Optimal dynamic output feedback control of Lipschitz nonlinear systems under input saturation

Abstract

In this study, the problem of optimal guaranteed cost control (OGCC) for nonlinear systems under input saturation is investigated. The purpose is to design a dynamic output feedback controller such that the closed-loop system is asymptotically stable and the upper bound of the cost function is minimized. Moreover, the designed controller ensures that the control signals do not exceed their permissible values. This leads to an optimization problem with bilinear matrix inequality (BMI) constraints. The BMI conditions are converted into the linear matrix inequality conditions by using some technical lemmas for straightforward computation of the controller matrices. The simulation results show the effectiveness and advantages of the proposed theoretical results.

Keywords

Guaranteed cost control linear matrix inequalities dynamic output feedback input saturation Lipschitz nonlinearity

1. Introduction

The input saturation is inevitable in many practical systems because of the safety constraints and physical limitations of actuators Sun and Zhang (2021). If the input saturation is neglected in control design, the designed controller may degrade the system performance or even lead to its instability Chen et al. (2016). For this reason, significant attention has been paid to control systems with input saturation and various theoretical results have been proposed in the literature Rehan et al. (2016); Alaviyan Shahri et al. (2018); Rahimi and Binazadeh (2020); Shahri et al. (2020); Yu et al. (2018); Ma et al. (2019); Yang and Li (2019); Mohammadi and Alfi (2019).

In Tarbouriech et al. (2006), a state feedback controller was designed for linear systems under input saturation by using a local generalized sector condition. A robust adaptive approach based on back-stepping for uncertain MIMO nonlinear systems with input constraints was provided in Chen et al. (2011). By using a parametric Lyapunov function, a state feedback controller is presented for the stabilization of discrete-time linear systems with saturation and input-delay in Zhou and Lin (2011). In Rehan (2013), local synchronization and anti-synchronization methods for Lipschitz nonlinear systems under disturbance and input saturation were proposed. In Rehan et al. (2020), the problem of observer-based stabilization for one-sided Lipschitz nonlinear systems in the presence of input saturation was addressed.

In practice, many systems are nonlinear. Most physical models satisfy a Lipschitz condition, at least locally Soleymani et al. (2020). The nonlinearities have to be considered in the control design. The controller design problem for Lipschitz nonlinear systems has drawn considerable attention in the past few decades Yu et al. (2020); Rehan et al. (2017); Saad et al. (2020); Yadegar et al. (2018); Zemouche et al. (2016).

In many practical applications, it is desirable to design a controller which ensures not only the asymptotically stability of the closed-loop system but also achieves an adequate level of performance Huang (2018). One way to address this problem is the so-called guaranteed cost control (GCC) Xie and Soh (1993). This approach is a useful tool to design stabilizing controllers that provides an upper bound on a prescribed performance index Khongja et al. (2018). Recently, many GCC results have been reported in the literature for linear systems Xie and Lam (2018); Paszke et al. (2006); Thuan et al. (2012); Yue et al. (2006) and nonlinear ones Amato et al. (2014); Zhang et al. (2019b, 2019a); Chen and Liu (2005). For instance, the problem of state feedback GCC for nonlinear systems with interval time-varying delays was addressed in Niamsup and Phat (2015). The robust GCC problem for quadrotor UAV with uncertainties was studied in Xu et al. (2017). The GCC method based on adaptive dynamic programming for uncertain nonlinear systems was proposed in Yang et al. (2016). In Du et al. (2021), the GCC was considered for discrete-time switched systems in the presence of parametric uncertainties.

To the best of the authors’ knowledge, less attention has been paid to the GCC problem for Lipschitz nonlinear systems in the presence of input saturation. In general, this issue leads to an optimization problem with bilinear matrix inequality (BMI) constraints, which is a non-convex problem and difficult to solve numerically. This non-convexity is because of the coupling between the controller gains and the Lyapunov matrices. We demonstrate first that the problem of the optimal guaranteed cost control (OGCC) synthesis is expressed in terms of BMIs. In order to derive design conditions in terms of LMIs, some technical lemmas are used to linearize the bilinear terms which unavoidably emerge in OGCC design for Lipschitz nonlinear systems under input saturation. By solving the LMI optimization problem, the dynamic output feedback controller (DOFC) is designed.

The main contributions of this study are summarized as follows:

Since the information of system states is difficult to obtain, the dynamic output feedback is considered to stabilize the Lipschitz nonlinear systems.

The upper bound of a given quadratic cost function is minimized by GCC technique.

The proposed method ensures that the control signals do not exceed the limits of the actuators, and thus no saturation effect is observed.

All the design conditions are formulated in terms of LMIs that can easily be solved by using standard software packages.

Finally, the results of the proposed method are compared with the LQR method for the linear model that is known to be an optimal solution. This comparison shows that the value of the cost function using the proposed method is very close to the value of the LQR cost function.

The current study is arranged as follows. The problem formulation is given in Section 2. The main results are presented in Section 3. The simulation results are provided in Section 4. Finally, a conclusion is drawn in Section 5.

2. Problem formulation

Consider a class of nonlinear systems given by

\begin{array}{l} {\dot{x}}_{p} (t) = A_{p} x_{p} (t) + B_{p} sat (u) + H_{p} f_{p} (x_{p}), \\ y (t) = C_{p} x_{p} (t), \end{array}

(1)

where

x_{p} \in ℝ^{n_{p}}, y \in ℝ^{n_{y}}, u \in ℝ^{n_{u}}

, and

f_{p} \in ℝ^{n_{f}}

denote the state, the output, the input, and the nonlinear dynamics vectors, respectively. sat(u) is a saturation function defined as

sat (u_{i}) = sign (u_{i}) \min {{\bar{u}}_{i}, | u_{i} |}, i = 1,2, \dots, n_{u}

(2)

where

{\bar{u}}_{i}

is the saturation bound of the i-th control input.

