State Compensation Linearization and Control

There are currently two main linearization methods: Jacobian linearization (i.e., linearization by Taylor’s expansion) and feedback linearization.

In this part, the additive state decomposition is applied to linearization, which leads to a new linearization method named state compensation linearization. For the original nonlinear system (8), denoted as $A$, the primary system is chosen as

which is a linear system and denoted as $B$. Here, $\mathbf{A}_{1}=\left.\frac{\partial\mathbf{f}}{\partial\mathbf{x}}\right|_{\mathbf{x}=\mathbf{0,u}=\mathbf{0}}\in\mathbf{\mathbb{R}}^{n\times n},\mathbf{B}_{1}=\left.\frac{\partial\mathbf{f}}{\partial\mathbf{u}}\right|_{\mathbf{x}=\mathbf{0,u}=\mathbf{0}}\in\mathbf{\mathbb{R}}^{n\times m}$, which are the same as in (9). Then, the secondary system is determined by the original system and the primary system using the rule (6), namely $C=A-B$, that

which can be then written as

where

In order to accomplish linearization, the linear system (12) ($B=A-C$) can be obtained by compensating for the nonlinearity of the original system ($A$) using the secondary system ($C$) and its controller $\mathbf{u}_{\text{s}}=\mathbf{L}_{4}\left(\mathbf{x},\mathbf{x}_{\text{s}}\right)$ as shown in Fig. 1. The system from the input $\mathbf{u}_{\text{p}}$ to the state $\mathbf{x}_{\text{p}}$ is linear. In the subsystems (12) and (13), the secondary system (13) is a virtual exact system, which exists only in design. On the other hand, since the sum of the primary system and the secondary system is the original system, all the uncertainties, such as the higher-order unmodeled dynamics of the real system, are left in the primary system. After state compensation linearization, the controller for the linear primary system (12) can be designed as $\mathbf{u}_{\text{p}}=\mathbf{L}_{3}\left(\mathbf{x}_{\text{p}}\right)$.

Based on the above content, the following theorem about the conditions for state compensation linearization can be stated.

Theorem 1. System (8) can be linearized by state compensation linearization if

i). in the entire space $\mathbf{x}\in\mathbb{R}^{n},\mathbf{u}\in\mathbb{R}^{m}$, Assumption 1 is satisfied.

ii). it can be linearizable by Jacobian linearization.

Proof. First, if Condition i) is met, according to Lemma 1, additive state decomposition holds. Next, if system (8) can be linearizable by Jacobian linearization, then the matrices $\mathbf{A}_{1},\mathbf{B}_{1}$ can be obtained, and thus the primary system (12) can be determined. Then, according to additive state decomposition, the secondary system (13) is determined by the original system and the primary system using the rule (6), namely $C=A-B$. Next, one can design a controller $\mathbf{u}_{\text{s}}=\mathbf{L}_{4}\left(\mathbf{x},\mathbf{x}_{\text{s}}\right)$ to stabilize the secondary system. Consequently, by compensating for the nonlinearity of the original system using the secondary system and its controller, the linear system (i.e., the primary system (12)) can be obtained. $\square$

State compensation linearization no longer has multiple equilibrium points. Moreover, in state compensation linearization, ${\mathbf{x}}={\mathbf{x}_{\text{p}}}+{\mathbf{x}_{\text{s}},}$ there are no coordinate transformation or unit transformation. In some sense, $\mathbf{x}_{\text{p}}$ can be considered as a part of $\mathbf{x}$, and they have the same dimension and unit. Thus, the physical meaning of the states remains the same after state compensation linearization. This property of state compensation linearization is similar to that of Jacobian linearization. In contrast, the new states obtained from feedback linearization have different physical meanings with the original states because of coordinate transformation.

Remark 1. In state compensation linearization, one keeps both ‘linear primary’ and ‘nonlinear secondary’ component systems. Furthermore, it is not necessarily assumed that the secondary system is ‘small’ in some sense. The secondary system may be significant when the original system has a strong nonlinearity. However, what we often do is to stabilize the secondary system so that it does become ‘small’ eventually, leaving one free to work with the primary system. $A$ and $B$ are said to be equivalent if the nonlinear secondary system $A-B$ can be stabilized by a controller. Then, checking the equivalence means checking whether the system $A-B$ can be stabilized. In that case a nonlinear system $A$ is equivalent to the linear system $B$ for which $A-B$ can be stabilized. The difference between state compensation linearization and Jacobian linearization is that state compensation linearization does not throw the ‘error’ system in Jacobian linearization away.

