Consider a nonlinear system described by

where x ∈ ⁿis the state and u ∈^mis the controlled input. Assume that the vector field ƒ(x) and the matrix G(x) each consist of two components: a known nominal part and an unknown part. Therefore,

where ƒ₀ and G₀ characterize the known nominal plant, and ƒ*, G* represent the uncertainty. Later we will assume that the unknown portion satisfies a certain bounding condition.

Moreover, we assume that the uncertainty satisfies a so-called matching condition:

The matching condition implies that the uncertainty terms appear in the same equations as the control inputs u, and as a result they can be handled by the controller.

By substituting (5.88)-(5.89) and (5.90)-(5.91) in (5.87) we obtain

where comprises all the uncertainty terms, and is given by

The Lyapunov redesign method addresses the following problem: suppose that the equilibrium of the nominal model can be made uniformly asymptotically stable by using a feedback control law u = p₀(x). The objective is to design a corrective control function p*(x) such that the augmented control law u = p₀(x) + p*(x) is able to stabilize the system (5.92) subject to the uncertainty (x, u) being bounded by a known function.

Next, we consider the details of the Lyapunov redesign method, which is thoroughly presented for a more general case in [134]. We assume that there exists a control law u = p₀(x)such that x = 0 is a uniformly asymptotically stable equilibrium point of the closed-loop nominal system

We also assume that we know a Lyapunov function V₀(x) that satisfies

where α₁, α₂, α₃: ⁺¹ are strictly increasing functions that satisfy α_i(0) = 0 and α_i(r) → ∞ as r → ∞. These type of functions are sometimes called class K_∞ functions [134].

The uncertainty term is assumed to satisfy the bound

where the bounding function is assumed to be known a priori or available for measurement.

Now, we will proceed to the design of the corrective "control component" p*(x) such that u = p₀ + p* stabilizes the class of systems described by (5.92) and satisfying (5.95). The corrective control term is designed based on a technique following the nominal Lyapunov function V₀, which justifies the name Lyapunov redesign method.

Consider the same Lyapunov function V₀ that guarantees the asymptotic stability of the nominal closed-loop system, but now consider the time derivative of V₀ along the solutions of the full system (5.92). We have

where

which is a known function. By taking bounds we obtain

The second term of the right-hand side of (5.97) can be made zero if is selected as

Each component of the corrective control vector p*(x) is selected to be of the form p*(x) = , where the sign of p*(x) depends on the sign of _i(x) and, in fact, changes as _i(x)changes sign.

By substituting (5.98) in (5.97) we obtain the desired "stability" property

which implies that the closed-loop system is asymptotically stable.

The augmented control law u = p₀(x) + p*(x) is discontinuous since each element is discontinuous at _i(x) = 0. Moreover, the discontinuity jump can be of large magnitude if the uncertainty bound is large. As discussed earlier, discontinuities in the control law can cause chattering, therefore it is desirable to smooth the discontinuity and at the same time retain to some degree the nice stability properties of the original discontinuous control law.

This can be achieved by replacing (5.98) with

where ε > 0 is a small design constant. Note that as ε approaches zero, the tanh function converges to the discontinuous sgn( _i) function.

By substituting (5.99) in (5.97) we obtain

Using Lemma A.5.1 (see p. 397),

where = 0.2785. Since α₃ is a class_∞ function (strictly increasing), for any uniformly bounded function and for any r > 0, there exists an ε (sufficiently small), such that for x outside a region D_ε = {x V(x) ≤ r}. Therefore, the trajectory is convergent to the invariant set D_ε.

The following example illustrates the use of the Lyapunov redesign method.

■ EXAMPLE 5.9

Consider the nonlinear system

where is unknown but is known to satisfy the inequality

for some known bound . This second-order model represents a jet engine compression system with no-stall [139], which is based on the Galerkin approximation or the nonlinear PDE model [176]. The state x₁ corresponds to the mass flow and x₂ is the pressure rise.

The first step is to design the nominal control law u = p₀(x) for the case of = 0. This can be accomplished by feedback linearization (note that it can also be accomplished by the backstepping method). Consider the change of coordinates z = T(x) where

The dynamics in the z-coordinates are described by

where _z(z) = (x)_x=T -1_(z). A stabilizing nominal controller is given by

A nominal Lyapunov function associated with the above nominal controller is given by

whose time derivative is given by

Since by eqn. (5.96) (z) = 2(z₁ + z₂), the corrective feedback control law obtained using the Lyapunov redesign method is given by

where is the assumed bound on . The above control law can be made continuous using the following approximation