)H(s) could be regarded as: ① ② ③ ④ Then we have: 20dB/dec 0dB, 0o 100 10 1 0.1 ② －40dB/dec 20dB, 45o -20dB, -45o -40dB, -90o 40dB, 90o -80dB,-180o -60dB.-135o ① －20dB/dec －20dB/dec ④ -40dB/dec －40dB/dec ③

5.3.2 Facility method to plot the magnitude response of the Bode diagram
Summarizing example 5.3.1, we have the facility method to plot the magnitude response of the Bode diagram: 1) Mark all break frequencies in theω-axis of the Bode diagram. 2) Determine the slope of the L(ω) of the lowest frequency band (before the first break frequency) according to the number of the integrating elements: －20dB/dec for 1 integrating element －40dB/dec for 2 integrating elements … 3) Continue the L(ω) of the lowest frequency band until to the first break frequency, afterwards change the the slope of the L(ω) which should be increased 20dB/dec for the break frequency of the 1th-order differentiating element . The slope of the L(ω) should be decreased 20dB/dec for the break frequency of the Inertial element …

Plot the L(ω) of the rest break frequencies by analogy . Example 5.3.2 The Bode diagram is shown in following figure:

0dB, 0o 100 10 1 0.1 －20dB/dec －20dB/dec 20dB, 45o -20dB, -45o -40dB, -90o 40dB, 90o -80dB,-180o -60dB.-135o -100dB,-225o -120dB,-270o 1.25dB －60dB/dec There is a resonant peak Mr at:

5.3.3 Determine the transfer function in terms of the Bode diagram
The minimum phase system(or transfer function) Compare following transfer functions: We have: The magnitude responses are the same. But the net phase shifts are different when ω vary from zero to infinite. It can be illustrated as following: Sketch the polar plot:

The polar plot: Re Im Re Im Im Re phase shift －π phase shift 00 phase shift －π Re Im It is obvious: the net phase shifts of the G1(s) is named: the minimum phase transfer function . G1(jω) is the minimum when ω vary from zero to infinite. phase shift π

Determine the transfer function of the minimum phase systems in terms of the magnitude response Definition: A transfer function is called a minimum phase transfer function if its zeros and poles all lie in the left-hand s-plane. A transfer function is called a non-minimum phase transfer function if it has any zero or pole lie in the right-hand s-plane. Only for the minimum phase systems we can affirmatively determine the relevant transfer function from the magnitude response of the Bode diagram . 2. Determine the transfer function from the magnitude response of the Bode diagram . Example 5.3.3

－40dB/dec －20dB/dec 0dB, 0o 100 10 1 0.1 Example 5.3.4 0dB 100 10 1 0.1 0.5 200 20dB/dec －20dB/dec 20dB

0dB 100 10 1 0.1 0.5 200 20dB/dec －20dB/dec 20dB Example 5.3.5 0dB 100 10 1 0.1 －20dB/dec －60dB/dec 8.136 dB 20 dB

0dB 100 10 1 0.1 －20dB/dec －60dB/dec 8.136 dB 20 dB For the non-minimum phase system we must combine the magnitude response and phase response together to determine the transfer function.

0dB, 0o 100 10 1 0.1 －180o －90o －20dB/dec Example 5.3.6 All satisfy the magnitude response But

5.4 The Nyquist-criterion
A method to investigate the stability of a system in terms of the open-loop frequency response. 5.4.1 The argument principle(Cauchy’s theorem) Assume: Make : Note: si→ the zeros of the F(s), also the roots of the 1+G(s)H(s)=0

5.4.1 The argument principle
Now we consider the net phase shift if s travels 360o along a closed path Γ of the s-plane in the clockwise direction shown in Fig S-plane Im Re Similarly we have: Fig

If Z zeros and P poles are enclosed by Γ , then: It is obvious that path Γ can not pass through any zeros si or poles pj . Then we have the argument principle: If a closed path Γ in the s-plane encircles Z zeros and P poles of F(s) and does not pass through any poles or zeros of F(s) , when s travels along the contour Γ in the clockwise direction, the corresponding F(s) locus mapped in the F(s)-plane will encircle the origin of the F(s) plane N = P-Z times in the counterclockwise direction, that is: N = P － Z

here: N —— number of the F(s) locus encircling the origin of the F(s)-plane in the counterclockwise direction. P —— number of the zeros of the F(s) encircled by the path Γ in the s-plane. Z —— number of the poles of the F(s) encircled by the path Nyquist criterion Re Im S-plane Fig If we choose the closed path Γ so that the Γ encircles the entire right hand of the s-plane but not pass through any zeros or poles of F(s) shown in Fig The path Γ is called the Nyquist-path.

When s travels along the the Nyquist-path:
Nyquist criterion Im S-plane When s travels along the the Nyquist-path: Re Because the origin of the F(s)-plane is Fig equivalent to the point (-1, j0) of the G(jω)H(jω)-plane, we have another statement of the argument principle: When ω vary from - (or 0) →+  , G(jω)H(jω) Locus mapped in the G(jω)H(jω)-plane will encircle the point (-1, j0) in the counterclockwise direction: here: P — the number of the poles of G(s)H(s) in the right hand of the s-plane. Z — the number of the zeros of F(s) in the right hand of the s-plane.

