Abstract

The paper explores the optimal vibration control design problem for a half-car suspension working on in-vehicle networks in delta domain. First, the original suspension system with ECU-actuator delay and sensor-ECU delay is modeled. By using delta operators, the original system is transformed into an associated sampled-data system with time delays in delta domain. After model transformation, the sampled-data system equation is reduced to one without actuator delays and convenient to calculate the states with nonintegral time delay. Therefore, the sampled-data optimal vibration control law can be easily obtained deriving from a Riccati equation and a Stein equation of delta domain. The feedforward control term and the control memory terms designed in the control law ensure the compensation for the effects produced by disturbance and actuator delay, respectively. Moreover, an observer is constructed to implement the physical realizability of the feedforward term and solve the immeasurability problem of some state variables. A half-car suspension model with delays is applied to simulate the responses through the designed controller. Simulation results illustrate the effectiveness of the proposed controller and the simplicity of the designing approach.

1. Introduction

In the past years, communication networks have been applied greatly and widely into the advanced vehicle systems, such as electronic control units (ECUs), sensors, and actuators which are all connected over the high-speed in-vehicle networks (IVNs), for example, Local Interconnect Network, Controller Area Network, Media Oriented Systems Transport [1–3], and so forth. However, in this kind of IVNs, two important issues emerge for the controller design problems. One is the network-induced delay issue. Over communication networks, the time delays generated between sensor-controller, controller-actuator, ECU computation delay, and so forth are unavoidably encountered. As well known, even a small time delay can make the systems disastrously unstable or generate oscillations [4–6]. So, this issue should be taken into account when we design a controller for a system over networks. The next issue is the system modeling problem. Actually, signals processed by microprocessors are digital, and most of those produced by sensors or put into actuators are analogs. Thus, the continuous-time plant is combined with a discrete-time controller, where and converts are used to combine these two different signals. Therefore, a sampled-data system is more appropriate for the reality of networked-control system. In previous studies, there are two main approaches to get the sampled-data systems, those are, indirect approach of continuous-time domain and direct approach of discrete-time domain. However, the former is only suitable for simple control algorithms, and the latter has two drawbacks: one is that the discrete-time model is unable to approach to its corresponding continuous-time one as the sampling frequency increases; and the other is that the discretized system can cause oscillations and unstable phenomenon as the sampling frequency increases.

Consequentially, we present two strategies to deal with these above-mentioned issues. First, we introduce the delta operator approach to build a sampled-data model for the networked-control system due to its accurate approximation to the continuous-time model under rapid sampling conditions [7–11]. This advantage can be demonstrated by citing an instance.

Consider a continuous-time system consisted of in the state-space representation. The associated discrete-time system can be got as with , , and the sampling period. Alternatively, the associated sampled-data system in delta domain is with , , and a unit matrix. Apparently, , and , while .

From this instance, it is clear that the discrete-time model cannot approximate to its original continuous-time one when sampling fast. On the contrary, the delta-domain sampled-data one enables approximating to the continuous accurately. See that this is our reason to apply the delta operators approach modeling a sampled-data system for an IVNs-based suspension.

The other contribution of this paper is developing the model transformation method [11–13] to solve optimal vibration control (OVC) design problem for sampled-data systems with time delays in delta domain. The model transformation method is based on the finite spectrum assignment methodology [14], and we have developed it solving the OVC design problems for time-delay systems in continuous-time domain or in discrete-time domain. It was proved able to transform the original time-delay system into a delay-free one such that the original solution problem is reduced from an infinite-dimension space to a finite-dimension space, and so that the controller design problem is greatly simplified [11–14].

In brief, as a result, the obtained control law in this paper consists of a feedforward term and some control memory terms which can be stored in memories beforehand. Hence, the persistent road excitation suffered by suspension can be reduced via the feedforward term, and the time-delay effect of the system can be compensated by the control memory terms. At last, we carry out some simulations to validate the effectiveness and the simplicity of the designed OVC comparing with the open-loop system (OLS).

The paper’s organization displays as follows. After this introduction of Section 1, in Section 2, the system description and problem formulation have been done. In Section 3, the OVC is designed accompanied with the observer-based control law design. Numerical examples are simulated in Section 4. Concluding remarks are given in Section 5.

2. Problem Statement

2.1. System Modeling

Consider a four-degree-of-freedom half-car model (refered to in [15]) where the suspension motion is determined by the following dynamic equations: through Newton-Euler method. In this half-car suspension model, the sprung mass and unsprung one are separated by spring, damper, and actuator, which are placed in parallel. The tire of the vehicle is modeled as a spring. Vertical motion and pitch motion of the sprung mass are considered, as well as the vertical motion of the front unsprung mass and the rear one .

Consider it working on an IVNs-based environment, as depicted in Figure 1, where , , and denote the system state, control input, and measured output, respectively; is the road excitation input; , are constant ECU-actuator delay and sensor-ECU delay, respectively, (actuator delay and sensor delay for short) which are always assumed to be known and constant in such IVNs environment.

