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Chapter 8 Adaptive Control

Chapter 8 Adaptive Control. Many dynamic systems to be controlled have constant or slowly-varying uncertain parameters.

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Chapter 8 Adaptive Control

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  1. Chapter 8 Adaptive Control

  2. Many dynamic systems to be controlled have constant or slowly-varying uncertain parameters. The Basic Idea in Adaptive Control To estimate the uncertain plant parameters (or, equivalently, the corresponding controller parameters) on-line based on the measured system signals,and use the estimated parameters in the control input computation.

  3. 8.1 Basic Concepts in Adaptive Control

  4. 8.1.1 Why Adaptive Control？ -The basic objective of adaptive control is to maintain consistent performance of a system in the presence of uncertainty or unknown variation in plant parameters. -Since such parameter uncertainty or variation occurs in many practical problems, adaptive control is useful in many industrial contexts.

  5. 8.1.2 What Is Adaptive control？ -An adaptive controller differs from an ordinary controller in that: 1. Controller parameters are variable. 2. A mechanism for adjusting these parameters online based on the signals in the system.

  6. Two main approaches for constructing adaptive controllers: -Model-reference adaptive control (MRAC) method . -Self-tuning (ST) method.

  7. MODEL-REFERENCE ADAPTIVE CONTROL (MRAC) -Composed of four parts: A plant containing unknown parameters. A reference model for compactly specifying the desired output of the control system. A feedback control law containing adjustable parameters. An adaptation mechanism for updating the adjustable parameters.

  8. Figure 8.3 ：A model-reference adaptive control system

  9. Plant The plant is assumed to have a known structure, although the parameters are unknown.

  10. Figure 8.3 ：A model-reference adaptive control system

  11. Reference model -Used to specify the ideal response of the adaptive control system to the external command that the adaptation mechanism should seek in adjusting the parameters. -The choice of the reference model is part of the adaptive control system design.

  12. Two Requirements on the Choice or Reference Model -Reflect the performance specification in the control tasks such as rise time, settling time, overshoot or frequency domain characteristics. -The ideal behavior should be achievable for the adaptive control system.

  13. Figure 8.3 ：A model-reference adaptive control system

  14. Controller • -The controller is usually parameterized by a number of adjustable parameters. • -The controller should have perfect tracking capacity. • -When the plant parameters are exactly known, the corresponding controller parameters should make the plant output identical to that of the reference model. When the plant parameters are not known, the adaptation mechanism will adjust the controller parameters so that perfect tracking is asymptotically achieved.

  15. Figure 8.3 ：A model-reference adaptive control system

  16. Adaptation -Main issue in adaptation design: To synthesize an adaptation mechanism which will guarantee that the control system remains stable and the tracking error converges to zero as the parameters are varied.

  17. Example 8.1：MRAC CONTROL OF AN UNKNOWN MASS 1. a motor force u, the plant dynamics being 2. Reference model

  18. 3. Control law: if the mass m is known exactly, Assume the mass is not known exactly 4. Adaptation update

  19. 5. Stability and convergence analysis Consider the following Lyapunov function candidate, -Using Barbalat’s Lemma, one can easily show that s converges to zero. -The convergence of s to zero implies that of the position tracking error and the velocity tracking error .

  20. Figure 8.4：Tracking performance and Parameter Estimation for an Unknown Mass, r(t)=0

  21. Figure 8.5：Tracking Performance and Parameter Estimation for an Unknown Mass, r(t)=sin4t

  22. (8.15) Lemma 8.1：Consider two signals e and  related by the following dynamic equation where e(t) is a scalar output signal, H(p) is a strictly positive real transfer function, k is an unknown constant with known sign, (t) is a m1 vector function of time, v(t) is a measurable m1 vector.

  23. (8.16) If the vector  varies according to  being a positive constant, then e(t) and (t) are globally bounded. if v is bounded,then e(t)0 as t

  24. SELF-TUNING CONTROLLERS (STC) • Self-tuning Controller • -If the plant parameters are not known, it is intuitively • reasonable to replace them by their estimated values, • as provided by a parameter estimator. • -A controller obtained by coupling a controller • with an online (recursive) parameter estimator is • called a self-tuning controller.

  25. Figure 8.6：A self-tuning controller

  26. Certainty Equivalence • -The controller parameters are computed from the estimates of the plant parameters as if they were the true plant parameters.

