A New Alternating Predictive Observer Approach for Higher Bandwidth Control of Dual-Rate Dynamic Systems

Dual-rate dynamic systems consisting of a sensor with a relatively slow sampling rate and a controller/actuator with a fast updating rate widely exist in control systems. The control bandwidth of these dual-rate dynamic systems is severely restricted by the slow sampling rate of the sensors, resulting in various issues like sluggish dynamics of the closed-loop systems, poor robustness performance. A novel alternating predictive observer (APO) is proposed to significantly enhance the control bandwidth of a generic dual-rate dynamic systems. Specifically, at each fast controller/actuator updating period, we will first implement the prediction step by using the system model to predict the system output, generating a so-called virtual measurement, when there is no output measurement during the slow sampling period. Subsequently, the observation step is carried out by exploiting this virtual measurement as informative update. An APO-based output feedback controller with a fast updating rate is developed and rigorous stability of the closed-loop system is established. The superiority of the proposed method is demonstrated by applying it to control a permanent magnet synchronous motor system.

than the sampling rate of the sensor. The reason is that the actual measurement is usually generated at a relatively slow rate due to limitations of sensor hardware, so a fast updating controller/actuator is required to improve control performance and reduce possible vibration excitation [7]. The slow sampling rate significantly reduces the control bandwidth, resulting in significant performance degradation such as poor tracking control accuracy and dynamic disturbance rejection performance [8].
In view of the above background, some significant efforts have devoted to improve the control performance of dualrate dynamic systems. The most promising mechanism to address this problem is to exploit state interpolation over the sampling interval [9], [10]. For discrete-time dual-rate dynamic systems, the state interpolations can be obtained by using model iteration based on the measured states at the sampling instant [11], [12], [13]. While for continuous-time dual-rate dynamic systems, the pre-integration approach is usually exploited to generate continuous state information over the digital sampling interval such that the missing states over the sampling interval are recovered and can be utilized for controller design [14], [15], [16]. These state interpolation approaches work well in the case of full state measurements [17], [18]. However, when it comes to output-feedback control problems where a state observer is required, the updating rate of the state observer is consistence with the sensor measurement rate, which significantly restricts the control bandwidth [19], [20], [21]. We attempt to develop a new alternating prediction and observation mechanism that renders the updating rate of the observer as fast as the controller/actuator updating rate.
A novel alternating predictive observer (APO) will be proposed as a dynamic prediction method for state estimate and prediction over the sampling interval. At each fast controller/actuator updating instant, when there is no output measurement throughout the slow sampling period, we will first implement the prediction step by using the system dynamic model to predict the virtual system output, generating a so-called virtual measurement. Subsequently, the observation step is carried out by exploiting this virtual measurement as the informative update of the observer. These prediction and observation steps alternate at each fast actuation period. At each slow sampling instant, the real sensor measurement is used to correct the predicted and observed states. An APObased output feedback controller with a fast updating rate is further developed. The closed-loop system with this APO works like one that has a sensor with a sampling rate as fast as the actuation rate. Consequently, the bandwidth of the system is substantially enhanced by APO, which has the ability to predict the dynamics between the sampling interval. Rigorous stability of the closed-loop system is established and it has been shown that under some mild conditions, the state of the dual-rate system can converge asymptotically to a bounded region. Finally, the feasibility of the proposed method is validated by conducting simulation studies on a position control example of a permanent magnet synchronous motor.
II. PROBLEM FORMULATION Let N + stands for the set of positive integers and R denotes the set of real numbers. Given a real matrix A ∈ R n×m , A T denotes the transpose of A. Given a real symmetric matrix P = P T ∈ R n×n , λ max (P) and λ min (P) represent the maximum and minimum eigenvalues of matrix P, respectively.
We consider a class of continuous linear time-invariant dynamic systems as followṡ where x(t) ∈ R n x , u(t) ∈ R n u and y(t) ∈ R n y stand for the system state, the control input and the controlled/measurement output, respectively. A, B, and C are system matrices with compatible dimensions. Define the fast updating period of the controller as T. The sampling rate of the sensor is slower than the updating rate of the controller, and the sampling period is supposed to be MT with M > 1 and M ∈ N + . We define the control period and sampling period as T c := T and T s := MT c , respectively. The output measurement y(t) is collected by the sensor with a slow sampling period T s , which is depicted by y(t) = y(t k s ), ∀t ∈ [t k s , t k s +1 ) and t k s = k s T s (k s = 0, 1, 2, . . .) is the slow-rate sampling instant. The connection between the sampled-date measurement y(t) and control input u(t) is established. Suppose that the control input remains unchanged by using a zero-order holder (ZOH). The sampled-date control input in the slow sampling period t ∈ [t k s , t k s +1 ) is depicted by u(t) = u(t k ), ∀t ∈ [t k , t k+1 ) and t k = kT c (k = k s M, k s M + 1, . . . , k s M + M − 1) is the fastrate updating instant of the controller. The above description shows that the controller updates at intervals T c := T and the sensor samples at intervals T s := MT c . The research objective of this brief is to develop a new dualrate dynamic controller which renders the observer works with the same updating rate as the fast updating controller even in the presence of a slow output measurement rate. Due to the effectiveness in recovering the states, the proposed control method will substantially improve the control bandwidth.

