## 6 Controller Design  
In this section, the design process of the loop-shaping controller is described. The diagram of the loop-shaping controller is shown in Figure 23, where \(r\), \(e\), \(\overset{\wedge}{K}\), \(u\), \(d_{i}\), \(u_{t}\), \(G\), \(d\), \(y\), \(n\) are the desired reference signal, the control error between the desired reference and feedback signals, the controller, the controller output, the disturbance input, the control plant, the output disturbances, the output of the system and the measured noise, respectively.  
Figure 21: Verification of the identified EPS system.  
Figure 22: Correlation of the assist torque and driver torque at different vehicle speeds.  
The design technique of the loop-shaping control is based on certain functions used to evaluate the stability of the system, including the sensitivity function, complementary sensitivity function and loop-shaping gain. These functions are defined as follows:  
The sensitivity function: \(S=\frac{1}{1+G\overset{\wedge}{K}}\)  
The complementary sensitivity function: \(T=\frac{\overset{\wedge}{GK}}{1+G\overset{\wedge}{K}}\). The loop-shaping gain: \(L=G\overset{\wedge}{K}\)  
The transfer function of the closed-loop system, which has the following relationships, is shown in Figure 23:  
\[y=Tr-Tn+GSd_{i}+Sd \tag{31}\] \[u=\overset{\wedge}{K}Sr-\overset{\wedge}{K}Sn-Td_{i}-\overset{ \wedge}{K}Sd\] (32) \[u_{t}=\overset{\wedge}{K}Sr-\overset{\wedge}{K}Sn+Sd_{i}- \overset{\wedge}{K}Sd\] (33) \[e=Sr-Sn-GSd_{i}-Sd \tag{34}\]  
The quality objectives of the desired system requirement can be proposed based on Equations (32)-(34). For example, the sensitivity function (\(S\)) must be small to reduce the influence of the output disturbances. Similarly, the complementary sensitivity function (\(T\)) must also be small to reduce the influence of the measured noise on the output signal (\(y\)). The constraint between the sensitivity function (\(S\)) and complementary sensitivity function (\(T\)) is satisfied by \(S+T=1\), indicating that the sensitivity function and complementary sensitivity function cannot be reduced simultaneously. Hence, the system is less affected by disturbances (\(d\)) and (\(d_{i}\)). Therefore, \(|S|\) and \(|GS|\) or (\(|S|\) and \(|\overset{\wedge}{K}S|\)) must be small in the lower frequencies. Then, one obtains \(|G\overset{\wedge}{K}|-1\leq|1+G\overset{\wedge}{K}|\leq|G\overset{\wedge}{K}|+1\). Thus, if \(|G\overset{\wedge}{K}|>1\), \(\frac{1}{|G\overset{\wedge}{K}|+1}\leq|\frac{1}{|1+|G\overset{\wedge}{K}|}|\leq \frac{1}{|G\overset{\wedge}{K}|-1}\), or if \(|L|>1\), \(\frac{1}{|L|+1}\leq|S|\leq\frac{1}{|L|-1}\). From these facts, one obtains \(|S|\ll 1\Leftrightarrow|L|\gg 1\).  
In addition, if \(|L|\gg 1\), \(|GS|=\left|\frac{G}{1+G\overset{\wedge}{K}}\right|\approx\frac{1}{|K|}\), \(|\overset{\wedge}{K}S|=|\frac{\overset{\wedge}{K}}{1+G\overset{\wedge}{K}}| \approx\frac{1}{|G|}\).  
From the above analysis, it is reasonable to propose that the EPS system must satisfy the conditions \(|G\overset{\wedge}{K}|\gg 1\), \(|\overset{\wedge}{K}|\gg 1\) to ensure the quality objective in the low frequency domain (\(0\), \(\omega_{l}\)). The variable (\(\cdot\))\({}_{l}\) refers to the low frequency.  
To ensure stability and adequate disturbance rejection ability in the high-frequency domain (\(\omega_{l}\), \(\infty\)), the system must satisfy the conditions \(|G\overset{\wedge}{K}|\ll 1\), \(|\overset{\wedge}{K}|\ll M\). The loop-shaping gain technique is a trade-off technique between control performance and system stability. Hence, the system requirement must achieve high control performance in the lower frequency domain and system stability in the high-frequency domain, as illustrated in Figure 24. The diagram of the loop-shaping controller for the EPS system is shown in Figure 25.  
Figure 23: Loop-shaping controller diagram.