  27. Indirect & Direct Adaptive Control -Indirect adaptive control: Estimates the plant parameters then computes the controller parameters. -Direct adaptive control: Re-parameterize the plant model using controller parameters (which are also unknown, of course), and then use standard estimation techniques on such a model.

  28. However, considerable noise may be in the measurement ,and the acceleration may be close to zero. Example 8.2：Self-tuning control of the unknown mass The simplest estimation

  29. A better approach is to estimate the parameter using a least-squares approach, i.e., choosing the estimate in such a way that the total prediction error is minimal, the prediction error e is defined as

  30. The total error minimization can potentially average out the effects of measurement noise.

  31. P(t): estimation gain, its update is obtained by Then, the differentiation of Equation (8.11) can be written as

  32. leads to

  33. COMPARISONS BETWEEN MRAC AND ST METHODS • MRAC control and ST control arise from different perspectives: • the parameters in MRAC systems being updated to minimize the tracking errors between the plant output and reference model output. • the parameters in ST systems being updated to minimize the data-fitting error in input-output measurements.

  34. -ST controllers are more flexible because of the possibility of coupling various controllers with various estimators (i.e., the separation of control and estimation). -Stability and convergence of ST controllers are generally difficult to guarantee. -MRAC system: the stability and tracking error convergence are usually guaranteed.

  35. 8.1.3 How To Design Adaptive Controllers？ The design of an adaptive controller usually involves the following three steps： • Choose a control law containing variable parameters • Choose an adaptation law for adjusting those parameters • Analyze the convergence properties of the resulting control system

  36. 8.2 Adaptive Control of First-Order Systems -An illustrative example of how to design and analyze an adaptive control system.

  37. -The braking of an automobile, the discharge of an electronic flash. -The flow of fluid from a tank can be approximately represented by a first-order differential equation: where y is the plant output, u is its input, and and are constant plant parameters.

  38. Problem specification -The plant parameters and are assumed to be unknown. -The desired performance is specified by a first-order reference model where and are constant parameter, and r(t) is a bounded external reference signal. -Ensuring the stability of the reference model requires - is chosen >0

  39. -The reference model can be represented by its transfer function M where with p being the Laplace variable. Note that M is a SPR function.

  40. The Objective of the Adaptive Control Design -To formulate a control law, and an adaptation law, such that the resulting model following error y(t)-ymasymptotically converges to zero. -Required Assumption: the sign of the parameter b is known. This is a quite mild condition, which is often the simple physical knowledge that braking slows down the car.

  41. Choice of Control Law -The first step in the adaptive controller design: Choosing the control law to be where and are variable feedback gains. -With this control law, the closed-loop dynamics are

  42. -The reason for the choice of control law: it allows the possibility of perfect model matching. -If the plant parameters were known, the following ideal values for control parameters would lead to the closed-loop dynamics which is identical to the reference model dynamics, and yields zero tracking error.

  43. In our adaptive control problem, since and are unknown, the adaptation law will base on the tracking error to make y tend to asymptotically. The structure of the adaptive controller is illustrated in Figure 8.8.

  44. Choice of Adaptation Law -Choosing parameters for the adaptation law Let denote the tracking error. - Parameter errors: the difference between the controller parameter provided by the adaptation law and the ideal parameters.

  45. The dynamics of tracking error: This can be conveniently represented as with p denoting the Laplace operator.

  46. -Relation (8.27) between the parameter errors and tracking error is in the familiar form given by (8.15). Thus, Lemma 8.1 suggests the following adaptation law with  being a positive constant representing the adaptation gain. From (8.28), it is seen that determines the direction of the search for the proper controller parameters.

  47. Tracking Convergence Analysis With the control law and adaptation law chosen above, the Lyapunov function candidate can be easily shown to have the following derivative along system trajectories Thus, the adaptive control system is globally stable, i. e., the signal e, are bounded.

  48. -The globally asymptotic convergence of the tracking error is guaranteed by Barbalat’s lemma, because the boundedness of implies the boundedness of (according to (8.26)) and therefore the uniform continuity of

  49. Example 8.3：A first-order plant Consider the control of the unstable plantusing the previously designed adaptive controller. The plant parameters , are assumed to be unknown to the adaptive controller. The reference model is chosen to be i. e., -Choosing  = 2. Choosing initial values of to be 0, indicating no a priori knowledge. Setting

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