Intelligent FOPID and LQR Control for Adaptive a Quarter Vehicle Suspension System

Vehicle suspension's main task is to separate passenger and vehicular body interactions from oscillations generated by road abnormalities whilst nonetheless preserving continuous wheel-road contact [1]. The major performance requirements of an automotive suspension system are to provide good ride and handling characteristics when random disturbances from the environment and the driver’s maneuvers act upon the moving vehicle [2, 3]. The suspension systems used for improving the ride, it categorized into three types, that is passive suspension systems, semi-active, and active suspension systems [4]. One of the most terms used for the suspension systems is the term « suspension travel », and travel a vehicle's suspension has referred to the distance between the center of the wheel from full extension to full compression of the suspension, and we define it the inches of the wheel can move up and down [5, 6]. The more travel in the suspension, the more time sharks have to dampen or react to the terrain, thus transferring less of that harshness to driver and passengers, giving a smoother, more controlled ride [7]. Therese many works in the past decade were accomplished to design, model, and control the active suspension systems, Manurya, and Bhangal [8], used two control techniques PID and LQR control to suppress the vibration of the system and a comparison between passive and active suspension with road disturbances as input has been made, the authors found LQR control is better in suppressing the vibration as compared to PID control as well as passive suspension.

In reference of Mahmoodabadi and Nejadkourki [9], the authors M.J. Mahmoodabadi, and N. Nejadkouriki proposes an optimal fuzzy adaptative robust proportional-integral-derivative (PID) for a quarter car model with an active suspension system a fuzzy system consisting of the singleton fuzzifier, center average defuzzifier, and the product inference engine is applied to regulate the control parameters, the results show the dominance of the proposed active suspension system. Wang. H.P, and all in the study of Wang et al. [10], proposed the model-free fractional-order sliding mode control (MFFOSMC) and applied it for the nonlinear quarter car of an active suspension system to improve the ride comfort and keep acceptable limits for both suspension deflection and dynamic weel load. In the study Dong et al. and Izadkhah et al. [11, 12], the authors work for the fractional-order PID control of an active suspension, the results in two references show the performance of the active suspension system that uses the FOPID actuator is obviously superior to that of the active suspension system that uses a PID actuator and a passive suspension system.

In this work, we have studied a comparison between fractional-order proportional integral derivative controller tuned by particle swarm optimization, and LQR control, this study applied for the controllers for an active suspension system. The remainder of this paper was organized as follows. In section 2, the dynamic model of nonlinear quarter vehicle active suspension system. In section 3 detailed the design procedure of FOPID, and LQR control. Section 4 gives the simulation results based on Matlab/Simulink and the interpretation of these results. Finally, in section 5 the conclusion of the paper.

3.1 Fractional-order PID tuning by PSO algorithm

The fractional-order control turned into designed and analysed via pondlubny. The FOPID delivers more effectiveness than the traditional PID. Based on three parameters of the necessary controller, the FOPID has a further tuning parameter this is indispensable order and differential order µ [19, 20].

The differ-integral operator, denoted by, is a mixed differentiation-integration operator usually utilized in fractional calculus, this operator is a notation for taking both the fractional by-product and the fractional critical in a single expression and is described by [21, 22]:

$\alpha D_t^q=\left\{\begin{array}{lc}\frac{d^q}{d t^q} & q>0 \\ 1 & q=0 \\ \int_\alpha^t(d \tau)^{-q} & q<0\end{array}\right.$                     (3)

where, q is the fractional-order which can be a complex number, and a and t are the limits of operation.

There are some definitions for fractional derivatives, the commonly used definition is Grunwald-Letnikov [23], Riemann-Liouville [24], and Caputo definition [25]. The Grunwald-Letnikov Definity:

$\begin{aligned} & a D_t^q f(t)=\frac{d^q f(t)}{d(t-a)^q}  =\lim _{N \rightarrow \infty}\left[\frac{t-a}{N}\right]^{-q} \sum_{j=0}^{N-1}(-1)^j\left(\begin{array}{l}q \\ j\end{array}\right) f\left(t-j\left[\frac{t-a}{N}\right]\right)\end{aligned}$                             (4)

The Riemann-Liouville is given by definition:

