*3.5. H*<sup>2</sup> *and H*<sup>∞</sup> *Optimal State Feedback Designs*

We assume hereafter that the state is measurable and that the system (43) is fully controllable. This happens to be true when no loss of controllability arises from the introduction of the shaping filters (poles/zeros cancellations). Thus, state-feedback control laws of the form

$$\mu(t) = \mathbb{K} \mathbf{x}\_{\mathcal{S}}(t) \tag{45}$$

can be arbitrarily designed. In particular, the following closed-loop system results

$$\begin{cases}
\dot{\mathbf{x}}\_{\mathcal{S}}(t) = (A\_{\mathcal{S}} + B\_{\mathcal{S}^{\rm ul}}\mathbf{K})\mathbf{x}\_{\mathcal{S}}(t) + B\_{\mathcal{S}^{\rm ul}}\dot{\mathbf{z}}\_{r}(t) \\
\dot{\mathbf{z}}(t) = (\mathbf{C}\_{\mathcal{S}^{\rm z}} + D\_{\mathcal{S}^{\rm ul}}\mathbf{K})\mathbf{x}\_{\mathcal{S}}(t) + D\_{\mathcal{S}^{\rm zug}}\dot{\mathbf{z}}\_{r}(t)
\end{cases} \tag{46}$$

with

$$T(\mathbf{s}) = (\mathbf{C}\_{\mathcal{S}^2} + D\_{\mathcal{S}^{2u}} \mathcal{K}) (\mathbf{s}I - (A\_{\mathcal{S}} + B\_{\mathcal{S}^u} \mathcal{K}))^{-1} B\_{\mathcal{S}^{2v}} + D\_{\mathcal{S}^{2w}} \tag{47}$$

being the closed-loop transfer matrix between *z*˙(*t*) and *z*˜(*t*).

3.5.1. Optimal *H*∞ Control Synthesis

The *H*∞ optimal state-feedback control law

$$\boldsymbol{u}(t) = \mathbb{K}\_{\approx 0} \boldsymbol{x}\_{\mathcal{X}}(t) \tag{48}$$

that minimizes the *H*∞ norm of (47), that is

$$K\_{\otimes} := \arg\min\_{K} \|T(s)\|\_{H\_{\otimes}} \tag{49}$$

can be achieved by solving the following LMI optimization problem [14]

$$\begin{cases} \min\_{X, Y, \gamma} \\ \text{s.t.} \\ \begin{bmatrix} A\_{cl} + A\_{cl}^T & B\_{\mathcal{S}^{\text{uv}}} & (\mathbb{C}\_{\mathcal{S}^{\text{z}}} X + D\_{\mathcal{S}^{\text{z}\text{u}}} Y)^T \\ \ast & -\gamma I & D\_{\mathcal{S}^{\text{z}\text{uv}}}^T \\ \ast & \ast & -\gamma I \end{bmatrix} < 0 \\ X = X^T > 0 \end{cases} \tag{50}$$

where *Acl* := *AgX* + *BguY*. If the problem is feasible, the optimal *H*<sup>∞</sup> state-feedback gain is given by

$$K\_{\infty} = \mathcal{Y}X^{-1} \tag{51}$$

3.5.2. Optimal *H*<sup>2</sup> Control Synthesis

The *H*<sup>2</sup> optimal state-feedback control law

$$
\mu(t) = \mathcal{K}\_2 \mathbf{x}\_{\mathcal{S}}(t) \tag{52}
$$

that minimizes the *H*<sup>2</sup> norm of (47), that is

$$K\_2 := \arg\min\_K \|T(s)\|\_{H\_2}^2 \tag{53}$$

can be achieved by solving the following LMI optimization problem [14]

