n
0
β(kx0− x∗k, n)
kx(n) − x∗k
Figure 3.5.:Asymptotic stability (3.17)
n
0
β(kx0− x∗k, n)
kx(n) − x∗k
δ
Figure 3.6.:Practical asymptotic stability (3.20)
Theorem 3.21(Estimated practical stability region)
Let V is a practical control Lyapunov function and there exists some positive constant ϑ > 0. Assume that Y = V−1[0, %] := {x ∈ X|V (x) ≤ %} ⊆ X is a positive invariant set with
% > τ := α2 ◦ (α−13 (ϑ) + ϑ), and D := V−1[0, τ ]. Then the system (3.1) is practically
asymptotically stable on Y , and V (x) is a truncated control Lyapunov function on Y \ D with radius of ultimately boundedness δ = α−11 ◦ α2◦ (α−13 (ϑ) + ϑ).
The proof can be found in [133].
Remark 3.22(Finite time to the practical stability region)
From the practical control Lyapunov function(3.21), we have α3(kx − x∗k) ≥ ϑ. Based on
the results presented in Theorem 3.21, since D ⊆ Y ⊆ X, there exists a constant > 0, such that ∀x1 ∈ X \ Y and x2 ∈ Y \ D make α3(kx2 − x∗k) ≤ α3(kx1 − x∗k) −
holds. Then we can get −α3(kx1 − x∗k) + ≤ −α3(kx2 − x∗k) ≤ −ϑ. Last, we can
conclude V (x+1) − V (x1) ≤ −α3(kx1− x∗k) + ϑ ≤ −, ∀x1 ∈ X \ Y , i.e. for the initial
current value x0 ∈ X \ Y , it will be able to reach the region Y in a finite time. We call
TY := {n ∈ N|x(n) ∈ Y } is the minimum time to reach the region Y [134]. Similarly, we
call TD := {n ∈ N|kx(n) − x∗k ≤ δ} is the minimum time to reach the Bδ. From(3.21), we
know the system trajectories are steered to Bδ as behavior like strictly asymptotic stability on Y \ D, i.e. V (x(n + 1)) ≤ V (x(n)) holds for n ∈ [0, TD− 1] until reach the ball Bδand then
will stay inside the ball, i.e. V (x) ≤ τ . Thus, V (x(n + 1)) ≤ max{V (x(n)), τ }, ∀n ∈ N0.
3.4. Types of controller
After introducing the concepts of stability, in this section we review the development of controllers which will be applied to the manufacturing process with RMTs to accomplish capacity adjustment. A number of different controllers have been developed, such as PID, linear quadratic regulator (LQR), MPC, sliding model control, Lyapunov-based control, operator-based control, optimal control, adaptive control, etc. [48, 60, 135, 136, 137]. In
3.4. Types of controller 30
this thesis, we mainly concentrate on the former three and particularly emphasize the im- portance of MPC which will be applied throughout this thesis, cf., Section 2.6.
3.4.1. Proportional integral derivative
PID controller has been widely applied in industrial processes [135, 138]. This controller is usually designed in the frequency domain via Laplace transform instead of time domain. The implementation of PID feedback controller depends on the error e(·) derived from the difference between reference r(·) with output y(·), which in continuous time reveals
u(t) = kpe(t) + ki Z t 0 e(τ )dτ + kd d dte(t) (3.22) or in discrete time: u(k) = kpe(k) + ki k X i=1
e(i)T + kd(e(k) − e(k − 1)
T ) (3.23)
where T is sampling time. The P (Proportional) part of the controller, which is proportional to the current error, has the ability of amending the error. Typically, the higher the pro- portional constant is, the faster the response is. Yet, only using a P controller may result in steady-state error. To this end, the I part is required for eliminating the latter. How- ever, this will limit the response speed and affect the stability of the system. Generally, the speed of the response will be increased along with the decrease of integral gain. The D part has the capability to incorporate the change of the error and improve the stability of system by compensating the phase lag caused by I part. The combination of the respective parameters setting of this controller significantly influences the system performance, and may even render the system unstable if inappropriate parameters are selected. Thus, before a PID controller is implemented, the optimal combination of such parameters needs to be tuned repeatedly to achieve a desirable performance. The existing methods for parameter tuning include trial and error, process reaction curve technique, Ziegler-Nichols method, etc., [135]. This type of controller is well understood. Yet, it is typically applied in SISO systems and is not designed from the view of minimization of energy consumption. Cur- rently, a research focus on PID is the transition from the classical traditional integer order to fractional order, i.e. P IλDµ[139] to improve performance.
3.4.2. Linear qadratic regulator
In the field of optimal control theory, Pontryagins Maximum Principle (PMP) and Dynamic Programming (DP) are the two famous methods [46, 140]. The former is an extension of the well-known calculus of variations. A boundary value optimization problem with con- straints is numerically solved by an indirect method, which is termed as “first optimize,
3.4. Types of controller 31
then discretize”. The main drawback is that differential equations with strong nonlinear- ity are difficult to solve and are difficult to handle active inequality constraints [140]. The latter attempts to derive an explicit feedback law off-line. Based on Bellmans principle of optimality, a multi-step optimization problem over a long horizon is split up into many sub- problems. This method provides a comprehensive insight into the mathematical problem and considers all information concerning the optimal control problems. Optimality can be guaranteed by an exhaustive search of all the control and state grids. Yet, it provides an exact solution at the expense of computational complexity and may be difficult to be ap- plied in practice due to the curse of dimensionality. Thus, both approaches may be fine for planning, yet they are computationally intractable for real time control. However, if the cost function is quadratic and system is linear, then the infinite optimization problem can be simplified into solving an Riccati equation, which is known as linear quadratic regulator (LQR). A basic derivation for LQR feedback control law is given in Appendix A.3. LQR is widely applied in the field of optimal control and also provides a way for the design of ter- minal cost for stability purpose in the framework of MPC [141]. Unfortunately, this method cannot handle constraints.
