EASY5 Dynamic Simulation & Linear Control (.ez5, .ezm, .ezd, .eas)

The @modelscript/easy5 package (languages/easy5) provides native support for Boeing / MSC Software EASY5™ dynamic system simulation models, control block diagrams, and empirical table lookups. It delivers multi-syntax parsing, embedded Fortran user block (UG/UX) compilation via @modelscript/fortran, pre-flight static topological linting (M7001–M7012) including Tarjan algebraic cycle detection, direct continuous DAE arena lowering (IDaeBuilder), linear control system analysis (trim solver, numerical $(A,B,C,D)$ Jacobian extraction, QR eigenvalue/pole analysis, Bode frequency sweeps), multi-domain projections to Modelica 3.x and FMI SSP, and interactive AntV X6 / SVG schematic diagram generation.

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Supported File Specifications

Extension	Format	Key Structural Features
.ez5 / .ezm	Modern Model Definition	Tagged MODEL, COMPONENTS, XP, S0, CONNECTIONS, TABLES blocks with graphical layout attributes
.ezd / .eas	Classical Card Deck	Fixed-field component, parameter, and connection card images with continuation flags
.tab	Empirical Lookup Tables	1D and 2D tabular lookup arrays for nonlinear characteristics ($f(x)$ or $f(x,y)$)
.f / .for	User General (UG/UX)	Embedded or referenced Fortran 77/90 routines computing custom nonlinear dynamics

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Architecture & Compilation Pipeline

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1. Standard General Purpose (GP) Library Catalog

@modelscript/easy5 includes built-in operational descriptors for standard EASY5 General Purpose library blocks:

| Block ID | Name | Mathematical Transfer Function / Formulation | Key Parameters (XP) | States (S) | | :------- | :------------------- | :---------------------------------------------------------------------------------------------------- | :----------------------------------------- | :-------------------------------- | -------------------------- | ---- | | GA | Proportional Gain | $y(t)=K\cdot u(t)$ | K (Gain factor) | None | | SJ | Summing Junction | $y(t)=\sum _{i=1}^m{\text{SGN}}_i\cdot u_i(t)$ | SGN1, SGN2, SGN3 ($\pm 1.0$) | None | | IN | Integrator | $\dot{x}(t)=K\cdot u(t),y(t)=x(t)$ | GAIN, S0 (Initial state), UL, LL | 1 continuous state | | TF | Transfer Function | $Y(s)=\frac{b_m{s}^m+\cdots +b_0}{a_n{s}^n+\cdots +a_0}U(s)$ | NUM (array), DEN (array) | $n$ continuous states | | LA | First-Order Lead-Lag | $Y(s)=K\frac{{\tau }_{1}s+1}{{\tau }_{2}s+1}U(s)$ | GAIN, TAU1, TAU2 | 1 continuous state (${\tau }_2>0$) | | LM | Hard Limiter | $y(t)=max(u_{min},min(u_{max},u(t)))$ | UL (Upper limit), LL (Lower limit) | None | | DZ | Deadband | $y(t)=\{\begin{array}{ll}u-\delta & u>\delta \\ u+\delta & u<-\delta \\ 0& |u|\le \delta \end{array}$ | DB (Deadband half-width) | None | | HY | Hysteresis Switch | Relay with toggle thresholds $u_{on},u_{off}$ | U_ON, U_OFF, Y_HIGH, Y_LOW | None (Discrete memory) | | TD | Transport Delay | $y(t)=u(t-\tau )$ | TAU (Delay time in seconds) | Buffer ring | | SW | Controlled Switch | $y(t)=\{\begin{array}{ll}{u}_1(t)& u_{ctrl}\ge u_{thresh}\\ u_2(t)& u_{ctrl}<u_{thresh}\end{array}$ | THRESH (Switch threshold) | None | | FN | Table Function | $y(t)=\mathrm{interp}(u(t),X,Y)$ | TABLE_NAME, ORDER (1=linear, 2=spline) | None | | UG | User General | Custom dynamics specified in Fortran 77/90 | User-defined parameters | User-defined ($S(i),SD(i)$) |

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2. Pre-Flight Diagnostic Rules (lintEasy5)

ModelScript evaluates 12 formal pre-flight diagnostic rules before simulation, linearization, or projection:

12 rules4 warning7 error1 info Run with msx lint

Code	Rule	Severity	Message
M7001	unconnected-input-port	warning	Input port '<comp>.<port>' has no incoming connection.
M7002	csmf-algebraic-loop	error	Direct feedthrough algebraic loop detected with no continuous state or delay: <value>.
M7003	duplicate-component-id	error	Duplicate component identifier '<id>' in model.
M7004	uninitialized-state	info	Continuous state in component '<comp>' defaults to 0.0 with no explicit S0 specified.
M7005	improper-transfer-function	error	Transfer function in component '<comp>' is improper: numerator degree (<degNum>) exceeds denominator degree (<degDen>).
M7006	parameter-out-of-bounds	error	Parameter '<param>' in component '<comp>' is out of valid bounds: <reason>.
M7007	port-dimension-mismatch	error	Connection width mismatch from '<from>' (dim <dim1>) to '<to>' (dim <dim2>).
M7008	state-index-out-of-bounds	error	State index S(<idx>) in component '<comp>' exceeds allocated continuous states (<max>).
M7009	sample-rate-mismatch	warning	Sample rate mismatch between '<from>' (<rate1>s) and '<to>' (<rate2>s) without rate transition block.
M7010	user-fortran-syntax-error	error	User Fortran subroutine in '<comp>' has defect: <detail>
M7011	unstable-linear-pole	warning	Component '<comp>' has open-loop pole in right-half plane (Re = +<value>), indicating open-loop instability.
M7012	table-extrapolation-hazard	warning	Component '<comp>' references table '<tableName>' without bounded extrapolation protection.

