State‐limiting PID controller for a class of nonlinear systems with constant uncertainties

Proportional Integral Derivative (PID) controllers still represent the core control method for achieving output regulation of either linear or nonlinear systems in the majority of industrial applications. However, conventional PID control cannot guarantee specific state constraint requirements for the plant, when the system introduces uncertainties. In this paper, a novel nonlinear PID control that achieves output regulation and guarantees a desired state limitation below a given value for a wide class of nonlinear systems with constant uncertainties is proposed. Using nonlinear ultimate boundedness theory, it is shown that the proposed state‐limiting PID (sl‐PID) control maintains a given bound for the desired system states at all times, ie, even during transients, whereas an analytic method for selecting the controller gains is also presented to ensure closed‐loop system stability and convergence at the desired equilibrium. Two nonlinear engineering examples that include an electric motor and a dc/dc converter are investigated using the conventional PID and the proposed sl‐PID to validate the superiority of the proposed controller in achieving the desired output regulation with a given bounded state requirement.

While the aforementioned approaches offer global or semiglobal stability guarantees, they do not tackle the problem of constraint satisfaction (input or state constraints), which arises from safety requirements and actuator limitations in a real engineering system. Although the output regulation problem with input constraint satisfaction has been well studied and addressed via antiwindup methods [12][13][14][15][16] or bounded integral control, 17,18 a state constraint requirement often creates the need for more advanced control methods that change the traditional PID control architecture. Safety of operation and stability guarantees are essential in modern processes such as power networks, motor control applications, and chemical processes. In the former, Tilli and Conficoni 19 and Conficoni 20 list some of the potential pitfalls of not considering the power converter limits into the control strategy, ie, loss of stability, performance degradation, and operation outside the desired ranges because of system malfunctions. The challenge of limiting a system state increases when the plant introduces uncertainties in the model. For example, consider the one-dimensional system . x = −wx 3 + u, where w is a constant unknown uncertain parameter defined in the range w ∈ [w min , w max ]. The objective is to regulate state x toward a desired constant nonzero reference r while satisfying state constraint |x| ≤ x max , ∀t ≥ 0. In general, the conventional PI controller of the form u = k P (r − x) + , . = k I (r − x) can achieve the desired regulation but does not offer any guarantees when it comes to the desired state limitation. It can easily be shown that the solution may exhibit overshoots violating constraints. Therefore, an effective management of constraints may increase the operation ranges in both the transient and the steady-state regimes.
The problem of maintaining a given bound for the system states when the system is subject to disturbances or uncertainties, or when the state vector is (partially) unknown has inspired researchers since the late 1960s. In particular, when uncertain parameters or disturbances belong to some known compact set, several state-bounding methods have been developed to design simple time-varying sets (eg, ellipsoids and n-orthotopes) to guarantee that the state vector is constrained within these sets. [21][22][23] Most of these methods are designed for linear systems, whereas their extension to nonlinear systems is not trivial and still represents an active problem. 24,25 In the majority of the cases, the methodologies proposed rely on linearization around the state trajectory, which fails to produce reliable estimates when uncertainties are explicitly considered in the model. 26 For nonlinear systems with discrete dynamics, a design methodology for determining the controller parameters is presented in the work of Andonov et al, 25 where a desired system performance, including state limitation, is guaranteed. In a similar framework, techniques such as model predictive control (MPC), 27 for an excellent survey on its different properties, include the constraints into their formulation and aim to exploit the behavior of the system around the constraints. These methods, however, present limitations in their implementation because they require the solution of an optimization problem online. Jerez et al 28 propose an MPC controller for linear systems that operates at megahertz; however, this approach is aimed at linear system and quadratic performance objectives, which is at odds with nonlinear formulations of the problem. Overall, for nonlinear systems with continuous-time dynamics, the output regulation problem with guaranteed state limitation using the well-understood and widely applied PID control and without modifying the control concept is still an open problem.
To address this issue, in this paper, a novel nonlinear state-limiting PID (sl-PID) controller is proposed for a wide class of nonlinear systems with constant uncertainties to achieve output regulation with a desired state constraint satisfaction. The proposed approach can be applied to multi-input multi-output systems, and using nonlinear ultimate boundedness theory, it is analytically proven that a limitation on the desired system states can be guaranteed at all times, even during transients. Furthermore, a detailed methodology for selecting the proportional, integral, and derivative gains is provided to guarantee closed-loop system stability and convergence to the desired set point. Hence, the proposed sl-PID can replace the conventional PID control in applications where a state limitation is required, whereas the framework for selecting the controller gains presented in this paper can lead to a simple and effective design and implementation.
Overall, the novelties and contributions of this paper are summarized as follows: 1. The design of a novel nonlinear sl-PID controller for nonlinear systems with constant uncertainties capable of guaranteeing a desired state limitation on a number of the plant states. 2. Detailed stability analysis and design framework for the sl-PID controller states, and comparison with the conventional PID control scheme.
To better explain and validate the sl-PID control design procedure, we apply the proposed approach to two engineering examples: a motor control regulation problem and a dc/dc power converter. We compare the performance of the proposed algorithm to that of a conventional PID controller. This paper is structured as follows. In Section 2, some key theoretic concepts for nonlinear system analysis are presented. In Section 3, the problem under investigation is stated and an analysis of the conventional PID controller is provided. The main result of this paper is presented in Section 4, where the sl-PID controller is designed and analyzed. A detailed stability analysis and the design framework for selecting the controller gains are provided as well. The proposed controller is applied to two different engineering applications in Section 5 and compared to the conventional approach to validate its effectiveness in a practical implementation. Finally, concluding remarks are provided in Section 6.

