This document provides an overview of automatic control and mixed sensitivity H∞ control. It begins with background on automatic control, including its history, basic notions such as feedback and components, and types of controllers and control objectives. It then discusses mixed sensitivity H∞ control, including the motivation of robustness to uncertainty, basic notions such as the infinity norm and robust H∞ control objective. It also covers uncertainty models and loop shaping techniques. The document provides context and definitions regarding automatic control and mixed sensitivity H∞ control.
Optimal PID Controller Design for Speed Control of a Separately Excited DC Mo...ijscmcj
This paper presents a new approach to determine the optimal proportional-integral-derivative controller parameters for the speed control of a separately excited DC motor using firefly optimization technique. Firefly algorithm is one of the recent evolutionary methods which are inspired by the Firefly’s behavior in nature. The firefly optimization technique is successfully implemented using MATLAB software. A comparison is drawn from the results obtained between the linear quadratic regulator and firefly optimization techniques. Simulation results are presented to illustrate the performance and validity of the design method.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
PID Tuning using Ziegler Nicholas - MATLAB ApproachWaleed El-Badry
This is an unreleased lab for undergraduate Mechatronics students to know how to practice Ziegler Nicholas method to find the PID factors using MATLAB.
Optimal PID Controller Design for Speed Control of a Separately Excited DC Mo...ijscmcj
This paper presents a new approach to determine the optimal proportional-integral-derivative controller parameters for the speed control of a separately excited DC motor using firefly optimization technique. Firefly algorithm is one of the recent evolutionary methods which are inspired by the Firefly’s behavior in nature. The firefly optimization technique is successfully implemented using MATLAB software. A comparison is drawn from the results obtained between the linear quadratic regulator and firefly optimization techniques. Simulation results are presented to illustrate the performance and validity of the design method.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
PID Tuning using Ziegler Nicholas - MATLAB ApproachWaleed El-Badry
This is an unreleased lab for undergraduate Mechatronics students to know how to practice Ziegler Nicholas method to find the PID factors using MATLAB.
Optimal and pid controller for controlling camera’s position in unmanned aeri...Zac Darcy
This paper describes two controllers designed specifically for adjusting camera’s position in a small unmanned aerial vehicle (UAV). The optimal controller was designed and first simulated by using MATLAB technique and the results displayed graphically, also PID controller was designedand simulatedby using MATLAB technique .The goal of this paper is to connect the tow controllers in cascade mode to obtain the desired performance and correction in camera’s position in both roll and pitch.
Analysis & Control of Inverted Pendulum System Using PID ControllerIJERA Editor
This Analysis designs a two-loop proportional–integral–derivative (PID) controller for an inverted cart– pendulum system via pole placement technique, where the (dominant) closed-loop poles to be placed at the desired locations are obtained from an Linear quadratic regulator (LQR) design. It is seen that in addition to yielding better responses (because of additional integral action) than this LQR (equivalent to two-loop PD controller) design, the proposed PID controller is robust enough. The performance and of the PID compensation are verified through simulations as well as experiments.
Energy-Based Control of Under-Actuated Mechanical Systems - Remotely Driven A...Mostafa Shokrian Zeini
This presentation concerns the energy-based swing-up control for a remotely driven acrobot (RDA) which is a 2-link planar robot with the first link being underactuated and the second link being remotely driven by an actuator mounted at a fixed base through a belt.
Metal cutting tool position control using static output feedback and full sta...Mustefa Jibril
In this paper, a metal cutting machine position control have been designed and simulated using
Matlab/Simulink Toolbox successfully. The open loop response of the system analysis shows that the system needs
performance improvement. Static output feedback and full state feedback H 2 controllers have been used to increase
the performance of the system. Comparison of the metal cutting machine position using static output feedback and
full state feedback H 2 controllers have been done to track a set point position using step and sine wave input signals
and a promising results have been analyzed.
Linear quadratic regulator and pole placement for stabilizing a cart inverted...journalBEEI
The system of a cart inverted pendulum has many problems such as nonlinearity, complexity, unstable, and underactuated system. It makes this system be a benchmark for testing many control algorithm. This paper presents a comparison between 2 conventional control methods consist of a linear quadratic regulator (LQR) and pole placement. The comparison indicated by the most optimal steps and results in the system performance that obtained from each method for stabilizing a cart inverted pendulum system. A mathematical model of DC motor and mechanical transmission are included in a mathematical model to minimize the realtime implementation problem. From the simulation, the obtained system performance shows that each method has its advantages, and the desired pendulum angle and cart position reached.
Robust Fuzzy Output Feedback Controller for Affine Nonlinear Systems via T–S ...Mostafa Shokrian Zeini
This presentation concerns the design of a robust H_∞ fuzzy output feedback controller for a class of affine nonlinear systems with disturbance via Takagi-Sugeno (T–S) fuzzy bilinear model. The parallel distributed compensation (PDC) technique is utilized to design a fuzzy controller. The stability conditions of the overall closed loop T-S fuzzy bilinear model are formulated in terms of Lyapunov function via linear matrix inequality (LMI). The control law is robustified by H_∞ sense to attenuate external disturbance. Moreover, the desired controller gains can be obtained by solving a set of LMI.
