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Abstract

A variety of nonlinear control design methods have been proposed for controlling severe nonlinear processes over the past three decades. The vast majority of approaches take a nonlinear affine representation of the system dynamics. It appears that many system dynamics can also be represented by a state-dependent model structure. Control of state-dependent systems has been investigated resulting in design methodologies such as state-dependent Riccati equation approach, state-dependent parameter and proportional–integral–plus approach, and the nonlinear generalized minimum variance approach. This article describes yet another approach based on a receding horizon strategy. Important results on optimal control are obtained. Implementation issues are also discussed. The proposed approach is validated through its application to a 25-tray binary distillation column process.

Keywords

Nonlinear systems state-dependent systems receding horizon control

Introduction

Nonlinear systems theory has developed rapidly over recent decades including concepts such as zero dynamics and normal forms,¹ passivity and dissipativity,² and nonequilibrium theory.³ As a consequence, a number of nonlinear control design techniques have been well established, for example, feedback linearization,¹ recursive designs including backstepping and forwarding,⁴ energy-based control design for nonholonomic dynamical systems,⁵ and nonlinear model predictive control (NMPC),⁶ to name just a few. A central theme in nonlinear control design is to take advantage of every potential of the system to improve control quality, for example, in the energy-based control design, the Lagrangian/Hamiltonian structure is utilized to generate Lyapunov functions to guarantee system stability; (exact) feedback linearization cancels the nonlinear dynamics in the closed loop and then takes advantage of the already well-established linear design methodologies. Recently, another linearization technique is proposed that utilizes a quasi-linear state-dependent model structure;^7–9 the nonlinear system is represented by a linear state-space-like description but the system matrices (state, input, output, and feed-through matrices) are functions of states. At each sampling instant, the system matrices are constant and the nonlinear system becomes linear but is different from the Jacobian linearization. This state-dependent representation of nonlinear systems is very general that may even include chaotic processes and bilinear systems.^10,11 More importantly, systematic identification methods have been developed by Young and colleagues^12–14 in a series of papers.

Control design for state-dependent systems has attracted extensive research interests and several design methodologies have emerged: the state-dependent Riccati equation (SDRE) approach (see Banks et al.¹⁵ and references therein), the state-dependent parameter and proportional–integral–plus (SDP-PIP) control,^11,16 and the nonlinear generalized minimum variance (NGMV) control.¹⁷ In the SDRE approach, it involves in a parameterization to transform the original nonlinear system to a linear structure with state-dependent coefficient matrices, and then a standard linear quadratic regulator (LQR) problem is solved based on the “frozen” Riccati equation. The SDRE approach has been successfully applied to space systems,¹⁸ autopilot design,¹⁹ and robotics,²⁰ see also the comprehensive article.¹⁶ The SDP-PIP control is different from SDRE approach, in that it takes a special PIP control structure, which can be interpreted as an extension of conventional proportional–integral (PI)/proportional–integral–derivative (PID) methods.²¹ The PIP controller parameters are then determined by either linear quadratic (LQ) cost function²¹ or pole assignment.²² However, it is noted that in the former case, the PIP control design also results in the solution of an algebraic Riccati equation closely related with the SDRE approach. The SDP/PIP control has also found a wide variety of applications, for example, to handle large disturbances in a microclimate test chamber or nonlinearities for a full-scale vibrolance system on a construction site (see Taylor et al.²¹ and references therein). In the NGMV approach, however, the control law is obtained by the minimization of a generalized minimum variance (GMV)-type cost function. The formulation is based on an operator approach to nonlinear control and is a straightforward extension to the previous results.²³

In this note, the problem of controlling state-dependent systems is tackled using a receding horizon strategy. The presentation is structured as follows. In section “System representation and problem formulation,” the state-dependent system representation is described and the problem to be solved is formulated; the optimal control is then obtained for the receding horizon control strategy; and to enhance the system performance, the issue of integral action is also analyzed. It happens that the state-dependent structure possesses a flexible representation and this may have significant implication to implementation, and this issue is discussed in section “Exploring design flexibility.” Section “Design example” then provides a case study to validate the proposed design method. In section “Constraint Handling and Stability,” the problems of constraint handling and stability are briefly discussed, and finally, section “Conclusion” concludes the article.

