Constrained Linear Least Squares — Control Systems/Optimization

Constrained Linear Least Squares

Control Systems / Optimization

Solves min_x ½·||C·x − d||² subject to Aeq·x = beq, A·x ≤ b and lb ≤ x ≤ ub on every step – MATLAB's lsqlin. C and the constraint matrices are settings; the data d and the inequality right-hand side b arrive on ports. It is the quadratic program with H = C'C and f = −C'd, and it runs the same Goldfarb-Idnani dual active set as Quadratic Program: it starts at the unconstrained least-squares answer, adds the most violated constraint, and drops whichever active constraint's multiplier would turn negative first.

Ports

• d – the data, an [m,1] column with one entry per ROW of C.
• b – the right-hand side of A·x ≤ b, a [p,1] column with one entry per row of A. When A is empty this port is unused and must be connected to any [1,1] signal.
• x – the minimizer, [n,1], one entry per COLUMN of C.
• resnorm – ||C·x − d||², [1,1], as lsqlin's second output. It is computed from the x that comes out, on every exit flag.
• exitflag – [1,1], with MATLAB's values: 1 optimal, 0 the iteration limit was reached, −2 the constraints are infeasible. Read it before using x.

Parameters

• Options – empty (default), or the path of a Solver Options block (Home/Solver Options), MATLAB’s options argument: for the run, every option it sets replaces this block’s parameter of the same name; one it leaves at default, or one this block does not have, changes nothing.
• Coefficient Matrix C – the matrix, [m,n] with m ≥ n and at most 48 rows and 12 columns. It must have full column rank, so that the unconstrained problem – and with it the constrained one – has a unique answer; a C whose columns are numerically dependent is refused with that reason when the model is built. Its Householder QR is taken once, at config load.
• Inequality Matrix A – the inequality matrix, [p,n], or [] for none. Its right-hand side is the b port.
• Equality Matrix Aeq, Equality Vector beq – the equality constraints, [q,n] and [q,1], or [] for none. The rows of Aeq must be linearly independent. beq is a SETTING, not a port: only the inequality right-hand side moves per step.
• Lower Bounds lb, Upper Bounds ub – bounds, [n,1] each or [] for none; -Inf/Inf entries leave a variable unbounded on that side. Each finite bound becomes one more constraint row, and Aeq, A and the finite bounds together may not exceed 40 rows.
• Constraint Tolerance – a constraint counts as violated when it is breached by more than this. Default 1e-8, lsqlin's own.
• Maximum Iterations – the limit on solver passes, each one a small linear solve. 0 selects the active-set default, 10·(variables + constraints). When the limit is reached the exit flag is 0 and x is the last iterate, which satisfies the constraints the solver had made active but may violate others.
• 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, printed from the same program the block runs, so every target does the same arithmetic in the same order as the simulation. C, the constraint rows, (C'C)⁻¹, the map R⁻¹Q₁' that gives the unconstrained answer and (C'C)⁻¹ times the constraint normals are baked in as constants at export time; re-export after changing them.

The whole solve runs in every generated body, and its cost is bounded: at most Maximum Iterations passes, each a Cholesky of a K×K matrix in the active constraints, where K is the number of constraint rows. That loop is that length in the emitted code and exits early through a flag.

The three HDL targets carry the arithmetic in real and quantize only at the ports: simulation-only, not offered as synthesizable. A Cholesky has a division by a pivot and a square root, and the active set is a data-dependent branch, none of which belongs in a Q16.16 datapath.

Simulink bridge

None (Support::None). lsqlin is an Optimization Toolbox function and that toolbox ships no Simulink library at all, so there is no path a diagram could name. The bridge reports this block rather than dropping it silently, and it has no parity testbench; code export verification covers all ten languages.

Notes

• Algebraic: the answer depends on this step's d and b alone, and nothing is carried between steps – there is no warm start. It is piecewise affine in (d, b) rather than linear, so it carries no state space.
• ⚠ x is continuous in d and b while the exit flag is 1, which is why quantizing the two ports (as the HDL targets do) moves x smoothly rather than jumping. The flag itself is a discrete output: a problem sitting on the feasibility boundary, or on the iteration limit, can be flagged differently by a target that sees a quantized right-hand side.
• ⚠ An exit flag of −2 or 0 leaves x defined but not optimal: it is the iterate the solver had reached, which satisfies the constraints it had made active and may violate the rest. MATLAB's own x is different there, because a different method walks a different path.
• ⚠ Unconstrained, this is not the plain least-squares block: with A, Aeq, lb and ub all empty the answer is R⁻¹Q₁'d and the solver does nothing, which is the same number a QR solve would give – but the block still carries the whole active-set program into every export.
• The measured agreement with R2026a, the problem sets behind it and the emitted MATLAB body's own check are on the source banner.