Next, the applicability of the relay feedback is shown by identifying non-linear processes with static non-linearities. In addition, the necessary condition for the existence of the limit cycle for unstable processes is obtained for first and second order process models.
N OMENCLATURE
M ATHEMATICAL N OTATIONS
L IST OF P UBLICATIONS
C HAPTER 1
I NTRODUCTION
- Research Background
- Process identification
- Extension to nonlinear processes
- On-line automatic tuning of controllers
- Extension to cascade control systems
- Relay feedback for unstable processes
- Motivation
- Contributions of this Thesis
- Identification of linear processes using the half limit cycle data
- Identification of nonlinear processes with monotonic static gains
- On-line relay auto-tuning for stable processes
- On-line identification of cascade control systems based on half limit cycle data
- Extension of relay feedback technique for unstable processes with large time delay The applicability of the relay feedback test is extended for unstable processes with large
In order to provide effective assistance to users in the design of a cascade control system, an automatic relay feedback control technique was proposed in [53]. An improved analytical method [70] under relay feedback and limit cycle adaptation conditions was described for unsteady processes by means of an iterative numerical algorithm.
C HAPTER 2
I DENTIFICATION OF LINEAR PROCESSES USING THE HALF LIMIT CYCLE DATA
Introduction
Therefore, a relay with hysteresis is used and half a cycle of the sustained oscillatory output data is measured for the estimation of process model parameters. Illustrative examples are given to demonstrate the effectiveness and merits of the proposed identification methods.
Analytical expressions for the limit cycle waveform
Assume that there exists a symmetric limit cycle output with half period T as shown in Fig.
FOPDT
- Process identification
- FOPDT Process Model
- SOPDT Process Model
- Integrating SOPDT Process Model
- Issue of load disturbance and measurement noise
- Simulation study
- Summary
Substituting the above constants into (2.9), the expression for the limit cycle output of the SOPDT process model in the time interval t1 C HAPTER 3 I DENTIFICATION OF NONLINEAR Sung and Lee [30] used a nonlinear control strategy to compensate for the nonlinear dynamics of the Wiener process using a relay feedback test. Sung and Lee [35] used the approximate sinusoidal test signal to manipulate the output nonlinearity and static disturbances by setting different time lengths of the relay output. Park and Lee [36] have proposed a simple identification method to estimate the nonlinear static function of the Wiener model using two-step inputs with different widths and amplitudes and least squares optimization. Most of the above methods reported in the literature use an ideal relay instead of a hysteresis relay thus limiting the practical applicability of their techniques. A hysteresis relay is used to prevent false switching of the relay in case of output measurement noise. When T+ =T−, the nonlinear process is of Wiener type, otherwise it is of Hammerstein type. Thus, a relay with a hysteresis test is performed and (3.4) is used to determine the structure of the block-oriented nonlinear model. Here, the hypothesis of the unitary gain of Gp is not restrictive at all, since the static gain of a nonlinear process model is determined by the static nonlinearity; the gain of the linear subsystem is redundant and can be normalized. Here, the hypothesis of a uniform gain Gp is not at all restrictive, since the static gain of the nonlinear process model determines the static nonlinearity, the gain of the linear subsystem is redundant and can be normalized to 1 [78, 79]. Again, for Wiener-type processes where the static nonlinear gain is monotonic, the zero crossings of the activated process output y(t) and the linear block output v(t) are always symmetric with respect to time. These input-output pairs are adapted to estimate the order of the polynomial m as well as the constants αj,(j=1, .,m) of the nonlinear function described in (3.9). Finally, the steps for determining parameters of the Wiener-type process model are summarized below. A similar relation can be obtained for negative values of the process output (when the input is v1) as 3.26) Likewise, the output of the process at time t2 is zero as shown in Fig. From a period of limit cycle data, the two unknowns v1 and v2 are estimated using expressions (3.26) and (3.33). Based on the estimated results and the corresponding input values, the constants of the nonlinear function (3.16) are calculated. It should be noted that one can obtain a higher order polynomial model for the static nonlinear function by repeating the same procedure from step 1 with relay height reset. The relay hysteresis width is set to twice the noise standard deviation to protect against unwanted relay jitter. An auto-tuning test with relay heights h=±1 and hysteresis ε =±0.2 results in a sustained process output as shown in the figure. To verify the applicability of the identification method in real conditions, the parameters of the process model are evaluated against noise measurements. The noise reduction using the SG filter produces a smooth output signal without attenuation of the data characteristics, as shown in Figure 2. Real-time experimental study Summary C HAPTER 4 Recently, Tsay [46] has proposed on-line calculation rules by inserting a relay with a pure time delay in the loop to find the parameters of proportional-integral (PI) and lead compensators based on specified gain margins and phase. The applicability of the basic relay tuning method is extended in this chapter with an on-line method to tune PI controllers without breaking the closed-loop control. The proposed method is effective against many of the aforementioned limitations and also maintains the simplicity of the original relay-based method. Results from the simulation examples and real-time experiments are presented to demonstrate the simplicity and effectiveness of the proposed identification and control methods. An online tuning method in which a relay is connected in series with a controller was proposed by Majhi [45]. By placing a relay in parallel with a PI controller, to use the limit cycle oscillations induced in the steady state over a limited time duration, the current process dynamics were modeled by a low-order transfer function. Tuning formulas are