An Efficient Robust Servo Design for Non-Minimum Phase Discrete-Time Systems with Unknown Matched/Mismatched Input Disturbances

This paper presents an efficient proportional-plus-integral (PI) current-output observer-based linear quadratic discrete tracker (LQDT) design methodology for the non-minimum-phase (NMP) discrete-time system with equal input and output number, for which the minimalized dynamic system contains the unmeasurable system state and unknown external matched/mismatched input disturbances. Illustrative examples are given to demonstrate the effectiveness of the proposed approach.


Introduction
The unknown input observer (UIO) design methodology involves the state estimation for a dynamic system subject to unknown input excitation [1], in which it may contain internal uncertainties and exogenous loads that cannot be measured or inconvenient to measure. Several developed approaches to simultaneously estimate the system state and unknown input can be referred to [1][2][3][4] and in the literature therein. Nevertheless, most UIO design methodologies presented in the early literatures require that the transfer function from the unknown input to the system output is minimum-phase and of relative degree one. Recently, some design methodologies, which exempted the assumptions that the transfer function must be the minimum-phase (with respect to the relation between the unknown input to the system output, but not the relation between the control input to the system output) and unit-relative-degree constraints have been reported in [1,3,4] under some rank conditions. More precisely, in order for the design methodology to be feasible in [1,3], the dimensions of the unknown inputs/disturbances must be no greater than the output dimensions. In addition, the distribution matrix of unknown inputs/disturbances with some pre-specified rank conditions has be known a priori.
It is noticed that, in the real world, either the distribution matrix of unknown inputs/disturbances or the number of unknown inputs might be known, or both might be unknown. Even both are known, they might not satisfy the rank conditions. This implies that there can be more unknown inputs than both the control inputs and measured outputs. Also, noticed that when both the distribution matrix of unknown inputs/disturbances and the number of unknown inputs/ disturbances are unknown, it is difficult to precisely estimate the components of unknown inputs/disturbances. For a given Systems with Unknown Matched/Mismatched Input Disturbances controllable and observable continuous-time dynamic system with unknown mismatched inputs/disturbances, the theoretical design methodology has been investigated by She et al. [5], and it has been reported that, for all times, the output of the plant with mismatched disturbances would be identical to the output of the plant with matched disturbances, which were produced via an equivalent input disturbance (EID) entering into the plant through the control input channels. The above fact is called as the (continuous-time) EID principle. Thus, one can use a matched disturbance model as an equivalent of the mismatched disturbance model. As a result, the control objective for this case is to estimate and feedback the effects produced by the EID through the input channels to cancel the negative effects induced by the mismatched disturbance. More detailed comments on improving disturbance-rejection performance based on an EID principle can be found in [5,6]. Hence, based on the EID principle, a given NMP continuous-time plant with the mismatched non-minimum-phase disturbance model can be replaced by a given NMP continuous-time plant with the matched non-minimum-phase disturbance model. Nevertheless, neither the unknown input estimation nor the servo design methodology for the NMP plant has been fully addressed in [5,6].
Some new optimal proportional-integral-derivative (PID) filter-shaped PI feedback linear quadratic analog tracker/linear quadratic digital tracker (LQAT/LQDT) design methodologies for non-square NMP continuous-time and discrete-time transfer function matrices and their minimalized dynamic systems are proposed in the recent works [7] and [8], respectively. It is known that for the square NMP plant (with equal input and output number), it is difficult to find a non-singular square transformation matrix to ensure that the transformed closed-loop system becomes a minimum-phase (MP) one (in the sensor of 'control zero'). As a result, the PID-filter shaped PI feedback LQAT/LQDT design methodologies for the square NMP system are left for future research as shown in [7,8]. At this point, it must call the attention that the recently proposed approaches [7,8] still work well for the square and/or non-square MP plants with explicitly known disturbances, but not for the plants with unknown disturbances. An advanced design approach of observer-based optimal tracking controllers for time delay systems with external disturbances can be referred to the recent work in [9]. Also, new simultaneous state and output disturbance estimations for a class of switched linear systems with unknown inputs have been presented by Yang et al. [10]. Some precocious disturbance estimations and disturbance cancellation controller design methodologies can be referred to [11,12]. Nevertheless, the design methodologies for the NMP plant have not been addressed in [9][10][11][12].
In the real world, an unknown external disturbance usually occurs at the plant input, which would result in poor tracking performance. To overcome this issue, Chang [13] constructed the discrete-time proportional-integral observer (PIO) to develop a state and disturbance estimator for the discrete-time system with an unknown external disturbance to improve its tracking performance. Consequently, an advanced algorithm on the robust discrete-time output tracking controller design for NMP systems has been proposed by Chang et al. [14]. However, some restrictive conditions have been imposed on the existing design methodology in [14], namely: (i) The NMP plant has to be in square dimension; (ii) Only the matched input disturbance is considered; (iii) The variation of the disturbance in two consecutive sampling instants is not changed significantly; (iv) The model-following-based command generator which uses the constant input has to be constructed to generate the desired output; and (v) The use of the past output measurements to simultaneously estimate the system state and its equivalent input disturbance.
In this paper, for the NMP square strictly proper discrete-time transfer function matrix, for which the minimalized dynamic system with unmeasurable system state and its unknown external matched/mismatched input disturbances, we propose a PI current output-based observer (PICO) design methodology to simultaneously estimate the system state and its equivalent input disturbance. In the proposed approach, a variable input signal instead of the constant in [14] is used to generate a drastic command input. Then, a current-output observer-based LQDT with a high-gain property is developed to have the desired tracking performance. This paper is organized as follows. An efficient PI current-output observer-based LQDT for the NMP discrete-time system with equal input and output number, for which the minimalized dynamic system with unmeasurable system state and unknown external matched/mismatched input disturbances is presented in Sec. 2. Numerical simulations are given in Sec. 3 to demonstrate the effectiveness of the proposed approach. Finally, conclusion is given in Sec. 4.

