ME5401-Mini Project Solved

Over the past several years, government mandated regulations for diesel engine out emissions- specifically oxides of nitrogen (NOx) and particulate matter (PM) - for both on and off-road vehicles have become increasingly stringent, which is shown in Figure 1. This has resulted in many diesel engine manufacturers adding more complexity to the engine systems: more sensors to better measure and/or model what the engine conditions and emissions are at any given time as a function of operating conditions, and more actuators to allow for simultaneous control of additional variables and an increasing number of performance objectives.

  

Figure 1 United States Environmental Protection Agency (EPA) emission standards [1] 

As diesel engine emissions standards become increasingly stringent, one of the most commonly employed method of emissions reduction by engine manufacturers is active control of inducted air and recirculated exhaust gas (EGR). Commonly, actuators like EGR (Exhaust Gas Recirculation) and VGTs (Variable Geometry Turbochargers) are employed in order to manipulate the flow of gases through a diesel engine to achieve the desired reduction in NOx and PM emissions, as mandated by the US government other governments around the world. A diagram of a common air path layout involving this kind of control strategy is illustrated in Figure 2, where one can see the interactions and internal feedback caused by the EGR loop and turbochargers.

  

Figure 2 A typical diesel engine system diagram 

In practice, the movement of either one of these actuators has a direct impact on the percentage of EGR in the intake manifold as well as the amount of air that can be inducted into the cylinder, resulting in a complex, interactive, multivariable system. Another difficulty of the engine air path control is to balance the engine’s customer performance against the mandates of government agencies to reduce emissions, since the introduction of EGR into the air system has a negative impact on the overall fuel economy of the engine. In this mini project, we will model this engine air system as a MIMO linear system and try to control it using the techniques you have learned in Linear Systems.

This mini project is adapted from a Master thesis of Iowa State University [2]. To learn more details about the background of this project or the following modeling process, please refer to the complete thesis [2]. You can find its soft copy from NUS library.   

2 Modelling
For model-based control, the first step is to build an effective dynamic model for our target plant, i.e., the engine air system in this project. Usually, there are two classes of methods to build a dynamic model, either by first principles or by system identification. For many complex process control applications, it is typically difficult, laborious, and time-consuming to derive a dynamic model from first principle equations. Besides, the acquired non-linear system also requires a significant time for calibration and verification, as well as more advanced control technologies.  

Therefore, in the thesis [2], instead of developing a first principles non-linear state-space model of the engine air system, empirical data have been collected from the engine air system, and used to tune a linear fourth order state-space model using the commercially available System Identification Toolbox software of MATLAB. The linearization point selected is the Mode 1 engine certification speed/load operating condition, and this point was selected because it is the point that defines the engine’s power rating. The detailed procedures to model this engine system through system identification can be found in [2]. Here we only give the obtained state space model,

x         Ax Bu= +

                                                                                                                                        (1)

y        Cx= ,

where the manipulated inputs 𝑢 = [𝑢1, 𝑢2]𝑇 are the VGT Vane position 𝑉𝐺𝑇𝑣𝑎𝑛𝑒 and EGR valve position 𝐸𝐺𝑅𝑣𝑎𝑙𝑣𝑒 respectively, which as mentioned earlier are common actuators used on diesel engines for air system management and emissions reduction. The outputs 𝑦 = [𝑦1, 𝑦2]𝑇 represent the in-cylinder air/fuel ratio (AFR) and intake manifold EGR percentage respectively.  Please note that the outputs are not the ratios themselves. The output (or controlled) variables are proprietary modeled incylinder conditions, which correlate well to the previously mentioned incylinder air/fuel ratio (AFR) and intake manifold EGR percentage.  

Since empirical input/output response data were used to identify the model, the matrices are therefore all full (i.e. all non-zero elements, which is usually not the case for first principles based models) and the states have no direct physical meaning (which is possible as state-space representations are not unique). The obtained system matrices are given as below

  8.8487+a b− ,

        −              5

 −4.5740,



A=

3.7698,



                        a b−

        −8.5645−  ,

                       c d+ + 2

0.0564+  b ,

                    10+c



0.0165−  c d+ −5
−0.0399,

d

2.5010*  +5, c+5 c

16.1212−  , 5

8.3742,

0.0319

, −0.02
c d

−5.5500+  + ,

10

−4.3662,

a d

−18.2103+  + ,

b+ 4

−4.4331,









,
3.5846               

 a c− 

−1.1183− 

20 ,



4.4936                

 c+5  −7.7181*            b+5 

  
(2)
B=            1000+ 20a
                 4.4939,                 1.5985*  a+10

                                                             b+12
                 −1.4269,                     −0.2730        

                 −3.2988, −2.1932+100 10c d++5a ,         0.0370,             −0.0109,

           C =                                                                             
                 0.2922−  ab , −2.1506,                   −0.0104,    0.0163 

                            500                                                                       
where a, b, c, d represent the last four digits in your matriculation number. For example, if your matriculation number is A0162903M, then 𝑎 = 2, 𝑏 = 9, 𝑐 = 0, 𝑑 = 3, and three parameters in

(2) can be computed as follows

                                                   d+5                   3+5
2.5010* = 2.5010* = 4.0016, c+5 0+5 ab    2 9

                                 0.2922−      = 0.2922− = 0.2562,                      (3) 500 500 c d+ −5    0+ −3 5

0.0165−   = 0.0165− = 0.0184. 1000+ 20a         1000+ 20*2

Note that in the model (1), the four state variables x have no physical meaning since they are determined by system identification. Therefore, in reality these state variables cannot be measured directly. However, for the purpose of control system design practice in this mini project, we may assume the state variables can be measured with some sensors in specific questions of next part.  