derived for the PI controller based on the parametric model so that it preserves the drive from the large variation of control signals. The derivative of output y(t) will have a discontinuity in the output at time t1=t0+θ, from its zero crossing due to the non-monotonic characteristics of the limit cycle data. Now the remaining parameters of the process model are obtained using the limit cycle expressions (4.7) and (4.8). Using (4.11) and (4.12) the two unknown parameters k and τ of the process model are estimated. In practice, typical ranges for the values of the parameters are often known, and they can also be obtained using a simple descriptive function method, in which case the convergence is fast. Similarly, the left-hand side of (4.18) introduces another performance index involving the derivative of the error signal to characterize the proportional control action as . The two performance indices (4.19) and (4.20) are simplified by (4.21) and an explicit design formula for Kcis obtained as. 4.22) To obtain another explicit ratio for the controller gain, the concept of inertia index is used, which reflects how much the closed-loop reduces the measure of response time relative to that of the open-loop process [83]. If we take the average residence time of the open-loop process as−G′(0)/G(0) = (τ+θ)[39], then for the closed-loop transfer function. Thus, by equating the average dwell time for the open-loop and closed-loop transfer functions, another clear design rule is made for setting the integral gain in terms of the Kiin values of the parametric model. Then the above expression can be written in the magnitude of the frequency response, where ω is small as. M(Gp(jω))M(Ki) (4.27) It shows that the minimum value of integral gain can play an important role in the presence of load disturbances. The proposed tuning method has been applied to the different processes in the test batch given in [39] as representative of typical industrial processes. 4.3 (b) gives the Nyquist curves of the test processes showing the maximum increased sensitivity between 1.2 and 2.0. In addition, the proposed method results in the desired performance in less control signal variations with TV=1.370, while the values in the case of Majhi and Tan et al. are 2.071 and 4.152 respectively. Process measurements usually contain a strong oscillation, and therefore the SNR is usually high enough to justify the use of the proposed method to estimate the model parameters in the face of output measurement noise. Assuming that there is no measurement noise during identification, a FOPDT model was obtained using an on-line relay test, and subsequently new PI parameters estimated by the proposed method are given in the second row of Table 4.1. The estimated FOPDT model and the new PI gains are given in the third row of Table 4.1. In this subsection, a DC servo position control system is used to illustrate the proposed method. The programmable controller is designed using FPAA kit and can also be dynamically configured through RS-232. The desired input references of position in degree are calibrated with equivalent DC voltages. The closed-loop responses using the initial PI settings for the 90o position and using the updated PI settings for the new 150o position are shown in Fig. Summary C HAPTER 5 O N - LINE IDENTIFICATION OF CASCADE Although individual controller tuning has been automated in [53] and [56], the sequential nature of the tuning procedure remains unchanged. Tuning methods for parallel cascade control are discussed in [61, 62] with the process information assumed to be known in terms of the FOPDT model. This chapter discusses a new method for on-line autotuning of the cascade control system using a single relay feedback test. A simple on-line technique for automatic tuning of cascade control systems has been developed to tune both controllers simultaneously. C HAPTER 6 E XTENSION OF RELAY FEEDBACK It is often desirable to extend the applicability of the relay feedback test for unstable processes with a large time delay. If the amplitude of the output at time t1 is A1, it can be shown that the expression of A1 using (2.7) is too. The method explained in section 4.2 is used to derive the solution of the equation of state (6.21). Initially, the setting of an internal controller PD(Kc,Kd)= (1/k,0.95/k) relay experiment is performed to obtain the limit cycle parameters, such as the half period T and the amplitude of the positive peak of the output A1. C HAPTER 7 The online relay method is extended for the auto-tuning of the cascade control system. The approach reduces the time required for the relay test by simultaneously identifying both the inner and outer process dynamics. It would be interesting if a general method for the online automatic tuning of stable, integrative and unstable processes could be proposed. Using the proposed on-line approach for a series cascade structure, the analysis for a parallel cascade structure can be performed. A PPENDIX A For the time range t0≤t≤t1, where the input is v1, the state equation becomes A.1 to A.4) gives an expression for the initial state. Finally, it becomes the output expression for the asymmetric limit cycle waveform between t1 Shape characteristics of function f (v) Detailed derivation of the expression (6.30) Kaya, “Parameter estimation for integration of processes using relay feedback control under static load disturbances,” Ind. Cluett, "Recursive estimation of process frequency response and step response from relay feedback experiments," Automatica, vol Wang, "A systematic approach to on-line identification of second-order process model from relay feedback tests," AIChE, vol. Gao, “Alternative identification algorithms for obtaining a first-order stable/unstable process model from a single relay feedback test,” Ind.
PROCESSES WITH MONOTONIC STATIC GAINS
Introduction
Modelling of nonlinear processes
Estimation of Wiener model parameters
Estimation of Hammerstein model parameters
Simulation study
O N - LINE RELAY AUTO - TUNING FOR STABLE PROCESSES
Introduction
Analytical expressions for the limit cycle waveform
PI Controller
On-line estimation of process model parameters
Optimal PI controller design
Evaluation of the tuning rules
Simulation study
Real-time experiments
CONTROL SYSTEMS BASED ON HALF LIMIT CYCLE DATA
TECHNIQUE FOR UNSTABLE PROCESSES WITH LARGE TIME DELAY
C ONCLUSIONS AND F UTURE W ORK
S UPPLEMENTARY M ATERIALS
Detailed derivation of the expression (3.19)
R EFERENCES
