# Autonomous Mobile Robot Design

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1 Autonomous Mobile Robot Design Topic: Guidance and Control Introduction and PID Loops Dr. Kostas Alexis (CSE)

2 Autonomous Robot Challenges How do I control where to go?

3 Autonomous Mobile Robot Design Topic: Basic principles and prerequisites for control Dr. Kostas Alexis (CSE)

4 What is it all about? Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs, and how their behavior is modified by feedback.

5 What is it all about? Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs, and how their behavior is modified by feedback. To do that in a systematic way, we should develop a model-based approach on control synthesis.

6 1. Modeling of Dynamic Systems Generic form of dynamic system: x is the state vector, u is the input vector. Generic linear state space form: And given specific output vector:

7 State Space Representation Generic linear (time invariant) state space form: A is the system matrix (nxn), B is the input matrix (nxp), C is the output matrix (qxn), D is the feedforward matrix (qxp). In general a MIMO system (Multi-Input, Multi-Output) In fact a Linear Time Invariant system (LTI). More complex representations exist.

8 Transfer Function Representation Linear time invariant can be represented in the frequency domain using the Laplace Transform. Laplace Transform: Transfer function form to represent Single Input Single Output (SISO) relationships: And equivalently: p 1 p n correspond to the poles and z 1 z m to the zeros.

9 State Space to Transfer Function Transformation from state space to transfer function: Possibly leads to multiple transfer functions

10 Dynamic system example System ordinary differential equation: Or in state space form: Or in transfer function form:

11 2. Analysis of Dynamic Systems Time-Domain Response of an LTI system Frequency-Domain Response of an LTI system Stability of an LTI system

12 Time-Domain Response The time response of a linear dynamic system consists of the sum of the transient response which depends on the initial conditions and the steadystate response which depends on the system input. These correspond to the free (homogeneous or zero input) and the forced (inhomogeneous or non-zero input) solutions of the governing differential equations respectively.

13 Frequency-Domain Response All the examples presented in this tutorial are modeled by linear constant coefficient differential equations and are thus linear time-invariant (LTI). LTI systems have the extremely important property that if the input to the system is sinusoidal, then the steady-state output will also be sinusoidal at the same frequency but in general with different magnitude and phase. These magnitude and phase differences as a function of frequency comprise the frequency response of the system. The frequency response of a system can be found from the transfer function in the following way: create a vector of frequencies (varying between zero or "DC" to infinity) and compute the value of the plant transfer function at those frequencies. If G(s) is the open-loop transfer function of a system and ω is the frequency vector, we then plot G(jω) versus ω. Since G(jω) is a complex number, we can plot both its magnitude and phase (the Bode Plot) or its position in the complex plane (the Nyquist Diagram). Both methods display the same information in different ways.

14 Stability Bounded Input Bounded Output (BIBO) sense: a system is stable if the output remains bounded for all bounded (finite) inputs. Practically, this means that the system will not blow up while in operation. Using transfer functions: the system is stable if ALL of the poles are in the left complex plane. If any of them is on the right then the system is unstable. If any of them has zero real component, the system is critically stable.

15 Stability Bounded Input Bounded Output (BIBO) sense: a system is stable if the output remains bounded for all bounded (finite) inputs. Practically, this means that the system will not blow up while in operation. Using transfer functions: the system is stable if ALL of the poles are in the left complex plane. If any of them is on the right then the system is unstable. If any of them has zero real component, the system is critically stable. Stable region

16 Stability Bounded Input Bounded Output (BIBO) sense: a system is stable if the output remains bounded for all bounded (finite) inputs. Practically, this means that the system will not blow up while in operation. Using transfer functions: the system is stable if ALL of the poles are in the left complex plane. If any of them is on the right then the system is unstable. If any of them has zero real component, the system is critically stable. Unstable region

17 Stability Bounded Input Bounded Output (BIBO) sense: a system is stable if the output remains bounded for all bounded (finite) inputs. Practically, this means that the system will not blow up while in operation. Using transfer functions: the system is stable if ALL of the poles are in the left complex plane. If any of them is on the right then the system is unstable. If any of them has zero real component, the system is critically stable. Critical stability

18 Stability Bounded Input Bounded Output (BIBO) sense: a system is stable if the output remains bounded for all bounded (finite) inputs. Practically, this means that the system will not blow up while in operation. Using transfer functions: the system is stable if ALL of the poles are in the left complex plane. If any of them is on the right then the system is unstable. If any of them has zero real component, the system is critically stable. Stable region oscillations No oscillations oscillations

19 1 st Order Systems First order system representation: With pole a, time constant τ and dcgain k dc Time constant is the value that the system reaches 63% of its steady-state value.

