Chapter 3:
Analysis of closed-loop
systems
Control Automático
3º Curso. Ing. Industrial
Escuela Técnica Superior de Ingenieros
Universidad de Sevilla
Control of SISO systems
Control around an operation point (u0,y0)
Process
+
u0 u(t) -y(t) ∆u(t) Controller r(t) e(t)(Reference is not a deviation variable ) Automatic control Controller e(t) ∆u(t) Controller
How to chose the value of the controller parameters (Kp and Ti) in a way such that the closed-loop system has an appropriate performance?
Control of SISO systems
Heuristic design
Tuning based on experiments
Real system
Model
Design based on tables
Tuning based on a set of experiments and a table that determines the
value of each parameter
Zieger-Nichols (Chapter 6)
Mathematical design
The tuning is based on a mathematical analisys of the closed-loop system
and provides guaranteed properties
Transient response Steady state response Robustness
Analitic design techniques Root–locus design techniques Loop-shaping design techniques
Control of SISO systems
Process+
u0 u(t) -y(t) ∆u(t) Controller r(t) e(t) Closed-loop systemIt is hard to analize the dynamics of the closed-loop system if the process or the controller are nonlinear systems
The design techniques studied in this course are based on the linearized model of the system around a given operation point (u0,y0)
Incremental variables model
The linearized model can be compared with the incremental variables model
for a given operating point which is defined as follows:
System
+
-Incremental variables model (u0,y0)
Remark: The model depends on the operating point
Assumption: The initial state is the operation point
Incremental variables model
Linearized model around (u0,y0)
Linearized model
-∆u(t) Controller
e(t)
Analisys of the closed-loop system. In this case, both the controller and the system are LTI systems.
Assumption: Zero initial conditions.
Teoría de sistemas
Assumption: The properties of this (simplified) system are similar to the ones of the real closed-loop system if the trajectories are close to the operating point
- Speed of the transient
- Tracking of time-varying references - Disturbance rejection
Teoría de sistemas
TEMA 1. Introducción y fundamentos.
Sistemas dinámicos. Conceptos básicos. Ecuaciones y evolución temporal. Linealidad en los
sistemas dinámicos.
TEMA 2. Representación de sistemas.
Clasificación de los sistemas. Clasificación de comportamientos. Señales de prueba.
Descripción externa e interna. Ecuaciones diferenciales y en diferencias. Simulación.
TEMA 3. Sistemas dinámicos lineales en tiempo continuo.
Transformación de Laplace. Descripción externa de los sistemas dinámicos. Función de
transferencia. Respuesta impulsional. Descripción interna de los sistemas dinámicos.
TEMA 4. Modelado y simulación.
Modelado de sistemas. Modelado de sistemas mecánicos. Modelado de sistemas hidráulicos.
Modelado de sistemas eléctricos. Modelado de sistemas térmicos. Linealización de modelos no lineales. Modelos lineales. Álgebra de bloques. Simulación.
TEMA 5. Respuesta temporal de sistemas lineales.
Sistemas dinámicos lineales de primer orden. Ejemplos. Sistemas dinámicos lineales de
segundo orden. Respuesta ante escalón. Sistemas de orden n.
TEMA 6. Respuesta frecuencial de sistemas lineales.
Función de transferencia en el dominio de la frecuencia. Transformación de Fourier.
Representación gráfica de la función de transferencia. Diagramas más comunes. Diagrama de Bode.
TEMA 7. Estabilidad.
Estabilidad de sistemas lineales. Criterios relativos a la descripción externa de los sistemas
dinámicos. Criterio de Routh-Hurwitz. Criterio de Nyquist. Criterios relativos a la descripción interna.