T-S Fuzzy Observer and Controller of Doubly-Fed Induction GeneratorIJPEDS-IAES
This paper aims to ensure a stability and observability of doubly fed induction generator DFIG of a wind turbine based on the approach of fuzzy control type T-S PDC (Parallel Distributed Compensation) which determines the control laws by return state and fuzzy observers.First, the fuzzy TS model is used to precisely represent a nonlinear model of DFIG proposed and adopted in this work. Then, the stability analysis is based on the quadratic Lyapunov function to determine the gains that ensure the stability conditions.The fuzzy observer of DFIG is built to estimate non-measurable state vectors and the estimated states converging to the actual statements. The gains of observatory and of stability are obtained by solving a set of linear matrix inequality (LMI).Finally, numerical simulations are performed to verify the theoretical results and demonstrate satisfactory performance.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
MODELLING AND SIMULATION OF INVERTED PENDULUM USING INTERNAL MODEL CONTROLJournal For Research
The internal model control (IMC) philosophy relies on the internal model principle, which states that control can be achieved only if the control system encapsulates, either implicitly or explicitly, some representation of the process to be controlled. In particular, if the control scheme is developed based on an exact model of the process, then perfect control is theoretically possible. Transfer function of Inverted Pendulum is selected as the base of design, which examines IMC controller. Matlab/simulink is used to simulate the procedures and validate the performance. The results shows robustness of the IMC and got graded responses when compared with PID. Furthermore, a comparison between the PID and IMC was shows that IMC gives better response specifications.
Real Time Implementation of Fuzzy Adaptive PI-sliding Mode Controller for Ind...IJECEIAES
In this work, a fuzzy adaptive PI-sliding mode control is proposed for Induction Motor speed control. First, an adaptive PI-sliding mode controller with a proportional plus integral equivalent control action is investigated, in which a simple adaptive algorithm is utilized for generalized soft-switching parameters. The proposed control design uses a fuzzy inference system to overcome the drawbacks of the sliding mode control in terms of high control gains and chattering to form a fuzzy sliding mode controller. The proposed controller has implemented for a 1.5kW three-Phase IM are completely carried out using a dSPACE DS1104 digital signal processor based real-time data acquisition control system, and MATLAB/Simulink environment. Digital experimental results show that the proposed controller can not only attenuate the chattering extent of the adaptive PI-sliding mode controller but can provide high-performance dynamic characteristics with regard to plant external load disturbance and reference variations.
Controller design of inverted pendulum using pole placement and lqreSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Optimal and pid controller for controlling camera’s position in unmanned aeri...Zac Darcy
This paper describes two controllers designed specifically for adjusting camera’s position in a small unmanned aerial vehicle (UAV). The optimal controller was designed and first simulated by using MATLAB technique and the results displayed graphically, also PID controller was designedand simulatedby using MATLAB technique .The goal of this paper is to connect the tow controllers in cascade mode to obtain the desired performance and correction in camera’s position in both roll and pitch.
Analysis & Control of Inverted Pendulum System Using PID ControllerIJERA Editor
This Analysis designs a two-loop proportional–integral–derivative (PID) controller for an inverted cart– pendulum system via pole placement technique, where the (dominant) closed-loop poles to be placed at the desired locations are obtained from an Linear quadratic regulator (LQR) design. It is seen that in addition to yielding better responses (because of additional integral action) than this LQR (equivalent to two-loop PD controller) design, the proposed PID controller is robust enough. The performance and of the PID compensation are verified through simulations as well as experiments.
Energy-Based Control of Under-Actuated Mechanical Systems - Remotely Driven A...Mostafa Shokrian Zeini
This presentation concerns the energy-based swing-up control for a remotely driven acrobot (RDA) which is a 2-link planar robot with the first link being underactuated and the second link being remotely driven by an actuator mounted at a fixed base through a belt.
Metal cutting tool position control using static output feedback and full sta...Mustefa Jibril
In this paper, a metal cutting machine position control have been designed and simulated using
Matlab/Simulink Toolbox successfully. The open loop response of the system analysis shows that the system needs
performance improvement. Static output feedback and full state feedback H 2 controllers have been used to increase
the performance of the system. Comparison of the metal cutting machine position using static output feedback and
full state feedback H 2 controllers have been done to track a set point position using step and sine wave input signals
and a promising results have been analyzed.
Linear quadratic regulator and pole placement for stabilizing a cart inverted...journalBEEI
The system of a cart inverted pendulum has many problems such as nonlinearity, complexity, unstable, and underactuated system. It makes this system be a benchmark for testing many control algorithm. This paper presents a comparison between 2 conventional control methods consist of a linear quadratic regulator (LQR) and pole placement. The comparison indicated by the most optimal steps and results in the system performance that obtained from each method for stabilizing a cart inverted pendulum system. A mathematical model of DC motor and mechanical transmission are included in a mathematical model to minimize the realtime implementation problem. From the simulation, the obtained system performance shows that each method has its advantages, and the desired pendulum angle and cart position reached.
Robust Fuzzy Output Feedback Controller for Affine Nonlinear Systems via T–S ...Mostafa Shokrian Zeini
This presentation concerns the design of a robust H_∞ fuzzy output feedback controller for a class of affine nonlinear systems with disturbance via Takagi-Sugeno (T–S) fuzzy bilinear model. The parallel distributed compensation (PDC) technique is utilized to design a fuzzy controller. The stability conditions of the overall closed loop T-S fuzzy bilinear model are formulated in terms of Lyapunov function via linear matrix inequality (LMI). The control law is robustified by H_∞ sense to attenuate external disturbance. Moreover, the desired controller gains can be obtained by solving a set of LMI.