20 2 nd Order Systems First order system representation: Or in transfer function form: dcgain: damping ratio: (if ζ < 1 system is underdamped) natural frequency: poles/zeros:

21 2 nd Order Systems Pole map

22 2 nd Order Systems Time response

23 3. The Root Locus Method Consider a transfer function and a gain K: The closed-loop transfer function takes the form: Thus the closed-loop poles are the solutions of: And if we write: Then the poles are the solutions of:

24 Root Locus Method example Consider the transfer function:

25 Root Locus Method example Consider the transfer function: Use a single-gain Controller (K):

26 Root Locus Method example Consider the transfer function: Root locus for a gain K:

27 5. State Space Control Consider the system: How do we define observability? How do we define controllability? How do we design a simple controller to tune the dynamic response as desired? To place the poles where desired?

28 State Space system Stability Consider the system: Stability: The system is stable if the eigenvalues of matrix A are all with negative real part.

29 State Space system Observability Consider the system: Observability: The system is observable if the initial state x 0 can be determined from the system output, y(t), over some finite time t 0 < t < t f. For LTI systems, a system is observable if the observability matrix is of full rank.

30 Rank of a Matrix The rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This is the same as the dimension of the space spanned by its rows. Rank row = Rank column It is a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by A

31 Rank of a Matrix The rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This is the same as the dimension of the space spanned by its rows. Rank row = Rank column It is a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by A Rank = 2

32 State Space system Controllability Consider the system: Controllability: A system is controllable if there exists a control input, u(t), that transfers any state of the system to zero in finite time. For LTI systems, a system is controllable if the controllability matrix is of full rank.

33 Pole Placement Consider the system: Then the following structure is correct:

34 Pole Placement Assuming zero reference: The state space equations of the closed loop system take the form: The pole placement techniques is about how to find the gain matrix K such that the poles of the closed-loop system go to desired locations. This is under the assumption of an LTI system and infinite actuator dynamic range and bandwidth.

35 Autonomous Mobile Robot Design Topic: PID Flight Control for Multirotors Dr. Kostas Alexis (CSE)

36 MAV Dynamics To append the forces and moments we need to combine their formulation with Next step: append the MAV forces and moments

37 MAV Dynamics MAV forces in the body frame: Moments in the body frame:

38 MAV Dynamics MAV forces in the body frame: Moments in the body frame:

39 Control System Block Diagram There are simpler

40 Control System Block Diagram Simplified loop

41 Controlling a Multirotor along the x-axis Assume a single-axis case. The system has to coordinate its pitching motion and thrust to move to the desired point ahead of its axis. Roll is considered to be zero, yaw is considered to be constant. No initial velocity. No motion is expressed in any other axis. A system of only two degrees of freedom.

42 Controlling a Multirotor along the x-axis Simplified linear dynamics

43 Controlling a Multirotor along the x-axis Explanation of translation model

44 Controlling a Multirotor along the x-axis Simplified linear dynamics How does this system behave?

45 Controlling a Multirotor along the x-axis

46 Controlling a Multirotor along the x-axis

47 Controlling a Multirotor along the x-axis Simplified linear dynamics How to control this system? Most common way: use of PD controllers

48 Controlling a Multirotor along the x-axis Decoupled Control Structure

49 Controlling a Multirotor along the x-axis Decoupled Control Structure How to select the PD gains?

50 A mini introduction to PID Control PID = Proportional-Integral-Derivative feedback control It corresponds to one of the most commonly used controllers used in industry. It's success is based on its capacity to efficiently and robustly control a variety of processes and dynamic systems, while having an extremely simple structure and intuitive tuning procedures. Not comparable in performance with modern control strategies, but still the most common starting point How to select the PD gains?

51 Controlling a Multirotor along the x-axis MATLAB Implementation

52 Controlling a Multirotor along the x-axis MATLAB Implementation

53 Controlling a Multirotor along the x-axis

54 Controlling a Multirotor along the x-axis MATLAB Implementation

55 Controlling a Multirotor along the x-axis

56 Controlling a Multirotor along the x-axis MATLAB Implementation

57 Controlling a Multirotor along the x-axis

58 Real-life Limitations The control margins of the aerial vehicle have limits and therefore the PID controller has to be designed account for these constraints. The integral term needs special caution due to the often critically stable or unstable characteristics expressed by unmanned aircraft. With the exception of hover/or trimmed-flight, an aerial vehicle is a nonlinear system. As the PID is controller, it naturally cannot maintain an equally good behavior for the full flight envelope of the system. A variety of techniques such as Gain scheduling are employed to deal with this fact.

59 Autonomous Mobile Robot Design Topic: LQR Flight Control for Multirotors Dr. Kostas Alexis (CSE)

60 Controlling a Multirotor along the x-axis Assume a single-axis multirotor. The system has to coordinate its pitching motion and thrust to move to the desired point ahead of its axis. Roll is considered to be zero, yaw is considered to be constant. No initial velocity. No motion is expressed in any other axis. A system of only two degrees of freedom.

