ABSTRACT
A standard method for achieving robustness to state-space uncertainty is Lyapunov redesign; see [12]. In this method, one begins with a Lyapunov function for a nominal closed-loop system and then uses this Lyapunov function to construct a controller which guarantees robustness to given uncertainties. To illustrate this method, we consider the system,
x˙ = F(x)+G(x)u+Δ(x, t), (49.1) where F andG are known functions comprising the nominal system andΔ is an uncertain function known only to lie within some bounds. For example, we may know a function ρ(x) so that | Δ(x, t) |≤ ρ(x). A more general uncertainty Δ would also depend on the control variable u, but for simplicity we do not consider such uncertainty here. We assume that the nominal system is stabilizable, that is, that some state feedback unom(x) exists so that the nominal closed-loop system,
x˙ = F(x)+G(x)unom(x), (49.2) has a globally asymptotically stable equilibrium at x = 0. We also assume knowledge of a Lyapunov function V for this system so that
∇V (x) [F(x)+G(x)unom(x)] < 0 (49.3) whenever x = 0. Our task is to design an additional robustifying feedback urob(x) so that the composite feedback u = unom + urob robustly stabilizes the system (Equation 49.1), that is, guarantees stability for every admissible uncertaintyΔ. It suffices that the derivative ofV along closed-loop trajectories is negative for all such uncertainties. We compute this derivative as follows:
V˙ = ∇V (x) [F(x)+G(x)unom(x)]+∇V (x) [G(x)urob(x)+Δ(x, t)] (49.4) Can wemake this derivative negative by some choice of urob(x)? Recall from Equation 49.3 that the first
of the two terms in Equation 49.4 is negative; it remains to examine the second of these terms. For those values of x for which the coefficient ∇V (x) ·G(x) of the control urob(x) is nonzero, we can always choose the value of urob(x) large enough to overcome any finite bound on the uncertainty Δ and thus make the second term in Equation 49.4 negative. The only problems occur on the set where ∇V (x) ·G(x) = 0, because on this set
V˙ = ∇V (x) · F(x)+∇V (x) ·Δ(x, t) (49.5) regardless of our choice for the control. Thus to guarantee the negativity of V˙ , the uncertainty Δ must satisfy
∇V (x) · F(x)+∇V (x) ·Δ(x, t) ≤ 0 (49.6) at all points where ∇V (x) ·G(x) = 0. This inequality constraint on the uncertainty Δ is necessary for the Lyapunov redesign method to succeed. Unfortunately, there are two undesirable aspects of this necessary condition. First, the allowable size of the uncertainty Δ is dictated by F and V and can thus be severely restricted. Second, this inequality (Equation 49.6) cannot be checked apriori on the system (Equation 49.1) because it depends on the choice for V .
