Model predictive control (MPC) is one of the most successful approaches to controlling constrained processes. Obtaining an optimal controller design for such a critical problem is one of the main goals of this thesis.
Nonlinear System Identification
Brief Historical Background
By using non-orthogonal wavelet frames instead of traditional orthogonal wavelet basis functions, they compromise somewhat on orthogonality, but gain more freedom in the choice of the wavelet basis function. In [166], the wavelet frames are generated with a radial construction and are used to develop a wavelet network.
Wiener Model
In both structures, the linear dynamic part of the model must be suitably represented. The model should also have information about the dimension of the process with an orthogonal representation spanning a subspace.
Model Predictive Control
Basic MPC Controller
Using the quadratic performance measure (ie, based on the ℓ2-norm) results in an efficient optimization problem and simplified mathematical analysis. Nevertheless, a piecewise affine feedback law [10] or a time-varying feedback law [96] may exist for an infinite-horizon problem.
Stability of MPC
To overcome this bottleneck, the infinite horizon problem is approximated by a finite horizon problem with the inclusion of a terminal cost. In [34, 126], such terminal costs have been used together with the terminal constraint set and called it a quasi-infinite horizon MPC.
Robust Model Predictive Control
- Modeling Uncertainty
- Open-loop & Feedback robust MPC
- Feedback MPC using LMI
- Mixed H 2 / H ∞ Control
- Game theory based Controller Design
N is the draw domain of the robust open-loop MPC for the admissible control sequence set. When the players of the game have conflicting objectives, then the game is also a non-cooperative game.
MPC based on Wiener models
Objectives of the Thesis
One of the goals in the present thesis is to develop an explicit Nash game-based mixed H2/H∞. The Nash game-based mixedH2/H∞ MPC developed in the present thesis leads to the solution of such non-symmetric coupled algebraic Riccati equations (cAREs).
Outline of the Thesis
In this model, the linear dynamic part of the Wiener structure is formed by orthogonal Laguerre filters, and the static nonlinear part is formed by wavelet network structure. The performance of the proposed robust MPC design is compared with the mixed H2/H∞-based MPC design developed in [121].
Laguerre Representation
With the correct, in fact optimal, choice of the Laguerre pool, the modeling error can be reduced. Let li(k) represent the output of the ith-order filter in the Laguerre filter network.
Wavelet Transforms
Continuous Wavelet Transform
Wavelet Frames
Wavelet Network
Laguerre-Wavelet Network Model
Wiener Model Construction
With this selection of the pole of the Laguerre filter, the stability of the Laguerre model is ensured. Thus, the convergence of network training is affected by the choice of Laguerre parameters.
Model Training
The target data is chosen to contain both the dynamic and stable information of the process. The dynamic response of the model is again due to the Laguerre filter states; while the wavelet network is trained to capture the non-linear gain of the process.
Stability Analysis
Numerical Example
Bioreactor - A simulation case study
Product concentration (P) and the cell mass concentration (X) are measured process outputs, while dilution rate (D) and the feed substrate concentration (Sf) are the process inputs. The model is also validated for the steady state response versus that of the process.
Pasteurization process
It can be seen in Figure 2.5 that there is some misalignment of the plant model in the initial minutes, which is due to the misalignment of the initial conditions in the Laguerre part of the model. But again, the model could overcome the initial state mismatch error in a reasonable time due to the state feedback mechanism. Other discrepancies between the model and process measurement may be due to some other dynamics in the process that the model is not aware of.
Summary
The mixed H2/H∞ control design has its own importance in the control community since its inception in [14]. However, the development in this work is for discrete-time systems within the MPC framework. One such suitable way to obtain the optimal solution of the control problem is discussed in this work.
Problem Formulation
The feedback MPC in the present work is adopted by iteratively solving online an infinite open-loop optimal control problem with infinite horizon at any time, with the concept of feedback. So the optimization problem for MPC is a multi-objective feedback min-max problem to find a minimal control value (u∗k) relative to the optimal feedback policy relative to the worst-case disturbance, wk∗ (within a prescribed upper bound γ) . The concept of positive definiteness of a matrix is defined in this context in this thesis, unless its associated symmetry property is explicitly stated.
Solving the H 2 problem - Minimizing Player
Solving the H ∞ problem - Maximizing Player
Saddle point solution
Terminal Condition
Solving the cross-coupled-AREs
There is no solid analytical method to find the cARE solution, except those given in [58, 82], which are not perfect. The above input constraint condition is used in the optimization problem while solving for the cARE solution. Thus, this approach ultimately yields the desired result, ie. closed loop operation in terms of both its transient response and stability.
Numerical Examples
From the closed-loop performance of controllers A and B (See Figure 3.2) and their respective IAE values of 4.1086 and 5.8697, respectively, the mixedH2/H∞ MPC based on the Nash game (controller A) gives a better performance and is less conservative than the other. controller. This chapter is a note on robustness and robustness of H2/H∞ mixed model predictive controllers for linear state feedback systems addressed in Chapter 3. The robustness and robustness issues of multicriteria optimal control are addressed in this chapter using set of theoretical concepts.
![Figure 3.1: Comparison of the proposed controller A (solid) against the controller B (dash-dot) for the example used in [121].](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10474449.0/89.918.144.769.141.741/figure-comparison-proposed-controller-solid-controller-dash-example.webp)
Introduction
One of the milestones in the applicability of set theory concepts to robust control problems appeared in the paper [23], which influenced the famous stability concepts of Lyapunov functions with set theory ideas. The computation of the minimum robust invariant set and the maximum robust invariant set has been of great interest in the recent past [127]. It is worth noting the fact that the robust controller synthesis presented in Chapter 3 is not explicitly based on the concepts of set theory; but it could be easily understood with a better insight into the language of set theory due to its robustness, stability and convergence which.
