The theoretical framework allows the consideration of any linear time-invariant (LTI) control system subject to plant input constraints and substitutions. This formal synthesis problem addresses each of the identified performance objectives in a quantitative manner.
Chapter 1
Introduction
Motivation
- Constraints
- Model Uncertainty
An obvious practical concern is that the performance provided by the controller is insensitive to the system model used in the design. Of course, it is only necessary that the controller be insensitive to design model perturbations, which create physically plausible models.
Previous Work
- Constraints
- Model Uncertainty
For example, the problem of windup is clearly related to the problem of resetting the state of the regulator. The main disadvantage of the MPC approach is that it is not amenable to analysis and quantitative synthesis procedures.
Thesis Overview
A summary of the couclusioms from Part II and additional suggestions for further work are presented in Chapter 8. As detailed in [79], constraints play a major role in the functional specifications for the control system design.
Part I
Chapter 2
- Introduction
- Model Predictive Control
- Norms and Model Predictive Formulations
- Computational Aspects
- Implementation of Model Predictive Control in the Internal Model Control Structure
- Example
Any of these combinations can be used in the formulation of a predictive control algorithm model. The oo - oo algorithm is a more direct mathematical representation of the dominant control objective in process industries.
Chapter 3
Robust Model Predictive Control
- Introduction
- Formulation of the RMPC Problem
- Impulse Response Uncertainty
- An Example
- Conclusions
- Chapter 4
We define the nominal model as Hi(Q.) (i.e., the nominal model corresponds to the origin in the parameter space). In addition, each member of the finite subset will be linear in the decision variables.
Model Predictive Optimal Averaging Level Control
Introduction
In this paper we apply these ideas to the solution of the so-called "optimal average level control" problem (66). The objective in surge tank control is to effectively use the tank capacity to filter inflow disturbances and prevent them from propagating downstream. units.
Continuous Time Optimal Averaging Level Control
However, to ensure that the level constraints are not violated, B must be equal to the magnitude of the largest predicted step disturbance. Indeed, in this scheme, the MRCO is independent of the size of the disturbances being realized.
Discrete Time Optimal Flow Filtering
Model h
Model Predictive Formulation
- Constant Level Constraints
- Box Level Constraints
For large perturbations k* ~ P, the predicted future level will reach its limit at time k* and the optimal outlet current change is the same as in the infinite horizon case, .6.q;. In the following, we show how changing the level constraints provides integral action and MRCO optimal filtering of disturbances whose magnitude is above a specified threshold.
Time, minutes
Other Level Constraints
The linear program formulation outlined in the next section allows a very general specification of future level constraints. A fixed endpoint state included in the definition of the target area ensures zero steady state offset. Since the moving horizon implementation does not guarantee that the level does not leave the target area, such constraint sets are unlikely to provide any additional benefits.
25], the set of admissible predicted levels can be chosen so that the future level is within the target area, centered on a nominal level whose magnitude decreases in the future. In general, these more restrictive level constraints result in poorer flow filtering and boost performance compared to box constraints.
Implementation
Faults occur at d' when the output of the controller is not equal to the actual value implemented in the process and passed to the internal model. If readback is not used, failures d' can arise due to algorithmic (rounding) errors in the implementation of the optimal solution. While it is reasonable to suggest that if readback is not used, a direct implementation of the MPC scheme could be successful in practice, in the next section we discuss an implementation of the MPC controller that is guaranteed to be internally stable is.
4. 7 Stabilizing Embedded Feedback
Examples
The model predictive controller brings the level back to within 5% of nominal in 13.0 minutes, the OPC needs 46.8 minutes. Figures 7 a and b show the level and outlet flows corresponding to an inlet step disturbance of 10% for the model predictive (P = 35), discrete infinite horizon, and. However, the OPC has a much larger exhaust flow overshoot and the level and exhaust flow responses are much more oscillatory than for the model predictive controller.
Conclusions
Appendix C Proof of Theorem 3
Subscripts
Superscripts
Chapter 5
Conclusions and Suggestions for Further Work - Part I
Summary of Contributions
A significant result of this work is that a closed form solution to the constrained MPC optimization problem is provided. Although this analytical solution does not include more general MPC problems, 1t makes practical application of the constrained optimal control policy for level control completely trivial. Quantitative terms are derived to evaluate the impact on flow filtering of the integral action requirement.
Suggestions for Further Work
Instead of solving a large optimization problem online, the application of the optimal policy requires only the evaluation of a simple, nonlinear relationship between level and exhaust flow. It is therefore important that the complexity of the design is explicitly taken into account. In addition to more general characterizations of the uncertainty in the time domain, simplification of the resulting minimax problems and the development of algorithmic solution techniques for the formulation of the Robust Model Predictive Control are required.
Chapter 6
Robust Control of Processes Subject to Saturation
Nonlinearities
Introduction
However, this is not true for multiple-input-multiple-output (MIMO) systems, and there are few working schemes. Since a complete nonlinear robust control theory is not available and actuator saturations are relatively simple nonlinearities, a two-level decomposition of the design problem seems justified. If the device is not open-loop stable, there is always an external input that is "big enough" to keep the system going.
Windup
- Anti Windup
- Anti-Windup from a State Space Perspective
Another approach that has been proposed, which guarantees closed-loop stability when there is no model error, is the use of the Internal Model Control ([\IC) structure (see [68] and references therein) shown in Figure 3. this realization, the state of the controller, v, is driven (only) by the error signal and we can expect significant excitation due to saturations when A includes slow dynamics. To demonstrate the effectiveness of the saturation compensator, we will consider a simple SISO example.
