These notes were developed in the instruction of the MIT graduate course 13.49: Maneuvering and Controlling Surface and Underwater Vehicles.
Vectors
Definition
Vector Magnitude
Vector Dot Product
Vector Cross Product
Matrices
- Definition
- Multiplying a Vector by a Matrix
- Multiplying a Matrix by a Matrix
- Common Matrices
- Transpose
- Determinant
- Inverse
- Trace
- Eigenvalues and Eigenvectors
- Modal Decomposition
- Singular Value
The product of a diagonal matrix with another diagonal matrix is diagonal, and in this case the operation is commutative. The determinant of a square matrix A is a scalar equal to the volume of the parallelepiped enclosed by its constituent vectors.
Laplace Transform
- Definition
- Convergence
- Convolution Theorem
- Solution of Differential Equations by Laplace Transform 13
Before the first rotation, the body coordinate matches the inertial frame coordinate: ~xb0 = ~x. This represents the location of the original point in the body's completely transformed reference frame, i.e. ~xb3.
Differential Rotations
On a rotating body whose point of origin is fixed, the time rate of change of a constant radius vector is the cross product of the rotational velocity vector ~ω and the radius vector itself. The total velocity of the particle is equal to the velocity of the origin of the reference frame, plus one component.
Rate of Change of Euler Angles
Dead Reckoning
A common frame for ships, submarines, and other maritime vehicles has the body-referencing x-axis forward, the y-axis pointing to port (left), and the z-axis pointing upward. Because the body is moving relative to an inertial frame, the dynamics expressed in the body-referenced frame require additional attention.
Linear Momentum in a Moving Frame
Example: Mass on a String
Moving Frame Affixed to Mass
Rotating Frame Attached to Pivot Point
Stationary Frame
Angular Momentum
26 3 Inertial dynamics of the ship If we let M~ ={K, M, N} be the total moment acting on the body, i.e. the left side of equation 28, then the complete moment equations are the same.
Example: Spinning Book
For rotation about the x-axis, both coefficients of the differential equation are positive, and thus α > 0.
Parallel Axis Theorem
Basis for Simulation
The forces and moments on a vessel are complicated functions of many factors, including water density, viscosity, surface tension, pressure, vapor pressure, and body motions. This notation covers some cases where the formal Taylor series is meaningless, but the notation is still clear.
Surface Vessel Linear Model
With the state vector ~s = {v, r} and the external force/momentum vector F~ = {Y0, N0}, is a state-space representation of the oscillation/transition system. The last form of the equation is a standard where A represents the internal dynamics of the system and B is a gain matrix for the control and disturbance inputs.
Stability of the Sway/Yaw System
Namely, the added mass term −Yv˙ is of the order of the material mass m of the ship, and similarly Nr˙ ' −Izz. The second part of the second term contains mU, which has a large positive value, making stability generally critical for the (usually negative) Nv.
Basic Rudder Action in the Sway/Yaw Model
Adding Yaw Damping through Feedback
The stability coefficient C, derived from the addition of the control law δ=krr, where kr >0 is the feedback gain, is. 56) Yδ is small positive and Nδ is large and negative. A properly balanced vessel achieves stability with only zero rudder action, so a reasonable amount of control will ensure good maneuverability.
Heading Control in the Sway/Yaw Model
Response of the Vessel to Step Rudder Input
Phase 1: Accelerations Dominate
Phase 3: Steady State
Summary of the Linear Maneuvering Model
Stability in the Vertical Plane
Therefore, only three non-dimensional groups exist, and one is a unique function of the other two. The uncertainty about where hand d should be used, and the questionable importance of h/d as a group, are remnants of the position.
Common Groups in Marine Engineering
The effect of the π theorem lies primarily in reducing the number of parameters which must be considered independently to characterize a process. This applies to the geometry of the body and the kinematics of the flow, the surface roughness and all the relevant groups that control the fluid dynamics.
Similitude in Maneuvering
44 5 SIMILARITY When testing models, it is imperative to maintain as many of the nondimensional groups as possible in the full-scale system. For surface vessels and near-surface submarines, it is routine procedure to use turbulence stimulators to achieve flow that would normally occur with ship-scale Re.
Roll Equation Similitude
Kψ0 again depends strongly on U, since B and hare are fixed; thisKψ0 must be maintained in model tests. In model testing, the Froude number F r = √UgL, which scales the influence of surface waves, must be maintained between model and full-scale surface vessels.
Rotating Arm Device
Reynolds number Re = U Lν, which scales the effect of viscosity, does not need to be adjusted as long as the scale model achieves turbulent flow (supercritical Re). The slope of the curve at zero angle determines Yv and Nv respectively; Higher order terms can be generated if the points deviate from a straight line.