Assumption 1

It is assumed that the nonlinear function $f_{p} (x_{p}) : ℝ^{n_{x}} \to ℝ^{n_{f}}$ satisfies the following Lipschitz condition locally on a set $M \subset \in ℝ^{n_{x}}$ :

‖ f_{p} (x_{p}) - f_{p} ({\tilde{x}}_{p}) ‖ \leq ‖ L_{p} (x_{p} - {\tilde{x}}_{p}) ‖, \forall x_{p}, {\tilde{x}}_{p} \in M

(3)

where f (0) = 0, and L_p is a matrix with appropriate dimension.

We construct the following DOFC for the system (1)

\begin{array}{l} {\dot{x}}_{c} (t) = A_{c} x_{c} (t) + B_{c} e (t) \\ u (t) = C_{c} x_{c} (t) \end{array}

(4)

where

x_{c} \in ℝ^{n_{c}}

is the controller state vector, e(t) = r–y(t) denotes the closed-loop error signal, and r is a reference signal.

Assumption 2

In this study, we assume r = 0, but the framework of design controller can also be extended for r ≠ 0 with a coordinate shift.

Assumption 3

It is assumed that the controller order is equal to the plant order (i.e. n_c = n_p). Substituting the DOFC (4) into the system (1) leads to the following closed-loop system:

\begin{array}{l} {\dot{x}}_{p} (t) = A_{p} x_{p} (t) + B_{p} sat (C_{c} x_{c} (t)) + H_{p} f_{p} (x_{p}) \\ {\dot{x}}_{c} (t) = A_{c} x_{c} (t) - B_{c} C_{p} x_{p} (t) \end{array}

(5)

The polyhedron $V$ is defined as

V (C_{c}, \bar{u}) = {x_{c} (t) \in ℝ^{n_{p}} : | C_{c i} x_{c} (t) | \leq {\bar{u}}_{i}, i = 1, \dots, n_{u}}

(6)

where

C_{c i}

is the i-th row of C_c.

If $x_{c} (t) \in V (C_{c}, \bar{u})$ , the equation (5) leads to

\begin{array}{l} \dot{\bar{x}} (t) = \underset{\bar{A}}{\underset{⏟}{[\begin{matrix} A_{p} & B_{p} C_{c} \\ - B_{c} C_{p} & A_{c} \end{matrix}]}} \bar{x} (t) + \underset{\bar{H}}{\underset{⏟}{[\begin{matrix} H_{p} \\ 0 \end{matrix}]}} f_{p} (x_{p}) \\ y (t) = \underset{\bar{C}}{\underset{⏟}{[\begin{matrix} C_{p} & 0 \end{matrix}]}} \bar{x} (t) \end{array}

(7)

where

\bar{x} (t) = {[x_{p} {(t)}^{T} x_{c} {(t)}^{T}]}^{T}

Remark 1

In this study, the nonlinear term f_p(·) can be regarded as uncertainty with Lipschitz property. Now, we introduce some lemmas which are used to prove the main results.

Lemma 1

Let σ be a positive scalar, then the following inequality holds for all $κ \in ℝ$

- \frac{1}{σ} \leq κ^{2} σ - 2 κ

(8)

Proof of Lemma 1 The proof follows from the following inequality

0 \leq {(\frac{1}{\sqrt{σ}} - κ \sqrt{σ})}^{2}

(9)

Lemma 2

(Schur complement lemma) Boyd et al. (1994) For a given matrix X, the following conditions are equivalent

\begin{array}{l} • X = [\begin{matrix} X_{11} & X_{12} \\ * & X_{22} \end{matrix}] < 0 \\ • X_{22} < 0, X_{11} - X_{12} X_{22}^{- 1} X_{12}^{T} < 0 \\ • X_{11} < 0, X_{22} - X_{12}^{T} X_{11}^{- 1} X_{12} < 0 \end{array}

(10)

3. Main results

This section is devoted to design a DOFC for a class of nonlinear systems (1), such that the asymptotic stability of the closed-loop system is guaranteed and an upper bound of a quadratic performance index subjected to the input amplitude constraint is minimized.

An optimal method is proposed for designing the DOFC based on the minimization of the following quadratic performance index

J = \int_{0}^{\infty} (x_{p}^{T} (t) Q x_{p} (t) + u^{T} (t) ℛ u (t)) d t

(11)

where

Q

and

ℛ

are positive definite symmetric weighting matrices.

The equation (11) is compactly written as

J = \int_{0}^{\infty} {\bar{x}}^{T} (t) \hat{Q} \bar{x} (t) d t

(12)

where

\hat{Q} = [\begin{matrix} Q & 0 \\ 0 & C_{c}^{T} ℛ C_{c} \end{matrix}]

Theorem 1

The closed-loop system (7) is asymptotically stable and the upper bound of the cost function (11) is minimized, if there exist symmetric positive definite matrices $P_{11} \in ℝ^{n_{p} \times n_{p}}, P_{22} \in ℝ^{n_{p} \times n_{p}}$ , matrices $P_{12} \in ℝ^{n_{p} \times n_{p}}, A_{c} \in ℝ^{n_{p} \times n_{p}}, B_{c} \in ℝ^{n_{p} \times n_{y}}, C_{c} \in ℝ^{n_{u} \times n_{p}}$ , and positive scalars ɛ, γ such that the following conditions are feasible:

\begin{array}{l} \min γ \\ subject to \\ [\begin{matrix} Ω_{11} & Ω_{12} & P_{11} H_{p}^{} \\ * & Ω_{22} & P_{12}^{T} H_{p}^{} \\ * & * & - ε I \end{matrix}] < 0 \\ Ω_{11} = P_{11} A_{p} - P_{12} B_{c} C_{p} + A_{p}^{T} P_{11} - C_{p}^{T} B_{c}^{T} P_{12}^{T} + Q + ε L_{p}^{T} L_{p} \\ Ω_{12} = P_{11} B_{p} C_{c} + P_{12} A_{c} + A_{p}^{T} P_{12} - C_{p}^{T} B_{c}^{T} P_{22} \\ Ω_{22} = P_{12}^{T} B_{p} C_{c} + P_{22} A_{c} + C_{c}^{T} B_{P}^{T} P_{12} + A_{c}^{T} P_{22} + C_{c}^{T} ℛ C_{c} \end{array}

