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#![allow(non_snake_case)]
//! PID Controller Design — model a plant, tune gains, verify stability, generate code.
//!
//! This example walks through a real controls engineering workflow:
//!
//! 1. Model a DC motor as a second-order transfer function
//! 2. Design a PID controller with symbolic gains Kp, Ki, Kd
//! 3. Compute the closed-loop characteristic polynomial
//! 4. Analyze stability via the Routh-Hurwitz criterion
//! 5. Choose gains to place poles, verify stability
//! 6. Generate optimized Rust code for the controller
//!
//! Run with: `cargo run --example pid_controller`
use symplex::prelude::*;
fn main() {
println!("=== PID Controller Design for a DC Motor ===\n");
let ctx = Context::new();
symplex::syms!(ctx; s, t);
// ── 1. Plant model ─────────────────────────────────────────────
//
// DC motor: G(s) = K / (s(Js + b))
// K = motor torque constant
// J = moment of inertia
// b = damping coefficient
//
// With J=1, b=10, K=20 (normalized):
// G(s) = 20 / (s² + 10s)
println!("--- Plant Model ---");
let _plant_num = ctx.int(20);
let _plant_den = expr!(ctx, s ^ 2 + 10 * s);
println!("G(s) = 20 / (s² + 10s)");
println!(" Open-loop poles: s = 0, s = -10");
// ── 2. PID controller ──────────────────────────────────────────
//
// C(s) = Kp + Ki/s + Kd·s = (Kd·s² + Kp·s + Ki) / s
println!("\n--- PID Controller ---");
symplex::syms!(ctx; Kp, Ki, Kd);
let _pid_num = expr!(ctx, Kd * s ^ 2 + Kp * s + Ki);
let _pid_den = s.clone();
println!("C(s) = Kp + Ki/s + Kd·s");
println!(" = (Kd·s² + Kp·s + Ki) / s");
// ── 3. Closed-loop transfer function ───────────────────────────
//
// T(s) = C(s)G(s) / (1 + C(s)G(s))
//
// Numerator of C·G = 20(Kd·s² + Kp·s + Ki)
// Denominator of C·G = s·(s² + 10s) = s³ + 10s²
//
// Closed-loop char. poly = den(C·G) + num(C·G)
// = s³ + 10s² + 20·Kd·s² + 20·Kp·s + 20·Ki
// = s³ + (10 + 20·Kd)s² + 20·Kp·s + 20·Ki
println!("\n--- Closed-Loop Characteristic Polynomial ---");
let char_poly = expr!(ctx, s ^ 3 + (10 + 20 * Kd) * s ^ 2 + 20 * Kp * s + 20 * Ki);
println!("P(s) = {char_poly}");
// ── 4. Routh-Hurwitz stability analysis ────────────────────────
//
// For s³ + a₂s² + a₁s + a₀, Routh conditions:
// a₂ > 0: 10 + 20·Kd > 0
// a₀ > 0: 20·Ki > 0
// a₂·a₁ > a₀: (10 + 20·Kd)·(20·Kp) > 20·Ki
println!("\n--- Routh-Hurwitz Stability Conditions ---");
let a2 = expr!(ctx, 10 + 20 * Kd);
let a1 = expr!(ctx, 20 * Kp);
let a0 = expr!(ctx, 20 * Ki);
println!(" a₂ = {a2} > 0");
println!(" a₀ = {a0} > 0");
println!(" a₂·a₁ > a₀: ({a2})·({a1}) > {a0}");
let routh_product = (&a2 * &a1).expand();
println!(" i.e., {routh_product} > {a0}");
// ── 5. Choose gains and verify ─────────────────────────────────
//
// Pick Kp = 5, Ki = 2, Kd = 0.5
// a₂ = 10 + 10 = 20 > 0 ✓
// a₀ = 40 > 0 ✓
// a₂·a₁ = 20·100 = 2000 > 40 ✓
println!("\n--- Gain Selection: Kp=5, Ki=2, Kd=0.5 ---");
let _gains = [(&Kp, 5i64), (&Ki, 2i64), (&Kd, 1i64)]; // Kd = 1 for now
let char_concrete = char_poly