System representation and problem formulation

State-dependent model

The nonlinear state-dependent dynamics will be represented by the following state-space model

\begin{matrix} x_{t + 1} = A (x_{t}, u_{t}) x_{t} + B (x_{t}, u_{t}) u_{t} + E φ_{t} \\ y_{t} = C (x_{t}, u_{t}) x_{t} + D (x_{t}, u_{t}) u_{t} \end{matrix}

(1)

where $x_{t} \subseteq ℜ^{n}$ is the state vector, $u_{t} \subseteq ℜ^{m}$ is the control input, $y_{t} \subseteq ℜ^{p}$ is the system output, and $φ_{t}$ is the exogenous signal with zero mean and identity covariance. $A (x_{t}, u_{t})$ , $B (x_{t}, u_{t})$ , $C (x_{t}, u_{t})$ , and $D (x_{t}, u_{t})$ are the system matrices, “mimicking” the conventional linear state-space model while E is assumed to be a constant matrix. It is worth noting that in both SDRE and SDP/PIP approaches, $A (x_{t}, u_{t})$ , $B (x_{t}, u_{t})$ , $C (x_{t}, u_{t})$ , and $D (x_{t}, u_{t})$ matrices in the state-dependent model depend only on the states $x_{t}$ but here, they depend on input $u_{t}$ as well. This is not really a trivial step but it provides more design flexibility. This will be explored later.

To simplify the notation in the expressions for future prediction, $A_{t}$ is used to denote $A (x_{t}, u_{t})$ and similarly for other matrices. Then from equation (1), the future values of states and outputs can be computed as follows

\begin{matrix} x_{t + j} = A_{j}^{j} x_{t} + \sum_{i = 0}^{j - 1} A_{j}^{j - 1 - i} (B_{t = i} u_{t + i} + E φ_{t + i}) \\ y_{t + j} = C_{t + j} x_{t + j} + D_{t + j} u_{t + j} \end{matrix}

(2)

where the algebra of $A_{j}^{i}$ is defined as $A_{m}^{n} = A_{t + m - 1} A_{t + m - 2} \dots A_{t + m - n}$ and $A_{m}^{n - 1} = A_{t + m - 1} A_{t + m - 2} \dots A_{t + m - n + 1}$ with $A_{i}^{0} = I$ and $m \geq n$ .

Hence, the prediction of the output values $y_{t + i}$ for $i \in [0, N]$ can be written as

\begin{matrix} [\begin{matrix} y_{t} \\ y_{t + 1} \\ ⋮ \\ y_{t + N} \end{matrix}] = [\begin{matrix} C_{t} \\ C_{t + 1} A_{t} \\ C_{t + 2} A_{t + 1} A_{t} \\ \begin{matrix} ⋮ \\ C_{t + N} A_{N}^{N} \end{matrix} \end{matrix}] x_{t} \\ + [\begin{matrix} D_{t} & 0 & 0 & \dots & 0 & 0 \\ C_{t + 1} B_{t} & D_{t + 1} & 0 & 0 & \dots & 0 \\ C_{t + 2} A_{t + 1} B_{t} & C_{t + 2} B_{t + 1} & D_{t + 2} & 0 & \dots & ⋮ \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & 0 \\ C_{t + N} A_{N}^{N - 1} B_{t} & C_{t + N} A_{N}^{N - 2} B_{t + 1} & C_{t + N} A_{N}^{N - 3} B_{t + 2} & \dots & C_{t + N} B_{t + N - 1} & D_{t + N} \end{matrix}] [\begin{matrix} u_{t} \\ u_{t + 1} \\ u_{t + 2} \\ ⋮ \\ u_{t + N} \end{matrix}] \\ + [\begin{matrix} 0 & 0 & 0 & \dots & 0 & 0 \\ C_{t + 1} E & 0 & 0 & 0 & \dots & 0 \\ C_{t + 2} A_{t + 1} E & C_{t + 2} E & 0 & 0 & \dots & ⋮ \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & 0 \\ C_{t + N} A_{N}^{N - 1} E & C_{t + N} A_{N}^{N - 2} E & C_{t + N} A_{N}^{N - 3} E & \dots & C_{t + N} E & 0 \end{matrix}] [\begin{matrix} ϕ_{t} \\ ϕ_{t + 1} \\ ϕ_{t + 2} \\ ⋮ \\ ϕ_{t + N} \end{matrix}] \end{matrix}

