118 
 

 American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) 
ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

© Global Society of Scientific Research and Researchers  
http://asrjetsjournal.org/  

 

LQR and 𝐇𝐇𝟐𝟐 Controllers Design Using State Derivative 

Feedback for Multivariable Systems 

Hazem I. Alia*, Mohammed Z. M. Alib 

a,bDepartment of control and system engineering, University of Technology, Baghdad-10001, Iraq 
aEmail: hazemcontrol2001@yahoo.com 

bEmail: mohammedzuhair29@gmail.com 

 

 

Abstract  

This paper presents the design of LQR (linear quadratic regular) and H2 controller using state derivative 

feedback. This design is solvable for all controllable systems.  The state derivative feedback is used instead of 

state feedback in many mechanical systems because the main sensors of vibration are accelerometers. A 

multivariable active suspension system is used in this paper to show the effectiveness of the proposed 

controllers. The obtained results are compared to the same approaches when a state feedback is used. It is shown 

that the design using state derivative feedback can achieve a better performance. 

Keywords: LQR control; H2 control; state derivative feedback; multivariable systems; active suspension. 

 1. Introduction 

The state derivative feedback is very useful and essential for achieving a desired specification for some control 

problems. The motivation of using state derivative feedback comes from controlled vibration suspension of 

mechanical systems where the accelerometers represent the main sensors of vibration [1]. Different approaches 

that are based on state feedback have been extended to be designed using state derivative feedback. Linear 

quadratic regulator (LQR) is considered one of the well-known approaches that provide practical feedback 

gains. This method has adopted either feedback or derivative feedback controller and it provides a perfect 

stabilization for an active suspension system [2]. The LQR approach can achieve an acceptable performance of 

the system by minimizing the performance index [3]. Based on LQR, some of new control algorithms have been 

derived such as in [1,4].  

------------------------------------------------------------------------ 

* Corresponding author.  

http://asrjetsjournal.org/


American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 37, No  1, pp 118-128 

119 

The H2 is used to find the optimal gain matrices the achieve the desired performance. The H2 optimal control is 

used in the design of state feedback control by minimizing a quadratic performance index of the system and 

attenuating the effect of disturbances. Reference  [5] have used the state derivative feedback for direct algorithm 

for the pole placement for multi input linear system. Reference [6] used state derivative feedback for Pole-

placement for single input single output systems. Cardim and his colleagues [7] used state derivative feedback 

for linear control systems. Kataria and his colleagues [8] used state derivative feedback for Pole-placement 

problem. Wang and his colleagues [9] have used the state feedback H2 control with regional pole assignment. 

Reference [10] presented a technique based on state derivative for robust vibration control of dynamical 

systems. 

In this paper, the design of LQR and H2 controllers are presented using state derivative feedback. The proposed 

controllers are applied to a multivariable active suspension. 

2. Controllers Design 

In this section, the solutions of  LQR optimal control and H2 robust control using state derivative feedback are 

presented. 

2.1. LQR State Derivative Feedback Problem Formulation 

Consider a continuous, time-invariant, linear system: 

�̇�𝑥(𝑡𝑡)= 𝐴𝐴𝑥𝑥(𝑡𝑡) + 𝐵𝐵𝐵𝐵(𝑡𝑡)                                (1) 

The objective is to stabilize the system by means of a linear state derivative feedback expressed by:  

𝐵𝐵(𝑡𝑡) = −𝐾𝐾�̇�𝑥(𝑡𝑡)                                        (2) 

The control law in equation (2) is to stabilize the system with a desired performance. The closed-loop system 

dynamics is: 

�̇�𝑥(𝑡𝑡)=𝐴𝐴𝑐𝑐  𝑥𝑥(𝑡𝑡)                                           (3) 

where 

𝐴𝐴𝑐𝑐=(𝐼𝐼 + 𝐵𝐵𝐾𝐾)−1𝐴𝐴                                          (4) 

The stabilizing control with good dynamic behavior is achieved by minimizing a quadratic cost or performance 

index of the type [1]: 

𝐽𝐽(�̇�𝑥(𝑡𝑡),𝐵𝐵(𝑡𝑡)=∫ (�̇�𝑥(𝑡𝑡)𝑄𝑄𝑥𝑥(𝑡𝑡) + 𝐵𝐵𝑇𝑇(𝑡𝑡)𝑅𝑅𝐵𝐵(𝑡𝑡))𝑑𝑑𝑡𝑡∞
0                                            (5) 