Notation.
A finite set of natural numbers is denoted by n = {1, … , n}; the set of integers is denoted by ℤ.

PRELIMINARIES
Consider the nonlinear system where ∶ [0, ∞)× x × u → ℝ n is piecewise continuous in t and locally Lipschitz in x and u, with  x ⊂ ℝ n and  u ⊂ ℝ m being open domains containing their respective origin, and u(t) is a piecewise continuous bounded function.
Definition 1 (See the work of Khalil 1 ). System (1) is said to be locally input-to-state stable if there exist ∈ , ∈ , and positive constants k 1 , k 2 > 0 such that for any initial state x(t 0 ) with ||x(t 0 )|| < k 1 and any input u(t) with sup t≥t 0 ||u(t)|| < k 2 , the solution x(t) exists and satisfies for all t ≥ t 0 ≥ 0. It is said to be input-to-state stable if  x = ℝ n ,  u = ℝ m , and inequality (2) is satisfied for any initial x(t 0 ) and for any bounded input u(t).
Lemma 1 (See the work of Khalil 1 ). Suppose that, in some neighborhood of (x, u) = (0, 0), function f (t, x, u) is continuously differentiable and the Jacobian matrices x and u are bounded uniformly in t. If the unforced system has a uniformly asymptotically stable equilibrium point at the origin x = 0, then system (1) is locally input-to-state stable.

Consider the interconnected system
where f 1 and f 2 satisfy similar conditions to f in (1).

Lemma 2 (See the work of Khalil 1 ).
If , with x 2 as input, is locally input-to-state stable and the origin of . x 2 = 2 (t, x 2 ) is uniformly asymptotically stable, the origin of the interconnected system (4) is uniformly asymptotically stable.
Definition 2 ( -limit set 29 ). The -limit set of a subset  ⊂ ℝ n , denoted by (), for an autonomous system . x = (x) contains all points x ∈ ℝ n for which there exists a sequence of pairs where ∶ ℝ × ℝ n → ℝ n is the system flow. x = (x) is bounded and belongs to  x for t ≥ 0, then for any compact set  ⊂  x , its positive -limit set () is a nonempty, compact, connected, invariant set. Moreover, x(t) → () as t → ∞.