3.4.3. Nonlinear model predictive control
Model predictive control (MPC) is an control method based on optimization to generate a static state feedback for a possibly nonlinear dynamical system, which is approximately optimal with respect to a given performance index, subject to constraints and used as a closed loop implementation of optimal control [107]. Although the method is demanding from both a computational and modeling point of view, MPC has been widely applied for mechanical or chemical systems [136, 142], inventory management in supply chains [143, 144] and has grown mature over the last decades [107].
Linear MPC has been maturely developed and widely used in the process control for many years. The reasons can be summarized as follows. Firstly, the system identification is relatively easy to be derived based on the measurable process data. Secondly, the linear models exhibit desirable behavior in the neighborhood of the operating point. Due to that, some nonlinear models with weak nonlinearities can be transformed into linear models by linearizion for analysis and control. Yet, this method is not always feasible. For instance, batch processes are not staying at the steady-state but in a transient mode for a long time, often frequently influenced by the market demand. That leads to frequent changes in op- eration points. In this case, a gain scheduling technique can be adopted and the process operation will be divided into a set of operating regions. In each region, the model is iden- tified and the controller is automatically updated [142]. This seamless switching between linear controllers needs to be cautiously implemented. Note that although the system is linear, the closed-loop dynamic by MPC is actually nonlinear due to the constraints. The
3.4. Types of controller 32
Past Future
Past control input
Predicted control input
· · · k − 1 k k + 1 k + 2 · · · k + N
Measured output
Reference Predicted output
Prediction horizon
Figure 3.7.:Principle of Model predictive control
stability of closed-loop system cannot be simply analyzed by poles due to nonlinearity. Unlike linear model predictive control, nonlinear model predictive control (NMPC) is applicable for both linear and nonlinear systems, linear and nonlinear constraints, and typ- ically requires to solve a nonconvex optimization problem. Since almost all systems in na- ture are essentially nonlinear, we employ NMPC as a general method. The work principle of this algorithm is intuitively described with Figure 3.7.
For the given performance index ` : X × U → R≥0, the idea of MPC is to approximate
the solution of the infinite horizon optimal control problem
J∞(x0, u) = ∞
X
k=0
`(x(k), u(k)) (3.24)
subject to the dynamics (3.1) and constraints x ∈ X, u ∈ U. The optimal value function corresponding to (3.24) is given by V∞(x0) = infu∈UJ∞(x0, u). Based on the dynamic
programming principle we obtain
V∞(x0) = inf
u∈U{`(x0, u) + V∞(f (x0, u))}
and can derive an optimal feedback control law
κ∞(x(n)) = argmin
u∈U
{`(x(n), u) + V∞(f (x(n), u)}
by using Bellman’s principle of optimality. Since this optimal control problem is typically computationally intractable and may not be analytically solved for nonlinear systems and / or in the presence of inequality constraints [145], MPC circumvents solving Bellman’s opti- mality equation by truncating the infinite optimization problem with a fixed finite receding time window and iteratively solving an open-loop optimization problem with a given initial conditions, a three step procedure arises: In the first step, the current state of the system is
3.4. Types of controller 33
derived. Thereafter, a computationally complex truncated optimal problem min JN(x0, u) = N −1 X k=0 `(x(k), u(k)) (3.25) subject to x(k + 1) = f(x(k), u(k)), x(0) = x0 x(k) ∈ X ∀k ∈ {0, . . . , N } u(k) ∈ U ∀k ∈ {0, . . . , N − 1}
with finite prediction horizon N is solved to obtain a corresponding optimal control se- quence. To solve such problems, direct approaches like SQP, IP Methods or heuristics are well established. Moreover, in contrast to the PMP and DP, the NMPC shows reduced com- putation burden regarding warm start. After solving this optimal control problem (3.25), an optimal open-loop control sequence u∗(0), u∗(1), · · · u∗(N − 1)and the associated tra-
jectory x∗(1), · · · x∗(N ) are obtained. For simplicity of exposition, we assume that the
minimizer u?(·) := argmin
u∈UJN(x0, u) of (3.25) is unique. In the third and last step,
only the first element of this control sequence is applied as a feedback control law (i.e.,
κN(x) = u?(0)), rendering the procedure to be iteratively applicable. Then, for a given
initial value x0 = x(0), we obtain the closed-loop solution
x(n + 1) = f (x(n), κN(x(n))), x(0) = x0, n ∈ N0. (3.26)
An illustrated example concerning the open-loop and closed-loop trajectory is given in
n 0 1 2 3 4 5 21 21.5 22 22.5 23 23.5 24 24.5 25 N=3 Open-loop Closed-loop
Figure 3.8.:Difference between open-loop and closed-loop trajectory