(13)

[\begin{matrix} x_{p} {(0)}^{T} & x_{c} {(0)}^{T} \end{matrix}] [\begin{matrix} P_{11} & P_{12} \\ * & P_{22} \end{matrix}] [\begin{matrix} x_{p} (0) \\ x_{c} (0) \end{matrix}] \leq γ

(14)

[\begin{matrix} - P_{11} & - P_{12} \\ * & - P_{22} + {\bar{u}}_{i}^{- 2} γ C_{c i}^{T} C_{c i} \end{matrix}] \leq 0, i = 1, \dots, n_{u}

(15)

[\begin{matrix} P_{11} & P_{12} \\ * & P_{22} \end{matrix}] > 0

(16)

Then, the DOFC (4) ensures $| u_{i} (t) | \leq {\bar{u}}_{i}$ .

Proof of Theorem 1

Consider a Lyapunov function given by

V (\bar{x}) = {\bar{x}}^{T} (t) P \bar{x} (t)

(17)

where P is partitioned as

P = [\begin{matrix} P_{11 (n_{p} \times n_{p})} & P_{12 (n_{p} \times n_{p})} \\ * & P_{22 (n_{p} \times n_{p})} \end{matrix}]

Then, the time derivative of (17) becomes

\dot{V} (\bar{x}) = {\bar{x}}^{T} (t) ({\bar{A}}^{T} P + P \bar{A}) \bar{x} (t) + f_{p}^{T} (x_{p}) {\bar{H}}^{T} P \bar{x} (t) + {\bar{x}}^{T} (t) P \bar{H} f_{p} (x_{p})

(18)

An upper bound for $\dot{V} (\bar{x})$ is considered as

\dot{V} (\bar{x}) < - (x_{p}^{T} (t) Q x_{p} (t) + u^{T} (t) ℛ u (t))

(19)

From (19), we have

\begin{matrix} {\bar{x}}^{T} (t) ({\bar{A}}^{T} P + P \bar{A} + \hat{Q}) \bar{x} (t) + f_{p}^{T} (x_{p}) {\bar{H}}^{T} P \bar{x} (t) \\ + {\bar{x}}^{T} (t) P \bar{H} f_{p} (x_{p}) < 0 \end{matrix}

(20)

The matrices $P \bar{A}$ and $P \bar{H}$ are obtained as follows

\begin{matrix} P \bar{A} = [\begin{matrix} ϕ_{11} & ϕ_{12} \\ ϕ_{21} & ϕ_{22} \end{matrix}], P \bar{H} = [\begin{matrix} P_{11} H_{p} \\ P_{12}^{T} H_{p} \end{matrix}] \\ ϕ_{11} = P_{11} A_{p} - P_{12} B_{c} C_{p} \\ ϕ_{12} = P_{11} B_{p} C_{c} + P_{12} A_{c} \\ ϕ_{21} = P_{12}^{T} A_{p} - P_{22} B_{c} C_{p} \\ ϕ_{22} = P_{12}^{T} B_{p} C_{c} + P_{22} A_{c} \end{matrix}

(21)

Equivalently, from (20), we can write

η^{T} (t) Ψ η (t) < 0

(22)

where

η (t) = {[x_{p}^{T} (t) x_{c}^{T} (t) f_{p}^{T} (x_{p})]}^{T}

and

Ψ = [\begin{matrix} ϕ_{11} + ϕ_{11}^{T} + Q & ϕ_{12} + ϕ_{21}^{T} & P_{11} H_{p} \\ * & ϕ_{22} + ϕ_{22}^{T} + C_{c}^{T} ℛ C_{c} & P_{12}^{T} H_{p} \\ * & * & 0 \end{matrix}]

Based on Assumption 1, the following inequality holds

0 \leq ε x_{p}^{T} (t) L_{p}^{T} L_{p} x_{p} (t) - ε f_{p}^{T} (x_{p}) f_{p} (x_{p})

(23)

where ɛ is an arbitrary positive scalar.

By adding the inequality (23) to (22), one can get the inequality (13).

Integrating the both sides of (19) from 0 to ∞ yields

\begin{array}{l} V (\bar{x} (\infty)) - V (\bar{x} (0)) < \\ - \int_{0}^{\infty} (x_{p}^{T} (t) Q x_{p} (t) + u^{T} (t) ℛ u (t)) d t = - J \end{array}

(24)

The asymptotic stability of (7) implies that $V (\bar{x} (\infty)) = 0$ . Then

J < V (\bar{x} (0)) = {\bar{x}}^{T} (0) P \bar{x} (0) \leq γ

(25)

Minimization of γ leads to minimization of the upper bound of the cost function $J$ .