PROBLEM DEFINITION
Consider the nonlinear plant of the form .
where (x, z) ∈ ℝ n+m , u = (u 1 , … , u n ) ∈ ℝ n and = ( 1 , … , n ) ∈ ℝ n are the state, input, and output vectors, respectively; similarly, w ∈ ℝ l is a vector containing unknown constant parameters. The functions f i (·, ·), g i (·), q(·, ·, ·, ·), and h i (·) are C 1 in (x, z, u) and continuous in w for (x, z) ∈  ⊂ ℝ n+m , u ∈ ℝ n , and w ∈  w ⊂ ℝ l , where  and  w are open connected sets. Note that although the dynamics for the states x i are given in the control-affine form (5b), the dynamics for the state vector z can have the generic nonlinear form (5a).
The main goal is to design a controller that regulates the output ∈ ℝ n to a constant reference r ∈  r ⊂ ℝ n , where r = (r 1 , … , r n ) and  r is an open connected set, while guaranteeing constraint satisfaction on the states x i , ∀i = 1, 2, … , n, such that for some x max i > 0, the flow satisfies We invoke a set of assumption to make our formulation consistent. The first of these is related to the output behavior of the system.

Assumption 1 (State measurement). The output function
The next assumptions are concerned with the nature of w ∈ ℝ l , which represent model uncertainties.

Assumption 2.
The function g ∶ ℝ l → ℝ n is positive, ie, g i (w) > 0 for i ∈ n . Remark 1. Assumption 1 states that state x i can be either measured or analytically calculated from the output y i to be used in the control design. Assumption 2 is concerned with the controllability of system (5b) and states that the sign of g i (w) does not change independently of the values of uncertain parameter w. Assumption 3 (Bounded parameter uncertainty). Vector w ∈ ℝ l lies in a known compact and convex set, ie, w ∈ .
, and two class  functions 1i and 2i such that Although the plant dynamics and above assumptions can seem restrictive, it should be mentioned that (i) several engineering applications are described by (5b)-(5c), eg, power electronic converters 12 and (ii) a wide class of nonlinear systems can be brought in the form of (5b)-(5c) using partial feedback linearization. We are interested in the effect that exogenous inputs and set points have on the equilibria of the open-loop dynamics. We exploit the structure of these dynamics, mainly the fact that the system can be viewed as an interconnection between . x = (x, w) + g(w)u and . z = q(z, x, w, u). We first characterize the steady state behavior of the upper system.

Proposition 1. Suppose Assumptions 1 to 3 hold. There exists a continuous function i
Proof. The proof is obtained by construction. Following Assumption 1, the output functions h i (·) admit a differen- On the other hand, Assumption 2 guarantees the existence of g i (w) −1 ≤ ∞, which is used to define the mapping When the value w ∈ is fixed, then i (·, ·) assigns to each set point a steady-state pair in an injective way; as a direct consequence of this, it is possible to find an interval such that For the proposed setting, both (x, u) can be considered as inputs to system (5a). A direct consequence of Proposition 1 is that for any compact set  × , the steady-state pairs also lie in a compact set; from Assumption 3, the uncertain parameters lie in a compact set , and  can be chosen such that |x i | ≤ x max i for all i ∈ n . Both the uncertain parameters and the set points can be realized via signal generators, 30 which allow us to characterize the steady-state locus, from constant set points and parameters to periodic ones, in terms of -limit sets. 29 Setting a compact set  ⊂ ℝ m containing the trajectories of (5a), Lemma 3 defines the steady-state behavior of the open-loop dynamics as ( ×  × ).

Conventional PID control
Based on the previous conditions, the desired regulation scenario can be achieved using a conventional PID controller of the form where k Pi , k Ii > 0, k Di ≥ 0, and at the desired equilibrium point, there is e i = u ie . The closed-loop system takes the form where i ∈ n . For system (9b)-(9c), stability of the equilibrium point (x e , e ), where = ( 1 … , n ), can be analyzed by investigating the Jacobian matrices around the equilibrium point which can be proven to be Hurwitz for a suitable selection of the proportional gain k Pi , ie, and for any k Ii > 0 and k Di ≥ 0. Now, by setting v = v e +ṽ, where v = (x, ), v e = (x e , e ),ṽ = (x,̃), and z = z e +z, then (9b)-(9a) can be rewritten in the generic form where the desired equilibrium has been shifted to the origin.