Remark 2. In state compensation linearization, one often needs to construct a stabilizing controller for a nonlinear secondary system. Although designing a stabilizing controller may cost some effort, the benefit is also attractive. The nonlinearity in the secondary system can be better considered, and thus better performance can be expected. Besides, because all the uncertainties and the initial value are considered in the primary system, the stabilizing control problem for the secondary system is a stabilizing control problem of an exact nonlinear system with zero initial states, which is often not complicated.

Although the linearized systems (12) and (9) have the same $\mathbf{A}_{1},\mathbf{B}_{1}$, the state and input in (12) are $\mathbf{x}_{\text{p}}$ and $\mathbf{u}_{\text{p}}$ instead of the real state and input $\mathbf{x}$ and $\mathbf{u}$ in (9).

Under some specific conditions, state compensation linearization can be transformed into Jacobian linearization. Considering the original system (8) with $\mathbf{d}=\mathbf{0}$. In state compensation linearization, let $\mathbf{x}_{\text{s}}=\mathbf{0}$, $\mathbf{u}_{\text{s}}=\mathbf{0}$, then the resulting linearized system is the same as that of Jacobian linearization, namely $\mathbf{\dot{x}=A}_{1}\mathbf{x}+\mathbf{B}_{1}\mathbf{u}$.

State compensation linearization seems similar to Jacobian linearization. The difference is that the latter drops the higher-order small nonlinear terms directly and Taylor’s expansion holds locally, which determines its approximate and local properties. In contrast, state compensation linearization uses a compensation strategy, which makes state compensation linearization extend to the whole space and become a kind of accurate linearization method. This property is similar to that of feedback linearization. Thus, state compensation linearization combines the advantages of the two linearization methods in some sense.

Along with state compensation linearization, an original control task can be assigned to the linearized primary system and nonlinear secondary system by dividing the original control task into two simpler control tasks. From this perspective, additive state decomposition is also a task-oriented problem decomposition that offers a general way to decompose an original system into several subsystems in charge of simpler subtasks. Thanks to task decomposition, one can design controllers for simpler subtasks more easily than designing a controller for the original task. These controllers can finally be combined to accomplish the original control task. In the following, state compensation linearization applications to nonlinear control design are discussed.

Consider the nonlinear system (8), state compensation linearization has been done in Subsection III-B. Then, the control objective is to make $\mathbf{x}\left(t\right)\rightarrow\mathbf{0}$ or $\mathbf{x}\left(t\right)\rightarrow\mathcal{B}\left(\mathbf{0}_{n\times 1},\sigma\right)$¹¹1$\mathcal{B}\left(\mathbf{o},\sigma\right)\triangleq\left\{\mathbf{x}\in\mathbb{R}^{n}\left|\left\|\mathbf{x}-\mathbf{o}\right\|\leq\sigma\right.\right\},$ and $\boldsymbol{\xi}\left(t\right)\rightarrow\mathcal{B}\left(\mathbf{o},\sigma\right)$ signifies $\underset{\mathbf{y}\in\mathcal{B}\left(\mathbf{o},\sigma\right)}{\min}$ $\left\|\boldsymbol{\xi}\left(t\right)-\mathbf{y}\right\|\rightarrow 0.$ as $t\rightarrow\infty$.

It should be pointed out that the primary system and the secondary system are virtual systems, which do not exist in practice and just exist in the analysis. Thus, an observer is necessary to provide the variables of the primary system and the secondary system for the later control design. By using the state compensation linearization method, the unknown disturbance and the initial value are assigned in the primary system, and the secondary system is an exact nonlinear system. Thus, the following observer is built based on the secondary system to observe states [19],[21].

Theorem 2. Suppose that $\mathbf{A}_{1}$ is stable, and an observer is designed to estimate $\mathbf{x}_{\text{p}}$, $\mathbf{x}_{\text{s}}$ in (12) and (13) as

where $\mathbf{\hat{x}}_{\text{s}}$ and $\mathbf{\hat{x}}_{\text{p}}$ are the estimation of $\mathbf{x}_{\text{s}}$ and $\mathbf{x}_{\text{p}}$. Then $\hat{\mathbf{x}}_{\text{p}}=\mathbf{x}_{\text{p}}$, $\mathbf{\hat{x}}_{\text{s}}=\mathbf{x}_{\text{s}}$.