61 Controlling a Multirotor along the x-axis Simplified linear dynamics How does this system behave?

62 Optimal Model-Based Control Use model knowledge to design optimal control behaviors. The employed model must be simultaneously sufficiently accurate but also simple enough to enable efficient control computation. Optimal control can support linear and nonlinear systems as well as systems subject to state, output and input constraints. Further extensions (e.g. for hybrid systems) also exist. Established method for unconstrained linear systems regulation: Linear Quadratic Regulator (LQR) Generalization: Linear Quadratic Gaussian (LQG) control

63 Linear Quadratic Regulator Consider the system and suppose we want to design state feedback control u=fx to stabilize the system. The design of F is a trade-off between the transit response and the control effort. The optimal control approach to this design trade-off is to define the performance index (cost functional): and search for the control u=fx that minimizes this index. Q is an nxm symmetric positive semidefinite matrix and R is an mxm symmetric positive definite matrix.

64 Linear Quadratic Regulator The matrix Q can be written as Q=M T M, where M is a pxn matrix, with p n. With this representation: Where z=mx can be viewed as a controlled input. Optimal Control Problem: Find u(t)=fx(t) to maximize J subject to the model: Since J is defined by an integral over, the first question we need to address is: Under what conditions will J exist and be finite?

65 Write J as: Linear Quadratic Regulator is a monotonically increasing function of t f. Hence, as either converges to a finite limit or diverges to infinity. Under what conditions will be finite?

66 Linear Quadratic Regulator Recall that (A,B) is stabilizable if the uncontrollable eigenvalues of A, if any, have negative real parts. Notice that (A,B) is stabilizable if (A,B) is controllable or Definition: (A,C) is detectable if the observable eigenvalues of A, if any, have negative real parts. Lemma 1: Suppose (A,B) is stabilizable, (A,M) is detectable, where Q=M T M, and u(t)=fx(t). Then, J is finite for every if and only if:

67 Remarks: Linear Quadratic Regulator The need for (A,B) to be stabilizable is clear, for otherwise there would be no F such that: To see why detectability of (A,M) is needed, consider: (A,B) is controllable, but (A,M) is not detectable The control is clearly optimal and results in a finite J but it does not stabilize the system because

68 Linear Quadratic Regulator Lemma 2: For any stabilizing control u(t)=fx(t), the cost given by: where W is a symmetric positive semidefinite matrix that satisfies the Lyapunov equation: Remark: The control u(t)=fx(t) is stabilizing if:

69 Linear Quadratic Regulator Theorem: Consider the system: and the performance index: where Q=M T M, R is symmetric and positive definite, (A,B) is stabilizable, and (A,M) is detectable. The optimal control is: where P is the symmetric positive semidefinite solution of the Algebraic Riccati Equation (ARE):

70 Remarks: Linear Quadratic Regulator Since the control is stabilizing: The control is optimal among all square integratable signals u(t), not just among u(t)=fx(t) The Ricatti equation can have multiple solutions, but only one of them is positive semidefinite.

71 Linear Quadratic Regulator Typical penalty matrices selection: t si is the desired settling time of xi x imax is a constraint on x i u imax is a constraint on u i ρ is chosen to tradeoff regulation versus control effort

72 Linear Quadratic Regulator LQR Loop (F=K)

73 MATLAB Design Example Recall:

74 MATLAB Design Example

75 MATLAB Design Example

76 MATLAB Design Example [K,S,e] = dlqr(a,b,q,r,n)

77 MATLAB Design Example

78 MATLAB Design Example

79 MATLAB Design Example

80 MATLAB Design Example

81 MATLAB Design Example

82 MATLAB Aircraft Pitch Example For a step reference of 0.2 radians, the design criteria are the following. 1. Overshoot less than 10% 2. Rise time less than 2 seconds 3. Settling time less than 10 seconds 4. Steady-state error less than 2%

83 MATLAB Aircraft Pitch Example For a step reference of 0.2 radians, the design criteria are the following. 1. Overshoot less than 10% 2. Rise time less than 2 seconds 3. Settling time less than 10 seconds 4. Steady-state error less than 2%

84 MATLAB Aircraft Pitch Example

85 MATLAB Aircraft Pitch Example

86 MATLAB Aircraft Pitch Example

87 MATLAB Aircraft Pitch Example

88 MATLAB Aircraft Pitch Example

89 MATLAB Aircraft Pitch Example

90 MATLAB Aircraft Pitch Example

91 Code Examples and Tasks tlab/control-systems/gain-scheduled-three-loop-aircraftautopilot tlab/control-systems/lqr tlab/control-systems/pid-cruise-control tlab/control-systems/pid

92 How does this apply to my project? To control the state of the robot and make it follow the desired trajectory.

93 Find out more &section=controlstatespace

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