Qingbin Gao

Investigation of Local Stability Transitions in the Spectral Delay Space and Delay Space

The stability boundaries of LTI time-delayed systems with respect to the delays are studied in two different domains: (i) delay space (DS) and (ii) spectral delay space (SDS), which contains pointwise frequency information as well as the delay. SDS is the preferred domain due to its advantageous boundedness properties and simple construct of stability transition boundaries. These transitions at the mentioned boundaries, however, present some conceptual challenges in SDS. This transition property enables us to extract the corresponding local stability variation properties in the DS, while it does not have any implication in the preferred SDS. The novel aspect of the investigation is to introduce a comparison mechanism between these two domains, DS and SDS, from the stability transition perspective. Interestingly, we are able to prove their equivalency, which provides complementary insight to the parametric stability variations. [DOI: 10.1115/1.4027171]

Combination of sign inverting and delay scheduling control concepts for multiple-delay dynamics

Abstract In this paper we treat a novel combination of two controversial control concepts, Sign Inverting Control (SIC) and Delay Scheduling (DS), for systems with multiple independent and large delays. SIC suggests the inversion of the control polarity and DS prolongs the existing delays. The combined scheme functions with a single requirement that, the union of the control schemes provides a larger stable operating region than each of its components does, in the domain of the delays. The critical knowledge that is needed to execute such a unified control strategy is the crisp description of these stable regions for each time-delayed control scheme. This need can be fulfilled using the recent Cluster Treatment of Characteristic Roots (CTCR) paradigm, which establishes the stable regions exhaustively and non-conservatively. The resulting options in selecting operating modes render more robust control performance against much larger delay variations than each of the schemes. We also investigate the disturbance rejection speeds within these enlarged stable regions in order to improve the control performance even further. Such multi-faceted paradoxical combinations provide previously-unexplored tools to control designers. Experimental validations of these novel concepts are presented on a simple setup with a single-axis manipulator.

Bounds of imaginary spectra of LTI systems in the domain of two of the multiple time delays

The stability of linear time invariant (LTI) systems with independent multiple time delays and the cluster treatment of characteristic roots (CTCR) paradigm are investigated from a new perspective. It is known that for such systems, all the imaginary characteristic roots can be detected completely on a small set of hypersurfaces in the domain of the delays (Sipahi and Olgac, 2006a). They are called kernel hypersurfaces (KH). The complete description of KH is the only prerequisite for the CTCR stability assessment procedure. As the number of delays increases, however, their evaluation becomes infeasible. Instead, we present a procedure to extract the 2-D cross-sections of these hypersurfaces in the domain of any two of the delays by fixing the remaining delays. In the 2-delay domain of interest, the exact upper and lower bounds of the imaginary spectra are determined. For this, a combination of half-angle tangent representation of the characteristic equation and the Dixon resultant theory is used as the main contributions of this paper. The complete KH are obtained by sweeping the root crossing frequency in this interval. Using this knowledge CTCR creates the cross-section of the stability map in the domain of the two arbitrarily selected delays. We demonstrate the effectiveness of this methodology over an example case study with three independent delays and two commensurate ones.

Critical Effects of the Polarity Change in Delayed States Within an LTI Dynamics With Multiple Delays

This note concerns the polarity inversion of the delayed states in linear time-invariant multiple time delay systems (LTI-MTDS). To start with, for such systems the assessment of asymptotic stability is complicated due to the infinite dimensionality introduced by the delays. It is shown that the mentioned polarity inversion influences the respective delay-dependent stability maps in interesting ways. There are some intriguing invariant characteristics and correspondences between the original and inverted systems, which form the primary contributions of this note. These findings are shown to be the enabling features to relate the stability maps of the two systems. The Cluster Treatment of Characteristic Roots (CTCR) paradigm is utilized to identify these features. An example case study is provided to illustrate the claims.

Stability analysis for LTI systems with multiple time delays using the bounds of its imaginary spectra

Abstract The stability of linear time invariant (LTI) systems with multiple independent time delays and the cluster treatment of characteristic roots (CTCR) paradigm are investigated from a new perspective. Any delay composition that results in an imaginary characteristic root lies either on a small number of kernel hypersurfaces (KH) or their infinitely many offspring hypersurfaces (OH). The complete description of KH is the only prerequisite for the CTCR-based stability assessment procedure. As the number of delays increases, however, the determination of these KH becomes computationally costly. Instead, we present a practical procedure to extract the 2-D cross-sections of the KH set in the domain of the two arbitrarily selected delays. First, we determine the exact upper and lower bounds of the imaginary spectra in this 2-D cross-section of interest. This process starts with a half-angle tangent representation of the characteristic equation followed by the Dixon resultant operation. Based on the full knowledge of KH, the CTCR paradigm creates the complete stability map in the domain of these two arbitrarily selected delays. We demonstrate the effectiveness of this methodology over an example case study.