3 Control System Design
After all, we get a linear state space model (1) for the engine air system. In the following, different control strategies will be explored to achieve control of this system. We will target both the regulation and set point tracking problems. The initial condition for this system is assumed to be

x0 =0.5, -0.1, 0.3, -0.8T .  

3.1 Design specifications  
The transient step response performance specifications for all the outputs y in state space model

(1) are as follows:  

1)     The overshoot is less than 10%.

2)     The 2% settling time is less than 20 seconds.

Note: (a) This transient response is checked by giving a step reference signal for each input channel, i.e., [1, 0] and [0, 1], with zero initial conditions; (b) For all the following task 1) to 5), your control system should satisfy this performance specification and you are supposed to finish the required investigation for each task as well.  

3.2 Tasks
Your study should include, but not limited to

1)        Assume that you can measure all the four state variables, design a state feedback controller using the pole place method, simulate the designed system, check the step responses and show all the four state responses to non-zero initial state with zero external inputs. Discuss effects of the positions of the poles on system performance, and also monitor control signal size. In this step, both the disturbance and set point can be assumed to be zero. (15 points)

2)        Assume that you can measure all the four state variables, design a state feedback controller using the LQR method, simulate the designed system, check the step responses and show all the state responses to non-zero initial state with zero external inputs. Discuss effects of weightings Q and R on system performance, and also monitor control signal size. In this step, both the disturbance

and set point can be assumed to be zero. (15 points)

3)        Assume you can only measure the two outputs. Design a state observer, simulate the resultant observer-based LQR control system, monitor the state estimation error, investigate effects of observer poles on state estimation error and closed-loop control performance. In this step, both the disturbance and set point can be assumed to be zero. (15 points)

4)        Design a decoupling controller with closed-loop stability and simulate the step response of the resultant control system to verify decoupling performance with stability. In this question, the disturbance can be assumed to be zero. Is the decoupled system BIBO stable? Is it internally stable in the sense of Lyapunov? (20 points)

5)        In an application, the operating set point for the two outputs is ysp = [0.4, 0.8] .T  

Assume that you only have two sensors to measure the output. Design a controller such that the plant (the diesel engine system) can operate around the set point as close as possible at steady state even when step disturbances are present at the plant input. Plot out both the control and output signals. In your simulation, you may assume the step disturbance of magnitude 𝑤 =

[0.3, 0.2]𝑇 takes effect from time 𝑡𝑑 = 10𝑠 afterwards. (20 points)

6)        To make things more interesting, suppose we intend to regulate the four state variables directly instead of the two outputs. Our target is to maintain the states 𝑥 around a given set point 𝑥𝑠𝑝 =

[0, 0.5, −0.4, 0.3]𝑇 at steady state. Is it possible? In this question, you may assume all the state variables can be measured and there are no disturbances. (10 points)

(a)   If your answer is YES, please detail your control system design strategy to ensure x to be 𝑥𝑠𝑝 at steady state and demonstrate its effectiveness through simulation.

(b)   If your answer is NO, explain why. In such a case, we may only want to keep the state variables at steady state close enough to the set point 𝑥𝑠𝑝. However, in practice, we usually place different emphasis on the exactness of the four state variables. To address our purpose quantitively, we aim to minimize the following objective function

                                                                        1              TW x x( s − sp),                                       (4)

  J x( s) = (x xs − sp) 2

where 𝑊 = 𝑑𝑖𝑎𝑔(𝑎 + 1, 𝑏 + 1, 𝑐 + 1, 𝑑 + 1)  is a diagonal weight matrix and 𝑥𝑠  is the state vector at steady state. Here, a, b, c, d are still the last four digits in your matriculation number, as defined above. Please detail your control system design strategy to minimize the objective 𝐽(𝑥𝑠) at steady state and demonstrate its effectiveness through simulation.  

 

Note that there are no unique answers to all the above design questions. For the tasks in our project, you can assume that the control input is unlimited. However, in practice all the physical actuators can only provide a limited drive capacity. You need to make your own judgement assuming you are the engineer responsible for the control system design in the real world. There are three major factors you should consider when you design and justify your controller:

▪ Speed --- Transient response

▪ Accuracy --- Steady state error

▪ Cost ---- Size of the control signals

 

Please do follow the design procedures you have learned in linear systems to solve all the above questions.