# WUSTL ESE 543 - Lecture 1 and 2 rev - Copy (131 pages)

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## Lecture 1 and 2 rev - Copy

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- Pages:
- 131
- School:
- Washington University in St. Louis
- Course:
- Ese 543 - Control Systems Design by State Space Methods

**Unformatted text preview:**

ESE 543 Control System Design Using State Space Methods Class Outline 1 Review of classical methods 2 System modeling and analysis 3 System structural properties 4 State Feedback design 5 Output feedback design 5 Multivariable frequency domain analysis 6 Robustness Theory 7 Optimal Control Text Modern Control Theory W L Brogan Text Robust and Adaptive Control with Aerospace Applications Lavretsky and Wise K A Wise Alt Optimal Control Anderson and Moore 2 Topics 1 System modeling and analysis Linearization State space systems Linear vector spaces Basis vectors Gram Schimdt orthogonalization Transformations mappings canonical forms Range space null space rank Eigenvalues eigenvectors Singular values singular vectors State transition matrix Solution to state equations 3 K A Wise Topics cont 2 System structural properties Controllability Observability Duality Degree of controllability observability Poles zeros MIMO transmission zeros Stability Lyapunov stability Regions of asymptotic stability 4 K A Wise Topics cont 3 Feedback design Pole placement Stabilizability Observer feedback Detectability Pole placement using observer feedback Linear Quadratic Regulator Theory Robust Servomechanism Projective Control Theory 5 K A Wise Topics cont 4 Multivariable frequency domain analysis Multivariable Nyquist theorem Relationship between SISO and MIMO stability analysis Singular value frequency response analysis 5 Optimal Control Linear quadratic regulator 6 Robustness Theory Uncertainty modeling Small gain theorem 6 K A Wise Grading Policy Homework Computer Projects Midterm Exam Final Exam 30 20 25 25 Matlab Boeing Phone 314 232 4549 Cell 636 866 9162 Email kevin a wise boeing com wiseka wustl edu 7 K A Wise Lecture 1 Robust and Adaptive Control Challenges 4th Generation Escape System X 45A J UCAS Nonlinear Aero Large Uncertainties Nonlinear Control Effectors Limited Actuation Vulture II Solar Eagle X 36 Tailless Agility Research Aircraft Unstable In Multiple Axes Non minimum Phase K A Wise 9 X 45A X 45C at Edwards AFB 10 K A Wise Phantom Eye 11 K A Wise Basic Control Theory Control Theory Basics Differential Equations Dynamics Dynamical Systems Linearization Stability Feedback Control Time Domain Frequency Domain Methods For Compensator Design 13 K A Wise Control System Design Control Design Problem Design control inputs to produce satisfactory output response in the presence of disturbances and plant uncertainties w Disturbances Control Inputs u P Output Variables y Plant Model 14 K A Wise Basic Control Theory Steps to a good design Implementation Signal Constraints Robustness Sensitivity Disturbance Rejection Transient Behavior Steady State Accuracy Stability Done System Model Derive and or model all relevant dynamics Analytical or experimental Open Loop Poles and Zeros Form transfer functions or state space models Determine open loop properties poles and zeros Determine control design requirements Close Loop Poles Design compensator to achieve desired pole locations Analysis Of The Closed Loop System Analyze system response in time domain simulation Determine stability robustness properties in frequency domain Bode Nyquist 15 K A Wise Control Theory Basics Classical Linear Modern 1st Principles ODE s Physics n Dimensional Dynamics Infinite Dimensional Nonlinear Analysis Feedback Linearization Backstepping Adaptive Control Modeling and Simulation Commercial Tools That Support Engineering Product Development 16 K A Wise Closed Loop Feedback Control e r u K s G s y SISO Controller Plant Scalar Variables K s G s Transfer Functions Given Derive G s Design K s such that y t follows r t in a prescribed way r e y u Command Reference Signal Error Plant Output Plant input Want all internal signals to remain bounded Control Law The algorithm used to produce the control input u SISO vs MIMO and combinations SIMO MISO etc 17 K A Wise Plant Models Actuators Plant Dynamics Sensors G s Create model the differential equations for the systems components and dynamics Linearize the equations at an operating condition for analysis and control design 18 K A Wise Open Loop vs Closed Loop Open Loop Closed Loop No Feedback Feedback If everything is known perfectly may work Adds robustness Errors asymptotically converge to zero How to Design 1 2 3 4 5 6 Establish performance and robustness specs Describe process using ODE s simulation Design control law classical modern optimal nonlinear Perform trade studies performance versus robustness Implement and test Errors cost Field and test Errors cost 19 K A Wise Mathematical Foundation di t 1 v t Ri t L i d dt C F t m d 2 x t dt 2 b dx t dt kx t n th Order Differential Equations an d n x t dt n a dx t dt a0 x t F t 20 K A Wise Classical Control Theory Transfer Functions Block Diagrams Root Locus PID Control Controller Type Bode Plot Nyquist Plot Lead Lag Compensation Laplace Transform Review Operational method for solving differential equations s complex variable Rectangular j Representations of a complex number Polar A A 2 2 tan 1 a bj Ae j 1 A j e 1 2 c dj B Be j 2 L f t Laplace Transforms L e at x ax 0 0 0 at st e e dt f t e st dt s a t e 1 s a t e dt s a s a 0 1 sX s x 0 aX s X s x 0 x t e at x 0 s a Transfer Functions n m y a2 y a1 y a0 y u b2u b1u b0u The transfer function is the ratio d s Y s n s U s y u u 1 s y Y s U s y ay u n s d s u 1 s a Zeros Points where F s 0 Zeros z1 Poles Points where F s Poles p1 p2 p2 F s s z1 s p1 s p2 2 Always the same number of zeros as poles y Block Diagrams BLOCK DIAGRAMS AND TRANSFER FUNCTIONS Classical control theory typically refers to working with block diagrams Block diagrams are a method for describing the algebra associated with modeling system dynamics They are used to represent complex physical system and are the primary tool for showing process and control system models They decompose the system down into distinct elements and models that support design analysis and simulation In a block diagram each block typically has one input and one output Control System Model e r u K s G s Controller Plant Command K A Wise Feedback Path y Control System Block Diagrams e u K s e u K s u G s Controller Transfer Function Plant e r K s y Plant u G s y Controller Plant Scalar Variables K s G s Transfer Functions K A Wise y G s u Plant Transfer Function Closed Loop Transfer Function y G s u u K s e Plant Controller Substitute u into the expression for y y G s K s e G s K s e K s G s e e r y Error Substitute e into the expression for

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