Proof. Subtracting (16) from (13) results in $\dot{\tilde{\mathbf{x}}}_{\text{s}}=\mathbf{A}_{1}\tilde{\mathbf{x}}_{\text{s}}$ and $\tilde{\mathbf{x}}_{\text{s}}\left(0\right)=\mathbf{0}$, where $\tilde{\mathbf{x}}_{\text{s}}\triangleq\mathbf{x}_{\text{s}}-\hat{\mathbf{x}}_{\text{s}}$. Then, considering that $\mathbf{A}_{1}$ is stable, $\tilde{\mathbf{x}}_{\text{s}}=\mathbf{0}$, which implies that $\mathbf{\hat{x}}_{\text{s}}=\mathbf{x}_{\text{s}}$. Consequently, by (15), it can be obtained that $\mathbf{\hat{x}}_{\text{p}}=\mathbf{x-\hat{x}}_{\text{s}}=\mathbf{x}_{\text{p}}$. $\square$

Remark 3. The designed observer is an open-loop observer, in which $\mathbf{A}_{1}$ must be stable. Otherwise, state feedback is needed to obtain a stable $\mathbf{A}_{1}$. If a closed-loop observer is adopted here, it will be rather difficult to analyze the stability of the closed-loop system consisting of a nonlinear controller and the observer because separation principle does not hold in nonlinear systems.

Remark 4. The initial values $\mathbf{x}_{\text{s}}\left(0\right)$ and $\mathbf{\hat{x}}_{\text{s}}\left(0\right)$ are both assigned as $\mathbf{x}_{\text{s}}\left(0\right)=\mathbf{0}$ and $\mathbf{\hat{x}}_{\text{s}}\left(0\right)=\mathbf{0}$. If there is an initial value measurement error, it will be assigned to and considered in the primary system in the form of disturbances.

Based on state compensation linearization, the original system (8) is decomposed into two systems: an LTI system including all disturbances as the primary system, together with the secondary system whose equilibrium point is the origin. Since the states of the primary system and the secondary system can be observed, the original stabilizing problem for the system (8) is correspondingly decomposed into two problems: a stabilizing problem for the nonlinear secondary system (for the nonlinearity compensation objective) and a stabilizing problem for an LTI primary system (for the control objective). Because the primary system is linear, the classical frequency-domain and time-domain methods can be applied, such as lead-lag compensation. Therefore, the original problem turns into two simpler subproblems. The final objective is achieved once the objectives of the two subproblems are achieved.

• •

  Problem 1 Consider the secondary system (13). Design the secondary controller as

  $$\mathbf{u}_{\text{s}}=\mathbf{L}(\mathbf{x},\mathbf{x_{\text{s}}})$$

  (18)

  such that $\mathbf{x}_{\text{s}}\left(t\right)\rightarrow\mathbf{0}$ as $t\rightarrow\infty$, where $\mathbf{L}(\cdot,\cdot)$ is a nonlinear function.

After the compensation of the secondary system and its controller (18), one can get a linear system (primary system). Next, just a stabilizing problem for a linear system needs to be considered.

• •

  Problem 2 Consider the primary system (12). Design the primary controller as

  $$\mathbf{u}_{\text{p}}=\mathcal{L}^{-1}(\mathbf{H}(s)\mathbf{K}\mathbf{x_{\text{p}}}(s))$$

  (19)

  such that $\mathbf{x}_{\text{p}}\left(t\right)\rightarrow\mathbf{0}$ or $\mathbf{x}_{\text{p}}\left(t\right)\rightarrow\mathcal{B}\left(\mathbf{0}_{n\times 1},\sigma\right)$ as $t\rightarrow\infty$, where H(s) is a stable transfer function, $\mathcal{L}^{-1}$ denotes the inverse Laplace transformation, and state feedback matrix $\mathbf{K\in\mathbb{R}^{\text{$m\times n$}}}$.

Remark 5. In the controller (18), the state feedback matrix $\mathbf{K}$ can be determined by the linear quadratic regulator (LQR) method or the “PID Tuner App” in the Matlab software. Besides, one can use some kind of robust methods to construct a controller for the primary system.

The controller of the original system can be obtained by combing the controllers of the primary system and the secondary system. With the solutions to the two problems in hand, we can state

Theorem 3. For system (8) under Assumptions 1-3, suppose (i) $\mathbf{A}_{1}$ is stable; (ii) Problems 1-2 are solved; (iii) the controller for system (8) is designed as

Then, the state of system (8) satisfies $\mathbf{x}\left(t\right)\rightarrow\mathbf{0}$ or $\mathbf{x}\left(t\right)\rightarrow\mathcal{B}\left(\mathbf{0}_{n\times 1},\sigma\right)$ as $t\rightarrow\infty$.