Current-Output Observer-Based LQDT for NMP Discrete-Time System with an Unknown Disturbance
Comparison with the results developed in [14], some extensions over [14] are summarized as follows: (i) The use of the past output measurements to estimate the system state with unknown matched input disturbances developed in [14] has been extended to the use of the current output measurements to estimate the system state and the EID principle for the unknown matched/mismatched input disturbances, so that the tracking performance can be rapidly improved; (ii) The restriction on the constant command input of reference model presented in [14] has been relaxed to time-varying command input of reference model, so that a more robust and flexible tracker can be achieved; (iii) The closed-loop poles of the observer error dynamic system presented in this paper are optimally assigned inside a circle with a pre-specified radius α ( 0 1 ), α < ≤ but not just lying inside the unit circle ( 1 α = ) presented in [14], so that the transient response of the designed servo can be significantly improved.
To present the discrete-time EID principle, let the dynamic and ( ) provided that ( ) ( ) The objective of this paper is to design (i) a current-output-based state and disturbance observer and (ii) the observer-based optimal tracker, which would ensure that the controlled system in (1)-(4) has a desired tracking performance for a given arbitrary reference trajectory with some drastic variations. In this regards, the following assumptions are pre-assumed throughout the paper.
det( ) CH ≠ This condition implies that the dynamic system (1)-(4) has no transmission zeros lying at 'one', which often arises in the response to the robust tracking problem.
The design procedure of the PICO-based LQDT for the square NMP systems with equal input and output number and unknown matched/mismatched input disturbances is described in the following.
Lemma 1 [14]: Suppose that ( ) d k is bounded and smooth and Assumption 3 holds, then ( ) d k satisfies the following equations: Proof of Lemma 1: See [14,15]. Then, we offer the following design steps, and each step is developed based on the insights of classical control theory.
Step 1: Specify the structure of the current output-based state estimator and disturbance observer.
Construct a current output-based observer, which uses previous and the current measurements of the output ( ( ), y k ( 1), ( 2), ... y k y k − − ) to predict state and disturbance, without requiring the information on ( ) f k , as follows where ˆ( ) 1, 2 i = are to be designed. Physical interpretation of (11) associated with (10) is given by Step 2: Construct error dynamic equations for state estimator and disturbance observer designs.
i) Derive the state estimation error dynamic equations. According to Lemma 1, one can obtain and derive ( ) are disturbance estimation error, output estimation error, and state estimation error, respectively.
To have a desired observer gain L in terms of 1 2 ( , , ) o L L L for the current output-based observer in (8)- (12) such that the closed-loop observer error dynamic system poles are optimally assigned inside a circle with a pre-specified radius α Then, solving the following steady-state algebraic Riccati equation, yields the desired observer gain as where the weighting matrices  (23) is then applied to the original observer system in (8)- (12), which results in the closed-loop characteristic equation The equality in (24) implies that the eigenvalues of ( ) G LC − are equal to those of ( ) G LC − ɶ ɶ multiplied by the factor α . Clearly, it is desirable to choose a small value of α , which would speed up the convergence of the estimation errors [16].
Step 4: Specify the structure of the proportional-plus-integral (PI) model-following servo control design.
i) Realize the command generator tracker.
To proceed the PICO-based tracker design, let the reference output ( ) p m y k ∈ ℜ be generated by the following square/non-square minimum phase reference model and expect the controlled system is obtained via solving the following Riccati equation for servo control design [8,16] ( ) ii) Perform the proportional model-following servo control design. If Assumption 3 holds, system (1)-(4) has no transmission zeros lying at 'one', then one can obtain (representing ( ) X z and ( ) U z as the weighted feedbacks of ( ) (matching the dimension of the Rosenbrock system matrix of the pre-assigned MP system in (25)-(26) with the one of the controlled NMP system in (1)-(4), without alternating transmission zeros of the pre-assigned MP system in (25)-(26) By taking the inverse z-transform, from (40) one has Error dynamic equations for the PI model-following servo control design can then be formulated as Step 5: Perform the PICO-based optimal LQDT design. By neglecting the last two terms in the right hand side of (42), replacing the unknown ( ) d k by the estimated ˆ( ) d k , replacing the unmeasurable ( ) x k by the estimated ˆ( ) x k , and performing the transformations , in which 0 1 β < ≤ , the optimal control law is given by in which z P is the positive definite solution of the following algebraic Riccati equation

