Synopsis

This post demonstrates a powerful, integrated workflow using Maple and MapleSim to transition seamlessly from the analytical derivation of a control law to its validation in a high-fidelity, nonlinear plant model. We'll design a state-feedback controller for an inverted pendulum system, showcasing how symbolic mathematics and multi-domain physical modeling complement each other for robust engineering design.

The Challenge: Bridging the Gap Between Theory and Practice

Control theory often begins with a simplified, linearized plant model. Deriving controllers symbolically in Maple provides unparalleled insight into parameter relationships and stability conditions. However, the final test is its performance on a more realistic model incorporating friction, saturation, and other nonlinearities. Manually recreating models and translating equations between tools is error-prone and time-consuming.

The Integrated Workflow: A Two-Stage Approach

Stage 1: Analytical Design in Maple

We start in Maple with a simplified, linear state-space model of an inverted pendulum on a cart.

with(LinearAlgebra): with(ControlDesign): with(DynamicSystems):

# Define system parameters (mass, length, gravity, etc.)
params := {M=5.0, m=1.0, l=0.5, g=9.81, b=0.1}:

# Derive linearized state-space matrices A and B symbolically
# ... (symbolic derivation code here) ...
A_sub := eval(A, params);
B_sub := eval(B, params);

# Design a Linear-Quadratic Regulator (LQR) for optimal performance
Q := DiagonalMatrix([10, 1, 5, 1]); # State weighting
R := Matrix([[1]]);                 # Control effort weighting
K := LQR(A_sub, B_sub, Q, R);

# Analyze closed-loop performance
sys_cl := StateSpace(A_sub - B_sub.K, B_sub, C, D);
poles := Eigenvalues(sys_cl:-A);

Maple's strength here is the symbolic derivation of the A and B matrices and the tunable, algorithmic design of the gain matrix `K`. We gain exact analytical insight before any numerical simulation.

Stage 2: High-Fidelity Validation in MapleSim

Instead of manually rebuilding the system, we export our controller gains directly to a pre-built, high-fidelity MapleSim model.

1.  The plant is modeled in MapleSim's multi-domain environment using 1D mechanical components, capturing full nonlinear dynamics, joint friction, and actuator limits.
2.  We use a Custom Component block in MapleSim. In its Equation View, we reference the gain matrix `K` computed in Maple via a parameter file import or a shared Maple worksheet connection.

# Inside MapleSim Custom Component Equation Tab
parameters := GetParameters(); # Gets K from linked Maple engine
u := -parameters.K . x; # State-feedback control law

3.  We then run simulations to test performance under realistic conditions—varying initial angles, adding disturbance forces, and observing control effort saturation.

Results & Visualization

The post would include comparative plots generated from both tools:

• A Maple plot of the symbolic system's eigenvalues migrating with changing Q weights.
• A MapleSim 3-D animation of the pendulum stabilizing.
• Overlaid response plots from the linear (Maple) and nonlinear (MapleSim) simulations, clearly showing the impact of unmodeled dynamics and the controller's robustness.

Why This Workflow is Transformative

Single Source of Truth: The control law `K` is derived once in Maple and deployed directly, eliminating manual translation errors.

Iterative Design Speed: You can rapidly adjust your `Q` and `R` weights in Maple, recompute `K`, and immediately re-simulate the full nonlinear system in MapleSim.

Deep Insight + High Confidence: Maple provides the "why" (pole placement, sensitivity analysis), while MapleSim provides the "how it really performs" under realistic constraints.

Product Tips & Techniques Used

• Maple: `ControlDesign` Toolbox for `LQR()` design, symbolic linearization, and parametric analysis.
• MapleSim: Using Custom Components with Maple code embedding, and leveraging parameter binding to create a dynamic link between the analytical and physical models.
• General: Utilizing the Maple-MapleSim connectivity to pass parameters and results fluidly.