Piecewise Expression — Control Systems/Symbolic

Piecewise Expression

Control Systems / Symbolic

Evaluates a conditional expression of the input at every sample: the first condition that holds selects its value, and the otherwise value answers when none does. This is MATLAB's piecewise, written in MATLAB's own syntax:

piecewise(x < 0, 0, x < 1, x^2, 2*x - 1)

The text is read once, when the configuration loads. Every value, and each side of every condition, goes through the symbolic engine exactly as a Symbolic Expression's does, so a constant such as pi/4 is folded once rather than recomputed per sample. What runs per sample is an if / else-if / else chain with one plain expression per branch, and only the branch that is taken is evaluated.

The factory setting above is a smooth ramp: 0 below zero, a parabola on [0, 1) and the line 2x − 1 above it. The pieces meet with the same value and the same slope at both joins, so the block out of the library is a worked example of a piecewise function with no jump.

Ports

• x – the variables, a vector of n entries (a column [n,1] or a row [1,n]) where n is the number of names in Variables: entry k is the k-th name. A Mux in front builds it from scalar signals; with one variable the input is a scalar.
• y – the selected value, a scalar [1,1].

Parameters

• Expression – ONE call, piecewise(cond1, value1, cond2, value2, ..., otherwise): conditions and values alternating, and the otherwise value last. A condition compares two expressions with < <= > >= == ~= and combines comparisons with & (and), | (or) and ~(...) (not). A chain such as -1 < x <= 1 is a range, as MATLAB's symbolic reader has it: every adjacent pair must hold. A value, and each side of a comparison, may use numbers, the declared variables, pi, + - * / ^, parentheses and the functions sin cos tan sec csc cot asin acos atan acot sinh cosh tanh asinh acosh atanh exp log log2 log10 sqrt abs sign heaviside.
• Variables – the names of the input's entries, in order, separated by spaces or commas: x y. Each is a MATLAB identifier, none may repeat, and none may be a function name or pi, e, i, j, Inf, NaN or eps. At most 12.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. Each carries the chain as that language's own if statement – if/elif/else, if/elseif/else/end, IF/ELSIF/ELSE/END_IF, if/elsif/else/end if, if ... begin/end – with one assignment per branch and every constant folded to 17 significant digits, and in each of them only the branch taken is evaluated. The three HDL targets are simulation-only real arithmetic, quantized to Q16.16 only at the ports: a comparison of arbitrary expressions is not a fixed-point datapath.

Simulink bridge

None (Support::None). piecewise is a Symbolic Math Toolbox function, and that toolbox ships no Simulink library at all, so there is no library path a diagram could name; the bridge reports this block rather than dropping it, and it therefore has no parity testbench. Code export verification covers it across all ten languages. No configuration crosses, including "Sampling Time (s)".

Notes

• ⚠ The otherwise value is REQUIRED here, where MATLAB makes it optional. Without one MATLAB answers NaN wherever no condition holds, and a NaN is not a value the HDL and PLC targets can carry, so a piecewise that ends on a value's condition is refused with that reason. To reproduce MATLAB's answer exactly, make the otherwise value something the rest of the diagram can recognise.
• The first condition that holds wins, as in MATLAB: conditions may overlap, and their ORDER is part of the expression.
• ⚠ A chain is a RANGE here, as it is in MATLAB's symbolic reader – and not as in MATLAB's numeric one, which reads -1 < x < 1 as (-1 < x) < 1. -1 < x < y < 1 means all three adjacent comparisons hold. A chain holding == or ~= is refused: MATLAB accepts one and then cannot evaluate it.
• && and || are refused, because MATLAB refuses them on a symbolic condition ("Conversion to logical from sym is not possible"); write & and |. And ~ must negate a condition: MATLAB reads ~x > 1 as (~x) > 1, so write ~(x > 1).
• A condition is not a number and a number is not a condition: a value such as (x > 0)*x and a condition such as x & y are refused, as is a piecewise nested inside another expression – add a branch instead.
• ⚠ A jump is a jump in every target: where the pieces do not meet, the output steps as the input crosses the boundary, and the three HDL targets – which see the input quantized at the port – can land on the other side of it near the join. That is the function, not the export.
• Comparisons follow IEEE rules: any comparison with a NaN input is false, so a NaN input falls through to the otherwise value.
• Algebraic, with no state: the output depends only on the current input.
• No state space: a branch is not linear, so model reduction correctly declines the block.