Stochastic tracking control of multivariable nonlinear systems subject to external disturbances

Abstract

By incorporating a disturbance rejection control law with receding horizon optimization technique, this paper synthesizes a stochastic tracking nonlinear model predictive control (STNMPC) framework for multivariable nonlinear systems subject to external disturbances. In the control scheme, a cancellation strategy combined with a feedforward prefilter is adopted to drive the expected value of the output to the reference signal by compensating the undesired components in the feedback loop. Given the statistics of the stochastic disturbances, covariance evolution is involved to describe the propagation of uncertainties through the system model and a time-varying feedback control law is designed to minimize the covariance. By employing the strong optimizing property of model predictive control (MPC), a stochastic MPC (SMPC) framework is formulated. By using the Schur complement, the nonlinear dependence of covariance evolution on time-varying feedback control gains is removed and the cost function is reformulated as a linear one with linear matrix inequalities, which resolves the implementation issue existing in the numerical problem. To guarantee the convergence and recursive feasibility of the SMPC framework, terminal constraint with some tricks are applied, for example, Lyapunov equation. The well-established theories and tricky techniques allow to cast the SMPC framework as a semidefinite programming, which reduces the computational cost and yields a feasible numerical solution. Numerical examples are provided to demonstrate the effectiveness of the proposed control method, comparisons with some adaptive control algorithm shows the superiority of the proposed approach.

A model‐and data‐driven predictive control approach for tracking of stochastic nonlinear systems using Gaussian processes

“…This can be in part explained by the difficulty of propagating stochastic uncertainties through a nonlinear system model without being prohibitively expensive. This also explains why the author's previous work 23 separates the uncertainty compensation control law from the finite‐horizon OCP and takes no account of the chance constraints.…”

“…A disadvantage of the PCE-based SNMPC is that the computational cost scales exponentially with the number of uncertain parameters. 23,39 To address these concerns, two tractable approximations to the SNMPC problems with the incorporation of unscented Kalman filter (UKF) 40 and Gaussian process (GP) 40 were proposed. Taking advantage of the UKF which estimates and propagates the entire conditional distribution of the states over the prediction horizon by taking into account state estimation errors, the UKF-SNMPC reformulates the model cost and constraint functions by using the mean and covariance information, which yields a tractable control framework for handling nonlinear stochastic dynamic optimization (SDO) problems.…”

“…SMPC exploits the probability distribution of stochastic uncertainties and allows admissible constraint violations at a prescribed probabilistic threshold, which reformulates the hard constraints on the system variables as stochastic ones, this is referred to as chance constraints or probabilistic constraints. By exploiting the statistics including the mean and variance information, SMPC fully determines and propagates the states over the prediction horizon and addresses the constraints and objectives in a probabilistic sense, which alleviates the conservatism mentioned above by seeking a trade‐off between the risk of constraint violation and closed‐loop control performance 22,23 . However, many existing SMPC results focus on linear systems, 24‐33 stochastic NMPC (SNMPC) has received relatively little attention.…”

“…In PCE‐based SNMPC, orthogonal polynomials are employed to describe the mappings between variables/parameters and the states. A disadvantage of the PCE‐based SNMPC is that the computational cost scales exponentially with the number of uncertain parameters 23,39 . To address these concerns, two tractable approximations to the SNMPC problems with the incorporation of unscented Kalman filter (UKF) 40 and Gaussian process (GP) 40 were proposed.…”

“…By exploiting the statistics including the mean and variance information, SMPC fully determines and propagates the states over the prediction horizon and addresses the constraints and objectives in a probabilistic sense, which alleviates the conservatism mentioned above by seeking a trade-off between the risk of constraint violation and closed-loop control performance. 22,23 However, many existing SMPC results focus on linear systems, [24][25][26][27][28][29][30][31][32][33] stochastic NMPC (SNMPC) has received relatively little attention. This can be in part explained by the difficulty of propagating stochastic uncertainties through a nonlinear system model without being prohibitively expensive.…”