Model Predictive Control of a Car Active Seat Suspension System

Parrot Mambo mini-drone is a readily available commercial quadrotor platform to understand and analyze the behavior of a quadrotor both in indoor and outdoor applications. This study evaluates the performance of three alternative controllers on a Parrot Mambo mini-drone in an interior environment, including Proportional–Integral–Derivative (PID), Linear Quadratic Regulator (LQR), and Model Predictive Control (MPC). To investigate the controllers’ performance, initially, the MATLAB®/Simulink™ environment was considered as the simulation platform. The successful simulation results finally led to the implementation of the controllers in real-time in the Parrot Mambo mini-drone. Here, MPC surpasses PID and LQR in ensuring the system’s stability and robustness in simulation and real-time experiment results. Thus, this work makes a contribution by introducing the impact of MPC on this quadrotor platform, such as system stability and robustness, and showing its efficacy over PID and LQR. All three controllers demonstrate similar tracking performance in simulations and experiments. In steady state, the maximal pitch deviation for the PID controller is 0.075 rad, for the LQR, it is 0.025 rad, and for the MPC, it is 0.04 rad. The maximum pitch deviation for the PID-based controller is 0.3 rad after the take-off impulse, 0.06 rad for the LQR, and 0.17 rad for the MPC.

Model predictive control (MPC) is a multi-objective control technique that can handle system constraints. However, the performance of an MPC controller highly relies on a proper prioritization weight for each objective, which highlights the need for a precise weight tuning technique. In this paper, we propose an analytical tuning technique by matching the MPC controller performance with the performance of a linear quadratic regulator (LQR) controller. The proposed methodology derives the transformation of a LQR weighting matrix with a fixed weighting factor using a discrete algebraic Riccati equation (DARE) and designs an MPC controller using the idea of a discrete time linear quadratic tracking problem (LQT) in the presence of constraints. The proposed methodology ensures optimal performance between unconstrained MPC and LQR controllers and provides a sub-optimal solution while the constraints are active during transient operations. The resulting MPC behaves as the discrete time LQR by selecting an appropriate weighting matrix in the MPC control problem and ensures the asymptotic stability of the system. In this paper, the effectiveness of the proposed technique is investigated in the application of a novel vehicle collision avoidance system that is designed in the form of linear inequality constraints within MPC. The simulation results confirm the potency of the proposed MPC control technique in performing a safe, feasible and collision-free path while respecting the inputs, states and collision avoidance constraints.

Considering the variability of driving conditions in mining areas, existing control strategies are difficult to meet the comprehensive performance requirements of mining dump trucks in the longitudinal, lateral, and vertical directions. Longitudinal, lateral, and vertical (LLV) coordinated control of active hydro-pneumatic suspension system based on model predictive control (MPC) is constructed in this paper. The vehicle dynamic response under random road surface input based on wheelbase characteristics is established, and the rationality of the active hydro-pneumatic suspension LLV coordinated control strategy based on MPC is analyzed. Handling stability is taken as the overall control objective for active hydro-pneumatic suspension on C-class road surfaces. The dynamic tire loads of the six wheels of the mining dump truck are reduced by 25.8%, 29.1%, 30.6%, 27.6%, 29.9%, and 28.1%, respectively, in the unloaded state, while the longitudinal, lateral, and vertical body accelerations have not deteriorated. Under the E-class road surface, the overall control objective of the mining dump truck is comfort, and the longitudinal, lateral, and vertical accelerations in the unloaded state have been optimized by 34.6%, 31.4%, and 34.1%, respectively.

Electrical drives play an important role in the implementation of tasks for increasing productivity in various industries, for automation and complex mechanization of production processes. In this paper three strategies; a classical Proportional-Integral-Derivative (PID), a Linear Quadratic Regulator (LQR) and a Model Predictive Control (MPC) are designed and implemented for the DC motor speed control. All these strategies are compared on the basis of output response, steady state error, controller efforts, and constraints handling for the reference tracking. Performed simulations shows that though the controller efforts are less in PID and LQR but MPC outperforms for reference tracking and constraints handling.