III. CONTROLLER DESIGN
We first construct a sampled-date state observer for state reconstruction of system (1), which is given bẏ where ζ (t) is the estimate of x(t) and L ∈ R n x ×n y is the observer gain. It should be highlighted that the inputs of the observer (2), namely u(t k ) and y(t k ), provide informative updates. However, these two inputs exhibit a considerable rate gap in the dualrate scenario. To be specific, only provides update at the slow sampling instant in the sense that we can only access the measurement at time instant t k = k s T s = k s MT, e.g., a time sequence of {0, MT, 2MT, . . .}.
To solve this problem, we propose a new manner to add multiple virtual sampling points with period T during the slow sampling period T s = MT. Based on the above principles, a new sampled-date APO is designed aṡ where ζ(t) is the estimate of x(t) and L ∈ R n x ×n y is the observer gain,ȳ(t k ) is a mixed actual/virtual measurement and where y(t k ) and y p (t k ) stand for actual measurement and virtual measurement, respectively. The actual output y(t k ) in the sampling instant k = k s M is obtained by the real sensor measurement. An important concept of virtual output measurement is newly defined in this brief to facilitate the APO design.
To be specific, the virtual output measurement y p (t k ) in nonsampling instant, e.g., k = k s M of the sensor is inferred and obtained by using system dynamic model, the control input and the estimated state in the last fast-rate updating instant t k−1 . We use the integrations of (1) to obtain (5), the output measurement prediction y p (t k ) is obtained and given by It can be seen from (3) and (4) that at the actual sampling instant, APO uses the actual measurement y(t k ) of sensor, which can effectively improve the observation effect and make the observation effect close to the actual measurement to achieve the correction effect. Then the continuous-time APO (3) with a sampler provides the same state estimate in sampling instant t k as the following discrete-time observer where F = eÂ T , G = T 0 eÂ τ dτ B and H = T 0 eÂ τ dτ L. Since the discrete-time APO (7) and the sampled-data APO (3) will produce exactly the same estimate ζ(t k ), we will use the discrete-time APO (7) for the design of the output feedback controller and practical implementation, while adopting the sampled-data APO (3) for stability analysis to restore a more realistic sample-date system. Consequently, the fast updating output feedback control law is constructed as where K ∈ R n u ×n x is the feedback control gain.
To show the essential differences between the proposed APOBC approach and the existing paradigm of interpolationbased dual-rate control approaches, a schematic diagram showing the signal flow and updating rate of observer, predictor and sensor measurement is given in Fig. 1. As clearly shown in Fig. 1, the updating period of the state observer is MT for the traditional approach, while it is merely T for the proposed APOBC approach. This indicates that the APOBC approach can significantly enhance the sampling efficiency in a soft manner, and consequently exhibits great potential to enhance the control bandwidth of the dual-rate systems.

IV. STABILITY ANALYSIS
The dynamics of the dual-rate closed-loop dynamic systems consisting of (1) are formulated in this subsection. It should be pointed out that, according to (4), we divide the analysis into two cases.

