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  1. Chapter 5 Adaptive Pole Placement Control

  2. Table of continents Introduction Simple APPC Schemes: Without Normalization APPC Schemes: Polynomial Approach APPC Schemes: State-Space Approach Adaptive Linear Quadratic Control (ALQC) Stabilizability Issues and Modified APPC Robust APPC Schemes Case Study

  3. Introduction In this chapter, pole placement schemes are introduced, where we don’t need the assumption of minimum phase by considering control schemes that change the poles of the plant and do not involve unstable zero-pole cancellations. These schemes applicable to both minimum- and nonminimum-phase LTI plants. The combination of a pole placement control law with a parameter estimator or an adaptive law leads to an adaptive pole placement control (APPC) scheme that can be used to control a wide class of LTI plants with unknown parameters.

  4. Introduction The APPC schemes may be divided into two classes: 1-Indirect APPCschemes, where the adaptive law generates online estimates of the coefficients of the plant transfer function which are then used to calculate the parameters of the pole placement control law by solving a certain algebraic equation. 2-Direct APPC, where the parameters of the pole placement control law are generated directly by an adaptive law without any intermediate calculations that involve estimates of the plant parameters.

  5. Introduction The direct APPC schemes are restricted to scalar plants and to special classes of plants where the desired parameters of the pole placement controller can be expressed in the form of the linear or bilinear parametric models. The indirect APPC schemes, on the other hand, are easy to design and are applicable to a wide class of LTI plants that are not required to be minimum-phase or stable. The main drawback of indirect APPC is the possible loss of stabilizability of the estimated plant based on which the calculation of the controller parameters is performed. This drawback can be eliminated by modifying the indirect APPC schemes at the expense of adding more complexity.

  6. Introduction Because of its flexibility in choosing the controller design methodology (state feedback, compensator design, linear quadratic, etc.) and adaptive law, indirect APPC is the most general class of adaptive control schemes. This class also includes indirect MRAC as a special case. Indirect APPC schemes have also been known as self-tuning regulators in the literature of adaptive control to distinguish them from direct MRAC schemes.

  7. Simple APPC Schemes In this section we use several examples to illustrate the design and analysis of simple APPC schemes. Scalar Example: Adaptive Regulation Consider the scalar plant where a and b are unknown constants, and the sign of b is known. The control objective is to choose u so that the closed-loop pole is placed at , where is a given constant, y and u are bounded, and y(t) converges to zero as t∞. If a and b were known and b≠0, then the control law

  8. Simple APPC Schemes Scalar Example: Adaptive Regulation would lead to the closed-loop plant i.e., the control law, changes the pole of the plant from but preserves the zero structure. This is in contrast to MRC, where the zeros of the plant are canceled and replaced with new ones. Now we consider the case where a and b are unknown. As in the MRAC case, we use the CE approach to form APPC schemes as follows: We use the same control law as but replace the unknown controller parameter k* with its online estimate k.

  9. Simple APPC Schemes Scalar Example: Adaptive Regulation The estimate k may be generated in two different ways: 1-Direct, where k is generated by an adaptive law, 2-Indirect, where k is calculated from provided where are the online estimates of a and b, respectively. We consider each design approach separately.

  10. Simple APPC Schemes Scalar Example: Adaptive Regulation Direct adaptive regulation In this case the time-varying gain k in the control law is updated directly by an adaptive law. The adaptive law is developed as follows: B-SSPM or B-DPM

  11. Simple APPC Schemes Scalar Example: Adaptive Regulation Barbalat's lemma Summary: The signals boundedness and regulation of y(t)0is guaranteed. But, pole placement at is not guaranteed. To achieve such a pole placement result, we need to show that For parameter convergence, however, y is required to be persistently exciting (PE), which is in conflict with the objective of regulating y to zero. The conflict between PI and regulation or control is well known in adaptive control and cannot be avoided in general.

  12. Simple APPC Schemes Scalar Example: Adaptive Regulation Indirect adaptive regulation In this case the time-varying gain k in the control law is calculated by using the algebraic equation SSPM

  13. Simple APPC Schemes Scalar Example: Adaptive Regulation Since condition cannot be guaranteed, we modify the adaptive laws using the projection techniques as: The indirect adaptive pole placement scheme has the same stability properties as the direct one. In the direct case we are estimating only one parameter, whereas in the indirect case we are estimating two.