Planar-Motion Mechanism
The test reveals whether the ship has a memory effect, manifested as a hysteresis in the deflection rate r. But if the corrective action causes the ship to turn all the way to the left, the same memory effect can occur.
Zig-Zag Maneuver
For example, suppose that the first 15◦ rudder deflection causes the vessel to turn right, but that the yaw rate at rudder zero, heading negative, is still to the right. The vessel has "stuck" here and will require a negative rudder action to pull out of the turn.
Circle Maneuver
- Drift Angle
- Speed Loss
- Heel Angle
- Heeling in Submarines with Sails
In summary, the vessel initially rolls into the turn, but then out of the turn in the steady state. Because |Yv˙| > m, the vessel rolls under sail, the keel out of turn.
Munk Moment
A symmetrically streamlined body at zero angle of attack experiences only a drag force which has the shape. Note that the A subscript will be used to indicate the zero angle of attack ratio; also the sign of FA is negative because it opposes the x-axis of the vehicle.
Separation Moment
Net Effects: Aerodynamic Center
The pointAC has an intuitive explanation: it is the location on the hull where Fn would act to create the total moment. Therefore, if the origin of the vehicle is ahead of AC, the net moment stabilizes.
Role of Fins in Moving the Aerodynamic Center
To solve this explicitly, let n denote the (positive) distance at which the fins are located after Fn; this is probably a small number as both effects usually operate close to the stern. Note that if the fins are located in front of the vortex force Fn, i.e. lf n <0,lf decreases, but since AC is applied to the fins, there is no net increase in stability.
Aggregate Effects of Body and Fins
The slender-body theory is accurate for small ratios d/L, except near the ends of the body. The added mass is equal to the mass of the water displaced by the cylinder.
Kinematics Following the Fluid
The idea of slender body theory, under these assumptions, is to think of the body as a longitudinal stack of thin sections, each having an easily calculated added mass. Many formulas for added mass of two-dimensional sections as well as for simple three-dimensional bodies can be found in the books of Newman and Blevins.
Derivative Following the Fluid
Differential Force on the Body
Total Force on a Vessel
Total Moment on a Vessel
The Munk moment (exact result) can therefore be used to correct the second term in the slender body approximation above Mw. As with buoyancy, Mw and Mα are closely related, depending only on the orientation of the body frame with respect to the flow.
Relation to Wing Lift
Convention: Hydrodynamic Mass Matrix A
The blunt rear causes lift as a product of added mass effects, and can be accurately modeled with slender body theory. As the angle of attack increases, the approximations in the fin and slender body analysis will break down.
Jorgensen’s Formulas
In the limit of an angle of attack of 90◦ , vortex shedding occurs as if from a bluff body, and there is little axial convection. Furthermore, we assumed that L >> D, which is also not a constraint in the complete equations.
Hoerner’s Data: Notation
Finally, we note that the second term in CM vanishes when xm =xc, that is, when the moment is related to the center of the planform area. As written, the moment coefficient is negative if the moment destabilizes the body, while CN is always positive.
Slender-Body Theory vs. Experiment
When comparing the two body shapes we see that the moment at the nose is much more stable (positive) for the body with a blunt trailing edge. The dependence of the lift force on the stubby tail is not difficult to see using slender body theory.
Slender-Body Approximation for Fin Lift
The theory of lifting surfaces focuses on the Kutta condition, which requires that streamlines of fluid particles do not wrap around the trailing edge of the surface, but instead rejoin streamlines from the other side of the wing at the trailing edge. Since the separation point at the front of the section rotates with the angle of incidence, it is clear that the fluid must travel faster over one side of the surface than the other.
Three-Dimensional Effects: Finite Length
The lifting surface is nominally an extrusion of a simplified cross-section: the cross-section has a rounded leading edge, a sharp trailing edge and a smooth surface. Circulation is the integral of the velocity around the cross section and the lifting surface requires circulation to satisfy the Kutta condition.
Ring Fins
In many ships and marine vehicles, an engine (diesel or gas turbine, say) or an electric motor drives the propeller through a connection of shafts, reducers and bearings, and the effects of each part are important in the response of the netting system. . An approximation of the transient behavior of a system can be made using the quasi-static assumption.
Steady Propulsion of Vessels
Basic Characteristics
We call J =Op/npD the entrainment ratio of the propene when exposed to a water velocity up; note that in the wake of the vessel, Up may not be the same as the vessel's U. A typical wake fraction of e.g. 0.1 indicates that the input speed seen by the propeller is only 90% of the vessel's speed.