Note that $\dot{V} (\bar{x} (t)) < 0$ implies that $V (\bar{x} (t)) < V (\bar{x} (0))$ . Hence, considering (17), we have

x^{T} (t) P x (t) < x^{T} (0) P x (0)

(26)

From (25) and (26), we get

\frac{1}{γ} {\bar{x}}^{T} (t) P \bar{x} (t) < \frac{1}{γ} {\bar{x}}^{T} (0) P \bar{x} (0) \leq 1

(27)

On the other hand, the condition $| u_{i} (t) | \leq {\bar{u}}_{i}$ can be written as

u_{i}^{T} (t) u_{i} (t) \leq {\bar{u}}_{i}^{2} \Rightarrow {\bar{u}}_{i}^{- 2} u_{i}^{T} (t) u_{i} (t) \leq 1

(28)

Therefore, if the following inequality is satisfied

{\bar{u}}_{i}^{- 2} u_{i}^{T} (t) u_{i} (t) \leq \frac{1}{γ} {\bar{x}}^{T} (t) P \bar{x} (t)

(29)

the condition

| u_{i} (t) | \leq {\bar{u}}_{i}

is guaranteed.

The inequality (29) can be rewritten as

{\bar{u}}_{i}^{- 2} x_{c}^{T} (t) C_{c i}^{T} C_{c i} x_{c} (t) - \frac{1}{γ} {\bar{x}}^{T} (t) P \bar{x} (t) \leq 0

(30)

which is equivalent to the following condition

\begin{matrix} {\bar{x}}^{T} (t) ([\begin{matrix} 0 & 0 \\ * & {\bar{u}}_{i}^{- 2} γ C_{c i}^{T} C_{c i} \end{matrix}] - [\begin{matrix} P_{11} & P_{12} \\ * & P_{22} \end{matrix}]) \bar{x} (t) \end{matrix} \leq 0

(31)

Remark 2

The conditions (13) and (15) are BMIs because of the coupling between the controller gains and Lyapunov matrices. It is well-known that optimization problems with BMIs are non-convex and NP-hard in general Toker and Ozbay (1995), and hence computationally intractable in theory. Next, the optimization problem is transformed into a set of LMIs that are numerically tractable by any convex optimization software.

Theorem 2. The closed-loop system (7) is asymptotically stable and the upper bound of the cost function (11) is minimized, if for given positive scalars ɛ, κ, there exist symmetric positive definite matrices $R \in ℝ^{n_{p} \times n_{p}}, S \in ℝ^{n_{p} \times n_{p}}$ , matrices $K \in ℝ^{n_{u} \times n_{p}}, L \in ℝ^{n_{p} \times n_{y}}, E \in ℝ^{n_{p} \times n_{p}}, N \in ℝ^{n_{p} \times n_{p}}$ , and a positive scalar γ such that the following LMIs are feasible

\begin{array}{l} \min γ \\ subject to \\ [\begin{matrix} Φ_{11} & Φ_{12} & H_{p} & ε R L_{p}^{T} & R & K^{T} \\ * & Φ_{22} & S H_{p} & ε L_{p}^{T} & I & 0 \\ * & * & - ε I & 0 & 0 & 0 \\ * & * & * & - ε I & 0 & 0 \\ * & * & * & * & - Q^{- 1} & 0 \\ * & * & * & * & * & - ℛ^{- 1} \end{matrix}] < 0 \\ Φ_{11} = A_{p} R + R A_{p}^{T} + B_{p} K + K^{T} B_{p}^{T} \\ Φ_{12} = A_{p} + E^{T} \\ Φ_{22} = A_{p}^{T} S + S A_{p} - L C_{p} - C_{p}^{T} L^{T} \end{array}

(32)

[\begin{matrix} - γ & x_{p}^{T} (0) & x_{p}^{T} (0) S + x_{c}^{T} (0) N \\ * & - R & - I \\ * & * & - S \end{matrix}] \leq 0

(33)

[\begin{matrix} - R & - I & K_{i}^{T} \\ * & - S & 0 \\ * & * & {\bar{u}}_{i}^{2} (κ^{2} γ - 2 κ) \end{matrix}] \leq 0, i = 1, \dots, n_{u}

(34)

[\begin{matrix} R & I \\ * & S \end{matrix}] > 0

(35)

Then, the controller parameters are obtained as

{\begin{cases} A_{c} = N^{- 1} (E + L C_{p} R - S A_{p} R - S B_{p} K) M^{- T} \\ B_{c} = N^{- 1} L \\ C_{c} = K M^{- T} \end{cases}

(36)

where the controller matrices (36) ensure

| u_{i} (t) | \leq {\bar{u}}_{i}

Proof of Theorem 2

By applying the Schur complement lemma to (13), we obtain

\begin{array}{l} [\begin{matrix} {\hat{Ω}}_{11} & Ω_{12} & P_{11} H_{p} & ε L_{p}^{T} & I & 0 \\ * & {\hat{Ω}}_{22} & P_{12}^{T} H_{p} & 0 & 0 & C_{c}^{T} \\ * & * & - ε I & 0 & 0 & 0 \\ * & * & * & - ε I & 0 & 0 \\ * & * & * & * & - Q^{- 1} & 0 \\ * & * & * & * & * & - ℛ^{- 1} \end{matrix}] < 0 \\ {\overset{⌢}{Ω}}_{11} = P_{11} A_{p} - P_{12} B_{c} C_{p} + A_{p}^{T} P_{11} - C_{p}^{T} B_{c}^{T} P_{12}^{T} \\ Ω_{12} = P_{11} B_{p} C_{c} + P_{12} A_{c} + A_{p}^{T} P_{12} - C_{p}^{T} B_{c}^{T} P_{22} \\ {\overset{⌢}{Ω}}_{22} = P_{12}^{T} B_{p} C_{c} + P_{22} A_{c} + C_{c}^{T} B_{P}^{T} P_{12} + A_{c}^{T} P_{22} \end{array}

(37)

We define P and P ⁻¹ as follows Scherer et al. (1997)

P = [\begin{matrix} P_{11} & P_{12} \\ * & P_{22} \end{matrix}] = [\begin{matrix} S & N \\ * & W \end{matrix}]

(38)

P^{- 1} = [\begin{matrix} R & M \\ * & T \end{matrix}]

(39)

Note that PP ⁻¹ = I leads to

M = (I - R S) N^{- T}

(40)

By defining

Γ_{1} = [\begin{matrix} R & I \\ M^{T} & 0 \end{matrix}], Γ_{2} = [\begin{matrix} I & S \\ 0 & N^{T} \end{matrix}]

(41)

we conclude that

P Γ_{1} = Γ_{2}

(42)

Therefore, one can get

Γ_{1}^{T} P Γ_{1} = Γ_{1}^{T} Γ_{2} = [\begin{matrix} R & I \\ I & S \end{matrix}]

(43)

Note that $Γ_{1}^{T} P Γ_{1} > 0$ implies that P > 0.

