Applied Nonlinear Control Theory

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Applied Nonlinear Control Theory
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Topics

• Basic concepts
• Controllability
• Reachability
• Stability
• Convergence
• Different types of systems
• Fully-actuated systems (robot arms)
• Under-actuated systems (acrobot robot arm with
  missing actuators)
• Non holonomic systems (mobile robots)
• Non holonomic systems without drift (slowly
  moving mobile robots)
• Applied control theory
• Linear systems
• Affine nonlinear systems, fully-actuated
• Affine nonlinear systems, under-actuated
• Drift-free nonlinear systems

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Fully-Actuated Robots

• Robots with enough actuators
• As many actuators as dimension of configuration
  space
• Examples of fully-actuated robots
• Industrial robots
• Generally six joints, six actuators
• Most machines
• As many actuators as there are degrees of freedom
• Examples of under-actuated robots

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Acrobot
Is an acrobot controllable?
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Basic Concepts
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Recap State Space

• State (mechanical systems)
• q describes the configuration (position) of the
  system
• x describes the state of the system
• Phase Portrait

(assuming holonomic)
Trajectory
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Controllability

• The system is controllable if, for any two
  points x0, x1, there exists a time T and an
  admissible control defined on 0, T such that
  for x(0) x0, we have x(T) x1.

• Tests for controllability
• Linear systems, well-known
• Nonlinear systems, hard
• Drift-free, affine control systems, doable

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Controllability
Affine system

• The system is controllable if, for any two
  points x0, x1, there exists a time T and an
  admissible control defined on 0, T such that
  for x(0) x0, we have x(T) x1.

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Controllability
The system is said to be small time locally
controllable (STLC) at x0, if given an open
subset V in Rn, and x0, x1 in V, if for all
positive T, there exists an admissible control
such that the system can be steered from x0 to x1
with x(t) staying inside V for all time.
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Reachability and Controllability

• Reachable set for a given time, t
• Reachable set in time T

The interior of the reachable set is not empty
for all neighborhoods V of x0, if and only if the
system is STLC at x0.
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Stability in the sense of Lyapunov

• If a disturbance at time t0 of a trajectory
  produces changes that remain permanently bounded,
  the motion is said to be stable in the sense of
  Lyapunov.

t
x2
x1
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Lyapunov Stability
(or stability in the sense of Lyapunov)
Note We have replaced 2n by n for convenience

• Equilibrium point
• A point where f(x) vanishes
• Translate the origin to the equilibrium point of
  interest
• Without loss of generality, let 0 be an
  equilibrium point
• f(0) 0
• The equilibrium solution or the null solution is
• x(t) 0, t gt t
• Local and global stability
• Recall that stability refers to small
  perturbations and therefore is intrinsically a
  local property
• However, can establish global stability for
  special cases

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Stability of the Equilibrium Point
x2

• Stability in the sense of Lyapunov
• x0 is stable if and only if for any e gt 0, there
  exists a d(e) gt 0 such that
• Asymptotic Stability
• A stable equilibrium point x 0 is asymptotically
  stable if for all t0, there exists d (t0) gt 0
  such that

x1
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Exponential Stability

• Asymptotic stability does not say anything about
  rate of convergence.
• Uniform asymptotic stability
• Exponential stability
• The asymptotically stable equilibrium point x0
  is an exponentially stable equilibrium point if
  there exists m, a gt 0 such that
• for all x0 in some ball around 0.

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Linear Autonomous (Time-Independent) Systems

• Local Stability implies Global Stability
• Global Asymptotic Stability
• Lyapunov Stability, not Global Asymptotic
  Stability
• Unstable

if and only if the real parts of all eigenvalues
are negative
if and only if the real parts of all eigenvalues
are non positive, and zero eigenvalue is not
repeated
if and only if there is one eigenvalue whose real
part is positive
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Linear Autonomous Systems

• Equations
• Solution

Exponential of a matrix
Eigenvalues and Eigenvectors
Matrix of eigenvectors
Matrix of eigenvalues
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Control System Design
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Proportional plus Derivative Control for Simple
Systems

• Kinematic Model
• Want x to follow trajectory xdes(t)
• Dynamic Model
• Want x to follow trajectory xdes(t)