  14. Simple APPC Schemes Scalar Example: Adaptive Tracking Let us consider the same plant The control objective is choosing the plant input u so that the closed-loop pole is at tracks the reference signal where is a known constant. tracking error dynamics

  15. Simple APPC Schemes Scalar Example: Adaptive Tracking Control law The updating of k1 and k2 may be done directly, or indirectly Direct adaptive tracking In this approach we develop an adaptive law that updates k1 and k2 directly. B-SSPM

  16. Simple APPC Schemes Scalar Example: Adaptive Tracking by following the usual arguments as in last section direct adaptive proportional plus integral (API) controller

  17. Simple APPC Schemes Scalar Example: Adaptive Tracking The same approach may be repeated when is a known bounded signal with known . In this case the following adaptive control scheme may be used.

  18. Simple APPC Schemes Scalar Example: Adaptive Tracking Indirect adaptive tracking In this approach, we use the same control law as in the direct case, i.e., with calculated using the equations

  19. Simple APPC Schemes Scalar Example: Adaptive Tracking similar arguments

  20. APPC Schemes: Polynomial Approach Let us consider the SISO LTI plant where is proper and is a monic polynomial. The control objective is to choose the plant input so that the closed-loop poles are assigned to those of a given monic Hurwitz polynomial This polynomial referred to as the desired closed-loop characteristic polynomial, is chosen based on the closed-loop performance requirements. To meet the control objective, we make the following assumptions about the plant:

  21. APPC Schemes: Polynomial Approach Assumptions P1 and P2 allow Zp, Rp, to be non-Hurwitz, in contrast to the MRC case where Zp is required to be Hurwitz. So the MRC problem is a special case of the general pole placement problem. A*(s) restricted to have Zp as a factor. We can also extend the pole placement control (PPC) objective to include tracking, where yp is required to follow a certain class of reference signals ym, by using the internal model principle as follows: The reference signal is assumed to satisfy where known as the internal model of ym, is a known monic polynomial of degree q with all roots in Re[s]≤0 and with no repeated roots on the jw-axis.

  22. APPC Schemes: Polynomial Approach The internal model is assumed to satisfy For example and, according to P3, should not have s or as a factor. APPC scheme is based on three steps: 1- Develop a control law in the known parameter case. 2- Design an adaptive law to estimate the plant parameters online. The estimated plant parameters are then used to calculate the controller parameters at each time t. 3- Replacing the controller parameters in Step 1 with their online estimates.

  23. APPC Schemes: Polynomial Approach Step 1. PPC for known parameters We consider the control law where P(s), L(s) are polynomials (with L(s) monic) of degree q+n-1 , n-1, respectively, chosen to satisfy the polynomial equation solution Sylvester equation

  24. APPC Schemes: Polynomial Approach Sylvester equation

  25. APPC Schemes: Polynomial Approach Therefore, the control objective is met. control law

  26. APPC Schemes: Polynomial Approach Step 2. Estimation of plant polynomials Using the results of Chapter 2, for the plant Using the results of Chapter 2, for the plant and estimate the plant as

  27. APPC Schemes: Polynomial Approach Step 3. Adaptive control law are calculated. control law:

  28. APPC Schemes: Polynomial Approach Theorem: Assume that the estimated plant polynomials are strongly coprime at each time t. Then all the signals in the closed loop plant are uniformly bounded, and the tracking error converges to zero asymptotically with time.

  29. APPC Schemes: Polynomial Approach Example: Consider the plant Step 1. PPC for known parameters Let and

  30. APPC Schemes: Polynomial Approach Step 2. Estimation of plant parameters Step 3. Adaptive control law Using CE approach by replacing the unknown parameters in the control law with their online estimates, i.e.,

  31. APPC Schemes: State-Space Approach Consider the SISO LTI plant where Gp(s) is proper and Rp(s) is monic. The control objective is the same and the same assumptions PI, P2, P3 apply. In this section, we consider a state-space approach to meet the control objective. We first solve the problem assuming known plant parameters. We then use the CE approach to replace the unknown parameters in the control law with their estimates calculated at each time t based on the plant parameter estimates generated by an adaptive law. The steps followed are presented below.