This study examines implementation of a Model Predictive Controller (MPC) to a new concept in active suspension design. Active and passive components are placed in series to mitigate both high and low frequency disturbance inputs at the tire-road interface. This is modelled using an additional mass spring damper tuned to regulate high frequency inputs, leaving the active components to respond to low frequency inputs. A generic half car model for such a system is developed and subjected to various disturbance inputs at constant velocity and output to verify the system dynamics. Inputs include step, multimodal, and random disturbances as well as a step input that returns to zero. These trials serve as a baseline to evaluate the performance of the passive suspension as well as a Model Predictive Controller. Current research that uses MPC in active suspension design focuses heavily on the traditional half car model with 4 DOF[4]. MPC is applied to this new 6 DOF model and incorporates preview information into the controller response for each of the test cases. The cost function for the MPC places penalties on the translational and rotational position and velocity of the chassis relative to a reference state that is based on each disturbance profile. Parameters of interest are driver absorbed power due to both linear and rotational movement of the chassis. The results for each test case demonstrate the utility of MPC. For every response, there is a decrease in the absorbed power due to rotational and linear sources on the magnitude of 98–100%. The incorporation of preview information also removed the rotation of the chassis for each test case by placing a heavy weight upon its movement. For the step input, the controller reduced the peak rate of change of the chassis by 71.4%. For the multi-mode input, the low frequency sinusoidal inputs showed a dramatic reduction in vertical displacement in the steady state behavior as the MPC will produce an output that is tuned to cancel the disturbance. The high frequency effects are also effectively removed by the passive components of the suspension. This ability to mitigate both sources of disturbance is a marked advantage of the double-stacked suspension design. MPC allowed for the overall reduction of chassis movement by 54.0% with preview information. This improvement is due to the ability of the double stacked suspension with MPC to use the additional degrees of freedom to attenuate disturbances at more than one frequency. The random input demonstrates the ability of the controller to maintain a smooth chassis trajectory even with a chaotic road profile. Finally, the step up-down input demonstrates the ability of the controller to use other components of the suspension system to mitigate a disturbance in order to keep the chassis stable. These results demonstrate that preview information can be used to take full advantage of double stacked, active suspensions and further enhance mobility over different kinds of terrain. Future work includes investigating the effectiveness of other predictive control methods such as two-point boundary value problem or dynamic programming, optimizing the weights used, or adding constraints to the model.

The original MPC(Model Predictive Control) algorithm cannot be applied to open loop unstable systems, because the step responses of the open loop unstable system never reach steadystates. So when we apply MPC to the open loop unstable systems, first we have to stabilize them by state feedback or output feedback. Then the stabilized systems can be controlled by MPC. But problems such as valve saturation may occur because the manipulated input is the summation of the state feedback output and the MPC output. Therefore, we propose Quadratic Dynamic Matrix Control(QDMC) combined with state feedback as a new method to handle the constraints on manipulated variables for multivariable unstable processes. We applied this control method to a single-input-single-output unstable nonlinear system and a multi-input-multi-output unstable system. The results show that this method is robust and can handle the input constraints explicitly and also its control performance is better than that of others such as well tuned PI control. Linear Quadratic Regulator (LQR) with integral action.

This paper presents a method to design a path tracking controller with a constraint on tire slip angles under low-friction road conditions. On a low-friction road surface, a lateral tire force is easily saturated and decreases as a tire slip angle increases by a large steering angle. Under this situation, a path tracking controller cannot achieve its maximum performance. To cope with this problem, it is necessary to limit tire slip angles to a value where the maximum lateral tire force is achieved. The most commonly used controllers for path tracking, linear quadratic regulator (LQR) and model predictive control (MPC), are adopted as a controller design methodology. The control inputs of LQR and MPC are front and rear steering angles and control yaw moment, which have been widely used for path tracking. The constraint derived from tire slip angles is imposed on the steering angles of LQR and MPC. To fully verify the performance of the path tracking controller with the constraint on tire slip angles, a simulation is conducted on vehicle simulation software. From the simulation results, it is shown that the path tracking controller with the constraint on tire slip angles presented in this paper is quite effective for path tracking on low-friction road surface.