5.4.2 Nyquist-criterion If the systems are stable, should be Z = 0, then we have: The sufficient and necessary condition of the stability of the linear systems is : When ω vary from - (or 0) →+  , the G(jω)H(jω) Locus mapped in the G(jω)H(jω)-plane will encircle the point (-1, j0) as P (or P/2) times in the counterclockwise direction. ——Nyquist criterion Here: P — the number of the poles of G(s)H(s) in the right hand of the s-plane. Discussion : i) If the open loop systems are stable, that is P = 0, then: for the stable open-loop systems, The sufficient and necessary condition of the stability of the closed-loop systems is : When ω vary from - (or 0) →+  , the G(jω)H(jω) locus mapped in the G(jω)H(jω)-plane will not encircle the point (-1, j0).

5.4.2 Nyquist-criterion ii) Because that the G(jω)H(jω) locus encircles the point (-1, j0) means that the G(jω)H(jω) locus traverse the left real axis of the point (-1, j0) , we make: G(jω)H(jω) Locus traverses the left real axis of the point (-1, j0) in the counterclockwise direction —“positive traversing”. G(jω)H(jω) Locus traverses the left real axis of the point (-1, j0) in the clockwise direction —“negative traversing”. Then we have another statement of the Nyquist criterion: The sufficient and necessary condition of the stability of the linear systems is : When ω vary from - (or 0) →+  , the number of the net “positive traversing” is P (or P/2). Here: the net “positive traversing” —— the difference between the number of the “positive traversing” and the number of the “negative traversing” .

5.4.2 Nyquist-criterion Example 5.4.1
The polar plots of the open loop systems are shown in Fig.5.4.3, determine whether the systems are stable. Re Im (-1, j0) (2) P=0 Re Im (-1, j0) (1) P=2 stable stable Re Im (-1, j0) (4) P=0 Re Im (-1, j0) (3) P=2 unstable unstable Fig.5.4.3

5.4.2 Nyquist-criterion Note:
the system with the poles (or zeros) at the imaginary axis Example 5.4.2 There is a pole s = 0 at the origin in this system, but the Nyquist path can not pass through any poles of G(s)H(s). We consider a semicircular detour around the pole (s = 0) repre-sented by setting Idea: Im at the s = 0 point we have: Re －2 －1 Fig

5.4.2 Nyquist-criterion It is obvious that there is a phase saltation of the G(jω)H(jω) at ω=0, and the magnitude of the G(jω)H(jω) is infinite at ω=0. Fig Re Im Im Re (-1, j0) Fig.5.4.5 In terms of above discussion , we can plot the system’s polar plot shown as Fig The closed loop system is unstable.

Nyquist criterion in the Bode diagram
Example 5.4.3 Determine the stability of the system applying Nyquist criterion. Solution Fig.5.4.6 Im Re (-1, j0) Similar to the Example 5.4.2, the system’s polar plot is shown as Fig The closed loop system is unstable. 5.4.3 Application of the Nyquist criterion in the Bode diagram

5.4.3 Application of the Nyquist criterion in the Bode diagram
G(jω)H(jω) locus traverses the left real axis of the point (-1, j0) in G(jω)H(jω)-plane → L(ω)≥0dB and φ(ω) =－180o in Bode diagram (as that mentioned in 5.4.2). We have the Nyquist criterion in the Bode diagram : The sufficient and necessary condition of the stability of the linear closed loop systems is : When ω vary from 0→+  , the number of the net “positive traversing” is P/2. Here: the net “positive traversing” —the difference between the number of the “positive traversing” and the number of the “negative traversing” in all L(ω)≥0dB ranges of the open-loop system’s Bode diagram. “positive traversing” — φ(ω) traverses the “-180o line” from below to above in the open-loop system’s Bode diagram; “negative traversing” — φ(ω) traverses the “-180o line” from above to below.

5.4.3 Application of the Nyquist criterion in the Bode diagram
Example 5.4.4 The Bode diagram of a open-loop stable system is shown in Fig.5.4.7, determine whether the closed loop system is stable. Solution Because the open-loop system is stable, P = 0 . 0dB, －180o －20 －40 －60 －270o －90o Fig.5.4.7 L(ω) φ(ω) In terms of the Nyquist criterion in the Bode diagram: The number of the net “positive traversing” is 0 ( = P/2 = 0 ). The closed loop system is stable .

5.4.4 Nyquist criterion and the relative stability (Relative stability of the control systems)
In frequency domain, the relative stability could be described by the “gain margin” and the “phase margin”. 1. Gain margin Kg 2. Phase margin γc 3. Geometrical and physical meanings of the Kg and γc

5.4.4 Nyquist criterion and the relative stability
The geometrical meanings is shown in Fig Re －1 Im γc 1/Kg stable Critical stability unstable Fig The physical signification : Kg— amount of the open-loop gain in decibels that can be allowed to increase before the closed-loop system reaches to be unstable. For the minimum phase system: Kg>1 the closed loop system is stable . γc —amount of the phase shift of G(jω)H(jω) to be allowed before the closed-loop system reaches to be unstable. For the minimum phase system: γc>0 the closed loop system is stable .

Attention : For the linear systems: The changes of the open-loop gain only alter the magnitude of G(jω)H(jω). The changes of the time constants of G(s)H(s) only alter the phase angle of G(jω)H(jω). Example 5.4.5 The open loop transfer function of a control system is: (1) Determine Kg and γc when K =1 and τ =1. (2) Determine the maximum K and τ based on K = 1 and τ = 1.

Solution (1) Determine Kg and γc ( K =1 and τ =1) (2)

Example 5.4.6 The G(jω)H(jω) polar plot of a system is shown in Fig (1) Determine Kg (2) Determine the stable range of the open loop gain. Solution Re Im (-1, j0) 0.8 1.5 2 G(jω)H(jω) Fig.5.4.9 (1) Determine Kg (2) Determine the stable range of the open loop gain.