Generated from languages/easy5/src/lints/index.ts. All languages →

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3. Direct Continuous DAE Arena Lowering (lowerEasy5ToDae)

ModelScript's data-oriented WebAssembly linear-memory DAE arena compiles EASY5 models directly into struct-of-arrays without intermediate TypeScript AST objects:

• Continuous states $S(i)$ become continuous variables with derivative bindings $\text{der}(x_i)=f_i(x,u)$.
• Initial states $S_0$ become start attributes on continuous state variables.
• Summing junctions, gains, limiters, and lookup functions become lowered arena expression IDs (ExprId) and equality equations (EqId).
• Embedded Fortran blocks evaluate expressions directly mapped into DAE equation nodes.

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4. Linear Control Systems Engine (linear/)

The linear control package provides exact numerical state-space modeling for control design and frequency domain verification:

State-Space Extraction ($A,B,C,D$)

Linearizes the continuous non-linear system around an operating point $(x_0,u_0)$:

$$\dot{x}=Ax+Bu,y=Cx+Du$$

Where:

$$A_{ij}=\frac{\partial {\dot{x}}_i}{\partial x_j},B_{ik}=\frac{\partial {\dot{x}}_i}{\partial u_k},C_{pj}=\frac{\partial y_p}{\partial x_j},D_{pk}=\frac{\partial y_p}{\partial u_k}$$

Computed via central finite differences:

$$\frac{\partial f_i}{\partial x_j}\approx \frac{f_i(x_0+\epsilon e_j,u_0)-f_i(x_0-\epsilon e_j,u_0)}{2\epsilon }$$

Equilibrium Operating Point Trim Solver

Solves $\dot{x}(x,u)=0$ using Levenberg-Marquardt damped least-squares:

$$(J^{T}J+\lambda I)\mathrm{\Delta }x=-J^T\dot{x}(x,u)$$

Eigenvalue & Stability Analysis

Computes closed-loop poles $s_i={\sigma }_i\pm j{\omega }_i$ using the QR algorithm with Wilkinson shifts:

• Natural frequency: ${\omega }_{n,i}=\sqrt{{\sigma }_i^2+{\omega }_i^2}$
• Damping ratio: ${\zeta }_i=-\frac{{\sigma }_i}{{\omega }_{n,i}}$
• Stability criterion: ${\sigma }_i\le 0$ for all $i$

Bode Frequency Response Sweeps

Solves the complex resolvent system $(j\omega I-A)X(j\omega )=B$ at logarithmically spaced frequencies:

$$G(j\omega )=CX(j\omega )+D$$

• Magnitude (dB): $20{\log }_{10}|G(j\omega )|$
• Phase (degrees): $\mathrm{atan}2(\mathrm{Im}G(j\omega ),\mathrm{Re}G(j\omega ))\cdot \frac{180}{\pi }$

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5. Cross-Language Projections

Projection to Modelica 3.x (easy5ToModelica)

Translates EASY5 continuous control blocks into canonical Modelica.Blocks continuous models:

// Modelica projection generated from EASY5 model: FlightController
model FlightController
  // Component instances
  Modelica.Blocks.Math.Gain GA01(k = 5.0);
  Modelica.Blocks.Math.Add SJ01;
  Modelica.Blocks.Continuous.Integrator IN01(k = 1.0, y_start = 0.0);
  Modelica.Blocks.Continuous.TransferFunction LA01(b = {0.1, 1.0}, a = {0.02, 1.0});
  Modelica.Blocks.Nonlinear.Limiter LM01(uMax = 10.0, uMin = -10.0);
equation
  connect(GA01.y, SJ01.u1);
  connect(SJ01.y, IN01.u);
  connect(IN01.y, LA01.u);
  connect(LA01.y, LM01.u);
end FlightController;

Projection to FMI SystemStructure (easy5ToSsp)

Exports complete multi-component topologies as FMI 2.0/3.0 SystemStructure.ssd XML containers for co-simulation orchestration.

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6. Command-Line Interface (msx easy5)

# Model inspection and topological summary
msx easy5 summary controller.ez5

# Pre-flight linter (including algebraic loop check)
msx easy5 lint controller.ez5
msx lint controller.ez5 --language=easy5

# Numerical linearization to A, B, C, D state-space matrices
msx easy5 linearize controller.ez5

# Bode plot logarithmic frequency response calculation
msx easy5 bode controller.ez5 --min-freq=0.01 --max-freq=1000 --num-points=60

# Translation to Modelica 3.x
msx easy5 to-modelica controller.ez5 --out=Controller.mo

# Export to FMI SystemStructure.ssd
msx easy5 to-ssp controller.ez5 --out=SystemStructure.ssd

# Export SVG schematic diagram
msx easy5 diagram controller.ez5 --out=schematic.svg