The modern era vehicles generally have an Cruise Control (ACC) system installed in them. Intended to assist drivers, some researchers have worked on multi-objectives like road safety improvement and driving process enhancement; hence, approaches like Linear Quadratic Regulator (LQR) and Model Predictive Control (MPC) are primarily used in the literature. As indicated in various literature, ACC equipped with MPC can be designed with multi-objectives and constraints for improved and reliable performance of automated vehicles. Therefore, ACC system based on MPC with an optimization known as Coati Optimization Algorithm (COA), i.e., MPC-COA, is introduced in this manuscript. The proposed COA for ACC provides necessary stability and maintains a safer distance between the ACC equipped host vehicle and the former target vehicle. The proposed MPC-COA technique is simulated on the MATLAB platform, and the results based on performance metrics like desired acceleration input, velocity error, distance error, and acceleration of ACC equipped host vehicle are compared with those of the conventional LQR and MPC controller. By this introduced strategy, the acceleration, the velocity, and the inter-vehicle distance error are regulated optimally. Consequently, the convergence speed is enhanced, and the optimal solution is maintained without deviation in the desired performances. The controller is further evaluated for different prediction horizons(N).

In this study, we investigate a robust H∞ controller for a quarter-car model of an active inerter-based suspension system under parameter uncertainties and road disturbance. Its main objective is to improve the inherent compromises between ride quality, handling performance, suspension stroke, and energy consumption. Inerters have been extensively used to suppress unwanted vibrations from various kinds of mechanical structures. The advantage of inerter is that the realized ratio of equivalent mass (inertance relative to the mass of the primary structure) is greater than its actual mass ratio, resulting in higher performance for the same effective mass. First, the dynamics and state space of the active inerter-based suspension system were achieved for the quarter-car model with parameter uncertainties. In order to attain the defined objectives, and ensure that the closed-loop system achieves the prescribed disturbance attenuation level, the Lyapunov stability function, and linear matrix inequality (LMI) techniques have been utilized to satisfy the robust H∞ criterion. Furthermore, to limit the gain of the controller, some LMIs have been added. In the case of feasibility, sufficient LMI conditions by solving a convex optimization problem afford the stabilizing gain of the robust state-feedback controller. According to numerical simulations, the active inerter-based suspension system in the presence of parameter uncertainties and external disturbance performs much better than both a passive suspension with inerter and active suspension without inerter.

Conventional linear controllers with active damping (AD) cannot stabilise LCL-filters for certain ranges of filter parameters due to resonances. This limits the design space of the filter design and can lead to sub-optimal designs in terms of filter volume and efficiency. When model predictive control (MPC) with a long-prediction horizon is applied, the inherent damping capability of MPC can stabilise the system despite the resonance phenomenon and expands the design space. In this paper, the limited design space due to insufficient damping of linear controllers with AD is analyzed based on capacitor current feedback AD. Exemplary LCL-filter optimization solutions and the resulting Pareto fronts are presented to illustrate the achiev-able volume and/or loss reductions with the extended design space. A novel MPC formulation is proposed to adopt explicit MPC and to enable a real-time implementation even at a high switching frequency. Simulation results validate the presented analysis.

International audience

Model predictive control (MPC) is a powerful control method that handles dynamical systems with constraints. However, solving MPC iteratively in real time, i.e., implicit MPC, remains a computational challenge. To address this, common solutions include explicit MPC and function approximation. Both methods, whenever applicable, may improve the computational efficiency of the implicit MPC by several orders of magnitude. Nevertheless, explicit MPC often requires expensive pre-computation and does not easily apply to higher-dimensional problems. Meanwhile, function approximation, although scales better with dimension, still requires pre-training on a large dataset and generally cannot guarantee to find an accurate surrogate policy, the failure of which often leads to closed-loop instability. To address these issues, we propose a triple-mode hybrid control scheme, named Memory-Augmented MPC, by combining a linear quadratic regulator, a neural network, and an MPC. From its standard form, we derive two variants of such hybrid control scheme: one customized for chaotic systems and the other for slow systems. The proposed scheme does not require pre-computation and is capable of improving the amortized running time of the composed MPC with a well-trained neural network. In addition, the scheme maintains closed-loop stability with any neural networks of proper input and output dimensions, alleviating the need for certifying optimality of the neural network in safety-critical applications. Note to Practitioners—This article was motivated by the need to reduce the amortized cost of MPC in repetitive industrial robotic applications, where long-term operational cost is important and safety is critical. Examples of such applications include factory robotic arm manipulation and fixed-route quadcopter payload transport. Unlike explicit MPC or function approximation, our approach does not require any pre-computation or pre-training. Rather, it attains task proficiency over time by learning a surrogate neural network on the spot and by gradually replacing the costly MPC with the more efficient surrogate model so long as safety permits. Consequently, the proposed scheme incurs a learning cost during the initial phase of the deployment but usually becomes more adept on the task afterwards, leading to amortized efficiency.