(3)

Y_{N} = C_{N} x_{t} + V_{NN} U_{N} + Q_{NN} Ψ_{N}

(4)

where

Y_{N} = [\begin{matrix} y_{t} \\ y_{t + 1} \\ ⋮ \\ y_{t + N} \end{matrix}]

U_{N} = [\begin{matrix} u_{t} \\ u_{t + 1} \\ ⋮ \\ u_{t + N} \end{matrix}]

Ψ_{N} = [\begin{matrix} φ_{t} \\ φ_{t + 1} \\ ⋮ \\ φ_{t + N} \end{matrix}]

C_{N} = [\begin{matrix} C_{t} \\ C_{t + 1} A_{t} \\ C_{t + 2} A_{t + 1} A_{t} \\ ⋮ \\ C_{t + N} A_{N}^{N} \end{matrix}]

V_{NN} = [\begin{matrix} D_{t} & 0 & 0 & \dots & 0 & 0 \\ C_{t + 1} B_{t} & D_{t + 1} & 0 & 0 & \dots & 0 \\ C_{t + 2} A_{t + 1} B_{t} & C_{t + 2} B_{t + 1} & D_{t + 2} & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & 0 \\ C_{t + N} A_{N}^{N - 1} B_{t} & C_{t + N} A_{N}^{N - 2} B_{t + 1} & C_{t + N} A_{N}^{N - 3} B_{t + 2} & \dots & C_{t + N} B_{t + N - 1} & D_{t + N} \end{matrix}]

Q_{NN} = [\begin{matrix} 0 & 0 & 0 & \dots & 0 & 0 \\ C_{t + 1} E & 0 & 0 & 0 & \dots & 0 \\ C_{t + 2} A_{t + 1} E & C_{t + 2} E & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & 0 \\ C_{t + N} A_{N}^{N - 1} E & C_{t + N} A_{N}^{N - 2} E & C_{t + N} A_{N}^{N - 3} E & \dots & C_{t + N} E & 0 \end{matrix}]

Reference model

The reference signal is assumed to be generated by the following linear time-invariant stochastic system

\begin{matrix} x_{t + 1}^{r} = A_{r} x_{t}^{r} + B_{r} ω_{t}^{r} \\ r_{t} = C_{r} x_{t}^{r} \end{matrix}

(5)

where $A_{r}$ , $B_{r}$ , $C_{r}$ , and $D_{r}$ are constant matrices; $ω_{t}^{r}$ is the noise signal and it is assumed to have zero mean with identity covariance. The future trajectory of the reference is then readily obtained

R_{N} = C_{N}^{R} x_{t}^{r} + Q_{NN}^{R} Ω_{N}

(6)

where the signal vectors are

R_{N} = [\begin{matrix} r_{t} \\ r_{t + 1} \\ ⋮ \\ r_{t + N} \end{matrix}]

C_{N}^{R} = [\begin{matrix} C_{r} \\ C_{r} A_{r}^{1} \\ ⋮ \\ C_{r} A_{r}^{N} \end{matrix}]