Optimal sign inverting control for time-delayed systems, a concept study with experiments

An intriguing control logic, sign inverting control (SIC) is considered for control systems with delayed feedback. It starts with a nominal control law formulated for non-delayed case and simply inverts the sign of the control gains for some surprising benefits when used with the delays. This operation sounds paradoxical as the sign inversion potentially harms the stability of the non-delayed dynamics. However, SIC with large delays may yield some complementary benefits to the nominal control logic from delay robustness perspective. The main question we address in this paper is ‘How to select the nominal control law so that such a contribution can be (a) feasible, (b) optimal in some sense?’ A structured methodology is proposed to achieve this, starting with a linear quadratic regulator based controller. A single scaling factor on the corresponding control gains is used for one-dimensional optimisation. Experimental validation of the concept of this optimal SIC procedure is also reported on a single-axis manipulator.

An effective algorithm to achieve accurate sinusoidal amplitude control with a low-resolution encoder

This paper presents a novel control methodology which uses an extremely low-resolution encoder to achieve a desired harmonic trajectory. The particular application of interest is a cellular microinjection technology called the Ros-Drill (Rotationally Oscillating Drill) for ICSI (Intra-Cytoplasmic Sperm Injection). It is an inexpensive set-up, which creates high-frequency (e.g. 500 Hz) and small-stroke (e.g. 0.2 degree) rotational oscillations at the tip of an injection pipette mimicking a harmonic motion profile. Such a motion control procedure presents no particular difficulty when it uses motion sensors with appropriate resolution. However, size, costs and accessibility of technology on hardware components for our charter severely constrain the sensory capabilities. The main objective is to achieve rotational amplitudes with peak-to-peak stroke of 0.4 degree and frequency of 500 Hz, using an encoder with resolution of 0.09 degree. This low-resolution encoder (which provides only 4 encoder steps for the entire stroke) presents two complications, a) large quantization errors at and between two successive measurements; b) stochasticity of actual amplitude for the desired discrete encoder readings. This paper proposes a method which enables accurate peak-to-peak stroke control using the same hardware, by essentially converting a stochastic stroke control problem into a deterministic dwell time balancing operation. The study describes a control strategy consisting of two-stage tuning: 1) coarse tuning which provides a 4-step desired encoder readings in peak-to-peak swings, 2) fine tuning which adapts the control gains to achieve the actual peak-to-peak stroke based on the deterministic relationships between the oscillation amplitudes and the encoder readings. Simulations and experiments are provided to validate the proposed method.

Some critical properties of sign inverting control for LTI systems with multiple delays

A novel control logic, sign inverting control (SIC) is proposed for linear time-invariant multiple time delay systems (LTI-MTDS). SIC simply inverts the sign of a nominal control logic which is formulated for non-delayed dynamics using any known technique. The main objective of the inversion is to increase the delay robustness of the system. Three necessary and sufficient conditions are provided for SIC to be a viable option. This inversion method gains its strength from a recent paradigm called the Cluster Treatment of Characteristic Roots (CTCR). CTCR provides the necessary exact and exhaustive declaration of the stable regions in the domain of the delays for such systems. As the highlight of the paper, the spectral and root tendency invariance properties between the nominal and the SIC applied systems are presented with detailed proofs. Finally, we demonstrate an unexpected feature by an example, that SIC may also be applied to systems which are closed-loop non-delayed unstable for both control schemes.

Equivalency of Stability Transitions Between the SDS (Spectral Delay Space) and DS (Delay Space)

It has been shown that the stability of LTI time-delayed systems with respect to the delays can be analyzed in two equivalent domains: (i) delay space (DS) and (ii) spectral delay space (SDS). Considering a broad class of linear time-invariant time delay systems with multiple delays, the equivalency of the stability transitions along the transition boundaries is studied in both spaces. For this we follow two corresponding radial lines in DS and SDS, and prove for the first time in literature that they are equivalent. This property enables us to extract local stability transition features within the SDS without going back to the DS. The main advantage of remaining in SDS is that, one can avoid a non-linear transition from kernel hypercurves to offspring hypercurves in DS. Instead the potential stability switching curves in SDS are generated simply by stacking a finite dimensional cube called the building block (BB) along the axes. A case study is presented within the report to visualize this property.Copyright

Differentiability of imaginary spectra and determination of its bounds for multiple-delay LTI systems

Several new procedures are presented to assess the stability of linear time invariant (LTI) systems with multiple time delays. The skeleton of the methods is on a paradigm named the cluster treatment of characteristic roots (CTCR). For such systems, all the imaginary characteristic roots can be determined completely on a finite set of hypersurfaces, called kernel hypersurfaces (KH), in the delay domain. The entire KH is the mere prerequisite to implement CTCR. However, as the number of delays increases, it becomes computationally prohibitive to calculate the KH in the entire delay domain. Alternatively, we explore the 2-D cross-section of these KH in the space of any two of the delays. First we investigate the bounds of the imaginary spectra as a novelty here. For this, the proof of the differentiability of the crossing-frequency variations is provided. Another novelty of the paper appears at the declaration of the KH. For this we utilize the 3-D “Building Block” and “Spectral Delay Space” concepts. The effectiveness of these procedures is shown over an example case study.