Improving Local Priority Hysteresis Switching Logic Convergence

Local priority hysteresis switching logic is associated with adaptive control convergence when using an infinite set of candidate parameters via constraints added to the switching scheme. In this paper, we reevaluate these constraints on the basis of the persistent excitation assumption. This makes room for the adaptive control to converge to its optimum, resulting in improved performance. Unconstrained local priority hysteresis switching logic is investigated, and global convergence conditions are proposed. This paper expands on the preliminary version of a conference paper [1] by adding numerical simulation examples to validate both the application and the advan-tage of the theory.


Introduction
One of the earliest definitions of the term "adaption" was introduced by Drenick and Shahbender [2] in 1957: "Adaptive systems in control theory are control systems that monitor their own performance and adjust their parameters in the direction of better performance." Adaptive control ensures the satisfactory performance of a closed-loop system by switching among a set of candidate controllers K when no single controller k can achieve the targeted performance objectives.
Two techniques have generally been used to achieve this goal: unfalsified adaptive control [3] [4] [5] and multiple model adaptive control [6] [7] [8]. In both cases, the switching process is supervised by a unit that provides the best controller k for the feedback loop from the controller set K based on the plant Intelligent Control and Automation input/output data and performance criterion ( ) t µ . Figure 1 shows the general architecture of an adaptive control system.
One serious challenge associated with adaptive control switching systems is a type of instability called chattering. Chattering means the switching system will cycle among two or more candidate controllers without ever converging. Various techniques can help to avoid chattering for a finite and infinite candidate controller set. Using a continuum of candidate controllers instead of a finite set allows for greater flexibility for the adaptive control system to manage uncertainty [9] [10] [11] [12].
One technique involves the hysteresis switching algorithm reported by Morse and Middleton in [13] [14]. Morse and his co-workers demonstrated adaptive control convergence for a finite controller set. Also, Hespanha et al. [9] [10] and Stefanovic et al. [11] proved adaptive control convergence for an infinite controller set.
These studies contributed to ensuring adaptive control switching system convergence by adding constraints to the switching scheme that required strictly positive hysteresis and local priority constants.
Unfortunately, however, such constraints in the switching scheme can hinder the adaptive control system in achieving optimality.
This paper contributes to the body of knowledge by easing the local priority hysteresis switching logic constraints in switching schemes based on the persistent excitation assumption. Easing these constraints is necessary to achieve the objective of high performance. This paper is organized as follows. In Section 2, preliminary facts are given. Section 3 reviews definitions of local priority hysteresis switching logic. Section 4 contains the main results. Simulation examples are shown in Section 5. The conclusion and avenues for future work are provided in Section 6.
is said to be equi-quasi-positive definite (EQPD) in k if for some continuous monotonic function , , , Under the persistent excitation assumption and for a sufficient length of time t, the function ( ) p µ τ is a uniformly convex function in p. Therefore, if the system is persistently excited, the monitoring signal will become uniformly convex.
The definition of persistent excitation (PE) is critical in adaptive scheme studies that seek parameter convergence; see, for example, [16]- [21].

Local Priority Hysteresis Switching Logic
In this section, we present notations and definitions of local priority hysteresis switching logic. The intent is to introduce a switching scheme that can be applied when the unknown parameters of the system belong to a continuum set. Due to the differences between finite and infinite sets of candidate controllers, an infinite set of candidate controllers (typically, a continuum of controllers) can result in a better environment that facilitates the feasibility assumption (i.e., there is a controller in the candidate controller set can satisfy the adaptive control performance).
The supervisory control approach [22] [23] is used to achieve or maintain a desired performance level in a closed-loop system via switching through the given set of candidate controllers. The basic idea in selecting a controller strategy is to determine which nominal process model is associated with the smallest monitoring signals " ( ) p t µ ". Then, the corresponding candidate controller can be selected.
Assume a linear single-input and single-output (SISO) finite-dimension uncertain process  shown in Figure 2.  is assumed to be a stabilizable and observable model with control input signals u and measured output signals y.
The supervisor contains three subsections, as shown in Figure 2.
1) Multi-estimator Σ  -is a dynamic system with inputs of u and y and outputs of the signals p y , p ∈  .  is a compact subset of a finite-dimension normalized linear vector space.
2) Monitoring signal generator Σ  -is a dynamic system with inputs of estimation errors p p e y y = − and outputs of monitoring signals p where p µ is defined as the integral norms of the estimation errors, called monitoring signals.
3) Switching logic Σ  -involves a switched system with inputs of the monitoring signals p µ and outputs of the parameters that optimize the performance criterion p , which is defined as follows.
The state-space equations for the three subsystems are described in [10]; the multi-estimator Σ  can be stated as follows. x where x  is the estimated state that is assumed to be available for the controller at all times and A  is a stable matrix.
The matrix p c is designed based on each p ∈  so that p c exists and is unique (see reference [22]; Section IV). Moreover, for  to form a continuum, p c is assumed to be linearly dependent on p, which ensures tractability (see reference [22]; Section XI). Therefore, the matrix p c can be written as follows.
For a SISO system, A is a nonzero matrix with a dimension n n × , p is 1 n × unknown process parameters, and b is a 1 n × vector.
In [9], the candidate controller set = { } : p k p∈  is selected so that each p ∈  , p k is a controller stabilizing all the process models in  is any element of  . We assume that feasibility holds. A set D γ is defined as follows.

