Tangent Hyperplanes II: Dynamics
Linearization Along a Reference Trajectory
Suppose we have a reference trajectory \((x^*(t), u^*(t))\) satisfying:
\[ \dot{x}^*(t) = f(x^*(t), u^*(t)). \]
Now consider a perturbed trajectory \(x(t) = x^*(t) + \delta x(t)\) driven by a perturbed control \(u(t) = u^*(t) + \delta u(t)\). The perturbed system satisfies:
\[ \dot{x}^*(t) + \delta\dot{x}(t) = f(x^*(t) + \delta x(t), u^*(t) + \delta u(t)). \]
Subtracting the reference trajectory equation and expanding the right-hand side to first order using the Taylor series:
\[ f(x^* + \delta x, u^* + \delta u) = f(x^*, u^*) + \frac{\partial f}{\partial x}\bigg|_{(x^*,u^*)} \delta x + \frac{\partial f}{\partial u}\bigg|_{(x^*,u^*)} \delta u + R_2(\delta x,\delta u), \]
where \(R_2(\delta x,\delta u)\) collects terms of second order and higher. Dropping that residual gives the variational equation:
\[ \delta\dot{x}(t) = A(t)\delta x(t) + B(t)\delta u(t) \]
where the Jacobian matrices are defined as:
\[ A(t) = \frac{\partial f}{\partial x}\bigg|_{(x^*(t),u^*(t))} \]
\[ B(t) = \frac{\partial f}{\partial u}\bigg|_{(x^*(t),u^*(t))}. \]
These are \(n \times n\) and \(n \times m\) matrices, respectively, evaluated along the reference trajectory at each time \(t\).
Moving Tangent Spaces
The tangent space at the reference state \(x^*(t)\) is the space in which this linearized perturbation system evolves. As the system evolves, the reference state moves along the trajectory, and the tangent space moves with it. At each instant \(t\), the tangent hyperplane is:
\[ H_{x^*(t)} = x^*(t) + T_{x^*(t)} M. \]
The perturbation \(\delta x(t)\) lives in this moving linear subspace. The matrices \(A(t)\) and \(B(t)\) describe how perturbations grow or shrink within this space.
Superposition: Exact, Not Approximate
Here is the crucial property: superposition holds exactly for the variational equation. If two perturbations solve the same linearized dynamics, their sum solves the same linearized dynamics with summed inputs.
If \(\delta x_1(t)\) is a solution to \(\delta\dot{x} = A(t)\delta x + B(t)\delta u_1\) and \(\delta x_2(t)\) is a solution to \(\delta\dot{x} = A(t)\delta x + B(t)\delta u_2\), then \(\delta x_1(t) + \delta x_2(t)\) is exactly a solution to \(\delta\dot{x} = A(t)\delta x + B(t)(\delta u_1 + \delta u_2)\).
The statement is exact for the differential equation after the residual \(R_2\) has been removed. It is not an exact statement about finite perturbations of the original nonlinear system.
The State Transition Matrix
Solutions to the linearized system can be expressed using the state transition matrix (also called the fundamental matrix) \(\Phi(t, t_0)\), which satisfies:
\[ \dot{\Phi}(t, t_0) = A(t)\Phi(t, t_0), \quad \Phi(t_0, t_0) = I \]
A solution to the linearized system with control input is:
\[ \delta x(t) = \Phi(t, t_0)\delta x(t_0) + \int_{t_0}^{t} \Phi(t, \tau) B(\tau) \delta u(\tau) \, d\tau \]
This is the exact solution of the variational equation. Its accuracy as a prediction of the original nonlinear trajectory is limited by the omitted residual \(R_2\).
Limitations: Finite Perturbations and Nonlinearity
The linearized system is exact infinitesimally. For finite perturbations, the story is more complicated:
Higher-Order Terms Neglected: The error from dropping \(R_2(\delta x,\delta u)\) is non-zero for finite perturbations. Under bounded second derivatives, the leading residual scales quadratically with perturbation size.
Superposition Breaks Down: Superposition holds exactly for the linearized system. But when you add the higher-order correction terms back in, two small perturbations no longer combine linearly β there are interaction terms.
Validity Range: The linearized system is trustworthy only when \(\|\delta x(t)\|\) and \(\|\delta u(t)\|\) remain small. In nonlinear systems, small initial perturbations can grow exponentially. Eventually, the linearization fails.
Looking Ahead
The fact that we can extract a linear system from nonlinear dynamics β and that this linear system is exact infinitesimally β opens the door to using powerful control tools. In Part 3, we will see how differential dynamic programming and iterative linear-quadratic regulation exploit this fact to design nonlinear controllers.