Ω_{N} = [\begin{matrix} ω_{t} \\ ω_{t + 1} \\ ⋮ \\ ω_{t + N} \end{matrix}]

and the matrix

Q_{NN}^{R} = [\begin{matrix} 0 & 0 & 0 & \dots & 0 & 0 \\ C_{r} B_{r} & 0 & 0 & 0 & \dots & 0 \\ C_{r} A_{r} B_{r} & C_{r} B_{r} & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & 0 \\ C_{r} A_{r}^{N - 1} B_{r} & C_{r} A_{r}^{N - 2} B_{r} & C_{r} A_{r}^{N - 3} B_{r} & \dots & C_{t + N} B_{r} & 0 \end{matrix}]

Problem formulation and solution

To formulate the problem, the performance index that is commonly used in the conventional generalized predictive control (GPC) will be minimized

J = E {\sum_{i = 0}^{N} ({e_{t + i}}^{T} e_{t + i} + λ_{i}^{2} {u_{t + i}}^{T} u_{t + i}) | t}

(7)

where $e_{t + i} \equiv r_{t + i} - y_{t + i}$ ; $E {\cdot | t}$ is the conditional expectation operator, conditioned on measurements up to time t; N is the prediction horizon; and $λ_{j}$ denotes the control signal weighting. It is noted that it has been assumed that the control horizon is equal to prediction horizon to simplify the analysis.

Using the notations in equation (4), the performance index (7) can now be rewritten as

J = E {{(R_{N} - Y_{N})}^{T} (R_{N} - Y_{N}) + {U_{N}}^{T} Λ^{2} U_{N} | t}

(8)

where $Λ^{2} = diagonal {\begin{matrix} λ_{0}^{2} & λ_{1}^{2} & \dots & λ_{N}^{2} \end{matrix}}$ .

To minimize J, the procedure for optimizing the deterministic signals can be utilized, and this results in setting the gradient of the cost function (with respect to control vector $U_{N}$ ) to zero. Then, the optimality condition is

({V_{NN}}^{T} V_{NN} + Λ^{2}) U_{N} = {V_{NN}}^{T} (C_{N}^{R} x_{t}^{r} - C_{N} x_{t})

(9)

In order for the optimal control signal vector $U_{N}$ to be calculable, an additional assumption on the last control $u_{t + N}$ must be made. Here, it will be assumed that the last control increment is zero or $u_{t + N} = u_{t + N - 1}$ . Then, at each time instant, the matrix $V_{NN}$ can be obtained using the “frozen” state-dependent model, and the optimal control signal vector $U_{N}$ is then

U_{N} = {({V_{NN}}^{T} V_{NN} + Λ^{2})}^{- 1} {V_{NN}}^{T} (C_{N}^{R} x_{t}^{r} - C_{N} x_{t})

(10)

These results can be summarized as follows.

Proposition 1: optimal receding horizon control law for state-dependent systems

For the state-dependent system model (1) and reference model (5), the optimal control signal vector $U_{N}$ that optimizes the finite horizon quadratic performance index (7) or (8) is given by equation (10); the optimal control signal $u_{t}$ applied at time instant t is obtained using the receding horizon strategy, for example, first element of $U_{N}$ . The algorithm for the calculation of $U_{N}$ can be developed in the following steps:

Initialization;

At time instant t, measure or estimate the current states $x_{t}^{r}$ and $x_{t}$ ;

Load the control signal vector $[\begin{matrix} u_{t - 1} & u_{t} & \dots & u_{t + N - 1} \end{matrix}]^{T}$ obtained from the previous iteration and remove the first element $u_{t - 1}$ , which has been used in the previous iteration as the optimal control input. The remaining signal vector $[\begin{matrix} u_{t} & u_{t + 1} & \dots & u_{t + N - 1} \end{matrix}]^{T}$ , together with the current state $x_{t}$ and the model (1) can be used to calculate the optimal prediction $[\begin{matrix} x_{t + 1} & x_{t + 2} & \dots & x_{t + N} \end{matrix}]^{T}$ ;