Utility companies seek to increase energy efficiency and productivity and try to reduce peak loads. This often involves consumer-side demand management in residential areas using dynamic time-of-use (ToU) tariff. Such strategies work if the consumer-side response is at least partly automated using some real-time optimization strategy. Our paper proposes a consumer-side optimization and control framework for scheduling the electric appliances in a smart household and preserving a thermal comfort level through an electric heating system. Our framework consists of two optimization components interacting with each other. The first optimization component schedules the home appliances based on a mixed integer programming approach. An electric vehicle (EV) is considered as a special home appliance with an energy storage capability. The second optimization component is the model predictive control (MPC) strategy for the electric heating system, such that the input constraints are defined by the scheduling results of the first component. Due to outside temperature variations, the input constraints may impede the MPC to maintain the required thermal comfort, which triggers a rescheduling event for the first component. The efficiency of the framework is presented in multiple simulations for scenarios with different consumer behaviors.

On a flat city road, the specificity of the suspension poses a challenge. Sport utility vehicles (SUVs) have less flexibility compared to regular cars, despite providing a firm grip on asphalt and concrete. Constant adjustments to the center of gravity of SUVs are necessary. Surprisingly, they are less reliable and stable in urban settings due to this characteristic. In this study, a mathematical model of the suspension system of SUVs based on the Newtonian approach is introduced and validated with data from experiments carried out by the MTS machine system on the mono-tube (oil/air) mixed damper element. The model accurately predicts the performance of this damper, commonly used in SUVs, across various operating conditions, including different frequencies. A coil spring element, serving as a passive suspension unit with this damper, is also tested experimentally under similar conditions. By integrating passive suspension elements with the active actuator, the proposed modified design reduces the power consumption needed for the actuator to function and ensures a certain level of reliability. The effectiveness and performance of the modified active suspension system in comparison to the traditional passive suspension system are assessed using three different strategies: hybrid PID-LQR, linear quadratic regulator (LQR), and proportional-integral-derivative (PID). The genetic algorithm is utilized to determine the optimal parameter values for each controller by minimizing a cost function, maximizing performance, and minimizing energy consumption. Simulation results demonstrate that the active suspension system controlled by the PID-LQR controller offers significantly improved ride comfort and vehicle stability compared to other systems. This suggests that the performance of the active suspension system is greatly enhanced by the combined application of the hybrid PID-LQR controller compared to other systems.

This paper proposes a solution method for the discrete-time long-term dynamic portfolio optimization problem with state and asset allocation constraints. We use the ideas of Model Predictive Control (MPC) to solve the constrained stochastic control problem. MPC is a solution technique which was developed to solve constrained optimal control problems for deterministic control applications. MPC solves the optimal control problem with a receding horizon where a series of consecutive open-loop optimal control problems is solved. The aim of this paper is to develop an MPC approach to the problem of long-term portfolio optimization when the expected returns of the risky assets are modeled using a factor model based on stochastic Gaussian processes. We prove that MPC is a suboptimal control strategy for stochastic systems which uses the new information advantageously and thus is better than the pure optimal open-loop control. For the open-loop optimal control optimization, we derive the conditional portfolio distribution and the corresponding conditional portfolio mean and variance. The mean and the variance depend on future decision about the asset allocation. For the dynamic portfolio optimization problem, we consider constraints on the asset allocation as well as probabilistic constraints on the attainable values of the portfolio wealth. We discuss two different objectives, a classical mean–variance objective and the objective to maximize the probability of exceeding a predetermined value of the portfolio. The dynamic portfolio optimization problem is stated, and the solution via MPC is explained in detail. The results are then illustrated in a case study.