  32. APPC Schemes: State-Space Approach Step 1. PPC for known parameters We start by the tracking error Filtering each side with where is an arbitrary monic Hurwitz polynomial of degree q. So the tracking problem has converted to the regulation problem of choosing up to regulate e1 to zero. state-space realization

  33. APPC Schemes: State-Space Approach Step 1. PPC for known parameters where and are the coefficient vectors of the polynomials and . We consider the feedback control law: where ê is the state of the Luenberger observer and are solutions to the polynomial equations

  34. APPC Schemes: State-Space Approach Step 1. PPC for known parameters Theorem: The PPC law guarantees that all signals in the closed-loop plant are bounded and e1 converges to zero exponentially fast.

  35. APPC Schemes: State-Space Approach Step 2. Estimation of plant parameters Using the same techniques as last section to estimate the plant parameters and generating by where

  36. APPC Schemes: State-Space Approach Using the CE approach Step 3. Adaptive control law where are the coefficient of the polynomials

  37. APPC Schemes: State-Space Approach Step 3. Adaptive control law where is the coefficient vector of and is calculated at each time t by solving the polynomial equation

  38. APPC Schemes: State-Space Approach Example: Consider the plant Step 1. PPC for known parameters

  39. APPC Schemes: State-Space Approach Example: The poles of the observer are chosen to be at s = — 10, — 10, Step 2. Estimation of plant parameters Independent of the choice of the control law, the same adaptive law used in last example is employed. Step 3. Adaptive control law Replacing the unknown plant parameters with their online estimates:

  40. APPC Schemes: State-Space Approach Example:

  41. APPC Schemes: State-Space Approach Theorem: Assume that the polynomials are strongly coprime at each time t. Then all the signals in the closed-loop APPC scheme are uniformly bounded, and the tracking error e1 converges to zero asymptotically with time.

  42. Adaptive Linear Quadratic Control (ALQC) Another method for PPC problem is using an optimization technique by minimizing a certain cost function that reflects the performance of the closed loop system. As we established before, the tracking problem can be converted to the regulation problem of the system The desired to meet the objective is chosen as the one that minimizes the quadratic cost where >0 is a weighting coefficient to be designed, penalizes the level of the control input signal.

  43. Adaptive Linear Quadratic Control (ALQC) The optimum control that minimizes J is where satisfies the algebraic equation Riccati equation The importance of the linear quadratic (LQ) control design is that the resulting closed-loop system has good robustness properties. As before, since the state e may not be available for measurement. Therefore, the control law is improved as (*)

  44. Adaptive Linear Quadratic Control (ALQC) Theorem: The LQ control law (*) guarantees that all the eigenvalues of are in R[s]<0, all signals in the closed-loop plant are bounded, and e1(t)0 exponentially fast. As before, we can use the CE approach to form the adaptive control law where are as defined in last section and generated using the same adaptive law and P(t) is calculated by solving the Riccati equation

  45. Adaptive Linear Quadratic Control (ALQC) Example: Let us consider the same plant as in last examples, i.e., The control objective is to choose up, so that the closed-loop poles are stable and yp tracks the reference signal ym =1. The problem is converted to a regulation problem by considering the tracking error equation where as before. The state-space representation of the tracking error equation is given by

  46. Adaptive Linear Quadratic Control (ALQC) The observer equation is as before, i.e., where is chosen so that the observer poles are equal to the roots of The control law, is given by where P satisfies the Riccati equation and  > 0 is a design parameter to be chosen.

  47. Adaptive Linear Quadratic Control (ALQC) Now replace the unknown a, b with their estimates generated by the mentioned adaptive law to form the adaptive control law:

  48. Adaptive Linear Quadratic Control (ALQC) Theorem: Assume that the polynomials are strongly coprime at each time t. Then the ALQC scheme described by guarantees that all signals in the closed-loop plant are bounded and the tracking error e1 converges to zero.

  49. Stabilizability Issues and Modified APPC The main common drawback of the APPC schemes is that the adaptive law cannot guarantee that the estimated plant parameters or polynomials satisfy the appropriate controllability or Stabilizability condition at each time. Loss of Stabilizability or controllability may lead to computational problems and instability. We first demonstrate a scalar example that the estimated plant could lose controllability. Then describe briefly some of the approaches described in the literature to deal with the Stabilizability problem. Simple Example Consider the first-order plant where b≠0 is an unknown constant.

  50. Stabilizability Issues and Modified APPC The control objective is to choose u such that y,uL, and y(t)0. If b were known, then the control law would meet the control objective exactly. When b is unknown, we use the CE control law where is the estimate of b at time t, generated online by an appropriate adaptive law,