Solution for Steady Conditions
Engine/Motor Models
Unsteady Propulsion Models
- One-State Model: Yoerger et al
- Two-State Model: Healey et al
- Force Balance
- Critical Angle
We use curved axial coordinates, which we take as zero at the end of the cable; up along the cable is the positive direction. In the case of a very heavy cable, δ is large, and the linear approximation of the square root.
Linearized Dynamics
Derivation
We now make trigonometric substitutions in the weighting expressions, leaving φ 'φ¯ to calculate the resistance and substituting the constitutive (Hooke's) law. The high-frequency movements of the vessel in the horizontal plane are completely missed by the vehicle, while the low-frequency movements appear slowly and only after a large delay.
Damped Axial Motion
The maximum voltage is ¯T+|T˜|and must be less than the working load of the cable. Normally this is problematic at the top of the cable, where the static voltage is highest.
Cable Strumming
Vehicle Design
The drag point must be longitudinally forward of the center of gravity, so that the vehicle enters the water fins first, and self-stabilizes with U >0. Partial fraction expansions change the form of Y(s) so that the simple transform pairs can be used to find the time domain output signals.
Partial Fractions: Unique Poles
Solving linear time-invariant systems with the Laplace Transform method will generally create a signal containing the (factorized) form. 166) Although we are now talking about the signal Y(s), we will see later that dynamical systems are described in the same format: in that case we call the impulse response G(s) a transfer function. We must have m < n; if not, we need to divide the numerator by the denominator if necessary to find a simple form.
Example: Partial Fractions with Unique Real Poles
Partial Fractions: Complex-Conjugate Poles
Example: Partial Fractions with Complex Poles
Stability in Linear Systems
Stability ⇐⇒ Poles in LHP
General Stability
Plants, Inputs, and Outputs
It is the basic concept of controller design that a set of input variables act through a given "plant" to create an output.
The Need for Modeling
Nonlinear Control
Representing Linear Systems
Standard State-Space Form
Converting a State-Space Model into a Transfer Function 102
If u is the output of the controller, and e is the error signal it receives, this control law has the form. In other words, the proportional part of this control law will create a control action that scales linearly with the error – we often think of this as a spring-like action.
Example: PID Control
Proportional Only
Proportional-Derivative Only
Note that if the mass had very large natural damping, a negative kd could be used to cancel out some of its effects and speed up the system response. However, very large values of kp will also increase the resonant frequency ωd, which is unacceptable.
Proportional-Integral-Derivative
Heuristic Tuning
Block Diagrams of Systems
Fundamental Feedback Loop
Block Diagrams: General Case
Primary Transfer Functions
Matrix Exponential
Definition
Modal Canonical Form
This is called the modal canonical form, since the states are simply modal amplitudes. These states are separated in Λ but can be connected via input (V−1B) and output (CV) mappings.
Modal Decomposition of Response
Forced Response and Controllability
Plant Output and Observability
In particular, the absolute values of the transfer functions are replaced by the largest singular values of the transfer matrices. A design based on singular values is the idea of L2-control or LQG/LTR, which will be introduced in the next lectures.
Roots of Stability – Nyquist Criterion
Mapping Theorem
Since the zeros of F(s) are in fact the poles of the closed-loop transfer function, e.g. S(s), stability requires that there are no zeros of F(s) in the right half s-plane. In other words, stability requires that the number of unstable poles inF(s) equals the number of CCW circlings of the origin, as s sweeps around the entire right-half s-plane.
Nyquist Criterion
The requirement that the number of poles in P(s)C(s) exceed the number of zeros means that at high frequencies P(s)C(s) always decays so that the loci go to the origin. Referring to the multivariable definition of S(s), we should count the rounds for the function [det(I+P(s)C(s))−1] instead of P(s)C(s).
Robustness on the Nyquist Plot
It is a necessary and sufficient condition that the number of unstable poles in the loop transfer function P(s)C(s) corresponds to an equal number of CCW revolutions of the critical point (−1 + 0j). Because the path taken in the plane includes negative frequencies (i.e. the negative imaginary axis), the loci of P(s)C(s) appear as complex conjugates – the plot is symmetrical about the real axis.
Design for Nominal Performance
Design for Robustness
One of the most useful descriptions of model uncertainty is the multiplicative uncertainty:. 179) Here P(s) represents the nominal plant model used in the design of the control loop, and ˜P(s) is the actual, disturbed plant. The disturbance is of the multiplicative type, ∆(s)W2(s)P(s), where ∆(s) is an unknown but stable function of frequency for which |∆(s)| ≤ 1.
Robust Performance
Implications of Bode’s Integral
The Recipe for Loopshaping
When a control system involves multiple inputs and outputs, the ideas of scalar looping can be adapted using the singular value. Useful properties and relationships for the singular value are found in the MATHEMATICS FACTS section.