Assumption 5.
The dynamics (11) are locally input-to-state stable whenṽ is considered as the input.
Based on the above assumption, when asymptotic stability at the equilibrium point v e of (9b)-(9c) is guaranteed by the controller gains k pi , then according to Lemma 2, the equilibrium point (x e , e , z e ) of the closed-loop system (9b)-(9a) is asymptotically stable.
Although the desired equilibrium point can be proven to be asymptotically stable using a conventional PID controller, it is not guaranteed that |x i | ≤ x max i , ∀t ≥ 0. This state constraint is crucial in several practical examples, such as power electronic converters and electromechanical systems, 12,19 where the current, voltage, or speed is required to remain bounded below a given value at all times to avoid damaging devices. To overcome this problem, a nonlinear sl-PID controller is proposed in the sequel.

MAIN RESULT
In this section, we state the main contribution of this paper: the sl-PID. The motivation behind this approach is to guarantee constraint satisfaction for a PID scheme without the need of saturation units or antiwindup schemes. The formulation follows a similar pattern to that of the classic PID controller in terms of design and analysis. The resulting nonlinear controller attains both desired properties of constraint satisfaction and closed-loop stability.

Proposed sl-PID control design
Consider the open-loop dynamics (5) subject to Assumptions 1 to 4; the control objective, as in the case of conventional PID, is to steer a number of states, x ∈ ℝ n , to predefined constant set points r ∈ ℝ n , which correspond to equilibrium triplets (x e , z e , u e ) such that |x e i | ≤ x max i , and z e ∈ ( ×  × ). Assumption 1 guarantees the availability of the state for feedback purposes; hence, the proposed novel nonlinear sl-PID controller is where M i , k Pi , k Ii > 0, k Di ≥ 0. By substituting the proposed controller (12a)-(12b) into the plant dynamics (5b)-(5c), the closed-loop system becomes where i ∈ n . This resulting closed-loop dynamics, indeed, satisfies the desired state-limiting property under a suitable choice of controller gains k Pi > 0, k Ii > 0, and k Di > 0. The following proposition gives guidelines for choosing K Pi and M i to achieve the desired objective. where <k Pi ≤ k Pi − i for any arbitrarily small i > 0.
Proof. Consider a continuously differentiable function V i (x i , w) for system (13b) satisfying conditions (7a)-(7c) of Assumption 4. Then, Because (15) can be rewritten as which, according to Theorem 4.18 of Khalil, 1 proves that the solution x i (t) is uniformly ultimately bounded. In particular, considering a positive constant > 0 such that Note that if the initial state x i (0) satisfies then the solution x(t) will remain in this range for all future time, ie, Hence, if M i and k Pi satisfy (14), then which verifies the state-limiting property of the proposed controller.
Consider the typical zero initial condition of the integral state i , ie, i (0) = 0; then, the controller state remains, following the equilibria of (13c), within interval i → 0, which means that i will converge to the upper or lower limit (± 2 ) independently of the term r i − h i (x i ). This also implies an inherent antiwindup property of the proposed sl-PID because the integration slows down near the two limits of i without the need for adding any antiwindup mechanisms, which often result in changes of the original controller dynamics, as in the conventional antiwindup PID control design. Similarly, if initially i (0) is selected in the range [− 2 + k , + 2 + k ], for all k ∈ ℤ, then it will remain within this range thereafter. This allows us to characterize the region of attraction (see Figure 1 and Section 4.2) for the composite dynamics (x, ) as for any k ∈ ℤ determined by the initial conditions of the integrator. An additional consequence of using the proposed control law is that the sets  i,k are positively invariant for the closed-loop dynamics, formalized in the following proposition.
that satisfies (7a)-(7c) with respect tox i . Then, the problem can be transformed into a problem with symmetric Hence, the sl-PID can take the form with i dynamics (12b) and guarantee the desired state limitation, following the same analysis provided in Proposition 2.

