Introduction to Sliding Mode Control.pdf

Cleveland State University Mechanical Engineering Department

Introduction to Sliding Mode Control Prof. Richter

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Intuitive Idea ⊲ Intuitive Idea

Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

Thermostat control: define a “performance surface” as s , T − Td = 0 Design a controller that forces temperature to reach the performance surface and remain there (or close), regardless of disturbances Disturbances: outside temperature, drafts, people and equipment inside...

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Switched control: thermostat ⊲ Intuitive Idea


Thermal system dynamics are like: T˙ + τ1 T = u + d(t) (d is the disturbance) Use the switching law u = −η sign(s) (too hot? make it cold!; too cold? make it hot!) *Simulation here* Ideally, this law has the properties that: 1. T reaches T d in finite time (not asymptotic convergence) 2. The set s = 0 is invariant (once there, can’t escape) 3. By definition, once s = 0, T is insensitive to d (total disturbance rejection)

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Differential (equations) with discontinuous right-hand sides ⊲ Intuitive Idea


Simulation and discussion on s˙ = −a sign(s) + bδ(t) with a > b > 0 and |δ(t)| ≤ 1.

Standard ODE theory fails (lack of a Lipschitz condition) Filippov and Utkin proposed constructions to generate solutions Modern approach: differential inclusions (Aubin and Cellina) Replace x˙ = f (x) by x˙ ∈ co(fi (x)) (convex combination: αf1 (x) + (1 − α)f2 (x))

Key observation: If a > b, s will reach zero in finite time and remain there despite δ(t).

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Sliding Mode Control and Relay Control Systems ⊲ Intuitive Idea


Sliding Mode Control (SMC) developed in the USSR from the more general variable structure control and work in relay controls in the early 1950’s. Research by Emel’yanov, Filippov, Tsypkin and others published much work in Avtomatika i Telemekhanika, a controls journal established in 1936. The journal and its English translation Automation and Remote Control are still active (Springer).

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Sliding Mode Control Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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Utkin is credited with bringing these ideas to the West, where they continue to be developed. Very active field of research, many proven applications. Recent research led by Spurgeon, Edwards and Zinober (UK), Shtessel, Drakunov and Utkin (US), Sira-Ramirez (Mexico). SMC is known for its robustness and ease of application. It has been used as the basis to develop state estimators (observers) for nonlinear systems, paralleling what is achievable with a Kalman filter. SMC (VSC) is also used for robust trajectory control of robots and robust impedance control.

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Sliding Mode Estimation (project topic?) Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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On Sliding Observers for Nonlinear Systems, J.-J. E. Slotine, J. K. Hedrick and E. A. Misawa, J. Dyn. Sys., Meas., Control 109(3),1987, 245-252 Nonlinear Observers: A State-of-the-Art Survey, E. A. Misawa; J. K. Hedrick, J. Dyn. Sys., Meas., Control. 1989; 111(3):344-352. Sliding Mode Observers - A Survey, S.K. Spurgeon, International Journal of Systems Science, v. 39 N8, 2008, pp 751-764

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SMC Tracker: Double-Integrator System Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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Controlled plant: y¨ = u + δ where δ is a composite disturbance term (could contain unmodeled dynamics). We only assume that |d(t)| ≤ D for some known D. Objective: Given a desired trajectory y d (t), make y(t) → y d (t) regardless of d(t). Define the performance surface s = 0 with s = e˙ + λe with e = y d − y. If we can achieve s(t) = 0 after some t = tr > 0, we obtain nice error decay dynamics and total insensitivity to d(t). 8 / 35

Double-Integrator System... Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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s = 0 is also called sliding manifold and sliding surface. Every SMC design must achieve three things: 1. Finite-time reaching (sliding surface is attractive): s = 0 must be reached from any e(0) and e(0) ˙ in a certain time tr , even with disturbances. 2. Sliding mode invariance (once s = 0 is reached, s = 0 is maintained), even with disturbances. 3. System dynamics in the sliding mode must be stable (in this case this is guaranteed by e˙ + λe = 0). A popular finite-time reaching condition: ss˙ < 0 whenever s 6= 0 9 / 35