From $[\begin{matrix} u_{t} & u_{t + 1} & \dots & u_{t + N - 1} \end{matrix}]^{T}$ , $[\begin{matrix} x_{t + 1} & x_{t + 2} & \dots x_{t + N} \end{matrix}]^{T}$ , and the measurement $x_{t}$ , together with the assumption $u_{t + N} = u_{t + N - 1}$ , the system matrices in $V_{NNu}$ can be calculated;

Solve the optimal control law (10), which provides a new updated control signal vector $[\begin{matrix} u_{t} & u_{t + 1} & \dots & u_{t + N - 1} \end{matrix}]^{T}$ ready for the next time instant (t + 1);

Apply the receding horizon principle, for example, only apply the first element $u_{t}$ of $U_{N}$ .

Proof

The first part of the proposition can be proved by collecting the results up to (10). Then, by assuming $u_{t + N} = u_{t + N - 1}$ , it is seen that both $V_{NN}$ and $C_{N}$ can be calculated by recursively computing the states from the system dynamics and the input vector. The algorithm developed in the proposition then provides the optimal control signal.▪

Integral action

Integral action can cause detrimental effects on closed-loop system performance, leading to windup phenomenon. This has sparked extensive investigation to prevent the windup problems called Anti-Windup control. However, in many cases, integral action is often required for benefits, for example, eliminating the steady-state error. It is therefore necessary to develop a mechanism to include integral action into the state-dependent receding horizon control framework. The following simple result provides an answer.

Proposition 2: receding horizon control of state-dependent systems with integral action

The integral action can be included by augmenting the state-dependent model (1) as follows

\begin{matrix} z_{t + 1} = \bar{A} (x_{t}, u_{t}) z_{t} + \bar{B} (x_{t}, u_{t}) Δ u_{t} + E φ_{t} \\ y_{t} = \bar{C} (x_{t}, u_{t}) x_{t} + \bar{D} (x_{t}, u_{t}) Δ u_{t} \end{matrix}

(11)

where $z_{t} = [\begin{matrix} x_{t} \\ u_{t - 1} \end{matrix}]$ , $\bar{A} (x_{t}, u_{t}) = [\begin{matrix} A (x_{t}, u_{t}) & B (x_{t}, u_{t}) \\ O & I \end{matrix}]$ , $\bar{B} (x_{t}, u_{t}) = [\begin{matrix} B (x_{t}, u_{t}) \\ I \end{matrix}]$ , $\bar{C} (x_{t}, u_{t}) = [\begin{matrix} C (x_{t}, u_{t}) & D (x_{t}, u_{t}) \end{matrix}]$ , and $\bar{D} (x_{t}, u_{t}) = D (x_{t}, u_{t})$ .

Then, the application of the algorithm developed in Proposition 1 naturally results in the integral action.

Proof

It is seen that (11) is obtained by defining $u_{t} = u_{t - 1} + Δ u_{t}$ as an additional state equation. Utilizing the algorithm in Proposition 1 will then give the increment of the optimal control $Δ U_{N} = [\begin{matrix} Δ u_{t} Δ u_{t + 1} & \dots & Δ u_{t + N} \end{matrix}]^{T}$ . Hence is the integral action.▪

It remains to see that it is very convenient to introduce integral action as it can be obtained by a simple augmentation of system dynamics.

Remark

This “augmentation” idea is very important, and in many situations, it can also be utilized to handle some constraints such as finite memory structure with respect to input and output (see Ahn et al.²⁴)

Exploring design flexibility

In both SDRE and SDP/PIP approaches, the system matrices of the state-dependent model depend only on the state $x_{t}$ . It has been pointed out that it is not a trivial step to allow $A (x_{t}, u_{t})$ , $B (x_{t}, u_{t})$ , $C (x_{t}, u_{t})$ , and $D (x_{t}, u_{t})$ matrices depend on not only the state $x_{t}$ but also the input $u_{t}$ as this can provide further design flexibility. To see this, utilize the following identify for a $(n \times 1)$ vector x

\frac{x^{T} x}{‖ x ‖_{2}^{2}} = I

(12)