Re Im (-1, j0) 0.8 1.5 2 G(jω)H(jω) Fig.5.4.9

5.5 System analysis based on the frequency response
Performance specifications in the frequency domain 1. For the closed loop systems The general frequency response of a closed loop systems is shown in Fig (1) Resonance frequency ωr: ω A(ω) A(0) 0.707A(0) Fig ωr ωb Mr (2) Resonance peak Mr : (3) Bandwidth ωb:

2. For the open loop systems
(1) Gain crossover frequency ωc: For the unity feedback systems, ωc≈ ωb , because: (2) Gain margin Kg: (3) Phase margin γc:

Generally Kg and γc could be concerned with the resonance peak Mr : Kg and γc ↑ —— Mr ↓. ωc could be concerned with the resonance frequency ωr and bandwidth ωb : ωc↑ —— ωr and ωb↓. 5.5.2 Relationship of the performance specifications between the frequency and time domain The relationship between the frequency response and the time response of a system can be expressed by following formula: But it is difficult to apply the formula .

(1) Bandwidth ωb(or Resonance frequency ωr)  Rise time tr
5.5.2 Relationship of the performance specifications between the frequency and time domain (1) Bandwidth ωb(or Resonance frequency ωr)  Rise time tr Generally ωb(or ωr )↑—— tr ↓ because of the “time scale” theorem: So ωb(or ωr )↑—— tr ↓ alike : ωc↑—— tr ↓ because of ωc≈ ωb . For the large ωb , there are more high-frequency portions in c(t), which make the time response to be faster.

(2) Resonance peak Mr  overshoot σp%
5.5.2 Relationship of the performance specifications between the frequency and time domain (2) Resonance peak Mr  overshoot σp% Normally Mr ↑ —σp% ↑ because of the large unbalance of the frequency signals passing to c(t) . Kg and γc ↓ —σp% ↑is alike because of Kg and γc ↓—Mr ↑. Some experiential formulas:

5.5.2 Relationship of the performance specifications between the frequency and time domain
(3) A(0) → Steady state error ess So for the unity feedback systems:

(4) Reproductive bandwidth ωM → accuracy of Reproducing r(t)
5.5.2 Relationship of the performance specifications between the frequency and time domain (4) Reproductive bandwidth ωM → accuracy of Reproducing r(t) ω A(ω) A(0) 0.707A(0) Fig ωr ωb Mr △ ωM Reproductive bandwidth ωM : for a given ωM , △↓—higher accuracy of reproducing r(t) . for a given △, ωM ↑ —higher accuracy of reproducing r(t) . Demonstration

For the frequency spectrum of r(t) shown in Fig.5.5.3 .
5.5.2 Relationship of the performance specifications between the frequency and time domain For the frequency spectrum of r(t) shown in Fig ω Fig ωM 5.5.3 Relationship of the performance specifications between the frequency and the time domain: for the typical 2th-order system For the typical 2th-order system:

5.5.3 Relationship of the performance specifications between the frequency and the time domain: for the typical 2th-order system We have:

“three frequency band” theorem The performance analysis of the closed loop systems according to the open loop frequency response. 1. For the low frequency band the low frequency band is mainly concerned with the control accuracy of the systems. The more negative the slope of L(ω) is , the higher the control accuracy of the systems. The bigger the magnitude of L(ω) is, the smaller the steady state error ess is. 2. For the middle frequency band The middle frequency band is mainly concerned with the transient performance of the systems. ωc↑—tr ↓; Kg and γc ↓—σp% ↑

5.5.4 “three frequency band” theorem
The slope of L(ω) in the middle frequency band should be the –20dB/dec and with a certain width . 3. For the high frequency band The high frequency band is mainly concerned with the ability of the systems restraining the high frequency noise. The smaller the magnitude of L(ω) is, the stronger the ability of the systems restraining the high frequency noise is. Example 5.5.1 ω 0dB Fig Ⅰ Ⅱ －40 －20 Compare the performances between the system Ⅰand system Ⅱ

Solution : ω 0dB Fig Ⅰ －40 －20 essⅠ> essⅡ trⅠ > trⅡ σpⅠ% =σpⅡ% The ability of the system Ⅰ restraining the high frequency noise is stronger than system Ⅱ Ⅱ L(ω) ω 0.1 1 －20dB/dec －40dB/dec Fig Example 5.5.2 For the minimum phase system, the open loop magnitude response shown as the Fig Determine the system’s parameter to make the system being the optimal second-order system and the steady-state error ess< 0.1. Solution :

5.6 Frequency response of the closed loop systems
How to obtain the closed loop frequency response in terms of the open loop frequency response. 5.6.1 The constant M circles: How to obtain the magnitude frequency response of the closed loop systems in terms of the open loop frequency response…… (refer to P495) The constant N circles: How to obtain the phase frequency characteristic of the closed loop systems in terms of the open loop frequency response…… (refer to P496) 5.6.3 The Nichols chart: How to obtain the closed loop frequency response in terms of the open loop frequency response…… (refer to P496)

Chapter 5 Frequency Response Methods

Chap 6 The Compensation of the linear control systems

Chap 6 The Compensation of the linear control systems
§6-1 Introduction 6.1.1 definition of compensation types of compensation §6-2 The basic controller operation analysis 6.2.1 PI D controller ---active compensation 6.2.2 phase-lead controller 6.2.3 phase-lag controller phase lag-lead controller §6-3 Cascade compensation method of Root loci §6-4 Cascade compensation method of frequency- Domain §6-5 Feedback compensation passive compensation controller

6.1 Introduction 6.1.1 What is compensation or correction of a control system ? solution

6.1 Introduction This closed-loop system can be stable. We make the system stable by increasing a component. This procedureis called the compensation or correction. Definition of the compensation: increasing a component ,which makes the system’s performance to be improved, other than only varying the system’s parameters, this procedure is called the compensation or correction of the system.