G P G P G P H H P H R H P G Q
Finally, the structure of the PICO-based LQDT for the square system with an unknown external disturbance is shown in Figure 1.  T is assumed to be sufficiently small.

Illustrative Examples
In this section, three numerical examples are given to illustrate the proposed approach.
Example 1: Consider the practical discrete-time non-minimum-phase MIMO flight control model considered in [16] with unknown external disturbances in (3) The design procedure is given as follows.
Step 1: Perform the optimal linear quadratic observer design.
Choose an appropriate weighting matrix pair } ± ± The performance of the proposed discrete-time current output-based estimator is depicted in Figure 2. It shows that the system state ( ) x k and unknown external disturbance ( ) d k are well estimated, which implies ˆ( ) ( ) y t y t → and Step 2: Construct the command reference model and solutions of undetermined parameter matrices.
The reference model is given by with the eigenvalues 0.0200, 0.0500, 0.0100, 0.0500 { } and a finite control zero 3 10 − , so that ( ) ( ), Step 3: Realize the PICO-based optimal LQDT design. Realize the optimal control law  In Figure 4, we compare the estimation and tracking performances of two observers obtained by the proposed approach and by Chang et al. method in [14], where the closed-loop eigenvalues of the observer error-dynamics system in [14] The comparison shows that our proposed method obviously outperforms that in [14].
where ( ) n f k ∈ ℜ has more unknown inputs than the control inputs and measured outputs and the distribution matrix of unknown inputs ( D ) is unknown. Then, apply the proposed approach to the system of interest. The tracking responses and control inputs of the closed-loop system are shown in Figure 5. Figure 5 demonstrates that although the system has not only unknown mismatched disturbances but also an unstable zero, parts of the system output ( ) y k well track the corresponding command inputs of ( ) m y k . However, for the third component, there is still room for improvement. Besides, the comparisons on the estimation performances and tracking responses of two observers depicted in Figure 6 show that our proposed method obviously outperforms that in [14].
For comparison, we apply the traditional optimal LQT integrated with the linear quadratic analog observer to the system to have( ) ( ) , , , }. z z z z The observer gain o L is obtained based on the optimal linear quadratic regulator for 6 4 10 o Q I = and 3 .
Simulation result is given in Figure 7, which demonstrates a poor tracking performance due to the nature of a NMP system, and it cannot be improved just by the traditional state-feedback/output feedback control. To overcome this difficulty, the design procedure is demonstrated as follows.
Step 1: Perform the optimal linear quadratic observer design.
Choose an appropriate weighting matrix pair , it follows that the separation principle is valid and the stability is guaranteed.
The tracking response and control input of the closed-loop system are shown in Figure 9, which demonstrate that although the system is in the presence of disturbance and has an unstable zero, the system output ( ) y k well tracks the reference trajectory ( ) m y k . We compare the estimation performances between the proposed approach and that in Chang et al. method [14], where the closed-loop eigenvalues of the observer error-dynamics system in [14]  Comparison on the estimation performances and tracking responses of these two observers depicted in Figure 10 reveals that our proposed method obviously outperforms that in [14]. This is because of the fact that the proposed approach ensures the relative stability in both the unknown input estimation and the tracking performance. The newly developed current-output observer integrated with the UID estimator-based servo design methodology can be applied to improve the performance of the discrete-time system with an unknown disturbance.