[latexpage]Assign properties to $\Delta$-blocks

With the property ChannelClass you can associate a $\Delta$ -block with the plant $M$ as uncertainty, performance, or control channel.

Class	Description
Uncertainty channel	Specify if a $\Delta$ -block is an uncertainty with: – ChannelClass=’U’ (Default)
Performance channel	Specify if a $\Delta$ -block is a performance channel with: – ChannelClass=’P’
Control channel	Specify if a $\Delta$ -block is an control channel with: – ChannelClass=’C’

With the properties InputChannel and OutputChannel you can specify to which input- and output channels the $\Delta$ -block is connected to $M$ . There are three options as specified next.

\[\begin{array}{c} InputChannel=[C_x,\ldots,C_y]\\OutputChannel=[C_v,\ldots,C_w]\end{array}\]

Next to the previous properties, you can also specify the nature of the uncertainties via the following properties.

Property	Description
LinNonlin	Specify whether the uncertainty is linear (‘L’) or nonlinear (‘NL’). Default: ‘L’
TimInvTimeVar	Specify whether the uncertainty is time-invariant (‘TI’) or time-varying (‘TV’). Default: ‘TI’
StaticDynamic	Specify whether the uncertainty is static (‘S’) or dynamic (‘D’). Default: ‘S’
Structure	Specify whether the uncertainty is diagonal (‘D’) or full-block (‘FB’). Default: ‘D’

In addition, you can specify the characteristics of the uncertainties via the following properties.

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Option/Example	Description
Option 1	For a single repeated SISO uncertainty block one should specify: $\begin{array}{c} InputChannel=[C_x,\ldots,C_y]\\OutputChannel=[C_v,\ldots,C_w]\end{array}$ with integers $C_x$ , $C_y$ , $C_v$ , $C_w$ . Note: The order of the channels is not important.
Example 1	For a scalar uncertainty that is repeated 3 times, and which is connected with the first 3 in- and output channels of $M$ , one should specify: $\begin{array}{c} InputChannel=[1:3]\\OutputChannel=[1:3]\end{array}$
Option 2	For a single repeated MIMO uncertainty block one should specify: $\begin{array}{c} InputChannel=\left[\begin{array}{ccc}C_{\delta_{x_1}&\cdots&C_{\delta_{y_1}\\ \vdots&\vdots&\vdots\\ C_{\delta_{x_N}}&\cdots&C_{\delta_{y_N}}\end{array}\right]\\ OutputChannel=\left[\begin{array}{ccc}C_{\delta_{v_1}&\cdots&C_{\delta_{w_1}\\ \vdots&\vdots&\vdots\\ C_{\delta_{v_N}}&\cdots&C_{\delta_{w_N}}\end{array}\right]$ where again the entries are integers.
Example 2	For a full-block uncertainty of dimension $2\times3$ that is repeated twice and which is respectively connected with the first 4 and 6 in- and output channels of $M$ , one should specify: $\begin{array}{c} InputChannel=[1,2;3,4]\\OutputChannel=[1:3;4:6]\end{array}$
Option 3	For multiple repeated SISO or MIMO uncertainty blocks one should proceed as before with the sole difference that each row should be specified as the element of a cell: $\begin{array}{c} InputChannel=\{row_1,\ldots,row_N\},\\OutputChannel=\{row_1,\ldots,row_N \} .\end{array}$
Example 3	For two scalar uncertainties that are repeated 2 and 3 times respectively, and which are connected with the first 5 in- and outputs of $M$ , one should specify $\begin{array}{c} InputChannel=\{[1,2],[3:5]\}\\OutputChannel=\{[1,2],[3:5]\}\end{array}$

Property	Description
Bounds	Specify (if any) the bounds on an uncertain parameter. For a single parameter and for multiple parameters respectively specify: – $Bounds=[-b,b]$ , $b\geq0$ – $Bounds=\{[-b_1,b_1],\ldots,[-b_N,b_N]\}$ , $b_j\geq0$
RateBounds	Specify (if any) the rate bounds on an uncertain parameter. For a single parameter and for multiple parameters respectively specify: – $RateBounds=[-b,b]$ , $b\geq0$ – $RateBounds=\{[-b_1,b_1],\ldots,[-b_N,b_N]\}$ , $b_j\geq0$
NormBounds	Specify (if any) the norm bound on an uncertainty/ nonlinearity: – $NormBound=b$ , $b\geq0$
SectorBounds	Specify (if any) the sector bounds on an uncertainty/ nonlinearity. For a single sector constraint and for multiple sector constraints respectively specify: – $SectorBounds=[a,b]$ , $a\leq0\leq b$ – $SectorBounds=\{[a_1,b_1],\ldots,[a_N,b_N]\}$ , $a_j\leq0\leq b_j$
SlopeBounds	Specify (if any) the slope bounds on an uncertainty/ nonlinearity. For a single slope constraint and for multiple slope constraints respectively specify: – $SlopeBounds=[a,b]$ , $a\leq0\leq b$ – $SlopeBounds=\{[a_1,b_1],\ldots,[a_N,b_N]\}$ , $a_j\leq0\leq b_j$
Polytope	Specify (if any) the generator points (i.e., the convex hull) of an uncertainty block: $Polytope=\left[\begin{array}{ccc}\delta_1^{p_1}& \cdots& \delta_N^{p_1}\\ \vdots&\cdots&\vdots\\ \delta_1^{p_M}& \cdots& \delta_N^{p_M} \end{array}\right]$
UncertaintyMap	Specify (if any) the uncertainty map of a parametric full-block uncertainty: $\Delta(\delta)=\delta_1T_1+\cdots+ \delta_NT_N.$ Here the matrices $T_1,\ldots,T_N$ , which should all have the same dimension, must be specified as a cell array $T=\left\{T1,\ldots,T_N\right\}.$
DelayTime/ DelayType	Specify (if any) the maximum delay of the uncertain delay and the type of delay operator (see link for further details): – $DelayTime=b$ , $b\geq0$ – $DelayType=c$ , $c\in\{1,2\}$
Passive	Specify (if any) whether the uncertainty is passive: – Passive=’Passive’
Odd	Specify (if any) whether a nonlinearity is ‘Odd’: – ‘yes’ – ‘no’ (default)
PerfMetric	Specify the performance metric. Respectively induced $L_2$ -gain, $H_2$ , generalized $H_2$ , passive performance, and some invariance options as further specified here: – PerfMetric=’L2′ – PerfMetric=’H2′ – PerfMetric=’genH2′ – PerfMetric=’Passive’ – PerfMetric=’e2x’ – PerfMetric=’e2p’ – PerfMetric=’x2p’