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2002 | Buch

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Positive 1D and 2D Systems

verfasst von: Tadeusz Kaczorek, DSc

Verlag: Springer London

Buchreihe : Communications and Control Engineering

Enthalten in: Professional Book Archive

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In the last decade a dynamic development in positive systems has been observed. Roughly speaking, positive systems are systems whose inputs, state variables and outputs take only nonnegative values. Examples of positive systems are industrial processes involving chemical reactors, heat exchangers and distillation columns, storage systems, compartmental systems, water and atmospheric pollution models. A variety of models having positive linear system behaviour can be found in engineering, management science, economics, social sciences, biology and medicine, etc. The basic mathematical tools for analysis and synthesis of linear systems are linear spaces and the theory of linear operators. Positive linear systems are defined on cones and not on linear spaces. This is why the theory of positive systems is more complicated and less advanced. The theory of positive systems has some elements in common with theories of linear and non-linear systems. Schematically the relationship between the theories of linear, non-linear and positive systems is shown in the following figure Figure 1.

Inhaltsverzeichnis

Frontmatter

1. Positive matrices and graphs

Abstract

Let R ^nxm be the set of nxm matrices with entries from the real field R.

Tadeusz Kaczorek

2. Continuous-time and discrete-time positive systems

Abstract

Consider the linear continuous-time system described by the equations

$$ \dot x = Ax + Bu,x(0 = x_0 ) $$

(2.1a)

$$ y = Cx + Du $$

(2.1b)

where $ x = x(t) \in R^n $ is the state vector at the instant $ t,u = u(t) \in R^m $ is the input vector, $ y = y(t) \in R^p $ is the output vector, $ A \in R^{nxn} ,B \in R^{nxm} ,C \in R^{pxn} ,D \in R^{pxm} . $

Tadeusz Kaczorek

3. Reachability, controllability and observability of positive systems

Abstract

Consider a discrete-time (internally) positive system described by the equation

$$ x_{i + 1} = Ax_i + Bu_i i \in Z_ + $$

(3.1)

where $ x_i \in R^n $ is the state vector, $ u_i \in R^m $ is the input vector and $ A \in R_ + ^{nxm} , $ $ B \in R_ + ^{nxm} . $

Tadeusz Kaczorek

4. Realisation problem of positive 1D systems

Abstract

Consider a discrete+time (internally) positive system described by the equations

$$ x_{i + 1} = Ax_i + Bu_i $$

(4.1a)

$$ y_i = Cx_i + Du_i $$

(4.1b)

where ${x_i} \in R_ + ^n,{u_i} \in R_ + ^m,{y_i} \in R_ + ^p$ are the state, input and output vectors, respectively and $ A \in R_ + ^{nxn} ,B \in R_ + ^{nxm} ,C \in R_ + ^{pxn} ,D \in R_ + ^{pxm} . $

Tadeusz Kaczorek

5. 2D models of positive linear systems

Abstract

Consider the 2D Roesser model described by the equations [1,2]

$$ \left[ {\begin{array}{*{20}c} {x_{i + 1,j}^h } \\ {x_{i,j + 1}^v } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {A_{11} } & {A_{12} } \\ {A_{21} } & {A_{22} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {x_{ij}^h } \\ {x_{ij}^v } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\bar B_1 } \\ {\bar B_2 } \\ \end{array} } \right]u_{ij} $$

(5.1a)

$$ y_{ij = \left[ {\begin{array}{*{20}c} {C_1 } & {C_2 } \\ \end{array} } \right]} \left[ {\begin{array}{*{20}c} {x_{ij}^h } \\ {x_{ij}^v } \\ \end{array} } \right] + Du_{ij} i,j \in Z_ + $$

(5.1b)

where $x_{ij}^h \in {R^{{n_1}}}$ , and $ x_{ij}^v \in R^{n2} $ are the horizontal and vertical state vectors at the point $ (i,j) \in Z_ + xZ_ + ,u_{ij} \in R^m $ and $ y_{ij} \in R^p $ are the input and output vectors, respectively and ${A_{11}} \in {R^{{n_1} \times {n_1}}},{A_{12}} \in {R^{{n_1} \times {n_2}}},{A_{21}} \in {R^{{n_2} \times {n_1}}},{A_{22}} \in {R^{{n_2} \times {n_2}}},{\bar B_1} \in {R^{{n_1} \times m}},{\bar B_2} \in {R^{{n_2} \times m}},{C_1} \in {R^{p \times {n_1}}},{C_2} \in {R^{p \times {n_2}}}D \in {R^{p \times m}}.$

Tadeusz Kaczorek

6. Controllability and minimum energy control of positive 2D systems

Abstract

Consider the positive 2D Roesser model [1]

$$ \left[ {\begin{array}{*{20}c} {x_{i + 1,j}^h } \\ {x_{i,j + 1}^v } \\ \end{array} } \right] = A\left[ {\begin{array}{*{20}c} {x_{ij}^h } \\ {x_{ij}^v } \\ \end{array} } \right] + Bu_{ij} $$

(6.1a)

$$ y_{ij} = C\left[ {\begin{array}{*{20}c} {x_{ij}^h } \\ {x_{ij}^v } \\ \end{array} } \right] + Du_{ij} ,i,j \in Z_ + $$

