This course serves to teach traditional concepts and recent advances in estimation, and to relate these concepts to modern dynamic systems found in aerospace disciplines. This course stresses modeling of physical problems into mathematical terms. Examples will be given from both spacecraft and aircraft systems.
Control of dynamic systems by digital computer. Characterization of discrete-time systems, discrete state space, Z transforms, time domain analysis of discrete-time control systems. Effects of sampling time. Discrete root locus. Frequency domain methods for compensator design. Laboratory experiences in the computer control of electromechanical systems with C/C++ programming, LabView and programmable logic controls (PLCs).
Text: Katsuhiko Ogata, “Discrete-Time Control Systems”, Prentice Hall
Continuous vs. Digital Systems
Laplace Transforms and Transfer Functions
Block Diagrams, Root Locus and Frequency Response Design
Feedback Control
Digitization and Effect of Sampling
Discrete System Analysis
Linear Difference Equation
The Discrete Transfer Function
The z-Transform – key basic principles
z-Transform Input and Response modeling
Final Value and Other Response Tests
Essential Properties of z-Transormation
Sampled Data Systems
Design of Discrete-Time Control Systems by Transforms Techniques
System Specifications
Root Locus in z-plane
Steady State Error Analysis
Frequency Response Methods (Bode plots & Nyquist plots)
This course serves to teach traditional concepts and recent advances in estimation, and to relate these concepts to modern dynamic systems found in aerospace disciplines. This course stresses modeling of physical problems into mathematical terms. Examples will be given from both spacecraft and aircraft systems.
This course develops the mathematical techniques for the analysis of systems in the time domain. An introduction to state space concepts and coordinate transformations is given, followed by solutions to state space equations and advanced topics such as controllability and observability. This course is very mathematical, but real-world examples will be given to bridge the gap between theory and practice.
TEXT: “Linear System Theory and Design,” Third Edition, by C.-T. Cheng, Oxford University Press, New York, NY, 1999.
State Space Modeling
Matrix representation
Definitions and Connections
Similarity Transformations
Jordan-Block Form
Multi-Input-Multi-Output Systems
Transmission Zeros
Linear Dynamical Equations
Solution of a Dynamical Equation
Impulse Response Matrices
Forced and Unforced Systems
Transition Matrix Properties
Stability of Linear Systems
Bounded-Input-Bounded-Output Stability
Asymptotic Stability
Lyapunov Stability Criterion
Controllability and Observability
Control and Observer Canonical Forms
Feedback Structures
Linear Systems Analysis
Controllability and Observability Grammians
Linearization of Systems
Superposition Concepts
Advanced Topics (time permitting)
Discrete-Time Systems
Control Issues
System Realizations
Years Taught: Fall ’12, Fall ’11, Fall ’10, Fall ’09, Fall ’08, Fall ’07, Fall ’06, Fall ’05, Fall ’04, Fall ’03, Fall ’02, Fall ’01
This course gives the fundamental aspects of guidance, navigation and control of dynamical systems. This course stresses modeling of physical problems into mathematical terms.
TEXT: “Inertial Navigation Systems with Geodetic Applications,” by C. Jekeli, Walter de Gruyter, Berlin, 2001.
Particle Kinematics
Vector Description of Particle Motion
Angular Velocity of Rotation of Coordinate Systems
Differentiating Vectors Seen From Other Frames
Direction Cosine Matrix (Euler Angles)
Orbital Dynamics
Newton’s Law
Kepler’s Equation
Orbit Propagation and External Disturbances
Geodetics and Reference Frames
Geodesy of the Earth
Transformations Between Reference Frames
Global Positioning System (GPS)
Constellation Overview
Pseudorange Measurements
Clock Bias and Other Error Sources
Position Determination Using GPS
Linear and Nonlinear Least Squares
Geometric Dilution of Precision (GDOP)
Inertial Navigation Sensors (INS)
Fundamental Equations
Combined GPS/INS
Guidance and Control Concepts
Hohmann Transfer
Launch to Rendezvous
Years Taught: Spring ’06, Spring ’05, Spring ’04, Spring ’03, Spring ’02
Introduces the concepts of spacecraft orbital mechanics and attitude dynamics. Orbital mechanics is the study of the positional motion, while attitude dynamics describes the orientation of the spacecraft. Topics include: review of rotational kinematics and dynamics, orbital mechanics, gravity turn and trajectory optimization, orbit lifetimes, three-body problem, orbit perturbations, orbit determination, spacecraft dynamics, spinning and three-axis stabilized spacecraft, and attitude determination.