Substituting (38) into (37), and pre- and post-multiplying by $diag (Γ_{1}^{T}, I, I, I, I)$ and its transpose yields

\begin{array}{l} [\begin{matrix} Σ_{11} & Σ_{12} & H_{p} & ε R L_{p}^{T} & R & M C_{c}^{T} \\ * & Σ_{22} & S H_{p} & ε L_{p}^{T} & I & 0 \\ * & * & - ε I & 0 & 0 & 0 \\ * & * & * & - ε I & 0 & 0 \\ * & * & * & * & - Q^{- 1} & 0 \\ * & * & * & * & * & - ℛ^{- 1} \end{matrix}] < 0 \\ Σ_{11} = R A_{p}^{T} + M C_{c}^{T} B_{p}^{T} + A_{p} R + B_{p} C_{c} M^{T} \\ Σ_{12} = A_{p} + R A_{p}^{T} S + M C_{c}^{T} B_{p}^{T} S - R C_{p}^{T} B_{c}^{T} N^{T} + M A_{c}^{T} N^{T} \\ Σ_{22} = A_{p}^{T} S - C_{p}^{T} B_{c}^{T} N^{T} + S A_{p} - N B_{c} C_{p} \end{array}

(44)

By applying the change of variables (36) to (44), we get the LMI (32).

Using the Schur complement lemma, the inequality (25) can be written as

[\begin{matrix} - γ & {\bar{x}}^{T} (0) P \\ * & - P \end{matrix}] \leq 0

(45)

Then, pre- and post-multiplying both sides of (45) by $diag (I, Γ_{1}^{T})$ and its transpose, we have

\begin{matrix} [\begin{matrix} - γ & {\bar{x}}^{T} (0) Γ_{2} \\ * & - Γ_{1}^{T} Γ_{2} \end{matrix}] \leq 0 \end{matrix}

(46)

which is equivalent to LMI (33).

Applying the Schur complement lemma to (31) results in

[\begin{matrix} - P & [\begin{matrix} 0 \\ C_{c i}^{T} \end{matrix}] \\ * & - {\bar{u}}_{i}^{2} γ^{- 1} \end{matrix}] \leq 0

(47)

From Lemma 1, substituting the upper bound of −γ⁻¹ into (47) yields

[\begin{matrix} - P & [\begin{matrix} 0 \\ C_{c i}^{T} \end{matrix}] \\ * & {\bar{u}}_{i}^{2} (κ^{2} γ - 2 κ) \end{matrix}] \leq 0

(48)

Performing a congruence transformation to (48) by $diag (Γ_{1}^{T}, I)$ and its transpose, one can get the LMI (34). This concludes the proof of Theorem 2.

Remark 3

From the practical point of view, the system states may not be available for implementation in many real applications. To this end, in this study, a DOFC is designed to overcome the difficulty in achieving all the information of system states.

Remark 4

If x_c(0), which is the coefficient of N in LMI (33), is zero, N = I can be chosen for simplicity.

Remark 5

The selection of $Q$ and $ℛ$ depends on the designer’s experience. For the sake of simplicity, the weighting matrices can be chosen in diagonal form. The existence conditions of the OGCC for linear systems under input saturation are provided in the following corollary.

Corollary 1

The closed-loop system (7) with f_p(x_p) = 0 is asymptotically stable and the upper bound of the cost function (11) is minimized, if for given positive scalar κ, there exist symmetric positive definite matrices $R \in ℝ^{n_{p} \times n_{p}}, S \in ℝ^{n_{p} \times n_{p}}$ , matrices $K \in ℝ^{n_{u} \times n_{p}}, L \in ℝ^{n_{p} \times n_{y}}, E \in ℝ^{n_{p} \times n_{p}}, N \in ℝ^{n_{p} \times n_{p}}$ , and a positive scalar γ such that the following LMIs are feasible

\begin{array}{l} \min γ \\ subject to \\ (33), (34), (35), a n d \\ [\begin{matrix} Φ_{11} & Φ_{12} & R & K^{T} \\ * & Φ_{22} & I & 0 \\ * & * & - Q^{- 1} & 0 \\ * & * & * & - ℛ^{- 1} \end{matrix}] < 0 \\ Φ_{11} = A_{p} R + R A_{p}^{T} + B_{p} K + K^{T} B_{p}^{T} \\ Φ_{12} = A_{p} + E^{T} \\ Φ_{22} = A_{p}^{T} S + S A_{p} - L C_{p} - C_{p}^{T} L^{T} \end{array}

(49)

where the controller matrices (36) ensure

| u_{i} (t) | \leq {\bar{u}}_{i}

4. Simulation results

In this section, three examples are given to illustrate the efficacy and validity of the proposed theoretical results. The LMI-based optimization problems are solved by YALMIP toolbox Lofberg (2004) with the MOSEK solver ApS (2019). In the examples, the weighting matrices are set as $Q = I_{n_{p}}$ and $ℛ = I_{n_{u}}$ .