Compensator: 6.1 Introduction
The compensator is an additional component or circuit that is inserted into a control system to compensate for a deficient performance. Types of the compensation

6.1 Introduction (1) Cascade(or series) compensation
(2) Feedback compensation (3) Both series and feedback compensation (4) Feed-forward compensation (1) Cascade(or series) compensation Features : simple but the effects to be restricted.

- 6.1 Introduction - (2) Feedback compensation
R(s) C(s) - Features: complicated but noise limiting, the effects are more than the cascade compensation. - R(s) C(s) (3) Both cascade and feedback compensation Features: have advantages both of cascade and feedback compensation.

- + 6.1 Introduction (4) Feed-forward compensation For input
R(s) C(s) + - F(s) For input For disturbance(voice) Features: theoretically we can make the error of a system to be zero and no effects to the transient performance of the system. Demonstration:

- + 6.1 Introduction For input
R(s) C(s) + - For input But no effect to the characteristic equation: 1+ G10G20 = 0 Question: actually the could not be easy implemented especially maybe the G20 is variable.

- + 6.1 Introduction For disturbance(voice)
C(s) F(s) R(s) - + For disturbance(voice) Also no effect to the characteristic equation: 1+ G10G20 = 0 Question: actually the could not be easy implemented especially maybe the G20 is variable. And the F(s) could not be easy measured.

- 6.1 Introduction example For the system shown in Fig.6.1.7: Where: +
C(s) N(s) R(s) - E(s) GCR + Fig.6.1.7 For the system shown in Fig.6.1.7: Determine GCR and GCN , make E(s) to be zero. Where: Solution : Thinking: if r(t) = n(t) = t , Determine GCR and GCN , make ess to be zero — as a exercise.

6.2 Operation analysis of the basic compensators
Active Compensation PID controller － active “compensator”. Transfer function:

PD controller C(s) G(s) R(s) －＋

Effects of PD controller: 1) PD controller does not alter the system type; 2) PD controller improve the system’s stability (to increase damping and reduce maximum overshoot); 3) PD controller reduce the rise time and settling time; 4) PD controller increase BW(Band Width) and improve GM(Kg),PM(γc), and Mr . － bring in the noise !

PI controller C(s) G(s) R(s) －＋

Effects of PI controller: 1) Increase the system’s type－clear the steady-state error ; 2) reduce BW(Band Width) and GM(Kg), PM(γc) and Mr ; beneficial to the noise limiting , not beneficial to the system’s stability.

Transfer function: PID controller G(s) R(s) C(s) - + PID controller have advantages both of PI and PD.

Circuits of PID _ _ + + PD controller PI controller _ + PID controller
ur u0 PD controller R2 _ + C R1 ur u0 PI controller R2 _ + C1 R1 ur u0 PID controller C2 R2

For example: Disk driver control system

solution How to get？ Shown in 6.3 detail.

6.2.2 Passive compensation controllers
Types of passive compensation controller

Zero and pole

Effects are similar to PD.
Bode plot z p Compensation ideal: make ωm to be ωc ! Effects are similar to PD.

Circuit of the phase-lead controller

Zero and pole

Effects are similar to PI.
Bode plot Effects are similar to PI. Compensation ideal: Make 1/τto be in the lower frequency-band and far from ωc !

Circuit of the Phase-lag controller

Zero and pole

Bode plot Effects are similar to PID. Compensation ideal: First make the phase-lag compensation－to satisfy ess and compensate a part of γc . second make the phase-lead compensation－to satisfy the transitional requirements.

Circuit of the Phase lag-lead controller

Comparing active compensation controllers and passive compensation controllers

6.3 Cascade compensation by Root loci method
6.3.1 Phase-lead compensation (P569) Example 6.3.1: Fig.6.3.1 solution The root loci of the system shown in Fig.6.3.1 Analysis: unstable. phase-lead compensation

Fig.6.3.2

There are two approaches to determine zc and pc . Fig.6.3.3 (1) Maximum α method (2) Method based on the open-loop gain

For this example we choose the Maximum α method: Fig.6.3.3 In terms of the sine’s law:

Root locus of the compensted system
6.3 Cascade compensation by Root loci method The root locus of the compensated system is shown in Fig.6.3.4 Steps of the cascade phase-lead Compensation: Fig.6.3.4 (1) Determine the dominant roots based on the performance specifications of the system: (2) plot the root locus of the system and analyze what compensation device should be applied. Root locus of the compensted system

(3) Determine the angle φc to be compensated: (4) calculate θ andγ: (5) calculate zc and pc In terms of the sine’s law : (6) plot the root locus of the compensated system and make validity check.

