自适应控制与非线性控制-国外讲义.pdf-资料库

95c84def-0831-4aed-a181-a81ae41d1215.pdf-第1页.png

第1页 / 共125页

95c84def-0831-4aed-a181-a81ae41d1215.pdf-第2页.png

第2页 / 共125页

95c84def-0831-4aed-a181-a81ae41d1215.pdf-第3页.png

第3页 / 共125页

95c84def-0831-4aed-a181-a81ae41d1215.pdf-第4页.png

第4页 / 共125页

95c84def-0831-4aed-a181-a81ae41d1215.pdf-第5页.png

第5页 / 共125页

95c84def-0831-4aed-a181-a81ae41d1215.pdf-第6页.png

第6页 / 共125页

95c84def-0831-4aed-a181-a81ae41d1215.pdf-第7页.png

第7页 / 共125页

95c84def-0831-4aed-a181-a81ae41d1215.pdf-第8页.png

第8页 / 共125页

ECE 517: Nonlinear and Adaptive Control Fall 2013 Lecture Notes Daniel Liberzon November 20, 2013

2 Disclaimers DANIEL LIBERZON Don’t print future lectures in advance as the material is always in the process of being updated. You can consider the material here stable 2 days after it was presented in class. These lecture notes are posted for class use only. This is a very rough draft which contains many errors. I don’t always give proper references to sources from which results are taken. A lack of reference does not mean that the result is original. In fact, all results presented in these notes (with possible exception of some simple examples) were borrowed from the literature and are not mine.

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 1.2 Motivating example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Course logistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6 9 2 Weak Lyapunov functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 2.2 2.3 LaSalle and Barbalat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Connection with observability . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Back to the adaptive control example . . . . . . . . . . . . . . . . . . . . . . 15 3 Minimum-phase systems and universal regulators . . . . . . . . . . . . . . . . . . . . 17 3.1 Universal regulators for scalar plants . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.1 3.1.2 3.1.3 The case b > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 General case: non-existence results . . . . . . . . . . . . . . . . . . . 20 Nussbaum gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Relative degree and minimum phase . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.1 Stabilization of nonlinear minimum-phase systems . . . . . . . . . . 27 3.3 Universal regulators for higher-dimensional plants . . . . . . . . . . . . . . . 29 4 Lyapunov-based design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.1 Control Lyapunov functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.1 Sontag’s universal formula . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Back to the adaptive control example . . . . . . . . . . . . . . . . . . . . . . 38 5 Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.1 5.2 Integrator backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Adaptive integrator backstepping . . . . . . . . . . . . . . . . . . . . . . . . . 44 6 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.1 Gradient method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3

4 DANIEL LIBERZON 6.2 6.3 6.4 6.5 Parameter estimation: stable case . . . . . . . . . . . . . . . . . . . . . . . . 49 Unstable case: adaptive laws with normalization . . . . . . . . . . . . . . . . 54 6.3.1 6.3.2 6.3.3 6.3.4 Linear plant parameterizations (parametric models) . . . . . . . . . 57 Gradient method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Least squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Suﬃciently rich signals and parameter identiﬁcation . . . . . . . . . . . . . . 66 Case study: model reference adaptive control . . . . . . . . . . . . . . . . . . 70 6.5.1 6.5.2 Direct MRAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Indirect MRAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 7 Input-to-state stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 7.1 Weakness of certainty equivalence . . . . . . . . . . . . . . . . . . . . . . . . 78 7.2 Input-to-state stability and stabilization . . . . . . . . . . . . . . . . . . . . . 80 7.2.1 ISS backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.3 Adaptive ISS controller design . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.3.1 7.3.2 Adaptive ISS backstepping . . . . . . . . . . . . . . . . . . . . . . . 90 Modular design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 8 Stability of slowly time-varying systems . . . . . . . . . . . . . . . . . . . . . . . . . 94 8.1 8.2 8.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Application to adaptive stabilization . . . . . . . . . . . . . . . . . . . . . . . 98 Detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 9 Switching adaptive control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9.1 The supervisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 9.1.1 9.1.2 9.1.3 Multi-estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Monitoring signal generator . . . . . . . . . . . . . . . . . . . . . . . 107 Switching logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 9.2 9.3 Example: linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Modular design objectives and analysis steps . . . . . . . . . . . . . . . . . . 112 9.3.1 9.3.2 Achieving detectability . . . . . . . . . . . . . . . . . . . . . . . . . 115 Achieving bounded error gain and non-destabilization . . . . . . . . 117 10 Singular perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