Then, the state-dependent model (1) can be rewritten as

x_{t + 1} = \hat{B} (x_{t}, u_{t}) + E φ_{t} with \hat{B} (x_{t}, u_{t}) \equiv A (x_{t}, u_{t}) x_{t} \frac{u_{t}^{T}}{{‖ u_{t} ‖}_{2}^{2}} + B (x_{t}, u_{t})

(13-1)

Or alternatively

x_{t + 1} = \hat{A} (x_{t}, u_{t}) + E φ_{t} with \hat{A} (x_{t}, u_{t}) \equiv A (x_{t}, u_{t}) + B (x_{t}, u_{t}) u_{t} \frac{x_{t}^{T}}{{‖ x_{t} ‖}_{2}^{2}}

(13-2)

This effectively simplifies the original model (1) to $A (x_{t}, u_{t}) = 0$ or $B (x_{t}, u_{t}) = 0$ , respectively. Similar operations can be made to simplify the output matrix $C (x_{t}, u_{t})$ or feed-through matrix $D (x_{t}, u_{t})$ to be zero. This further flexibility can be exploited, for example, to reduce computational complexity. To see this, recall that in the design algorithm presented in the last section, the optimal control law (10) must be solved during each sampling instant. In equation (10), the matrix $V_{NN}$ and the vector $C_{N}$ are defined as

V_{NN} = [\begin{matrix} D_{t} & 0 & 0 & \dots & 0 & 0 \\ C_{t + 1} B_{t} & D_{t + 1} & 0 & 0 & \dots & 0 \\ C_{t + 2} A_{t + 1} B_{t} & C_{t + 2} B_{t + 1} & D_{t + 2} & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & 0 \\ C_{t + N} A_{N}^{N - 1} B_{t} & C_{t + N} A_{N}^{N - 2} B_{t + 1} & C_{t + N} A_{N}^{N - 3} B_{t + 2} & \dots & C_{t + N} B_{t + N - 1} & D_{t + N} \end{matrix}]

and

C_{N} = [\begin{matrix} C_{t} \\ C_{t + 1} A_{t} \\ C_{t + 2} A_{t + 1} A_{t} \\ ⋮ \\ C_{t + N} A_{N}^{N} \end{matrix}]

respectively.

Hence, either the input matrix $B (x_{t}, u_{t}) = 0$ or the output matrix $C (x_{t}, u_{t}) = 0$ will result in $V_{NN}$ being diagonal; consequently, the optimal control signal $u_{t}$ can be obtained from the first line of the optimality condition (9), without solving the optimal control law (10). Indeed, the optimal control signal $u_{t}$ is

u_{t} = {({D_{t}}^{T} D_{t} + λ_{0}^{2})}^{- 1} {D_{t}}^{T} (C^{r} x_{t}^{r} - C_{t} x_{t})

(14)

And this is equivalent to the minimization of a GMV type performance index

J = E {{e_{t}}^{T} e_{t} + λ_{0}^{2} {u_{t}}^{T} u_{t} | t}

(15)

Hence, the receding horizon control law now reverts to a GMV control law, revealing their close relationship in the minimization of a finite horizon quadratic performance index. This result is summarized as follows.

Proposition 3: equivalence to GMV control

For the state-dependent system (1) with either the input matrix $B (x_{t}, u_{t}) = 0$ or the output matrix $C (x_{t}, u_{t}) = 0$ , the optimality condition (9) for optimal receding horizon control with performance index (7) is equivalent to the optimality condition (14) for GMV control with performance index (15).

Proof

The result can be easily proved by following the results leading to (15).▪

This is an important result for a state-dependent system as its representation can often be expressed in an infinite number of ways. This design flexibility can be exploited to substantially reduce the computational complexity. The resulting control can be even more effective with a dynamic weighting by defining $λ_{0}^{2} = F (z^{- 1})$ ( $z^{- 1}$ is the backward shift operator). This shall be elaborated in the following design example. Similarly, the state matrix $A (x_{t}, u_{t}) = 0$ leaves only the first element in $C_{N}$ being nonzero while substantially simplifying the expression for $V_{NN}$ , resulting in fast computation of the optimal control law.