T-S Fuzzy Observer and Controller of Doubly-Fed Induction GeneratorIJPEDS-IAES
This paper aims to ensure a stability and observability of doubly fed induction generator DFIG of a wind turbine based on the approach of fuzzy control type T-S PDC (Parallel Distributed Compensation) which determines the control laws by return state and fuzzy observers.First, the fuzzy TS model is used to precisely represent a nonlinear model of DFIG proposed and adopted in this work. Then, the stability analysis is based on the quadratic Lyapunov function to determine the gains that ensure the stability conditions.The fuzzy observer of DFIG is built to estimate non-measurable state vectors and the estimated states converging to the actual statements. The gains of observatory and of stability are obtained by solving a set of linear matrix inequality (LMI).Finally, numerical simulations are performed to verify the theoretical results and demonstrate satisfactory performance.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
MODELLING AND SIMULATION OF INVERTED PENDULUM USING INTERNAL MODEL CONTROLJournal For Research
The internal model control (IMC) philosophy relies on the internal model principle, which states that control can be achieved only if the control system encapsulates, either implicitly or explicitly, some representation of the process to be controlled. In particular, if the control scheme is developed based on an exact model of the process, then perfect control is theoretically possible. Transfer function of Inverted Pendulum is selected as the base of design, which examines IMC controller. Matlab/simulink is used to simulate the procedures and validate the performance. The results shows robustness of the IMC and got graded responses when compared with PID. Furthermore, a comparison between the PID and IMC was shows that IMC gives better response specifications.
Real Time Implementation of Fuzzy Adaptive PI-sliding Mode Controller for Ind...IJECEIAES
In this work, a fuzzy adaptive PI-sliding mode control is proposed for Induction Motor speed control. First, an adaptive PI-sliding mode controller with a proportional plus integral equivalent control action is investigated, in which a simple adaptive algorithm is utilized for generalized soft-switching parameters. The proposed control design uses a fuzzy inference system to overcome the drawbacks of the sliding mode control in terms of high control gains and chattering to form a fuzzy sliding mode controller. The proposed controller has implemented for a 1.5kW three-Phase IM are completely carried out using a dSPACE DS1104 digital signal processor based real-time data acquisition control system, and MATLAB/Simulink environment. Digital experimental results show that the proposed controller can not only attenuate the chattering extent of the adaptive PI-sliding mode controller but can provide high-performance dynamic characteristics with regard to plant external load disturbance and reference variations.
Controller design of inverted pendulum using pole placement and lqreSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
International Journal of Automatic Control System
publishes original research articles and review papers, both experimental and theoretical in origin. Paper published in the international Journal of automatic control systems covers all major topics under automatic control systems including embedded control systems, real time systems, digital and analogue control.
Comparison of backstepping, sliding mode and PID regulators for a voltage inv...IJECEIAES
In the present paper, an efficient and performant nonlinear regulator is designed for the control of the pulse width modulation (PWM) voltage inverter that can be used in a standalone photovoltaic microgrid. The main objective of our control is to produce a sinusoidal voltage output signal with amplitude and frequency that are fixed by the reference signal for different loads including linear or nonlinear types. A comparative performance study of controllers based on linear and non-linear techniques such as backstepping, sliding mode, and proportional integral derivative (PID) is developed to ensure the best choice among these three types of controllers. The performance of the system is investigated and compared under various operating conditions by simulations in the MATLAB/Simulink environment to demonstrate the effectiveness of the control methods. Our investigation shows that the backstepping controller can give better performance than the sliding mode and PID controllers. The accuracy and efficiency of the proposed backstepping controller are verified experimentally in terms of tracking objectives.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Automatic control and mixed sensitivity Hinf control
1. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Automatic control and mixed sensitivity H∞
control
René Galindo Orozco
FIME - UANL
CINVESTAV-Monterrey, 20 de Febrero de 2008
2. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Contents
1 Automatic control
1 Basic notions
2 Degree of mechanization
3 History
4 Class of systems
2 Mixed sensitivity H∞ control
1 Motivation
2 Basic notions
3 Background
4 Loop-shaping
5 Mixed sensitivity
6 Standard solutions
7 Parity interlacing property
1 Mixed sensitivity H∞ control in a
non conventional scheme
1 A mixed sensitivity problem
2 Direct solutions
3 Controllers
4 Tuning procedure
5 Benchmark of a mechanical
system
6 Conclusions
3. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Basic notions
Automatic control [Wikipedia]
A research area and theoretical base for mechanization and
automation , employing methods from mathematics and
engineering
Mechanization
Machinery to assist human operators with the physical
requirements
Automation (ancient Greek: = self dictated) [Salvat
encyclopedia]
Control systems for industrial machinery and
processes, replacing human operators
Reduces the need for human sensory and mental
requirements
4. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Basic notions
Automatic control [Wikipedia]
A research area and theoretical base for mechanization and
automation , employing methods from mathematics and
engineering
Theory that deals with influencing the behavior of dynamic
systems
Mechanization
Machinery to assist human operators with the physical
requirements
Automation (ancient Greek: = self dictated) [Salvat
encyclopedia]
Control systems for industrial machinery and
processes, replacing human operators
Reduces the need for human sensory and mental
requirements
5. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Basic notions
Components [Wikipedia]
u (t) -
f ( )
O& O
O
%
#&
O
"
O
"
-y (t)
?
d (t)
System , set of interacting
entities, real or abstract,
forming an integrated whole
Sensor , measure some physical state
Controller , manipulates u (t) to obtain the desired y (t)
Actuator effect a response under the command of the controller
Reference or set point , a desired y (t)
6. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Controller improved by J. Watt
7. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Basic notions
Control objectives
1 Regulation. Ex. controller improved by Watt
lim
t!∞
x (t) ! 0, or lim
t!∞
[x (t) xd] ! 0, xd 2 <
2 Tracking or servo. Ex. radar
lim
t!∞
x (t) ! xd (t)
3 Model matching
4 Input / output decoupling
5 Disturbance rejection or attenuation, etc.
8. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Basic notions
Type of controller
Open-loop controller
-r
K(s) -L
di
?u- P∆ (s) -L
do
? -y
Can not compensate d (t), ∆ and noise
Closed-loop controller
r
-
L
-
e
K(s) -
L
di
?u
- P∆ (s) -
L
do
?
