Self Conditioned 3D — Control Systems/Gain Scheduling

3D Self-Conditioned

Control Systems / Gain Scheduling

The Aerospace Blockset's 3D Self-Conditioned [A(v),B(v),C(v),D(v)]: the Self-Conditioned Controller with its four matrices scheduled on three variables, v1, v2 and v3. Its states stay consistent with what the actuator is really doing:

dx/dt = H(v)·u_meas + (B(v) − H(v)·D(v))·y + (A(v) − H(v)·C(v))·x
u_dem = C(v)·x + D(v)·y

While the actuator delivers what was demanded, u_meas = u_dem, the H terms cancel and the block is the scheduled controller [A, B, C, D]; when it saturates or is switched out, H pulls the states toward values consistent with the real input. H is placed at every breakpoint triple – the eigenvalues of A_ijl − H_ijl·C_ijl are the poles you give – and then interpolated like the matrices: each of A, B, C, D and H is the trilinear blend of the eight stored around (v1, v2, v3), exactly as in the 3D Controller, and each variable clamps outside its breakpoints.

Ports

• y – the controller input, an [m,1] column, m being the column count of the B and D matrices.
• v1 – the first scheduling variable, a scalar (the Simulink block labels it AoA).
• v2 – the second scheduling variable, a scalar (labelled AoS there).
• v3 – the third scheduling variable, a scalar (labelled Mach there).
• u_meas – the actuator's measured output, [1,1]: the value the plant actually received.
• u_dem – the demanded actuator command, [1,1].

Parameters

• A Matrices – the n×n A matrix for every breakpoint triple, stacked vertically with the first variable varying fastest and the third slowest: a [P₁·P₂·P₃·n, n] matrix, with n from 1 to 8. From a MATLAB array A(:,:,i,j,l) the config is reshape(permute(A,[1 3 4 5 2]),[],size(A,2)).
• B Matrices – the n×m B matrices, stacked the same way.
• C Matrices – the 1×n C rows, stacked: [P₁·P₂·P₃, n]. ⚠ One row each: with a single controller output the gain H that places n poles is unique, so this block and Simulink compute the same H. With several outputs there are infinitely many, and Simulink picks one by a robustness criterion this block does not reproduce.
• D Matrices – the 1×m D rows, stacked: [P₁·P₂·P₃, m].
• Breakpoints v1 – the P₁ values of v1, strictly increasing, at least two.
• Breakpoints v2 – the P₂ values of v2, strictly increasing, at least two.
• Breakpoints v3 – the P₃ values of v3, strictly increasing, at least two.
• Initial State – x at the start of the run: a scalar for every state, or an [n,1] column. Default 0.
• Poles of A-H*C – n real, distinct values, the eigenvalues H places at every breakpoint triple (the Simulink parameter vec_w, default [-5 -2]). Repeated poles are refused, exactly as Simulink's place refuses them, and so is a breakpoint whose (A, C) pair is not observable. Complex poles cannot be written in a real matrix and are not offered.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

The defaults are a second-order example with two breakpoints and Simulink's own default poles on a 2×2×2 grid; the Simulink block has no literal matrix defaults (its dialog names workspace variables).

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. The gains H are placed at export time and baked in with the matrices; a chain of comparisons per variable assigns its interval and fraction, a nested chain on the three intervals assigns the blended A, B, C, D and H, and the core forms A − H·C and B − H·D from them each sample, as the Simulink block does – so no generated core places a pole. The state integrates with forward Euler at the block's period. All ten are rendered from the one description of the arithmetic the block's own simulation runs.

The three HDL targets are simulation-only: the fraction divides by a signal difference, so they compute in real arithmetic and quantize only at the ports. VHDL keeps the arithmetic in a function of its own.

Simulink bridge

None (Support::None). The counterpart is aerolibschedule/3D Self-Conditioned [A(v),B(v),C(v),D(v)] in the Aerospace Blockset, and this block reproduces it exactly – but its four matrix parameters are five-dimensional arrays, one slice per breakpoint triple, and neither MATLAB's matrix literal syntax nor an ICore matrix config has that form. So the schedule cannot cross in either direction; the block is reported rather than exported without its matrices. It therefore has no parity testbench; code export verification covers it.

Notes

• Stateful: n states. Direct feedthrough when some D row is nonzero, and also when the C rows differ across the grid (u_dem = C(v)·x then moves with v at the same instant). u_meas reaches only the state, but this block declares feedthrough for the whole block, so a loop from u_dem back to u_meas through a static actuator model (a Saturation alone) is reported as an algebraic loop when either holds; an actuator with dynamics, or a Memory in that path, opens it.
• Between breakpoints the eigenvalues of A(v) − H(v)·C(v) are near the given poles, not on them: H is interpolated, not re-placed, exactly as the Simulink block does.
• Each breakpoint search is inclusive at the bottom of each interval: a variable exactly on an interior breakpoint starts the next interval, with fraction 0.