A state-compensation extended state observer for model predictive control

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Abstract

Motion control in absence of human involvement is difficult to realize for autonomous vessels because there usually exist environmental disturbances and unmeasurable states at the same time. A discrete-time model predictive control (MPC) approach based on a state-compensation extended state observer (SCESO) is proposed to achieve more precise control performance with state estimations and disturbance rejections simultaneously. The main idea is that lumped disturbances encompassing nonlinear dynamics and external disturbances are handled as two parts, unlike the standard extended state observer (ESO). Particularly, the nonlinear terms are compensated by estimated states and the external disturbances are considered as extended states and attenuated by the traditional ESO strategy. Assuming that the lumped disturbances are constant over the prediction horizon, the prediction model is linearized to save computational time since after linearization the online MPC optimization problems are solved as quadratic programming problems instead of nonlinear programming problems. The convergence of the proposed SCESO estimation errors to zero is proved even when the disturbances keep variable. Two case studies involving a numerical example and ship heading control have been conducted to verify the effectiveness of the proposed control method.

Introduction

In a real system, there usually exist external disturbances, nonlinear dynamics, and unmeasurable states that bring challenges to the controller design for such a system [6], [19], [41], [42]. Autonomous vessels have been encountering these challenges in motion control, e.g., path following or trajectory tracking [20], [45], because wind, current or waves always exist and heading acceleration is hardly measured directly [26]. To obtain reliable performance, a controller needs to reject the effect of disturbances and use as precise as possible state information. Observers are often utilized to estimate states or disturbances that are unknown. For the estimation of states, state observers, e.g., Luenberger observers [43], Kalman filter-based estimators [35], and sliding observers [31], have been used widely. For the estimation of disturbances, disturbance observers (DOB) have been developed and applied in industry [5], [41]. A review of DOB-based control (DOBC) methods can be found in [6]. Generally, states and disturbances need to be estimated at the same time. Unknown input observers (UIO) [4], [15], and extended state observers (ESO) [13], [14], can deal with state and disturbance estimation problems simultaneously [15]. UIO has been used widely in fault diagnosis and isolation [4], [27], [28], [37]. ESO was first proposed for active disturbance rejection control by Han [13]. After that, ESO has been used and discussed widely with applications of active disturbance rejection control [11], [30], [32], [33]. ESO is actually the same as UIO if the assumptions of disturbances for UIO and ESO are consistent [6].

Different from most existing observers, the ESO adds another state to a system instead of reducing the system order [38] and requires the least amount of system information [46]. ESO-based control has also been applied as composite control combined with feedback and disturbance compensation [19], predictive functional control [21], and sliding mode control [39]. Moreover, the ESO itself has been improved both in the aspects of practical applications and theories. A linear ESO method was presented to simplify the implementation of ESO for engineers in [9]. For a rigorous proof of the ESO convergence, a high gain approach was used to eliminate the influence of uncertainties for nonlinear extended state observers [12]. Furthermore, an extended high-gain state observer was proposed to deal with a class of nonlinear uncertain systems in which known nonlinear terms were used in the observer design [8]. A generalized ESO was represented to deal with a system model that did not satisfy the standard chain form [19]. However, there are some defects for ESO. For instance, the ESO estimated errors cannot be guaranteed to converge to zero unless under the assumptions that the disturbances are constant [19]. Generally, the errors can only be guaranteed to be within the vicinity of zero with bounded disturbances [9], [12].

Model predictive control (MPC) is an important advanced control method in industrial areas due to its optimized control performance and the ability of considering various kinds of constraints explicitly [22], [42]. An MPC based controller relies on state measuring, disturbance estimating and an accurate prediction model [47]. DOB based MPC methods have been developed and proved to be effective by compensating the effect of unknown disturbances and uncertainties recently [18], [40], [41], [42], [47]. Considering that the ESO is a combination of DOB and state observer, it is reasonable to utilize the ESO to estimate unmeasurable or costly-measured states and unknown disturbances at the same time for a MPC based controller. A simplified MPC method, i.e., predictive functional control (PFC), is employed with the ESO for speed control of permanent magnet synchronous motor servo system [21]. One deficiency of the proposed method in [21] is that the ESO based feedforward control law is designed separately and is not taken into account in the receding optimization process of PFC. Another deficiency is that there is no strict proof of the convergence of ESO estimation errors [41]. Note that MPC is usually implemented as a digital control because an analog circuit hardly deals with online linear programming, quadratic programming, or nonlinear programming problems [36]. For the digital control, system and observer models should be discretized and the control input should be updated during the sampling interval [7].

