Research Article | Open Access
Chu-Tong Wang, Jason S. H. Tsai, Chia-Wei Chen, You Lin, Shu-Mei Guo, Leang-San Shieh, "An Active Fault-Tolerant PWM Tracker for Unknown Nonlinear Stochastic Hybrid Systems: NARMAX Model and OKID-Based State-Space Self-Tuning Control", Journal of Control Science and Engineering, vol. 2010, Article ID 217515, 27 pages, 2010. https://doi.org/10.1155/2010/217515
An Active Fault-Tolerant PWM Tracker for Unknown Nonlinear Stochastic Hybrid Systems: NARMAX Model and OKID-Based State-Space Self-Tuning Control
An active fault-tolerant pulse-width-modulated tracker using the nonlinear autoregressive moving average with exogenous inputs model-based state-space self-tuning control is proposed for continuous-time multivariable nonlinear stochastic systems with unknown system parameters, plant noises, measurement noises, and inaccessible system states. Through observer/Kalman filter identification method, a good initial guess of the unknown parameters of the chosen model is obtained so as to reduce the identification process time and enhance the system performances. Besides, by modifying the conventional self-tuning control, a fault-tolerant control scheme is also developed. For the detection of fault occurrence, a quantitative criterion is exploited by comparing the innovation process errors estimated by the Kalman filter estimation algorithm. In addition, the weighting matrix resetting technique is presented by adjusting and resetting the covariance matrix of parameter estimates to improve the parameter estimation for faulty system recovery. The technique can effectively cope with partially abrupt and/or gradual system faults and/or input failures with fault detection.
One of the major challenges in the identification of nonlinear stochastic systems is the choice of model representation and structure. Generally speaking, polynomial expressions are extensively used to represent nonlinear dynamic systems when the response of a system is dominated by nonlinear characteristics, so it is usually necessary to use an adequate nonlinear model to analyze and synthesize nonlinear systems. The Nonlinear AutoRegressive Moving Average with eXogenous inputs (NARMAX) model was first introduced and rigorously derived by [1, 2], which provides a unified representation for a wide class of nonlinear stochastic systems. The initial parameter of NARMAX model is important to reduce the time of the identifying process; therefore observer/Kalman filter identification (OKID)  is applied to estimate the initial parameter of NARMAX model in this paper.
Over the past decades, the state-space self-tuning control (STC) methods initially introduced in [4, 5] have been shown to be effective in designing advanced adaptive controllers for linear multivariable stochastic systems . In those approaches, the illustrious Kalman state-estimation algorithm  has been embedded into an online parameter estimation algorithm. They utilize state-space self-tuners based on innovation models, where (i) the equivalent internal states can be estimated successively; (ii) the stable/unstable and minimum/nonminimum-phase multivariable systems can be controlled accurately; (iii) the self-tuners are simple, reliable, and robust; (iv) the adaptive Kalman gain can subsequently be obtained.
In the state-space self-tuning control of a class of multivariable stochastic systems, it is often required to transform the state equation in an observer block companion form to the equivalent state equation in a controller block companion form. However, this method involves the products of high-dimensional system matrices; hence, it may introduce large computational errors and should be avoided. In this paper, the suboptimal tracker is directly designed in an observer block companion form. One point must be noticed that the state-space self-tuning control scheme for nonlinear stochastic hybrid systems proposed by Guo et al.  estimates the system parameters at every sampling instant, and then an adaptive controller could be designed based on the parameter estimated at every sampling instant. The framework of the state-space STC seems to agree with that of the active fault tolerance in a real time. For faulty system recovery, the modified Kalman filter estimation algorithm can be used to estimate system parameters, instead of using the recursive extended-least-squares (RELSs) algorithm in the conventional STC scheme. An important procedure in the proposed fault-tolerant control (FTC) is to determine the initializations of the covariance matrices of process noises, measurement noises, and parameter estimation errors and to reset these matrices for improvement of the parameter estimation  whenever the unanticipated faults happen. The fault detection is decided by the given preset threshold. As for the faults, both abrupt and gradual faults are considered in this paper.