Text: Schaub, H., and Junkins, J.L, “Analytical Mechanics of Space Systems” 2nd Edition, AIAA Educational Series, Reston, VA 2009
Orbital Dynamics
Introduction and Vectors
Introduction to Orbital Dynamics
Ellipses, Kepler and Newton
Angular Momentum & Newton’s Law
The Two-Body Problem
Kepler’s Equation
Orbital Coordinate Systems
Lagrange/Gibbs F and G Solution
Spacecraft Formation Flying
Orbita Determination
Earth-Satellite Operations
Attitude Dynamics
Transport Theorem
Newtonian Mechanics
Angular Momentum & Kinetic Energy
Attitude Matrix
Attitude Representations
Attitude Kinematics
Rotational Dynamics
Inertia and Kinetic Energy
Euler’s Rotational Equations
Energy/Momentum Integrals and Attitude Control
Attitude Determination and Feedback Attitude Control
Spacecraft Missions
Years Taught: Spring ’07, Spring ’08, Spring’09, Spring ’10, Spring ’11, Spring ’12, Spring ’13, Spring ’14, Spring ’15
The goal for the solution of Kepler’s equation is to determine the eccentric anomaly accurately, given the mean anomaly and eccentricity.
M = E – e sin(E)
This study presents a new approach to solve this very well-documented problem. We focus on demonstrating the procedure for both the elliptical and hyperbolic cases. It is found that by means of a series approximation, an angle identity, the application of Sturm’s theorem and an iterative correction method, the need to evaluate transcendental functions or store tables is eliminated. A 15th-order polynomial is developed through a series approximation of Kepler’s equation. Sturm’s theorem is used to prove that only one real roots exists for this polynomial for the given range of mean anomaly and eccentricity. An initial approximation for this root is found using a 3rd-order polynomial. Then a one time generalized Newton-Raphson correction is applied to obtain accuracies to the level of around 10-15 rad for the elliptical case and 10-13 rad for the hyperbolic case, which is near machine precision.
Research is being performed to develop a new approach for robust adaptive control. This approach, called Model-Error Control Synthesis (MECS), uses an optimal real-time nonlinear estimator to determine model-error corrections to the control input. The estimator determines the model error using a one time-step ahead approach. Control compensation is achieved by using the estimated model-error as a signal synthesis adaptive correction to the nominal control input so that maximum performance is achieved in the face of extreme model uncertainty and disturbance inputs. The MECS approach has many significant advances, including:
The determined model is a natural by-product of the state estimator
Model parameters need not be updated in order to achieve robust performance
It can easily handle both time-varying parameter changes in the model and unmodeled disturbance inputs
This work is sponsored by National Science Foundation. Results from research can be obtained from:
[1] Kim, J.-R., and Crassidis, J.L., “Spacecraft Attitude Control Using Approximate Receding-Horizon Model-Error Control Synthesis,” AIAA Journal of Guidance, Control, and Dynamics, Vol. 29, No. 5, Sept.-Oct. 2006, pp. 1023-1031.
[2] Alsuwaidan, B., and Crassidis, J.L., “Robust Aircraft Longitudinal Control Using Model-Error Control Synthesis,” AIAA Guidance, Navigation, and Control Conference, Keystone, CO, Aug. 2006, AIAA Paper #2006-6050.
[3] Kim, J.-R., and Crassidis, J.L., “Spacecraft Attitude Control Using Approximate Receding-Horizon Model-Error Control Synthesis,” AIAA Guidance, Navigation, and Control Conference, San Francisco, CA, Aug. 2005, AIAA Paper #2005-6178.
[4] Kim, J.-R., and Crassidis, J.L., “Fundamental Relation of Approximate Receding-Horizon Optimization to the Simple Mass-Spring-Damper System,” AIAA Guidance, Navigation, and Control Conference, Austin, TX, Aug. 2003, AIAA Paper #2003-5563.
[5] Kim, J.-R., and Crassidis, J.L., “Limit-Cycle Oscillation Control of Aeroelastic System Using Model-Error Control Synthesis,” AIAA Guidance, Navigation, and Control Conference, Austin, TX, Aug. 2003, AIAA Paper #2003-5506.
[6] Kim, J.-R., and Crassidis, J.L., “Robust Spacecraft Attitude Control Using Model-Error Control Synthesis,” AIAA Guidance, Navigation, and Control Conference, Monterey, CA, Aug. 2002, AIAA Paper #02-4576.
[7] Kim, J.-R., and Crassidis, J.L., “Model-Error Control Synthesis Using Approximate Receding-Horizon Control Laws,” AIAA Guidance, Navigation, and Control Conference, Montreal, CA, Aug. 2001, AIAA Paper #01-4220.