General Approach
1. Define error, e(t)xdes(t)- x(t)
2. Want e(t) to converge exponentially to zero.
or
Note control law has feedforward and feedback
(PD) term.
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PID Servo (Feedback) Control

• Single-input, single-output (SISO), linear system
• (1)
• desired trajectory xd(t)
• actual trajectory x(t)
• error e(t) xd(t)-x(t)
• PID feedback control scheme
• (2)
• From (1) and (2), we get

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PID control schemes

• PID feedback control
• (2)
• PID feedback control with feedforward
• (4)

PID control with feed- forward
PID control
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Model based control

• Disadvantages of PID control schemes
• performance will depend on the model
• need to tune gains to maximize performance
• Model based control schemes

model based
model based
PD feedback feedforward

• Two parts of a model based scheme
• model based part
• cancel the dynamics of the system
• specific to the model
• servo based part
• use PID or PD with feedforward to drive errors to
  zero
• independent of the model of the system

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Model based control

• Model
• Model based control law
• Performance

model based
PID control with feedforward
Model based control (servoPD)
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Model based control

• Advantage
• decomposes the control law into
• model-dependent part (depends on the knowledge of
  the model)
• model-independent part (servo control, gains are
  independent of the model)
• Disadvantage
• Model based control law (based on estimates of
  model parameters)

Ideal performance Actual performance
1. Error term will not go exponentially to
zero 2. Right hand side is a forcing function
driving the error away from zero
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Model based control
Imperfect model, 10 errors in parameters
Perfect model

• Not all is lost however
• Treat the right hand side as a perturbation or a
  disturbance force fp
• If
• we can prove that the error e(t) is also
  bounded

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Robot Dynamics

• Dynamic model
• H is the positive definite, n by n inertia
  matrix
• h is the n-dimensional vector of Coriolis and
  centripetal forces
• g is the n-dimensional vector of gravitational
  forces
• J(q) is the 6n manipulator Jacobian matrix
• F is the 6-dimensional vector of forces and
  moments exerted by the end effector on the
  environment
• t is the n-dimensional vector of actuator
  forces and torques

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Robot Control

• Two general schemes
• Point to Point control
• Trajectory control
• Actuators
• motors/actuators are torque controlled
• Dynamic control
• We will assume torque controlled motors
• typical of brushless dc motors, ac servo motors
  (increasingly used in robots today), and direct
  drive motors
• What should the torques be to drive a manipulator
  along a trajectory?

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SISO nonlinear systems

• Single link with a rotary joint in the vertical
  plane
• Dynamic model
• Control law
• model based
• servo part is PD
• perfect model results in the error being driven
  exponentially to zero

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Robot control

• n degree of freedom robot manipulator (MIMO
  nonlinear)
• Dynamic model
• Control law
• model based
• servo part is PD
• perfect model results in the error being driven
  exponentially to zero

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Industrial Robots

• n degree of freedom robot manipulator
• Dynamic model
• Two types of control laws are generally used
• PID control law
• PID control law with gravity compensation
• It can be shown that PID control laws always
  result in a stable system! However, no
  performance bounds can be established.

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Mobile Robots
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Dynamic Model of a Mobile Robot
inputs
Outputs?
Underactuated!
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Mobile robots are STLC!

• To understand this, need following background
  from geometric control theory
• Lie brackets of vector fields
• Distribution
• Involutive distributions
• Involutive closure of distributions
• Chows theorem

MLS, Chapter 7 or Sastry, Chapter 11
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Kinematic Model of a Mobile Robot
Y
(x1, y1)
f1
C1
l
q
C
y
C2
f2
C2
(x2, y2)
X
x
v, w are inputs (x, y, q) are outputs

Please note change in notation (q, f, l, r, 1, 2)
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Kinematic Model of a Front-Wheel Steered Mobile
Robot
Rear wheel drive car model
Y
f
q
y
d
Differential drive model
X
x
Please note change in notation (q, f, l, r, 1, 2)
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STLC, but how to control them?
y
x
W
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Input/Output Linearization

• Control using kinematic models
• Trajectory following
• Track (xdes(t), ydes(t))
• Path following
• Follow yfdes(x), or fdes(x, y)0
• Maintain speed, vdes

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Assumptions 1. Kinematic model (inertial effects
are negligible) 2. O and C are the same point
(easily relaxable)
u
g(x)