Difference between revisions of "State Feedback"

From FBSwiki
Jump to: navigation, search
(Chapter Summary: amsmath)
Line 30: Line 30:
 
<ol>
 
<ol>
 
<li> <p> A linear system with dynamics
 
<li> <p> A linear system with dynamics
<center>
+
<center><amsmath>
{|
+
\begin{aligned}
|-
+
  \dot x &= A x + B u &\quad x &\in R^n, u \in R \\
| <math> \dot x = A x + B u </math>
+
  y &= C x + D u &\quad y &\in R
| &nbsp;&nbsp;
+
\end{aligned}
| <math> x \in R^n, u \in R </math>
+
</amsmath></center>
|-
+
| <math> y = \left. C x + D u \right.</math>
+
| &nbsp;&nbsp;
+
| <math> y \in R</math>
+
|}
+
</center>
+
 
is said to be ''reachable'' if we can find an input <math>u(t)</math> defined on the interval <math>[0, T]</math> that can steer the system from a given final point <math>x(0) = x_0</math> to a desired final point <math>x(T) = x_f</math>.
 
is said to be ''reachable'' if we can find an input <math>u(t)</math> defined on the interval <math>[0, T]</math> that can steer the system from a given final point <math>x(0) = x_0</math> to a desired final point <math>x(T) = x_f</math>.
 
</p>
 
</p>
  
 
<li><p>The ''reachability matrix'' for a linear system is given by
 
<li><p>The ''reachability matrix'' for a linear system is given by
<center><math>
+
<center><amsmath>
 
W_r = \left[\begin{matrix} A & AB & \cdots & A^{n-1}B \end{matrix}\right].
 
W_r = \left[\begin{matrix} A & AB & \cdots & A^{n-1}B \end{matrix}\right].
</math></center>
+
</amsmath></center>
 
A linear system is reachable if and only if the reachability matrix <math>W_r</math> is invertible (assuming a single intput/single output system).  Systems that are not reachable have states that are constrained to have a fixed relationship with each other.
 
A linear system is reachable if and only if the reachability matrix <math>W_r</math> is invertible (assuming a single intput/single output system).  Systems that are not reachable have states that are constrained to have a fixed relationship with each other.
 
</p>
 
</p>
  
 
<li><p>A linear system of the form
 
<li><p>A linear system of the form
<center><math>
+
<center><amsmath>
 
\frac{dz}{dt}=
 
\frac{dz}{dt}=
 
   \left[\begin{matrix}
 
   \left[\begin{matrix}
Line 64: Line 58:
 
     1 \\ 0 \\ 0 \\ \vdots \\ 0  
 
     1 \\ 0 \\ 0 \\ \vdots \\ 0  
 
   \end{matrix}\right] u  
 
   \end{matrix}\right] u  
</math></center>
+
</amsmath></center>
 
is said to be in ''reachable canonical form''.  A system in this form is always reachable and has a characteristic polynomial given by
 
is said to be in ''reachable canonical form''.  A system in this form is always reachable and has a characteristic polynomial given by
<center><math>
+
<center><amsmath>
 
   \det(sI-A) = s^n+a_1 s^{n-1} + \cdots + a_{n-1}s + a_n,
 
   \det(sI-A) = s^n+a_1 s^{n-1} + \cdots + a_{n-1}s + a_n,
</math></center>
+
</amsmath></center>
 
A reachable linear system can be transformed into reachable canonical form through the use of a coordinate transformation <math>z = T x</math>.   
 
A reachable linear system can be transformed into reachable canonical form through the use of a coordinate transformation <math>z = T x</math>.   
 
</p>
 
</p>
  
 
<li><p>A state feedback law has the form
 
<li><p>A state feedback law has the form
<center><math>
+
<center><amsmath>
\left. u = -K x + K_r r \right.
+
  u = -K x + k_r r
</math></center>
+
</amsmath></center>
 
where <math>r</math> is the reference value for the output.  The closed loop dynamics for the system are given by
 
where <math>r</math> is the reference value for the output.  The closed loop dynamics for the system are given by
<center><math>
+
<center><amsmath>
\dot x = (A - B K) x + B K_r r.
+
\dot x = (A - B K) x + B k_r r.
</math></center>
+
</amsmath></center>
 
The stability of the system is determined by the stability of the matrix <math>A - BK</math>.  The equilibrium point and steady state output (assuming the systems is stable) are given by
 
The stability of the system is determined by the stability of the matrix <math>A - BK</math>.  The equilibrium point and steady state output (assuming the systems is stable) are given by
<center><math>
+
<center><amsmath>
     x_e = -(A-BK)^{-1} B K_r r \qquad y_e = C x_e.
+
     x_e = -(A-BK)^{-1} B k_r r \qquad y_e = C x_e.
</math></center>
+
</amsmath></center>
Choosing <math>K_r</math> as
+
Choosing <math>k_r</math> as
<center><math>
+
<center><amsmath>
   K_r = {-1}/\left(C (A-BK)^{-1} B\right).
+
   k_r = {-1}/\left(C (A-BK)^{-1} B\right).
</math></center>
+
</amsmath></center>
 
gives <math>y_e = r</math>.</p>
 
gives <math>y_e = r</math>.</p>
  
 
<li><p>If a system is reachable, then there exists a feedback law of the form
 
<li><p>If a system is reachable, then there exists a feedback law of the form
<center><math>
+
<center><amsmath>
\left. u = -K x + K_r r \right.
+
  u = -K x + k_r r
</math></center>
+
</amsmath></center>
 
the gives a closed loop system with an arbitrary characteristic polynomial.  Hence the eigenvalues of a reachable linear system can be placed arbitrarily through the use of an appropriate feedback control law.
 
the gives a closed loop system with an arbitrary characteristic polynomial.  Hence the eigenvalues of a reachable linear system can be placed arbitrarily through the use of an appropriate feedback control law.
 