Example 1

Consider a linear system given by the following state-space representation

\begin{array}{l} {\dot{x}}_{p} (t) = [\begin{array}{l} - 2 & 0 \\ - 1 & 0 \end{array}] x_{p} (t) + [\begin{matrix} - 4 \\ 0 \end{matrix}] u (t) \\ y (t) = [\begin{matrix} 0 & 1 \end{matrix}] x_{p} (t) \end{array}

For κ = 0.45, the optimization problem of Corollary 1 is solved for different saturation levels.

The controller matrices are computed as

For $\bar{u} = 1$

\begin{array}{l} A_{c} = [\begin{array}{l} - 3.8713 & - 3.1558 \\ 2.3142 & 1.0460 \end{array}], B_{c} = [\begin{matrix} - 1.4404 \\ 0.7306 \end{matrix}], \\ C_{c} = [\begin{matrix} - 0.4256 & - 0.5311 \end{matrix}] \end{array}

For $\bar{u} = 2$

\begin{matrix} A_{c} = [\begin{matrix} - 4.7206 & - 3.4935 \\ 1.4878 & 0.1242 \end{matrix}], B_{c} = [\begin{matrix} - 0.4829 \\ - 0.0804 \end{matrix}], \\ C_{c} = [\begin{matrix} - 0.7141 & - 0.8130 \end{matrix}] \end{matrix}

Let the initial conditions be x_p (0) = [1,−1]^T and x_c (0) = [−1,−1]^T. The system state responses and control signals are presented in Figures 1–3. Figures 1 and 2 show the asymptotic stability of the closed-loop system under input saturation. As is evident from Figure 3, the blue and red dash-dotted lines do not exceed saturation levels $\bar{u} = 1$ and $\bar{u} = 2$ , respectively.

The performance of the proposed method is compared with the full-state feedback linear quadratic regulator (LQR) controller. The LQR is an optimal controller used for linear systems and can minimize the cost function Liu et al. (2014). The comparative results are listed in Table 1. As can be seen from the table, the values of the cost function in the proposed control scheme are very close to the LQR cost function value.

It is worth pointing out that the proposed controller design considers the actuator saturation and uses output feedback so as to avoid the dependence on the knowledge of the system states. It is observed from Table 1 that the performance of the proposed controller can be improved by increasing the saturation level.

Table 1.

Comparison of the cost function values of Example 1.

	Corollary 1 $(\bar{u} = 1)$	Corollary 1 $(\bar{u} = 2)$	LQR	Open-loop system
$J$	2.1492	2.0411	2.0307	$\infty$

Example 2

Consider a single-link flexible joint manipulator, whose dynamics are represented by Raghavan and Hedrick (1994); Ahmad et al. (2016)

{\begin{cases} {\dot{θ}}_{m} = ω_{m} \\ {\dot{ω}}_{m} = \frac{k}{J_{m}} (θ_{l} - θ_{m}) - \frac{C_{v f}}{J_{m}} ω_{m} + \frac{K_{τ}}{J_{m}} u \\ {\dot{θ}}_{l} = ω_{l} \\ {\dot{ω}}_{l} = - \frac{k}{J_{l}} (θ_{l} - θ_{m}) - \frac{m g h}{J_{l}} \sin (θ_{l}) \end{cases}

(50)

where θ_m and θ_l represent the rotation angles of the motor and the link, respectively, and ω_m and ω_l are their angular velocities. Using the parameters in Table 2, the system (50) can be reformulated as

\begin{array}{l} {\dot{x}}_{p} (t) = [\begin{matrix} 0 & 1 & 0 & 0 \\ - 48.6 & - 1.25 & 48.6 & 0 \\ 0 & 0 & 0 & 1 \\ 19.5 & 0 & - 19.5 & 0 \end{matrix}] x_{p} (t) + [\begin{matrix} 0 \\ 21.6 \\ 0 \\ 0 \end{matrix}] u (t) \\ + [\begin{matrix} 0 \\ 0 \\ 0 \\ - 3.33 \end{matrix}] \sin (x_{p 3}) \\ y (t) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}] x_{p} (t) \end{array}

The saturation level is set to $\bar{u} = 2$ . Let the initial conditions be x_p (0) = [−1,1,0,0]^T and x_c (0) = [0,0,0,0]^T. Applying Theorem 2 and Remark 4 with κ = 0.1, ɛ = 5, the controller matrices are computed as

\begin{array}{l} A_{c} = [\begin{matrix} - 1254 & 1244.9 & 10.144 & - 55.203 \\ - 1165.7 & 1145.6 & 8.975 & - 61.496 \\ - 2.165 & - 44.786 & 1.4811 & 44.895 \\ - 0.4607 & 0.5058 & - 0.9615 & - 3.9286 \end{matrix}] \\ B_{c} = [\begin{matrix} - 554.13 & - 130.78 \\ - 436.51 & 3.607 \\ 575.83 & 570.34 \\ 38.848 & - 0.8491 \end{matrix}] \\ C_{c} = [\begin{matrix} 0.0102 & 0.0238 & 0.0338 & 0.3226 \end{matrix}] \end{array}

The designed controller ensures that the closed-loop system is asymptotically stable with a minimum upper bound of the cost function (11). Moreover, the control signals remain within the considered bounds. The simulation results are provided in Figures 4-6. The system states under controller (4) converge to the origin rapidly, as shown in Figures 4 and 5. As expected from Figure 6, the control signal does not exceed the limitation of the actuator. The comparison of the cost function values is given in Table 3. The results verify the performance improvement of the closed-loop system compared to the open-loop system.

Table 2.

Manipulator parameters.

