A: SISO Feedback Control A.1 Internal Stability And Youla .
Robust and Optimal Control, Spring 2015Instructor: Prof. Masayuki Fujita (S5-303B)A: SISO Feedback ControlA.1 Internal Stability and Youla Parameterization[SP05, Sec. 3.2, 4.1.5, 4.7, 4.8]A.2 Sensitivity and Feedback Performance[SP05, Sec. 2.2, 5.2]A.3 Loop Shaping[SP05, Sec. 2.4, 2.6]Reference:[SP05] S. Skogestad and I. Postlethwaite,Multivariable Feedback Control; Analysis and Design,Second Edition, Wiley, 2005.
Internal StabilityGang of Four[AM08, p. 317]SensitivityComplementary SensitivityLoad SensitivityNoise Sensitivity[AM08] K.J. Astrom and R.M. Murray, Feedback Systems,Princeton University Press, 20082
Internal Stability[SP05, Ex. 4.16](p. 144)Comp. SensitivitySensitivityStable?Load SensitivityNoise SensitivityStep ResponseUnstableTime [s]Time [s]Time [s]Time [s]3
Internal Stability[SP05, Theorem 4.6] (p. 145)The feedback system in the above figure is internally stableif and only if all “Gang of Four () ” are stableWell-posedness:(Gang of Four: well-defined and proper)C. DesoerC.A. Desoer and W.S. Chan,Journal of the Franklin Institute,300 (5-6) 335-351, 19754
Youla Parameterization ( Parameterization)Case 1: Stable Plant[SP05, p. 148] Internal Model Control (IMC) StructureAll Stabilizing Controllers:-parameterGang of Four： Proper Stable Transfer Function5
Youla ParameterizationCase 2: Unstable Plant[SP05, p. 149]Coprime Factorization [SP05, p. 122]Coprime: No common right-half plane(RHP) zeros： Proper Stable Transfer Functions[SP05, Ex. 4.1](*)Bezout Identity[SP05, Ex.][Ex.]: Coprime： Proper Stable Transfer Functions： (*)： Integer： Integer6
Youla ParameterizationCase 2: Unstable Plants[SP05, p. 149]A Stabilizing Controller[SP05, Ex.]All Stabilizing ControllersGang of FourAffine Functions of7
Sensitivity and Feedback PerformanceDisturbance AttenuationOpen-loopClosed-loop: Sensitivitysmall: good Feedback Performance8
Insensitivity to Plant Variations[SP05, p. 23]small : good Feedback Performance9
Benefits of Feedback Disturbance Attenuation Insensitivity to Plant Variations Stabilization (Unstable Plant) Linearizing Effects Reference Tracking: smallTwo-degrees-of-freedom Control Feedback Feedforward10
Waterbed Effects[SP05, p. 167]There exists a frequency range over which the magnitude of the sensitivityfunction exceeds 1 if it is to be kept below 1 at the other frequency range.[dB]100 10100Frequency [rad/s] 101[SP05, Ex., p. 170](unstable)11
Maximum Peaks ofandSensitivity[SP05, p. 36]Complementary Sensitivity: Maximum Peak Magnitude of: Maximum Peak Magnitude of: Bandwidth Frequency of: Bandwidth Frequency of12
Loop ShapingLoop Transfer FunctionSensitivity:Comp. Sensitivity: ConstraintlargesmallsmallsmallLoop ShapingClosed-loopOpen LoopStability, Performance, Robustness13
Loop Transfer Function[SP05, Ex. 2.4] (p. 34)Gain Crossover FrequencyStability Margins [SP05, p. 32]Gain MarginPhase MarginTime Delay MarginStability Margin[SP05, Ex. 2.4] (p. 34)14
Frequency Domain Performance[SP05, Ex. 2.4] (p. 34)Maximum Peak Criteria[SP05, p. 36][Ex.][Ex.]15
Bode Gain-phase Relationship [SP05, p. 18](minimum phase systems)Slope of the Gain Curve atSteep Slope: Small Phase Margin[SP05, Ex., p. 20]0-2-1-216
Fundamental Limitations[SP05, pp. 183]Bound on the Crossover FrequencyRHP (Right half-plane) ZeroFast RHP Zeros ( large): Loose Restrictions ImworseSlow RHP Zeros ( small): Tight RestrictionsTime Delay0betterRezUnstable zeroStep ResponseFrequency [rad/s]Time [s]
Fundamental Limitations[SP05, pp. 192, 194]Bound on the Crossover FrequencyRHP (Right half-plane) PoleSlow RHP Poles ( small): Loose RestrictionsFast RHP Poles ( large): Tight RestrictionsIm0betterworse RepUnstable poleFrequency [rad/s]Poles on imaginary axis
SISO Loop Shaping[SP05, pp. 41, 42, 343]PerformanceRobust Stability( Roll-off)Loop Shaping Specifications Gain Crossover Frequency Shape of System Type, Defined as the Number of Pure Integrators in Roll-off at Higher Frequencies19
Step response analysis/Performance criteriaRise timeSettling timePeak timeOvershootError toleranceFirst-order System10.90.10Second-order SystemRise timeRise timeSettling timeSettling timeOvershootOvershootPeak Time[QZ07] L. Qiu and K. Zhou (2007) Introduction to Feedback Control, Prentice Hall.20