Substituting equation (2) into 𝐽𝐽, the performance index is: 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 37, No  1, pp 118-128 

120 

𝐽𝐽 =∫ (�̇�𝑥𝑄𝑄𝑥𝑥 + (𝐾𝐾�̇�𝑥)𝑇𝑇𝑅𝑅(𝐾𝐾�̇�𝑥))𝑑𝑑𝑡𝑡 = ∫ �̇�𝑥𝑇𝑇(𝑄𝑄 + 𝐾𝐾𝑇𝑇𝑅𝑅𝐾𝐾)�̇�𝑥𝑑𝑑𝑡𝑡∞
0

∞
0                           (6) 

Suppose that a constant positive semidefinite symmetric matrix 𝑃𝑃 that satisfy equation (6) can be obtained, thus 

�̇�𝑥𝑇𝑇(𝑄𝑄 + 𝐾𝐾𝑇𝑇𝑅𝑅𝐾𝐾)�̇�𝑥=− 𝑑𝑑
𝑑𝑑𝑑𝑑

(𝑥𝑥𝑇𝑇𝑃𝑃𝑥𝑥) = −�̇�𝑥𝑇𝑇𝑃𝑃𝑥𝑥 − 𝑥𝑥𝑇𝑇𝑃𝑃�̇�𝑥                                        (7) 

then, equation (7) can be rewritten as: 

�̇�𝑥𝑇𝑇(𝑄𝑄 + 𝐾𝐾𝑇𝑇𝑅𝑅𝐾𝐾)�̇�𝑥 = −�̇�𝑥𝑇𝑇(𝑃𝑃𝐴𝐴𝑐𝑐−1 + 𝐴𝐴𝑐𝑐−𝑇𝑇𝑃𝑃)�̇�𝑥                                                     (8) 

where 

𝐴𝐴𝑐𝑐−1 = 𝐴𝐴−1(𝐼𝐼 + 𝐵𝐵𝐾𝐾) = 𝐴𝐴−1 + 𝐴𝐴−1𝐵𝐵𝐾𝐾                                                             (9) 

Comparing both sides of  equation (8), 

𝑃𝑃𝐴𝐴𝑐𝑐−1 + 𝐴𝐴𝑐𝑐−𝑇𝑇𝑃𝑃 + 𝐾𝐾𝑇𝑇𝑅𝑅𝐾𝐾 + 𝑄𝑄 = 0                                                                     (10) 

where 

𝐴𝐴𝑐𝑐−𝑇𝑇 = 𝐾𝐾𝑇𝑇𝐵𝐵𝑇𝑇𝐴𝐴−𝑇𝑇 + 𝐴𝐴−𝑇𝑇                                                                                   (11) 

Substituting equation (9) and (11) in equation (10), 

𝑃𝑃(𝐴𝐴−1 + 𝐴𝐴−1𝐵𝐵𝐾𝐾) + (𝐾𝐾𝑇𝑇𝐵𝐵𝑇𝑇𝐴𝐴−𝑇𝑇 + 𝐴𝐴−𝑇𝑇)𝑃𝑃 + 𝐾𝐾𝑇𝑇𝑅𝑅𝐾𝐾 + 𝑄𝑄 = 0                         (12) 

then, equation (12) can be rewritten as:   

𝑃𝑃𝐴𝐴−1 + 𝑃𝑃𝐴𝐴−1𝐵𝐵𝐾𝐾 + 𝐾𝐾𝑇𝑇𝐵𝐵𝑇𝑇𝐴𝐴−𝑇𝑇𝑃𝑃 + 𝐴𝐴−𝑇𝑇𝑃𝑃 + 𝐾𝐾𝑇𝑇𝑅𝑅𝐾𝐾 + 𝑄𝑄 = 0                           (13) 

   Since 𝑅𝑅 is positive-definite symmetric matrix, then                                   

𝑅𝑅 = 𝑇𝑇𝑇𝑇𝑇𝑇                                                                                                           (14) 

where 𝑇𝑇 is a nonsingular matrix. Substituting equation (14)  in equation (13), yields: 