-
y
?L dm
6
1
6
Feedback on the performance allows the controller to dynamically
compensate for d (t), ∆ and noise
9. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Basic notions
Feedback [Wikipedia]
Basic mechanism by which systems maintain their equilibrium or
homeostasis
Types of feedback
Negative, tends to reduce output,
Positive, tends to increase output, or
Bipolar.
10. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
por flotante
4.pdf
11. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Kalman decomposition
12. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Degree of mechanization
[Salvat encyclopedia]
13. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
History
[Wikipedia]
Ktesibios, -270 Float regulator for a water clock
Philon, -250 Keep a constant level of oil in a lamp
In China, 12th cen-
tury
South-pointing chariot used for navigational
purposes
14th century Mechanical clock
1588 Mill-hopper, a device which regulated the flow
of grain in a mill
C. Drebbel, 1624 An automatic temperature control system for a
furnace
P. de Fermat, 1600’s Minimum-time principle in optics
D. Papin, 1681 A safety valve for a pressure cooker
Bernoulli, 1696
Principle of Optimality in connection with the
Brachistochrone Problem
T. Newcomen, 1712 Steam engine
E. Lee, 1745 Fantail for a windmill
14. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
curve
7.jpg
t12 =
R P2
P1
s
1 + (y0)2
2gy
dx
Brachistochrone curve
15. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
J. Brindley, 1758 Float valve regulator in a steam boiler
I.I. Polzunov, 1765 A float regulator for a steam engine
Pontryagin, Boltyan-
sky, Gamkrelidze, and
Mishchenko 1962
On/off relay control as optimal control
L. Euler (1707-1783)
Calculus of variations . System moves in such
a way as to minimize the time integral of the
difference between the kinetic and potential
energies
W. Henry, 1771 Sentinel register
Bonnemain, 1777 A temperature regulator suitable for indus-
trial use
J. Watt, 1788,
#Industrial revolu-
tion
Centrifugal flyball governor
A.-L. Breguet, 1793 A closed-loop feedback system to synchro-
nize pocket watches
16. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
R. Delap, M. Murray, 1799 Pressure regulator
Boulton, Watt, 1803 Combined a pressure regulator with a
float regulator
I. Newton (1642-1727),
G.W. Leibniz (1646-1716),
brothers Bernoulli (late
1600’s, early 1700’s), J.F.
Riccati (1676-1754)
Infinitesimal calculus
G.B. Airy, 1840 A feedback device for pointing a tele-
scope. Discuss the instability of closed-loop
systems . Analysis using differential equa-
tions
J.L. Lagrange (1736-
1813), W.R. Hamilton
(1805-1865)
Motion of dynamical systems using differ-
ential equations
C. Babbage, 1830 Computer principles
J.C. Maxwell, 1868, I.I.
Vishnegradsky, 1877,
"Prehistory
Analyzed the stability of Watt’s flyball
governor, Re froots (G (s))g < 0
17. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
E.J. Routh, 1877, A.
Hurwitz, 1895
Determining when a characteristic equation
has stable roots. Generalized the results of
Maxwell for linear systems
A.B. Stodola, 1893 Included the delay of the actuating mecha-
nism. System time constant
Lyapunov, 1892 Stability of nonlinear differential equations
using a generalized notion of energy
O. Heaviside, 1892-
1898
Operational calculus. The transfer function
P.-S. de Laplace
(1749-1827), J. Fourier
(1768-1830), A.L.
Cauchy (1789-1857)
Frequency domain approach
Wright Brothers, 1903,
"Primitive period
Successful test flights
C.R. Darwin Feedback over long time periods is responsi-
ble for the evolution of species
18. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
E.A. Sperry, 1910
Gyroscope
J. Groszkowski
Describing function approach
H. S. Black, 1927 Apply negative feedback to electrical amplifiers
H. Nyquist, 1930s Regeneration theory for the design of stable am-
plifiers. Nyquist stability criterion for feedback
systems
A. Einstein The motion of systems occurs in such a way as
to maximize the time, in 4-D space-time
H.W. Bode, 1938
Magnitude and phase frequency response plots.
Closed-loop stability using gain and phase margin
N. Minorsky, 1922 Proportional-integral-derivative (PID ) con-
troller. Nonlinear effects in the closed-loop
system
A. Rosenblueth
and N. Wiener,
1943
Set the basis for cybernetics
19. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
plot
9.jpg
Nyquist plot
20. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
V. Volterra, 1931 Explained the balance between two populations
of fish using feedback
H.L. Házen, 1934 Theory of Servomechanisms
A.N. Kolmogorov,
1941
Theory for discrete-time stationary stochastic
processes
N. Wiener, 1942 Statistically optimal filter for stationary
continuous-time signals
A.C. Hall, 1946 Confront noise effects in frequency-domain
N.B. Nichols, 1947
Nichols Chart
Ivachenko, 1948
Relay control
W.R. Evans, 1948
Root locus technique
J. R. Ragazzini,
1950s Digital control and the z-transform
Tsypkin, 1955 Phase plane for nonlinear controls
21. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
J. von Neumann, 1948 Construction of the IAS stored-program
Sperry Rand, 1950 Commercial data processing machine, UNI-
VAC I
R. Bellman, 1957 Dynamic programming to the optimal con-
trol of discrete-time systems
L.S. Pontryagin, 1958 Maximum principle
C.S. Draper, 1960,
"Classical period
#Modern period
Inertial navigation system
V.M. Popov, 1961
Circle criterion for nonlinear stability analysis
Kalman, 1960’s Linear quadratic regulator (LQR ). Discrete
and continuous Kalman filter . Linear algebra
and matrices. Internal system state
G. Zames, 1966, I.W.
Sandberg, K.S. Naren-
dra, Goldwyn, 1964,
C.A. Desoer, 1965
Nonlinear stability
22. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
W. Hoff, 1969 Microprocessor
J.R. Ragazzini, G.