Parameter	Description	Value (unit)
J _m	Motor inertia	3.7 × 10⁻³ (kgm²)
J _ℓ	Link inertia	9.3 × 10⁻³ (kgm²)
M	Pointer mass	2.1 × 10⁻¹ (kg)
K	Torsional spring constant	1.8 × 10⁻¹(Nm/rad)
C _vf	Viscous friction coefficient	4.6 × 10⁻²(Nm/V)
K _τ	Amplifier gain	8.0 × 10⁻²(Nm/V)

Table 3.

Comparison of the cost function values of Example 2.

	Theorem 2 $(\bar{u} = 2)$	Open-loop system
$J$	9.2217	26.052

Example 3

Consider the following unstable nonlinear system described by Ekramian et al. (2013)

\begin{array}{l} {\dot{x}}_{p} (t) & = [\begin{matrix} - 2 & 3 \\ 3 & 1 \end{matrix}] x_{p} (t) + [\begin{matrix} 0 \\ 1 \end{matrix}] u (t) + [\begin{matrix} 0 \\ 1 \end{matrix}] \sin (x_{p 1}) \\ y (t) & = [\begin{matrix} 0 & 1 \end{matrix}] x_{p} (t) \end{array}

The saturation level is selected as $\bar{u} = 2$ . The initial conditions are chosen as x_p (0) = [1,−1]^T and x_c (0) = [−1,−1]^T. By solving the optimization problem of Theorem 2 with κ = 0.3 and ɛ = 12, the controller matrices are calculated as

\begin{array}{l} A_{c} & = [\begin{matrix} - 0.82849 & 19.565 \\ - 1.8331 & - 28.44 \end{matrix}], B_{c} = [\begin{matrix} 24.764 \\ - 27.183 \end{matrix}], \\ C_{c} & = [\begin{matrix} 1.6116 & - 3.2909 \end{matrix}] \end{array}

It should be noted that the open-loop system is unstable as shown in Figure 7(a). After applying the controller, the closed-loop system is stabilized and the system states converge to zero rapidly (Figure 7(b)). From Figure 8, it is observed that the control signal does not exceed the given saturation bound. The comparative results are presented in Table 4. The results of the comparison show the desired performance of the proposed control method in stabilizing Lipschitz nonlinear systems.

Table 4.

Comparison of the cost function values of Example 3.

	Theorem 2 $(\bar{u} = 2)$	Open-loop system
$J$	0.6479	$\infty$

5. Conclusion

In this study, the OGCC problem for Lipschitz nonlinear systems is studied. This problem is often formulated as an optimization problem with BMI constraints. Based on some technical lemmas and LMI tools, the OGCC synthesis is translated into an LMI optimization problem. By solving the LMI optimization problem, the DOFC is designed. The designed controller minimizes the upper bound of the performance index and ensures the control signals remain within the saturation limits. The obtained results demonstrate the effectiveness and superiority of the proposed control scheme.

Footnotes

Declaration of Conflicting Interests

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

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Majid Shahbazzadeh

Homa Salehifar

S Jalil Sadati

References

Ahmad

Rehan

Hong

(2016) Observer-based robust control of one-sided Lipschitz nonlinear systems. ISA Transactions 65: 230–240.

Alaviyan Shahri

Alfi

Tenreiro Machado

(2018) Robust stability and stabilization of uncertain fractional order systems subject to input saturation. Journal of Vibration and Control 24(16): 3676–3683.

Amato

Colacino

Cosentino

, et al. (2014) Guaranteed cost control for uncertain nonlinear quadratic systems. In: 2014 European Control Conference (ECC), Strasbourg, France, 24–27 June 2014. IEEE, pp. 1229–1235.

ApS (2019) The MOSEK Optimization Toolbox for MATLAB Manual. Available at: http://docs.mosek.com/9.0/toolbox/index.html.

Boyd

El Ghaoui

Feron

, et al. (1994) Linear Matrix Inequalities in System and Control Theory. Philadelphia: SIAM.

Chen

Liu

(2005) Fuzzy guaranteed cost control for nonlinear systems with time-varying delay. IEEE Transactions on Fuzzy systems 13(2): 238–249.

Chen

Ren

(2011) Adaptive tracking control of uncertain MIMO nonlinear systems with input constraints. Automatica 47(3): 452–465.

Chen

Shi

Nan

, et al. (2016) Adaptive neural dynamic surface sliding mode control for uncertain nonlinear systems with unknown input saturation. International Journal of Advanced Robotic Systems 13(5): 1729881416657750.

Zhao

Qiao

, et al. (2021) Guaranteed cost stabilization control of discrete‐time switched systems. IET Control Theory & Applications 15(3): 404–415.

10.

Ekramian

Sheikholeslam

Hosseinnia

(2013) Dynamic gain observer design for Lipschitz nonlinear systems subjected to noise. International journal of systems science 44(11): 2094–2103

11.

Huang

(2018) Optimal guaranteed cost control of uncertain non‐linear systems using adaptive dynamic programming with concurrent learning. IET Control Theory & Applications 12(8): 1025–1035.

12.

Khongja

Botmart

Niamsup

, et al. (2018) Guaranteed cost control for exponential stability of a nonlinear system with mixed time-varying delays in states and controls. Advances in Difference Equations 2018(1): 1–23.

13.

Liu

Zhang

, et al. (2014) A control method to make LQR robust: A planes cluster approaching mode. International Journal of Control, Automation and Systems 12(2): 302–308.

14.

Lofberg

(2004) YALMIP: A toolbox for modeling and optimization in MATLAB. In: 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No. 04CH37508), Taipei, Taiwan, 2–4 September 2004. IEEE, pp. 284–289.

15.

Görges

, et al. (2019) Event-triggered feedback control for discrete-time piecewise affine systems subject to input saturation. Nonlinear Dynamics 95(3): 2353–2365.

16.