𝑃𝑃𝐴𝐴−1 + 𝐴𝐴−𝑇𝑇𝑃𝑃+(𝑇𝑇𝐾𝐾 + 𝑇𝑇−𝑇𝑇𝐵𝐵𝑇𝑇𝐴𝐴−𝑇𝑇𝑃𝑃)𝑇𝑇(𝑇𝑇𝐾𝐾 + 𝑇𝑇−𝑇𝑇𝐵𝐵𝑇𝑇𝐴𝐴−𝑇𝑇𝑃𝑃) − 𝑃𝑃𝐴𝐴−1𝐵𝐵𝑇𝑇−1𝑇𝑇−𝑇𝑇𝐵𝐵𝑇𝑇𝐴𝐴−𝑇𝑇𝑃𝑃 + 𝑄𝑄 = 0    (15)  The 

minimization of  𝐽𝐽 requires the minimization of the following: 

�̇�𝑥𝑇𝑇(𝑇𝑇𝐾𝐾 + 𝑇𝑇−𝑇𝑇𝐵𝐵𝑇𝑇𝐴𝐴−𝑇𝑇𝑃𝑃)𝑇𝑇(𝑇𝑇𝐾𝐾 + 𝑇𝑇−𝑇𝑇𝐵𝐵𝑇𝑇𝐴𝐴−𝑇𝑇𝑃𝑃)�̇�𝑥                                               (16) 

Since the last expression is nonnegative, the minimum occurs when it is zero, then  



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 37, No  1, pp 118-128 

121 

𝑇𝑇𝐾𝐾 = −𝑇𝑇−𝑇𝑇𝐵𝐵𝑇𝑇𝐴𝐴−𝑇𝑇𝑃𝑃                                                                                         (17) 

The optimal gain matrix  𝐾𝐾  is:  

𝐾𝐾 = −𝑅𝑅−1𝐵𝐵𝑇𝑇𝐴𝐴−𝑇𝑇𝑃𝑃                                                                                           (18) 

Finally, the optimal stabilizing control law is given by:           

𝐵𝐵(𝑡𝑡) = 𝑅𝑅−1𝐵𝐵𝑇𝑇𝐴𝐴−𝑇𝑇𝑃𝑃�̇�𝑥(𝑡𝑡)                                                                                  (19) 

The matrix 𝑃𝑃 in equation (19) must satisfy equation (13) or the following algebraic Riccati equation (ARE):  

𝑃𝑃𝐴𝐴−1 + 𝐴𝐴−𝑇𝑇𝑃𝑃 − 𝑃𝑃𝐴𝐴−1𝐵𝐵𝑅𝑅−1𝐵𝐵𝑇𝑇𝐴𝐴−𝑇𝑇𝑃𝑃 + 𝑄𝑄 = 0                                                (20) 

2.2. 𝑯𝑯𝟐𝟐 State Derivative Feedback Problem Formulation 

Consider a linear time invariant system expressed by: 

�̇�𝑥(𝑡𝑡) = 𝐴𝐴𝑥𝑥(𝑡𝑡) + 𝐵𝐵1𝑑𝑑(𝑡𝑡) + 𝐵𝐵2𝐵𝐵(𝑡𝑡)                                                                     (21) 

𝑒𝑒(𝑡𝑡) = 𝐶𝐶1�̇�𝑥(𝑡𝑡) + 𝐷𝐷12𝐵𝐵(𝑡𝑡)                                                                                   (22) 

𝑍𝑍(𝑡𝑡) = �̇�𝑥(𝑡𝑡)                                                                                                         (23) 

The following assumptions are made: 

1.   The system matrix 𝐴𝐴 is of full rank. 

2.   (𝐴𝐴, 𝐵𝐵1) and (𝐴𝐴, 𝐵𝐵2) are stabilizable. 

3.   (𝐶𝐶1, 𝐴𝐴) is detectable. 