6.3.2 Phase-lag compensation using the root locus (P577)
Example 6.3.2: -10 Fig.6.3.5 Solution: The root locus of the system is shown in Fig

The detail of the root-loci is shown in Fig Fig 6.3.6

Fig 6.3.6

Validate…… Steps of the cascade phase-lag Compensation: (1) Determine the dominant roots based on the performance specifications of the system: (2) plot the root locus of the system and analyze what compensation device should be applied. If the phase-lag Compensation be applied:

(5) plot the root locus of the compensated system and make validity check. 6.3.3 Phase lag-lead compensation by the root locus method Basic ideal:

6.3.3 Phase lag-lead compensation by the root locus method
First: make the phase-lead compensation－to satisfy the transitional requirements. Second: make the phase-lag compensation－to satisfy ess requirements. Exercise: Make compensation using PD and PI for example and example 6.3.2

6.4 Cascade compensation by frequency response method
Phase-Lead Compensation using Bode diagram

6.4 Cascade compensation by frequency response
method

Example: 6.4 Cascade compensation by frequency response method
solution: －40dB/dec Fig.6.4.1

6.4 Cascade compensation by frequency response method
－40dB/dec Fig.6.4.1 Exercise: Make validity check for this example. 2. Make compensation using PD for this example.

6.4.2 Phase-Lag Compensation using Bode diagram
Example: 2 6.32 －20dB/dec －40dB/dec －900 －1800 －20lgβ solution: Fig.6.4.2

6.32 －20dB/dec －40dB/dec －900 －1800 －20lgβ Fig.6.4.2 Validate……

Steps of the phase-lag compensation: Exercise: 1. Make validity check for this example. 2. Make phase-lag compensation for γc=50o and Kv=20.

6.4.3 Phase-Lag-lead Compensation using Bode diagram
First: make the phase-lag compensation－to satisfy ess and compensate a part of γc . Second: make the phase-lead compensation－to satisfy the requirementsγc and ωc etc. 6.4.4 Compensation according to the desired frequency response Example 10 100 －20dB/dec －40dB/dec Fig.6.4.3 solution

6.4.4 Compensation according to the desired frequency response
In terms of the desired frequency response we have: 10 100 －20dB/dec －40dB/dec Fig.6.4.3

Exercise: 10 100 －20dB/dec －40dB/dec 1 Fig

6.5 Feedback compensation
6.5.1 The configuration of the Feedback compensation R(s) C(s) － G’0(s) R(s) C(s) － G’20(s) Fig 6.5.2 The basic Feedback compensators

6.5.3 Function of the feedback:
Decrease the time constant of the encircled elements → Quicken the response of the encircled elements－may be; For example 2. Impair(weaken) the influences of the disturbance to the encircled elements. 3. make the performance of the encircled elements to be desired .

6.5.4 The design procedure of the feedback compensator
1. Design the desired characteristics, such as the desired Bode diagram, of the encircled elements in terms of the system’s analysis. 2. Choose the appropriate feedback compensators to get the desired characteristics. R(s) C(s) － G’20(s) Fig For the system shown in Fig , G20=10/s2, the desired G’20(jω) shown in Fig determine the Gc. Example solution 1 10 －20dB/dec －40dB/dec 0.1 Fig

Chap7 Nonlinear Control system
7.1 Introduction 7.2 Describing function 7.3 Method of the phase locus

Chap7 Nonlinear Control system
7.1 Introduction 7.1.1 What is the nonlinearity ? 7.1.2 What is the nonlinear control system? 7.1.3 The typical nonlinearities. 7.1.4 The speciality of the nonlinear systems 7.1.5 Analysis method of the nonlinear systems

7.1 Introduction 7.1.1 What is the nonlinearity ?
The “output” varying is not proportional to the “input” varying for a device. The characteristic of the nonlinear device can not be described by the linear differential equation. Types of the nonlinearity: (1) Essential nonlinearity The nonlinearity y=f(x) can not be expressed as the Talor series expansion in all x. ( 2) Nonessential nonlinearity The y=f(x) can be expressed as the Talor series expan-sion in all x.

7.1 Introduction 7.1. 2 What is the nonlinear control system?
If a control system include one or more nonlinear cha-racteristic element or link , the system is named as the nonlinear control system. 7.1.3 The typical nonlinearities (1) Saturation nonlinearity

7.1.3 The typical nonlinearities
Actual examples: saturation characteristic of the amplifier;valve journey; power limit etc. (2) Dead zone nonlinearity Actual examples: Insensitive zone of the measure system; Turn on characteristic of the diode etc.

(3) Relay nonlinearity Actual examples: relay, switch etc. Several special relay nonlinearity:

Approximate relay nonlinearity (m=1) Ideal relay nonlinearity (4) Backlash hysteresis loop(clearance) nonlinearity Actual example: gear backlash

(5) changeable gain nonlinearity

Satisfy superposition theorem.
The characteristics of the nonlinear systems (distinguishing features with linear system) Linear system characteristics Nonlinear system 1 Satisfy superposition theorem. Not satisfy superposition theorem. 2 Stability is only related to the system parameters. Stability is related to system input, initial state, parameters, structure etc. 3 Have two kind of work states: stable and unstable. Have stable, unstable and self-oscillation. 4 The form of the output is the same as input. The form of the output is different from the input.

7.1.4 The analysis and design methods of the nonlinear systems
Classical ① Phase plane method ② Describing function method ③ Computer and intelligence → modern

Review of last class: 2. The typical nonlinearity. 3. The characteristics of the nonlinear control system. 4. The analysis methods of the nonlinear control system: 1. What is the nonlinearity, nonlinear control system? Classical methods:

7.2 Describing function method of the nonlinear system analysis
Four items: What is the describing function? How to get the describing function? How to analyze a nonlinear system by describing function? 4. Attentions and development (modeling) (analysis and design) (Put forwarded by P.J.Daniel, In 1940) 7.2.1 What is the describing function? 1. Basic idea For the nonlinear system

7.2.1 What is the describing function?
Fig.7.7 Typical structure of the nonlinear systems N G(s) H(s) nonlinear linear a sinusoidal input, y(t), maybe it is not a sinusoidal but a periodic function, can be expressed as a Fourier series:

Discuss: i) For the symmetry nonlinearity: ii) the harmonic of y(t) could be neglected, then: and output frequency is equal to input frequency approximately.