Proof. It is easy to obtain that $\mathbf{\hat{x}}_{\text{p}}=\mathbf{x}_{\text{p}}$ and $\mathbf{\hat{x}}_{\text{s}}=\mathbf{x}_{\text{s}}$ with the observer (16)-(17). Then, since Problem 1 is solved, $\mathbf{x}_{\text{s}}\left(t\right)\rightarrow\mathbf{0}$ as $t\rightarrow\infty$. Furthermore, $\mathbf{x}_{\text{p}}\left(t\right)\rightarrow\mathbf{0}$ or $\mathbf{x}_{\text{p}}\left(t\right)\rightarrow\mathcal{B}\left(\mathbf{0}_{n\times 1},\sigma\right)$ as $t\rightarrow\infty$ with Problem 2 solved. By additive state decomposition, we have $\mathbf{x=\mathbf{x}}_{\text{p}}+\mathbf{x}_{\text{s}}\rightarrow\mathbf{0}$ or $\mathbf{x}\left(t\right)\rightarrow\mathcal{B}\left(\mathbf{0}_{n\times 1},\sigma\right)$ as $t\rightarrow\infty$. $\square$

In this new framework, the original problem is simplified, and different control methods are allowed to be applied to the systems. The closed-loop system is shown in Fig. 2. The controller based on state compensation linearization includes three parts: the controller of the primary system, the controller of the secondary system and the observer. In practice, the primary system and the secondary system do not need to have physical status. They are only the models existing in the design.

Remark 6. The basic idea of the state-compensation-linearization-based tracking control framework is similar to that of the state-compensation-linearization-based stabilizing control framework. The most significant difference is that a tracking control problem rather than a stabilizing control problem needs to be specified to the linear primary system.

Consider a bilinear system as follows

where $x,y,u,d\in\mathbb{R}$ represent the state, output, input and disturbance, respectively. It is assumed that $d\neq 0$. The tracking control goal is to make $y\rightarrow y_{\text{d}}$ as $t\rightarrow\infty$.

Consider an NMP system with input saturation as follows

with

The saturation function sat$\left(\cdot\right)$ is defined as

The system state is measurable. The control objective is to design $u$ such that $y$ tracks $y_{\text{d}}$. Next, SCLC is adopted. System (25) can be decomposed into a linear primary system

and a nonlinear secondary system

where let $u_{\text{p}}=u$. According to additive state decomposition, it follows

Then, the linear system (26) ($B=A-C$) can be obtained by compensating for the nonlinearity of the original system (25) ($A$) using the secondary system ($C$). If the controller for the primary system (26) is well designed, sat$\left(u_{\text{p}}\right)\rightarrow u_{\text{p}}$, $u_{\text{p}}\rightarrow u_{\text{d}}$, which also implies $u\rightarrow u_{\text{d}}$. Then, sat$\left(u_{\text{p}}\right)-u\rightarrow 0$. In this case, since $\mathbf{A}$ is stable, the secondary system (27) can stabilize itself. The remaining problem is to design a tracking controller for (26) without saturation. By using state compensation linearization and the remaining tracking controller for (26), $y\rightarrow y_{\text{d}}$ can hold globally. Let $\mathbf{x}_{0}=\mathbf{0}$ and

With a PID controller (the PID parameters are $k_{\text{p}}=-0.5,k_{\text{i}}=-1.3,k_{\text{d}}=0$, which is tuned by the “PID Tuner App” in the Matlab software) designed for the primary system (26), the simulation result is displayed in Fig. 5(a). The simulation result of JLC is shown in Fig. 5(b) for comparison. For the SCLC, the system undergoes input saturation from $3.3$s to $11.5$s (7.2s duration); for the JLC, with the same PID controller, the system undergoes input saturation from $3.5$s to $16.2$s (12.7s duration). It can be seen that the SCLC can exit saturation earlier than the JLC and can achieve better tracking performance. In terms of FLC, because the saturation function is irreversible, the real input $u$ cannot be obtained. Moreover, because system (25) is an NMP system, FLC may lead to an unstable controller.

Consider a mismatched nonlinear system as follows

where $\mathbf{x=}\left[x_{1}\ x_{2}\right]^{\text{T}}$is the state, $u$ is the input, $\mathbf{x}_{0}$ is the initial state value, and $\mathbf{d=}\left[d_{1}\ d_{2}\right]^{\text{T}}$represents the unknown uncertainties.