4.   All state derivative measurements are possible. 

The objective of this work is to obtain a scalar state derivative feedback control law described by:  

𝐵𝐵(𝑡𝑡) = −𝐾𝐾�̇�𝑥(𝑡𝑡)                                                                                                  (24) 

Assuming that 𝑑𝑑(𝑡𝑡) is the white noise vector with unit intensity, then [11]: 

‖𝑇𝑇𝑒𝑒𝑑𝑑‖𝐻𝐻2
2 = 𝐸𝐸(𝑒𝑒𝑇𝑇(𝑡𝑡)𝑒𝑒(𝑡𝑡))                                                                                   (25) 

where 𝑇𝑇𝑒𝑒𝑑𝑑 represents the overall transfer function 𝑑𝑑(𝑡𝑡) to 𝑒𝑒(𝑡𝑡), then  



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 37, No  1, pp 118-128 

122 

𝑒𝑒𝑇𝑇𝑒𝑒 = �̇�𝑥𝑇𝑇𝐶𝐶1𝑇𝑇𝐶𝐶1�̇�𝑥 + 2�̇�𝑥𝑇𝑇𝐶𝐶1𝑇𝑇𝐷𝐷12𝐵𝐵 + 𝐵𝐵𝑇𝑇𝐷𝐷12𝑇𝑇𝐷𝐷12𝐵𝐵                                               (26) 

The minimization of ‖𝑇𝑇𝑒𝑒𝑑𝑑‖𝐻𝐻2
2  is equivalent to the solution of the stochastic regulator problem by setting: 

𝑄𝑄 = 𝐶𝐶1𝑇𝑇𝐶𝐶1 , 𝑁𝑁 = 𝐶𝐶1𝑇𝑇𝐷𝐷12 , 𝑅𝑅 = 𝐷𝐷12𝑇𝑇𝐷𝐷12 

then 

𝐸𝐸�𝑒𝑒𝑇𝑇(𝑡𝑡)𝑒𝑒(𝑡𝑡)� = 𝐽𝐽��̇�𝑥(𝑡𝑡),𝐵𝐵(𝑡𝑡)� = ∫ (∞
0 �̇�𝑥𝑇𝑇(𝑡𝑡)𝑄𝑄𝑥𝑥(t) + 2�̇�𝑥𝑇𝑇(t)𝑁𝑁𝐵𝐵(t) + 𝐵𝐵𝑇𝑇(t)𝑅𝑅𝐵𝐵(t))𝑑𝑑𝑡𝑡     (27) 

and 

𝐽𝐽��̇�𝑥(𝑡𝑡), 𝑣𝑣(𝑡𝑡)� = ∫ (∞
0 �̇�𝑥𝑇𝑇(𝑡𝑡)𝑄𝑄𝑚𝑚�̇�𝑥(t) + 𝑣𝑣𝑇𝑇(t)𝑅𝑅𝑣𝑣(t))𝑑𝑑𝑡𝑡                                          (28) 

where  

𝑄𝑄𝑚𝑚 = 𝑄𝑄 − 𝑁𝑁𝑅𝑅−1𝑁𝑁𝑇𝑇                                                                                              (29) 

𝑣𝑣(t) = 𝐵𝐵(t) + 𝑅𝑅−1𝑁𝑁𝑇𝑇�̇�𝑥(𝑡𝑡)                                                                                    (30) 

Consequently, the system in equation (21) will be rewritten as: 

�̇�𝑥(𝑡𝑡) = 𝐴𝐴𝑚𝑚𝑥𝑥(𝑡𝑡) + 𝐵𝐵1𝑑𝑑(𝑡𝑡) + 𝐵𝐵2𝑣𝑣(𝑡𝑡)                                                                     (31) 

where  

𝐴𝐴𝑚𝑚 = 𝐴𝐴 − 𝐵𝐵2𝑅𝑅−1𝑁𝑁𝑇𝑇                                                                                            (32) 

In term of 𝑣𝑣(𝑡𝑡) and from equation (30), the optimal state derivative feedback is:  

𝑣𝑣(𝑡𝑡) = −𝐾𝐾𝑚𝑚�̇�𝑥(𝑡𝑡)                                                                                                  (33) 

where  

𝐾𝐾𝑚𝑚 = 𝐾𝐾 − 𝑅𝑅−1𝑁𝑁𝑇𝑇                                                                                                (34) 

Substitute equation (33) in equation (31), the system equation will be:  

�̇�𝑥(𝑡𝑡) = 𝐴𝐴𝑚𝑚𝑥𝑥(𝑡𝑡) + 𝐵𝐵1𝑑𝑑(𝑡𝑡) − 𝐵𝐵2K𝑚𝑚�̇�𝑥(𝑡𝑡) = 𝐴𝐴𝑛𝑛𝑥𝑥(𝑡𝑡) + 𝐵𝐵1𝑑𝑑(𝑡𝑡)                             (35) 

where  

𝐴𝐴𝑛𝑛 = (𝐼𝐼 + 𝐵𝐵2K𝑚𝑚)−1𝐴𝐴𝑚𝑚                                                                                       (36) 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 37, No  1, pp 118-128 

