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Research Papers

Stochastic Approximation Approach to Design of Linear Controllers for Tracking Systems With Asymmetric Saturation

[+] Author and Article Information
P. T. Kabamba

Aerospace Department,
University of Michigan,
Ann Arbor, MI 48109

S. M. Meerkov

EECS Department,
University of Michigan,
Ann Arbor, MI 48109
e-mail: smm@umich.edu

H. R. Ossareh

EECS Department,
University of Michigan,
Ann Arbor, MI 48109
e-mail: hossareh@ford.com

1Deceased.

2Present address: Ford Research and Advanced Engineering, Ford Motor Company, Dearborn, MI 48124.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received October 23, 2014; final manuscript received July 13, 2015; published online October 5, 2015. Assoc. Editor: Dejan Milutinovic.

J. Dyn. Sys., Meas., Control 137(12), 121011 (Oct 05, 2015) (16 pages) Paper No: DS-14-1432; doi: 10.1115/1.4031460 History: Received October 23, 2014; Revised July 13, 2015

Reference-tracking closed-loop systems with saturating actuators often operate in asymmetric regimes. This is because reference signals cause the operating points away from the point of saturation symmetry (even if the actuator itself is symmetric, i.e., odd, function). Stability analysis and stabilizing controller design for asymmetric systems can be carried out using the same techniques as those for the symmetric case. In contrast, currently available methods for controller design in the framework of reference tracking are not applicable to asymmetric systems. The goal of this paper is to develop such a method for single-input single-output (SISO) plants having no poles in the open right-side plane. The approach is based on a global quasi-linearization technique referred to as stochastic linearization, which approximates the saturation function by an equivalent gain and equivalent bias. The main qualitative result obtained is that the asymmetry leads to a constant disturbance acting at the input of the plant. The quantitative results are analytical expressions for this disturbance and the ensuing steady-state tracking errors. It is shown that these errors exhibit a behavior incompatible with the linear control theory. Specifically, they may be increasing or nonmonotonic functions of the controller gain. In view of this fact, the paper develops a time-domain technique for linear tracking controller design based on two loci: the saturating root locus (to account for dynamics) and the saturating tracking error locus (to accounts for statics). Methods for sketching these loci are provided and applied to controllers design.

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References

Figures

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Fig. 2

Stochastic linearization of asymmetric nonlinearity

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Fig. 3

Control-theoretic representations of stochastically linearized asymmetric nonlinearity. (a) Coupled representation and (b) decoupled representation.

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Fig. 4

Measure of asymmetry, A, as a function of μu and σu. (a) A(μu) and (b) A(σu).

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Fig. 5

Equivalent gain, N, and decoupled bias, md, as a function of measure of asymmetry, A. (a) N(A) and (b) md(A).

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Fig. 6

Coupled bias, |mc|, as a function of |A|

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Fig. 8

Stochastically linearized system in the coupled representation

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Fig. 9

Stochastically linearized system in the decoupled representation

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Fig. 10

Measure of asymmetry, A, as a function of average value of the reference, μr

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Fig. 11

Equivalent gain, N, and decoupled bias, md, as a function of A. (a) N(A) and (b)md(A).

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Fig. 12

AS-root locus of Example 1 with β=0.92

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Fig. 14

Sketch of the TE locus for system (27) with μr=1, α=−0.5, and β=1.5

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Fig. 15

The TE locus for Example 1

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Fig. 16

Measure of asymmetry, A, as a function of controller gain, K

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Fig. 17

Tracking error distortion, δ, as a function of measure of asymmetry, A

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Fig. 18

Equivalent gain, Keff, and tracking error distortion, δ, as a function of measure of asymmetry, A. (a) Keff(A) and (b) δ(A).

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Fig. 19

AS-root locus of Example 1 with β=1.3

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Fig. 20

The TE locus for Example 1 with β=0.92 and β=1.3

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Fig. 21

Responses of the system of Example 1 with β=0.92 and β=1.3. (a) β=0.92 and (b) β=1.3.

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Fig. 22

Performance loci of Example 2. (a) AS-root locus and (b) TE locus.

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Fig. 23

Responses of the system of Example 2 with K = 1.3 and K = 22. (a) K = 1.3 and (b) K = 22.

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Fig. 24

The step tracking nonlinear system

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Fig. 25

AS-root locus of Example 3

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Fig. 26

Responses of the system of Example 3. (a) Random reference tracking and (b) step tracking with precompensator.

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Fig. 27

Responses of the system of Example 3 without the precompensator. (a) System output and (b) controller output and plant input.

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Fig. 28

Performance loci of Example 4. (a) AS-root locus and (b) TE locus.

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Fig. 29

Responses of the system of Example 4. (a) Random reference tracking and (b) step tracking with precompensator.

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