Franklin, L.A. Zadeh,
C.E. Shannon, 1950’s, E.I.
Jury, 1960, B.C. Kuo, 1963
Theory of sampled data systems
Åström, Wittenmark,
1984
Industrial process control
Gelb, 1974 Digital filtering theory
H.H. Rosenbrock, 1974,
A.G.J. MacFarlane, I.
Postlethwaite, 1977
Extend frequency-domain techniques to
multivariable systems. Characteristic lo-
cus, diagonal dominance and the inverse
Nyquist array
I. Horowitz, 1970’s Quantitative feedback theory
J. Doyle, G. Stein, M.G.
Safonov, A.J. Laub, G.L.
Hartmann, 1981
Singular value plots in robust multivariable
design
23. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Class of systems
8
>>>>>>>>>>>>>><
>>>>>>>>>>>>>>:
Non-causal or
anticipative or predictive
f
Causal or
non-anticipative
8
>>>>>>>>>><
>>>>>>>>>>:
Stochastic
with noise
f
Deterministic
8
>>>>>><
>>>>>>:
Static or
without memory
f
Dynamic
8
>><
>>:
Distributed
Parameters
f
Lumped
Parameters
f
8
>><
>>:
Non-linear f
Linear
8
<
:
Discrete f
Continuous
Time varying
Time invariant
24. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Motivation
Robust : some property is preserved
Uncertainty ∆
x (t0)
u (t)
; y (t)
∆ always exists due to frequency dependent elements,
unmodeled dynamics and failures
Infinity norm kargk∞
kargk∞ := sup
w
σ (arg)
25. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Motivation
Robust : some property is preserved
Uncertainty ∆
x (t0)
u (t)
; y (t)
∆ always exists due to frequency dependent elements,
unmodeled dynamics and failures
Zames in 1981 describes ∆ (s) in the frequency domain as
classical control
Infinity norm kargk∞
kargk∞ := sup
w
σ (arg)
26. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Motivation
Robust : some property is preserved
Uncertainty ∆
x (t0)
u (t)
; y (t)
∆ always exists due to frequency dependent elements,
unmodeled dynamics and failures
Zames in 1981 describes ∆ (s) in the frequency domain as
classical control
Infinity norm kargk∞
kargk∞ := sup
w
σ (arg)
kargk∞ is “good” for specifying the ∆ level and the effect of
kd (t)k2 < ∞ over ky (t)k2
27. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Basic notions
Robust H∞ control objective
rLe
K(s)
-
w- G (s)
z-
- e∆ (s)
?
1
?
Uncertainty and kw (t)k2 attenuation
in a bandwidth, over kz (t)k2,
guaranteeing stability
Minimize,
J := kz (t)k2 :=
Z ∞
∞
z2
(t) dt
1/2
For linear time invariant systems, minimize
J := kTzew (s)k∞ := supew(t):kew(t)k 1 σ (Tzew), or J := σ (Tzew (s))
28. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Basic notions
Uncertainty models8
>>>>>>>>>>>>>>>>>><
>>>>>>>>>>>>>>>>>>:
Structured or parametric
Finite number of uncertainty parameters
Ex. diag k∆1 (s)k∞ , ..., ∆q (s) ∞
diag m1, ..., mq
Non structured
The frequency response is in a set 8w
Ex.: a) phase and gain margins
b) k∆ (s)k∞ m
u - P(s) -L
-y
d
?
- ∆a
29. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
D.C. Youla, H. A.
Jabr, J.J. Bongiorno;
1976
Explicit formula for the optimal controller based
on a least-square Wiener-Hopf minimization of a
cost functional
C.A. Desoer, R. Liu,
J. Murray, R. Saeks;
1980
Controllers placing the feedback system in a ring
of operators with the prescribed properties. The
plant is modeled as a ratio of two operators in that
ring
M. Vidyasagar, H.
Schneider, B.A.
Francis, 1982
Necessary and sufficient conditions for a given
transfer function matrix to have a coprime factor-
ization . Characterization of all stabilizing compen-
sators
C.N. Nett, C.A. Ja-
cobson, M. J. Balas;
1984
Give explicit formulas for a doubly coprime frac-
tional representation of the transfer function in
state-space
K. Glover, D. Mc-
Farlane, 1989
An optimal stability margin. Characterization
of all controllers satisfying a suboptimal stabil-
ity margin, in state-space
30. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Loop-shaping
d (t), usually in w < wl
σ (P∆ (s)) " and any phase of P∆ (s), in w > wh =)worst ∆ (s) in
w > wh
σ (arg) gives a measure of the “gain” of ∆ (s) or d (t)
Stabilize the system under ∆ (s) ,
¯σ (To (s)) # ()
_
σ (Lo (s)) # , in w > wh
Regulation of y (t) under d (t) ,
_
σ(So (s)) # () σ (Lo (s)) " , in w < wl
9
>>>>>>>>=
>>>>>>>>;
Loop-shaping
31. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Loop-shaping
W2 (s) and W1 (s) high and low pass weightings
wl and wh depends on
the specific application
the knowledge of d (t) and ∆ (s)
the Bode Phase-Gain Relation
32. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Mixed sensitivity
r
-
L
-
e
K(s) -
L
di
?u
- P∆ (s) -
L
do
?
