Consider the nonlinear system (5.1), where in this case the control objective is to design a control law such that the state x(t) tracks a desired vector signal x_d (t). Let the tracking error be denoted by e(t) = x(t) x_d (t). If a tracking controller is designed based on a linearization valid at some operating point x_e ∈ ⁿ, then as x_d (t) moves away from the equilibrium point, the state x(t) will try to follow it. However, as the distance between x(t) and x_eincreases, the linear approximation may become increasingly inaccurate. As the accuracy of the linear approximation decreases, the designed linear controller may become unsuitable, thus possibly forcing x(t) even further away from the equilibrium x_e.

The tracking objective is in general more suitably addressed by a control law that is designed based on linearization about the desired trajectory x_d (t). Obviously, linearization around x_d (t) assumes that this signal is available a priori. If x_d (t) is not available but needs to be generated online possibly by an outer-loop controller, then small-signal linearization can be performed around a nominal trajectory x*(t), which is available a priori. Associated with a nominal trajectory x*(t) is a nominal control signal u*(t) and initial conditions such that x*(t) satisfies

Let . Then

Using the Taylor series expansion of around (x*, u*) we obtain

where F represents the higher-order terms of the Taylor series expansion. Since F contains the higher-order terms, it satisfies

In other words, as and become small, F goes to zero faster than Using the Taylor series expansion, (5.10) can be rewritten

In a linear approximation the higher-order terms are ignored. Hence, the small-signal linearization of (5.1) around the nominal trajectory x*(t) is given by

where z is the state of the linear model and the matrices A(t) : [0, ∞) ^{n x n} and B(t) : [0, ∞) ^{n x m} are given by

Now, suppose we select the control law as. The closed-loop dynamics for the linear system are given by

If the pair (A(t), B(t)) is uniformly completely controllable, then there exists K(t) such that the closed-loop system (5.13) is asymptotically stable; therefore, z(t) → 0, which implies that x(t) → x*(t). If the nominal trajectory x*(t) coincides with the desired vector signal x_d (t), then we achieve asymptotic convergence of the tracking error to zero.

Clearly, the above stability arguments were based on the linear model. Applying the same control law to the nonlinear system we have

which implies

Again, linearizing the closed-loop system around x = x* = x_d, u = u* yields

Therefore, applying the linear control law to the nonlinear system yields a locally asymptotically stable closed-loop system. In this case, locality is defined relative to the nominal trajectory (i.e., x(t) – x*(t)sufficiently small for all t > 0).

If the nonlinear system has an output function then, again, we can proceed to obtain the C(t) and D(t) matrices. Linearization of the nonlinear system (5.5)-(5.6) around a nominal trajectory x*(t) produces a linear model of the form

Therefore, we see that linearizing around a trajectory yields similar results as linearizing around an equilibrium point, with the key difference that in the former case the linear model is time-varying.

Next we present an example of linearizing around a nominal trajectory to illustrate the concepts introduced in this subsection.

■ EXAMPLE 5.2

A simple model of a satellite of unit mass moving in a plane can be described by the following equations of motion in polar coordinates [225]:

where, as shown in Figure 5.2, r(t) is the radius from the origin to the mass, θ(t) is the angle from a reference axis, u₁(t) is the thrust force applied in the radial direction, u₂(t) is the thrust force applied in the tangential direction, and Β is a constant parameter. With zero thrust forces (i.e., u₁(t) = 0 and u₂(t) = 0), the resulting solution can take various forms (ellipses, parabolas, or hyperbolas) depending on the initial conditions. In this example, we consider a simple circular trajectory with constant angular velocity (i.e., r(t) and are both constant). It is easy to verify that with zero thrusts forces and the initial conditions the resulting nominal trajectory is r*(t) = r₀ and θ*(t) = w₀t + θ₀. The objective is to linearize the model around this nominal trajectory.

To construct the state equation representation, let
The equations of motion in the state coordinates are given by

Figure 5.2: Point mass satellite moving in a planar gravitational orbit.

The nominal trajectory is described by

If we define , then the small-signal linearized system (around the nominal trajectory) is given by