$\begin{array}{r}a D_t^q f(t)=\frac{d^q f(t)}{d(t-a)^q} \frac{1}{\Gamma(n-q)} \frac{d^n}{d t^n} \int_0^t(t  -\tau)^{n-q-1} f(\tau) d \tau\end{array}$                    (5)

With n is the first integral which is not less than q i.e n-1≤q≤n and Γ is Gamma function definition by equation follows:

$\Gamma(z)=\int_0^{\infty} t^{z-1-t} e^{-t} d t$                                            (6)

The Caputo fractional derivative gives by:

$L\left\{\alpha D_t^q f(t)\right\}=S^q F(S)-\sum_{k=0}^{n-1} S^k 0 D_t^{q-k-1} f^k(0)$                               (7)

With: $n-1 \leq q \leq n \in \mathbb{N}$

The mathematical illustration of the FOPID Controller is as given [26]:

$G(s)=K_p+\frac{K_i}{S^\lambda}+K_d S^\mu$                               (8)

The method used for tuning FOPID is PSO algorithms this technique is inspired by the social behavior of birds and fish, it initializes by n-swarm of particles and defined by the position of the particle of a swarm, and velocity of a particle, at iteration, are defined as following equation [27, 28]:

$\begin{aligned} V_{i j}(i+1)=w V_{i j} & +C_1 r_1\left(\text { Pbest }_{i j}-X_{i j}\right) +C_2 r_2\left(\text { Gbest }-X_{i j}\right)\end{aligned}$                    (9)

With:

$w=w_{\max }\left(\frac{\text { iter }}{\text { maxiter }}\right)\left(w_{\max }-w_{\min }\right)$                                                (11)

where, i=1, 2,.., n, and j is a search space r₁ and r₂ is a random number (0,1). C₁ called cognitive parameters pulls each particle towards a local best position. C₂ called social parameters pulls the particle towards a global best position, w_max is final weight and w_min initial weight, maxiter is a maximum iteration number, iter is a current iteration number.

The steps of particle swarm algorithm for optimization the FOPID shown in the points follows:

• Step 01: Define objective function;
• Step 02: Define PSO parameters;
• Step 03: Initialization of position and velocity;
• Step 04: Define function Evolution;
• Step 05: Update Pbest and Gbest;
• Step 06: Compute velocity and position handling boundary;
• Step 07: Stope while getting the best value.

The parameters of FOPID tuned by particle swarm optimization algorithm (PSO) are illustrated in Table 2:

Table 2. Parameters of fractional-order PID controller (FIPID)

3.2 The linear quadratic regulator (LQR) control

An advantage of the linear quadratic regulator or quadratic optimal control method over the pole-placement method is that the former provides a systematic way of computing the state feedback control gain matrix [29].

We Shall now consider the optimal regulator problem that given the system equation [29-31]:

Determine the matrix K of the optimal control vector:

The state variable feedback configuration is proven below in Figure 2:

2.png

Figure 2. State variable feedback configuration

So, as to minimize the performance index:

$J=\int_0^{\infty}\left(X^T Q X+U^T R U\right) d t$                                          (14)

where, Q and R is a position definite Hermitian or real symmetric.

The design steps are follows [32]:

Solve the following equation for the matrix P [If a position-definite matrix P (n x n) matrix exists (Sure systems might not have a position definite matrix P), the system is stable, or matrix A-BK is stable]:

Substitute this matrix P into the following equation, the resulting matrix K is the optimal matrix:

$K=R^{-1} B^T P$                                (16)

Q and R taking by:

$Q=\left[\begin{array}{cccc}10^{-2} & 0 & 0 & 0 \\ 0 & 10^{-9} & 0 & 0 \\ 0 & 0 & 10^{-3} & 0 \\ 0 & 0 & 0 & 10^{-4}\end{array}\right], R=\left[\begin{array}{cc}0.001 & 0 \\ 0 & 0.12\end{array}\right]$

The values of obtained feedback gain matrix of LQR:

$K=\left[\begin{array}{cccc}0.0010 & -0.0001 & -0.0001 & 0.0010 \\ 0.2372 & -0.3078 & 0.0878 & 0.3082\end{array}\right]$