NONLINEAR AND ADAPTIVE CONTROL 5 10.1 Unmodeled dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 10.2 Singular perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 10.3 Direct MRAC with unmodeled dynamics . . . . . . . . . . . . . . . . . . . . 123 11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6 1 Introduction DANIEL LIBERZON The meaning of “nonlinear” should be clear, even if you only studied linear systems so far (by exclusion). The meaning of “adaptive” is less clear and takes longer to explain. From www.webster.com: Adaptive: showing or having a capacity for or tendency toward adaptation. Adaptation: the act or process of adapting. Adapt: to become adapted. Perhaps it’s easier to ﬁrst explain the class of problems it studies: modeling uncertainty. This includes (but is not limited to) the presence of unknown parameters in the model of the plant. There are many specialized techniques in adaptive control, and details of analysis and design tend to be challenging. We’ll try to extract fundamental concepts and ideas, of interest not only in adaptive control. The presentation of adaptive control results will mostly be at the level of examples, not general theory. The pattern will be: general concept in nonlinear systems/control, followed by its application in adaptive control. Or, even better: a motivating example/problem in adaptive control, then the general treatment of the concept or technique, then back to its adaptive application. Overall, the course is designed to provide an introduction to further studies both in nonlinear systems and control and in adaptive control. 1.1 Motivating example Example 1 Consider the scalar system ˙x = θx + u where x is state, u is control, and θ is an unknown ﬁxed parameter. A word on notation: There’s no consistent notation in adaptive control literature for the true value of the unknown parameters. When there is only one parameter, θ is a fairly standard symbol. Sometimes it’s denoted as θ∗ (to further emphasize that it is the actual value of θ). In other sources, p∗ is used. When there are several unknown parameters, they are either combined into a vector (θ, θ∗, p∗, etc.) or written individually using diﬀerent letters such as a, b, and so on. Estimates of the unknown parameters commonly have hats over them, e.g., ˆθ, and estimation errors commonly have tildas over them, e.g., ˜θ = ˆθ − θ. Goal: regulation, i.e., make x(t) → 0 as t → ∞. If θ < 0, then u ≡ 0 works.

NONLINEAR AND ADAPTIVE CONTROL If θ > 0 but is known, then the feedback law u = −(θ + 1)x gives ˙x = −x ⇒ x → 0. (Instead of +1 can use any other positive number.) But if (as is the case of interest) θ is unknown, this u is not implementable. −→ Even in this simplest possible example, it’s not obvious what to do. Adaptive control law: ˙ˆθ = x2 u = −(ˆθ + 1)x Here (1) is the tuning law, it “tunes” the feedback gain. Closed-loop system: ˙x = (θ − ˆθ − 1)x ˙ˆθ = x2 7 (1) (2) Intuition: the growth of ˆθ dominates the linear growth of x, and eventually the feedback gain ˆθ + 1 becomes large enough to overcome the uncertainty and stabilize the system. Analysis: let’s try to ﬁnd a Lyapunov function. If we take V := x2 2 then its derivative along the closed-loop system is ˙V = (θ − ˆθ − 1)x2 and this is not guaranteed to be negative. Besides, V should be a function of both states of the closed-loop system, x and ˆθ. Actually, with the above V we can still prove stability, although analysis is more intricate. We’ll see this later. Let’s take V (x, ˆθ) := x2 2 + (ˆθ − θ)2 2 (3) The choice of the second term reﬂects the fact that in principle, we want to have an asymptotically stable equilibrium at x = 0, ˆθ = θ. In other words, we can think of ˆθ as an estimate of θ. However, the control objective doesn’t explicitly require that ˆθ → θ.

8 DANIEL LIBERZON With V given by (3), we get ˙V = (θ − ˆθ − 1)x2 + (ˆθ − θ)x2 = −x2 (4) Is this enough to prove that x(t) → 0? Recall: Theorem 1 (Lyapunov) Let V be a positive deﬁnite C1 function. If its derivative along solutions satisﬁes (5) ˙V ≤ 0 everywhere, then the system is stable. If ˙V < 0 (6) everywhere (except at the equilibrium being studied), then the system is asymptotically stable. If in the latter case V is also radially unbounded (i.e., V → ∞ as the state approaches ∞ along any direction), then the system is globally asymptotically stable. From (4) we certainly have (5), hence we have stability (in the sense of Lyapunov). In particular, both x and ˆθ remain bounded for all time (by a constant depending on initial conditions). On the other hand, we don’t have (6) because ˙V = 0 for (x, ˆθ) with x = 0 and ˆθ arbitrary. It seems plausible that at least convergence of x to 0 should follow from (4). This is indeed true, but proving this requires knowing a precise result about weak (nonstrictly decreasing) Lyapunov functions. We will learn/review such results and will then ﬁnish the example. Some observations from the above example: • Even though the plant is linear, the control is nonlinear (because of the square terms). To analyze the closed-loop system, we need nonlinear analysis tools. • The control law is dynamic as it incorporates the tuning equation for ˆθ. equation “learns” the unknown value of θ, providing estimates of θ. Intuitively, this • The standard Lyapunov stability theorem is not enough, and we need to work with a weak Lyapunov function. As we will see, this is typical in adaptive control, because the esti- mates (ˆθ) might not converge to the actual parameter values (θ). We will discuss techniques for making the parameter estimates converge. However, even without this we can achieve regulation of the state x to 0 (i.e., have convergence of those variables that we care about). So, what is adaptive control? We can think of the tuning law in the above example as an adaptation block in the overall system—see ﬁgure. (The diagram is a bit more general than what we had in the example.)

资料库

自适应控制与非线性控制-国外讲义.pdf

相关推荐

课程资源

热门标签

最新资料