This paper is organized as follows. In Section 2, the self-tuning control of stochastic systems based on NARMAX model is introduced. Section 3 proposes an online identification and observer with OKID for self-tuning control. Section 4 discusses the NARMAX model-based state-space self-tuner for unknown nonlinear stochastic hybrid systems. In Section 5, a fault-tolerant scheme is proposed by modifying the conventional state-space self-tuning control approach for unknown multivariable stochastic systems. Finally, two illustrative examples are shown in Section 6.
2. Self-Tuning Control Scheme Based on NARMAX Model
2.1. Digital Linear Suboptimal Tracker Design
Consider the underlying linear discrete-time system:where , and are the state variable, control input, and control output, respectively. stands for the sampling period. For simplifying equations and causing no ambiguity, will be replaced by in many equations of this paper.
It is desired to minimize the following quadratic cost function: where , and is a reference input. The superscript designates the transpose of a matrix or vector. For the linear discrete-time system (1a) and (1b), the digital linear suboptimal tracker design is given by where = for the tracking purpose, and is the positive definite and symmetric solution to the following Riccati equation: If the discrete-time system is time-varying, the digital tracker gains would also be time-varying.
2.2. Design of PWM Controller from PAM Controller
In general, there exist two types of digital controllers: the pulse-amplitude-modulated (PAM) controller and the pulse-width-modulated (PWM) controller [10, 11]. The PAM controller, which produces a series of piecewise-constant continuous pulses having variable amplitude and fixed width, is commonly utilized in digital control of all types. The PWM controller, which produces a series of discontinuous pulses with a fixed amplitude and variable width, has become popular in industry for on-off control of DC power converters and stepper motors (widely used in robotics), satellite station-keeping (with on-off reaction jets), and so forth. Since the conventional direct digital design approach takes into account only the sampling instants of the continuous-time system, the resulting PAM and PWM controllers could produce degradation in the intersample behavior of the closed-loop sampled-data system. Hence, we focus on the digital approach for the development of the PAM and PWM controllers.
The PAM controlled sampled-data system is described by where is the th column of and is the th component of the PAM input vector. The corresponding discrete-time model of the system (6) with a zero-order hold is whereHere denotes the identity matrix with the same size as .
Let a PWM controlled sampled-data system be represented as where the PWM controller is Here , , and are the amplitude of the component of the input vector, the firing delay, and the firing duration in the PWM mode at the th time step, respectively. The graphical illustration of PAM and PWM inputs is shown in Figure 1. The values of and can be determined as follows: Therefore, the aforementioned PAM controller (3) can be equivalently transferred into the corresponding PWM controller (9) by (10).
2.3. The Structure of the State-Space STC
The structure of the state-space STC scheme includes a parameter and state estimator and a controller design. A typical state-space STC structure is illustrated in Figure 2.
Under this framework, the parameters and states of the unknown system are estimated from the consecutive control input data and the system output data . To be useful in STC the parameter estimation scheme should be iterative, allowing the estimated model of the controlled system to be updated at each sample interval until the control goal is achieved. In this scheme new input/output data become available at each sample interval. First, assume that the parameter estimate is obtained based on the past information up to the time step . This could be used to yield an estimate of the output at the current time step . The estimate of the current output is then compared with the observed current output to generate a prediction error. This in turn generates an update to the model which corrects to the new value . Depending on the parameter estimate , appropriate controllers can be designed. The reference input and the designed adaptive controller generate real-time control actions for unknown dynamic systems. Notice that if the control input is persistently excited, then the convergence to the true system parameters is guaranteed .
2.4. NARMAX Model for Self-Tuning Control Scheme
The ARMAX model-based state-space self-tuning control has been represented in . Nevertheless, it is well known that the NARMAX model is a general and natural representation of nonlinear systems; however, the NARMAX model-based state-space self-tuner for fault-tolerant control has not been developed in literature. Naturally, the objective of this paper is to extend the ARMAX model-based state-space self-tuner to the NARMAX model-based state-space self-tuner for a high-performance tracking control.
