## CDC Program | Wednesday December 11,

Book Description In control theory, sliding mode control SMC is a nonlinear control method that alters the dynamics of a nonlinear system by application of a discontinuous control signal that forces the system to slide along a cross-section of the system's normal behaviour. Key features: Covers new concepts, new models and new methodologies with theoretical significance in system analysis and control synthesis Includes recent advances in Markovian jump systems, switched hybrid systems, singular systems, stochastic systems and time-delay systems Includes solved problems Introduces advanced techniques Sliding Mode Control of Uncertain Parameter-Switching Hybrid Systems is a comprehensive reference for researchers and practitioners working in control engineering, system sciences and applied mathematics, and is also a useful source of information for senior undergraduate and graduates studying in these areas.

A two-degree-of-freedom model of suspension system is established, as is shown in Fig. Based on the model in Fig. For the controller design problems of suspension systems, it is widely accepted that the evaluating vehicle suspension performances such as the ride comfort represented by the vertical body acceleration a z , suspension deflection f d , and road holding, i. It indicates that the optimal control objectives are to minimize the value of the control output vector in order to secure vehicle ride comfort and handling stability.

The characteristic equation of the homogeneous system of differential Equation 1 can be derived after substituting Equation 4 into Equation 1 , as is shown in 5. According to the non-zero real solution conditions of the differential Equation 1 , we can obtain that Equation 5 is always equal to zero. Based on the Lyapunov stability analysis [ 24, 27 ], the necessary and sufficient condition of the time-delay system is that all the roots of the characteristic Equation 5 is negative real part or the conjugate complex root with negative real part.

In order to solve the critical time delay of suspension instability, Equation 7 should satisfy the condition that no complex roots exist to the characteristic equation.

Time Delay Systems and Inverse Response Systems

Meanwhile, it will make it feasible that Equation 9 has no real roots when choosing some appropriate suspension parameters in a finite frequency domain. Thus, there is not a. We can simultaneously obtain conclusions that the sprung mass m 2 has a great impact on the time delay of a suspension system and that the critical time delay will increase with increasing m 2. There are four curves that denote the critical delay boundary curve of the instability for a suspension system within the non-full time delay stable region 0. Specially, when the value of c o is small enough, the value of c s is big enough, and c s is greater than c o , the semi-active suspension is going to rapidly enter into the full stability region of all delays.

In addition, if the value of c o is great enough, the suspension system enters into the full-time-delay instability area, i. It can be observed from Fig. The critical time delay of instability shows a certain amount of nonlinear stochastic instability, which allows the actual value of the time delay to easily exceed the stability margin of the minimum threshold delay and usually causes an unexpected situation of instability and also results in periodic wheel jumping.

Furthermore, the suspension system motion can become distorted due to the shift change of the damping coefficients under the On-state and Off-state. It also reveals an extreme sensitivity to the perturbation of the chaotic instability phenomenon, which will deteriorate vehicle handling stability, driving safety, and comfort.

### Unsere Leistungen

In order to analyze the frequency response of a semi-active suspension with time delay uncertainties, a Fourier transformation is conducted on Equation 1 , and then, transfer functions of the vehicle body acceleration a z and the tire dynamic load t l corresponding to the road excitation are to be obtained, as shown in Equation The acceleration and dynamic load of the tire amplitude frequency characteristic performance are shown in Figs.

From Figs. Furthermore, the time delay will cause a small augmentation of the resonance frequency of the first and the second order vibration mode. Furthermore, the critical time delay usually causes periodic wheel jumping and seriously influences ride comfort. The block diagram of the smith predictive compensation control system with time delay uncertainties is shown in Fig. Through a mathematical transformation, the proposed control scheme can be changed into Fig. Moreover, the total transfer function of the fuzzy-smith controller is shown in Equation 13 after compensating for time delay, and the control block diagram that is similar to Equation 13 can be seen as Fig.

In the equivalent block diagram shown in Fig. In this section, the Fuzzy-Smith controller design is presented for semi-active suspension systems with time delay based on the integration of the smith predictor control and fuzzy logic control. The structure of the Fuzzy-Smith controller is shown in Fig.

## Methodologies for Control of Jump Time-Delay Systems

The approach makes use of the Smith predictive control to compensate for the time delay and fuzzy control to overcome the challenging issues of model uncertainties. Thus, the fuzzy-smith controller can be tuned on-line by fuzzy logics to improve the control accuracy and stability of a time-delay suspension system. E and EC are used to denote the fuzzy set of the two input variables e and ec , respectively.

In addition, the control force f t is selected as the output to regulate the sprung mass acceleration, U is the fuzzy output sets after fuzzy reasoning, and u is the practical and accurate fuzzy output sets after computing and conversing. The fuzzy reasoning algorithm based on the Mandani method is used in this paper, and the weight value is fixed; thus, the center of gravity method is adopted for the defuzzifier of the controller.

However, establishing the fuzzy control sets and all the membership functions are very difficult. Furthermore, it is also a difficult problem that the fuzzy control list is determined according to expert experience in control practices. For these problems, adopted ways are auto-adjusting the proportion factor, quantization factor, or changing the fuzzy relation matrix directly.

In order to analyze the dynamic characteristics of the semi-active suspension in the time domain, the road profile of random white noise is selected as the excitation sources waveform. This paper outlined the random white noise through the integral method, in which the mathematical formula of road roughness is expressed as Equation We use the fuzzy controller to adjust and control the damping force, and simultaneously, a smith predicator is added to the fuzzy controller in series, which is also used to analyze the impact of time delay uncertainties on vehicle dynamic performances.

After theoretical calculation, the two order frequency of the simulation test system is 0. In this paper, we establish the simulation block diagram of the skyhook on-off control semi-active suspension with time delay based on the theoretical results in Section 4, as is shown in Fig.

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1. Mechanics of Non-Homogeneous and Anisotropic Foundations;
2. Research Summary.
3. Introduction!
4. Methodologies for Control of Jump Time-Delay Systems : Magdi S. Mahmoud : !
5. Sliding Mode Control of Uncertain Parameter-Switching Hybrid Systems [Book];
6. Analysis and Synthesis for Jumping Time-Delay Systems using Transformation Methods.
7. Methodologies for Control of Jump Time-Delay Systems.