-
y
?L dm
6
1
6
Robust stability
By the small gain theorem if e∆ (s)
∞
< 1, stability is guaranteed if,
W2 (s) Tu∆y∆
(s) ∞
< 1
Robust performance
kW1 (s) So (s)k∞ < 1, minK(s) ke (t)k2
So (s) = Tdoy (s) = Ter (s) = (I + P (s) K (s)) 1
: output sensitivity
P∆ (s) : uncertain plant
33. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Mixed sensitivity
Minimize
kW1 (s) So (s)k∞ and W2 (s) Tu∆y∆
(s) ∞
in the frequency range in which kd (t)k2 and k∆ (s)k∞ are
significative by a stable K (s) designed for P (s), guaranteeing robust
performance and stability, i.e. minimize,
J1 :=
W1 (s) So (s)
W2 (s) Tu∆y∆
(s) ∞
Uncertainty model Tu∆y∆
(s)
Additive K (s) So (s)
Multiplicative at the output To (s) := So (s) P (s) K (s)
Feedback at the input So (s) P (s)
34. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Standard solutions
General Standard H∞ Optimal Problem
[Doyle, 1981], [Glover, 1984], [Francis, Doyle, 1987], [Chiang, Safonov,
1997]
K (s) stabilizing P (s), and minimizing, J := sup
w:kwk2
2 k
kz (t)k2,
J = kTzw (s)k∞
[Nett, Jacobson, Balas, 1984]
Formula for the YJBK-parametrization,
using static state feedback to stabilize P (s)
=)
Recursive
procedures
35. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Parity interlacing property
P (s) 2 L
pxm
∞ is strongly stabilizable ()
the unstable poles of P (s) between every
even real and unstable zeros of P (s), is even
9
=
;
) 9K (s) 2 RH∞
Strong stability )
8
<
:
For loop breaking
For closed-loop bandwidth "
36. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Problem
J1 is transformed into [Galindo, Malabre, Kuˇcera, 2004]:
J2 :=
Sol
Tu∆y∆h ∞
R (s) is fixed solving a MSP without an augmented system
Sol and Tu∆y∆h becomes real matrices
J2 involves the simultaneous minimization of kSolk∞ and Tu∆y∆h ∞
,
min
K(s)
kSolk∞
subject to kSolk∞ = Tu∆y∆h ∞
that is equivalent to minimize the Lagrange function [Galindo,
Herrera, Martínez, 2000],
f := kSolk∞ η kSolk∞ Tu∆y∆h ∞
η Lagrange multiplier
37. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Direct solutions
[Galindo, Sanchez, Herrera, 2002]
Suppose that det f(s + a) In R (s)g is a Hurwitz polynomial, R (s) 2 <H∞
Define
X (s) = eX (s) = aIn + A 2 <H∞
Y (s) = eY (s) = In 2 <H∞
eNp (s) = Np (s) =
1
s + a
In 2 <H∞
eDp (s) = Dp (s) =
1
s + a
(sIn A) 2 <H∞
NpD 1
p =
1
s + a
1
s + a
(sIn A)
1
XNp + YDp = (aIn + A)
1
s + a
+
1
s + a
(sIn A) = In
eNp (s), Np (s), eDp (s) and Dp (s) are of low order ) less
computational effort
38. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Direct solutions
[Galindo, Sanchez, Herrera, 2002]
Then, a proper stabilizing K (s) 2RH∞ is:
K (s) = A + [(s + a) In R (s)] 1
[(s + a) aIn + R (s) s]
and,
kSolk∞ = 1
a2 k(aIn Rl) Ak∞
kKhSohk∞ = kA + aIn + Rhk∞
kTohk∞ = 1
wh
kA + aIn + Rhk∞
kSohPhk∞ = 1
wh
kSolk∞ # by a ", and kTohk∞ # by wh "
For P (s) strictly proper Toh = Loh
Select rIn for Rh and Rl, and r < a,
kSolk∞ =
a r
a2
kAk∞
A solution of kTohk∞ = kSolk∞ for Tu∆y∆h = SohPh is,
re = a 1
a
wh kAk∞
(1)
39. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Direct Solutions
9 [ a, a] in which kTohk∞ # (") and kSolk∞ " (#) as linear functions of
r [Galindo, Malabre, Kuˇcera, 2004],
-
a
6
1
wh
kA + aInk∞
1
wh
kA + 2aInk∞
1
a kAk∞
@
@
@
@
@
@
@
@
@
@
@
@
a r
kSolk∞
kTohk∞
40. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Direct solutions
[Galindo, Malabre, Kucera, 2004]
Let rIn be for R (s) 2 <H∞
Then, a value for r is:
r = a 1
γmina
(wh + 1) kAk∞
where
γmin = [1 + λmax (YX)]1/2
being Y and X the solutions of the Riccati equations
ATX + XA X2 + In = 0
AY + YAT Y2 + In = 0
The optimal value for r lies in,
r 2 [rb, a]
and a lower bound rb for r is:
rb =
a (wh a) kAk∞ a2
wh kAk∞ + a2
lim
wh !0
rb = (a + kAk∞), lim
wh !∞
rb = a
41. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Let rIn be for R (s) 2 <H∞
Then, an optimal value for r is:
re =
b2 b1
m1 m2
where
b1 :=
1
a
kAk∞ , m1 :=
1
a2
kAk∞
b2 :=
1
wh
kA + aInk∞
m2 :=
1
awh
(kA + 2aInk∞ kA + aInk∞)
Moreover
kSolk∞ =
kA + 2aInk∞ kAk∞
wh kAk∞ + a (kA + 2aInk∞ kA + aInk∞)
42. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Controllers