It means: We can describe the nonlinear components by the frequency response like as that we did in chapter 5. So we have: 2. Definition of the describing function The describing function N(X) of the nonlinear element is: the complex ratio of the fundamental component of the output y(t) and the sinusoidal input x(t), that is: Here:

Because the describing function actually is the linearized “frequency response” → “harmonic linearization”, we can analyze the nonlinear systems like as that we did in chapter 5. 7.2.2 How to get the describing function? 1. Steps (1) Input a sinusoid signal x(t) to the nonlinear elements:

7.2.2 How to get the describing function?
(2) Solve y(t) and obtain the fundamental component of y(t). (3) Calculate the describing function N(X) according to following formula: 2. Examples

Example 7.1 The mathematical description of a nonlinear device is: Determine the describing function of the device. Solution

Determine the describing function of the saturation nonlinearity. Example 7.2 Solution

Determine the describing function of the dead zone nonlinearity. Example 7.3 solution

3. The describing function of some typical nonlinearity M －M Relay M －M h saturation k dead zone

3. The describing function of some typical nonlinearity
k h backlash hysteresis

4. characters (1) For the “single value” nonlinearity the describing function must be a “real number”. such as the dead zone, saturation and the ideal relay nonlinearity etc. (2) The describing function satisfy the superposition principle (nonlinearity not). For example: + + =

1. Review of Nyquist criterion
7.2.3 Stability analysis of the nonlinear system by describing function unstable Re －1 Im γc 1/Kg stable Critical stability Fig 1. Review of Nyquist criterion G(s) Fig For the linear system:

2. Compare the nonlinear system with the linear system
7.2.3 Stability analysis of the nonlinear system by describing function 2. Compare the nonlinear system with the linear system nonlinear system N(x) G(s) Fig Linear system Transfer function of the system: Characteristic equation: A point A curve

3. Stability analysis of the nonlinear system
(For example the minimum phase system) (1) (2) (3) (1) (3) (2)

3. Stability analysis of the nonlinear system
(For example the minimum phase system) Graphical explanation is shown as following: Re Im Intersect with －1/N(A) (self-oscillation) do not circle －1/N(X) (stable) circle －1/N(X) (unstable) X increasing direction

4. Self-oscillation of the nonlinear system
7.2.3 Stability analysis of the nonlinear system by describing function 4. Self-oscillation of the nonlinear system A special motion of the nonlinear system: Re Im Self-oscillation System will be at a continuous oscillation, which has a constant amplitude and frequency, when the system come under a light disturbance. B: unstable self-oscillation point→－1/N(X)enter unstable zone from stable zone. A: stable self-oscillation point→－1/N(X) enter stable zone from unstable zone.

Example: (a graduate examination)
G(s) 1 -1 Solution: The system is equivalent to the Fig G(s) N(X)

Stability analysis: Graphical explanation is shown as Fig.7.2.3.6
Exercise: for this example, if: Fig Re Im Stability analysis: (1) (2)

7.2.4 Attentions and development
(1) Using the describing function to analyze the nonlinear system, the Linear parts of the system must be provided with a good charact-eristic of the low-pass filter→so that the harmonics produced by the nonlinear element can be neglected. (2) Generally the describing function method can only be used for analyzing the stability and self-oscillation of the nonlinear systems, not the stead-state error and transient specifications. 2. development Modern analysis and design method of the nonlinear systems: Computer simulation and intelligent design.

Four items: 7.3 Phase plane method 7.3.1 What is the Phase plane
It is a kind of graphic method to solve first and second order differential equation, put forward by Poincare In 1885. Four items: What is the Phase plane？ How to plot the Phase plane ？ How to analyze the nonlinear systems by the Phase plane method. Attentions and development. 7.3.1 What is the Phase plane

7.3.1 What is the Phase plane Phase plane graph Increasing t direction Always to be clockwise Fig.7.3.1 Graphic expression

7.3.2. plotting method of the phase locus
1. Analytic method

1. Analytic method Example 7.3.1: m=1 x k=1 Stable position Solution:

- 7.3.2. plotting method of the phase locus Example 7.3.2:
r =0 e y c - For the system: Plot the phase loci of the system: Solution:

r The phase loci of the system is shown in following figure→self-oscillation.

2. Graphic method---isoclinal method Example 7.3.3:

The values are shown in following table for different
solution The values are shown in following table for different －2 －1 －0.5 0.5 1 2 We can plot the isoclinals like as following figure:

(1) Plot the isoclinals for different α.
(2) Plot the corresponding tangents of the phase loci in each isoclinals. (3) Plot the phase loci starting at the initial states

Attentions:

1. Singularity points of the phase locus
Analysis of the phase plane 1. Singularity points of the phase locus

1. Singularity points of the phase locus
(2) Types of the singularity points Characteristic equation: According to the position of s1,2 in s-plane, there are six types of the singularity points:

7.3.3 Analysis of the phase plane
2. limit cycle A kind of phase locus with the closed loop form →Corresponding to the self-oscillation.

Example 7.3.4: Solution Linearize the nonlinear differential equation to determine the types of the singularity points:

The phase loci plotted by the isoclinal method are shown in following figure:

3. How to get the time response x(t) from the phase locus The graphical expression: We can get the time response curve x(t) from the phase locus to analyze the time specifications, such as the rise time tr, Settling time ts etc. , of the nonlinear systems.