(6.1b)

where $ x_{ij}^h \in R_ + ^{n_1 } $ and $ x_{ij}^v \in R_ + ^{n_2 } $ are the horizontal and vertical state vectors respectively at the point $ (i,j),u_{ij} \in R_ + ^m $ and $ y_{ij} \in R_ + ^p $ are the input and output vectors respectively, and

$$\begin{array}{*{20}{c}} {A = \left[ {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}} \\ {{A_{21}}}&{{A_{22}}} \end{array}} \right] \in R_ + ^{n \times n},B = \left[ {\begin{array}{*{20}{c}} {{B_1}} \\ {{B_2}} \end{array}} \right] \in R_ + ^{n \times m},} \\ {C = [\begin{array}{*{20}{c}} {{C_1}}&{{C_2}] \in R_ + ^{p \times n},D \in R_ + ^{p \times m}} \end{array}(n = {n_1} + {n_2})} \end{array}$$

(6.2)

Boundary conditions for Equation (6.1 a) have the form

$$ x_{0j}^h \in R_ + ^{n_1 } ,j \in Z_ + andx_{i0}^v \in R_ + ^{n_2 } ,i \in Z_ + $$

(6.3)

The transition matrix T_ij of Equation (6.1) is defined as follows

$$ T_{ij} = \left\{ {\begin{array}{*{20}c} {I_n (identitymatrix)fori = j - 0} \\ {\left[ {\begin{array}{*{20}c} {A_{11} } & {A_{12} } \\ 0 & 0 \\ \end{array} } \right]fori = 1,j = 0and\left[ {\begin{array}{*{20}c} 0 & 0 \\ {A_{21} } & {A_{11} } \\ \end{array} } \right]fori = 0,j = 1} \\ {T_{10} T_{i - 1,j} + T_{01} T_{i,j - 1} fori \geqslant 0,j \geqslant 0(i + j > 0)} \\ {T_{ij} = 0(zeromatrix)fori < 0or/andj < 0} \\ \end{array} } \right. $$

(6.4)

From Equation (6.4) it follows that if $ A \in R_ + ^{nxn} , $ then $ T_{ij} \in R_ + ^{nxn} $ for all $ i,j \in Z_ + . $

Tadeusz Kaczorek

7. Realisation problem for positive 2D systems

Abstract

Consider the 2D Roesser model

$$ \left[ {\begin{array}{*{20}c} {x_{i + 1,j}^h } \\ {x_{i,j + 1}^v } \\ \end{array} } \right] = A\left[ {\begin{array}{*{20}c} {x_{ij}^h } \\ {x_{ij}^v } \\ \end{array} } \right] + Bu_{ij} $$

(7.1a)

$$ y_{ij} = C\left[ {\begin{array}{*{20}c} {x_{ij}^h } \\ {x_{ij}^v } \\ \end{array} } \right] + Du_{ij} i,j \in Z_ + $$

(7.1b)

where $ x_{ij}^h \in R^{n_1 } $ and $ x_{ij}^v \in R^{n_2 } $ are the horizontal and vertical state vectors at the point $ (i,j),u_{ij} \in R^m $ and $ y_{ij} \in R^p $ are input and output vectors respectively, and

$$A = \left[ {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}} \\ {{A_{21}}}&{{A_{22}}} \end{array}} \right],B = \left[ {\begin{array}{*{20}{c}} {{B_1}} \\ {{B_2}} \end{array}} \right],C = \left[ {\begin{array}{*{20}{c}} {{C_1}}&{{C_2}} \end{array}} \right]$$

$$c A_{11} \in R^{n_1 xn_1 } ,B_1 \in R^{n_1 xm} ,C_1 \in R^{pxn_1 } $$

$$ A_{22} \in R^{n_2 xn_2 } ,B_2 \in R^{n_2 xm} ,C_2 \in R^{pxn_2 } $$

The transfer matrix of Equation (7.1) is given by

$$ T(z_1 ,z_2 ) = C\left[ {\begin{array}{*{20}c} {I_{n_1 } z_1 - A_{11} } & { - A_2 } \\ { - A_{21} } & {I_{n_2 } z_2 - A_{22} } \\ \end{array} } \right]^{ - 1} B + D $$

(7.2)

By Theorem 5.1 the model in Equation (7.1) is positive if and only if

$$ A \in R_ + ^{nxn} ,B \in R_ + ^{nxm} ,C \in R_ + ^{pxn} ,D \in R_ + ^{pxm} ,n = n_1 + n_2 $$

(7.3)

Tadeusz Kaczorek

Backmatter

Titel: Positive 1D and 2D Systems
verfasst von: Tadeusz Kaczorek, DSc
Verlag: Springer London
Electronic ISBN: 978-1-4471-0221-2
Print ISBN: 978-1-4471-1097-2
DOI: https://doi.org/10.1007/978-1-4471-0221-2

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Positive matrices and graphs

2. Continuous-time and discrete-time positive systems

3. Reachability, controllability and observability of positive systems

4. Realisation problem of positive 1D systems

5. 2D models of positive linear systems

6. Controllability and minimum energy control of positive 2D systems

7. Realisation problem for positive 2D systems

Backmatter