Remark

The issue of nonuniqueness can play an important role in simplifying the computation, but it may also influence the controllability of the parameterized pair ( $A (x_{t}, u_{t})$ , $B (x_{t}, u_{t})$ ), although the presence or lack of controllability of the pair does not imply on the controllability of the original system dynamics. See previous studies^18,25,26 for detailed discussion.

Design example

Consider a 25-tray binary distillation column process described by the following model²⁵

\begin{matrix} y_{t} = 0.757 y_{t - 1} + 0.243 u_{t - 1} \\ g (x) = 1.04 x - 14.11 x^{2} - 16.72 x^{3} + 562.7 x^{4} \end{matrix}

(16)

where y (%) is the top column composition and u (mol/min) is the reflux flow rate. Both of them are defined as deviations from their nominal values.²⁷

Define $x_{t} = y_{t}$ , then equation (16) can be written in a state-dependent representation as follows

\begin{matrix} x_{t + 1} = 0.757 x_{t} + 0.243 \\ (1.04 - 14.11 u_{t} - 16.72 u_{t}^{2} + 562.7 u_{t}^{3}) u_{t} \\ y_{t} = x_{t} \end{matrix}

(17)

It has been shown in Cheng and Chiu²⁸ that a PI controller with parameters $K_{C} = 1.24$ and $K_{I} = 2.5$ can provide stable response for the set-point changes covering the entire operating space $y \in [- 0.0817, 0.009]$ .

The objective now is to design a receding horizon controller to achieve good tracking response over the whole operating range $y \in [- 0.0817, 0.009]$ . From section “System representation and problem formulation,” it is clear that we only need to designate the prediction horizon N and the corresponding control weighting parameters ${\begin{matrix} λ_{0} & λ_{1} & \dots & λ_{N} \end{matrix}}$ . Now choose prediction horizon $N = 10$ with equal weighting control action $λ_{0} = λ_{1} = \dots = λ_{N} = 0.005$ to track the upper output limit while with $λ_{0} = λ_{1} = \dots = λ_{N} = 0.58$ for tracking the lower output limit. The resulting performance is shown in Figure 1, also shown is the performance of the PI controller.

Figure 1.

PI and receding horizon control performance: (a) output y and (b) control u.

The performance for varying prediction horizon N is shown in Figure 2. In this study, the control weighting parameters $λ_{i} = 0.005$ ( $i \in [0, N]$ ) are fixed for all the cases N = 5, 10, and 50 in tracking the upper output limit; $λ_{i} = 0.58$ ( $i \in [0, N]$ ) for N = 10 and 50, while $λ_{i} = 0.48$ ( $i \in [0, N]$ ) for N = 5 in tracking the lower output limit. It is seen that the performance for varying prediction horizon does not differ much and remains comparable with that of PI controller. In all the cases, the receding horizon controllers take action before the future set-point change, resulting in smooth transition for both control and output tracking around the set-point changing point.

Figure 2.

Controller performance for varying prediction horizon N.

Now rewrite the state-dependent model (17) as follows

\begin{array}{l} x_{t + 1} = [0.757 + 0.243 (1.04 - 14.11 u_{t} - 16.72 u_{t}^{2} + 562.7 u_{t}^{3}) u_{t} / x_{t}] x_{t} \\ y_{t} = x_{t} \end{array}

(18)

