Hybrid Schemes for Adaptive Control Strategies

The purpose of this chapter is to redesign the standard adaptive control schemes by using hybrid structure composed by Model Reference Adaptive Control (MRAC) or Adaptive Pole Placement Control (APPC) strategies, associated to Variable Structure (VS) schemes for achieving non-standard robust adaptive control strategies. The both control strategies is now on named VS-MRAC and VS-APPC. We start with the theoretical base of standard control strategies APPC and MRAC, discussing their structures, as how their parameters are identified by adaptive observers and their robustness properties for guaranteeing their stability. After that, we introduce the sliding mode control (variable structure) in each control scheme for simplifying their design procedure. These design procedure are based on stability analysis of each hybrid robust control scheme. With the definition of both hybrid control strategies, it is analyzed their behavior when controlling system plants with unmodeled disturbances and parameter variation. It is established how the adaptive laws compensates these unmodeled dynamics. Furthermore, by using simple systems examples it is realized a comparison study between the hybrid structures VS-APPC and VSMRAC and the standard schemes APPC and MRAC. As the hybrid structures use switching laws due to the sliding mode scheme, the effect of chattering is analyzed on the implementation and consequently effects on the digital control hardware where sampling times are limiting factor. For reducing these drawbacks it is also discussed possibilities which kind of modifications can employ. Finally, some practical considerations are discussed on an implementation on motor drive systems.


Introduction
The purpose of this chapter is to redesign the standard adaptive control schemes by using hybrid structure composed by Model Reference Adaptive Control (MRAC) or Adaptive Pole Placement Control (APPC) strategies, associated to Variable Structure (VS) schemes for achieving non-standard robust adaptive control strategies. The both control strategies is now on named VS-MRAC and VS-APPC. We start with the theoretical base of standard control strategies APPC and MRAC, discussing their structures, as how their parameters are identified by adaptive observers and their robustness properties for guaranteeing their stability. After that, we introduce the sliding mode control (variable structure) in each control scheme for simplifying their design procedure. These design procedure are based on stability analysis of each hybrid robust control scheme. With the definition of both hybrid control strategies, it is analyzed their behavior when controlling system plants with unmodeled disturbances and parameter variation. It is established how the adaptive laws compensates these unmodeled dynamics. Furthermore, by using simple systems examples it is realized a comparison study between the hybrid structures VS-APPC and VSMRAC and the standard schemes APPC and MRAC. As the hybrid structures use switching laws due to the sliding mode scheme, the effect of chattering is analyzed on the implementation and consequently effects on the digital control hardware where sampling times are limiting factor. For reducing these drawbacks it is also discussed possibilities which kind of modifications can employ. Finally, some practical considerations are discussed on an implementation on motor drive systems.

Variable Structure Model Reference Adaptive Controller (VS-MRAC)
The VS-MRAC was originally proposed in (Hsu et al., 1989) and extensively discussed in (Hsu et al., 1994). The main features of this control scheme are the robustness of parameters uncertainties and unmodeled disturbances, as well as good transitory response. Consider the following first order plant The control objective is to force () yt to asymptotically track the reference output signal, () m yt , by regulating 0 e to be zero, while keeping all the closed-loop signals uniformly bounded. The control law used for accomplished this is which is the same as used in traditional model reference adaptive control. However, instead of the integral adaptive laws for the controller parameters, switching laws are proposed in order to improve the system transient performance and its robustness.
which means that our control objective is achieved, i.e., the closed-loop system behaves like the open-loop reference model. Consequently, the control law equation can be rewritten as ** 2 1 y ur θθ + = .
Analyzing (1) and (2) in the time domain, we get Adding and subtracting terms related to the ideal control parameters in (4), we have in which terms 1 θ and 2 θ are deviations of ideal controller parameters 1 θ and 2 θ .
Substituting the resulting equation (11) in (7), we can rewrite this equation as which results in From (6), the model input r can be defined as Therefore, using (11) and (15) in (8), we get Finally, comparing (14) and (16) due to the condition (5), we have the desired controller parameters www.intechopen.com (18) The above desired controller parameters assure that plant output converges to its reference model, because p b and p a are known. This design criteria is named as The Matching Conditions. However, our interests are concerned with unknown plant parameters or with known plant parameters with uncertainties, which require the use of adaptive laws for adjusting controller parameters. Derivating the output error equation given in (3), and using the condition (5), with equations (8), (16) and (19), we get which can be rearranged as Thus, Now, consider the Lyapunov function candidate given by 2 00 and its respective first time derivative 00 0 () Ve ee = . (24) By substituting (22) in (24), we obtain the following equation that can be rewritten as Using the switching laws, we obtain, If the conditions which guarantees that 0 0 e = is a globally asymptotically stable (GAS) equilibrium point, because (30) is a negative definite function.

