HOVHANNISYAN T.N.

 

ON STABILITY OF SINGLE-INPUT SINGLE-OUTPUT L1 ADAPTIVE CONTROL SYSTEMS

 

Some issues concerning the stability of SISO (i.e. having the one input and output) adaptive control systems for rejecting of external disturbances are discussed. Based on the properties of positive real transfer functions, it is shown that such systems are stable for arbitrary large values of the adaptation gain, even in the case of systems with right half plane zeros.

 Keywords: single input single output (SISO) control system, adaptive control, reference model, stability, positive real system.

 

The paper examines application of  adaptive control to SISO control systems [2].  adaptive control was developed to address some of the deficiencies apparent in Model Reference Adaptive Control (MRAC), as loss of robustness in the presence of fast adaptation [3, 4].

As a basic model of linear one-dimensional systems with constant parameters let  consider the system that can be expressed in the following standard state-space form: 

                                                          (1)

where  is an -dimensional state vector;  and  are input and output scalar variables;  is an   constant  matrix, and  respectively are -dimensional  constant row-  and  column-vectors. In what follows, we will assume that system completely controllable and observable, strictly stable,  and, maybe, with Right Half Plane (RHP) transmission zeros.

The SISO system (1) can also be described in the operator form by the   scalar strictly proper rational functions in complex variable .

Generally, the transfer function  is connected with the function, vectors  in (1) by the formula [5]

                                                        (2)

where  is an identity function. 

In this article, we will adhere to the  architecture with state predictor and low-pass filter presented in [2]. Let an one-dimensional strictly stable SISO system be described in state-space by the following equations:

                           (3)

where  is a time-dependent scalar function of unknown external bounded () disturbances that should be rejected by adaptive control, and all other variables and vectors have the dimensions as in (1).

The state predictor has the same structure as the system in (3):

                               (4)

and the only difference is that the unknown disturbance vector  is replaced by its estimate .

The disturbance rejection process is governed by the following adaptation law [1]

,                                                           (5)

where is the prediction error,  () is the solution of the Lyapunov equation

                                                (6)

for an arbitrary symmetric positive definite function  (), and the positive scalar  is called the adaptation gain [1].

The control signal  of the system is given in operator form as

,                                         (7)

where  is an  one-dimensional reference signal,  is an static gain, and  is the transfer function of a low-pass filter. In the simplest case, the function  satisfying the DC gain condition  = 1. Its state-space realization assumes zero initialization.

The block diagram of the control system with the state predictor (4), the adaptive disturbance rejection law (5), and control signal  (7), is shown in Figure 1.

Fig 1.  Block diagram of the adaptive SISO control system with the state

predictor and the adaptive disturbance rejection law (5).

 Thus, the architecture of the discussed adaptive control system represents a linear SISO system with integral feedback and therefore can be investigated by the methods and approaches of linear one variable feedback control [5, 6]. It should be noted that due to the adopted  scheme with the state predictor, the transfer function  in the control signal  (7) is not present in the disturbance rejection law (5). Let us consider in more detail the structure and performance characteristics of the adaptive system in Figure 1. Toward that end, we introduce the following transfer function

                                                 (8)  

relating the input to the system (1) with the state vector , and the corresponding (the same) function  for the state predictor. Then, the block diagram in Figure 1 can be recast to an equivalent form in Figure 2.

Fig 2.  Equivalent block diagram of the adaptive system in Fig 1.

 Based on the block diagram in Figure 2 and equations (2) by considering that , it is easy to derive the following function equations of the adaptive system with the state predictor:

 ,      (9)

where

;      .                           (10)

As can be seen from (10), the output signal of the system  consists of two components generated, respectively, by the input reference signal  and by the disturbance . Since the adaptive SISO system in Figures 1 and 2 is linear, the superposition principle holds and, according to the equation (9), the dynamics of the system can be represented by equivalent block diagram in Figures 3.

Let us proceed to the stability analysis of the adaptive system in Figure 1.

Fig3.  Equivalent block diagram of the adaptive system

with respect to the disturbance .

The block diagram in Figure 3 contains a negative feedback loop with the open-loop transfer function  (10) and the following closed-loop transfer function:

.             (11)

The characteristic equation of that system is     

  ,                            (12)

and, clearly, the poles of the adaptive system depend on the adaptation gain , which can be considered as the gain of the open-loop transfer function  (10).

Note now that, allowing for (8), the system with the transfer function  in (10) can be written in state-space form as

                                             (13)

where

.                                                       (14)

Taking into account the form of the function  (13) and recalling the Kalman-Yakubovich lemma [2-4], we come to a conclusion that  belongs to the so-called Positive Real (PR) transfer functions and is positive semi-definite (for brevity we shall write ) for all real , for which  is not a pole of any element of .

Example.  Consider a one-dimensional SISO system whit integrating chain and with the transfer function

                                                                           (15)

 

for which the transfer function of equivalent adaptive system will be:

 

                                                      (16)

 

The Nyquist plots, as well as the root loci of the of the above function  (16) are shown in Figure 4.

  

                                        (a)                                           (b)

Fig 4.  Nyquist plots (a)  and root loci  (b) of  .

As can be seen from the graphs in Figure 4, the one-dimensional adaptive system with the transfer function  is stable for any value of the adaptation gain .

References

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