Stability analysis
Although the desired x i state limitation is guaranteed, the stability of the desired equilibrium point (x e i , e i ) with x e ∈  x , still needs to be investigated. Following Proposition 1, the map (x e , u e ) = (r, w) for a constant w ∈ defines injectively a steady-state pair, which allows us to characterize the equilibrium points.
Proof. The steady-state locus for (13b) and (13c) for constant set points is given by the solutions of the system of nonlinear equations which can be translated into the set for k ∈ ℤ. As a result, the steady-state locus can be characterized completely by two types of behavior: when e i = ± 2 +k , the state satisfies i (x e i , w)−k Pi g i (w)x e i +M i g i (w) = 0, and on the other hand, when x e i = h −1 i (r i ), the equilibrium for the integrator state can be obtained as ) .
This implies that the following inequality holds: which in turn allows us to define two functions A direct consequence of the previous result is the existence of equilibrium points inside the region of attraction  i,k : Corollary 1. The set  i,k for all i ∈ n and any k ∈ ℤ contains three equilibrium points: For the desired equilibrium point , ∀i ∈ n , the Jacobian matrix of system (13b)-(13c) is Thus, the equilibrium point (x e , e ) will be asymptotically stable when and for any k Ii > 0 and k Di ≥ 0. Hence, by selecting k Pi according to (19), then M i can be chosen to satisfy (14). For the remaining dynamics, similarly to the analysis of the conventional PID controller, (13a) can be rewritten in the generic form with the desired equilibrium being shifted at the originz = 0. As in Section 3.1, if Assumption 5 holds for system (20), according to Lemma 2, the equilibrium point (x e , e , z e ) of the closed-loop system (13b)-(13a) is asymptotically stable. The asymptotic stability of the desired equilibrium point (x e , e ) has been proven in a neighborhood of the equilibrium point, as in the case of the conventional PID control when system (5a)-(5c) is nonlinear. However, in the proposed approach, a detailed methodology for selecting the sl-PID controller gains can be applied to prove that the asymptotic stability holds in the entire constrained range , as explained in the following theorem.