Figure 1 

Suspension system working on IVNs.

With the purpose of replacing (1) into the state-space representation, define the state, control, and disturbance vectors as the controlled output vector as and the measured output vector as Together with the associated dynamic equations and the definitions (2)–(4), the system (1) thus is rewritten in the state-space representation as with , , , and as the state vector, control output, measurement vector, and controlled output, respectively, , , , , , and as the real constant matrices of appropriate dimensions, and as the initial state vector.

Consequently, the sampled-data system form of the system (6) can be got by applying the sampled-data controller where there is a zero-order holder with as the sampling times, as the state on time of , and as a constant data controller gain, denoting the actuator delay and the sensor delay with and . Notice that the delays are expressed in either integers or nonintegers. Then, the sampled-data system of (1) can be described by where and . Here, the triple is assumed to be completely controllable and observable.

Further, the sampled-data system (8) will be converted to delta domain and without actuator delays. There are two situations that should be considered concerning the nonintegral part of the time delay: and . However, the derivation procedures of these two situations are similar. So, for the sake of simplicity, the derivation procedure of the former situation will be presented in what follows.

Introducing the delta operator as and letting and discretize the sampled-data system (8) in the delta-domain form Noting that the system (10) is the delta-domain sampled-data system with actuator delays, for the convenience of calculation, it will be transformed without actuator delays using the model transformation approach. From the first difference equation in (10), it follows the analytical expression of state response where Consequently, define a new state vector as Equation (10) yields the analytical expression of state response for the transformed system which produces its state equation in delta-domain form Through the same way, defining the new measured and controlled outputs with , the output equations of the transformed system are got as follows Then, the state equation (15) and the output equation (17) consist the transformed equivalent sampled-data system in delta domain without actuator delays:

Furthermore, the road excitation should be described by an exosystem in delta domain in order to employ the state-space representation designing the OVC.

2.2. Road Disturbance Modeling

According to ISO 2631 standards, the road displacement power spectral density (PSD) is usually approximately represented in the formulation of with as the spatial frequency, as the road roughness constant, and as the sort of the road as shown in Table 1.

Due to the low-pass-filter characteristic of vehicle tires and suspension, the road displacement can be approximately simulated by a finite Fourier series sum where are the amplitudes, are the frequencies with the time frequency internal , are the random phases which follow a uniform distribution in , is a constant horizontal velocity, is the given road segment length, and positive integer limits the considered frequency band.

Letting the disturbance state vector the road velocity then is described by the exosystem with in which and represent the zero matrix and the -order identity matrix, respectively. Using delta operator (9), exosystem (22) is transformed into the delta-domain form where .

Hence, both the original system and the exosystem are transformed into the delta-domain sampled-data systems so that we would design the OVC for them.

2.3. Problem Formulation

The principal variables for the evaluation of the suspension system are sprung mass acceleration and determining the ride comfort, suspension deflection indicating the limit of vehicle body motion, and tire deflection ensuring the road holding ability. The purpose is to reduce the acceleration of vehicle body and decrease the dynamic tire forces for improving the road holding ability and the stability of vehicles facing road excitation.

In practice, control and controlled output are unable synchronously zero so that the general infinite-horizon performance index is not convergent. In this case, an average infinite-horizon performance index could be chosen as with , as positive semidefinite matrices and as a positive definite matrix, assuming that with an arbitrary matrix and the triple completely controllable and observable.

Remark 1. When , the delta-domain sampled-data system is described by with The derivation procedure of this situation is similar to that of . It is omitted for the simplification reason.

3. Optimal Vibration Controller Design

3.1. Sampled-Data OVC Design

Theorem 2. Consider the optimal vibration control problem described by the time-delay sampled-data system (10) under disturbance (24) respecting the average performance index (25). The optimal vibration control law is existent and unique and given by where is the unique positive definite solution of Riccati equation: is the unique solution of Stein equation:

Proof. According to Pontryagin’s minimum principle, the optimal control problem combined by the system (18) and the performance index (25) results in the two-point boundary value problem in delta-operator form: which yieldswith the optimal control law From (31) or (32a) and (32b) it is clear that and are the linear relationship, so denote the costate vector Consequently, on one hand, together with (34) and (32a), it gives On the other hand, from (34) and (32b), the following equation holds: Substituting (35) into (36) yields Due to and arbitrarily satisfying (37), it results in Riccati equation (29) and Stein equation (30).
Further, the uniqueness is to be proved. According to linear optimal regulator theory, there exists a unique positive definite solution for Riccati equation (26) and the closed-loop system of (18) is asymptotically stable, which implies that matrix is Hurwitz, that is, for . Moreover, since road disturbances are persistent and not asymptotically stable, this means that for . Therefore, from (39) and (40), the following inequality holds: As a result, Stein equation (30) has the unique solution matrix [16]. The uniqueness of and leads to the uniqueness of OVC (28). This ends the proof.