123 

Substitute equation (33) in equation (28), the objective function will be:  

𝐽𝐽�x(𝑡𝑡), 𝑣𝑣(𝑡𝑡)� = ∫ (∞
0 �̇�𝑥𝑇𝑇(𝑡𝑡)(𝑄𝑄𝑚𝑚 + 𝐾𝐾𝑚𝑚𝑇𝑇𝑅𝑅𝐾𝐾𝑚𝑚)𝑥𝑥(t))𝑑𝑑𝑡𝑡                                           (37) 

Suppose that, it can be found a constant positive simidefinite symmetric 𝑃𝑃 that satisfy equation (37), 

�̇�𝑥𝑇𝑇(𝑡𝑡)(𝑄𝑄𝑚𝑚 + 𝐾𝐾𝑚𝑚𝑇𝑇𝑅𝑅K𝑚𝑚)�̇�𝑥(t) = − 𝑑𝑑
𝑑𝑑𝑑𝑑
�𝑥𝑥𝑇𝑇(𝑡𝑡)𝑃𝑃𝑥𝑥(𝑡𝑡)� = −�̇�𝑥𝑇𝑇(𝑡𝑡)𝑃𝑃𝑥𝑥(𝑡𝑡) − 𝑥𝑥𝑇𝑇(𝑡𝑡)𝑃𝑃�̇�𝑥(𝑡𝑡)          (38) 

Therefore, the performance index can be obtained as: 

𝐽𝐽��̇�𝑥(𝑡𝑡), 𝑣𝑣(𝑡𝑡)� = ∫ (∞
0 �̇�𝑥𝑇𝑇(𝑡𝑡)𝑄𝑄𝑚𝑚�̇�𝑥(t) + (𝐾𝐾𝑚𝑚�̇�𝑥(t))𝑇𝑇𝑅𝑅(𝐾𝐾𝑚𝑚�̇�𝑥(t)))𝑑𝑑𝑡𝑡 = −𝑥𝑥𝑇𝑇(𝑡𝑡)𝑃𝑃𝑥𝑥(𝑡𝑡)|0∞ = −𝑥𝑥𝑇𝑇(∞)𝑃𝑃𝑃𝑃(∞) +

𝑥𝑥𝑇𝑇(0)𝑃𝑃𝑥𝑥(0)                                                                        (39) 

Assume that the closed loop system is asymptotically stable, then 𝑥𝑥(∞)  → 0. Therefore the performance index 

can be obtained in terms of initial conditions and matrix 𝑃𝑃 as:   

𝐽𝐽 = 𝑥𝑥𝑇𝑇(0)𝑃𝑃𝑥𝑥(0)                                                                                                 (40) 

From equation (35), the following relationship can be obtained: 

𝑥𝑥(𝑡𝑡) = 𝐴𝐴𝑛𝑛−1�̇�𝑥(𝑡𝑡) + 𝐵𝐵1𝑑𝑑(𝑡𝑡)                                                                                  (41) 

where 

𝐴𝐴𝑛𝑛−1 = 𝐴𝐴𝑚𝑚−1(𝐼𝐼 + 𝐵𝐵2𝐾𝐾𝑚𝑚)                                                                                        (42) 

Then equation (38) can be rewritten as: 

�̇�𝑥𝑇𝑇(𝑡𝑡)(𝑄𝑄𝑚𝑚 + 𝐾𝐾𝑚𝑚𝑇𝑇𝑅𝑅𝐾𝐾𝑚𝑚)�̇�𝑥(t) = −�̇�𝑥𝑇𝑇(𝑡𝑡)(𝑃𝑃𝐴𝐴𝑛𝑛−1 + 𝐴𝐴𝑛𝑛−𝑇𝑇𝑃𝑃)�̇�𝑥(t)                                 (43) 

By comparing the two sides of equation (43), we obtain: 

𝑃𝑃𝐴𝐴𝑛𝑛−1 + 𝐴𝐴𝑛𝑛−1𝑃𝑃 + 𝐾𝐾𝑚𝑚𝑇𝑇𝑅𝑅𝐾𝐾𝑚𝑚 + 𝑄𝑄𝑚𝑚 = 0                                                                    (44) 