Theorem 1. Suppose that Assumptions 1 to 4 hold. The desired equilibrium point (x e , e ) of the closed-loop system
, ∀i ∈ n , is asymptotically stable and every solution ( , ∀i = n converges to (x e i , e i ) as t → ∞ when k Pi , k Ii , and k Di ≥ 0 satisfy Proof. Because (21) is satisfied for all |x i | < x max i , then (19) holds true, and as a result, the desired equilibrium point is asymptotically stable. The extension of this result to the entire bounded range follows from an analysis of the closed-loop dynamics (13b)-(13c), which describe a second-order system for each i ∈ n of the form .
First, we ensure that no limit cycles exist within the constraint set (x i , i ) ∈  i,k ; this assertion follows from the Bendixon theorem 31 such that the divergence of the vector field does not vanish nor change sign, ie,̃i Computing the derivatives yields the expression Therefore, no limit cycles will exist in the bounded range if From (21) and (22), the above expression does not change sign nor vanish. This condition implies from Lemma 3 that any bounded trajectory (x i (t), i (t)) within  i,k will converge to the omega limit set ( i,k ) of (24), which cannot include a limit cycle. However, except from the desired equilibrium point (x e , e ), which is asymptotically stable, according to Corollary 1, there exist two more equilibrium points in the bounded range, ie, and . The proof that these two equilibrium points are unstable follows by contradiction. In particular, consider that for as t → ∞. However, for a point in  i,0 arbitrarily close to the equilibrium point ( , because Assumption 1 holds in  i,0 , then h(x i ) < h(x e i ) and x i < x e i hold. As a result, . i < 0, which implies that i will decrease and diverge from e i = 2 , ie, consequently, the solution (x i (t), i (t)) will diverge from taking into account (13b) and Corollary 1, leading to a contradiction. Hence, the equilibrium point is unstable. A similar procedure holds for . As a result, the desired equilibrium point ( represents the only positive limit point in the omega limit set ( i,0 ); thus, every solution (x i (t), i (t)) starting in the range (x i (0), i (0)) ∈  i,0 , ∀i ∈ n , converges to (x e i , e i ) as t → ∞ when k Pi , k Ii , and k Di ≥ 0 satisfy (21)- (22), thus completing the proof.
Although the condition of the integral gain selection, ie, condition (22), might seem complicated to verify, it should be noted that it can be significantly simplified for the case of the sl-PI controller, as given in the following remark.
Remark 4. For the design of a sl-PI controller, the derivative gain is zero, ie, k Di = 0, and hence, condition (22) is simplified to the expression: Remark 5. From the closed-loop dynamics, the set point r i acts as a bifurcation parameter in the sense that its value changes the dynamic properties of the system. For any |h −1 i (r i )| < x max i , following Corollary 1, there are three distinct equilibrium points: two unstable at the boundary of  i,k and one stable in the interior of  i,k . As the set point approaches h i (x max i ), the stable set point also approaches the boundary of  i,k ; eventually, when |r i | ≥ h i (x max i ), there are only two equilibrium points.
To summarize the design procedure of the proposed sl-PID controller, the following steps can be followed.
1. Check that the system under investigation is in form of (5a)-(5c) or whether it can be brought into this form using partial feedback linearization. 2. Check that Assumptions 1 to 5 hold. 3. Select k Pi to satisfy (21) and k Di ≥ 0. 4. Select M i according to (14). 5. Select k Ii satisfying (22). 6. Design the sl-PID controller according to (12a)-(12b).
Although, in this paper, the state constraint |x i | ≤ x max i is the main requirement for the closed-loop system equipped with the proposed sl-PID, it is worth mentioning the proposed design in the PI control format, ie, sl-PI controller, can additionally guarantee an input constraint of the form |u i | ≤ u max i , u max i > 0 when required. This is explained in the following remark.
The above remark and the additional conditions for M i and k Pi can be easily obtained by taking into account the control structure (12a), the desired input and state constraints, and M i > 0. It should be noted that even though the sl-PI controller can be designed based on Remark 6 to guarantee both state and input constraints, the selection of the controller parameters may lead to a slow response in a real application because the upper limits for M i and k Pi have been obtained to guarantee the desired input constraint for the worst-case scenario, ie, when |x i | ≤ x max i and | sin( i )| ≤ 1.