In this article, an improved ESO based MPC approach for a discrete-time prediction model is proposed to achieve more precise control performance while estimating states and rejecting disturbances. The main idea is that lumped disturbances encompassing nonlinear dynamics and external disturbances are handled separately, which is different from the standard ESO. In this method, the nonlinear terms are compensated by estimated states, and external disturbances are considered as extended states and estimated by the proposed ESO. The lumped disturbances are considered constant in the prediction horizon, which makes a nonlinear programming problem become a quadratic programming problem. The external disturbances consist of high order polynomials as in [17]. Different from the proof in [19], the convergence of the proposed ESO is proved when the disturbances keep variable. A numerical example is conducted to prove the advancement of the proposed method compared with the previous method in [19], and the proposed method is applied for the vessel heading control in presence of disturbances and unknown states in comparison with PID (proportional-integral-derivative) method.

The remainder of this article is organized as follows. Generalized ESO is introduced in Section 2. In Section 3, an improved ESO based on the generalized ESO, i.e., state-compensation ESO (SCESO), is proposed with continuous-time and discrete-time forms. In Section 4 and 5, an SCESO based MPC scheme is elaborated on and relevant stability is analyzed. Then, a numerical example and a ship heading control case are studied in Section 6. Conclusions and future research are presented in Section 7.

Section snippets

Generalized extended state observer

However, a different system, for instance second-order system (2), is not consistent with the standard integral chain form as (1), and the channel of lumped disturbances is also different from the channel of input in system (2) [19]. {x˙1=x12x2+f(x1,x2,d(t),t)x˙2=x1+x2+u

Considering that systems not satisfying standard ESO systems, like system (2), can not be dealt with by normal ESO methods, a new system form for a generalized ESO is proposed in [19]: {x˙=Ax+Bu+Dfd(x,d(t),t)y=Cx,where xRn×1

State-compensation extended state observer

For Lemma 2, the assumptions are not always satisfied if the lumped disturbances contain a nonlinear term w(x), i.e., fd(x, d(t), t) ≠ f(d(t), t), or if lumped disturbances are not constant at the steady state, or if there exist constraints on system inputs. To handle systems where the assumptions for Lemma 2 are not satisfied, in this section, an improved generalized ESO, i.e., state-compensation extend state observer (SCESO), is proposed. A continuous-time SCESO is developed firstly, then a

SCESO based model predictive control

MPC is used to design a controller to optimize the tracking performance considering system constraints. The prediction model is based on the nominal dynamics that are updated by the SCESO observer (8). Without loss of generality, the control objective is to achieve xxo where xo is the objective and stable state vector. Therefore, the cost function J(k) at instant time k is defined as follows: J(k)=i=1NP{x˜(k+i)xoQ2+u˜(k+i1)R2},where Q ≥ 0 and R > 0 are the weighting matrix and

Stability analysis for observer

Assumption 1

a) The order of external disturbances in (7), i.e., q, is known; b) fd(x, d(t), t) is differentiable on t; c) w¯(xf)(k)2 and w¯(x^f)(k)2 are bounded.

Assumption 2

The ith eigenvalue λi(i=1,,n+q+1) of Afce satisfies |λi| < 1 where Afce=AfcLfcCfc.

Theorem 1

If the Assumptions 1 and 2 are satisfied, the observer error ef(k) of SCESO is bounded, whereef(k)=xf(k)x^f(k).

Proof

Based on (15) and (14), ef is obtained: ef(k+1)=Afceef(k)+Δw¯(k),where Δw¯(k)=Dfc[w¯(xf(k))w¯(x^f(k))]. With |λi| < 1, let a Lyapunov function set

Case study

To verify the effectiveness of the proposed SCESO approach, simulation experiments are conducted. Firstly, a numerical example in [19] is presented and the control performance with the proposed method is compared with the control performance with the method proposed in [19], then an application of the proposed control and observer strategy to ship heading control is presented.

Conclusions and future research

Considering that unmeasurable states and external disturbances usually exist in dynamic systems, it is difficult to control these systems without the information of system states and external disturbances, especially in the presence of system nonlinearities. To acquire accurate state information and attenuate the effect of disturbances is important to improve control performance. A state-compensation extended state observer based model predictive control method is proposed to obtain better

Acknowledgment

This research is supported by the China Scholarship Council under Grant 201506950053, the National Natural Science Foundation of China (NSFC) Project (No. 61273234), the Natural Science Foundation of Hubei Province Project (No. 2015CFA111), and the Project of Ministry of Transport, China (No. 2015326548030).

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