Define $A (x_{t}, u_{t}) \equiv 0.757 + 0.243 (1.04 - 14.11 u_{t} - 16.72 u_{t}^{2} + 562.7 u_{t}^{3}) u_{t} / x_{t}$ , and then the state-dependent model (18) has a zero input matrix $A (x_{t}, u_{t}) = 0$ . From section “Exploring design flexibility,” it is known that the optimal control comes from the equivalent minimization of a GMV-type performance index and the optimal control signal $u_{t}$ is further given in equation (14); since $D_{t}$ should be non-zero, the output in equation (18) should be reformulated with a nonzero direct through term. There is infinite number of ways to achieve this. The simplest one is

y_{t} = (1 - u_{t} / x_{t}) x_{t} + u_{t}

(19)

with $C (x_{t}, u_{t}) = (1 - u_{t} / x_{t})$ and $D (x_{t}, u_{t}) = 1$ . This further simplifies the computation of the optimal control signal $u_{t}$ as it can now be spelled out directly as

u_{t} = \frac{c^{r} x_{t}^{r} - x_{t}}{λ_{0}^{2}}

(20)

This is simply a static-state feedback controller with a constant gain $1 / λ_{0}^{2}$ . It can be shown that this controller stabilizes the closed-loop system for $| λ_{0} | \in [1 / \sqrt{2}, \infty)$ but it does not provide good performance. Closed-loop performance can be substantially improved by allowing $λ_{0}$ being a dynamic filter, for example, $λ_{0} = F (z^{- 1})$ . Indeed, this strategy is often utilized in the GMV control with dynamic weightings to provide additional design flexibility.

Now choose a simple dynamic weighting $F (z^{- 1}) = (1 - z^{- 1}) / (1 - 0.7 z^{- 1})$ , this results in the following performance shown in Figure 3, also shown is the performance of the PI controller. It is seen that the tracking performance of the GMV control is comparable with that of PI control, but with the benefit of having a considerably small control signal. This shows that design flexibility can indeed be exploited to facility control design by augmenting the state-dependent model into state- and input-dependent systems.

Figure 3.

PI and generalized minimum variance control performance: (a) output y and (b) control u.

Constraint handling and stability

When calculating the current optimal control signal, it is seen from Proposition 1 that all the predicted control from previous iteration must be utilized. Hence, starting from a feasible solution, for example, satisfying the constraints, feasibility cannot be guaranteed at the next iteration. As a consequence, constraints will be handled implicitly via numerical experiments or simulation studies in the proposed framework. This is not a deficiency compared with the methods that explicitly incorporate constrains in the design phase, as it can reduce the computational complexity once a design satisfying the constraints is verified.

In fact, computational burden in NMPC also comes from trying to provide an a priori stability guarantee as in infinite-horizon NMPC,²⁹ quasi-infinite horizon NMPC,³⁰ contractive NMPC,³¹ approximation NMPC,³² sparse numerical approach NMPC,³³ and so on. Excellent reviews of existing stability-guaranteed NMPC techniques can be found in previous studies.^6,34,35 These researches have substantially improved the understanding of stability issues in NMPC, but the resulting approaches can be difficult to implement in certain circumstances.^31,33 Furthermore, the results are obtained for the regulation problem and do not readily be generated to the tracking problem, which is more frequently encountered in practice. In fact, no method exists for global tracking with guaranteed stability. This justifies the proposed framework where stability is checked through simulations.

Conclusion

Although lack of features such as constraints handling and stability guarantee (both are achieved through simulation), the method presented here does provide an easy-to-use methodology to deal with a very general class of nonlinear systems. Within the state-dependent system literature, the approach presented here represents a new and powerful design methodology, different from the usual LQ and pole-assignment framework.

Footnotes

Academic Editor: Hamid Reza Shaker

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are grateful for the financial support of the Natural Science Foundation of Jiangsu Province (no. BK20140829) and the Fundamental Research Funds for the Central Universities (no. NS2016024).

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Control of nonlinear state-dependent systems using a receding horizon strategy

Abstract

Keywords

Introduction

System representation and problem formulation

State-dependent model

Reference model

Problem formulation and solution

Proposition 1: optimal receding horizon control law for state-dependent systems

Proof

Integral action

Proposition 2: receding horizon control of state-dependent systems with integral action

Proof

Remark

Exploring design flexibility

Proposition 3: equivalence to GMV control

Proof

Remark

Design example

Constraint handling and stability

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References