The expression of the NARMAX model proposed here for the m-input p-output system is given by Here is the th output at time step (); is the control input; notation denotes the component of the prediction error . When the true system output is corrupted by white or colored noise, knowledge of the past values of the modeling error is required to estimate unknown parameters. However, the modeling error is, in general, an unobservable error process, since it depends on the unknown noise. A number of estimation procedures exist, nevertheless, which replace the modeling error by an estimate, usually taken to be the prediction error or the residual. In this paper the RELS algorithm will be utilized to estimate unknown parameters of the NARMAX model; hence, the prediction error will be instead of the modeling error. In addition, , , and are the maximum lag indices of , , and , respectively. is the unknown parameter vector for the th output. denotes the number of . The vector is the linear or nonlinear function of ( related to the output. Although may be linear or nonlinear polynomial in terms of the past values of , the whole model is nonlinear. may include any linear or nonlinear variables such as terms raised to an integer of power (e.g., , ), products of present and past inputs (e.g., ), products of past outputs (e.g., ), or cross-terms (e.g., ).
To determine the unknown parameter vector , the standard RELS algorithm can be applied to the linear-in-the-parameter model (11). This RELS algorithm is given by Here is referred to as the forgetting factor to discount the old measurements with the initial condition and the updating factor , while recovers the original RELS algorithm without forgetting where all data were weighted equally no matter how far back in the past. The adjustment of is a tradeoff between high robustness against noises (large ) and fast tracking capability (small ). The prediction error is the difference between the measured output and the one-step-ahead prediction of made at time step based on the model corresponding to the estimate .
If the prediction error is small, the estimate is “good” and should not be modified very much. The term is the parameter estimation error covariance matrix with , where is called the identity matrix of order , and is a sufficiently large positive number in order that the parameter estimate can quickly jump away from .
Owing to various kinds of combinations of () for , many classes of NARMAX models can be chosen. To simplify the whole control scheme, it is desired to choose some simple structures of dynamic nonlinear models. Thus, the state-space STC scheme with the NARMAX model for nonlinear stochastic systems can work fast and more precisely.
3. The Fast Online Identification and Observer Design Based on OKID
In this section, we slightly modify the basic structure of discrete-time state-space model, which is useful for the hybrid state-space self-tuning control law design for both linear and nonlinear systems. Here we consider a class of multivariable nonlinear systems. Once having the parameter estimate from the standard RELS algorithm, the NARMAX model for the STC scheme can accurately approximate the responses of the nonlinear system. Moreover, the initial parameter of the NARMAX model will affect the convergent speed of RELS process. In order to get a suitable initial parameter to shorten the transient process of RELS, one could apply the observer/Kalman filter identification (OKID) to evaluate it here.
The regression vector in (11) is composed ofwhere denotes and denotes . They are not independent factors, so it is difficult to design a digital controller directly from the STC scheme with the NARMAX model. For this reason, one could apply the optimal linearization to the NARMAX model to configure a linear discrete-time state-space observer so as to design a digital linear controller of the STC scheme. Thus the well-designed control law could make the output of the unknown system track exactly the prespecified reference .
3.1. Preliminary of Discrete State-Space Observer
A preliminary structure of the discrete state-space observer of the linear system is presented in . Consider the linear discrete-time stochastic system characterized by where , , and are system, input, and output matrices, respectively; , and are state, input, and output vectors, respectively; and are assumed to be stationary white noise processes with zero mean values and the covariance matrix: where is termed the expected value of a random sequence . and are nonnegative and positive definite symmetric matrices for all , respectively. Due to the assumption that both white noise processes are stationary, the covariance matrix of the process noise, , and the covariance matrix of the measurement noise, , are constant; otherwise, they might change with each time step or measurement. The symbol is the Kronecker delta function, which is 1 for and 0 for . The system (14) is said to be in the block observable form, if the following observability matrix is of full rank.
Note that the observability index of is , if it is an integer (otherwise, it is undefined). This constraint means that the Kronecker indices of the system are all such integer that satisfies . When this system (14) is block observable, it can be transformed into the block observable companion form as follows: where in which and stand for the null matrix and identity matrix, respectively.
