The equations of motion, linearized around the upright position ($\theta \approx 0$), yield an $A$ matrix that couples these states. Notice the magic: The $A$ matrix will have entries showing that the angle $\theta$ influences the cart acceleration $\ddotp$ (via $A_2,3$), and the cart acceleration influences the pole’s angular acceleration $\ddot\theta$ (via $A_4,2$). This coupling is invisible in a SISO transfer function but explicit in state-space.
$$x = [p, \dotp, \theta, \dot\theta]^T$$ Control System Design An Introduction To State-space Methods
For a system with $n$ states, construct the controllability matrix: $$ \mathcalC = [B \quad AB \quad A^2B \quad ... \quad A^n-1B] $$ If $\mathcalC$ has full row rank (rank $n$), the system is controllable. The equations of motion, linearized around the upright
Several software tools are available for state-space control design, including: $$x = [p, \dotp, \theta, \dot\theta]^T$$ For a
If you are designing a control system today, here is the state-space workflow:
At its core, state-space design is a mathematical model of a physical system as a set of input, output, and state variables related by first-order differential equations.
If you want to minimize something (like fuel consumption or time), state-space is the foundation for advanced techniques like LQR (Linear Quadratic Regulator). 3. Key Concepts in State-Space Design
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