Robust Nonlinear Control Design

The main purpose of every feedback loop, created by nature or designed by engineers, is to reduce the effect of uncertainty on vital system functions. Indeed, feedback as a design paradigm for dynamic systems has the potential to counteract uncertainty. However, dynamic systems with feedback (closed-loop systems) are often more complex than systems without feedback (open-loop systems), and the design of feedback controllers involves certain risks. Feedback can be used for stabilization, but inappropriately designed feedback controllers may reduce, rather than enlarge, regions of stability. A feedback controller that performs well on a linearized model may in fact drastically reduce the stability region of the actual nonlinear system.

Significant advances in the theory of linear robust control in recent years have led to powerful new tools for the design and analysis of control systems. A popular paradigm for such theory is depicted in Figure 3.1, which shows the interconnection of three system blocks G, K, and Δ. The plant G relates a control input u and a disturbance input v to a measurement output y and a penalized output z. The control input u is generated from the measured output y by the controller K. All uncertainty is located in the block Δ which generates the disturbance input v from the penalized output z. The plant G, which is assumed to be linear and precisely known, may incorporate some nominal plant as well as frequency-dependent weights on the uncertainty Δ (for this reason G is sometimes called a generalized plant). Once G is determined, the robust stabilization problem is to construct a controller K which guarantees closed-loop stability for all systems Δ belonging to a given family of admissible (possibly nonlinear) uncertain systems.

We have just established that the existence of a robust control Lyapunov function (rclf) is equivalent to robust stabilizability. This result lays the foundation for the design methods to be developed in the remainder of this book. The design tasks facing us are: Methods for constructing rclf’s will be presented in Chapter 5. In this chapter we address the second task, which is much easier than the first.

Thus far our path has brought us to a base camp at which the design of a robustly stabilizing control law appears deceptively simple. All we need to do is find a robust control Lyapunov function (rclf); the remaining task of selecting a control law to make the Lyapunov derivative negative is straightforward. As we demonstrated in Chapter 4, explicit formulas are available for control laws which are optimal with respect to meaningful cost functionals.

Our robust stabilization results thus far were obtained under the assumption of perfect state feedback. In particular, in Chapter 5 we gave a constructive proof of the fact that every nonlinear system in strict feedback form admits a robust control Lyapunov function (rclf) and is therefore robustly stabilizable with perfect state measurements. In this chapter, we show that such systems remain robustly stabilizable when the state measurement is corrupted by disturbances (such as sensor noise). To be precise, we show that strict feedback systems can be made (globally) input-to-state stable (cf. Definition 3.3) with respect to additive state measurement disturbances.

The controllers we have designed thus far have been static (memoryless) and have employed state feedback either perfect or corrupted by additive measurement disturbances. Potential advantages of dynamic over static feedback have been explored in the nonlinear control literature, and several paradigms for dynamic feedback design have been introduced. Among them, paradigms for dynamic feedback linearization and disturbance decoupling [60, 111] are being developed elsewhere and will not be pursued here.

Thus far we have represented system uncertainty by a disturbance input w allowed to have an arbitrarily fast time variation. Its only constraint was the pointwise condition w ∈ W where W was some known set possibly depending on the state x and control u. We now address a more specific situation in which our system contains some uncertain nonlinearity φ(x). Suppose that all we know about φ is a set-valued map Φ(x) such that φ(x) ∈ Φ(x) for all x ∈ χ. We could assign w ≔ φ(x) and W(x) ≔ Φ(x) and proceed as in Chapters 3–6, but we would be throwing away a crucial piece of information about the uncertainty φ, namely, that φ(x) does not explicitly depend on time t. Our goal in this chapter is to illustrate how we can take advantage of this additional information to design less conservative robust controllers.