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The above reaching condition can be achieved by selecting s˙ = −η sign(s) Note that this also guarantees invariance, if sign(0)=0 (Matlab’s choice). Other finite-time reaching laws exist (terminal attractors). The derivative of s is s˙ = e¨ + λe˙ = −η sign(s) But e¨ = y¨d − y¨ = −u − δ + y¨d . Suppose for now that δ is known or estimated accurately. Then the SMC law becomes u = y¨d (t) + λy˙ d (t) − λy˙ − δ + η sign(s)

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Since δ is unknown, do not include it in the control law: u = y¨d (t) + λy˙ d (t) − λy˙ + η sign(s) Verification of the reaching condition: ss˙ = s(δ − η sign(s)) Suppose η is chosen appropriately (η > D). Then, if s > 0 ss˙ = s(δ − η) but |δ| < D, so −D ≤ δ ≤ D, therefore δ − η ≤ D − η < 0, since −η < −D. So we conclude ss˙ < 0. The case s < 0 is handled similarly. Since ss˙ < 0 for s 6= 0, the region s = 0 must be invariant. 11 / 35

Notes Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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1. This control law has many advantages relative to PID. This includes its simplicity (no differentiation, no integration) and its total insensitivity to disturbances and other model errors contained in d 2. The implementation requires only position and velocity feedback and knowledge of the trajectory to be followed (feedforward term) 3. The signum (sign) function cannot be implemented (just like a relay without a deadzone). It results in an infinitely fast switching action across s = 0 called chattering. It will be replaced by a continuous approximation. 4. The saturation function sat coincides with the signum function for |s| > φ. Otherwise, sat(s/φ) = s/φ.

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Saturation Function Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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sat (s/φ) =

sign(s) |s| > φ s/φ |s| ≤ φ

(1)

The slope of the approximation can be adjusted with a small positive number φ (boundary layer parameter ). 1. With this approximation, the boundary layer defined by |s| ≤ φ can be shown to be attractive and invariant 2. Design tradeoff: chattering intensity vs. tracking accuracy. 3. Sometimes a good compromise cannot be found. This is one reason for the development of high-order sliding mode controls.

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Double-Integrator: Phase Plane Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

Reaching and Sliding Phases 8

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6

4 Reaching

edot

2

0

−2

Sliding

−4

S=0 −6 −0.1

0

0.1

0.2

0.3

0.4

e

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Example: Independent-Joint SMC Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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The SMC tracker can be used to achieve trajectory tracking of robots even when inertial and friction parameters are subject to large uncertainties. As an example, we design an IJ-SMC for the cart-pendulum system. This controller has been successfully implemented for CSU’s hip robot: http://academic.csuohio.edu/richter_h/lab/ccfrobot/

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Multivariable State-Space SMC Regulator Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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Plant: x˙ = (A + ∆A)x + (B + ∆B)u + Γw where x ∈ Rn , u ∈ Rm , w ∈ Rh . Objective: Bring the state to the origin from any initial condition and achieve robust regulation, despite model uncertainties and disturbances. Consider the sliding hyperplane s = Gx with G an m-by-n matrix to be determined. We assume that GB is invertible (transversality condition). As in the tracking case, s = 0 must be reached in finite time and it must be invariant, and the sliding mode (dynamics under s = 0) must be stable. 16 / 35

Matched vs. Unmatched Uncertainties Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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Combine uncertainties and disturbances into one term: x˙ = Ax + Bu + d(x, u, t) If the generalized uncertainty can be expressed as d(x, u, t) = Bζ(t, x, u) for some new uncertainty vector ζ(t, x, u), then the uncertainties are said to be matched. (they enter the system through the same “doors” as the control input, so total rejection is possible).

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Matched vs. Unmatched Uncertainties Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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Unmatched uncertainties, even if measured or accurately estimated, cannot be cancelled by u, but their effect can be reduced through proper design (see Khalil, Spurgeon and Edwards). For ∆A = ∆B = 0, the matching condition reduces to Γ ∈ col (B) (each column of Γ must be a linear combination of those of B) For m = h = 1, Γ must be parallel to B for the disturbance to be matched.