Closed-loop transfer function
Laplace transform (assuming zero initial conditions)
Remark: From now on, the variables “y” and “u” denote incremental values; that is, the deviation of the input and the output from the operating point
Linear time invariant systems (LTI)
Properties used: Linearity, transform of the time derivative
Proportional term
Controller
e(t)
u(t)
The value of the input u(t) is proportional to the error
Transfer function C(s) E(s) U(s) Time domain Frequency domain Desing parameter: Kp
Integral term
Controller
e(t)
u(t)
The value of the input u(t) is proportional to the error and its integral
Transfer function C(s) E(s) U(s) Temporal domain Frequency domain Desing parameter: Kp, Ti
Lag compensation net
Controller with properties similar to the PI
Derivative term
Controller e(t) u(t) Transfer function C(s) E(s) U(s) Time domain Frequency domain Design parameters: Kp, TdLead compensator net
The value of the input u(t) is proportional to the error and its time derivative
Controller with properties similar to the PD
PID controller
Controller
e(t) u(t)
Widely used in industry
PID controller
Controller e(t) u(t) Transfer function C(s) E(s) U(s) Time domain Frequency domain Design parameter: Kp, Td, TiLead-lag compensator
The value of the input u(t) is proportional to the error, its time derivative and its integral
Index
Closed Loop Transfer Function
Tuning a Controller
Stability analysis of system
Steady-state response of a closed-loop systems
Transient response of a stable system.
Closed-loop transfer function
Block algebra (Ogata 3.3, Tema 3, Teoría de sistemas)+ -S1(s) S2(s) S3(s) = S1(s)-S2(s) S1(s) S2(s) = S1(s) S3(s) = S1(s)
Signal sum Signal bifurcation
G(s)
S1(s) S2(s)=G(s)S1(s)
Closed-loop transfer function
G(s) -Y(s) C(s) E(s) U(s) R(s)R(s) Gbc(s) Y(s) Gbc(s) models how the ouput of the closed-loop system reacts to changes in the references The properties of a controller are defined based on the response of the closed-loop system
Other transfer functions
C(s) G(s) H(s) R(s) Ym(s) U(s) Y(s) Sensor dynamics C(s) G(s) H(s) R(s) Ym(s) U(s) Y(s) Gd(s) D(s) Disturbances -+ + + +-Index
Closed Loop Transfer Function
Tuning a Controller
Stability analysis of system
Steady-state response of a closed-loop systems
Transient response of a stable system.
Controller tuning
Obtain a set of controller parameter (C(s)) in order to guarantee that the closed-loop systems satisfies a given set of conditions (specifications)
Specifications
Stability
Raise time (step reference change) Steady state error
Specifications on the linearized model are
relevant to the behaviour of the real
closed-loop system
Mathematical design
Tuning based on the anlisys of the closed-loop dynamics (which depend on the controller parameters)
Gbc(s) is not defined if parameters of C(s) are not fixed
Example
Closed-loop system
Closed-loop response? Depends on the value of Kp Three poles that depend on Kp
Static gain depens on Kp
Reference signal: Step change
Example
Kp=0.1
Kp=1
Kp=10
Example
0 10 20 30 40 50 60 0 0.5 1 Kp = 0.1, Td = 0, 1/Ti = 0 y(t) 0 10 20 30 40 50 60 0 0.05 0.1 u(t) 0 10 20 30 40 50 60 0 0.5 1 e(t) 0 10 20 30 40 50 60 0 20 40 ∫ 0 t e( τ )d τRespuesta del sistema en BC
0 10 20 30 40 50 60 0 0.5 1 1.5 Kp = 1, Td = 0, 1/Ti = 0 y(t) 0 10 20 30 40 50 60 −0.5 0 0.5 1 u(t) 0 10 20 30 40 50 60 −0.5 0 0.5 1 e(t) 0 10 20 30 40 50 60 0 5 10 ∫ 0 t e( τ )d τ
Respuesta del sistema en BC
0 10 20 30 40 50 60 0 1 2 Kp = 10, Td = 0, 1/Ti = 0 y(t) 0 10 20 30 40 50 60 −10 0 10 u(t) 0 10 20 30 40 50 60 −1 0 1 e(t) 0 10 20 30 40 50 60 −0.5 0 0.5 1 ∫ 0 t e( τ )d τ