. Defining the state estimation error of APO as ξ(t) = x(t) − ζ(t) and substituting the control law (8) into (1) and (3) leads to the following closed-loop system
which can be further rewritten as x(t) ξ(t) Case 2: t ∈ [t k , t k+1 )(k = k s M). Similar to the deviation process in Case 1, the dual-rate closed-loop dynamic system in Case 2 is the same as (10) without y p (t k )−y(t k ). Consequently, Case 2 can be considered as a special scenario of Case 1 when the prediction error y p (t k )−y(t k ) is zero and the system is less restricted under Case 2.
V. APPLICATION OF A AC SERVO SYSTEM The position control-oriented mathematical model of a permanent magnet synchronous motor (PMSM) system is described as follows where i q is q axis currents, ω is electrical angular velocity, θ is electrical angular position. The electrical and mechanical parameters R = 0.54 is stator resistance, f = 0.61 Wb is permanent magnet flux, L q = 0.0096 H represent q axis inductors, p = 4 is number of pole pairs, B ω = 0.0001 Nm/rad s −1 is the viscous friction coefficient, J = 0.016 kgm 2 is rotational inertia, and K T = 3.66 Nm/A is the torque constant, respectively. Define the state vector as x(t) = [θ(t), ω(t),ω(t)] T , and the control input as u(t) = u q (t). The PMSM model (23) is rewritten as the same as (1) and , and C = 1 0 0 . Suppose that the reference position of the motor system is given by y d (t) = sin(σ t) and σ is a real number. A slightly modified tracking controller based on the proposed APOBC approach is designed as u(t) = Kx(t) +ȳ d (t), To highlight the benefit of the proposed dual-rate control with APO (DRC-APO), quantitative comparisons with two existing popular control approaches (i.e., single-rate control (SRC) and pure model interpolation (DRC-PMI)) for dualrate dynamic systems, where SRC represents the case when both the observer and controller update in a period of MT, while DRC-PMI indicates that the observer is updated with a period of MT while the controller is updating with a faster period of T by exploiting model-based pure state interpolation. Different from these two existing control approaches, both the controller and observer update with a fast updating period of T for the proposed DRC-APO. The output measurement is only available during actual slow sampling instant t = k s MT. To demonstrate the improvement of dynamic control performance of the proposed APOBC approach, two simulation scenarios with position references y d = sin(0.5t) and y d = sin(1.5t) are selected, where the state observation curve, output tracking curve and error curve are shown in Figs. 2 and 3. Because the system under SRC is unstable, it is not plotted in figures. As clearly shown by the simulation results, compared with SRC and DRC-PMI, DRC-APO delivers a  To further demonstrate the claimed benefits of the proposed APOBC approach, a group of simulations with different harmonic frequencies of the position references ranging from 0.5 rad/s to 3 rad/s, have been carried out for position tracking performance comparisons among the three controllers. For the sake of illustration, we set the maximum tracking error ratio which is defined as the ratio of the maximum tracking error to the amplitude of the reference position output. Then the maximum tracking errors ratio are shown in Table I for comparisons among SRC, DRC-PMI and DRC-APO. To conclude, the proposed APOBC approach has the advantages of simple implementation, better dynamic and static tracking performance, and higher control bandwidth.

VI. CONCLUSION
In this brief, a new alternating predictive observer (APO) based control (APOBC) approach has been proposed to significantly improve the control performance of a class of dual-rate dynamic systems. A promising feature within the proposed APOBC method is that updating rate of the observer can be set as fast as the controller/actuator updating rate even the sampling rate of the sensor is relatively slow. To be specific, the APOBC updates the output information of the state observer by exploiting several virtual sampling points during the slow sampling period when there is no actual sensor measurement. The asymptotic stability of the resultant control system under APOBC has been established under some standard conditions by using Lyapunov stability theory. Finally, simulation studies on a position tracking control system has been conducted, which show that the proposed APOBC approach can achieve much better dynamic and static tracking control performance as well as much higher control bandwidth. This brief is a basic version. APO is only a preliminary version at present, and the proposed APO will be validated against the background of more complex and practical systems in the future.