Frequency domain precision analysis and design of sliding mode observers
Introduction
The idea of using a dynamical system, which is called observer, to obtain estimates of the system states from measurable system variables was proposed by Luenberger [1]. The observer dynamics are driven by the control and by the difference between the output of the observer and the output of the plant. In SM observers, this difference is maintained equal to zero by means of SM organized in the observer loop. The control should be designed to provide the existence of SM in the observer dynamical system. SM observers were analyzed in a number of publications (see, for example, respective chapters of textbooks [2], [3] and recent tutorials [4], [5]).
However, only ideal SM in the observer dynamical system was analyzed in [2], [3], [4], [5]. To the best of our knowledge, only in [6] for the first time the mechanism of generation of the observation error was found and analyzed through a frequency-domain approach. The proposed approach was based on the locus of a perturbed relay system (LPRS) method [7] and the method of analysis of SM systems with parasitic dynamics [8]. Further development of the approach of [6] was presented in [9] for second-order SM observers. Publication [10] provides some results that can be considered as experimental verifications of the approach of [6], [9]. However, in publication [6], the proposed frequency-domain approach was just outlined. The present paper further develops this approach extending it to the observer design problem and providing an example of its application to DC motor speed observation. This new development is aimed at reaching out to the area of engineering design of SM observers.
The objective of the present paper is, therefore, to extend the approach of [6] to design-related problems, in particular, to investigation of precision dependence on observer design parameters (execution period of the algorithm and values of the observer gain matrix). We aim to show that the observation precision has complex dependence on the SM algorithm parameters. We investigate that dependence and provide a model that allows for computing the observation error. The paper is organized as follows. At first, the problem formulation is considered. Then a frequency-domain approach to SM observer analysis is presented, and a model that provides precision of observation is given. After that, considering the example of DC motor speed observation, an investigation of the dependence of the observation precision on the values of the algorithm design parameters is done, and the design is outlined.
Section snippets
Problem formulation
Consider an n-dimensional version of the observer proposed in [2]. Let the linear plant, the states of which are supposed to be observed, be the nth order dynamical system:where x∈Rn is the state vector, y∈R1 is the measurable system output, A∈Rn×n, B∈Rn×1, and C∈R1×n are the state matrix, the input matrix, and the output matrix of the plant, respectively. The pair (C,A) is assumed to be observable.
The SM observer can be designed in the same form as the original system (1) and (2)
The concepts of the locus of a perturbed relay system (LPRS) approach
In [7], [8], the LPRS was introduced as a method of analysis and design of relay servo systems having a linear plant (Fig. 2). Let us call the part of the relay servo system that is given by the linear differential equations the linear part. With respect to the SM observer, the linear part will be the one given by Eqs. (3), (4).
The LPRS was defined as a complex function J(ω) of the frequency ω as follows:where t=0 is the time of the switch of the relay
Analysis and characteristics of SM observer performance
With the representation of the SM observer as a relay servo system, we formulate performance measures of the observer. Using the LPRS method and the concept of the equivalent gain, we obtain a linear model of the plant-observer dynamics for average (on the period of chattering) motions. We characterize the precision of observation by the output error and by the state observation errors . We note that the output error and state observation errors are not equal to zero [6], [9]
Selection of matrix L and observer precision
Apparently, selection of matrix L must have an effect on observer precision, which follows from formulas (12), (15), and (16). Yet, the choice of matrix L is limited by the conditions of the existence of the sliding mode in the observer dynamics and other conditions that were discussed above. We now transform the condition of the existence of the sliding mode. Assume the absence of parasitic dynamics and the autonomous mode (u(t)≡0, which leads to y(t)≡0). We transform the condition of
Example of SM observer precision analysis
Consider an example of estimation of DC motor speed and acceleration from the measurement of armature current. The motor model is given by the diagram (Fig. 5).
In Fig. 5, Ωm is a motor speed, ia is an armature current, Ea is an armature voltage (control), L is an armature inductance, R is an armature resistance, Jt is a moment of inertia of rotor and load, Bt is a friction coefficient, Ke is a back e.m.f. coefficient, and Kt is a torque constant. We transform the original model of the motor
Conclusion
Analysis of a SM observer is done above as of a relay feedback–feedforward system. The frequency-domain model of observation precision is obtained via application of the LPRS method. It is found that the dynamical performance of the SM observer, which translates into observation precision, is not ideal. Because of the existence of parasitic dynamics in the observer loop (time delay due to discrete implementation of the algorithm) there always exists a non-zero observation error—even after the
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