Respuesta del sistema en BC
0 10 20 30 40 50 60 −10 0 10 20 Kp = 15, Td = 0, 1/Ti = 0 y(t) 0 10 20 30 40 50 60 −200 0 200 u(t) 0 10 20 30 40 50 60 −20 −10 0 10 e(t) 0 10 20 30 40 50 60 −5 0 5 10 ∫ 0 t e( τ )d τ
Example
0 10 20 30 40 50 60 0 0.5 1 Kp = 0.1, Td = 0, 1/Ti = 0 y(t) 0 10 20 30 40 50 60 0 0.05 0.1 u(t) 0 10 20 30 40 50 60 0 0.5 1 e(t) 0 10 20 30 40 50 60 0 20 40 ∫ 0 t e( τ )d τExample
0 10 20 30 40 50 60 0 0.5 1 1.5 Kp = 1, Td = 0, 1/Ti = 0 y(t) 0 10 20 30 40 50 60 −0.5 0 0.5 1 u(t) 0 10 20 30 40 50 60 −0.5 0 0.5 1 e(t) 0 10 20 30 40 50 60 0 5 10 ∫ 0 t e( τ )d τExample
0 10 20 30 40 50 60 0 1 2 Kp = 10, Td = 0, 1/Ti = 0 y(t) 0 10 20 30 40 50 60 −10 0 10 u(t) 0 10 20 30 40 50 60 −1 0 1 e(t) 0 10 20 30 40 50 60 −0.5 0 0.5 1 ∫ 0 t e( τ )d τExample
0 10 20 30 40 50 60 −10 0 10 20 Kp = 15, Td = 0, 1/Ti = 0 y(t) 0 10 20 30 40 50 60 −200 0 200 u(t) 0 10 20 30 40 50 60 −20 −10 0 10 e(t) 0 10 20 30 40 50 60 −5 0 5 10 ∫ 0 t e( τ )d τTypes of behaviour
Unit step response
Clasification of the output signal ∆y(t) depending on the input signal
Unit Step (the most widely used).
Ramp.
Sinusoidal.
Provides information about the dynamic proterties of the system
Model expressed in error variables (u0,y0)
Assumption: Initial conditions in the operating point.
Unit Step: Behaviours:
• Overdamped • Underdamped • Unstable
Types of behaviour
Overdamped 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Step Response Time (sec) Amplitude Delay L Gain K Raise Time ts L ts KRaise Time: Time to reach 63% of the steady state value.
Delay: Time of reaction for the output with respect to a change in the input.
Gain: Quotient of the output and the input values.
Types of behaviour
Underdamped K te tp ts Delay L Gain K Raise Time ts Peak Time tp Settling Time te Overshoot MpRaise Time: Time to reach
the steady state value for the first time.
Peak Time: Time to reach the
maximum value.
Settling Time: Time to
confine the output within a band of 5% arounf the steady state value.
Overshoot: Percentage
Increment of the peak value with respect to the steady state value..
Types of behaviour
Index
Closed Loop Transfer Function
Tuning a Controller
Stability analysis of system
Steady-state response of a closed-loop systems
Transient response of a stable system.
Stability
(Chapter 7. Stability)Stability Criterion:
Gbc(s) is stable if and only if all poles are located on the left-half complex plane.
Closed loop poles are the roots of (depend on C(s))
A closed-loop system might become unstable if the controller is not properly designed.
The controller design must guarantee closed loop stability Example: Kp=15
Stability
Analitic Procedure (Try & Error)
• Evalute closed loop poles for every combination of the controller parameters (Kp, Td, Ti) using the model of the system.
Routh-Hurwitz stability criterion
• A tool to evaluate if a polinomial has roots on the right-half complex plane. • The method prevents form computing the whole set of roots of a higher
order polinomial
• It can be used to evaluate stability conditions
Routh-Hurwitz Stability Criterion
Allows to determine if there exists a root in the right-half complex plane
Important: Note the notation
1 – If there exists a negative parameter, then the polinomial has at least one root in the right-half plane.
2 – Build the Routh-Hurwitz table. If there exists a negative component on the first column, then the polinomial has at least one root in the right-half plane.