Preliminaries
Invariant Sets
Moreover, it is of natural interest to determine which subset of the given set conforms to the input constraints. However, states can only target a target located in the home neighborhood. The set C∞(Ω)\ S∞(Ω,T) includes all initial states from which it is not possible to force the states of the system into the stable region S∞(Ω,T) (and thus to T) [90].
Lyapunov theory
The relationship between the Lyapunov function and positive invariant sets for dynamical systems is discussed in great detail in [26]. But if the system has the form xk+1 = f(xk, uk, wk), where uk ∈ U is disturbed by some bounded disturbance wk ∈ W, then for such a system the corresponding Lyapunov control function is called robust control Lyapunov function. We can easily conclude that when there is no disturbance affecting the system (W = ∅), but the control input has a constraint uk ∈ U 6= ∅, the robust Lyapunov control function in (4.13) reduces to a simple Lyapunov control function with cV = 0.
Lyapunov function and Invariant set
Proposition 4.3 If the closed-loop system xk+1 = f(xk, κk(xk)) has a Lyapunov function Vk(xk) ∈ F with xk ∈ X, 0 ∈ int(X) and F is further invariant ‡, then the origin is asymptotically stable with an attractive pool (containing) X. In general, the robust asymptotic stability of a closed-loop system on int(X) can be guaranteed if [70]. For a closed-loop system to be robustly stable, it is enough if the states of the system reach an invariant set with positive control, i.e. T, for some ¯k and lie inside the set for all k≥k >¯ 0.
Feasibility and continuity implies robust stability
Continuity of MPC control law and Value function
Theorem 4.6 [55] Let Ξ be a locally compact metric space, and let F : Ξ ; Rn are locally Lipschitz continuous with non-empty closed convex values. If there exists a robust control Lyapunov function Vk(xk) for the system Σ such that ∆LfV(·) +αV is locally Lipschitz, then Σ is robustly stabilized via a local Lipschitz feedback control lawµ. Then it follows from assumption A2 and statement 4.7 that there exists a locally feasible check uk ∈ U such that uk = κk(xk) when Vk(xk) ≥ ck.
Nominal and Robust Feasibility
Proof From theorem 4.8 it is known that D is convex in uat points where ck = Vk(xk). For the feasible input set of the MPC controller design, the closed-loop system is now given asxk+1 =f(xk,U)⊕W. Proposition 4.8 (i) The feedback MBK is strongly feasible if the feasible set is a positive invariant set for the closed-loop system xk+1=f(xk, κ(xk)); (ii) The feedback MBK is strong and strongly feasible if the feasible set is control-invariant for the closed-loop system xk+1 =f(xk, uk)+wk metuk∈ U andwk ∈Wand the closed-loop.
Input-to-State Stability (ISS) for MPC
To achieve a better performance, as stated in Corollary 4.1, it is expected that the norm of the matrix Q−1 must be lowered and the closed-loop solutionφ(·) reaches a minimal invariant setX∞. Thus, the choice of controller gain at each time point is a function of the system response. Thus, if the choice of controller gain depends only on the system response and not on any a priori estimate of the worst possible disturbance, then the solution to the control problem will be less conservative.
So the control input lies in the linear saturation region of the actuator, let L(U) represent the linear saturation region of the actuator. To establish robust stability, when the control input hits actuator saturation, there must first be the existence of the control input at ∂U. The control input uk ∈∂U which is an acceptable control such that the response of the system is Ext(Rk) is called Extreme Control.
Simulation Results
The value of γ = 7, which is smaller than that of Orukpe et al., reaffirms the analysis presented earlier in this chapter. In addition, the overall performance of the closed-loop system using the proposed control algorithm (γ = 2.4) is much better than that of Orukpe et al. (γ =√. 43). The algorithm of Orukpe et al. it constrains the control signal to work well within the linear range of actuator nonlinearity.
Summary
Solving the H 2 problem
This part of the control problem finds an optimal controller such that y7→u whose realization is given by. The above two equations represent an output-feedback controller, the first is an observer and the second is a state controller.
Solving the H ∞ Problem
Thus equations and (5.36) are the three coupled Riccati algebraic equations (3-cAREs) of the optimal output feedback control problem. The coupling is attributed to the presence of the H∞ solution term, Hk, in eqns.
Saddle point solution
Solving the 3-cAREs
Successive Linearisation Model
Automatic Differentiation
Symbolic differentiation methods turn a function derivative into a single long expression at the point where the derivative of the function is to be evaluated. In operator overloading, the existing classes or types are redefined, initialized with appropriate values for the function and its derivative, call the function, and find the values of the derivatives. The function is evaluated based on its forward derivative, starting from the input variable (or independent variable).
Numerical Example
Bioreactor Control
It could be observed that there is not much distortion in the performance of the LWN model due to successively linearized deviation approach. For example, the maximum specific growth rate of microbes (µm) is taken to be 0.48 in the present nominal case. This will cause a shift in the nominal operating point of the process and therefore introduces plant model mismatch.
Comparison of Output Feedback MPC Schemes
Fragility
Current measured value of the process is used as starting condition for the above purposes. Assume that there exists a control invariant set Θ⊆ X∞, near the origin, such that 0∈int(Θ). A detailed MATLAB code for Controller-B is added in Appendix C for the benefit of the reader.
This small change in the regulator parameter causes the closed-loop system to become unstable (Figure 5.6). ANSWER The following sources are listed in the appropriate places in the present version of the assignment.
