TRANSTATION

Chapter 3&4: 線形システムの時間応答&状態フィードバックによる制御

Asymptotic stability and BIBO (Bounded-Input Bounded-Output) stability

The asymptotic stability of system is equivalent to BIBO stability, if there are no common factors, zeros and poles cannot be canceled.

-> Example:

$P : \{\begin{array}{l} \dot{x} (t) = A x (t) + b u (t) \\ y (t) = c x (t) \end{array}, A = [\begin{matrix} - 2 & 0 \\ 0 & 1 \end{matrix}], b = [\begin{matrix} 1 \\ 0 \end{matrix}], c = [\begin{matrix} 1 & 0 \end{matrix}] (1)$

Eigenvalues of $A$ are -2, 1. Therefore system equation (1) is not asymptotic stable . However, transfer function of equation (1) is:

$P (s) = c {(s I - A)}^{- 1} b = \frac{s - 1}{(s + 2) (s - 1)} = \frac{1}{s + 2} (2)$

System equation (1) is BIBO stable . No matter how state $x_{2} (t)$ vibrates unstably, there is no effect on $y (t) = x_{1} (t)$ . That is, $y (t) = x_{1} (t)$ will not disperse at all.

*If $c$ is set to be ${[\begin{matrix} 0 & 1 \end{matrix}]}^{- 1}$ , transfer function will turn out to be zero. That means the system is uncontrollable.

Transition matrix, Heaviside's method and inverse matrix

For state function as $\dot{x} (t) = A x (t)$ , the root of which may be obtained by Laplace transform ^[ref0.1] as ^pp.40 :

$\begin{array}{rcl} s X (s) - x (0) & = & A X (s) \\ X (s) & = & \frac{x (0)}{s I - A} \\ x (t) & = & e^{A t} x (0) \end{array} (2.1)$

The definition of transition matrix $e^{A t}$ based on Taylor's series is:

$e^{A t} : = I + t A + \frac{t^{2}}{2!} A^{2} + \dots + \frac{t^{k}}{k!} A^{k} + \dots (3)$

The most common measure (some other method called diagonalization , see pp.44-45, another using solution of ODE, see [ref0.2] ) to work out transition matrix is Heaviside's method , which is

$e^{A t} = L^{- 1} [{(s I - A)}^{- 1}] (4)$

By the way, the inverse of matrix under third order may be derived as ^[ref1] :

${[\begin{matrix} a & b \\ c & d \end{matrix}]}^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}] (5)$

For matrix above second order, see pp.19-20 Theorem 1.2 of The Schur Complement and Its Applications.pdf using Schur complement . Or using adjoint matrix ^[ref2] .

-> Example (pp. 53):

Second order system with eigenvalue of state matrix $A$ as conjugate complex number $λ = α \pm j β (α < 0, β > 0)$ can be written as

$\begin{matrix} P : \{\begin{cases} \dot{x} (t) = A x (t) + b u (t), x (0) = x_{0} \\ y (t) = c x (t) \end{cases} \\ A = [\begin{matrix} 0 & 1 \\ - (α^{2} + β^{2}) & 2 α \end{matrix}], b = [\begin{matrix} 0 \\ 1 \end{matrix}], c = [\begin{matrix} 1 & 0 \end{matrix}], x_{0} = [\begin{matrix} 1 \\ 0 \end{matrix}] \end{matrix} (6)$

Via equation (5) , the transition matrix may be derived as:

$\begin{array}{rcl} {(s I - A)}^{- 1} & = & {[\begin{matrix} s & - 1 \\ α^{2} + β^{2} & s - 2 a \end{matrix}]}^{- 1} = \frac{1}{{(s - α)}^{2} + β^{2}} [\begin{matrix} s - 2 α & 1 \\ - (α^{2} + β^{2}) & s \end{matrix}] \\ = & \frac{s - α}{{(s - α)}^{2} + β^{2}} [•] + \frac{β}{{(s - α)}^{2} + β^{2}} [•] \\ = & \frac{s - α}{{(s - α)}^{2} + β^{2}} [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] + \frac{β}{{(s - α)}^{2} + β^{2}} \frac{1}{β} [\begin{matrix} - α & 1 \\ - (α^{2} + β^{2}) & α \end{matrix}] \end{array} (7)$