There are special rules to deal with degenerate cases (See Chapter 7)
Example
Closed loop system
The poles are the solution of the following equation (depends on Kp)
Example
Closed loop system
The poles are the solution of the following equation (depend on Kp y Ti)
Not vey useful for multiple parameters
Index
Closed Loop Transfer Function
Tuning a Controller
Stability analysis of systems
Steady-state response of a closed-loop system
Transient response of a stable system.
Steady State Response
Analysis of system response as time tends to infinity(We assume the closed loop system is stable)
Steady state error
Final Value Theorem (property of Laplace transform)
Important: Depends on R(s)
Error for Step input
0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1Error for a constant steady-state input
Position error constant
All stable systems have bounded steady-state errors
Error for Ramp input
Steady state error for ramp input
Velocity Error
Bounded Velocity Error ⇔ Null position Error (C(s)G(s) has at least one integrator)
For the velocity error to be null (the system reached the reference)
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
Error for parabolic input
Steady state error for parabolic input
Acceleration error constant
Bounded acceleration error ⇔ Null position error ⇔ Null velocity error (C(s)G(s) has at least two integrators)
For the parabolic error to be null (The system reaches the reference)
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Error Table
Type of a system = Number of integrators
0 1 2 Step 0 0 Ramp 0 Parabolic
Error
Type
Example
Type I System
Proportional Controller
Proportional Controller affects the Bode Gain of the system, but can not change its Type.
Improves (quantitatively) steady state behaviour.
Example
0 10 20 30 40 50 60 0 0.5 1 1.5 Kp = 1, Td = 0, 1/Ti = 0 y(t) 0 10 20 30 40 50 60 −0.5 0 0.5 1 u(t) 0 10 20 30 40 50 60 −0.5 0 0.5 1 e(t) 0 10 20 30 40 50 60 0 5 10 ∫ 0 t e( τ )d τRespuesta del sistema en BC Position error. Constant reference (Step)
Example
Velocity error. Increasing reference (ramp)
0 10 20 30 40 50 60 0 20 40 60 Kp = 1, Td = 0, 1/Ti = 0 y(t) 0 10 20 30 40 50 60 0 5 10 u(t) 0 10 20 30 40 50 60 0 5 10 e(t) 0 10 20 30 40 50 60 0 100 200 300 ∫ 0 t e( τ )d τ
Example
Type I system
PI Controller
P controllers affect the Bode gain of the system and increases the system type
Improves (qualitatively) steady state behaviour
Depends on Kp and Ti
Example
Position error. Constant reference (Step)
0 10 20 30 40 50 60 0 0.5 1 1.5 Kp = 1, Td = 0, 1/Ti = 0.1 y(t) 0 10 20 30 40 50 60 −1 0 1 2 u(t) 0 10 20 30 40 50 60 −0.5 0 0.5 1 e(t) 0 10 20 30 40 50 60 0 1 2 3 ∫ 0 t e( τ )d τ
Example
Velocity error. Increasing reference (ramp)
0 10 20 30 40 50 60 0 20 40 60 Kp = 1, Td = 0, 1/Ti = 0.1 y(t) 0 10 20 30 40 50 60 0 5 10 u(t) 0 10 20 30 40 50 60 0 1 2 3 e(t) 0 10 20 30 40 50 60 0 50 100 ∫ 0 t e( τ )d τ
Respuesta del sistema en BC The integral term is
introduced to improve steady state response.
(It might unstabilize the system.
Ex. Try simulation with Kp=1, Ti=1)
Index
Closed Loop Transfer Function
Tuning a Controller
Stability analysis of systems
Steady-state response of a closed-loop system
Transient response of a stable system.
Transient Response
Response to unit step input
Classification of output signal ∆y(t) depending on the input signal.
Unit Step (the most widely used).
Ramp.
Sinusoidal.
Provides information about the dynamical properties of the system
Y(s) G(s)
-C(s) E(s) U(s) R(s)Response in y(t) when a reference r(t) is applied
Reference signal: Unit Step signal. Shows the speed of response of the system (In general the reference signal will be different than the unit step)
Transient Response
TEMA 5. Respuesta temporal de sistemas lineales.
