Much of the complexity that arises in the backstepping control laws that result from recursive application of Lemma 5.3.1 is due to the computation of the time derivative of the virtual control variables. The computation of these time derivatives becomes even more complex in applications where the functions ƒ and g are approximated online. This section presents an alternative formulation of the backstepping approach that decreases the algebraic complexity of the backstepping control law from that of eqn. (5.54).

Consider the second-order system

where x = [x₁x₂]^T ∈ ² is the state, x₂ ∈ ¹, and u is the scalar control signal. A region D is the specified operation region of the system. The functions ƒ_i, g_i for i = 1, 2 are known locally Lipschitz functions. The functions g_iare assumed to be nonzero for all x ∈ D. There is a desired trajectory x_1c (t), with derivative , both of which lie in a region D for t ≥ 0 and both signals are assumed known. Define the tracking errors

where x_2cwill be defined by the backstepping controller. Let

be a smooth feedback control and define the smooth positive definite function such that

where is positive definite in

To solve the tracking control problem for the system of eqns. (5.65)-(5.66) we use the following procedure:

1. Define
   
   
   where ξ₂ will be defined in step 3. The signal is filtered to produce the command signal x_2cand its derivative Such a filter is defined in Appendix A.4. Note that by the design of this command filter, the signal is bounded and small. Therefore, as long as g₁(x₁) is bounded, then ξ₁ is bounded because it is the output of a stable linear filter with a bounded input.
   
   
2. Define the compensated tracking errors as
   
   
   
3. Define
   
   
   
   where is filtered to produce is the control signal applied to the actual system. By the design of the command filter, the signal is bounded and small; therefore, if g₂(x) is bounded, then ξ₂ is the bounded output of a stable linear filter with a bounded input. If.

Figure 5.4: Diagram illustrating the command filter computations related to x₁. The nominal control block refers to eqn. (5.67). The diagram for x₂ would be similar.

Figure 5.4 displays a block diagram implementation of the above procedure. Note that is computed using , not . The quantity is available as the output of the filter in step 1. The quantity is not used in the control law. It is not directly available and is tedious to compute for higher order systems.

Given the above procedure, we now analyze the stability of the control law. The tracking error dynamics can be written as

As defined in (5.70) and (5.73), the variables ₁, ₂ represent the filtered effect of the errors and , respectively. The variables represent the compensated tracking errors, obtained after removing the corresponding unachieved portion of and . After some algebraic manipulation, the dynamics of the compensated tracking errors are described by

Consider the following Lyapunov function candidate

The time derivative of V along the solution of (5.77)-(5.78) is

where λ = 2min(k₁, k₂) > 0. The fact that shows that the origin of the system is exponentially stable. Therefore, we can summarize these results in the following theorem.

Lemma 5.3.2 Let the control law α₁ solve the tracking problem for system

with Lyapunov function V₁ satisfying (5.68). Then the controller of (5.69)-(5.73) solves the tracking problem (i.e., guarantees that x₁(t) converges to y_d (t)) for the system described by (5.65)-(5.66).

Note that this lemma can be applied recursively n – 1 times to address a system with n states. An example of this will be presented below. Note that the result guarantees desirable properties for the compensated tracking errors not the actual tracking errors . The difference between these two quantities is ξ_i, which is the output of the stable linear filter

The magnitude of the portion of the input defined by () determined by the design of the (i + 1)st command filter. This portion can be made arbitrarily small by appropriate design of the command filter. If the function g_iis bounded, then ξ_i is bounded. When r_iapproaches zero, then ξ_i → 0 and all i.

The goal of the derivation of this theorem was to avoid tedious algebraic manipulations involved in the computation of the backstepping control signal. Avoiding such computations will become increasingly important in backstepping approaches that include parameter adaptation.

In the following example, we return to the problem of Example 5.7 using Lemma 5.3.2.

■ EXAMPLE 5.8

From (5.61)-(5.63) and (5.69), we have that

and for i = 1, 2, 3 we have . Each pair and is the output of second-order, low-pass, unity-gain filter of Figure A.4 with input , respectively. If is used as the control signal, then and ξ₃ = 0. 

This example should be compared with Example 5.7. For an n-th order system, standard backstepping will require as controller inputs for i = 0, ... , n and will analytically compute . The command filtered approach will require as controller inputs only and will analytically compute only α_i. The tradeoff is that the command filtered approach will require n scalar filters for the ξ variables and n command filters.