Substituting equation (42) in equation (44), one can obtain:  

𝑃𝑃(𝐴𝐴𝑚𝑚−1(𝐼𝐼 + 𝐵𝐵2𝐾𝐾𝑚𝑚)) + (𝐴𝐴𝑚𝑚−1(𝐼𝐼 + 𝐵𝐵2𝐾𝐾𝑚𝑚))𝑇𝑇𝑃𝑃 + 𝐾𝐾𝑚𝑚𝑇𝑇𝑅𝑅𝐾𝐾𝑚𝑚 + 𝑄𝑄𝑚𝑚 = 0                     (45) 

then, equation (45) can be rewritten as: 

𝑃𝑃𝐴𝐴𝑚𝑚−1 + 𝐴𝐴𝑚𝑚−𝑇𝑇𝑃𝑃 + 𝑃𝑃𝐴𝐴𝑚𝑚−𝑇𝑇𝐵𝐵2𝐾𝐾𝑚𝑚 + 𝐾𝐾𝑚𝑚𝑇𝑇𝐵𝐵2𝑇𝑇𝐴𝐴𝑚𝑚−𝑇𝑇𝑃𝑃 + 𝐾𝐾𝑚𝑚𝑇𝑇𝑅𝑅𝐾𝐾𝑚𝑚 + 𝑄𝑄𝑚𝑚 = 0                     (46) 

Since 𝑅𝑅 is positive definite symmetric matrix, then 𝑅𝑅 = 𝑇𝑇𝑇𝑇𝑇𝑇, where 𝑇𝑇 is nonsingular matrix. Equation (46) can 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 37, No  1, pp 118-128 

124 

be rewritten as:   

𝑃𝑃𝐴𝐴𝑚𝑚−1 + 𝐴𝐴𝑚𝑚−𝑇𝑇𝑃𝑃 + 𝑃𝑃𝐴𝐴𝑚𝑚−1𝐵𝐵2𝐾𝐾𝑚𝑚 + 𝐾𝐾𝑚𝑚𝑇𝑇𝐵𝐵2𝑇𝑇𝐴𝐴𝑚𝑚−𝑇𝑇𝑃𝑃 + 𝐾𝐾𝑚𝑚𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝐾𝐾𝑚𝑚 + 𝑄𝑄𝑚𝑚 = 0                 (47) 

By reformulating equation (47), the following equation can be obtained: 

𝑃𝑃𝐴𝐴𝑚𝑚−1 + 𝐴𝐴𝑚𝑚−𝑇𝑇𝑃𝑃 (𝑇𝑇𝐾𝐾𝑚𝑚 + 𝑇𝑇−𝑇𝑇𝐵𝐵2𝑇𝑇𝐴𝐴𝑚𝑚−𝑇𝑇𝑃𝑃)𝑇𝑇(𝑇𝑇𝐾𝐾𝑚𝑚 + 𝑇𝑇−𝑇𝑇𝐵𝐵2𝑇𝑇𝐴𝐴𝑚𝑚−𝑇𝑇𝑃𝑃) − 𝑃𝑃𝐴𝐴𝑚𝑚−1𝐵𝐵2𝑅𝑅−1𝐵𝐵2𝑇𝑇𝐴𝐴𝑚𝑚−𝑇𝑇𝑃𝑃 + 𝑄𝑄𝑚𝑚 = 0     (48) 

The minimization of 𝐽𝐽 requires the minimization of 

�̇�𝑥𝑇𝑇(𝑇𝑇𝐾𝐾 + 𝑇𝑇−𝑇𝑇𝐵𝐵𝑇𝑇𝐴𝐴−𝑇𝑇𝑃𝑃)𝑇𝑇(𝑇𝑇𝐾𝐾 + 𝑇𝑇−𝑇𝑇𝐵𝐵𝑇𝑇𝐴𝐴−𝑇𝑇𝑃𝑃)�̇�𝑥                                                  (49) 

Since the last expression is nonnegative, the minimum occurs when it is zero  

𝑇𝑇𝐾𝐾𝑚𝑚 + 𝑇𝑇−𝑇𝑇𝐵𝐵2𝑇𝑇𝐴𝐴𝑚𝑚−𝑇𝑇𝑃𝑃 = 0                                                                                     (50) 