The system (17) can be represented by a state-space innovation model  as follows: Here the Kalman gain can be computed by the following algorithm : in which is the optimal estimate of given by the measurement data and control input data for ; is called the innovation process with zero mean and the covariance matrix .
If the pair is detectable and the pair is completely stabilizable for any satisfying with , then , where is the stationary error covariance matrix, so that (the stationary Kalman gain) as . Furthermore, the eigenvalues of are all inside the unit circle centered at the origin. Let be the backward time-shift operator and set . Then, the input-output relationship of the steady-state innovation representation of (19) can be rewritten as orwhere
Note that all the zeros of must be inside the unit circle centered at the origin. Notation means the determinant of . From (21b) we can easily see that (21a) is a multivariable ARMAX model. If these parameter matrices , , and in (17) are known, and the covariance matrix therein is available, then the recursive estimation algorithm (20) can be applied to determine the Kalman gain . Thus, the state can be optimally estimated by using the state-space innovation model (19). When the matrices and are unknown, and the covariance matrix is unavailable, the following model in the block observable form can be used in conjunction with the RELS algorithm to find the Kalman gain estimate and the state estimate . Here , and contain parameter estimates , and for = When all these parameter estimates converge to the true values, the state estimate of the state and the innovation process converge to the optimal state estimate and innovation process respectively. The linear regression model of (23) can be given in the following form: where
It is important to note that as long as the matrix is asymptotically stable, the boundedness of the noise sequences implies that the estimation error will always be bounded. Whenever , for , it designates a dead-beat-like property.
3.2. OKID Formulation
Consider again the linear discrete-time system represented by (1a) and (1b). When states of an observable system are inaccessible, an observer is usually applied to estimate the states from the information of input and output. Therefore, given the observer gain , the system (1a) can be rewritten as where
For a nonzero initial condition, , the approach should be taken. For this purpose, (26a)–(26d) are extended into the following form: Then, the output equation (1b) at time step is given by Hence, these output equations, for with , can be expressed as where
Note that the first term on the right-hand side of (29a) represents the effect of the preceding time steps, where is sufficiently small and all the states in are bounded; so (29a) can be approximated by neglecting the first term on the right-hand side, such that It has the following least-squares solution: where is the pseudoinverse of the matrix with . It clearly shows that once information of inputs and outputs is obtained, the observer Markov parameters for can be known.
The Hankel matrix  is defined by where and are sufficiently large positive integers. Furthermore, can be easily decomposed into where is the observability matrix defined by and is the controllability matrix given byFrom (33), one knows that When the observer Markov parameters are determined by using (31), the eigensystem realization algorithm (ERA) method is used to obtain the system realization estimates and the observer gain through the singular value decomposition (SVD) of the Hankel matrix. The ERA processes the SVD factorization of the block data matrix , started for that is, Here and are orthogonal matrices and is a block diagonal matrix of the following form: where contains monotonically nonincreasing entries Here some singular values are relatively small and negligible in the sense that they contain more noise information than system information. In order to construct the low-order observer of the system, define Hence, (36) can be rewritten as where and are composed of the first columns of and , respectively.
Similarly, applying (39) to (35b) gets Hence, one obtains In other words, the reduced model of order after deleting singular values is then considered as the robustly controllable and observable part of the realized system with an acceptable closed-loop performance. Observing (26b), (26c), (34a)–(34b), (39), and (41), the system realization estimates and the observer gain can be found out as follows: For system identification, SVD is very useful in determining the system order. In practice, the primary purpose of applying the OKID method is that the constructed observer satisfies the least-squares solution or acts the input-output map the same as a Kalman filter. If the data length is sufficiently long and the order of the observer is sufficiently large, the truncation error is negligible.