System and controller design
Consider first the case of a motor feeding a constant power load (CPL), ie, the load torque is represented as where P L > 0 is the power of the load, which is constant, and is the speed of the motor. Example of these loads includes machine tools and center winders in the industry. Hence, the motor dynamics are given in the form where J is the motor inertia, is the motor damping, and T is the electrical torque applied to the motor, representing the control input of the system. It is assumed that the motor parameters and the load are not accurately known but can vary around their nominal values, ie, they satisfy J ∈ [J n − ΔJ, J n + ΔJ] > 0, ∈ [ n − Δ , n + Δ ] > 0, and P L ∈ [P Ln − ΔP L , P Ln + ΔP L ] > 0 (Assumption 3). The main task is to regulate the speed of the motor to a desired reference ref , while guaranteeing that during the transient response, the speed never violates an upper limit, ie, | | ≤ max , ∀t ≥ 0. Note that because a CPL is connected at the motor, the motor speed should take strictly positive values, ie, consider that ≥ min > 0. The plant is in the form of (5b) with y = . Hence, g(w) = 1 J > 0 and h( ) = , ie, h = 1 > 0, which confirm that both Assumptions 1 and 2 hold. For system (28), consider the continuous and differentiable function V = 1 2 J 2 , which satisfies (7a). In addition, which verifies that (7b) and (7c) are also satisfied with c = − n + Δ and b 1 = b 2 = 1, yielding that Assumption 4 holds as well. As a result, both the conventional PID and the sl-PID can be applied to regulate the motor speed. For the conventional PID, the proportional gain k P can be selected according to (10), whereas the integral and derivative gains can take any positive values k I and k D . For the sl-PID design, the proportional gain should be selected according to (21) as Consequently, parameter M when chosen according to (14) satisfies for an arbitrarily small > 0 such that k P + n − Δ − > 0. The proportional and derivative gains can take any positive value k P and nonnegative value k D , respectively, whereas the integral gain can be selected according to condition (22), taking into account that h( ) = , ie, h = 1, 2 h 2 = 0: Based on the given ranges of the uncertain parameters and the proven limit of the state below max , which together with | ref | ≤ max yields max{| ref − |} = 2 max , then, the range of k I becomes Note that because the nominal power of the load P Ln is known and the motor speed is measured, an additional term P Ln can be introduced in the control input torque T together with the proposed sl-PID to reduce system nonlinearity. In this case, k P should satisfy k P > Δ P L 2 min − n + Δ and parameters M and k I can be defined accordingly. However, this special case is not applied in the simulation that follows in order to keep consistency with the generic case of the system and the control design procedure presented in the theory.

Simulation results
To validate the proposed sl-PID control performance and compare it with the conventional PID control, the motor system with parameters given in Table 1    . As it can be observed from the response in Figure 2A, both the conventional PID and the proposed sl-PID can achieve the desired regulation as expected from the theoretical analysis. However, during the transient, the conventional PID control forces the motor speed to exceed the upper limit max , whereas the proposed sl-PID maintains the desired state limitation at all times. Note, however, that for larger values of the proportional gain k P , the conventional PID may possibly maintain the speed to values lower than max . Nevertheless, the conventional PID gains are designed based on the linearized model, ie, the desired state limitation cannot be guaranteed for a generic nonlinear system with uncertainties, whereas the state limitation for the proposed sl-PID has been proven using ultimate boundedness theory for the generic nonlinear model. In addition, as it can be seen from the response of the input torque in Figure 2B, larger values of the proportional gain k P will lead to unrealistically large values of the torque (eg, higher than 160[N m]), whereas the proposed sl-PID results in a much smoother input response.