˙x (t) = Ax (t) + Bu (t)
Define v1 (t) := Bu (t) ) ˙x (t) = Ax (t) + v1 (t)
+
u (t) = BLv1 (t) ) ˙x (t) = Ax (t) + BBLv1 (t)
) E1 := BBL In
xd-L
- K1(s)
v1- BL -u
(sIn A) 1
B -L
d1
? -x
?L d2
6
1
6
A 2 <n n, B 2 <n m, C 2 <p n
v1 (t) the output of the precompensator K1 (s)
BL a left inverse of B
xd (t) the input reference
43. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Controllers
Let A, BBL, C ,
u (s) = BL
K1 (s) (xd (s) x (s))
Dual system AT, CT, BBL T
,
u (s) = CT
L
KT
2 (s) (y (s) by (s))
In original coordinates,
u (s) = K2 (s) CR
(y (s) by (s))
x
-
L ξ
- C - CR - K2(s) - BL -
v2
(sIn A) 1
B -
bx
6
1
6
CR a right inverse of C
v2 (t) the output of K2 (s)
bx (t) and by (t), the estimated state and output
45. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Controllers
Nominal plant P (s)
(A, B, C) a stabilizable and detectable realization of P (s) satisfying
the parity interlacing property
A =
A11 A12
A21 A22
B =
0
B1
C = C eC
B1 2 <m m non-singular
For P (s) proper, transform quadruples into and extended triples
[Basile, Marro, 1992]
46. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Controllers [Galindo, 2006]
exd-L e- K1(s)
v1- BL -
?
u
P(s) -L
d1
? -y
?L
?
d2
6
1
6
L
- (sIn A) 1
B
bx- C - 1 -L
CRK2(s)
v2
BL
6
bx (t) : estimated state
e1 (t) : deviation from the desired state trajectory xd (t)
exd (s) := Wr (s) xd (s) : filtered state reference
Satisfies the separation principle
Allows to get a stable H∞ compensator
bxss = xss, lim
ri!ai
xss ! xdss, Toh ! 0
The class of systems depends of the observability in closed loop
Some of the poles are fixed. For eC = 0, det (sIn m A11) must be
47. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Controllers
[Galindo, 2007]
exd-
L e1- K1(s)
v1 -
?
BL -
u
P(s) -
L
d1
?
-
y
?L
?
d2
6
1
6
L
- (sIn A) 1 bx- C - 1 -L
CR
e2
K2(s)
v2
6
A simplified version of the one of [Galindo, 2006]
The separation principle is not satisfied
Ki (s), become PI as ri ! ai, low complexity controllers
The closed loop poles depends on the selection of the free
parameters of BL and CR, and the rest s = ai, are stable poles
Some of the poles are fixed. For eC = 0, det (sIm A22) and for
48. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Tuning procedure
Tuning Procedure [Galindo, Malabre, Kuˇcera, 2004]
For a desired time response, attenuation of kd1 (t)k2, i.e., for a given a1
1 Select a2 = a1 > 0
2 Find the largest free parameters of BL and CR, and the lowest wh,
satisfying
a) The stationary state error specifications,
b) ri ai (1 ai), i = 1, 2,
c) and minimizing kE1 (ρ1A + In)k∞ and k(ρ2A + In) E2k∞
ρi :=
(ai ri) /a2
i if ri 6= ai
wl/a2
i if ri = ai
wl a fixed frequency in the low frequency bandwidth of Ki (s)
3 If possible select xd 2 Im B to assure that lim
ri!ai
bxss ! xss
4 If needed, use a pre-filter Wr (s) for the reference
49. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Controllers
[Galindo, 2007]
exd
- φ(s)
?- C -L e- K2(s)CR -L
- V(s) -u
P(s) -L
d1
? -y
?L d2
6
1
6
V (s) := σ (s) BLK1 (s) Γ 1 (s), Γ (s) := In + σ (s) K2 (s) CRC,
σ (s) := s+a1 r1
(s+a1)2 , Φ (s) = sIn A
For P∆ (s), we must satisfy also the Small Gain Theorem
Tu∆y∆
(s) does not depend on exd (t), indeed Tu∆y∆
(s) becomes
K (s) So (s), To (s) := P (s) K (s) So (s), and So (s) P (s), where
K (s) = V (s) K2 (s) CR
50. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Controllers
Assignment of part of the poles by a change of basis
Let a change of basis, [Galindo, 2007]
T =
Iq 0
T21 Iq
, T 1 =
Iq 0
T21 Iq
preserving the structure of B, where q m, and
A =
eA11
eA12
eA21
eA22
be partitioned accordingly with the block partition of T. So,
A = TAT 1 ="
eA11
eA12T21
eA12
T21
eA11 + eA21 T21
eA12 + eA22 T21 T21
eA12 + eA22
#
Then,
T21 = eAR
12
eA11 Λ11
assigns a desired dynamics Λ11 to A11, and,
T21 = Λ22
eA22
eAL
12
assigns a desired dynamics Λ22 to A22
51. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Mixed sensitivity
[Galindo, 2007] The norm-∞ of Tu∆y∆h is,
kKhSohk∞ = 1
wh
BLD1D2CR
∞
kTohk∞ = 1
w2
h
CBBLD1D2CR
∞
kSohPhk∞ = 1
wh
kCBk∞
Di := A + (ai + ri) In, i = 1, 2.
52. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Mixed sensitivity
[Galindo, Malabre, Kuˇcera, 2004] lim
s!0
lim
ρi!0
So (s)
∞
=
wl
a2
i
kAk∞
Robust stability is achieved
ai # , but the performance is ameliorated
wh " , but the high frequency bandwidth is decreased
Tuning Procedure
1 Look for the highest values of ai, i = 1, 2, satisfying stability
conditions, minimizing Tu∆y∆h ∞
and satisfying plant input
specifications
2 Fix the value of the free parameters of BL and CR
3 Select wh, satisfying stability conditions, stationary state error
specifications and minimizing Tu∆y∆h ∞
53. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Benchmark of a Mechanical System
-
u
m1
7 ! x1(t)
k
b
m2
7 ! x2(t)
-
d
Consider a model of a mechanical system,
˙x (t) = Ax (t) + Bu (t) + Ψd (t)
y (t) = Cx (t)
where x (t)T
:= x1 (t) x2 (t) ˙x2 (t) ˙x1 (t) ,
A =
2
6
6
6
4
0 0 0 1
0 0 1 0
k
m2
k
m2
b
m2
b
m2
k
m1
k
m1
b
m1
b
m1
3
7
7
7
5
, B =
2
6
6
4
0
0
0
1
m1
3
7
7
5 , Ψ =
2
6
6
4
0
0
1
m2
0
3
7
7
5
C = 0 1 0 0
54. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Benchmark of a Mechanical System
Non-collocated case,
The control input acts only on one uncertainty mass 0.1 m1 3
and the output is the position of m2
The nominal value of m1 = 1
d (t) unknown disturbance
k and b the elasticity and friction coefficients
m1 and m2 the mass
m2 = k = b = 1
55. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Benchmark of a Mechanical System
Let a := a1 = a2
So, r := r1 = r2 which implies K1 (s) = K2 (s)
(A, B, C) is a minimal realization and B has the desired structure
P (s) satisfies the parity interlacing property
eC = 0, det (sIm A22) = s + 1 is Hurwitz
A desired dynamics Λ22 =diagf 2, 2g, and T with q = 2 is
realized, getting,
A =
2
6
6
4
1 1 0 1
1 1 1 0
1 3 2 0
3 1 0 2
3
7
7
5 , B = B, C = C
det sIm A22 = s + 2 is Hurwitz
56. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Benchmark of a Mechanical System
Select a = 2 and,
BL = g1 0 0 1
CR = g2 1 0 g3
T
g1 = 1.6, g2 = 0.3, g3 = 0.55
CB = 0 =) kTohk∞ = 0 and kSohPhk∞ = 0,
wh kKhSohk∞ kKhSohk∞ kKhSohk∞
with r with rb with re
1 1.9 stable 0.05 unstable 0.909 unstable
3 0.675 stable 0.487 unstable 0.57 unstable
5 0.412 unstable 0.35 unstable 0.377 stable
10 0.209 unstable 0.195 stable 0.201 stable
12 0.175 unstable 0.165 stable 0.169 stable
15 0.14 unstable 0.134 stable 0.136 unstable
17 0.124 unstable 0.119 stable 0.121 unstable
18 0.117 unstable 0.113 stable 0.114 unstable
20 0.105 unstable 0.102 unstable 0.103 unstable
57. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Benchmark of a Mechanical System
Select wh = 3 rad/sec., wh = 17 rad/sec. and wh = 12 rad/sec.
=) r = 1.61, rb = 1.622, and re = 1.631
with r with rb with re
kE1 (ρ1A + In)k∞ 2.067 2.052 2.042
k(ρ2A + In) E2k∞ 1.627 1.625 1.623
The characteristic polynomials det (sI AK) of the overall
compensators are stables
Tol = CclA 1
cl Bcl =) exd (t) = (1/To2l) xd (t)
xd (t) = 0 yd 0 0
T
/2 Im B
58. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Benchmark of a Mechanical System
yd = 5, under d (t) = 0.1 (sin (10t) + sin (100t))
59. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Benchmark of a Mechanical System
yd = 5, under d (t) = 0.1 (sin (10t) + sin (100t))
60. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Benchmark of a Mechanical System
x2 (t) tracks the reference signal with r, rb and re, under the d (t)
and the variation of the parameter m1
d (t) remains as very small oscillations at y (t)
Sinusoidal functions of frequencies over wh = 3 rad/sec.,
wh = 17 rad/sec. and wh = 12 rad/sec. for r, rb and re, are well
attenuated at y (t)
Bigger time response with re and less control energy, the contrary
with r, and rb in the middle
Smooth control energy
As m1 ", more energy is required, and the peaks and frequency
of the oscillations decrease
61. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Conclusions
1 A methodology to design a mixed sensitivity H∞ compensator
for a LTI MIMO plant is proposed
2 A nominal compensator is designed for the nominal plant
solving a mixed sensitivity H∞ problem, in a non-conventional
observer-compensator scheme
3 A mixed sensitivity H∞ control law and necessary and sufficient
stability conditions are given
4 Good performance guaranteeing stability, in spite of the
uncertainties and of the external disturbances that are attenuated
5 The controllers with r, rb and re have good performance and their
selection depends on the desired time response and the plant
input specifications
6 An analytic or a numerical method replacing the tuning
procedure is still an open problem
62. Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions
Thank you