Design RelationsMaximum Peak Magnitude ofComplementarySensitivityPhase MarginBandwidthifif: Maximum Peak Magnitude of: Bandwidth Frequency ofif[FPN09] G.F. Franklin, J.D. Powell and A. E.-Naeini (2009) Feedback Control of DynamicSystems, Sixth Edition, Prentice Hall.21
Controllability analysis with SISO feedback control[SP05, pp. 206-209]Margin to stay within constraintsMargin for performanceMargin because of RHP-poleMargin because of RHP-zeroMargin because of frequencywhere plant hasphase lagMargin because of delayTypically, the closed-loop bandwidth of the spacecraft is an order of magnitudeless than the lowest mode frequency, and as long as the controller does not exciteany of the flexible modes, the sampling period may be selected solely based onthe closed-loop bandwidth.[Le10] W.S. Levine (Eds.) (2010) The Control Handbook, Second Edition: Control SystemFundamentals, Second Edition, CRC Press.22
RHP Poles/Zeros, Time Delays and SensitivityFor systems with a RHP pole p and RHP zero z(or a time delay τ ), any stabilizing controllergives sensitivity functions with the propertyp zM S sup S ( jω ) p zωM T sup T ( jω ) e pτωRHP pole and zero and time delay significantly limitthe achievable performance of a systemMSS ( jω )p zp zMTe pτT ( jω )9
RHP Poles/Zeros, Time Delays and SensitivityAll-pass system( p 1 , z b , τ )b sPap ( s ) s 1RHP pole/zero pairz / p 1 / 6 or 6 z / pThe zero and the pole must besufficiently far aparte sτPap ( s ) s 1RHP pole and time delaypτ 0.3The product of RHP pole and timedelay must be sufficiently small10allowable phase lag of Pap at ω gc : ϕl 90
A: SISO Feedback Control. A.2 Sensitivity and Feedback Performance. Reference: A.3 Loop Shaping [SP05] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control; Analysis and Design ,
5700 SISO 1 20.574 Only for Report Use PASS 802.11n(HT40) . 5290 SISO 1 81.927 Only for Report Use PASS 5530 SISO 1 83.990 Only for Report Use PASS . 5600 SISO 1 17.800 Only for Report Us
comparison between experimental results for the SISO and MIMO controller implementations. II. S YSTEM IDENTIFICATION AND A NALYSIS A. System Model Identification While first principles models for VCC systems are available, Rasmussen [7] verified that system identification (ID) techniques can be used to construct accurate, low-
RLTOOL Tutorial 7/30/2007 1. Introduction What is RLTOOL? RLTOOL is a tool in MATLAB, that provides a GUI for per-forming Root Locus analysis on Single Input Single Output (SISO) systems, which are the class of systems we cover in E105. RLTOOL is a component of the broader SISO Design Tool in MAT
magento survey magento feedback extension magento survey extension magento feedback magento feedback form magento quiz magento poll extension magento customer feedback magento customer feedback extension magento feedback
1. Ask for Feedback – on an ongoing basis. Feedback is so important that we have to ask for it if it does not occur naturally. Sometimes we do get feedback, but it is restricted to one aspect of our behavior, and we may have to ask for additional feedback. To ensure that we get feedback the way we’d most easily hear it, we can direct the giver
Relay feedback and multivariable control Abstract This doctoral thesis treats three issues in control engineering related to relay feedback and multivariable control systems. Linearsystems with relay feedback is the first topic. Such systemsare shown to exhibitseveral interesting behaviors.
INTRODUCTION TO FEEDBACK CONTRO L SYSTEMS 2 1 INTRODUCTION TO FEEDBACK CONTROL SYSTEMS 5 1.1 Objectives of feedback control 6 1.2 Need for feedback 7 1.3 Control system technology: actuators, sensors, controllers 8 1.4 Some applications 8 1.4.1 Water level regulator for a toilet tank 8 1.4.2 Single-link robot 9 1.4.3 Air pressure control in a .
lic perceptions of the criminal courts by focusing on a few basic topics. We begin by discussing where the courts fit in the criminal justice system and how the public perceives the courts. Next, attention shifts to the three activities that set the stage for the rest of the book: Finding the courthouse Identifying the actors Following the steps of the process As we will see .