$$\begin{cases} \min\_{\boldsymbol{X}, \boldsymbol{Y}, \boldsymbol{Q}, \boldsymbol{\nu}} \\ \text{s.t.} \\ \begin{bmatrix} A\_{cl} + A\_{cl}^T & B\_{\mathcal{S}^H} \\ \ast & -I \end{bmatrix} < 0 \\ \begin{bmatrix} \mathbf{X} & (\mathbf{C}\_{\mathcal{S}^T} \mathbf{X} + D\_{\mathcal{S}^{\text{null}}} \mathbf{Y})^T \\ \ast & Q \end{bmatrix} > 0 \\ \begin{bmatrix} \mathbf{X} = \mathbf{X}^T > 0 \\ Q = \mathbf{Q}^T > 0 \\ \text{Tr}(Q) \le \boldsymbol{\nu}\_t \end{cases} \end{cases} \tag{54}$$

where *Acl* := *AgX* + *BguY*. If the problem is feasible, the optimal *H*<sup>2</sup> state-feedback gain is given by

$$K2 = \mathcal{Y}X^{-1} \tag{55}$$

#### **4. Simulation Results**

Several Matlab/Simulink simulations have been undertaken for assessing the full-car *H*<sup>2</sup> and *H*<sup>∞</sup> MIPC approaches presented here and compare them with the quarter-car *H*∞ state-feedback solution described in [3]. The latter control law, designed for the suspension systems of a single wheel, will be then applied to all four suspension systems in a decentralized way.

The three control approaches, namely *H*<sup>∞</sup> decentralized, *H*<sup>∞</sup> centralized and *H*<sup>2</sup> centralized, will be compared on the same car, actuators and driving scenario with the available design knobs tuned to achieve the same Ride Index for the three control strategies.

In particular, three values of the Ride Index will be considered: RI = 0.25, RI = 0.47 and RI = 0.70 that, according to the ISO 2631 RI classification, correspond respectively to *not uncomfortable*, *a little uncomfortable* and *fairly uncomfortable* likely passengers reactions

Road's profiles complying with the ISO-8608 standard [15] have been used in the simulations. In particular, all simulations have been undertaken by assuming to drive on a **C** straight road at 70 Km/h. Figure 5 depicts the corresponding profiles

**Figure 5.** Road profiles *zr*1(*t*), *zr*2(*t*), *zr*3(*t*) and *zr*4(*t*) used in the simulations.

The same standard class-C vehicle and actuator considered in [3] (where all relevant parameters are listed) is used here.

The following dynamical weights have been used in the three control strategies

$$\mathcal{W}\_{\rm ai}(s) = \rho \, \mathcal{W}\_k(s) \quad \text{(defined in (26))}\tag{56}$$

$$\mathcal{W}\_{ul}(s) = \beta \frac{s+1}{s+10} \tag{57}$$

$$\mathcal{W}\_{\rm di}(\mathbf{s}) = \gamma \frac{10}{\mathbf{s} + 1000} \tag{58}$$

with *ρ*, *β*, *γ* and *α* in (29) as free design knobs. The fact that the regulation of the Ride Index can be simply achieved by tuning the few control design knobs testifies favorably on the flexibility of the proposed MIPC approach.

#### *4.1. Simulations for Ride Index = 0.25*

The values of design knobs values used in the RI = 0.25 simulations are reported in Table 1 whereas the plots of the acceleration *z*¨*s*1, actuation current *i*1(*t*) and the harvested electrical power *Pe*1(*t*) under the three control laws are reported in Figure 6

**RI = 0.25 (Excellent Comfort/Smallest Harvesting**) **Control** *ρ βγ α* H∞,*dec* 0.2067 100 10 30 H∞,*cen* 0.42 3.45 1 27 H2,*cen* 0.5 2 1 29

**Table 1.** Knobs tuning—RI = 0.25.

**Figure 6.** (**top**) Accelerations *z*¨*s*1(*t*), (**middle**) Actuation current *i*1(*t*), (**down**) Instantaneous harvested electrical power *Pe*1(*t*).

#### *4.2. Simulations for Ride Index = 0.47*

In Table 2 the design knobs values used in the RI = 0.47 simulations are reported while the plots of the corresponding acceleration *z*¨*s*1, actuation current *i*1(*t*)and the electrical power *Pe*1(*t*) under the three control laws are reported in Figure 7.


