Adaptive control deals with systems where some of the parameters are unknown or slowly time-varying. The basic idea behind adaptive control is to estimate the unknown parameters online using parameter estimation methods (such as those presented in Chapter 4), and then to use the estimated parameters, in place of the unknown ones, in the feedback control law. Most of the research in adaptive control has been developed for linear models, even though in the last decade or so there has been a lot of activity on adaptive nonlinear control as well. Even in the case of adaptive control applied to linear systems, the resulting control law is nonlinear. This is due to the parameter update laws, which render the feedback controller nonlinear.

There are two strategies for combining the control law and the parameter estimation algorithm. In the first strategy, referred to as indirect adaptive control, the parameter estimation algorithm is used to estimate the unknown parameters of the plant. Based on these parameter estimates, the control law is computed by treating the estimates as if they were the true parameters, based on the certainty equivalence principle [11]. In the second strategy, referred to as direct adaptive control, the parameter estimator is used to estimate directly the unknown controller parameters.

It is interesting to note the similarities and difference between so called robust control laws and adaptive control laws. The robust approaches, which were discussed in Section 5.4, treat the uncertainty as an unknown box where the only information available are some bounds. The robust control law is obtained based on these bounds, and in fact is designed to stabilizes the system for any uncertainty within the assumed bounds. As a result, the robust control law tends to be conservative and it may lead to large control input signals or control saturation. On the other hand, adaptive control assumes a special structure for the uncertainty where the nonlinearities are known but the parameters are unknown. In contrast to robust control, in adaptive control the objective is to try to estimate the uncertain (or time-varying) parameters to reduce the level of uncertainty.

In the next chapter, we will start investigating the adaptive approximation control approach where the uncertainty also includes nonlinearities that are estimated online. Hence, adaptive approximation based control can be viewed as an expansion of the adaptive control methodology where instead of having simply unknown parameters we have unknown nonlinearities.

Adaptive control is a well-established methodology in the design of feedback control systems. The first practical attempts to design adaptive feedback control systems go back as far as the 1950s, in connection with the design of autopilots [295]. Stability analysis of adaptive control for linear systems started in the mid-1960s [196] and culminated in 1980 with the complete stability proof for linear systems [69, 177, 180]. The first stability results assumed that the only uncertainty in the system was due to unknown parameters; i.e., no disturbances, measurement noise, nor any other form of uncertainty. In the 1980s, adaptive control research focused on robust adaptive control for linear systems, which dealt with modifications to the adaptive algorithms and the control law in order to address some types of uncertainties [119]. In the 1990s, most of the effort in adaptive control focused on adaptive control of nonlinear systems with some elegant results [139].

To illustrate the use of the adaptive control methodology we consider below two examples of adaptive nonlinear control.

■ EXAMPLE 5.11

In this example we consider the feedback linearization problem of Section 5.2 with unknown parameters. Consider the n-th order model

where θ₁, θ₂, θ₃ are unknown, constant parameters and ƒ₁ andƒ₂ are known functions. The objective is to design an adaptive controller such that y(t) = x₁(t) tracks a desired signal y_d (t). Let e = y – y_dbe the tracking error.

If θ₁, θ₂, θ₃were known and θ₃ ≠ 0 then the control law

would result in the following tracking error dynamics:

The coefficients {λ₁, λ₂, ... λ_n-1} would be selected such that the characteristic polynomial

has all its roots in the left-halt complex plane.

Since θ₁, θ₂, θ₃ are unknown, we replace them in the control law by their corresponding estimates , where it is assumed for the time being that for all t ≥ 0. The adaptive control law is given by

which yields the following tracking error dynamics

where . If we let then the tracking error dynamics can be written as

where A₀, B₀ is the canonical form pair given by

Since A₀is a stability matrix, there exists a positive definite matrix P such that

We choose the Lyapunov function

whose time derivative along the solution of the tracking error dynamics is given by

Therefore, we select the adaptive laws as follows:

Clearly, this results in

■ EXAMPLE 5.12

In this example we consider the backstepping control procedure of Section 5.3 for the case where there is an unknown parameter. Consider the second-order system

where θ is an unknown parameter and ƒ is a known function. The parameter estimate for θ is defined as is the parameter estimation error. The objective is to design an adaptive nonlinear tracking controller such that y = x₁ tracks a desired signal y_d(t).

We define the change of coordinates

where α is defined as

Now, consider the time derivative of the Lyapunov function candidate

where γ > 0 is the adaptive gain. We have

Based on the above derivative of the Lyapunov function, we select the update law for as follows:

Hence,

which implies that z1, z2 and are uniformly bounded and z1, z2both converge to zero.

Δ