System and controller design
Consider the dynamic equations of the dc/dc boost converter connected to a resistive load R, given as in Konstantopoulos and Zhong 32 : where V in > 0 is the dc input voltage, L is the converter inductance with a series resistance r, C is the converter capacitance, x = (i, v) is the state vector, and u is the control input describing the duty-ratio input of the converter. The dc/dc converter can achieve a higher dc voltage v at its output compared to the input voltage V in . The main task is to regulate the converter power P = V in i to a constant reference P ref , while maintaining a desired constraint |i| ≤ i max , where i max > 0 represents the maximum allowed current of the converter to avoid damage of the device. It is assumed that L, C, r, and R are not accurately known, ie, L ∈ [L n − ΔL, L n + ΔL] > 0, C ∈ [C n − ΔC, C n + ΔC] > 0, r ∈ [r n − Δr, r n + Δr] ≥ 0, and R ∈ [R n − ΔR, R n + ΔR] > 0, where L n , C n , r n , and R n are the corresponding nominal quantities, which are considered known together with the maximum deviations of the parameters (Assumption 3). By defining the control input u as when v ≥ V in > 0, which is a physical property of DC/DC boost converters, and replacing it in (29a)-(29b), the converter dynamics take the form which is in the form of (5b)-(5c) considering the control inputū. Note that g(w) = 1 L > 0 and h(i) = V in i, ie, h i = V in > 0, which confirm that both Assumptions 1 and 2 hold. Because (31a) is linear and time invariant, then by considering the quadratic function V = 1 2 Li 2 , which satisfies (7a), hence, (7b) and (7c) are also satisfied with c = −r n + Δr and b 1 = b 2 = 1, yielding that Assumption 4 holds as well. As a result, both the conventional PID controller and the proposed nonlinear sl-PID controller can be implemented to regulatē x to the desired unique equilibriumx e = (i e , v e ), where . Because of its physical properties, the boost converter output voltage is always higher than the input voltage, ie, v ≥ V in > 0. For any proportional gain k P > 0, the equilibrium point i e = i ref of (31a) will be asymptotically stable because conditions (10) and (19) will be satisfied for both the conventional and the proposed sl-PID controller. For the proposed controller, from (14), parameter M should satisfy for an arbitrarily small > 0 such that k P + r n − Δr − > 0. Finally, from (22), the integral gain k I should satisfy By setting i = i e +ĩ, v = v e +ṽ, and = e +̃, then (31b) can be expressed as Considering a set Dṽ forṽ, whereṽ > −v e in Dṽ, then both̃q̃v and̃q̃x are bounded in Dṽ wherex = (ĩ,̃). In addition, the unforced system, ie, forx = 0, has the form The Jacobian of system (36) results in < 0, and therefore, the origin of (36) is asymptotically stable. Then, according to Lemma 1, system (35) is locally input-to-state stable. As a result, the desired equilibrium point (i e , e , v e ) of system (31a)-(31b) with both the conventional PID and the proposed sl-PID controller will be asymptotically stable. In Figure 3, we illustrate the behavior of the system dynamics under the proposed control law. We can see that the vector field and any trajectory starting inside  1,0 remain within. This illustrates the nature of the equilibria (see Corollary 1); for those on the boundary of  1,0 , ie, and the closed-loop system is unstable and any trajectory starting arbitrarily closed to them diverges, whereas point (i e , e ) is stable.

Simulation results
To demonstrate the performance of the proposed nonlinear sl-PID controller in comparison with the conventional PID control, the dc/dc converter system of (29a)-(29b) was simulated using the parameters shown in Table 2. For the conventional PID controller, the integral gain is selected as k I = 200, whereas two different values are tested for the proportional gain k P = 0.4 and 0.6, with k D = 0. For the proposed nonlinear sl-PID controller gains, the design procedure mentioned in the previous subsection is followed, providing the selection k P = 25, k I = 2083 and k D = 0. As it is illustrated in Figure 4A, both the proposed and the conventional PID controllers (with both proportional gains)    Figure 4C. However, when the conventional PID controller is applied, the desired state constraint |i| ≤ i max is not guaranteed at all times because during the transient response, current i violates the desired maximum value ( Figure 4B). On the other hand, as expected from the theoretical analysis, the proposed nonlinear sl-PID controller leads the converter current to the desired regulation without violating the maximum bound. It should be highlighted that for a different choice of the proportional and integral gains, it is possible that the conventional PID controller can maintain the current below the maximum value for the given regulation scenario. However, there is no analytic method for calculating the gains and guaranteeing that |i| ≤ i max , ∀t ≥ 0, for different values of i ref or different load cases, opposed to the proposed approach, which guarantees the desired state constraint at all times. Furthermore, from the duty-ratio input performance shown in Figure 4D, it is clear that for a larger value of k P for the conventional PID controller, the input value will exceed the value of 1, which represents the physical limit of the converter. As a result, the proposed nonlinear PI controller offers a superior performance during transients and guarantees the desired state limitation for the generic nonlinear model, whereas the proposed analysis offers a rigorous methodology for the selection of the proportional and integral gains.

CONCLUSIONS
A novel nonlinear sl-PID controller was proposed in this paper to guarantee accurate output regulation and state constraint satisfaction for a wide class of nonlinear systems with constant uncertainties. Asymptotic stability of the desired equilibrium point and a given upper bound for the desired system states were analytically proven. A design procedure for the controller gains was also presented. The superiority of the proposed sl-PID controller compared with the conventional approach was demonstrated in two practical examples consisting of a motor with a constant power load and a dc/dc power electronic converter.