4. How to analyze the performance of the nonlinear systems from the phase locus (1) We can analyze the stability directly from the phase locus: the phase locus is convergent or divergent. (2) We can analyze the self-oscillation directly from the phase locus: the phase loci converge upon a limit circle. (3) We can transform the phase locus into the time response curve x(t) to analyze the rise time tr , settling time ts etc.. (4) Also we can analyze the steady state error, overshoot etc., directly from the phase locus. Example 7.3.5

7.3.3 Analysis of the phase plane solution
Singularity points: (0, 0), and a stable nodes in the plane. The phase locus is shown in following figure: (1) Stability: stable (2) Steady state error ess = 0 (3) Overshoot (1,0) (4) We can transform the phase locus into the time response curve e(t) to analyze the rise time tr , settling time ts etc.

7.3 Phase plane method 7.3.4 Attentions and development
Phase plane method is only used for analyzing or designing the 1th-order or 2th-order nonlinear systems. (2) Analyzing the nonlinear systems by phase plane method is more all-sided compare with the describing function method. but more complicated. (3) Also the phase plane method is used to analyze the stability of some intelligent control systems, such as the Fuzzy control systems. Exercise: For example 7.3.5, if the nonlinearity is: 0.2 2 (1) 0.5 1 (2)

Chapter 8 Discrete (Sampling) System
8.1 Introduction 8.2 Z-transform 8.3 Mathematical describing of the sampling systems 8.4 Time-domain analysis of the sampling systems 8.5 The root locus of the sampling control systems 8.6 The frequency response of the sampling control systems 8.7 The design of the “least-clap” sampling systems

8.1 Introduction x*(t) t x(t) Sampling x(t) Make a analog signal to be a discrete signal shown as in Fig.8.1 . t1 t2 t3 t4 t5 t6 x(t) —analog signal . x*(t) —discrete signal . 8.1.2 Ideal sampling switch —sampler Fig.8.1 signal sampling Sampler —the device which fulfill the sampling. Another name —the sampling switch — which works like a switch shown as in Fig.8.2 . T t x*(t) x(t) Some terms 1. Sampling period T— the time interval of the signal sampling: T = ti+1 － ti . Fig.8.2 sampling switch

8.1.3 Some terms 2. Sampling frequency ωs — ωs = 2π fs = 2π / T .
3. Periodic Sampling — the sampling period Ts = constant. 4. Variable period sampling — the sampling period Ts≠constant. 5. Synchronous sampling —not only one sampling switch in a system, but all work Synchronously. 6. Multi-rate sampling. 7. Opportunity(Random) sampling. We mainly discuss the periodic and synchronous sampling in chapter 8. Sampling (or discrete) control system There are one or more discrete signals in a control system — the sampling (or discrete) control system. For example the digital computer control system:

8.1 Introduction 8.1.5 Sampling analysis
A/D D/A computer process measure r(t) c(t) e(t) － e*(t) u*(t) u (t) Fig.8.3 computer control system Sampling analysis Expression of the sampling signal: It can be regarded as Fig.8.4:

8.1.5 Sampling analysis ＝ We have: × modulating pulse(carrier)
t x*(t) x(t) T δT(t) × ＝ modulating pulse(carrier) modulated wave Modulation signal Fig.8.4 sampling process We have:

8.1.5 Sampling analysis Only: could be reproduced
This means: for the frequency spectrum of x(t) shown in Fig.8.5, the frequency spectrum of x*(t) is like as Fig.8.6. Fig.8.6 Filter Fig.8.5 Only: could be reproduced

8.1 Introduction So we have:
Sampling theorem ( Shannon’s theorem) If the analog signal could be whole restituted from the sampling signal, the sampling frequency must be satisfied : 8.1.7 zero-order hold Usually the controlled process require the analog signals, so we need a discrete-to-analog converter shown in Fig.8.5. discrete-to-analog converter x*(t) xh(t) Fig.8.7 D/A convert

8.1.7 zero-order hold The ideal frequency response of the D/A converter is shown in Fig.8.8. ω A(ω) Fig.8.8 To put the ideal frequency response in practice is difficult, the zero -order hold is usually adopted. The action of the zero-order hold is shown in Fig.8.9. x*(t) x(t) xh(t) The mathematic expression of xh(t) : The unity pulse response of the zero-order hold is shown in Fig.8.10. Fig.8.9 T Fig.8.10 t g(t) The transfer function of the zero-order hold can be obtained from the unity pulse response:

8.1 Introduction 8.2 Z-transform 8.1.7 zero-order hold
Fig.8.11 The frequency response of the zero-order hold, which is shown in Fig.8.11, is: 8.2 Z-transform 8.2.1 Definition Expression of the sampled signal: Using the Laplace transform: Define:

8.2 Z-transform We have the Z-transform: 8.2.2 Z-transforms of some
common signals Table 8.1 The Z-transforms of some common signals is shown in table 8.1. 8.2.3 characteristics of Z- transform The characteristics of Z- transform is given in table 8.2.