Mohammadi

Alfi

(2019) Guaranteed cost control in delayed teleoperation systems under actuator saturation. Iranian Journal of Science and Technology, Transactions of Electrical Engineering 43(4): 827–835.

17.

Niamsup

Phat

(2015) State feedback guaranteed cost controller for nonlinear time-varying delay systems. Vietnam Journal of Mathematics 43(2): 215–228.

18.

Paszke

Gałkowski

Rogers

, et al. (2006) H_∞ and guaranteed cost control of discrete linear repetitive processes. Linear Algebra and Its Applications 412(2–3): 93–131.

19.

Raghavan

Hedrick

(1994) Observer design for a class of nonlinear systems. International Journal of Control 59(2): 515–528.

20.

Rahimi

Binazadeh

(2020) Robust model predictive control of heterogeneous time-delay multi-agent systems with polytopic uncertainties and input amplitude constraints. Journal of Vibration and Control 27(9–10): 1098–1112.

21.

Rehan

(2013) Synchronization and anti-synchronization of chaotic oscillators under input saturation. Applied Mathematical Modelling 37(10–11): 6829–6837

22.

Rehan

Ahmad

Hong

(2020) Novel results on observer-based control of one-sided Lipschitz systems under input saturation. European Journal of Control 53: 29–42.

23.

Rehan

Iqbal

Hong

(2016) Delay-range-dependent control of nonlinear time-delay systems under input saturation. International Journal of Robust and Nonlinear Control 26(8): 1647–1666.

24.

Rehan

Tufail

Ahn

, et al. (2017) Stabilisation of locally Lipschitz non‐linear systems under input saturation and quantisation. IET Control Theory & Applications 11(9): 1459–1466.

25.

Saad

Sellami

Garcia

(2020) Robust stabilization of one-sided Lipschitz nonlinear systems via adaptive sliding mode control. Journal of Vibration and Control 26(7–8): 399–412.

26.

Scherer

Gahinet

Chilali

(1997) Multiobjective output-feedback control via LMI optimization. IEEE Transactions on Automatic Control 42(7): 896–911.

27.

Shahri

ESA

Alfi

Machado

(2020) Lyapunov method for the stability analysis of uncertain fractional-order systems under input saturation. Applied Mathematical Modelling 81: 663–672.

28.

Soleymani

Nekoui

Moarefianpour

(2020) Robust adaptive unknown input observer design for a class of disturbed systems. Systems Science & Control Engineering 8(1): 249–257.

29.

Sun

Zhang

(2021) Guaranteed cost control of discrete-time switched saturated systems. Discrete & Continuous Dynamical Systems-B 26(8): 4515.

30.

Tarbouriech

Prieur

Da Silva

(2006) Stability analysis and stabilization of systems presenting nested saturations. IEEE Transactions on Automatic Control 51(8): 1364–1371.

31.

Thuan

Phat

Trinh

(2012) Dynamic output feedback guaranteed cost control for linear systems with interval time-varying delays in states and outputs. Applied Mathematics and Computation 218(21): 10697–10707.

32.

Toker

Ozbay

(1995) On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback. In: Proceedings of 1995 American Control Conference-ACC’95, Seattle, WA, 21–23 June 1995. IEEE, Vol. 4, pp. 2525–2526.

33.

Xie

Soh

(1993) Control of uncertain discrete-time systems with guaranteed cost. In: Proceedings of 32nd IEEE Conference on Decision and Control, San Antonio, TX, 15–17 December 1993. IEEE, pp. 56–61.

34.

Xie

Lam

(2018) Guaranteed cost control of periodic piecewise linear time-delay systems. Automatica 94: 274–282.

35.

Nian

Wang

, et al. (2017) Robust guaranteed cost tracking control of quadrotor UAV with uncertainties. ISA Transactions 69: 157–165.

36.

Yadegar

Afshar

Davoodi

(2018) Observer-based tracking controller design for a class of Lipschitz nonlinear systems. Journal of Vibration and Control 24(11): 2112–2119.

37.

Yang

(2019) Distributed iterative learning coordination control for leader–follower uncertain non-linear multi-agent systems with input saturation. IET Control Theory & Applications 13(14): 2252–2260.

38.

Yang

Liu

Wei

, et al. (2016) Guaranteed cost neural tracking control for a class of uncertain nonlinear systems using adaptive dynamic programming. Neurocomputing 198: 80–90.

39.

Liao

(2020) New results on observer-based robust preview tracking control for Lipschitz nonlinear systems. Journal of Vibration and Control 27(17–18): 2081–2096.

40.

Lam

Chan

(2018) T–S fuzzy-model-based output feedback tracking control with control input saturation. IEEE Transactions on Fuzzy Systems 26(6): 3514–3523.

41.

Yue

Peng

Tang

(2006) Guaranteed cost control of linear systems over networks with state and input quantisations. IEE Proceedings-Control Theory and Applications 153(6): 658–664.

42.

Zemouche

Rajamani

Boulkroune

, et al. (2016) H_∞ circle criterion observer design for Lipschitz nonlinear systems with enhanced LMI conditions.In: 2016 American Control Conference (ACC), Boston, MA, 6–8 July 2016. IEEE, pp. 131–136.

43.

Zhang

Liang

, et al. (2019a) Event-driven guaranteed cost control design for nonlinear systems with actuator faults via reinforcement learning algorithm. IEEE Transactions on Systems, Man, and Cybernetics: Systems 50(11): 4135–4150.

44.

Zhang

Xue

Wang

, et al. (2019b) Guaranteed cost control for a class of nonlinear discrete time-delay systems. IEEE Access 7: 130067–130073.

45.

Zhou

Lin

(2011) Parametric Lyapunov equation approach to stabilization of discrete-time systems with input delay and saturation. IEEE Transactions on Circuits and Systems I: Regular Papers 58(11): 2741–2754.