The rest of this paper is organized as follows. (3) It is shown that the approach proposed in this paper is effective and performs better with the faster response, smaller overshoot, stronger robustness and so on by a practical example. The existence of the controller can be expressed by the convex optimization algorithm. A new model predictive controller is designed under the model transformation by approximating the state delay, such that the robust asymptotical stability of the closed-loop system is guaranteed. (2) In the controller design process, for the state-feedback MPC design objective, we formulate an optimization problem over an infinite time horizon. (1) The uncertainty is supposed to be polytopic uncertainty type, and the state with unknown delay with both specified upper and lower bounds is handled by an approximated model. We summarize the main contributions of this paper as follows. Motivated by the above observation, the problem of MPC for time-varying delay systems with parameter uncertainties and input constraints is studied in this paper. However, since the stability is guaranteed under the fixed constant weighting matrix at all time, the method is very limited and the conservatism may be generated. The work in put forward an MPC method for time-varying state-delay systems with uncertainty and constrained control input. Recently, the authors in presented an improved delay-dependent robust MPC to reduce the conservatism, still with a known delay. However, it is proper only if the delay indices are known.
![mpc 2 norm state feedback solution mpc 2 norm state feedback solution](https://www.mdpi.com/mathematics/mathematics-06-00051/article_deploy/html/images/mathematics-06-00051-g001.png)
To mention a few, in, the authors proposed that the control strategy for uncertain systems could be developed into a delay system via the MPC. Many results about MPC technique for time-delay systems have been addressed. Therefore, considerable researchers have been attracted to study the robust control problem of constrained uncertain systems with state delays. Moreover, there exist some physical limits, for instance power limitations and value saturation, in many industrial processes, which result in constraints on input and output. The authors in designed a novel output-feedback controller for the suspension systems with input delay. The authors in discussed the problem of dissipativity analysis of stochastic neural networks systems of discrete-time form with time-varying and finite-distributed delays. References studied the networked control with time-delay investigated linear switched systems with time-varying delay. In addition, time-delay often appears in industrial processes, which results in degradation and instability in such systems. They are norm-bounded parameter uncertainty and polytopic parameter uncertainty. In the literatures, two kinds of parameter uncertainties are often included in the uncertain systems.
![mpc 2 norm state feedback solution mpc 2 norm state feedback solution](https://forces.embotech.com/Documentation/_images/Example1_Basic_MPC_Simulink_Final.png)
In practical control systems, parameter uncertainties cannot be avoided. The original MPC technique is aimed at solving an open-loop optimization problem with constraints at every sampling instant, implementing only the first control step of solutions. In, the authors gave us an overview of the origins of model predictive control and the recent results. It has been shown from that model predictive control (MPC) is an effective way to handle multivariable constrained control problems, which appear in the chemical process control, the petrochemical industries, gas pipeline, and so on. The ideas of model predictive control and receding horizon control have been developed since 1960s. Finally, the applicability of the presented results are demonstrated by a practical example. Under model transformation, a new model predictive controller is designed such that the robust asymptotical stability of the closed-loop system can be guaranteed. For the state-feedback MPC design objective, we formulate an optimization problem.
![mpc 2 norm state feedback solution mpc 2 norm state feedback solution](https://www.mathworks.com/help/examples/mpc/win64/ReviewModelPredictiveControllerForStabilityAndRobustnessExample_08.png)
A new model is proposed to approximate the delay. The time-varying delay is considered with both upper and lower bounds.
![mpc 2 norm state feedback solution mpc 2 norm state feedback solution](https://i1.rgstatic.net/publication/222682015_The_Explicit_Linear_Quadratic_Regulator_for_Constrained_Systems/links/5e6ba700458515e555792719/largepreview.png)
This paper investigates the problem of model predictive control for a class of nonlinear systems subject to state delays and input constraints.