**Table 2.** Knobs tuning—RI = 0.47.

**Figure 7.** (**top**) Accelerations *z*¨*s*1(*t*), (**middle**) Actuation current *i*1(*t*), (**down**) Instantaneous harvested electrical power *Pe*1(*t*).

#### *4.3. Simulations for Ride Index = 0.70*

Finally, in Table 3 the values of the design knobs used for the case RI = 0.70 are reported while the plots of the acceleration *z*¨*s*1, actuation current *i*1(*t*) and the harvested electrical power *Pe*1(*t*) under the three control laws are reported in Figure 8.

**Table 3.** Knobs tuning—RI = 0.70.



**Figure 8.** (**top**) Accelerations *z*¨*s*1(*t*), (**middle**) Actuation current *i*1(*t*), (**down**) Instantaneous harvested electrical power *Pe*1(*t*).

#### *4.4. Average Harvested Electrical Power*

Finally, next Table 4 summarize the average harvest electrical power during the simulations.

From the simulation it results that the H∞,*cen* and H2,*cen* regulators, designated on the basis of the full-car suspension model, guarantee good rider performance and higher levels of harvested energy with respect to the H∞,*dec* controller in all the situations tested. Moreover, the ride comfort performance of the H∞,*dec* regulator, designated on the basis of the quarter-car suspension model and implemented in a decentralized way, degrades remarkably at the increase of the energy harvesting requirements. This can be seen from the accelerations *z*¨*s*1(*t*) plots but similar conclusions can be drawn out by observing the sprung mass accelerations *z*¨*s*(*t*) and the pitch and roll angles evolutions (non reported here for space limitations but available in [13]).

As far as the amount of harvested energy during the simulations, Table 4 allows one to observe that the centralized H∞,*cen* and H2,*cen* regulators always are able to recover more energy than the decentralized H∞,*dec* controller, the more at lower values of the RI index. Moreover, it can be seen that the H2,*cen* regulator recovers the largest amount of electric power, while the H∞,*cen* regulator is more robust in rejecting the exogenous signals acting to the system.

**Table 4.** Average harvest electrical power. The percentages express the improvements with respect to the H∞,*dec* control achievements for the same RI.


#### **5. Conclusions**

The usage of regenerative suspension systems in modern electrical/hybrid cars could contribute to the vehicle's autonomy with a modest degradation to the usual ride comfort and road handling performance. The integration of such systems with other existing energy harvesting devices, such as regenerative brakes, may help the diffusion of this kind of cars, providing together a total amount of many tens/hundreds watts.

This paper has complemented the results achieved in [3] on the design of active control laws for the regulation of regenerative suspension systems by extending the scalar MIPC approach there presented for a quarter-car system to the general multivariable solution achieved on the basis of a full-car model. Moreover, both the *H*<sup>∞</sup> and *H*<sup>2</sup> state-feedback solutions have been considered based on a novel and *ad-hoc* state-space realization and have been shown to be enough flexible and powerful to easily trade-off amongst conflicting energetic and dynamic requirements.

From the simulations it results that the centralized solutions have to be preferred with respect to the decentralized one. In fact, from Table 4 it results that the improvements on the harvested energy are increasingly higher (up to 40% for the centralized *H*<sup>2</sup> and up to 20% for the centralized *H*<sup>∞</sup> optimal controllers) as the ride index RI decreases. This is especially important in the large part of cars usage, where maintaining low values of the ride index is of paramount importance for ensuring a good ride comfort. It is also found that the centralized *H*<sup>2</sup> control is able to gain the double of the energy harvested by the centralized *H*∞ control for the same value of the ride index.

An important contribution of this work is the demonstration that the coordination of the control actions achievable by a multivariable design has to be preferred than a simpler decentralized

implementation. The usage of modern *H*<sup>2</sup> and *H*<sup>∞</sup> multi-objective control design methodologies make the extra numerical complexity for the multivariable design negligible.

**Author Contributions:** Methodology, supervision and writing-review and editing A.C.; Investigation and formal analysis F.T.; Software, data curation, visualization and writing-original draft preparation P.V. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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