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Reaching Condition and Equivalent Control Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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As in the tracking problem, require −−−→ s˙ = −Ξ sign (s) −−−→ where Ξ is a diagonal matrix of positive gains and sign (s) is a vector whose components are sign(si ), i = 1, 2, ..m. If d were known, the hypothetical control input ueq required to achieve this condition could be found using s˙ = Gx˙ = G(Ax + Bu + d): ueq = −(GB)

−1

−−−→ (GAx + Ξ sign (s) + Gd)

This control input is known as equivalent control. Even when u is implemented without the d term, we can achieve s = s˙ = 0. ueq can be understood as the input making this possible. 19 / 35

Sliding Mode Stability Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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Supposing s = 0 has been reached, system dynamics under equivalent control become: x˙ = (A − B(GB)−1 GA)x − B(GB)−1 Gd + d With matched uncertainties, d = Bζ, so the terms containing d in the above equation cancel out, reducing the sliding dynamics to x˙ = (A − B(GB)−1 GA)x = Aeq x Note: 1. With matched uncertainties, total insensitivity is achieved. 2. System dynamics in the sliding mode are linear, and dictated by Aeq .

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Sliding Coefficient Selection Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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Since s = 0 introduces an algebraic dependence among states (Gx = 0), Aeq is singular. Sliding mode dynamics defined by the nonzero eigenvalues and corresponding eigenvectors. Aeq has a closed-loop state feedback form with a “funny” feedback gain: (A − BK), with K = (GB)−1 GA). For a stable sliding mode, G must be designed so that the nonzero eigenvalues of Aeq are stable. Basic method: transformation to “regular form” via QR decomposition (Edwards and Spurgeon) Utkin and Young’s LQ optimal selection Eigenstructure assignment (Spurgeon).

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Implemented Control Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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As done before, implement u without the disturbance term: u = −(GB)

−1

−−−→ (GAx + Ξ sign (s))

Suppose the generalized uncertainty (matched or unmatched) is bounded as ||d(t, x, u)|| ≤ ∆. It can be shown (Richter H., Advanced Control of Turbofan Engines, Springer, 2011) that choosing ηi > ||Gi ||∆ results in finite-time convergence to Gx = 0

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Normal Form Decomposition Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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A coordinate transformation z = T x can be applied so that the dynamics in the sliding regime reduce to z˙1 = (A11 − A12 Gz1 )z1 where A11 and A12 are n − m by n − m and n − m by m blocks of T AT −1 and Gz1 is designed by direct pole placement (see Richter’s book for details or Edwards and Spurgeon). The sliding coefficients are then taken as G = [Gz1 I]T where I is the m-by-m identity matrix.

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Example Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

Use the QR decomposition code to select G for a random third-order system with 1 input. Check that the random (A, B) is controllable!

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State-Space Output Tracking SMC - Outline Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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Plant: x˙ = Ax + Bu y = Cx Objective: Let y(t) → y d (t). Can we use s = y d − y? Two fundamental restrictions of SMC will be illustrated: 1. Unity relative degree requirement (1-SMC vs. high-order SMC) 2. Minimum-phase requirement Not much can be done about the second requirement, except selecting a new output to be tracked. SMC has been generalized to deal with the first requirement.

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Relative Degree Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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The relative degree is the number of differentiations required for the output (y) until the control (u) appears. In linear systems, this coincides with the difference between the number of poles and zeroes in G(s) = C(sI − A)−1 B A rigorous definition is available for nonlinear systems (Isidori). So far, we needed only one differentiation of s to reach u. What if more are needed?