The optimal gain matrix 𝐾𝐾𝑚𝑚 is: 

𝐾𝐾𝑚𝑚 = −𝑇𝑇−1𝑇𝑇−𝑇𝑇𝐵𝐵2𝑇𝑇𝐴𝐴𝑚𝑚−𝑇𝑇𝑃𝑃 = −𝑅𝑅−1𝐵𝐵2𝑇𝑇𝐴𝐴𝑚𝑚−𝑇𝑇𝑃𝑃                                                         (51) 

Finally, the optimal stabilizing control is: 

𝑣𝑣(𝑡𝑡) = −𝐾𝐾𝑚𝑚�̇�𝑥(𝑡𝑡) = 𝑅𝑅−1𝐵𝐵2𝑇𝑇𝐴𝐴𝑚𝑚−𝑇𝑇𝑃𝑃�̇�𝑥(𝑡𝑡)                                                                 (52) 

Substituting equation (52) in equation (30) yields: 

𝐵𝐵(𝑡𝑡) = 𝑅𝑅−1𝐵𝐵2𝑇𝑇𝐴𝐴𝑚𝑚−𝑇𝑇𝑃𝑃�̇�𝑥(𝑡𝑡) − 𝑅𝑅−1𝑁𝑁𝑇𝑇�̇�𝑥(𝑡𝑡)                                                              (53) 

then  

𝐾𝐾 = −𝑅𝑅−1[𝐵𝐵2𝑇𝑇(𝐴𝐴 − 𝐵𝐵2𝑅𝑅−1𝑁𝑁𝑇𝑇)−𝑇𝑇𝑃𝑃 − 𝑁𝑁𝑇𝑇]                                                         (54) 

The equations of the closed loop system using state derivative feedback H2 control are: 

�̇�𝑥(𝑡𝑡) = 𝐴𝐴𝑐𝑐𝑥𝑥(𝑡𝑡) + 𝐵𝐵1𝑑𝑑(𝑡𝑡) + 𝐵𝐵𝑐𝑐𝑟𝑟(𝑡𝑡)                                                                     (55) 

𝑦𝑦(𝑡𝑡) = 𝐶𝐶𝑥𝑥(𝑡𝑡)                                                                                                       (56) 

where  

𝐴𝐴𝑐𝑐 = (𝐼𝐼 + 𝐵𝐵2𝐾𝐾)−1(𝐴𝐴 − 𝐵𝐵2𝐶𝐶)                                                                              (57) 

𝐵𝐵𝑐𝑐 = (𝐼𝐼 + 𝐵𝐵2𝐾𝐾)−1𝐵𝐵2                                                                                            (58) 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 37, No  1, pp 118-128 

125 

3. Illustrative Example 

A multivariable active suspension system, shown in Figure 1 is used to show the effectiveness of the proposed 

controllers. The system dynamics can be represented by a state space model as [7]:  

⎣
⎢
⎢
⎡�̇�𝑥1(𝑡𝑡)
�̇�𝑥2(𝑡𝑡)
�̇�𝑥3(𝑡𝑡)
�̇�𝑥4(𝑡𝑡)⎦

⎥
⎥
⎤

=























−−

−−−−

ssss

cccc

m
b

m
b

m
k

m
k

M
b

M
bb

M
k

M
kk

2222

221221

1000
0100

⎣
⎢
⎢
⎡𝑥𝑥1

(𝑡𝑡)
𝑥𝑥2(𝑡𝑡)
𝑥𝑥3(𝑡𝑡)
𝑥𝑥4(𝑡𝑡)⎦

⎥
⎥
⎤

+























−

s

cc

m

MM
10

11
00
00

𝐵𝐵(𝑡𝑡)        (59) 

�𝑦𝑦1
(𝑡𝑡)

𝑦𝑦2(𝑡𝑡)� = 







0010
0001

⎣
⎢
⎢
⎡𝑥𝑥1

(𝑡𝑡)
𝑥𝑥2(𝑡𝑡)
𝑥𝑥3(𝑡𝑡)
𝑥𝑥4(𝑡𝑡)⎦

⎥
⎥
⎤
                                                                                      (60) 

where 𝑀𝑀𝑐𝑐  represents a car mass, 𝑚𝑚𝑠𝑠 represents the driver plus seat mass. The stiffness 𝑘𝑘1 and the damping 𝑏𝑏1 

represent the shock absorbers by which the vertical vibration caused by a street may be partially attenuated. The 

stiffness 𝑘𝑘2  and the damping b2 represent the car seat suspension elements by which the undesirable vibrations 

subjected to the driver can be reduced. The control inputs 𝐵𝐵1(𝑡𝑡) and 𝐵𝐵2(𝑡𝑡).can be changed to increase the 

damping of vibration of the masses 𝑀𝑀𝑐𝑐  and 𝑚𝑚𝑠𝑠. 