[8] Kim, J.-R., and Crassidis, J.L., “Linear Stability Analysis of Model Error Control Synthesis,” AIAA Guidance, Navigation, and Control Conference, Denver, CO, Aug. 2000, AIAA Paper #00-3963.
[9] Crassidis, J.L., “Robust Control of Nonlinear Systems Using Model Error Control Synthesis,” AIAA Journal of Guidance, Control, and Dynamics, Vol. 22, No. 4, July-Aug. 1999, pp. 595-601.
Systems comprised of a number of computing nodes all supporting a common mission. All nodes can be given the same task or asked to support the mission through a subset of tasks. Decentalized systems offer several advantages over their centralized counterparts including:
Each node does not require intrinsic knowledge of the system topology
Topology is variable. Nodes can be added or subtracted without significantly impacting the overall system
Less vulnerable to single-point failures
Computational load can be spread amongst nodes for efficiency
Decentralized systems also suffer from some disadvantages:
All nodes may not have access to all information contained within the system
Multiple nodes have access to the same information
Examples of decentralized systems include robotic swarms, cooperative UAVs, formation flying spacecraft and formation of synthetic apertures using multiple smaller sensors.
Decentralized Estimation
Decentralized Attitude Estimation
Decentralized estimation representes an important aspect of decentralized systems. Because various nodes within the system may not have access to all measurements within the system, the individual estimates will be suboptimal. In a perfect world there would exist a means to combine the estimates from all of the nodes into a single estimate which makes use of all system measurements and is optimal. Difficulty arises because some measurements may be used by more than one node leading the estimates from those nodes to be correlated. An example of such a system is the spacecraft attitude estimator shown. Because each extended Kalman filter (EKF) makes use of a common gyro measurement the two estimates will be correlated. The problem to be solved is how can we efficiently and consistently fuse the individual estimates from each EKF to obtain the best possible estimate?
Covariance Intersection
Covariance Intersection
An elegant solution for the fusion of estimates with unknown correlations is the Covariance Intersection (CI) algorithm [Julier and Uhlmann, 1997]. As the name suggests, the CI algorithm is routed in the fiundamentals of Gaussian intersection. Appreciation for the CI algorithm is gained by considering the fusion of two estimates in 2D. In 2D the covariance of a Gaussian random variable can be interpreted as an ellipse. If multiple estimates of the same 2D state vector are available, their covariance ellipses will intersect as shown. The CI algorithm expresses the covariance of the fused estimate as the smallest covariance ellipse which contains all of the intersection of the two individual covariances. Depending on the confidence in each estimate or a desired final statistic, each individual covariance can be weighted by a scalar value which leads the CI algorithm to include a family of solutions as shown.
Another elegant property of the CI algorithm is that, while being derived from a geometric viewpoint, it can also be derived from a matrix- and scalar-weighted optimization problem. This property allows the CI algorithm to be extended to the fusion of higher dimensional state vectors and an arbitrary number of estimates.
Covariance Intersection for Attitude Estimation
While the CI algorithm provides a meaningful and elegant solution for the problem of fusion under unknown correlations, the fusion of attitude states is not trivial. The main difficulty with fusing attitude estimates is the choice of attitude parametrization. The traditional attitude parametrization of choice for space applications, the quaternion, must satisfy a unit norm constraint. This leads the CI fused estimate to be the solution of a constrained optimization problem. For state vectors including only a single quaternion there are several solutions to the resulting optimization problem. For fusion of state vectors containing more than a single quaternion iterative solution methods are required which also seek to mitigate the increasing numeric instability of the problem statement.
References
Julier, S. J., and Uhlmann, J., K., “A Non-divergent Estimation Algorithm in the Presence of Unknown Correlations,” American Control Conference, Albuquerque, NM, IEEE Publications, Piscataway, NJ, June 1997, pp. 2369 – 2373.
Julier, S. J., and Uhlmann, J., K., General Decentralized Data Fusion with Covariance Intersection (CI) in Handbook of Multisensor Data Fusion. CRC Press. Boca Raton, FL., 2001, Chap. 12.
Shuster, M. D., “A Survey of Attitude Representations,” Journal of the Astronautical Sciences, Vol. 41, No. 4, Oct.-Dec. 1993, pp. 439-517.
Cheng, Y., Banas, W.D., and Crassidis, J.L., “Quaternion Data Fusion,” Itzhack Y. Bar-Itzhack Memorial Symposium on Estimation, Navigation, and Spacecraft Control, Haifa, Israel, Oct. 2012, Paper MoA1 .3.
