## 1 Introduction  
Mathematical models play a great role in adaptive control, online prediction and signal modeling [1; 2; 3; 4]. The system models of physical plants can be constructed from first principle, but this method cannot work well when the physical mechanism of the true plants is not very clear [5; 6]. Thus, data-driven-based black-box or gray-box state-space modeling can be used to approximate the law of motion [7; 8; 9]. The advantages of state-space modeling lie in that the model maps the relationship from inputs to outputs, and the states reflect the internal dynamic behavior of the plants. The estimation of state-space models may include both the unknown system parameters and the unmeasurable states [10; 11]. It is well known that the Kalman filtering method captures the optimal state estimator based on the measurement data [12; 13; 14]. However, the model parameters of the system are generally assumed to be known in advance, and the statistical characteristics of measurement noise and process noise have to be chosen, which are hard to satisfy in practice [15; 16; 17]. On the identification of state-space models, the classical identification methods are the prediction-error methods, which require adequate knowledge of the system structures and parameters [18; 19], and the subspace identification methods, which ignore the consistency of system parameter estimates [20; 21; 22]. Since the system states and parameters are involved in the state-space model, the simultaneous estimation of them is a feasible choice [23; 24]. In this aspect, [PERSON] and [PERSON] investigated the state and parameter estimation problem of the state-space model with bounded noise, providing a maximum estimator based on Bayesian theory [25]. By transforming the single-input multiple-output Hammerstein state-space model into the multivariable one, [PERSON] et al. used the Kalman smoother to estimate the system states and the expectation maximization algorithm to compute parameter estimates [26]. [PERSON] et al. presented a maximum likelihood-based identification algorithm for the Wiener nonlinear system with a linear state-space subsystem [27]. [PERSON] and [PERSON] applied the recursive least squares algorithm to non-uniformly sampled state-space systems with the states being available and combined the singular value decomposition with the hierarchical identification principle for the identification of the system with the states being unavailable [28]. These studies were mainly focused on the state-space system with white noise. In industrial processes, the observation data are often corrupted by colored noises [29; 30; 31]. As we have known, the parameter estimates generated by the recursive least squares algorithm for the system with colored noise are biased. To obtain the unbiased estimates, the bias compensation method was proposed [32]. The basic idea is to compensate the biased least squares estimate by adding a correction term [33; 34; 35]. [PERSON] investigated the parameter estimation of the multiple-input single-output linear system with colored noise and introduced a stable prefilter to preprocess the input data for the purpose of obtaining the unbiased estimate [36]. [PERSON] extended the bias compensation method to the errors-in-variables output-error system, where the input was contaminated by white noise and the output errors were modeled by colored noise [37]. On the identification of state-space models with colored noise, [PERSON] et al. applied the Kalman filter algorithm to estimate the system states and developed a filtering-based recursive least squares algorithm for observer canonical state-space systems with colored noise [38]. On the basis of the work in [38], this paper discusses the state and parameter identification of the state-space model with colored noise and presents a bias compensation-based identification algorithm for jointly estimating the system parameters and states based on the bias compensation. The main contributions of the paper are as follows.