Output
Relationship Between Saturation Compensators
This allows us to demonstrate, using the block diagram of Figure 2, that this anti-inrush compensation is a generalization of both classical anti-inrush and L\lC. As we saw in Section 2.1, the classical anti-windup scheme corresponds to R(s) =. 6.29) which is exactly the classic anti-windup strategy, "turn off the integrators during saturation," which has been used successfully for decades for PI controllers and open-loop stable plants. Classical anti-excitation avoids excitation only for PI controllers and is insufficient when K(.s) has poles in the right half-plane; IMC avoids windup only if the closed-loop dynamics are the same as the open-loop dynamics.
Multivariable Issues - Directionality
This can be seen in Figure 7c where we have included the closure offset, (6.27), in the nonlinear simulation. This can be achieved by inserting an additional block in the loop as in Figure 8. Implicit in our assumption that I< ( s) is appropriately designed for the linear plant is the assumption that the output of K(s), u, is in the right direction.
Analysis Theory
- Application of Analysis Theory
- Example 3
This conservatism and its impact on the analysis of saturation compensation is elaborated in the next section. With respect to the state space realization of Q(s), (6.63), this condition is equivalent to the eigenvalues of A - BD-1C must lie in the left half-plane. With directional compensation, we are guaranteed that the plant input will always be in the same direction as the regulator output, only the magnitude of the actual plant input can be affected by saturation.
Synthesis Methods for Saturating Systems
Solutions to this problem would provide an optimal linear compensator f{ (s), which, in connection with saturation compensation, would be guaranteed to be stable at saturation. In a more general setting, constraints could be included to provide not only stability margins but also minimum levels for secondary performance objectives. Although we do not have techniques to obtain an optimal solution, we can create suboptimal models.
Conclusions
Chapter 7
Multivariable Anti-Windup and
Bumpless Transfer: A General
Theory
7 .1 Introduction
- A Design Paradigm
- Contributions of This Work
- The AWBT Problem Statement
- The General Formulation of the Problem
- The General Interconnection Structure
Along the way we will use the theoretical framework to better understand some of the AWBT methods proposed in the literature. The new signal, um, is the measured or estimated value of the actual plant input u'. As in the error feedback example (Figure lb), the estimate of the plant input is made available to the compensated controller AWBT k (s), in this case as a component of the measurement vector y.
Admissible AWBT
The first condition ensures that the compensated AWBT controller, k (s), can be realized as a time-invariant linear system. The U (s) and V (s) blocks that define the compensated AWBT controller, k (.s), constitute a factorization of the idealized linear model, ]{ (s). However, in general, the AWBT implementation is not equivalent to the idealized linear model, even when there are no constraints and substitutions, since P:n =f O and . w z u' P Thus the performance of the AWBT implementation will be different from the idealized linear design for which T zw(s) is given by (5). The standard AWBT technique for SISO PI and PID controllers is known as anti-reset latching (39,15,6). The integral term of the PI controller is "reset" by the feedback of u u' through the ¼ block (it is generally assumed in anti-reset PI latch design that the measurement of u.' is correct). It is precisely these stability issues that have motivated all the AWBT analysis results available in the literature (for example, in section 7.7.1 we derive certain necessary conditions for the internal stability of the AWBT compensated system, shown in Figure 3. from I< (s) will not provide a physical interpretation of the controller's status. For causal linear time-invariant systems, L2e stability corresponds to the requirement that all system poles lie in the open left half-plane. In the rest of the paper, we will simply say that a system element, or card, is stable and means that it is L2e-stable. In other words, a system is internally stable if bounded signals injected at any point in the system give rise to bounded signals at all other points in the system (28). The examples presented here demonstrate the modeling process and indicate the flexibility of the conical sector model paradigm. We note that both the identity operator, I : u -+ u, and the null operator, 0 : u - 0, are contained in this conic sector model. To obtain a normalized conic sector model of this scheme, we first approximate Lhe iuJiviJual miu aud max selectors using (85)-(86) as iu Figure 13L. The meaning of Theorem 1 and its consequence is that stability of the idealized linear design need not imply stability of the AWBT implementation, even when N = I. In this case the stable factors form U ( s ) and V ( s ) a left co -prime factorization of the original idealized linear design, i.e. In addition to the necessary conditions outlined above, we want results that will guarantee nonlinear stability of the AWBT compensated system. This definition will be made more stringent with respect to the Hankel operator associated with an LTI system. The Hankel operator associated with the stable system, G(s), is of ~nite rank, equal to the McMillan rank (the order of a minimal realization) of G(s). The Hankel singular values are thus only functions of the input-output behavior of the system, not its realization. To ensure that the output of the AWBT-compensated controller, k(s), is independent of the previous input, we need Af em(k(s)) = 0. In addition to making K(s) memoryless so that switching between modes is handled smoothly, we require that when no throttling or substitution occurs (N = I), the performance of the idealized linear design is restored. Introducing the scalings, T and r-1 as in Figure 15, does not change the closed loop map from w to z.Special Cases of the General Framework
7 .3.4 Extended Kalman Filter
AWBT Objectives
Mathematical Preliminaries
Kcs)
Conic Sector Models of Limitations and Sub- stitutions
Stability Analysis
Mes)
Mode Switching Performance
7 .8.3 Dynamic Memory