Sistemas dinámicos lineales de primer orden. Ejemplos. Sistemas
dinámicos lineales de segundo orden. Respuesta ante escalón. Sistemas de orden n.
We are interested in the output y(t) as r(t) varies in time (Closed-loop behavior) The trnasient response of a LTI system depends on the closed-loop transfer function (Gbc(s))
One option: Try & Error
Given a system, simulate or aplly inverde Laplace transform
It is difficult to characterize propoerties as raise time or overshoot Identify the effect of parameters in the response.
First Order Systems
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• In practice, some poles have more influence in the response than others. These poles are called dominant poles
• The dominant poles are those yielding the slowest reponse
• The response speed is given by the exponent of the exponential terms (the real part of the pole). Remember:
Dominant Poles
Dominant dynamics: poles with the slowest response
In practice, the dominat poles are determined from their relative distance to the imaginary axis.
Re Im p1 p’1 p2 p’2 d2 d1 Re Im p1 p2 p’2 d2 d1 p1 is dominant if d2/d1>5
The static gain must remain the same
Dominant Poles
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Tiempo(s) 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 y(t)Effect of zeros in the output
0 0.5 1 1.5 0 1 2 3 4 5 6 Step Response Time (sec) A m p lit u d e 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 Step Response Time (sec) A m p lit u d eEffect of zeros in the output
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 6 Step Response Time (sec) A m p lit u d e y(t) dy(t)/dt yc(t) Qualitatively:Non-minimum phase zeros
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Step Response Time (sec) A m p lit u d e 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Step Response Time (sec) A m p lit u d e 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 Step Response Time (sec) A m p lit u d e 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 6 Step Response Time (sec) A m p lit u d e -20 -15 -10 -5 0 5 -1 0 1x
x
o
o
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o
Non-minimum phase zeros
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Step Response Time (sec) A m p lit u d e 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -1 -0.5 0 0.5 1 1.5 2 Step Response Time (sec) A m p lit u d e -20 -15 -10 -5 0 5 -1 0 1x
x
o
o
Dynamics cancellation
-7 -6 -5 -4 -3 -2 -1 0 -1 0 1x
o
x
The closer the zero is to the pole, the less it influences system response Affects the dominant dynamics (in transient regime)
Settling time is not significantly affected.
0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 Step Response Time (sec) A m p lit u d e
Design Hypothesis
Quick review of concepts
• Time response of first order systems • Time response of second order systems • Time response of higher order systems • Effect of zeros
It is difficult to obtain explicit results in general
Design Hypothesis
• Explicit expressions for the effect of controller parameters on the transient response for step inputs are required.
• A pair of conjugate complex poles dominate the closed loop response
• Zeros are difficult to deal with, in general. Not considered.
Two design tools:
• Root Locus design
Index
Closed Loop Transfer Function
Tuning a Controller
Stability analysis of system
Steady-state response of a closed-loop systems
Transient response of a stable system.
Poles and Zeros of Gbc(s)
Closed-loop transfer functionZeros of the closed-loop system The same zeros of the open-loop plant plus those of the controller
Poles of the closed-loop system Depend on the design parameter
• In some cases it is possible to get explicit expressions for 2nd order systems with P and PD controllers
• Not possible in general
Example
P Controller
The poles depend on Kp
Complex plane representation Root Locus
Example
−6 −5 −4 −3 −2 −1 0 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Kp=0.1 Kp=1 Kp=10 Kp=15Illustratuve example:
Magnetic levitation system
Description Value
Ball material Steel
Ball diameter 25 mm Coil diameter 80 mm Winding turns 2850 Resistence 22 Ω Inductance 277 mH a 1 kHz 442 mH a 120 kHz
Illustratuve example:
Magnetic levitation system
Nonlinear model of the system
2 2 X I k mg X m&& = − m : Ball mass g : Gravity constant
X : Distance between ball and coil (magnitudes to be controlled) I : Coil current (control action) K : constant coefficient
X Fm
Fg
System Linealization
We assume the operating point X0 with control action I0 and consider error variables
X X X I I I ∆ + = ∆ + = 0 0
The incremental variables depend on the
selected operating point.