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Example Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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Consider the triple-integrator plant x˙ 1 = x2 x˙ 2 = x3 x˙ 3 = u Let y = x1 . Define s = e˙ + λe, with e = xd1 − x1 . Here : s˙ = x ¨d1 + λ(x˙ d1 − x2 ) − x3 ... s¨ = x d1 + λ(¨ xd1 − x3 ) − u

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Twisting Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

The relative degree is 2. We can’t use s˙ = −η sign(s) because the control term does not appear. The reaching condition must now involve s¨. The twisting algorithm was pioneered by Levant.

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u = f (x) − r1 sign(s) − r2 sign(s) ˙ Arie Levant, Higher Order Sliding Modes, Differentiation and Output Feedback Control, Int. J. Control, 2003, v.76, n9, 924-941. 28 / 35

Super-Twisting Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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The super-twisting algorithm was pioneered by Moreno and Osorio. Here, the relative degree is one, but additional dynamics are introduced to move chattering action to the dynamic state, eliminating it from the final control input: http://www.intechopen.com/books/sliding-mode-control/ super-twisting-sliding-mode-in-motion-control-systems

The twisting algorithm has also been used for noise-free, but accurate numerical differentiation. Application of this algorithm to differentiation of biomechanical data can be an interesting term project.

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Minimum-Phase Restrictions Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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Given the state-space system (A, B, C, D), the zero dynamics correspond to system dynamics under the restriction y = 0. With transfer functions, the zero dynamics are defined by the location of the zeroes. Example: For the system x˙ 1 = x2 , x˙ 2 = −2x1 − 3x2 + u with output y = x1 + 2x2 , set y = y˙ = 0, find the hypothetical control required for y˙ = 0 and make substitutions to find the zero dynamics of this system. Then find the transfer function, look at the zero and compare.

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Nonlinear Zero Dynamics Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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Although there is no concept of ‘zeroes” in nonlinear systems, the notions of relative degree and zero dynamics still exist (see Isidori or Khalil). HW (doctoral): Find the zero dynamics for the system x˙ 1 = −x31 + u, x˙ 2 = −x1 + sin(x2 ) with output y = x2 . Are the zero dynamics stable? For SMC, let s = y = Gx and suppose the objective is to regulate y to zero. Then G is not a design choice any more. If Aeq = (A − B(GB)−1 GA) has an unstable sliding subspace, the problem is not solvable.

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State-Space SMC Tracking by Reference States Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

One way to circumvent relative degree and non-minimum phase limitations is to require the state to track a state reference trajectory: x(t) → xd (t). Then there is no output, so relative degree and zero dynamics become irrelevant. Full state feedback and a feedforward trajectory generator will be required.

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Plant: x˙ = Ax + Bu + Γδ Objective: Let x(t) → xd (t). Define e = xd − x and use s = Ge.

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SMC Tracking by Reference States Intuitive Idea Sliding Mode Control SMC Tracker: Double-Integrator System Multivariable State-Space SMC Regulator Relative Degree Minimum-Phase Restrictions State-Space SMC Tracking by Reference States

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As before, make

−−−→ s˙ = −Ξ sign (s)

The control required to maintain s˙ = 0 can be solved as ueq = (GB)

−1

−−−→ (Gx˙ − GAx − GΓδ + Ξ sign (s)) d

It can be shown (doctoral HW) that the tracking error dynamics in the sliding mode (s = 0) become e˙ = Aeq e + (I − B(GB)−1 G)(x˙ d − Axd ) + (I − B(GB)−1 G)Γδ If the disturbance is matched, δ = Bψ for some new disturbance ψ, and the last term cancels (total insensitivity).

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The term (I − B(GB)−1 G)(x˙ d − Axd ) suggests a way to determine the reference states (Misawa’s “consistent generation”). Suppose xd (t) = Axd (t) + Bv(t) for some arbitrary input v(t). Then error dynamics reduce to e˙ = Aeq e and asymptotic convergence of the error is guaranteed by making Aeq stable (by pole placement, for instance).

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The reference states can be chosen so that the output y = Cxd (t) matches a desired profile. The input v(t) is determined in a separate process, which can be offline or online. It can even be designed by open-loop optimization (time-optimal control, for instance). Example/HW: Design a tracking controller for some state space system (A, B, C, D).

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