Table 8.2

8.2.3 characteristics of Z-transform
Using the characteristics of Z-transform we can conveniently deduce the Z-transforms of some signals. Such as the examples shown in table 8.3: Table 8.3

8.2 Z-transform 8.2.4 Z-transform methods
1. Partial-fraction expansion approaches Example 8.1 2. Residues approaches

8.2.4 Z-transform methods Example 8.2 8.2.5 Inverse Z-transform
1. Partial-fraction expansion approaches

8.2.5 Inverse Z-transform Example 8.3 2. Power-series approaches

Inverse Z-transform 3. Residues approaches Example 8.5

8.3 Mathematical modeling of the sampling systems 8.3.1 Difference equation For a nth-order differential equation: Make:

8.3.1 Difference equation … Or : …

8.3.1 Difference equation A nth-order differential equation can be transformed into a nth-order difference equation by the backward or forward difference: To get the solution of the difference equation is very simple by the recursive algorithm. Fig.8.12 T Example 8.6 K = 10, T = 0.5s, r(t) = 1(t) Determine the output c*(mT). For the sampling system shown in Fig.8.12, Assume: Solution

8.3.1 Difference equation

8.3.1 Difference equation For K = 10, T = 0.5s, we have:
Consider e*(k) = r(k)－c(k) = 1－c(k): If c(0) = 0, applying the recursive algorithm we have: ……

8.3 Mathematical modeling of the sampling systems
Z-transfer (pulse) function Definition: Z-transfer (pulse) function — the ratio of the Z-transformation of the output signal versus input signal for the linear sampling systems in the zero-initial conditions, that is: 1. The Z-transfer function of the open-loop system G1(s) G2(s) T r(t) c(t) c*(t) G1(z)G2(z) R(z) C(z) G1(z) =Z [ G1(s)] G2(z) =Z [ G2(s)] T G1(s) r(t) G2(s) c(t) c*(t) G1G2(z) R(z) C(z) G1G2(z) =Z [ G1(s)G2(s) ]

8.3.2 Z-transfer (pulse) function
2. The z-transfer function of the closed-loop system G(s) r c － H(s) r c － G(s) H(s) r － c G(s) H(s) r G2(s) c － G1(s) H(s) r － G2(s) c G1(s) H(s)

8.3.2 Z-transfer (pulse) function
－ G3(s) c G2(s) H(s) G1(s) r － G2(s) c G1(s) H2(s) H1(s) r － G1(s) H2(s) G2(s) c H1(s) G3(s)

8.4 Time-domain analysis of the sampling systems 8.4.1 The stability analysis 1. The stability condition The characteristic equation of the sampling control systems: Suppose: In s-plane, α need to be negative for a stable system, it means: So we have: The sufficient and necessary condition of the stability for the sampling control systems is: The roots zi of the characteristic equation 1+GH(z)=0 must all be inside the unity circle of the z-plane, that is:

8.4.1 The stability analysis
Im critical stability z-plane The graphic expression of the stability condition for the sampling control systems is shown in Fig 1 Re Stable zone unstable zone 2. The stability criterion Fig.8.4.1 In the characteristic equation 1+GH(z)=0, substitute z with —— W (bilinear) transformation. We can analyze the stability of the sampling control systems the same as we did in chapter 3 (Routh criterion in the w-plane) .

8.4.1 The stability analysis
Determine K for the stable system. Example 8.7 Solution : make We have: 0 < K < 4.33. 8.4.2 The steady state error analysis The same as the calculation of the steady state error in Chapter 3, we can use the final value theorem of the z-transform:

8.4.2 The steady state error analysis
For the stable system shown in Fig.8.4.2 G(s) r c － e Fig.8.4.2

－ G (s) c Z.o.h T e Example 8.8 Z.o.h —Zero-order hold. 2) If r(t) = 1+t, determine ess=？ 1) Determine K for the stable system. Solution 1)

2)

8.4 Time-domain analysis of the sampling systems
8.4.3 The unit-step response analysis Im 1 Re Analyzing c(kT) we have the graphic expression of C(kT) is shown in Fig Fig.8.4.3

8.5 The root locus of the sampling control systems The plotting procedure of the root loci of the sampling systems are the same as that we introduced in chapter 4. But the analysis of the root loci of the sampling systems is different from that we discussed in chapter 4 (imaginary axis of the s-plane ←→ the unit circle of the z-plane). 8.6 The frequency response of the sampling control systems The analysis and design methods of the frequency response of the sampling systems are the same as that we discussed respect-ively in chapter 5 , chapter 6, only making: ←→ Here: v — the counterfeit frequency

8.7 The design of the “least-clap” sampling systems The transition process of the sampling control systems can be finished in the minimum sampling periods—the “least-clap” systems. 8.7.1 design of D(z) r － G (s) c D(z) Fig.8.6.1 e For the system shown in Fig We have:

8.7.1 design of D(z)

8.7.1 design of D(z) Proof: The responses of the “least-clap”
Fig.8.6.2 1 c*(t) t T 2T 3T The responses of the “least-clap” System are shown in Fig

For the system shown in Fig.8.6.3
8.7.1 design of D(z) Example 8.9 For the system shown in Fig.8.6.3 r － G (s) c Gh(s) D(z) e T Fig.8.6.3 u Determine D(z), make the system to be the “least-clap” system for r(t) = t. Solution

8.7 The design of the “least-clap” sampling systems
The realization of D(z) To illustrate by example 8.10 Example 8.10 To realize D(z) for Example 8.9 Solution We can program the computer in terms of above formula to realize D(z).

Exercises: p820～ E13.11; E13.14; P13.12; p13.18; AP13.2 In example 8.9, if and respectively for r(t)=1(t); t; t2 .

Modern Control Engineering

Similar presentations

Presentation on theme: "Modern Control Engineering"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Modern Control Engineering

Similar presentations

Presentation on theme: "Modern Control Engineering"— Presentation transcript:

Similar presentations

About project

Feedback