3.3. The Optimal Linearization Method
Many useful techniques for analysis and design of linear systems have been successfully developed and reported in literatures. Also, these techniques are easier to carry out performance analysis and controller design for linear systems than those for nonlinear systems. Unfortunately, most physical systems are nonlinear in nature. Therefore, it is desirable to find an effective linearization model for analyzing the dynamic characteristics of nonlinear systems . The optimal linearization was first proposed in literature  for continuous-time nonlinear systems followed by stabilizing controller design for uncertain nonlinear systems using fuzzy models. The proposed optimal linearization at the operating state, not necessarily the equilibrium state, yields the exact linear model. Also it yields the optimal linear model defined by some convex constraint optimization criterion in the vicinity of the operating state. Taylor expansion is a common approach used; however for linearization, a truncated Taylor expansion usually results in an affine rather than linear model due to the generally nonvanishing constant term.
Consider the class of nonlinear systems governed by where and are nonlinear with continuous partial derivatives with respect to the state variable at all steps , where is the state vector at time index , and is the control input vector at time index . It is desired to have an exact local linearization system at an operating state of interest, say ; namely, where and are matrices of appropriate dimensions depending on . The linearization of the nonlinear system (43) is commonly represented by the truncated Taylor expansion as where is an equilibrium point, , and . is the gradient of (evaluated at ) with respect to . One can then represent (45) in the following form: Clearly, (46) is an affine rather than linear model due to the generally nonvanished term . One exception is the trivial case where the equilibrium is zero, , which cannot, however, be ensured throughout a nonlinear control process.
Suppose that is the given operating state; it is not necessarily an equilibrium of the given system (43). The goal is to construct a local linear model in terms of and , so that it can well approximate the dynamical behaviors of the system in the vicinity of the operating state . In other words, one has Since the designed control input is arbitrary, the following equality must be held: so that (47) becomes quite simple; that is, To satisfy these, let denote the th row of the matrix . Then (49) can be represented as where is the component of . Next, expanding the left-hand side of (50) about and neglecting the second- and higher-order terms, one has Now, using (51), one can rewrite (52) as in which is arbitrary but should be “close” to so that the approximation is good. To determine a constant vector , such that it is “as close as possible” to and also satisfies , one may consider the following constrained minimization problem: subject to Notice that this is a convex constrained optimization problem; therefore, the first-order necessary condition for a minimum of is also sufficient, which is where is the Lagrange multiplier. It follows from (56) that Left multiplying (57) by , one obtains Substituting (58) into (57) gives It is easily verified that when , (57) yields
The controllability matrix for the nonlinear system (43) at the operating state is derived from the optimal linear model defined by (44), resulting in where and are constructed via the following rule: the columns of and are set to be zero whenever the components of and of are zero, respectively.
3.4. The Fast Online Identification and Observer of Unknown Nonlinear Stochastic Hybrid Systems
Through the optimal linearization methodology, the identified NARMAX model can be explicitly presented as the ARMAX model-based observer without any approximation at each sampling instant. Besides, based on OKID method, an initial parameter of RELS algorithm (12) for STC will be presented in this section.
3.4.1. The Online Identification and Observer Based on NARMAX Model through Optimal Linearization
By the proposed NARMAX model shown in (11) for the STC, the discrete-time state-space innovation model (19) is constructed to design the control input for simply controlling the unknown true nonlinear system. The elements , , and of the control gains and could be obtained from the discrete-time state-space innovation model. Since the NARMAX model is nonlinear, the discrete-time state-space innovation model is linear; for this reason the optimal linearization method will be applied to the NARMAX model, so as to obtain a linear ARMAX model at operating states of interest without any approximation. Besides, it is also the optimal one in the sense of minimizing the optimization problem given in (54).
In this paper, a suitable NARMAX model with -inputs, -outputs and, -time-steps could be selected as follows: where The prediction errors all have been identified from the NARMAX model based on the given control inputs and the output measurement data for and . This chosen NARMAX model (61) has the advantage that and are linear in all items of and , respectively.
To separate linear/nonlinear variables, this NARMAX model (61) can be rewritten as a linear-in-the-parameter expression as follows: where
Estimate the parameter vector by the RELS algorithm to make the output of the model approximate the true system output. Although only includes output delays, it is a nonlinear function of . Therefore, needs to be linearized by the optimal linearization method in order to find the suboptimal controller.
After combining (68) for , a disturbed output of the plant in the ARMAX model takes the following form: where