Full-State Feedback
Optimal controllers, i.e. controllers that are the best possible, according to some figure of merit, seem to generate only stabilizing controllers for MIMO plants. In this sense, optimal control solutions provide an automatic design procedure – we only need to decide which figure of merit to use.
Dynamic Programming
Suppose there are only three places to cross the Rocky Mountains, B1, B2, B3, and three places to cross the Mississippi River, C1, C2, C3.3 In notation, we say that the path from A to B1 is AB1. For example, from B1 to D there are three possible routes, and from level B to D there are a total of nine routes.
Dynamic Programming and Full-State Feedback
Now the control input u in the interval (t, t+δt) cannot affect V(x(t), u(t)), so inserting the above and canceling yes. The Q and R matrices must be set by the user and are the main "tuning knobs" for the LQR.
Properties and Use of the LQR
The expensive control solution places stable closed-loop poles at the mirror images of the unstable plant poles. The angular separation of n closed-loop coils on the arc is constant, and equal to 180◦/n.
Introduction
Problem Statement
Step 1: An Equation for ˙ Σ
To pass from the first right-hand side to the second, we note that the initial condition (0) is uncorrelated with W1T−W2THT. The fact that W1 and HW2 are uncorrelated leads to the third row and the final result follows from.
132 19 KALMAN FILTER Duality of Linear Quadratic Regulator and Kalman Filter Linear Quadratic Regulator Kalman Filter. Also note that the result is independent of the weight matrix W, which might as well be the identity.
Properties of the Solution
Therefore, the gain matrix H of the estimator can be written as a function of Σ. Plugging this back into equation 206, we obtain. 211) Equations 210 and 211 represent the practical solution to the Kalman filter problem, which minimizes the squared norm of the estimation error.
Combination of LQR and KF
It can be connected to take the tracking error e(s) = r(s)−y(s) as input, so that it is not limited to the regularization problem. In this case, ˆx no longer represents an estimated state, but rather an estimated state tracking error.
Proofs of the Intermediate Results
- Proof that E(e T W e) = trace(ΣW )
- Proof that ∂H ∂ trace( − ΛHCΣ) = − Λ T ΣC T
- Proof that ∂H ∂ trace( − ΛΣC T H T ) = − ΛΣC T
- Proof of the Separation Principle
Separation principle: The eigenvalues of the nominal closed-loop system consist of the eigenvalues of (A−HC) and the eigenvalues of (A−BK), separately. Second, we choose suitable parameters of the LQR design, so that the LQG compensator satisfies the approximation of Equation 215.
A Special Property of the LQR Solution
First, a KF design for H is performed so that the Kalman filter loop itself has good performance and robustness characteristics.
The Loop Transfer Recovery Result
Usage of the Loop Transfer Recovery
Three Lemmas
The process of generating useful models from observed data is the goal of system identification. It should be noted that the field of system identification is very rich and the methods are only a small part of what is available.
Visual Output from a Simple Input
The user then has the choice of either neglecting the nonlinearity, for example if the operating point is very precisely controlled, or developing a controller that specifically takes it into account. In any case, simulations with a nonlinear device should always be performed to assess the robustness of the control strategy.
Transfer Function Estimation – Sinusoidal Input
Note that the experimental magnitude and phase in this example deteriorate at the higher frequencies. This is a characteristic of almost all physical systems and an indicator that the plant model cannot be trusted above a certain frequency range.
Transfer Function Estimation – Broadband Input
Fourier Transform of Sampled Data
The Nyquist rate depends only on the sampling time step dt, but the frequency vector accompanying the DFT can be made arbitrarily long by increasing the number of data points. The magnitude of the DFT at a certain ω(m) is the actual average of the continuous frequencies from the vicinity of the point.
Estimating the Transfer Function
Smoothing windows are generally chosen to achieve a compromise between two opposing properties related to averaging: the width of the primary lobe (wide primary lobes yield frequencies belonging to adjacent bins ω(m− 1) and ω(m+ 1)), and the size of the sidelobes (high sidelobes bring frequencies far away from ω(m)). If possible, the DFT should be performed on a number of samples N that is a multiple of a large power of two.
Time-Domain Simulation
Better and better performance can be achieved by increasing the number of responses, and thus baselines and ranges. Here there are three equations and three unknowns, but the non-linearity of the equations leads to the non-uniqueness of the solution.
Global Positioning System (GPS)
Where is the aerodynamic center relative to the center of the fuselage and is that fuselage stable in pitch. Note that the input to the compensator is the error signal: e = r−u, where u is the vessel speed and r is the reference speed.
