Linear Systems Theory  

• ``Classical'' control theory is mainly concerned with solving linear differential and difference equations by transform/spectral/frequency-domain methods (Laplace and z-transform). It regards the system as a black box and only cares about the input-output relationship (system transfer function).  "Modern" control theory has developed over the last thirty years, and is based on the state variable (state space) approach. It treats the system as a white box and thus accurate modeling of the system is required.  An important aim of control theory is to allow synthesis of a control regime to achieve a desired result (and we may or may not have a cost function). 
• Feedback Control is an error-driven strategy; corrections are made on the basis of a difference between the system's current state, and the desired state. In the simplest case of Linear Feedback Control, the corrections are proportion to the magnitude of the difference, or error. This may not work very well for nonlinear plants, so it's common to multiply these control inputs by a gain matrix. The desired state acts as an attractor, and the system behaves as a simple spring. Springs are notorious for exhibiting oscillations, so it is common to include some damping terms.  Note that this is a completely model-free approach; there is no need to model the complex dynamical system; simply feedback the observations of the system and the control will tend to eliminate the errors. 
• Feedforward (model-based, indirect) Control takes an alternative approach; a model of the dynamics of the system is built, and the inverse dynamics are solved for the input. This method has the potential for very accurate control, but depends critically on the integrity of the model. 
• Most control architectures can be catagorized as direct or indirect: indirect control makes use of a model of the plant in designing the controller, direct control tries to optimize the outputs of the plant directly. In the case of indirect control, a further distinction can be made between on-line and off-line optimization. Off-line optimization allows us to learn a plant model by observing the plant's behavior under different conditions, and subsequently use that model to design a controller. In on-line optimization, the plant has to be controlled all the time, while the plant model is being learned. This is obviously a much more difficult problem, as the combined system is now time-variant. 
• If the system has a cost associated with the state and the control, then the problem becomes an optimal control problem.  Further, if the system evolves probabilistically, the problem becomes stochastic optimal control problem and we can only consider the expected cost, which is an expectation over the random state.
• For linear systems, we can use both the transform approach (frequency analysis) and the state-space approach (time-domain analysis).  But for nonlinear systems, we can only use time-domain analysis; frequency analysis is not applicable since the superposition principle does not apply and thus the output y(t) of an arbitrary input x(t) cannot be derived from system transfer function H(s) (or impulse response h(t)).  In linear system, the output Y(s) of an arbitrary input X(s) is Y(s) = X(s)* H(s) or y(t)= x(t) convolved with h(t).
• The time-domain approach vs. the frequency-domain approach
  • The time-domain approach is more general than the frequency-domain approach since it can be used for finite horizon and/or time-varying systems as well as transient analysis.
• Controllability (or controllability to the origin)
  • A state x_0 is called controllable if there exists an input that transfers the state of the system x(t) from x_0 to the zero state in some finite time T.
• Reachability (or controllability from the origin)
  • (Geometric)   A state x_1 is called reachable if there exists an input that transfers the state of the system x(t) from the zero state to x_1 in some finite time T.
  • (Mathematical rigor)   For linear time-invariant (LTI) systems, a system is said to be reachable if and only if its controllability matrix [B, AB, ..., A^{n-1} B] is of full rank, having rank n, where n is the number of rows in A or B.
  • (From the perspective of implementation structure)    The control variables can be explicitly or implicitly enforced on the inputs of all memory devices (delay elements).
  • Reachability and controllability guarantee only the ability of a system to transfer an initial state to a final state by some control input action over a finite time interval.  By themselves, reachability and controllability do not imply the capability of a system to follow some particular trajectory.
• Observability
• Stabilizability: The pair (A, B) is called stabilizable if all its uncontrollable eigenvalues are stable.
• Detectability
• What is a dynamic system?
  • Any system with memory can be called a dynamic system.
    • A system is memoryless if its output at any time depends only on the value of the input at that same time
    • A system has a memory if it is not memoryless.
  • A dynamic system can be characterized by differential equation (in continuous time) or difference equation (in discrete time).  Linear and nonlinear differential/difference equations characterize linear and nonlinear dynamic systems, respectively.
• In Matlab, DLYAP is Discrete Lyapunov equation solver.

Key Techniques: 

• Superposition principle for linear systems
  • In time domain, analyze the response of an LTI system by zero-input (with a certain initial state) and zero-state (with a certain control input), respectively.   Then sum up the responses of zero input and zero state, resulting in the response with the specific initial state and the specific control input.  
  • The response of an LTI system can also be obtained by solving differential/difference equation with homogeneous and particular solutions.
  • For a zero-state LTI system, it satisfies the superposition principle w.r.t. inputs.   For a nonzero-state LTI system, it satisfies the superposition principle for difference in the output (\delta y) w.r.t. difference in the input (\delta x).
  • In frequency domain, for a zero-state LTI system, it satisfies the superposition principle.
  • For a zero-state LTI system, the time response with convolution is isomorphic to the frequency response with product.
• Fourier transform is a linear operator, which maps a function in L^p space (a linear vector space) to another function in L^p space.
• PID (Proportional-plus-Integral-plus-Derivative) controller:
  • P: the correction is proportional to the error between the target state X_d(t) and the current state X(t).
  • I: the correction is proportional to the integral of the error between the target state and the current state.
    • The steady-state error will be compensated/removed by this correction.   Since the non-systematic error or random errors will be cancelled out in the integral over time, then the value of the integral is produced only by the systematic error (steady-state error).
  • D: the correction is proportional to the derivative of the error between the target state and the current state.
    • The transient response will be improved due to this correction, which introduces a damping effect.  The faster the state changes, the higher the rate is, which leads to a higher damping force.
  • Thus, the control r(t)= K_p * E(t) + K_i * \int_0^t E(t) dt + K_v * d E(t)/dt, where E(t)=X_d(t) - X(t).