Then with inverse Laplace transform via Laplace table, transition matrix may be derived as:

$e^{A t} = e^{α t} ([\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \cos (β t) + \frac{1}{β} [\begin{matrix} - α & 1 \\ - (α^{2} + β^{2}) & α \end{matrix}] \sin (β t)) (8)$

Matlab Exercise

MATLAB/Simulinkによる現代制御入門

Chapter 3&4: 線形システムの時間応答&状態フィードバックによる制御

Asymptotic stability and BIBO (Bounded-Input Bounded-Output) stability

The asymptotic stability of system is equivalent to BIBO stability, if there are no common factors, zeros and poles cannot be canceled.

-> Example:

Eigenvalues of $A$ are -2, 1. Therefore system equation (1) is not asymptotic stable . However, transfer function of equation (1) is:

System equation (1) is BIBO stable . No matter how state $x_{2} (t)$ vibrates unstably, there is no effect on $y (t) = x_{1} (t)$ . That is, $y (t) = x_{1} (t)$ will not disperse at all.

*If $c$ is set to be ${[\begin{matrix} 0 & 1 \end{matrix}]}^{- 1}$ , transfer function will turn out to be zero. That means the system is uncontrollable.

Transition matrix, Heaviside's method and inverse matrix

For state function as $\dot{x} (t) = A x (t)$ , the root of which may be obtained by Laplace transform ^[ref0.1] as ^pp.40 :

The definition of transition matrix $e^{A t}$ based on Taylor's series is:

The most common measure (some other method called diagonalization , see pp.44-45, another using solution of ODE, see [ref0.2] ) to work out transition matrix is Heaviside's method , which is

By the way, the inverse of matrix under third order may be derived as ^[ref1] :

For matrix above second order, see pp.19-20 Theorem 1.2 of The Schur Complement and Its Applications.pdf using Schur complement . Or using adjoint matrix ^[ref2] .

-> Example (pp. 53):

Second order system with eigenvalue of state matrix $A$ as conjugate complex number $λ = α \pm j β (α < 0, β > 0)$ can be written as

Via equation (5) , the transition matrix may be derived as:

Then with inverse Laplace transform via Laplace table, transition matrix may be derived as:

CONTENTS

Chapter 3&4: 線形システムの時間応答&状態フィードバックによる制御

ELSEWHERE

Matlab Exercise

MATLAB/Simulinkによる現代制御入門

Chapter 3&4: 線形システムの時間応答&状態フィードバックによる制御

Asymptotic stability and BIBO (Bounded-Input Bounded-Output) stability

The asymptotic stability of system is equivalent to BIBO stability, if there are no common factors, zeros and poles cannot be canceled.

-> Example:

Eigenvalues of A<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> are -2, 1. Therefore system equation (1) is not asymptotic stable . However, transfer function of equation (1) is:

Transition matrix, Heaviside's method and inverse matrix

The definition of transition matrix eAt<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mrow><mi mathvariant="bold-italic">A</mi><mi>t</mi></mrow></msup></math> based on Taylor's series is:

The most common measure (some other method called diagonalization , see pp.44-45, another using solution of ODE, see [ref0.2] ) to work out transition matrix is Heaviside's method , which is

By the way, the inverse of matrix under third order may be derived as [ref1] :

For matrix above second order, see pp.19-20 Theorem 1.2 of The Schur Complement and Its Applications.pdf using Schur complement . Or using adjoint matrix [ref2] .

-> Example (pp. 53):

Via equation (5) , the transition matrix may be derived as:

Then with inverse Laplace transform via Laplace table, transition matrix may be derived as:

CONTENTS

Chapter 3&4: 線形システムの時間応答&状態フィードバックによる制御

ELSEWHERE

Eigenvalues of $A$ are -2, 1. Therefore system equation (1) is not asymptotic stable . However, transfer function of equation (1) is:

The definition of transition matrix $e^{A t}$ based on Taylor's series is:

By the way, the inverse of matrix under third order may be derived as ^[ref1] :

For matrix above second order, see pp.19-20 Theorem 1.2 of The Schur Complement and Its Applications.pdf using Schur complement . Or using adjoint matrix ^[ref2] .