</p>
 
</p>
  
 
<li><p>''Integral feedback'' can be used to provide zero steady state error instead of careful calibration of the gain <math>K_r</math>.  An integral feedback controller has the form
 
<li><p>''Integral feedback'' can be used to provide zero steady state error instead of careful calibration of the gain <math>K_r</math>.  An integral feedback controller has the form
<center><math>
+
<center><amsmath>
  \left. u = - K (x - x_e) - K_i z + K_r r \right.
+
  u = - k_p (x - x_e) - k_i z + k_r r.
</math></center>
+
</amsmath></center>
 
where
 
where
 
<center><math>
 
<center><math>
 
\dot z = y - r
 
\dot z = y - r
 
</math></center>
 
</math></center>
is the integral error.  The gains <math>K</math>, <math>K_i</math> and <math>K_r</math> can be found by designing a stabilizing state feedback for the system dynamics augmented by the integrator dynamics.
+
is the integral error.  The gains <math>k_p</math>, <math>k_i</math> and <math>k_r</math> can be found by designing a stabilizing state feedback for the system dynamics augmented by the integrator dynamics.
 
</p>
 
</p>
  
 
<li><p>A ''linear quadratic regulator'' minimizes the cost function
 
<li><p>A ''linear quadratic regulator'' minimizes the cost function
<center><math>
+
<center><amsmath>
 
     \tilde J = \int_0^\infty \left(x^T Q_x x + u^T Q_u u \right)\,dt.
 
     \tilde J = \int_0^\infty \left(x^T Q_x x + u^T Q_u u \right)\,dt.
</math></center>
+
</amsmath></center>
 
The solution to the LQR problem is given by a linear control law of
 
The solution to the LQR problem is given by a linear control law of
 
the form
 
the form
<center><math>
+
<center><amsmath>
     \left. u = -Q_u^{-1} B^T P x \right.
+
     u = -Q_u^{-1} B^T P x.
</math></center>
+
</amsmath></center>
 
where <math>P \in R^{n \times n}</math> is a positive definite, symmetric
 
where <math>P \in R^{n \times n}</math> is a positive definite, symmetric
 
matrix that satisfies the equation
 
matrix that satisfies the equation
<center><math>
+
<center><amsmath>
   \left. P A + A^T P - P B Q_u^{-1} B^T P + Q_x = 0. \right.
+
   P A + A^T P - P B Q_u^{-1} B^T P + Q_x = 0.
</math></center>
+
</amsmath></center>
 
This equation is called the''algebraic
 
This equation is called the''algebraic
 
Riccati equation'' and can be solved numerically.
 
Riccati equation'' and can be solved numerically.

Revision as of 14:55, 3 January 2007

Prev: Linear Systems Chapter 6 - State Feedback Next: Output Feedback

This chapter describes how feedback can be used shape the local behavior of a system. The concept of reachability is introduced and used to investigate how to "design" the dynamics of a system through placement of its eigenvalues. In particular, it will be shown that under certain conditions it is possible to assign the system eigenvalues to arbitrary values by appropriate feedback of the system state.

Textbook Contents

State Feedback (pdf, 16Sep06)

  • 1. Introduction
  • 2. Reachability
  • 3. Stabilization by State Feedback
  • 4. State Feedback Design Issues
  • 5. Integral Action
  • 6. Linear Quadratic Regulators
  • 7. Further Reading
  • 8. Exercises

Lecture Materials

Supplemental Information

Chapter Summary

This chapter describes how state feedback can be used to design the (closed loop) dynamics of the system:

  1. A linear system with dynamics

    math

    is said to be reachable if we can find an input defined on the interval that can steer the system from a given final point to a desired final point .

  2. The reachability matrix for a linear system is given by

    math

    A linear system is reachable if and only if the reachability matrix is invertible (assuming a single intput/single output system). Systems that are not reachable have states that are constrained to have a fixed relationship with each other.

  3. A linear system of the form

    math

    is said to be in reachable canonical form. A system in this form is always reachable and has a characteristic polynomial given by

    math

    A reachable linear system can be transformed into reachable canonical form through the use of a coordinate transformation .

  4. A state feedback law has the form

    math

    where is the reference value for the output. The closed loop dynamics for the system are given by

    math

    The stability of the system is determined by the stability of the matrix . The equilibrium point and steady state output (assuming the systems is stable) are given by

    math

    Choosing as

    math
    gives .

  5. If a system is reachable, then there exists a feedback law of the form

    math

    the gives a closed loop system with an arbitrary characteristic polynomial. Hence the eigenvalues of a reachable linear system can be placed arbitrarily through the use of an appropriate feedback control law.

  6. Integral feedback can be used to provide zero steady state error instead of careful calibration of the gain . An integral feedback controller has the form

    math

    where

    is the integral error. The gains , and can be found by designing a stabilizing state feedback for the system dynamics augmented by the integrator dynamics.

  7. A linear quadratic regulator minimizes the cost function

    math

    The solution to the LQR problem is given by a linear control law of the form

    math

    where is a positive definite, symmetric matrix that satisfies the equation

    math

    This equation is called thealgebraic Riccati equation and can be solved numerically.

Exercises

Frequently Asked Questions

Additional Information