.subs(&Kp, &ctx.int(5))
.subs(&Ki, &ctx.int(2))
.subs(&Kd, &ctx.rational(1, 2));
let char_expanded = char_concrete.expand().eval();
println!("P(s) = {char_expanded}");
// Find the closed-loop poles
let poles = char_expanded.solve(&s);
println!("\nClosed-loop poles:");
match &poles {
Ok(roots) => {
let mut all_stable = true;
for (i, root) in roots.iter().enumerate() {
let val = root.eval_f64();
let stable = match &val {
Ok(v) => *v < 0.0,
Err(_) => {
// Complex root — check real part via eval_complex64
root.eval_complex64().map(|z| z.re < 0.0).unwrap_or(false)
}
};
if !stable {
all_stable = false;
}
let val_str = val
.map(|v| format!("{v:.4}"))
.unwrap_or_else(|_| format!("{root}"));
println!(
" p{} = {} {}",
i + 1,
val_str,
if stable { "✓" } else { "✗" }
);
}
println!(
"\nStability: {}",
if all_stable {
"STABLE — all poles in left half-plane"
} else {
"UNSTABLE — pole(s) in right half-plane"
}
);
}
Err(e) => println!(" Could not find poles analytically: {e}"),
}
// Verify Routh conditions numerically
println!("\nRouth-Hurwitz verification:");
let a2_val = a2.subs(&Kd, &ctx.rational(1, 2)).eval_f64().unwrap();
let a1_val = a1.subs(&Kp, &ctx.int(5)).eval_f64().unwrap();
let a0_val = a0.subs(&Ki, &ctx.int(2)).eval_f64().unwrap();
println!(" a₂ = {a2_val} > 0: {}", a2_val > 0.0);
println!(" a₀ = {a0_val} > 0: {}", a0_val > 0.0);
println!(
" a₂·a₁ = {} > a₀ = {}: {}",
a2_val * a1_val,
a0_val,
a2_val * a1_val > a0_val
);
// ── 6. Generate controller code ────────────────────────────────
//
// PID output: u(t) = Kp·e + Ki·∫e dt + Kd·de/dt
// Discrete approximation (for embedded):
// u[k] = Kp·e[k] + Ki·Ts·Σe + Kd·(e[k] - e[k-1])/Ts
//
// Generate the update equation as Rust code.
println!("\n--- Generated Controller Code ---");
// Build the PID output expression with concrete gains
symplex::syms!(ctx; error, integral, derivative);
let kp_val = ctx.rational(5, 1);
let ki_val = ctx.rational(2, 1);
let kd_val = ctx.rational(1, 2);
let pid_output = &kp_val * &error + &ki_val * &integral + &kd_val * &derivative;
let pid_simplified = pid_output.eval();
match pid_simplified.to_rust_fn("pid_update", &["error", "integral", "derivative"]) {
Ok(code) => {
println!("{code}");
}
Err(e) => {
println!(" Code generation error: {e}");
println!(" Expression: {pid_simplified}");
// Fall back to manual code
println!("\n // Manual implementation:");
println!(" fn pid_update(error: f64, integral: f64, derivative: f64) -> f64 {{");
println!(" 5.0 * error + 2.0 * integral + 0.5 * derivative");
println!(" }}");
}
}
// Compile and test numerically
if let Ok(f) = pid_simplified.compile(&["error", "integral", "derivative"]) {
println!("Numerical verification:");
// Step response: error=1.0, no integral or derivative yet
println!(" u(e=1.0, i=0, d=0) = {:.2}", f(&[1.0, 0.0, 0.0]));
// With integral buildup
println!(" u(e=0.5, i=2.0, d=-1) = {:.2}", f(&[0.5, 2.0, -1.0]));
// Steady state: error=0
println!(" u(e=0, i=3.0, d=0) = {:.2}", f(&[0.0, 3.0, 0.0]));
}
println!("\n✓ Done!");
}