The accelerations signals �̈�𝑥1(𝑡𝑡) and �̈�𝑥2(𝑡𝑡) are only available for feedback because they are measured by 

accelerometers sensors. Depending on their measured time derivatives, the velocities �̇�𝑥1(𝑡𝑡) and �̇�𝑥2(𝑡𝑡) can be 

estimated. Now, the accelerations and velocities signals are available and the proposed method can be used to 

solve the problem. 

 

Figure 1: Active suspension of a car seat [7]. 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 37, No  1, pp 118-128 

126 

The nominal system parameters are taken as follows [7]: 𝑏𝑏1(damping) = 4 × 103 Ns/m, 𝑏𝑏2(damper of the seat 

suspension) = 5 × 102 Ns/m, 𝑘𝑘1 (stiffness)= 4 × 104 N/m, 𝑘𝑘2 (stiffness)= 5 × 103 N/m, 𝑀𝑀𝐶𝐶 (mass of the 

car)= 1500 kg, 𝑚𝑚𝑠𝑠(mass of the driver) = 70 kg. 

3.1. LQR Controller Results 

Figure 2 shows the system states trajectories when state feedback LQR control and state derivative feedback 

LQR control are applied. It shows that the response obtained using state derivative feedback LQR control is fast 

with small oscillation amplitudes in comparison to that obtained using state feedback LQR control. The 

performance index weighting matrices Q and R for state feedback LQR control and state derivative feedback 

LQR control are chosen as  Q = diag{8 × 107, 0.568475001, 0.2 × 10−8, 0.8521111} and  R = diag{1, 1}. The 

resulting feedback gain matrices in cases, state feedback LQR control and state derivative feedback LQR control 

respectively are: 

𝐾𝐾 = 







−− 4995.68865.622854.04379.417

9456.182930.3438625.501726.935

 

𝐾𝐾 = 








×−×××
×−×−×−×

3333

3333

100072.0101273.0100629.0101000.1
100582.0104695.1102590.0106024.4

 

 

Figure 2: System trajectories using state feedback LQR control (dotted line) and state derivative feedback LQR 

control (solid line). 

 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 37, No  1, pp 118-128 

127 

3.2. 𝑯𝑯𝟐𝟐 Controller Results 

 Figure 4 shows the system states trajectories when state feedback H2 control and state derivative feedback H2 

control are applied. It shows that the response obtained using state derivative feedback H2 control is fast with 

small oscillation amplitudes in comparison to that obtained using state feedback H2 control. The performance 

index weighting matrices Q and R for state feedback H2 control and state derivative feedback H2 control are 

chosen as  Q = diag{100,100, 1, 1} and  R = diag{0.01, 100}. The resulting feedback gain matrices in cases, 

state feedback H2 control and state derivative feedback H2 control respectively are:                            

𝐾𝐾 = 








××××
××××

3333

3333

100001.0100012.0100011.0100021.0
101164.0109903.0103209.9104641.6

 

𝐾𝐾 = 








×−××−×−
×−×−××

3333

3333

100005.0100008.0100119.0100101.0
102162.0101078.1106049.9109414.2

 

Figure 4: System trajectories using state feedback H2 control (dotted line) and state derivative feedback H2 

control (solid line). 

4. Conclusion 

In this paper the LQR and  H2 controllers have been designed using state derivative feedback. The  H2 optimal 

control has been derived using state derivative feedback similar to LQR to find the optimal gain matrices that 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 37, No  1, pp 118-128 

128 

achieve the desired performance. The two designed approaches were applied to a multivariable active 

suspension system. It was found that the designed LQR and  H2 controllers using state derivative feedback can 

given a better performance in comparison to the same approaches using state feedback. 

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