( ) { }
: where ⋅ is a norm function in  and γ is a proper positive constant. The output of Σ  at each instance is ( )  ( ) ( ) .

L i i k t k t = (10)
A key result is the local priority hysteresis switching convergence lemma, which can be stated as follows. A concern with constant γ required by Lemma 2 is that the constant γ prevents the adaptive system from switching to a new parameter ( )

Main Results
The following lemmas and theorem establish that under the PE assumption, if the strictly positive local priority and hysteresis constants are eased in the local priority hysteresis switching logic convergence lemma, the optimal process parameter ( ) p t , as defined in (5), still converges as t → ∞ under the same conditions (i.e., monotonic monitoring signal and feasible system) given in [9] and with the identical monitoring signal.
Recall that the authors in [9] use the following monitoring signal. ( ) Then, The following lemmas will be used in the proof of the main result.
Also, since ( ) From (15) and from (16) Additionally, since the system is persistently excited, ( ) ( ) p t converges as t → ∞ to a point in the closure of the set  . Remark 1. (Performance Improvement) According to the certainty equivalence concept [25], "The nominal process model with the smallest performance criterion signal 'best' approximates the actual process, and therefore the candidate controller associated with that model can be expected to do the best job of controlling the process." The basic idea is to determine which nominal process model is associated with the smallest monitoring signals. Then, the corresponding candidate controller can be selected.
As shown in theorem 1, the approach introduced in this paper (which relies on easing the local priority hysteresis switching logic constraints) improves adaptive controller convergence. Based on certainty equivalence [25], this method also improves adaptive control performance.

Simulation Examples
In this section, two systems are examined to demonstrate the unconstrained  Figure 3 and Figure 4. The graph in Figure 3 shows that the  p → − ", which is the point when the error "e" reaches zero. The graph in Figure 4 shows the rapid and smooth convergence of the error and process parameters to optimal values. In both graphs, exact matching is achieved.
Example 2: In this example, a second-order system shows how the unconstrained performance criterion manages a system that exhibits a transient response.
Consider the model reference adaptive control in Figure 2 with a second-order The simulation starts with the initial process parameter set to 1 0.1 p = − . The results are shown in Figure 5 and Figure 6. The graph in Figure 5 shows that the error "e" reaches zero when the process parameter " 1 p " converges to −6.
The graph in Figure 6 shows the smooth convergence of the error and process parameter to the optimal values over a short time. In this example, the plant performance and error convergence results are satisfactory, and exact matching is achieved.

Conclusion
In this paper, we examined the local priority hysteresis switching logic and established performance criteria under which the hysteresis constant can be set to zero. The main results indicate that when the convergence lemma conditions  (i.e., monotonic monitoring signal and feasible system) hold, the PE assumption ensures convergence in local priority hysteresis switching logic without adding constraints to the switching logic. Easing these constraints improves adaptive control convergence, which results in improved performance.

Future Work
The quadratic model reference performance criterion in the example satisfies the EQPD condition, but it lacks the fading memory term, which may lead to difficulties for plants with unstructured uncertainty or plant/sensor noise. In those cases, a performance criterion with a fading memory term that satisfies the EQPD condition must be designed. Whether such a performance criterion exists remains an open question to address in future studies.