Variable Structure Adaptive Pole Placement Control (VS-APPC)
As the VS-MRAC, the VS-APPC is the hybrid control structure obtained from the association of Pole Placement Control (PPC) together with Variable Structure (VS). Therefore, the theoretical development of this section starts from PPC control scheme and then we introduce the VS concepts for achieving the proposed VS-APPC. Considering the single input/single output (SISO) LTI plant there are, as plant parameters, 2n elements, which are the coefficients of the numerator and denominator of transfer function () Gs . We can define the vector * θ as From this, the following constraints must be observed:

S1. ()
Rs is a monic polynomial whose degree n is known.
Assumptions (S1) and (S2) allow () Zs , () Rs to be non-Hurwitz in contrast to the MRC (Model Reference Control) case, where () Zs is required to be Hurwitz.
We can also extend the PPC scheme for including the tracking objective, where output y is required to follow a certain class of reference signals r , by using the internal model principle (Ioannou & Sun, 1996). The uniformly bounded reference signal is assumed to satisfy where () m Qs , the internal model of r , is a known monic polynomial of degree q with nonrepeated roots on the jω-axis and satisfies
Considering the control law given by and has order 21 nq +−. The objective now is chosen () is satisfied for a given monic Hurwitz polynomial * () As of degree 21 nq +−.  (38) and this solution is unique (Ioannou & Sun, 1996).
Using (38), the closed-loop is described by (39) Similarly, from the plant (31) and the control law (35) and (38), we obtain (40) Because r is uniformly bounded and

RsM s
As are proper with stable poles, y and u remain bounded whenever t →∞ for any polynomial () Ms of degree -1 nq + (Ioannou & Sun, 1996). Therefore, the pole placement objective is achieved by the control law (35) without having any additional restrictions in Qs . When 0 r = , (39) and (40) imply that y and u converge to zero exponentially fast. On the other hand, when 0 r ≠ , the tracking error eyr =− is given by In order to obtain zero tracking error, the equation above suggests the choice of () () Ms Ps = to cancel its first term, while the second term can be canceled by using (34). Therefore, the pole placement and tracking objective are achieved by using the control law www.intechopen.com which is implemented as shown in Fig. 1 using -1 nq + integrators for the controller realization. An alternative realization of (42) is obtained by rewriting it as where Λ is any monic Hurwitz polynomial of degree The PPC design supposes that the plant parameters are known, what not always is true or possible. Therefore, integral adaptive laws can be proposed for estimating these parameters and then used with PPC schemes. This new strategy is called Adaptive Pole Placement Controller (APPC), where the certainty equivalence principle guarantees that the output plant tracks the reference signal r , if the estimates converge to the desired values. In this section, instead of these traditional adaptive laws, switching laws will be used for the the first order plant case, according to (Silva et al., 2004).
A model for the plant may be written as where â and b are estimates for a and b , respectively (Ioannou & Sun, 1996).
We define the estimation error 0 e as 0ê y y =−, and with (46) and (47), we get where Choosing the following Lyapunov function candidate, we have 00 0 which can be rewritten using (49), Expanding the above equation with (50) and (51), and then using the switching laws, which guarantees that 0 0 e = is a globally asymptotic stable (GAS) equilibrium point.
Moreover, if we follow a similar procedure described in (Hsu & Costa, 1989), we can prove   Therefore, equations (60) and (61)

Application on a Current Control Loop of an Induction Machine
The current x voltage transfer function of the induction machine can be obtained from (62) and (63)

VS-MRAC Scheme
Consider that the linear first order plant of induction machine current-voltage transfer function s isdq W given by (67) which attends for the stability constraints that is the constant s b in (67) Introducing nominal values of controller parameters The input and output filters given by equation (76) are designed as proposed in (Narendra & Annaswamy, 1989). The filter parameter Λ is chosen such that () m Ns is a factor of det( ) sI −Λ . Conventionally, these filters are used when the system plant is the second order or higher. However, it is used in the proposed controller to get two more parameters The block diagram of the VS-MRAC control algorithm is presented in Fig. 3. The proposed control scheme is composed by VS for calculating the controller parameters and a MRAC for determining the system desired performance. The VS is implemented by the block Controller Calculation, in which Equations (77) and (78) From this reference model, the nominal values can be determined by using equations (71) and (72)  It is important to highlight that choice criteria determines how fast the system converges to their references. Moreover, it also determines the level of the chattering verified at the control system after its convergence. As mentioned before the use of input and output filters are not required for control plant of fist order. They are used here for smoothing the control signal. Their parameters was determined experimentally, which results in . This solution is not unique and different adjust can be employed on these filters setup which addresses to different overall system performance.

VS-APPC Scheme
The first approach of VS-APPC in (Silva et al., 2004) does not deal with unmodeled disturbances occurred at the system control loop like machine fems. To overcome this, a modified VS-APPC is proposed here. Let us consider the first order IM current-voltage transfer function given by equation (67).  (86) is the current error that is calculated from the measured quantities issued by data acquisition plug-in board as described next. Therefore, to generate the output signal of the controllers it is necessary to solve the equations (91)-(93). p , 1 p and 0 p . The introduction of the IMP into the controller modeling avoids the use of stator to synchronous reference frame transformations. With this approach, the robustness for unmodeled disturbances is achieved.

Experimental Results
The performance of the proposed VS-MRAC and VS-APPC adaptive controllers was evaluated by experimental results. To realize these tests, an experimental platform composed by a microcomputer equipped with a specific data acquisition card, a control board, IM and a three-phase power converter was used. The data of the IM used in this platform, are listed in Table 1 Figure 6(b) shows the graphs of the reference phase current s sd i * superimposed by its corresponded machine phase current s sd i . In this result, it can be verified that the machine phase current converges to its reference current imposed by RFO vector control strategy. Similar to the results presented before, Fig. 7(a) presents the experimental results of reference phase current s sq i * superimposed by its estimation phase current ˆs sq i and Fig. 7(b) shows the reference phase current s sq i * superimposed by its corresponded machine phase current s sq i . These results show that the VS-APPC also demonstrates a good performance. In comparison to the VS-MRAC, the machine phase currents of the VS-APPC present small current ripple.