Difference between revisions of "State Feedback"
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<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 \\  
−  +  y &= C x + D u &\quad y &\in R  
−  +  \end{aligned}  
−  +  </amsmath></center>  
−  +  
−  +  
−  +  
−  +  
−  +  
−  </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  
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W_r = \left[\begin{matrix} A & AB & \cdots & A^{n1}B \end{matrix}\right].  W_r = \left[\begin{matrix} A & AB & \cdots & A^{n1}B \end{matrix}\right].  
−  </  +  </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  
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\frac{dz}{dt}=  \frac{dz}{dt}=  
\left[\begin{matrix}  \left[\begin{matrix}  
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1 \\ 0 \\ 0 \\ \vdots \\ 0  1 \\ 0 \\ 0 \\ \vdots \\ 0  
\end{matrix}\right] u  \end{matrix}\right] u  
−  </  +  </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><  +  <center><amsmath> 
\det(sIA) = s^n+a_1 s^{n1} + \cdots + a_{n1}s + a_n,  \det(sIA) = s^n+a_1 s^{n1} + \cdots + a_{n1}s + a_n,  
−  </  +  </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><  +  <center><amsmath> 
−  +  u = K x + k_r r  
−  </  +  </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><  +  <center><amsmath> 
−  \dot x = (A  B K) x + B  +  \dot x = (A  B K) x + B k_r r. 
−  </  +  </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><  +  <center><amsmath> 
−  x_e = (ABK)^{1} B  +  x_e = (ABK)^{1} B k_r r \qquad y_e = C x_e. 
−  </  +  </amsmath></center> 
−  Choosing <math>  +  Choosing <math>k_r</math> as 
−  <center><  +  <center><amsmath> 
−  +  k_r = {1}/\left(C (ABK)^{1} B\right).  
−  </  +  </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><  +  <center><amsmath> 
−  +  u = K x + k_r r  
−  </  +  </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><  +  <center><amsmath> 
−  +  u =  k_p (x  x_e)  k_i z + k_r r.  
−  </  +  </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>  +  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  
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\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.  
−  </  +  </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  
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−  +  u = Q_u^{1} B^T P x.  
−  </  +  </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><  +  <center><amsmath> 
−  +  P A + A^T P  P B Q_u^{1} B^T P + Q_x = 0.  
−  </  +  </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

Lecture MaterialsSupplemental Information

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

A linear system with dynamics
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 .
The reachability matrix for a linear system is given by
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.
A linear system of the form
is said to be in reachable canonical form. A system in this form is always reachable and has a characteristic polynomial given by
A reachable linear system can be transformed into reachable canonical form through the use of a coordinate transformation .
A state feedback law has the form
where is the reference value for the output. The closed loop dynamics for the system are given by
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
Choosing as
gives . If a system is reachable, then there exists a feedback law of the form
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.
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
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.
A linear quadratic regulator minimizes the cost function
The solution to the LQR problem is given by a linear control law of the form
where is a positive definite, symmetric matrix that satisfies the equation
This equation is called thealgebraic Riccati equation and can be solved numerically.
Exercises
Frequently Asked Questions